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theoremTheorem[section] lemma[theorem]Lemma definition[theorem]Definition proposition[theorem]Proposition corollary[theorem]Corollary example[theorem]Example # Fixpoint Theory - Upside Down Paolo Baldan Università di Padova, Italy<EMAIL_ADDRESS>, Richard Eggert Universität Duisburg-Essen, Germany<EMAIL_ADDRESS>, Barbara König Universität Duisburg-Essen, Germany<EMAIL_ADDRESS>and Tommaso Padoan Università di Padova, Italy<EMAIL_ADDRESS> ###### Abstract. Knaster-Tarski’s theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $\mathbb{M}^{Y}$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games. ###### Key words and phrases: Fixpoints, Knaster-Tarski theorem, MV-algebras, non-expansive functions, bisimilarity, stochastic games This work is supported by the MIUR project PRIN2017- ASPRA, Grant No. 201784YSZ5, and and the DFG projects BEMEGA and SpeQt. ## 1\. Introduction Fixpoints are ubiquitous in computer science as they allow to provide a meaning to inductive and coinductive definitions (see, e.g., [San:IntroBisCoind, NNH:PPA]). A monotone function $f:L\to L$ over a complete lattice $(L,\sqsubseteq)$, by Knaster-Tarski’s theorem [t:lattice-fixed- point], admits a least fixpoint $\mu f$ and greatest fixpoint $\nu f$ which are characterised as the least pre-fixpoint and the greatest post-fixpoint, respectively. This immediately gives well-known proof principles for showing that a lattice element $l\in L$ is _below_ $\nu f$ or _above_ $\mu f$ $\frac{l\sqsubseteq f(l)}{l\sqsubseteq\nu f}\qquad\qquad\frac{f(l)\sqsubseteq l}{\mu f\sqsubseteq l}$ On the other hand, showing that a given element $l$ is _above_ $\nu f$ or _below_ $\mu f$ is more difficult. One can think of using the characterisation of least and largest fixpoints via Kleene’s iteration. E.g., the largest fixpoint is the least element of the (possibly transfinite) descending chain obtained by iterating $f$ from $\top$. Then showing that $f^{i}(\top)\sqsubseteq l$ for some $i$, one concludes that $\nu f\sqsubseteq l$. This proof principle is related to the notion of ranking functions. However, this is a less satisfying notion of witness since $f$ has to be applied $i$ times, and this can be inefficient or unfeasible when $i$ is an infinite ordinal. The aim of this paper is to present an alternative proof rule for this purpose for functions over lattices of the form $L=\mathbb{M}^{Y}$ where $Y$ is a finite set and $\mathbb{M}$ is an MV-chain, i.e., a totally ordered complete lattice endowed with suitable operations of sum and complement. This allows us to capture several examples, ranging from ordinary relations for dealing with bisimilarity to behavioural metrics, termination probabilities and simple stochastic games. Assume $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ monotone and consider the question of proving that some fixpoint $a:Y\to\mathbb{M}$ is the largest fixpoint $\nu f$. The idea is to show that there is no “slack” or “wiggle room” in the fixpoint $a$ that would allow us to further increase it. This is done by associating with every $a:Y\to\mathbb{M}$ a function $f^{\\#}_{a}$ on $\mathbf{2}^{Y}$ whose greatest fixpoint gives us the elements of $Y$ where we have a potential for increasing $a$ by adding a constant. If no such potential exists, i.e. $\nu f^{\\#}_{a}$ is empty, we conclude that $a$ is $\nu f$. A similar function $f_{\\#}^{a}$ (specifying decrease instead of increase) exists for the case of least fixpoints. Note that the premise is $\nu f_{\\#}^{a}=\emptyset$, i.e. the witness remains coinductive. The proof rules are: $\frac{f(a)=a\qquad\nu f^{\\#}_{a}=\emptyset}{\nu f=a}\qquad\frac{f(a)=a\qquad\nu f_{\\#}^{a}=\emptyset}{\mu f=a}$ For applying the rule we compute a greatest fixpoint on $\mathbf{2}^{Y}$, which is finite, instead of working on the potentially infinite $\mathbb{M}^{Y}$. The rule does not work for all monotone functions $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$, but we show that whenever $f$ is non- expansive the rule is valid. Actually, it is not only sound, but also reversible, i.e., if $a=\nu f$ then $\nu f_{a}^{\\#}=\emptyset$, providing an if-and-only-if characterisation of whether a given fixpoint corresponds to the greatest fixpoint. Quite interestingly, under the same assumptions on $f$, using a restricted function $f_{a}^{}$, the rule can be used, more generally, when $a$ is just a _pre-fixpoint_ ($f(a)\sqsubseteq a$) and it allows to conclude that $\nu f\sqsubseteq a$. A dual result holds for _post-fixpoints_ in the case of least fixpoints. $\frac{f(a)\sqsubseteq a\qquad\nu f^{}_{a}=\emptyset}{\nu f\sqsubseteq a}\qquad\frac{a\sqsubseteq f(a)\qquad\nu f_{}^{a}=\emptyset}{a\sqsubseteq\mu f}$ As already mentioned, the theory above applies to many interesting scenarios: witnesses for non-bisimilarity, algorithms for simple stochastic games [condon92], lower bounds for termination probabilities and behavioural metrics in the setting of probabilistic [bblm:on-the-fly-exact-journal] and metric transition systems [afs:linear-branching-metrics] and probabilistic automata [bblmtv:prob-bisim-distance-automata]. In particular we were inspired by, and generalise, the self-closed relations of Fu [f:game-metrics-markov-decision], also used in [bblmtv:prob-bisim-distance-automata]. #### Motivating example. Consider a Markov chain $(S,T,\eta)$ with a finite set of states $S$, where $T\subseteq S$ are the terminal states and every state $s\in S\backslash T$ is associated with a probability distribution $\eta(s)\in\mathcal{D}(S)$.111By $\mathcal{D}(S)$ we denote the set of all maps $p:S\to[0,1]$ such that $\sum_{s\in S}p(s)=1$. Intuitively, $\eta(s)(s^{\prime})$ denotes the probability of state $s$ choosing $s^{\prime}$ as its successor. Assume that, given a fixed state $s\in S$, we want to determine the termination probability of $s$, i.e. the probability of reaching any terminal state from $s$. As a concrete example, take the Markov chain given in Fig. 1, where $u$ is the only terminal state. The termination probability arises as the least fixpoint of a function $\mathcal{T}$ defined as in Fig. 1. The values of $\mu\mathcal{T}$ are indicated in green (left value). Now consider the function $t$ assigning to each state the termination probability written in red (right value). It is not difficult to see that $t$ is another fixpoint of $\mathcal{T}$, in which states $y$ and $z$ convince each other incorrectly that they terminate with probability $1$, resulting in a vicious cycle that gives “wrong” results. We want to show that $\mu\mathcal{T}\neq t$ without knowing $\mu\mathcal{T}$. Our idea is to compute the set of states that still has some “wiggle room”, i.e., those states which could reduce their termination probability by $\delta$ if all their successors did the same. This definition has a coinductive flavour and it can be computed as a greatest fixpoint on the finite powerset $\mathbf{2}^{S}$ of states, instead of on the infinite lattice $[{0},{1}]^{S}$. We hence consider a function $\mathcal{T}_{\\#}^{t}:\mathbf{2}^{[{S}]^{t}}\to\mathbf{2}^{[{S}]^{t}}$, dependent on $t$, defined as follows. Let $[{S}]^{t}$ be the set of all states $s$ where $t(s)>0$, i.e., a reduction is in principle possible. Then a state $s\in[{S}]^{t}$ is in $\mathcal{T}_{\\#}^{t}(S^{\prime})$ iff $s\not\in T$ and for all $s^{\prime}$ for which $\eta(s)(s^{\prime})>0$ it holds that $s^{\prime}\in S^{\prime}$, i.e. all successors of $s$ are in $S^{\prime}$. The greatest fixpoint of $\mathcal{T}_{\\#}^{t}$ is $\\{y,z\\}$. The fact that it is not empty means that there is some “wiggle room”, i.e., the value of $t$ can be reduced on the elements $\\{y,z\\}$ and thus $t$ cannot be the least fixpoint of $f$. Moreover, the intuition that $t$ can be improved on $\\{y,z\\}$ can be made precise, leading to the possibility of performing the improvement and search for the least fixpoint from there. #### Contributions. In the paper we formalise the theory outlined above, showing that the proof rules work for non-expansive monotone functions $f$ on lattices of the form $\mathbb{M}^{Y}$, where $Y$ is a finite set and $\mathbb{M}$ a (potentially infinite) MV-algebra (Section 3 and Section 4). Additionally, given a decomposition of $f$ we show how to obtain the corresponding approximation compositionally (Section 5). Then, in order to show that our approach covers a wide range of examples and allows us to derive useful and original algorithms, we discuss various applications: termination probability, behavioural distances for metric transition systems and probabilistic automata, bisimilarity (Section 6) and simple stochastic games (Section 7). Proofs and further material can be found in the appendix. $\displaystyle\mathcal{T}:[0,1]^{S}\to[0,1]^{S}$ $\displaystyle\mathcal{T}(t)(s)=\left\\{\begin{array}[]{ll}1&\mbox{if $v\in T$}\\\ \sum\limits_{s^{\prime}\in S}\eta(s)(s^{\prime})\cdot t(s^{\prime})&\mbox{otherwise}\end{array}\right.$ $x$$\frac{1}{2}$/$1$$u$$1$/$1$$y$$0$/$1$$z$$0$/$1$$\frac{1}{3}$$\frac{1}{3}$$\frac{1}{3}$$1$$1$ Figure 1. Function $\mathcal{T}$ (left) and a Markov chain with two fixpoints of $\mathcal{T}$ (right) ## 2\. Lattices and MV-algebras In this section, we review some basic notions used in the paper, concerning complete lattices and MV-algebras [Mun:MV]. A preordered or partially ordered set $(P,\sqsubseteq)$ is often denoted simply as $P$, omitting the order relation. Given $x,y\in P$, with $x\sqsubseteq y$, we denote by $[{x},{y}]$ the interval $\\{z\in P\mid x\sqsubseteq z\sqsubseteq y\\}$. The _join_ and the _meet_ of a subset $X\subseteq P$ (if they exist) are denoted $\bigsqcup X$ and $\bigsqcap X$, respectively. A _complete lattice_ is a partially ordered set $(L,\sqsubseteq)$ such that each subset $X\subseteq L$ admits a join $\bigsqcup X$ and a meet $\bigsqcap X$. A complete lattice $(L,\sqsubseteq)$ always has a least element $\bot=\bigsqcup\emptyset$ and a greatest element $\top=\bigsqcap\emptyset$. A function $f:L\to L$ is _monotone_ if for all $l,l^{\prime}\in L$, if $l\sqsubseteq l^{\prime}$ then $f(l)\sqsubseteq f(l^{\prime})$. By Knaster- Tarski’s theorem [t:lattice-fixed-point, Theorem 1], any monotone function on a complete lattice has a least and a greatest fixpoint, denoted respectively $\mu f$ and $\nu f$, characterised as the meet of all pre-fixpoints respectively the join of all post-fixpoints: $\mu f=\bigsqcap\\{l\mid f(l)\sqsubseteq l\\}$ and $\nu f=\bigsqcup\\{l\mid l\sqsubseteq f(l)\\}$. Let $(C,\sqsubseteq)$, $(A,\leq)$ be complete lattices. A _Galois connection_ is a pair of monotone functions $\langle\alpha,\gamma\rangle$ such that $\alpha:C\to A$, $\gamma:A\to C$ and for all $a\in A$ and $c\in C$: $\alpha(c)\leq a$ iff $c\sqsubseteq\gamma(a)$. Equivalently, for all $a\in A$ and $c\in C$, (i) $c\sqsubseteq\gamma(\alpha(c))$ and (ii) $\alpha(\gamma(a))\leq a$. In this case we will write $\langle\alpha,\gamma\rangle:C\to A$. For a Galois connection $\langle\alpha,\gamma\rangle:C\to A$, the function $\alpha$ is called the left (or lower) adjoint and $\gamma$ the right (or upper) adjoint. Galois connections are at the heart of abstract interpretation [cc:ai-unified- lattice-model, CC:TLA]. In particular, when $\langle\alpha,\gamma\rangle$ is a Galois connection, given $f^{C}:C\to C$ and $f^{A}:A\to A$, monotone functions, if $f^{C}\circ\gamma\sqsubseteq\gamma\circ f^{A}$, then $\nu f^{C}\sqsubseteq\gamma(\nu f^{A})$. If equality holds, i.e., $f^{C}\circ\gamma=\gamma\circ f^{A}$, a condition sometimes referred to as $\gamma$-completeness, then greatest fixpoints are preserved along the connection, i.e., $\nu f^{C}=\gamma(\nu f^{A})$. Given a set $Y$ and a complete lattice $L$, the set of functions $L^{Y}=\\{f\mid f:Y\to L\\}$, endowed with pointwise order, i.e., for $a,b\in L^{Y}$, $a\sqsubseteq b$ if $a(y)\sqsubseteq b(y)$ for all $y\in Y$, is a complete lattice. In the paper we will mostly work with lattices of the form $\mathbb{M}^{Y}$ where $\mathbb{M}$ is a special kind of lattice with a rich algebraic structure, i.e. an MV-algebra [Mun:MV]. ###### Definition 2.1 (MV-algebra). An _MV-algebra_ is a tuple $\mathbb{M}=(M,\oplus,0,\overline{(\cdot)})$ where $(M,\oplus,0)$ is a commutative monoid and $\overline{(\cdot)}:M\to M$ maps each element to its _complement_ , such that for all $x,y\in M$ 1. (1) $\overline{\overline{x}}=x$ 2. (2) $x\oplus\overline{0}=\overline{0}$ 3. (3) $\overline{(\overline{x}\oplus y)}\oplus y=\overline{(\overline{y}\oplus x)}\oplus x$. We denote $1=\overline{0}$, multiplication $x\otimes y=\overline{\overline{x}\oplus\overline{y}}$ and subtraction $x\ominus y=x\otimes\overline{y}$. Note that by using the derived operations, axioms (2) and (3) above can be written as 1. (2) $x\oplus 1=1$ 2. (3) $(y\ominus x)\oplus x=(x\ominus y)\oplus y$ MV-algebras are endowed with a natural order. ###### Definition 2.2 (natural order). Let $\mathbb{M}=(M,\oplus,0,\overline{(\cdot)})$ be an MV-algebra. The _natural order_ on $\mathbb{M}$ is defined, for $x,y\in M$, by $x\sqsubseteq y$ if $x\oplus z=y$ for some $z\in M$. When $\sqsubseteq$ is total $\mathbb{M}$ is called an _MV-chain_. The natural order gives an MV-algebra a lattice structure where $\bot=0$, $\top=1$, $x\sqcup y=(x\ominus y)\oplus y$ and $x\sqcap y=\overline{\overline{x}\sqcup\overline{y}}=x\otimes(\overline{x}\oplus y)$. We call the MV-algebra _complete_ , if it is a complete lattice. This is not true in general, e.g., $([0,1]\cap\mathbb{Q},\leq)$. ###### Example 2.3. A prototypical example of an MV-algebra is $([0,1],\oplus,0,\overline{(\cdot)})$ where $x\oplus y=\min\\{x+y,1\\}$ and $\overline{x}=1-x$ for $x,y\in[0,1]$. This means that $x\otimes y=\max\\{x+y-1,0\\}$ and $x\ominus y=\max\\{0,x-y\\}$ (truncated subtraction). The operators $\oplus$ and $\otimes$ are also known as strong disjunction and conjunction in Łukasiewicz logic [m:lukasiewicz-mv]. The natural order is $\leq$ (less or equal) on the reals. Another example is $(\\{0,\dots,k\\},\oplus,0,\overline{(\cdot)})$ where $n\oplus m=\min\\{n+m,k\\}$ and $\overline{n}=k-n$ for $n,m\in\\{0,\dots,k\\}$. We are in particular interested in the case $k=1$. Both MV-algebras are complete and MV-chains. Boolean algebras (with disjunction and complement) also form MV-algebras that are complete, but in general not MV-chains. MV-algebras are the algebraic semantics of Łukasiewicz logic. They can be shown to correspond to intervals of the kind $[{0},{u}]$ in suitable groups, i.e., abelian lattice-ordered groups with a strong unit $u$ [Mun:MV]. We next review some properties of MV-algebras. They are taken from or easy consequences of properties in [Mun:MV] and will be used throughout the paper. [properties of MV-algebras] Let $\mathbb{M}=(M,\oplus,0,\overline{(\cdot)})$ be an MV-algebra. For all $x,y,z\in M$ 1. (1) $x\oplus\overline{x}=1$ 2. (2) $x\sqsubseteq y$ iff $\overline{x}\oplus y=1$ iff $x\otimes\overline{y}=0$ iff $y=x\oplus(y\ominus x)$ 3. (3) $x\sqsubseteq y$ iff $\overline{y}\sqsubseteq\overline{x}$ 4. (4) $\oplus$, $\otimes$ are monotone in both arguments, $\ominus$ monotone in the first and antitone in the second argument. 5. (5) if $x\sqsubset y$ then $0\sqsubset y\ominus x$; 6. (6) $(x\oplus y)\ominus y\sqsubseteq x$ 7. (7) $z\sqsubseteq x\oplus y$ if and only if $z\ominus x\sqsubseteq y$. 8. (8) if $x\sqsubset y$ and $z\sqsubseteq\overline{y}$ then $x\oplus z\sqsubset y\oplus z$; 9. (9) $y\sqsubseteq\overline{x}$ if and only if $(x\oplus y)\ominus y=x$; 10. (10) $x\ominus(x\ominus y)\sqsubseteq y$ and if $y\sqsubseteq x$ then $x\ominus(x\ominus y)=y$. 11. (11) Whenever $\mathbb{M}$ is an MV-chain, $x\sqsubset y$ and $0\sqsubset z$ imply $(x\oplus z)\ominus y\sqsubset z$ ###### Proof 2.4. The proof of properties (1), (2), (3), (4) can be found directly in [Mun:MV]. For the rest: 1. (5) Immediate consequence of (2). In fact, given $x,y\in M$, if we had $y\ominus x=0$ then by (2), $y=x\oplus(y\ominus x)=x\oplus 0=x$. 2. (6) Observe that $(x\oplus y)\ominus y=\overline{\overline{(x\oplus y)}\oplus y}=\overline{(\overline{x}\ominus y)\oplus y}=\overline{(y\ominus\overline{x})\oplus\overline{x}}\sqsubseteq\overline{\overline{x}}=x$, where the last inequality is motivated by the fact that $\overline{x}\sqsubseteq(y\ominus\overline{x})\oplus\overline{x}$ and point (3). 3. (7) The direction from left to right is an immediate consequence of (6). In fact, if $z\sqsubseteq x\oplus y$ then $z\ominus x\sqsubseteq(x\oplus y)\ominus x\sqsubseteq y$. The other direction goes as follows: if $z\ominus x\sqsubseteq y$, then – by monotonicity (4) – $(z\ominus x)\oplus x\sqsubseteq y\oplus x=x\oplus y$. The left hand side can be rewritten to $(x\ominus z)\oplus z\sqsupseteq z$. 4. (8) Assume that $x\sqsubset y$ and $z\sqsubseteq\overline{y}$. We know, by property (4) that $x\oplus z\sqsubseteq y\oplus z$. Assume by contradiction that $x\oplus z=y\oplus z$. Then we have $\displaystyle\overline{x}$ $\displaystyle\sqsubseteq$ [by properties (3) and (6)] $\displaystyle\sqsubseteq\overline{(x\oplus z)\ominus z}$ [since $x\oplus z=y\oplus z$] $\displaystyle\sqsubseteq\overline{(y\oplus z)\ominus z}$ [definition of $\ominus$] $\displaystyle=(\overline{y}\ominus z)\oplus z$ [since $z\sqsubseteq\overline{y}$ and property (2)] $\displaystyle=\overline{y}$ And with point (3) this is a contradiction. 5. (9) Assume $y\sqsubseteq\overline{x}$. We know $(x\oplus y)\ominus y\sqsubseteq x$. If it were $(x\oplus y)\ominus y\sqsubset x$, then $((x\oplus y)\ominus y)\oplus y\sqsubset x\oplus y$, with (8). Since the left-hand side is equal to $(y\ominus(x\oplus y))\oplus(x\oplus y)\sqsupseteq x\oplus y$, this is a contradiction. For the other direction assume that $(x\oplus y)\ominus y=x$. Hence we have $x=(x\oplus y)\ominus y=\overline{\overline{(x\oplus y)}\oplus y}$. By complementing on both sides we obtain $\overline{x}=\overline{(x\oplus y)}\oplus y$ which implies that $y\sqsubseteq\overline{x}$. 6. (10) Observe that, by (7), we have $\overline{y}\sqsubseteq\overline{x}\oplus(\overline{y}\ominus\overline{x})=\overline{x}\oplus(x\ominus y)=\overline{x\ominus(x\ominus y)}$. Therefore, by (3), $x\ominus(x\ominus y)\sqsubseteq y$, as desired. For the second part, assume if $y\sqsubseteq x$ and thus, by (3), $\overline{x}\sqsubseteq\overline{y}$. Using (2), we obtain $\overline{y}=\overline{x}\oplus(\overline{y}\ominus\overline{x})=\overline{x}\oplus\overline{y\oplus\overline{x}}=\overline{x}\oplus(x\ominus y)$. Hence $y=\overline{\overline{x}\oplus(x\ominus y)}=x\ominus(x\ominus y)$. 7. (11) We first observe that $x\sqsubseteq y\oplus(x\ominus y)$. This is a direct consequence of axiom (3) of MV-algebras and the definition of natural order. Second, in an MV-chain if $x,y\sqsupset 0$, then $x\ominus y\sqsubset x$. In fact, if $x\sqsubseteq y$ and thus $x\ominus y=0\sqsubset x$. If instead, $y\sqsubset x$ we have $0\sqsubset y$ and $x\ominus y\sqsubseteq 1\ominus y=\overline{y}$, hence by 2.3(8) it holds that $0\oplus(x\ominus y)\sqsubset y\oplus(x\ominus y)$. Recalling that $y\sqsubset x$ and thus by 2.3(2), $(x\ominus y)\oplus y=x$, we conclude $x\ominus y\sqsubset x$. Now $\displaystyle(x\oplus z)\ominus y$ $\displaystyle\sqsubseteq(x\oplus(z\ominus(y\ominus x))\oplus(y\ominus x))\ominus y$ [by first obs. above] $\displaystyle=(y\oplus(z\ominus(y\ominus x))\ominus y$ [since $x\sqsubseteq y$, by 2.3(2)] $\displaystyle\sqsubseteq z\ominus(y\ominus x)$ [by 2.3(6)] $\displaystyle\sqsubset z$ [by second obs. above, since $z\sqsupset 0$ and $y\ominus x\sqsupset 0$ by 2.3(5)] Note that we adhere to the following convention: whenever brackets are missing, we always assume that we associate from left to right. So $a\oplus b\ominus c$ should be read as $(a\oplus b)\ominus c$ and not as $a\oplus(b\ominus c)$, which is in general different. ## 3\. Non-expansive functions and their approximations As mentioned in the introduction, our interest is for fixpoints of monotone functions $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$, where $\mathbb{M}$ is an MV- chain and $Y$ is a finite set. We will see that for non-expansive functions we can over-approximate the sets of points in which a given $a\in\mathbb{M}^{Y}$ can be increased in a way that is preserved by the application of $f$. This will be the core of the proof rules outlined earlier. ### 3.1. Non-expansive functions on MV-algebras. For defining non-expansiveness it is convenient to introduce a norm. ###### Definition 3.1 (norm). Let $\mathbb{M}$ be an MV-chain and let $Y$ be a finite set. Given $a\in\mathbb{M}^{Y}$ we define its _norm_ as $\|\\!\|{a}\|\\!\|=\max\\{a(y)\mid y\in Y\\}$. Given a finite set $Y$ we extend $\oplus$ and $\otimes$ to $\mathbb{M}^{Y}$ pointwise. E.g. if $a,b\in\mathbb{M}^{Y}$, we write $a\oplus b$ for the function defined by $(a\oplus b)(y)=a(y)\oplus b(y)$ for all $y\in Y$. Given $Y^{\prime}\subseteq Y$ and $\delta\in\mathbb{M}$, we write $\delta_{Y^{\prime}}$ for the function defined by $\delta_{Y^{\prime}}(y)=\delta$ if $y\in Y^{\prime}$ and $\delta_{Y^{\prime}}(y)=0$, otherwise. Whenever this does not generate confusion, we write $\delta$ instead of $\delta_{Y}$. It can be seen that $\|\\!\|{\cdot}\|\\!\|$ has the properties of a norm, i.e., for all $a,b\in\mathbb{M}^{Y}$ and $\delta\in\mathbb{M}$, it holds that (1) $\|\\!\|{a\oplus b}\|\\!\|\sqsubseteq\|\\!\|{a}\|\\!\|\oplus\|\\!\|{b}\|\\!\|$, (2) $\|\\!\|{\delta\otimes a}\|\\!\|=\delta\otimes\|\\!\|{a}\|\\!\|$ and $\|\\!\|{a}\|\\!\|=0$ implies that $a$ is the constant $0$ (see 3.2 in the appendix). ###### Lemma 3.2 (properties of the norm). Let $\mathbb{M}$ be an MV-chain and let $Y$ be a finite set. Then $\|\\!\|{\cdot}\|\\!\|:\mathbb{M}^{Y}\to\mathbb{M}$ satisfies, for all $a,b\in\mathbb{M}^{Y}$, $\delta\in\mathbb{M}$ 1. (1) $\|\\!\|{a\oplus b}\|\\!\|\sqsubseteq\|\\!\|{a}\|\\!\|\oplus\|\\!\|{b}\|\\!\|$, 2. (2) $\|\\!\|{\delta\otimes a}\|\\!\|=\delta\otimes\|\\!\|{a}\|\\!\|$ and 3. (3) $\|\\!\|{a}\|\\!\|=0$ implies that $a$ is the constant $0$. ###### Proof 3.3. Concerning (1), let $\|\\!\|{a\oplus b}\|\\!\|$ be realised on some element $y\in Y$, i.e., $\|\\!\|{a\oplus b}\|\\!\|=a(y)\oplus b(y)$. Since $a(y)\sqsubseteq\|\\!\|{a}\|\\!\|$ and $b(y)\sqsubseteq\|\\!\|{b}\|\\!\|$, by monotonicity of $\oplus$ we deduce that $\|\\!\|{a\oplus b}\|\\!\|\sqsubseteq\|\\!\|{a}\|\\!\|\oplus\|\\!\|{b}\|\\!\|$. Concerning (2), note that $\displaystyle\|\\!\|{\delta\otimes a}\|\\!\|$ $\displaystyle=\max\\{\overline{\overline{\delta}\oplus\overline{a(y)}}\mid y\in Y\\}$ $\displaystyle=\overline{\min\\{\overline{\delta}\oplus\overline{a(y)}\mid y\in Y\\}}$ $\displaystyle=\overline{\overline{\delta}\oplus\min\\{\overline{a(y)}\mid y\in Y\\}}$ $\displaystyle=\overline{\overline{\delta}\oplus\overline{\max\\{a(y)\mid y\in Y\\}}}$ $\displaystyle=\overline{\overline{\delta}\oplus\overline{\|\\!\|{a}\|\\!\|}}$ $\displaystyle=\delta\otimes\|\\!\|{a}\|\\!\|\ $ Finally, point (3) is straightforward, since $0$ is the bottom of $\mathbb{M}$. Moreover, it is clearly monotone, i.e., if $a\sqsubseteq b$ then $\|\\!\|{a}\|\\!\|\sqsubseteq\|\\!\|{b}\|\\!\|$. We next introduce non-expansiveness. Despite the fact that we will finally be interested in endo-functions $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$, in order to allow for a compositional reasoning we work with functions where domain and codomain can be different. ###### Definition 3.4 (non-expansiveness). Let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a function, where $\mathbb{M}$ is an MV-chain and $Y,Z$ are finite sets. We say that it is _non-expansive_ if for all $a,b\in\mathbb{M}^{Y}$ it holds $\|\\!\|{f(b)\ominus f(a)}\|\\!\|\sqsubseteq\|\\!\|{b\ominus a}\|\\!\|$. Note that $(a,b)\mapsto\|\\!\|{a\ominus b}\|\\!\|$ is the supremum lifting of a directed version of Chang’s distance [Mun:MV]. It is easy to see that all non- expansive functions on MV-chains are monotone (see Lemma 3.5 in the appendix). Moreover, when $\mathbb{M}=\\{0,1\\}$, i.e., $\mathbb{M}$ is the two-points boolean algebra, the two notions coincide. ###### Lemma 3.5 (non-expansiveness implies monotonicity). Let $\mathbb{M}$ is an MV-chain and let $Y,Z$ be finite sets. Every non- expansive function $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ is monotone. ###### Proof 3.6. Let $a,b\in\mathbb{M}^{Y}$ be such that $a\sqsubseteq b$. Therefore, by 2.3(2), $a(y)\ominus b(y)=0$ for all $y\in Y$, hence $a\ominus b=0$. Thus $\|\\!\|{f(a)\ominus f(b)}\|\\!\|\sqsubseteq\|\\!\|{a\ominus b}\|\\!\|=0$. In turn this implies that for all $z\in Z$, $f(a)(z)\ominus f(b)(z)=0$. Hence 2.3(2), allows us to conclude $f(a)(z)\sqsubseteq f(b)(z)$ for all $z\in Z$, i.e., $f(a)\sqsubseteq f(b)$, as desired. The next lemma provides a useful equivalent characterisation of non- expansiveness. ###### Lemma 3.7 (characterisation of non-expansiveness). Let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a monotone function, where $\mathbb{M}$ is an MV-chain and $Y,Z$ are finite sets. Then $f$ is non- expansive iff for all $a\in\mathbb{M}^{Y}$, $\theta\in\mathbb{M}$ and $z\in Z$ it holds $f(a\oplus\theta)(z)\ominus f(a)(z)\sqsubseteq\theta$. ###### Proof 3.8. Let $f$ be non-expansive and let $a\in\mathbb{M}^{Y}$ and $\theta\in\mathbb{M}$. We have that for all $z\in Z$ $\displaystyle f(a\oplus\theta)(z)\ominus f(a)(z)\sqsubseteq$ $\displaystyle\quad\sqsubseteq\|\\!\|{f(a\oplus\theta)\ominus f(a)}\|\\!\|$ [by definition of norm] $\displaystyle\quad\sqsubseteq\|\\!\|{(a\oplus\theta)\ominus a}\|\\!\|$ [by hypothesis] $\displaystyle\quad\sqsubseteq\|\\!\|{\lambda y.\theta}\|\\!\|$ [by 2.3(6) and monotonicity of norm] $\displaystyle\quad=\theta$ [by definition of norm] Conversely, assume that for all $a\in\mathbb{M}^{Y}$, $\theta\in\mathbb{M}$ and $z\in Z$ it holds $f(a\oplus\theta)(z)\ominus f(a)(z)\sqsubseteq\theta$. For $a,b\in\mathbb{M}^{Y}$, first observe that for all $y\in Y$ it holds $b(y)\ominus a(y)\sqsubseteq\|\\!\|{b\ominus a}\|\\!\|$, hence, if we let $\theta=\|\\!\|{b\ominus a}\|\\!\|$, we have $b\sqsubseteq a\oplus\theta$ and thus, by monotonicity, $f(b)\ominus f(a)\sqsubseteq f(a\oplus\theta)\ominus f(a)$. Thus $\displaystyle\|\\!\|{f(b)\ominus f(a)}\|\\!\|\sqsubseteq$ $\displaystyle\quad\sqsubseteq\|\\!\|{f(a+\theta)\ominus f(a)}\|\\!\|=$ [by the observation above and monotonicity of norm] $\displaystyle\quad=\max\\{f(a+\theta)(z)\ominus f(a)(z)\|z\in Z\\}$ [by definition of norm] $\displaystyle\quad\sqsubseteq\theta$ [by hypothesis] $\displaystyle\quad=\|\\!\|{b\ominus a}\|\\!\|$ [by the choice of $\theta$] ###### Lemma 3.9 (composing non-expansive functions). Let $\mathbb{M}$ be an MV-chain and let $Y,W,Z$ be finite sets. If $g:\mathbb{M}^{Y}\to\mathbb{M}^{W}$ and $h:\mathbb{M}^{W}\to\mathbb{M}^{Z}$ are non-expansive then $h\circ g:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ is non- expansive. ###### Proof 3.10. Straightforward. We have for any $a,b\in\mathbb{M}^{Y}$ that $\displaystyle\|\\!\|{h(g(b))\ominus h(g(a))}\|\\!\|\sqsubseteq$ $\displaystyle\quad\sqsubseteq\|\\!\|{g(b)\ominus g(a)}\|\\!\|$ [by non- expansiveness of $h$] $\displaystyle\quad\sqsubseteq\|\\!\|{b\ominus a}\|\\!\|$ [by non-expansiveness of $g$] ### 3.2. Approximating the propagation of increases. Let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a monotone function and take $a,b\in\mathbb{M}^{Y}$ with $a\sqsubseteq b$. We are interested in the difference $b(y)\ominus a(y)$ for some $y\in Y$ and on how the application of $f$ “propagates” this difference. The reason is that, understanding that no increase can be propagated will be crucial to establish when a fixpoint of a non-expansive function $f$ is actually the largest one, and, more generally, when a (pre-)fixpoint of $f$ is above the largest fixpoint. In order to formalise the above intuition, we rely on tools from abstract interpretation. In particular, the following pair of functions, which, under a suitable condition, form a Galois connection, will play a major role. The left adjoint $\alpha_{a,\delta}$ takes as input a set $Y^{\prime}$ and, for $y\in Y^{\prime}$, it increases the values $a(y)$ by $\delta$, while the right adjoint $\gamma_{a,\delta}$ takes as input a function $b\in\mathbb{M}^{Y}$, $b\in[{a},{a\oplus\delta}]$ and checks for which parameters $y\in Y$ the value $b(y)$ exceeds $a(y)$ by $\delta$. We also define $[{Y}]_{a}$, the subset of elements in $Y$ where $a(y)$ is not $1$ and thus there is a potential to increase, and $\delta_{a}$, which gives us the least of such increases (i.e., the largest increase that can be used on all elements in $[{Y}]_{a}$ without “overflowing”). ###### Definition 3.11 (functions to sets, and vice versa). Let $\mathbb{M}$ be an MV-algebra and let $Y$ be a finite set. Define the set $[{Y}]_{a}=\\{y\in Y\mid a(y)\neq 1\\}$ (support of $\overline{a}$) and $\delta_{a}=\min\\{\overline{a(y)}\mid y\in[{Y}]_{a}\\}$ with $\min\emptyset=1$. For $0\sqsubset\delta\in\mathbb{M}$ we consider the functions $\alpha_{a,\delta}:\mathbf{2}^{[{Y}]_{a}}\to[{a},{a\oplus\delta}]$ and $\gamma_{a,\delta}:[{a},{a\oplus\delta}]\to\mathbf{2}^{[{Y}]_{a}}$, defined, for $Y^{\prime}\in\mathbf{2}^{[{Y}]_{a}}$ and $b\in[{a},{a\oplus\delta}]$, by $\alpha_{a,\delta}(Y^{\prime})=a\oplus\delta_{Y^{\prime}}\qquad\gamma_{a,\delta}(b)=\\{y\in[{Y}]_{a}\mid b(y)\ominus a(y)\sqsupseteq\delta\\}.$ ###### Lemma 3.12 (well-definedness). The functions $\alpha_{a,\delta}$, $\gamma_{a,\delta}$ from Def. 3.11 are well-defined and monotone. ###### Proof 3.13. The involved functions $\alpha_{a,\delta}$ and $\gamma_{a,\delta}$ are well- defined. In fact, for $Y^{\prime}\subseteq[{Y}]_{a}$, clearly $\alpha_{a,\delta}=a\oplus\delta_{Y^{\prime}}\in[{a},{a\oplus\delta}]$. Moreover, for $b\in[{a},{a\oplus\delta}]$ we have $\gamma_{a,\delta}(b)\subseteq[{Y}]_{a}$. In fact, if $y\not\in[{Y}]_{a}$ then $a(y)=1$, hence $b(y)=1$ and thus $b(y)\ominus a(y)=0\not\sqsupseteq\delta$, and thus $y\not\in\gamma_{a,\delta}(b)$. Moreover, they are clearly monotone. When $\delta$ is sufficiently small, the pair $\langle\alpha_{a,\delta},\gamma_{a,\delta}\rangle$ is a Galois connection. [Galois connection] Let $\mathbb{M}$ be an MV-algebra and $Y$ be a finite set. For $0\neq\delta\sqsubseteq\delta_{a}$, the pair $\langle\alpha_{a,\delta},\gamma_{a,\delta}\rangle:\mathbf{2}^{[{Y}]_{a}}\to[{a},{a\oplus\delta}]$ is a Galois connection. $\mathbf{2}^{[{Y}]_{a}}$$[{a},{a\oplus\delta}]$$\alpha_{a,\delta}$$\gamma_{a,\delta}$ ###### Proof 3.14. For all $Y^{\prime}\in\mathbf{2}^{[{Y}]_{a}}$ it holds $\gamma_{a,\delta}(\alpha_{a,\delta}(Y^{\prime}))=\gamma_{a,\delta}(a\oplus\delta_{Y^{\prime}})=Y^{\prime}$. In fact, for all $y\in Y^{\prime}$, $(a\oplus\delta_{Y^{\prime}})(y)=a(y)\oplus\delta$. Moreover, and by the choice of $\delta$ and definition of $[{Y}]_{a}$, we have $\delta\sqsubseteq\delta_{a}\sqsubseteq\overline{a(y)}$, by 2.3(9), we have $(a\oplus\delta_{Y^{\prime}})(y)\ominus a(y)=\delta$ hence $y\in\gamma_{a,\delta}(\alpha_{a,\delta}(Y^{\prime}))$. Conversely, if $y\not\in Y^{\prime}$, then $(a\oplus\delta_{Y^{\prime}})(y)=a(y)$, and thus $(a\oplus\delta_{Y^{\prime}})(y)\ominus a(y)=0\not\sqsupseteq\delta$. Moreover, for all $b\in[{a},{a\oplus\delta}]$ we have $\alpha_{a,\delta}(\gamma_{a,\delta}(b))=a\oplus\delta_{\gamma_{a,\delta}(b)}\sqsubseteq b$ In fact, for all $y\in Y$, if $y\in\gamma_{a,\delta}(b)$, i.e., $\delta\sqsubseteq b(y)\ominus a(y)$ then $(a\oplus\delta_{\gamma_{a,\delta}(b)})(y)=a(y)\oplus\delta\sqsubseteq a(y)\oplus(b(y)\ominus a(y))=b(y)$, by 2.3(2). If instead, $y\not\in\gamma_{a,\delta}(b)$, then $(a\oplus\delta_{\gamma_{a,\delta}(b)}(b))(y)=a(y)\sqsubseteq b(y)$. Whenever $f$ is non-expansive, it is easy to see that it restricts to a function $f:[{a},{a\oplus\delta}]\to[{f(a)},{f(a)\oplus\delta}]$ for all $\delta\in\mathbb{M}$. ###### Lemma 3.15 (restricting non-expansive functions to intervals). Let $\mathbb{M}$ be an MV-chain, let $Y,Z$ be finite sets $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non-expansive function. Then $f$ restricts to a function $f_{a,\delta}:[{a},{a\oplus\delta}]\to[{f(a)},{f(a)\oplus\delta}]$, defined by $f_{a,\delta}(b)=f(b)$. ###### Proof 3.16. Given $b\in[{a},{a\oplus\delta}]$, by monotonicity of $f$ we have that $f(a)\sqsubseteq f(b)$. Moreover, $f(b)\sqsubseteq f(a\oplus\delta)\sqsubseteq f(a)\oplus\delta$, where the last passage is motivated by 3.7. In the following we will simply write $f$ instead of $f_{a,\delta}$. Given an MV-chain $\mathbb{M}$ and a finite set $Y$, we first observe that each function $b\in\mathbb{M}^{Y}$ can be expressed as a suitable sum of functions of the shape $\delta_{Y^{\prime}}$. ###### Lemma 3.17 (standard form). Let $\mathbb{M}$ be an MV-chain and let $Y$ be a finite set. Then for any $b\in\mathbb{M}^{Y}$ there are $Y_{1},\ldots,Y_{n}\subseteq Y$ with $Y_{i+1}\subseteq Y_{i}$ for $i\in\\{1,\ldots,n-1\\}$ and $\delta^{i}\in\mathbb{M}$, $0\neq\delta^{i}\sqsubseteq\overline{\bigoplus_{j=1}^{i-1}\delta^{j}}$ for $i\in\\{1,\ldots,n\\}$ such that $b=\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}$ and $\|\\!\|{b}\|\\!\|=\bigoplus_{i=1}^{n}\delta^{i}$. where we assume that an empty sum evaluates to $0$. ###### Proof 3.18. Given $b\in\mathbb{M}^{Y}$, consider $V=\\{b(y)\mid y\in Y\\}$. If $V$ is empty, then $Y$ is empty and thus $b=1_{Y}$, i.e., we can take $n=1$, $\delta^{1}=1$ and $Y_{1}=Y$. Otherwise, if $Y\neq\emptyset$, then $V$ is a finite non-empty set. Let $V=\\{v_{1},\ldots,v_{n}\\}$, with $v_{i}\sqsubseteq v_{i+1}$ for $i\in\\{1,\ldots,n-1\\}$. For $i\in\\{1,\ldots,n\\}$ define $Y_{i}=\\{y\in Y\mid v_{i}\sqsubseteq b(y)\\}$. Clearly, $Y_{1}\supseteq Y_{2}\supseteq\ldots\supseteq Y_{n}$. Moreover let $\delta^{1}=v_{1}$ and $\delta^{i+1}=v_{i+1}\ominus v_{i}$ for $i\in\\{1,\ldots,n-1\\}$. Observe that for each $i$, we have $v_{i}=\bigoplus_{j=1}^{i}\delta^{i}$, as it can easily shown by induction. Hence $\delta^{i+1}=v_{i+1}\ominus v_{i}=v_{i+1}\ominus\bigoplus_{j=1}^{i}\delta^{i}\sqsubseteq 1\ominus\bigoplus_{j=1}^{i}\delta^{i}=\overline{\bigoplus_{j=1}^{i}\delta^{i}}$. We now show that $b=\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}$ by induction on $n$. • If $n=1$ then $V=\\{v_{1}\\}$ and thus $b$ is a constant function $b(y)=v_{1}$ for all $y\in Y$. Hence $Y_{1}=Y$ and thus $b=\delta^{1}_{Y}=\delta^{1}_{Y_{1}}$, as desired. * • If $n>1$, let $b^{\prime}\in\mathbb{M}^{Y}$ defined by $b^{\prime}(y)=b(y)$ for $y\in Y\backslash Y_{n}$ and $b^{\prime}(y)=v_{n-1}$ for $y\in Y_{n}$. Note that $\\{b^{\prime}(y)\mid y\in Y\\}=\\{v_{1},\ldots,v_{n-1}\\}$. Hence, by inductive hypothesis, $b^{\prime}=\bigoplus_{i=1}^{n-1}\delta^{i}_{Y_{i}}$. Moreover, $b^{\prime}(y)=b\oplus\delta^{n}_{Y_{n}}$, and thus we conclude. Finally observe that the statement requires $\delta^{i}\neq 0$ for all $i$. We can enjoy this property by just omitting the first summand when $v_{1}=0$. ###### Example 3.19. We illustrate the definitions with small examples whose sole purpose is to get a better intuition. Consider the MV-chain $\mathbb{M}=[0,1]$, a set $Y=\\{y_{1},y_{2},y_{3},y_{4}\\}$ and a function $a\colon Y\to[0,1]$ with $a(y_{1})=0.2$, $a(y_{2})=0.4$, $a(y_{3})=0.9$, $a(y_{4})=1$. In this case $\delta_{a}=0.1$ and $[{Y}]_{a}=\\{y_{1},y_{2},y_{3}\\}$. Choose $\delta=0.1$ and $Y^{\prime}=\\{y_{1},y_{3}\\}$. Then $\alpha_{a,\delta}(Y^{\prime})$ is a function that maps $y_{1}\mapsto 0.3$, $y_{2}\mapsto 0.4$, $y_{3}\mapsto 1$, $y_{4}\mapsto 1$. We keep $\delta=0.1$ and consider a function $b\colon Y\to[0,1]$ with $b(y_{1})=0.3$, $b(y_{2})=0.45$, $b(y_{3})=b(y_{4})=1$. Then $\gamma_{a,\delta}(b)=\\{y_{1},y_{3}\\}$. (See Fig. 2 for a visual representation.) ${\color[rgb]{0.09,0.45,0.27}\definecolor[named]{pgfstrokecolor}{rgb}{0.09,0.45,0.27}a}\ =\ \raisebox{-30.0pt}{\scalebox{0.65}{\begin{picture}(0.0,0.0)\includegraphics{./galois- a- base.pdf}\end{picture}\begin{picture}(3250.0,1524.0)(429.0,-2227.0)\put(631.0,-2131.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{1}$}}}}} \put(1081.0,-2131.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{2}$}}}}} \put(1531.0,-2131.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{3}$}}}}} \put(1981.0,-2131.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{4}$}}}}} \put(3016.0,-1141.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$\delta=$}}}}} \end{picture}}}$ $\alpha_{a,\delta}\colon$ $Y^{\prime}=\\{y_{1},y_{3}\\}\quad\mapsto\quad\raisebox{-30.0pt}{\scalebox{0.65}{\begin{picture}(0.0,0.0)\includegraphics{./galois- a-plus- base.pdf}\end{picture}\begin{picture}(1824.0,1569.0)(439.0,-2272.0)\put(676.0,-2176.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{1}$}}}}} \put(1126.0,-2176.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{2}$}}}}} \put(1576.0,-2176.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{3}$}}}}} \put(2026.0,-2176.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{4}$}}}}} \end{picture}}}$ $\gamma_{a,\delta}\colon$ ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}=\raisebox{-30.0pt}{\scalebox{0.65}{\begin{picture}(0.0,0.0)\includegraphics{./galois- b- base.pdf}\end{picture}\begin{picture}(1844.0,1479.0)(429.0,-2182.0)\put(676.0,-2086.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{1}$}}}}} \put(1126.0,-2086.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{2}$}}}}} \put(1576.0,-2086.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{3}$}}}}} \put(2026.0,-2086.0){\makebox(0.0,0.0)[b]{\smash{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}$y_{4}$}}}}} \end{picture}}}\quad\mapsto\quad Y^{\prime}=\\{y_{1},y_{3}\\}$ Figure 2. Visual representation of $\alpha_{a,\delta}$ and $\gamma_{a,\delta}$ As mentioned before, a crucial result shows that for all non-expansive functions, under the assumption that $Y,Z$ are finite and the order on $\mathbb{M}$ is total, we can suitably approximate the propagation of increases. In order to state this result, a useful tool is a notion of approximation of a function. ###### Definition 3.20 ($(\delta,a)$-approximation). Let $\mathbb{M}$ be an MV-chain, let $Y$, $Z$ be finite sets and let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non-expansive function. For $a\in\mathbb{M}^{Y}$ and any $\delta\in\mathbb{M}$ we define $f_{a,\delta}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{f(a)}}$ as $f_{a,\delta}^{\\#}=\gamma_{f(a),\delta}\circ f\circ\alpha_{a,\delta}$. Given $Y^{\prime}\subseteq[{Y}]_{a}$, its image $f_{a,\delta}^{\\#}(Y^{\prime})\subseteq[{Z}]_{f(a)}$ is the set of points $z\in[{Z}]_{f(a)}$ such that $\delta\sqsubseteq f(a\oplus\delta_{Y^{\prime}})(z)\ominus f(a)(z)$, i.e., the points to which $f$ propagates an increase of the function $a$ with value $\delta$ on the subset $Y^{\prime}$. ###### Example 3.21. We continue with Example 3.19 and consider the function $f\colon[0,1]^{Y}\to[0,1]^{Y}$ with $f(b)=b\ominus 0.3$ for every $b\in[0,1]^{Y}$, which can easily be seen to be non-expansive. We again consider $a\colon Y\to[0,1]$ and $\delta=0.1$ as in Example 3.19, and $Y^{\prime}=\\{y_{1},y_{2},y_{3}\\}$. The maps $a$, $\alpha_{a,\delta}(Y^{\prime})$, $f(a)$ and $f(\alpha_{a,\delta}(Y^{\prime}))$ are given in the table below and we obtain $f_{a,\delta}^{\\#}(Y^{\prime})=\gamma_{f(a),\delta}(f(\alpha_{a,\delta}(Y^{\prime})))=\\{y_{2},y_{3}\\}$, that is only the increase at $y_{2}$ and $y_{3}$ can be propagated, while the value of $y_{1}$ is too low and $y_{4}$ is not even contained in $[{Y}]_{a}$, i.e. the domain of $f_{a,\delta}^{\\#}$, since its value is already $1.0$ and there is no slack left. \| $y_{1}$ \| $y_{2}$ \| $y_{3}$ \| $y_{4}$ ---\|---\|---\|---\|--- $a$ \| 0.2 \| 0.4 \| 0.9 \| 1.0 $\alpha_{a,\delta}(Y^{\prime})$ \| 0.3 \| 0.5 \| 1.0 \| 1.0 $f(a)$ \| 0.0 \| 0.1 \| 0.6 \| 0.7 $f(\alpha_{a,\delta}(Y^{\prime}))$ \| 0.0 \| 0.2 \| 0.7 \| 0.7 In general we have $f_{a,\delta}^{\\#}(Y^{\prime})=Y^{\prime}\cap\\{y_{2},y_{3}\\}$ if $\delta\leq\delta_{a}=0.1$, $f_{a,\delta}^{\\#}(Y^{\prime})=Y^{\prime}\cap\\{y_{2}\\}$ if $0.1<\delta\leq 0.6$ and $f_{a,\delta}^{\\#}(Y^{\prime})=\emptyset$ if $0.6<\delta$. We now show that $f_{a,\delta}^{\\#}$ is antitone in the parameter $\delta$, a non-trivial result. [anti-monotonicity] Let $\mathbb{M}$ be an MV-chain, let $Y$, $Z$ be finite sets, let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non-expansive function and let $a\in\mathbb{M}^{Y}$. For $\theta,\delta\in\mathbb{M}$, if $\theta\sqsubseteq\delta$ then $f_{a,\delta}^{\\#}\subseteq f_{a,\theta}^{\\#}$. ###### Proof 3.22. Let $Y^{\prime}\subseteq[{Y}]_{a}$ and let us prove that $f_{a,\delta}^{\\#}(Y^{\prime})\subseteq f_{a,\theta}^{\\#}(Y^{\prime})$. Take $z\in f_{a,\delta}^{\\#}(Y^{\prime})$. This means that $\delta\sqsubseteq f(a\oplus\delta_{Y^{\prime}})(z)\ominus f(a)(z)$. We have $\displaystyle\delta\sqsubseteq f(a\oplus\delta_{Y^{\prime}})(z)\ominus f(a)(z)$ [by hypothesis] $\displaystyle=f(a\oplus\theta_{Y^{\prime}}\oplus(\delta\ominus\theta)_{Y^{\prime}})(z)\ominus f(a)(z)$ $\displaystyle=f(a\oplus\theta_{Y^{\prime}}\oplus(\delta\ominus\theta)_{Y^{\prime}})(z)\ominus f(a\oplus\theta_{Y^{\prime}})(z)\oplus f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ $\displaystyle\sqsubseteq\|\\!\|{f(a\oplus\theta_{Y^{\prime}}\oplus(\delta\ominus\theta)_{Y^{\prime}})\ominus f(a\oplus\theta_{Y^{\prime}})}\|\\!\|\oplus f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ [by definition of norm and monotonicity of $\oplus$] $\displaystyle\sqsubseteq\|\\!\|{a\oplus\theta_{Y^{\prime}}\oplus(\delta\ominus\theta)_{Y^{\prime}}\ominus(a\oplus\theta_{Y^{\prime}})}\|\\!\|\oplus f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ [by non-expansiveness of $f$ and monotonicity of $\oplus$] $\displaystyle\sqsubseteq\|\\!\|{(\delta\ominus\theta)_{Y^{\prime}}}\|\\!\|\oplus f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ $\displaystyle\sqsubseteq(\delta\ominus\theta)\oplus f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ [by definition of norm] If we subtract $\delta\ominus\theta$ on both sides, we get $\delta\ominus(\delta\ominus\theta)\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$, and, as above, since, by 2.3(10), $\delta\ominus(\delta\ominus\theta)=\theta$ we conclude $\theta\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)$ which means $z\in f_{a,\theta}^{\\#}(Y^{\prime})$. Since $f_{a,\delta}^{\\#}$ increases when $\delta$ decreases and there are finitely many such functions, there must be a value $\iota_{a}^{f}$ such that all functions $f_{a,\delta}^{\\#}$ for $0\sqsubset\delta\sqsubseteq\iota_{a}^{f}$ are equal. The resulting function will be the approximation of interest. We next show how $\iota_{a}^{f}$ can be determined. We start by observing that for each $z\in[{Z}]_{f(a)}$ and $Y^{\prime}\subseteq[{Y}]_{a}$ there is a largest increase $\theta$ such that $z\in f_{a,\theta}^{\\#}(Y^{\prime})$. [largest increase for a point] Let $\mathbb{M}$ be a complete MV-chain, let $Y$, $Z$ be finite sets, let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non- expansive function and fix $a\in\mathbb{M}^{Y}$. For all $z\in[{Z}]_{f(a)}$ and $Y^{\prime}\subseteq[{Y}]_{a}$ the set $\\{\theta\in\mathbb{M}\mid z\in f_{a,\theta}^{\\#}(Y^{\prime})\\}$ has a maximum, that we denote by $\iota_{a}^{f}(Y^{\prime},z)$. ###### Proof 3.23. Let $V=\\{\theta\in\mathbb{M}\mid z\in f_{a,\theta}^{\\#}(Y^{\prime})\\}$. Expanding the definition we have that $V=\\{\theta\in\mathbb{M}\mid\theta\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(z)\ominus f(a)(z)\\}$. If we let $\eta=\sup V$, for all $\theta\in V$, since $\theta_{Y^{\prime}}\sqsubseteq\eta_{Y^{\prime}}$, clearly, by monotonicity $\theta\sqsubseteq f(a\oplus\eta_{Y^{\prime}})(z)\ominus f(a)(z)$ and therefore, by definition of supremum, $\eta\sqsubseteq f(a\oplus\eta_{Y^{\prime}})(z)\ominus f(a)(z)$, i.e., $\eta\in V$ is a maximum, as desired. ###### Lemma 3.24. Let $\mathbb{M}$ be an MV-chain, let $Y$, $Z$ be finite sets and let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non-expansive function. Let $a\in\mathbb{M}^{Y}$. For $b\in[{a},{a\oplus\delta}]$, let $b\ominus a=\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}$ be a standard form for $b\ominus a$. If $\gamma_{f(a),\delta}(f(b))\not=\emptyset$ then $Y_{n}=\gamma_{a,\delta}(b)$ and $\gamma_{f(a),\delta}(f(b))\subseteq f_{a,\delta^{n}}^{\\#}(Y_{n})$. ###### Proof 3.25. By hypothesis $\gamma_{f(a),\delta}(f(b))\not=\emptyset$. Let $z\in\gamma_{f(a),\delta}(f(b))$. This means that $\delta\sqsubseteq f(b)(z)\ominus f(a)(z)$. First observe that $\displaystyle\delta\sqsubseteq f(b)(z)\ominus f(a)(z)$ [by hypothesis] $\displaystyle\quad\sqsubseteq\|\\!\|{f(b)\ominus f(a)}\|\\!\|$ [by definition of norm] $\displaystyle\quad\sqsubseteq\|\\!\|{b\ominus a}\|\\!\|$ [by non- expansiveness of $f$] $\displaystyle\quad\sqsubseteq\delta$ [since $b\in[{a},{a\oplus\delta}]$] Hence $\|\\!\|{f(b)\ominus f(a)}\|\\!\|=\delta=\|\\!\|{b\ominus a}\|\\!\|=\bigoplus_{i=1}^{n}\delta^{i}$. Also observe that, since $\delta^{n}\neq 0$, we have $(b\ominus a)(z)=\delta$ iff $z\in Y_{n}$. In fact, if $z\in Y_{n}$ then $z\in Y_{i}$ for all $i\in\\{1,\ldots,n\\}$ and thus $(b\ominus a)(z)=\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}(z)=\bigoplus_{i=1}^{n}\delta^{i}=\delta$. Conversely, if $z\not\in Y_{n}$, then $(b\ominus a)(z)\sqsubseteq\bigoplus_{i=1}^{n-1}\delta^{i}\sqsubset\delta$. In fact, $0\sqsubset\delta^{n}$ and $\bigoplus_{i=1}^{n-1}\delta^{i}\sqsubseteq\overline{\delta^{n}}$. Thus by Lemma 2.3(8), $\bigoplus_{i=1}^{n-1}\delta^{i}\sqsubset\delta^{n}\oplus\bigoplus_{i=1}^{n-1}\delta^{i}=\bigoplus_{i=1}^{n}\delta^{i}=\delta$. Hence $Y_{n}=\gamma_{a,\delta}(b)$. Let us now show that $\gamma_{f(a),\delta}(f(b))\subseteq f_{a,\delta^{n}}^{\\#}(Y_{n})$. Given $z\in\gamma_{f(a),\delta}(f(b))$, we show that $z\in f_{a,\delta^{n}}^{\\#}(Y_{n})$. Observe that $\displaystyle\delta\sqsubseteq f(b)(z)\ominus f(a)(z)=$ [by hypothesis] $\displaystyle=f(a\oplus(b\ominus a))(z)\ominus f(a)(z)=$ [by 2.3(2), since $a\sqsubseteq b$] $\displaystyle=f(a\oplus\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}})(z)\ominus f(a)(z)=$ [by construction] $\displaystyle=f(a\oplus\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}))(z)\ominus f(a\oplus\delta^{n}_{Y_{n}})(z)\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by 2.3(2), since $f(a\oplus\delta^{n}_{Y_{n}})(z)\sqsubseteq f(a\oplus\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}})(z)$] $\displaystyle\sqsubseteq\|\\!\|{f(a\oplus\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}})\ominus f(a\oplus\delta^{n}_{Y_{n}})}\|\\!\|\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by definition of norm and monotonicity of $\oplus$] $\displaystyle\sqsubseteq\|\\!\|{a\oplus\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}\ominus(a\oplus\delta^{n}_{Y_{n}})}\|\\!\|\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by non-expansiveness of $f$ and monotonicity of $\oplus$] $\displaystyle=\|\\!\|{a\oplus\delta^{n}_{Y_{n}}\oplus\bigoplus_{i=1}^{n-1}\delta^{i}_{Y_{i}}\ominus(a\oplus\delta^{n}_{Y_{n}})}\|\\!\|\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by algebraic manipulation] $\displaystyle\sqsubseteq\|\\!\|{\bigoplus_{i=1}^{n-1}\delta^{i}_{Y_{i}}}\|\\!\|\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by 2.3(6) and monotonicity of norm] $\displaystyle\sqsubseteq\bigoplus_{i=1}^{n-1}\delta^{i}\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by 3.2(1) and the fact that $\|\\!\|{\delta^{i}_{Y_{i}}}\|\\!\|=\delta^{i}$] $\displaystyle=(\delta\ominus\delta^{n})\oplus f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$ [by construction, since $\delta^{n}=\overline{\bigoplus_{i=1}^{n-1}\delta^{i}}$] If we subtract $\delta\ominus\delta^{n}$ on both sides, we get $\delta\ominus(\delta\ominus\delta^{n})\sqsubseteq f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$, i.e., since, by 2.3(10), $\delta\ominus(\delta\ominus\delta^{n})=\delta^{n}$ we conclude $\delta^{n}\sqsubseteq f(a\oplus\delta^{n}_{Y_{n}})(z)\ominus f(a)(z)$. Hence $z\in\gamma_{f(a),\delta^{n}}(f(\alpha_{a,\delta^{n}}(Y_{n}))=f_{a,\delta^{n}}^{\\#}(Y_{n})$, which is the desired result. We can then provide an explicit definition of $\iota_{a}^{f}$ and of the approximation of a function. [$a$-approximation for a function] Let $\mathbb{M}$ be a complete MV-chain, let $Y,Z$ be finite sets and let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non- expansive function. Let $\iota_{a}^{f}=\min\\{\iota_{a}^{f}(Y^{\prime},z)\mid Y^{\prime}\subseteq[{Y}]_{a}\ \land\ z\in[{Z}]_{f(a)}\ \land\ \iota_{a}^{f}(Y^{\prime},z)\neq 0\\}\cup\\{\delta_{a}\\}$. Then for all $0\neq\delta\sqsubseteq\iota_{a}^{f}$ it holds that $f_{a,\delta}^{\\#}=f_{a,\iota_{a}^{f}}^{\\#}$. The function $f_{a,\iota_{a}^{f}}^{\\#}$ is called the _$a$ -approximation_ of $f$ and it is denoted by $f_{a}^{\\#}$. ###### Proof 3.26. Since $\delta\sqsubseteq\iota_{a}^{f}$, by 3.21 we have $f_{a,\delta}^{\\#}\supseteq f_{a,\iota_{a}^{f}}^{\\#}$. For the other inclusion let $Y^{\prime}\subseteq[{Y}]_{a}$. We have $f_{a,\delta}^{\\#}(Y^{\prime})=\\{z\in[{Z}]_{f(a)}\mid f(a\oplus\delta_{Y^{\prime}})(z)\ominus f(a)(z)\sqsupseteq\delta\\}$ by definition. Assume that there exists $z\in f_{a,\delta}^{\\#}(Y^{\prime})$ where $f(a\oplus(\iota_{a}^{f})_{Y^{\prime}})(z)\ominus f(a)(z)\not\sqsupseteq\iota_{a}^{f}$. But this is a contradiction, since $\iota_{a}^{f}$ is the minimum of all such non-zero values. We next show that indeed, for all non-expansive functions, the $a$-approximation properly approximates the propagation of increases. [approximation of non-expansive functions] Let $\mathbb{M}$ be a complete MV- chain, let $Y,Z$ be finite sets and let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a non-expansive function. Then for all $0\sqsubset\delta\in\mathbb{M}$: 1. (1) $\gamma_{f(a),\delta}\circ f\subseteq f^{\\#}_{a}\circ\gamma_{a,\delta}$ 2. (2) for $\delta\sqsubseteq\delta_{a}$: $\delta\sqsubseteq\iota_{a}^{f}$ iff $\gamma_{f(a),\delta}\circ f=f^{\\#}_{a}\circ\gamma_{a,\delta}$ $\textstyle{[a,a\oplus\delta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\gamma_{a,\delta}}$$\scriptstyle{\sqsubseteq}$$\textstyle{\mathbf{2}^{[{Y}]_{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\\#}_{a}}$$\textstyle{[f(a),f(a)\oplus\delta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{f(a),\delta}}$$\textstyle{\mathbf{2}^{[{Z}]_{f(a)}}}$ ###### Proof 3.27. 1. (1) Let $b\in[{a},{a\oplus\delta}]$. First note that whenever $\gamma_{f(a),\delta}(f(b))=\emptyset$, the desired inclusion obviously holds. If instead $\gamma_{f(a),\delta}(f(b))\not=\emptyset$, let $b\ominus a=\bigoplus_{i=1}^{n}\delta^{i}_{Y_{i}}$ be a standard form with $\delta^{n}\neq 0$. First observe that, by 3.24, we have $Y_{n}=\gamma_{a,\delta^{n}}(b)$ and $\gamma_{f(a),\delta}(f(b))\subseteq f_{a,\delta^{n}}^{\\#}(Y_{n}).$ (2) For all $z\in f_{a,\delta^{n}}^{\\#}(Y_{n})$, by definition of $\iota_{a}^{f}(Y_{n},z)$ we have that $0\sqsubset\delta_{n}\sqsubseteq\iota_{a}^{f}(Y_{n},z)$, therefore $\iota_{a}^{f}\sqsubseteq\iota_{a}^{f}(Y_{n},z)$. Moreover, $z\in f_{a,\iota_{a}^{f}(Y_{n},z)}^{\\#}(Y_{n})\subseteq f_{a,\iota_{a}^{f}}^{\\#}(Y_{n})=f_{a}^{\\#}(Y_{n})$, where the last inequality is motivated by 3.21 since $\iota_{a}^{f}\sqsubseteq\iota_{a}^{f}(Y_{n},z)$. Therefore, $f_{a,\delta^{n}}^{\\#}(Y_{n})\subseteq f_{a}^{\\#}(\gamma_{a,\delta}(b))$, which combined with (2) gives the desired result. 2. (2) For (2), we first show the direction from left to right. Assume that $\delta\sqsubseteq\iota_{a}^{f}$. By (a) clearly, $\gamma_{f(a),\delta}\circ f(b)\subseteq f^{\\#}_{a}\circ\gamma_{a,\delta}(b)$. For the converse inclusion, note that: $\displaystyle f_{a}^{\\#}(\gamma_{a,\delta}(b))$ [by definition of $f_{a}^{\\#}$] $\displaystyle\quad=f_{a,\iota_{a}^{f}}^{\\#}(\gamma_{a,\delta}(b))\subseteq$ [by 3.21, since $\delta\sqsubseteq\iota_{a}^{f}$] $\displaystyle\quad\subseteq f_{a,\delta}^{\\#}(\gamma_{a,\delta}(b))$ [by definition of $f_{a,\delta}^{\\#}$] $\displaystyle\quad=\gamma_{f(a),\delta}(f(\alpha_{a,\delta}(\gamma_{a,\delta}(b))))$ [since $\alpha_{a,\delta}\circ\gamma_{a,\delta}(b)\sqsubseteq b$] $\displaystyle\quad\subseteq\gamma_{f(a),\delta}(f(b))$ as desired. For the other direction, assume $\gamma_{f(a),\delta}\circ f(b)=f_{a}^{\\#}\circ\gamma_{a,\delta}(b)$ holds for all $b\in[a,a\oplus\delta]$. Now, for every $Y^{\prime}\subseteq[{Y}]_{a}$ we have $f_{a,\delta}^{\\#}(Y^{\prime})=\gamma_{f(a),\delta}\circ f\circ\alpha_{a,\delta}(Y^{\prime})=f_{a}^{\\#}\circ\gamma_{a,\delta}\circ\alpha_{a,\delta}(Y^{\prime})$. We also have $\gamma_{a,\delta}\circ\alpha_{a,\delta}(Y^{\prime})=Y^{\prime}$ (see proof of 3.13), thus $f_{a,\delta}^{\\#}(Y^{\prime})=f_{a}^{\\#}(Y^{\prime})$. For any $\delta$ with $\iota_{a}^{f}\sqsubset\delta\sqsubseteq\delta_{a}$ there exists $Y^{\prime}\subseteq[{Y}]_{a}$ and $z\in[{Z}]_{f(a)}$ with $z\in f_{a}^{\\#}(Y^{\prime})$ but $z\notin f_{a,\delta}^{\\#}(Y^{\prime})$, by definition of $\iota_{a}^{f}$. Therefore $\delta\sqsubseteq\iota_{a}^{f}$ has to hold. Note that if $Y=Z$ and $a$ is a fixpoint of $f$, i.e., $a=f(a)$, then condition (1) above corresponds exactly to soundness in the sense of abstract interpretation [cc:ai-unified-lattice-model]. Moreover, when $\delta\sqsubseteq\delta_{a}$ and thus $\langle\alpha_{a,\delta},\gamma_{a,\delta}\rangle$ is a Galois connection, $f_{a,\delta}^{\\#}=\gamma_{a,\delta}\circ f\circ\alpha_{a,\delta}$ is the best correct approximation of $f$. In particular, when $\delta\sqsubseteq\iota_{a}^{f}$, such a best correct approximation is $f_{a}^{\\#}$, the $a$-approximation of $f$, i.e., it becomes independent of $\delta$, and condition (2) corresponds to ($\gamma$-)completeness [GRS:MAIC] (see also Section 2). ## 4\. Proof rules In this section we formalise the proof technique outlined in the introduction for showing that a fixpoint is the largest and, more generally, for checking over-approximations of greatest fixpoints of non-expansive functions. ### 4.1. Proof rules for fixpoints Consider a monotone function $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ for some finite set $Y$. We first focus on the problem of establishing whether some given fixpoint $a$ of $f$ coincides with $\nu f$ (without explicitly knowing $\nu f$), and, in case it does not, finding an “improvement”, i.e., a post- fixpoint of $f$, larger than $a$. We first prove a technical lemma. ###### Lemma 4.1. Let $\mathbb{M}$ be a complete MV-chain, $Y$ a finite set and $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ be a non-expansive function. Let $a\in\mathbb{M}^{Y}$ be a pre-fixpoint of $f$, let $f_{a}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Y}]_{f(a)}}$ be the $a$-approximation of $f$ (3.25). Assume $\nu f\not\sqsubseteq a$ and let $Y^{\prime}=\\{y\in[{Y}]_{a}\mid\nu f(y)\ominus a(y)=\|\\!\|{\nu f\ominus a}\|\\!\|\\}$. Then for all $y\in Y^{\prime}$ it holds $a(y)=f(a)(y)$ and $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})$. ###### Proof 4.2. Let $\delta=\|\\!\|{\nu f\ominus a}\|\\!\|$. Assume $\nu f\not\sqsubseteq a$, i.e., there exists $y\in Y$ such that $\nu f(y)\not\sqsubseteq a(y)$. Since the order is total, this means that $a(y)\sqsubset\nu f(y)$. Hence, by 2.3(5), $\nu f(y)\ominus a(y)\sqsupset 0$. Then $\delta=\|\\!\|{\nu f\ominus a}\|\\!\|\sqsupset 0$. Moreover, for all $y\in Y^{\prime}$, $\overline{a(y)}=1\ominus a(y)\sqsupseteq\nu f(y)\ominus a(y)=\delta$. First, observe that $\nu f\sqsubseteq a\oplus\delta,$ (3) since for all $y\in Y$ $\nu f(y)\ominus a(y)\sqsubseteq\delta$ by definition of $\delta$ and then (3) follows from 2.3(7). Concerning the first part, let $y\in Y^{\prime}$. Since $a$ is a pre-fixpoint, $f(a)(y)\sqsubseteq a(y)$. Assume by contradiction that $f(a)(y)\sqsubset a(y)$. Then we have $\displaystyle f(a\oplus\delta)(y)=$ [by 2.3(2), since $f$ is monotone and thus $f(a)\sqsubseteq f(a\oplus\delta)$] $\displaystyle=f(a)(y)\oplus(f(a\oplus\delta)(y)\ominus f(a)(y))$ [since $f$ is non-expansive, by 3.7, hence $f(a\oplus\delta)(y)\ominus f(a)(y)\sqsubseteq\delta$] $\displaystyle\sqsubseteq f(a)(y)\oplus\delta$ [by $f(a)(y)\sqsubset a(y)$, $\delta\sqsubseteq\overline{a(y)}$ and 2.3(6)] $\displaystyle\sqsubset a(y)\oplus\delta$ [by 2.3(2) since $a(y)\sqsubseteq\nu f(y)$ and $\delta=\nu f(y)\ominus a(y)$] $\displaystyle=\nu f(y)$ $\displaystyle=f(\nu f)(y)$ [since $\nu f\sqsubseteq a\oplus\delta$ (3) and $f$ monotone] $\displaystyle\sqsubseteq f(a\oplus\delta)(y)$ i.e., a contradiction. Hence it must be $a(y)=f(a)(y)$. For the second part, in order to show $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})$, we let $b=\nu f\sqcup a$. By using (3) we immediately have that $b\in[{a},{a\oplus\delta}]$. We next prove that $Y^{\prime}=\gamma_{a,\delta}(b)$. We show separately the two inclusions. If $y\in Y^{\prime}$ then $a(y)\sqsubset\nu f(y)$ and thus $b(y)=a(y)\sqcup\nu f(y)=\nu f(y)$ and thus $b(y)\ominus a(y)=\nu f(y)\ominus a(y)=\delta$. Hence $y\in\gamma_{a,\delta}(b)$. Conversely, if $y\in\gamma_{a,\delta}(b)$, then $a(y)\sqsubset\nu f(y)$. In fact, if it were $a(y)\sqsupseteq\nu f(y)$, then, by definition of $b$ we would have $b(y)=a(y)$ and $b(y)\ominus a(y)=0\not\sqsupseteq\delta$. Therefore, $b(y)=\nu f(y)$ and thus $\nu f(y)\ominus a(y)=b(y)\ominus a(y)\sqsupseteq\delta$, whence $y\in Y^{\prime}$. We can now conclude. In fact, since $f$ is non-expansive, by 3.26(1), we have $\gamma_{f(a),\delta}(f(b))\subseteq f^{\\#}_{a}(Y^{\prime}).$ Moreover $Y^{\prime}\subseteq\gamma_{f(a),\delta}(f(b))$. In fact, let $y\in Y^{\prime}$, i.e., $y\in[{Y}]_{a}$ and $\delta\sqsubseteq b(y)\ominus a(y)$. Since $a(y)=f(a)(y)$, we have that $y\in[{Y}]_{f(a)}$. In order to conclude that $y\in\gamma_{f(a),\delta}(f(b))$ it is left to show that $\delta\sqsubseteq f(b)(y)\ominus f(a)(y)$. We have $\displaystyle f(b)(y)\ominus f(a)(y)$ $\displaystyle=f(b)(y)\ominus a(y)$ [since $y\in Y^{\prime}$] $\displaystyle=f(\nu f\sqcup a)(y)\ominus a(y)$ [definition of $b$] $\displaystyle\sqsupseteq(f(\nu f)(y)\sqcup f(a)(y))\ominus a(y)$ [properties of $\sqcup$] $\displaystyle=(\nu f(y)\sqcup a(y))\ominus a(y)$ [since $\nu f$ fixpoint and $y\in Y^{\prime}$] $\displaystyle=b(y)\ominus a(y)$ [definition of $b$] $\displaystyle\sqsupseteq\delta$ [since $y\in Y^{\prime}$] Combining the two inclusions, we have $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})$, as desired. Observe that when $a$ is a fixpoint, $[{Y}]_{a}=[{Y}]_{f(a)}$ and thus the $a$-approximation of $f$ (3.25) is an endo-function $f_{a}^{\\#}:[{Y}]_{a}\to[{Y}]_{a}$. We have the following result, which relies on the fact that due to 3.26 $\gamma_{a,\delta}$ preserves fixpoints (of $f$ and $f_{a}^{\\#}$). [soundness and completeness for fixpoints] Let $\mathbb{M}$ be a complete MV- chain, $Y$ a finite set and $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ be a non- expansive function. Let $a\in\mathbb{M}^{Y}$ be a fixpoint of $f$. Then $\nu f_{a}^{\\#}=\emptyset$ if and only if $a=\nu f$. ###### Proof 4.3. Let $a$ be a fixpoint of $f$ and assume that $a=\nu f$. For $\delta=\iota_{a}^{f}\sqsubseteq\delta_{a}$, according to 3.13, we have a Galois connection: $\mathbf{2}^{[{Y}]_{a}}$$[{a},{a+\delta}]$$\alpha_{a,\delta}$$\gamma_{a,\delta}$$f^{\\#}_{a}$$f_{a,\delta}$ Since $a$ is a fixpoint, then $[{Y}]_{f(a)}=[{Y}]_{a}$ and, by 3.26(2), $\gamma_{a,{\delta}}\circ f=\gamma_{f(a),{\delta}}\circ f=f_{a}^{\\#}\circ\gamma_{a,{\delta}}$. Therefore by [CC:TLA, Proposition 14], $\nu f^{\\#}_{a}=\gamma_{a,{\delta}}(\nu f)$. Recall that $\gamma_{a,{\delta}}(\nu f)=\\{y\in Y\mid{\delta}\sqsubseteq\nu f(y)\ominus a(y)\\}$. Since $a=\nu f$ and ${\delta}\sqsupset 0$, we know that $\gamma_{a,{\delta}}(\nu f)=\emptyset$ and we conclude $\nu f^{\\#}_{a}=\emptyset$, as desired. Conversely, in order to prove that if $\nu f^{\\#}_{a}=\emptyset$ then $a=\nu f$, we prove the contrapositive. Assume that $a\neq\nu f$. Since $a$ is a fixpoint and $\nu f$ is the largest, this means that $a\sqsubset\nu f$ and thus $\|\\!\|{\nu f\ominus a}\|\\!\|\neq 0$. Consider $Y^{\prime}=\\{y\in[{Y}]_{a}\mid\nu f(y)\ominus a(y)=\|\\!\|{\nu f\ominus a}\|\\!\|\\}\neq\emptyset$. By 4.1, $Y^{\prime}$ is a post-fixpont of $f_{a}^{\\#}$, i.e., $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})$, and thus $\nu f^{\\#}_{a}\supseteq Y^{\prime}$ which implies $\nu f^{\\#}_{a}\neq\emptyset$, as desired. Whenever $a$ is a fixpoint, but not yet the largest fixpoint of $f$, from the result above $\nu f^{\\#}_{a}\neq\emptyset$. Intuitively, $\nu f^{\\#}_{a}$ is the set of points where $a$ can still be “improved”. More precisely, we can show that $a$ can be increased on the points in $\nu f^{\\#}_{a}$ producing a post-fixpoint of $f$. In order to determine how much $a$ can be increased we proceed similarly to what we have done for defining $\iota_{a}^{f}$ (Lemma 3.25), but restricting the attention to $\nu f_{a}^{\\#}$ instead of considering the full $[{Y}]_{a}$. ###### Definition 4.4 (largest increase for a subset). Let $\mathbb{M}$ be a complete MV-chain and let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ be a non-expansive function, where $Y$ is a finite set and let $a\in\mathbb{M}^{Y}$. For $Y^{\prime}\subseteq Y$, we define $\delta_{a}(Y^{\prime})=\min\\{\overline{a(y)}\mid y\in Y^{\prime}\\}$ and $\iota_{a}^{f}(Y^{\prime})=\min\\{\iota_{a}^{f}(Y^{\prime},y)\mid y\in Y^{\prime}\\}$. Note that the increase $\iota_{a}^{f}$, used in 3.25 for defining the $a$-approximation $f_{a}^{\\#}$, is $\iota_{a}^{f}=\iota_{a}^{f}([{Y}]_{a})$. ###### Example 4.5. We intuitively explain the computation of the values in the definition above. Let $g\colon[0,1]^{Y}\to[0,1]^{Y}$ with $g(b)=b\oplus 0.1$, where the set $Y$ and the function $a\in[0,1]^{Y}$ are as in Example 3.19. Let $Y^{\prime}=\\{y_{1},y_{2}\\}$. Then $\delta_{a}(Y^{\prime})=0.6$ and $\iota_{a}^{g}(Y^{\prime})=0.5$, i.e., since $g$ adds $0.1$, we can propagate an increase of at most $0.5$. We next prove that when $a\in\mathbb{M}^{Y}$ is a fixpoint of $f$ and $Y^{\prime}=\nu f_{a}^{\\#}$, the value $\iota_{a}^{f}(Y^{\prime})$ is the largest increase $\delta$ below $\delta_{a}(Y^{\prime})$ such that $a\oplus\delta_{Y^{\prime}}$ is a post-fixpoint of $f$. [from a fixpoint to larger post-fixpoint] Let $\mathbb{M}$ be a complete MV- chain, $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ a non-expansive function, $a\in\mathbb{M}$ a fixpoint of $f$, and let $Y^{\prime}=\nu f_{a}^{\\#}$ be the greatest fixpoint of the corresponding $a$-approximation. Then $\iota_{a}^{f}(Y^{\prime})\sqsubseteq\delta_{a}(Y^{\prime})$. Moreover, for all $\theta\sqsubseteq\iota_{a}^{f}(Y^{\prime})$ the function $a\oplus\theta_{Y^{\prime}}$ is a post-fixpoint of $f$, while for $\iota_{a}^{f}(Y^{\prime})\sqsubset\theta\sqsubseteq\delta_{a}(Y^{\prime})$, it is not. ###### Proof 4.6. We first prove that $\iota_{a}^{f}(Y^{\prime})\sqsubseteq\delta_{a}(Y^{\prime})$. Observe that for all $y\in Y^{\prime}$ and $\delta\in\mathbb{M}$, if $y\in f_{a,\delta}^{\\#}(Y^{\prime})$, by definition of $f_{a,\delta}^{\\#}$, it holds that $\delta\sqsubseteq f(a\oplus\delta_{Y^{\prime}})(y)\ominus f(a)(y)=f(a\oplus\delta_{Y^{\prime}})(y)\ominus a(y)\sqsubseteq 1\ominus a(y)=\overline{a(y)}$, where the second equality is motivated by the fact that $a$ is a fixpoint. Therefore for all $y\in Y^{\prime}$ we have $\max\\{\delta\in\mathbb{M}\mid y\in f_{a,\delta}^{\\#}(Y^{\prime})\\}\sqsubseteq\overline{a(y)}$ and thus $\iota_{a}^{f}(Y^{\prime})=\min_{y\in Y^{\prime}}\max\\{\delta\in\mathbb{M}\mid y\in f_{a,\delta}^{\\#}(Y^{\prime})\\}\sqsubseteq\min_{y\in Y^{\prime}}\overline{a(y)}=\delta_{a}(Y^{\prime})$, as desired. Given $\theta\sqsubseteq\iota_{a}^{f}(Y^{\prime})$, let us prove that $a\oplus\theta_{Y^{\prime}}$ is a post-fixpoint of $f$, i.e., $a\oplus\theta_{Y^{\prime}}\sqsubseteq f(a\oplus\theta_{Y^{\prime}})$. If $y\in Y^{\prime}$, since $\theta\sqsubseteq\iota_{a}^{f}(Y^{\prime})$, by definition of $\iota_{a}^{f}(Y^{\prime})$, we have $\theta\sqsubseteq\max\\{\delta\in\mathbb{M}\mid y\in f_{a,\delta}^{\\#}(Y^{\prime})\\}$ and thus, by antimonotonicity of $f_{a,\delta}^{\\#}$ with respect to $\delta$, we have $y\in f_{a,\theta}^{\\#}(Y^{\prime})$. This means that $\theta\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(y)\ominus f(a)(y)=f(a\oplus\theta_{Y^{\prime}})(y)\ominus a(y)$, where the last passage uses the fact that $a$ is a fixpoint. Adding $a(y)$ on both sides and using Lemma 2.3(2), we obtain $a(y)\oplus\theta\sqsubseteq(f(a\oplus\theta_{Y^{\prime}})(y)\ominus a(y))\oplus a(y)=f(a\oplus\theta_{Y^{\prime}})(y)$. Since $y\in Y^{\prime}$, $(a\oplus\theta_{Y^{\prime}})(y)=a(y)\oplus\theta$ and thus $(a\oplus\theta_{Y^{\prime}})(y)\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(y)$, as desired. If instead, $y\not\in Y^{\prime}$, clearly $(a\oplus\theta_{Y^{\prime}})(y)=a(y)=f(a)(y)\sqsubseteq f(a\oplus\theta_{Y^{\prime}})(y)$, where we again use the fact that $a$ is a fixpoint and monotonicity of $f$. Lastly, we have to show that if $\iota_{a}^{f}(Y^{\prime})\sqsubset\theta\sqsubseteq\delta_{a}(Y^{\prime})$, then $a\oplus\theta_{Y^{\prime}}$ is a not a post-fixpoint of $f$. By definition of $\iota_{a}^{f}(Y^{\prime})$, from the fact that $\iota_{a}^{f}(Y^{\prime})\sqsubset\theta$, we deduce that $\max\\{\delta\in\mathbb{M}\mid y\in f_{a,\delta}^{\\#}(Y^{\prime})\\}\sqsubset\theta$ for some $y\in Y^{\prime}$ and thus $y\not\in f_{a,\theta}^{\\#}(Y^{\prime})$. By definition of $f_{a,\theta}^{\\#}$ and totality of $\sqsubseteq$, the above means $\theta\sqsupset f(a\oplus\theta_{Y^{\prime}})(y)\ominus f(a)(y)=f(a\oplus\theta_{Y^{\prime}})(y)\ominus a(y)$, since $a$ is a fixpoint of $f$. Since $\theta\sqsubseteq\delta_{a}(Y^{\prime})$, we can add $a(y)$ on both sides and, by Lemma 2.3(8), we obtain $a(y)\oplus\theta\sqsupset f(a\oplus\theta_{Y^{\prime}})(y)$. Since $y\in Y^{\prime}$, the left-hand side is $(a\oplus\theta_{Y^{\prime}})(y)$. Hence we conclude that indeed $a\oplus\theta_{Y^{\prime}}$ is not a post fixpoint. Using these results one can perform an alternative fixpoint iteration where we iterate to the largest fixpoint from below: start with a post-fixpoint $a_{0}\sqsubseteq f(a_{0})$ (which is clearly below $\nu f$) and obtain, by (possibly transfinite) iteration, an ascending chain that converges to $a$, the least fixpoint above $a_{0}$. Now check with 4.2 whether $Y^{\prime}=\nu f_{a}^{\\#}=\emptyset$. If so, we have reached $\nu f=a$. If not, $\alpha_{a,\iota_{a}^{f}(Y^{\prime})}(Y^{\prime})=a\oplus(\iota_{a}^{f}(Y^{\prime}))_{Y^{\prime}}$ is again a post-fixpoint (cf. 4.5) and we continue this procedure until – for some ordinal – we reach the largest fixpoint $\nu f$, for which we have $\nu f_{\nu f}^{\\#}=\emptyset$. In order to make the above approach as efficient as possible, the question naturally arises asking whether ${\iota_{a}^{f}(Y^{\prime})}$ is the largest possible increase such that $a\oplus(\iota_{a}^{f}(Y^{\prime}))_{Y^{\prime}}$ is again a post-fixpoint of $f$, even if we allow increases above $\delta_{a,Y^{\prime}}$. The answer is negative, however, while the set of increases below $\delta_{a,Y^{\prime}}$ which lead to post-fixpoints is downward-closed, as proved in 4.5, this is not the case for those above $\delta_{a,Y^{\prime}}$. This is shown later in Example 6.3, for the dual case of least fixpoints. We believe that a binary search bounded by $\delta_{a,\nu f_{a}^{\\#}}$ can be the most efficient way to find the largest propagation or at least some satisfying level of propagation. The search surely works thanks to the fact that the set of propagations below $\delta_{a,\nu f_{a}^{\\#}}$ is downward- closed. ### 4.2. Proof rules for pre-fixpoints Interestingly, the soundness result in 4.2 can be generalised to the case in which $a$ is a pre-fixpoint instead of a fixpoint. In this case, the $a$-approximation for a function $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ is a function $f_{a}^{\\#}:[{Y}]_{a}\to[{Y}]_{f(a)}$ where domain and codomain are different, hence it would not be meaningful to look for fixpoints. However, as explained below, it can be restricted to an endo-function. [soundness for pre-fixpoints] Let $\mathbb{M}$ be a complete MV-chain, $Y$ a finite set and $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$ be a non-expansive function. Given a pre-fixpoint $a\in\mathbb{M}^{Y}$ of $f$, let $[{Y}]_{a=f(a)}=\\{y\in[{Y}]_{a}\mid a(y)=f(a)(y)\\}$. Let us define $f^{}_{a}:[{Y}]_{a=f(a)}\to[{Y}]_{a=f(a)}$ as $f^{}_{a}(Y^{\prime})=f_{a}^{\\#}(Y^{\prime})\cap[{Y}]_{a=f(a)}$, where $f_{a}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Y}]_{f(a)}}$ is the $a$-approximation of $f$. If $\nu f_{a}^{}=\emptyset$ then $\nu f\sqsubseteq a$. ###### Proof 4.7. We prove the contrapositive, i.e., we show that $\nu f\not\sqsubseteq a$ allows us to derive that $\nu f^{}_{a}\neq\emptyset$. Assume $\nu f\not\sqsubseteq a$, i.e., there exists $y\in Y$ such that $\nu f(y)\not\sqsubseteq a(y)$. Since the order is total, this means that $a(y)\sqsubset\nu f(y)$. Hence, by 2.3(5), $\nu f(y)\ominus a(y)\sqsupset 0$. Then $\delta=\|\\!\|{\nu f\ominus a}\|\\!\|\sqsupset 0$. Consider $Y^{\prime}=\\{y\in Y_{a}\mid\nu f(y)\ominus a(y)=\|\\!\|{\nu f\ominus a}\|\\!\|\\}\neq\emptyset$. By 4.1, $Y^{\prime}$ is a post-fixpont of $f_{a}^{\\#}$, i.e., $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})$, and thus $Y^{\prime}\subseteq\nu f^{\\#}_{a}$. Moreover, for all $y\in Y^{\prime}$, $a(y)=f(a)(y)$, i.e., $Y^{\prime}\subseteq[{Y}]_{a=f(a)}$. Therefore we conclude $Y^{\prime}\subseteq f_{a}^{\\#}(Y^{\prime})\cap[{Y}]_{a=f(a)}=f_{a}^{}(Y^{\prime})$, i.e., $Y^{\prime}$ is a post-fixpoint also for $f_{a}^{}$, and thus $\nu f_{a}^{}\supseteq Y^{\prime}\neq\emptyset$, as desired. The reason why we can limit our attention to the set of points where $a(y)=f(a)(y)$ is as follows. Observe that, since $a$ is a pre-fixpoint and $\ominus$ is antitone in the second argument, $\nu f\ominus a\sqsubseteq\nu f\ominus f(a)$. Thus $\|\\!\|{\nu f\ominus a}\|\\!\|\sqsubseteq\|\\!\|{\nu f\ominus f(a)}\|\\!\|=\|\\!\|{f(\nu f)\ominus f(a)}\|\\!\|\sqsubseteq\|\\!\|{\nu f\ominus a}\|\\!\|$, where the last passage is motivated by non-expansiveness of $f$. Therefore $\|\\!\|{\nu f\ominus a}\|\\!\|=\|\\!\|{\nu f\ominus f(a)}\|\\!\|$. From this we can deduce that, if $\nu f$ is strictly larger than $a$ on some points, surely some of these points are in $[{Y}]_{a=f(a)}$. In particular, all points $y_{0}$ such that $\nu f(y_{0})\ominus a(y_{0})=\|\\!\|{\nu f\ominus a}\|\\!\|$ are necessarily in $[{Y}]_{a=f(a)}$. Otherwise, we would have $f(a)(y_{0})\sqsubset a(y_{0})$ and thus $\|\\!\|{\nu f\ominus a}\|\\!\|=\nu f(y_{0})\ominus a(y_{0})\sqsubset\nu f(y_{0})\ominus f(a)(y_{0})\sqsubset\|\\!\|{\nu f\ominus f(a)}\|\\!\|$. Completeness does not generalise to pre-fixpoints, i.e., it is not true that if $a$ is a pre-fixpoint of $f$ and $\nu f\sqsubseteq a$ then $\nu f^{}_{a}=\emptyset$. A pre-fixpoint might contain slack even though it is above the greatest fixpoint. A counterexample is in 6.14. ### 4.3. The dual view for least fixpoints The theory developed so far can be easily dualised to check under- approximations of least fixpoints. Given a complete MV-algebra $\mathbb{M}=(M,\oplus,0,\overline{(\cdot)})$ and a non-expansive function $f:\mathbb{M}^{Y}\to\mathbb{M}^{Y}$, in order to show that a post-fixpoint $a\in\mathbb{M}^{Y}$ is such that $a\sqsubseteq\mu f$ we can in fact simply work in the dual MV-algebra, ${\mathbb{M}}^{\mathit{op}}=(M,\sqsupseteq,\otimes,\overline{(\cdot)},1,0)$. Since $\oplus$ could be the “standard” operation on $\mathbb{M}$, it is convenient to formulate the conditions using $\oplus$ and $\ominus$ and the original order. The notation for the dual case is obtained from that of the original (primal) case, exchanging subscripts and superscripts. The pair of functions $\langle\alpha^{a,\theta},\gamma^{a,\theta}\rangle$ is as follows. Let $a:Y\to\mathbb{M}$ and $0\sqsubset\theta\in\mathbb{M}$. The set $[{Y}]^{a}=\\{y\in Y\mid a(y)\neq 0\\}$ and $\delta^{a}=\min\\{a(y)\mid y\in[{Y}]^{a}\\}$ The target of the approximation is $[{a},{a\otimes\theta}]$ in the reverse order, hence $[{a\otimes\theta},{a}]$ in the original order. Recall that $a\otimes\theta=\overline{\overline{a}\oplus\overline{\theta}}=a\ominus\overline{\theta}$. Hence we obtain $\mathbf{2}^{[{Y}]^{a}}$$[{a\ominus\overline{\theta}},{a}]$$\alpha^{a,\theta}$$\gamma^{a,\theta}$ For $Y^{\prime}\in\mathbf{2}^{[{Y}]^{a}}$ we define $\alpha^{a,\theta}(Y^{\prime})=a\otimes\theta_{Y^{\prime}}=a\ominus\overline{\theta}_{Y^{\prime}}$ Instead $\gamma^{a,\theta}(b)=\\{y\in Y\mid\theta\sqsupseteq b(y)\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}a(y)\\}$ where $\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}$ is the subtraction in the dual MV-algebra. Observe that $x\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}y=\overline{\overline{x}\otimes y}=\overline{x\oplus\overline{y}}=\overline{y\ominus x}$. Hence $\theta\sqsupseteq b(y)\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}a(y)$ iff $a(y)\ominus b(y)\sqsupseteq\overline{\theta}$. Thus for $b\in[{a\ominus\overline{\theta}},{a}]$ we have $\gamma^{a,\theta}(b)=\\{y\in Y\mid\theta\sqsupseteq b(y)\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}a(y)\\}=\\{y\in Y\mid a(y)\ominus b(y)\sqsupseteq\overline{\theta}\\}$. Let $f:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ be a monotone function. The norm becomes $\|\\!\|{a}\|\\!\|=\min\\{a(y)\mid y\in Y\\}$. Non-expansiveness (Def. 3.4) in the dual MV-algebra becomes: for all $a,b\in\mathbb{M}^{Y}$, $\|\\!\|{f(b)\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}f(a)}\|\\!\|\sqsupseteq\|\\!\|{b\mathrel{\mathchoice{{\ooalign{$\displaystyle\ominus$\cr\hfil$\displaystyle\div$\hfil\cr}}}{{\ooalign{$\textstyle\ominus$\cr\hfil$\textstyle\div$\hfil\cr}}}{{\ooalign{$\scriptstyle\ominus$\cr\hfil$\scriptstyle\div$\hfil\cr}}}{{\ooalign{$\scriptscriptstyle\ominus$\cr\hfil$\scriptscriptstyle\div$\hfil\cr}}}}a}\|\\!\|$, which in turn is $\min\\{\overline{f(a)\ominus f(b)}\mid y\in Y\\}\sqsupseteq\min\\{\overline{a(y)\ominus b(y)}\mid y\in Y\\}$ i.e., $\|\\!\|{f(a)\ominus f(b)}\|\\!\|\sqsubseteq\|\\!\|{a\ominus b}\|\\!\|$, which coincides with non-expansiveness in the original MV-algebra. Observe that, instead of taking a generic $\theta\sqsubset 1$ and then working with $\bar{\theta}$, we can directly take $0\sqsubset\theta$ and replace everywhere $\bar{\theta}$ with $\theta$. While the approximation of a function in the primal sense are called $f_{a}^{\\#}$, the approximations in the dual sense will be denoted by $f^{a}_{\\#}$. We can also dualise 4.5 and obtain that, whenever $a$ is a fixpoint and $Y^{\prime}=\nu f^{a}_{\\#}\neq\emptyset$, then $a\ominus\theta_{Y^{\prime}}$ is a pre-fixpoint, where $\theta=\iota_{a}^{f}(Y^{\prime})$ is suitably defined, dualising Definition. 4.4. ## 5\. (De)Composing functions and approximations Given a non-expansive function $f$ and a (pre/post-)fixpoint $a$, it is often non-trivial to determine the corresponding approximations. However, non- expansive functions enjoy good closure properties (closure under composition, and closure under disjoint union) and we will see that the same holds for the corresponding approximations. Furthermore, it turns out that the functions needed in the applications can be obtained from just a few templates. This gives us a toolbox for assembling approximations with relative ease. We start by introducing some basic functions, which will be used as the building blocks for the functions needed in the applications. ###### Definition 5.1 (basic functions). Let $\mathbb{M}$ be an MV-chain and let $Y$, $Z$ be finite sets. 1. (1) _Constant:_ For a fixed $k\in\mathbb{M}^{Z}$, we define $c_{k}:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ by $c_{k}(a)=k$ 2. (2) _Reindexing:_ For $u:Z\to Y$, we define $u^{}:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ by $u^{}(a)=a\circ u.$ 3. (3) _Min/Max:_ For $\mathcal{R}\subseteq Y\times Z$, we define $\min\nolimits_{\mathcal{R}},\max\nolimits_{\mathcal{R}}:\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ by $\min\nolimits_{\mathcal{R}}(a)(z)=\min\nolimits_{y\mathcal{R}z}a(y)\qquad\max\nolimits_{\mathcal{R}}(a)(z)=\max\nolimits_{y\mathcal{R}z}a(y)$ 4. (4) _Average:_ Call a function $p:Y\to\mathbb{M}$ a distribution when for all $y\in Y$, it holds $\overline{p(y)}=\bigoplus_{y^{\prime}\in Y\backslash\\{y\\}}p(y^{\prime})$ and let $\mathcal{D}(Y)$ be the set of distributions. Assume that $\mathbb{M}$ is endowed with an additional operation $\odot$ such that $(\mathbb{M},\odot,1)$ is a commutative monoid, for $x,y\in\mathbb{M}$, $x\odot y\sqsubseteq x$, and $x\odot y=0$ iff $x=0$ or $y=0$, and $\odot$ weakly distributes over $\oplus$, i.e., for all $x,y,z\in\mathbb{M}$ with $y\sqsubseteq\overline{z}$, $x\odot(y\oplus z)=x\odot y\oplus x\odot z$. For a finite set $D\subseteq\mathcal{D}(Y)$, we define $\mathrm{av}_{D}:\mathbb{M}^{Y}\to\mathbb{M}^{D}$ by $\mathrm{av}_{D}(a)(p)=\bigoplus_{y\in Y}p(y)\odot a(y)$ A particularly interesting subcase of (3) is when we take as relation the _belongs to_ relation $\in\ \subseteq Y\times\mathbf{2}^{Y}$. In this way we obtain functions for selecting the minimum and the maximum, respectively, of an input function over a set $Y^{\prime}\subseteq Y$, that is, the functions $\min\nolimits_{\in},\max\nolimits_{\in}:\mathbb{M}^{Y}\to\mathbb{M}^{\mathbf{2}^{Y}}$, defined as $\min\nolimits_{\in}(a)(Y^{\prime})=\min\limits_{y\in Y^{\prime}}a(y)\qquad\qquad\max\nolimits_{\in}(a)(Y^{\prime})=\max\limits_{y\in Y^{\prime}}a(y)$ Also note that in the definition of $\mathit{av}_{D}$, the operation $\odot$ is necessarily monotone. In fact, if $y\sqsubseteq y^{\prime}$ then, by Lemma 2.3(2), we have $y^{\prime}=y\oplus(y^{\prime}\ominus y)$. Therefore $x\odot y\sqsubseteq x\odot y\oplus x\odot(y^{\prime}\ominus y)=x\odot(y\oplus(y^{\prime}\ominus y))=x\odot y^{\prime}$, where the second passage holds by weak distributivity. The basic functions can be shown to be non-expansive. ###### Proposition 5.2. The basic functions from Def. 5.1 are all non-expansive. ###### Proof 5.3. * • _Constant functions:_ immediate. * • _Reindexing:_ Let $u:Z\to Y$. For all $a,b\in\mathbb{M}^{Y}$, we have $\displaystyle\|\\!\|{u^{}(b)\ominus u^{}(a)}\|\\!\|$ $\displaystyle=\max_{z\in Z}(b(u(z))\ominus a(u(z)))$ $\displaystyle\sqsubseteq\max_{y\in Y}(b(y)\ominus a(y))$ [since $u(Z)\subseteq Y$] $\displaystyle=\|\\!\|{b\ominus a}\|\\!\|$ [by def. of norm] * • _Minimum:_ Let $\mathcal{R}\subseteq Y\times Z$ be a relation. For all $a,b\in\mathbb{M}^{Y}$, we have $\|\\!\|{\min\nolimits_{\mathcal{R}}(b)\ominus\min\nolimits_{\mathcal{R}}(a)}\|\\!\|=\max_{z\in Z}(\min_{y\mathcal{R}z}b(y)\ominus\min_{y\mathcal{R}z}a(y))$ Observe that $\max_{z\in Z}(\min_{y\mathcal{R}z}b(y)\ominus\min_{y\mathcal{R}z}a(y))=\max_{z\in Z^{\prime}}(\min_{y\mathcal{R}z}b(y)\ominus\min_{y\mathcal{R}z}a(y))$ where $Z^{\prime}=\\{z\in Z\mid\exists\,y\in Y.\,y\mathcal{R}z\\}$, since on every other $z\in Z\backslash Z^{\prime}$ the difference would be $0$. Now, for every $z\in Z^{\prime}$, take $y_{z}\in Y$ such that $y_{z}\mathcal{R}z$ and $a(y_{z})=\min\limits_{y\mathcal{R}z}a(y)$. Such a $y_{z}$ is guaranteed to exist whenever $Y$ is finite. Then, we have $\displaystyle\max_{z\in Z^{\prime}}(\min_{y\mathcal{R}z}b(y)\ominus\min_{y\mathcal{R}z}a(y))$ $\displaystyle\sqsubseteq\max_{z\in Z^{\prime}}(b(y_{z})\ominus a(y_{z}))$ [$\ominus$ monotone in first arg.] $\displaystyle\sqsubseteq\max_{z\in Z^{\prime}}\|\\!\|{b\ominus a}\|\\!\|$ [by def. of norm] $\displaystyle=\|\\!\|{b\ominus a}\|\\!\|$ [$\|\\!\|{b\ominus a}\|\\!\|$ is independent from $z$] * • _Maximum:_ Let $\mathcal{R}\subseteq Y\times Z$ be a relation. For all $a,b\in\mathbb{M}^{Y}$ we have $\displaystyle\|\\!\|{\max\nolimits_{\mathcal{R}}(b)\ominus\max\nolimits_{\mathcal{R}}(a)}\|\\!\|$ $\displaystyle=\max_{z\in Z}(\max_{y\mathcal{R}z}b(y)\ominus\max_{y\mathcal{R}z}a(y))$ $\displaystyle\sqsubseteq\max_{z\in Z}(\max_{y\mathcal{R}z}((b(y)\ominus a(y))\oplus a(y))\ominus\max_{y\mathcal{R}z}a(y))$ [since $(b(y)\ominus a(y))\oplus a(y)=a(y)\sqcup b(y)$ and $\ominus$ monotone in first arg.] $\displaystyle\sqsubseteq\max_{z\in Z}((\max_{y\mathcal{R}z}(b(y)\ominus a(y))\oplus\max_{y\mathcal{R}z}a(y))\ominus\max_{y\mathcal{R}z}a(y))$ [by def. of $\max$ and monotonicity of $\oplus$] $\displaystyle\sqsubseteq\max_{z\in Z}\max_{y\mathcal{R}z}(b(y)\ominus a(y))\qquad\qquad\mbox{[by \autoref{le:mvprop}(\ref{le:mvprop:6})]}$ $\displaystyle\sqsubseteq\max_{z\in Z}\max_{y\mathcal{R}z}\|\\!\|{b\ominus a}\|\\!\|\qquad\qquad\mbox{[by def.\ of norm]}$ $\displaystyle=\|\\!\|{b\ominus a}\|\\!\|\qquad\qquad\mbox{[since $\|\\!\|{b\ominus a}\|\\!\|$ is independent]}$ * • _Average:_ We first note that, when $p:Y\to\mathbb{M}$, with $Y$ finite, is a distribution, then an inductive argument based on weak distributivity, allows one to show that for all $x\in\mathbb{M}$, $Y^{\prime}\subseteq Y$, $x\odot\bigoplus_{y\in Y^{\prime}}p(y)=\bigoplus_{y\in Y^{\prime}}x\odot p(y)$. For all $a,b\in\mathbb{M}^{Y}$ we have $\displaystyle\|\\!\|{\mathrm{av}_{D}(b)\ominus\mathrm{av}_{D}(a)}\|\\!\|$ $\displaystyle=\max_{p\in D}(\bigoplus_{y\in Y}p(y)\odot b(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y))$ $\displaystyle\sqsubseteq\max_{p\in D}(\bigoplus_{y\in Y}p(y)\odot((b(y)\ominus a(y))\oplus a(y))\ominus\bigoplus_{y\in Y}p(y)\odot a(y))$ [by monotonicity of $\odot,\oplus,\ominus$ and $(b(y)\ominus a(y))\oplus a(y)=a(y)\sqcup b(y)$] $\displaystyle=\max_{p\in D}(\bigoplus_{y\in Y}(p(y)\odot(b(y)\ominus a(y)))\oplus(p(y)\odot a(y))\ominus\bigoplus_{y\in Y}p(y)\odot a(y))$ [since $b(y)\ominus a(y)\sqsubseteq 1\ominus a(y)=\overline{a(y)}$, and $\odot$ weakly distributes over $\oplus$] $\displaystyle=\max_{p\in D}((\bigoplus_{y\in Y}p(y)\odot(b(y)\ominus a(y))\oplus\bigoplus_{y\in Y}p(y)\odot a(y))\ominus\bigoplus_{y\in Y}p(y)\odot a(y))$ $\displaystyle\sqsubseteq\max_{p\in D}\bigoplus_{y\in Y}p(y)\odot(b(y)\ominus a(y))\qquad\qquad\mbox{[by \autoref{le:mvprop}(\ref{le:mvprop:6})]}$ $\displaystyle\sqsubseteq\max_{p\in D}\bigoplus_{y\in Y}p(y)\odot\|\\!\|{b\ominus a}\|\\!\|\qquad\qquad\mbox{[by def.\ of norm and monotonicity of $\odot$]}$ $\displaystyle=\max_{p\in D}\|\\!\|{b\ominus a}\|\\!\|\odot\bigoplus_{y\in Y}p(y)\qquad\mbox{[since $p$ is a distribution and $\odot$ weakly distributes over $\oplus$]}$ $\displaystyle=\max_{p\in D}(\|\\!\|{b\ominus a}\|\\!\|\odot 1)\qquad\qquad\mbox{[since $p$ is a distribution over $Y$]}$ $\displaystyle=\|\\!\|{b\ominus a}\|\\!\|\qquad\qquad\mbox{[since $\|\\!\|{b\ominus a}\|\\!\|$ is independent from $p$]}$ The next result determines the approximations associated with the basic functions. ###### Proposition 5.4 (approximations of basic functions). Let $\mathbb{M}$ be an MV-chain, $Y,Z$ be finite sets and let $a\in\mathbb{M}^{Y}$. We define $\mathit{Min}_{a}=\\{y\in Y\mid a(y)\text{ minimal}\\}\qquad\mathit{Max}_{a}=\\{y\in Y\mid a(y)\text{ maximal}\\}$ * • _Constant:_ for $k:\mathbb{M}^{Z}$, the approximations $(c_{k})^{\\#}_{a}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{c_{k}(a)}}$, $(c_{k})_{\\#}^{a}:\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{c_{k}(a)}}$ are $(c_{k})^{\\#}_{a}(Y^{\prime})=\emptyset=(c_{k})_{\\#}^{a}(Y^{\prime})$ * • _Reindexing:_ for $u:Z\to Y$, the approximations $(u^{})^{\\#}_{a}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{u^{}(a)}}$, $(u^{})_{\\#}^{a}:\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{u^{}(a)}}$ are $(u^{})^{\\#}_{a}(Y^{\prime})=(u^{})_{\\#}^{a}(Y^{\prime})=u^{-1}(Y^{\prime})=\\{z\in[{Z}]_{u^{}(a)}\mid u(z)\in Y^{\prime}\\}$ • _Min:_ for $\mathcal{R}\subseteq Y\times Z$, the approximations $(\min\nolimits_{\mathcal{R}})_{a}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{\min_{\mathcal{R}}(a)}}$, $(\min\nolimits_{\mathcal{R}})^{a}_{\\#}:\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{\min_{\mathcal{R}}(a)}}$ are given below, where $\mathcal{R}^{-1}(z)=\\{y\in Y\mid y\mathcal{R}z\\}$: $(\min\nolimits_{\mathcal{R}})_{a}^{\\#}(Y^{\prime})=\\{z\in[{Z}]_{\min\nolimits_{\mathcal{R}}(a)}\mid\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}\\}$ $(\min\nolimits_{\mathcal{R}})^{a}_{\\#}(Y^{\prime})=\\{z\in[{Z}]^{\min\nolimits_{\mathcal{R}}(a)}\mid\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset\\}$ * • _Max:_ for $\mathcal{R}\subseteq Y\times Z$, the approximations $(\max\nolimits_{\mathcal{R}})_{a}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{\max_{\mathcal{R}}(a)}}$, $(\max\nolimits_{\mathcal{R}})^{a}_{\\#}:\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{\max_{\mathcal{R}}(a)}}$ are $(\max\nolimits_{\mathcal{R}})_{a}^{\\#}(Y^{\prime})=\\{z\in[{Z}]_{\max\nolimits_{\mathcal{R}}(a)}\mid\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset\\}$ $(\max\nolimits_{\mathcal{R}})^{a}_{\\#}(Y^{\prime})=\\{z\in[{Z}]^{\max\nolimits_{\mathcal{R}}(a)}\mid\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}\\}$ * • _Average:_ for a finite $D\subseteq\mathcal{D}(Y)$, the approximations $(\mathrm{av}_{D})_{a}^{\\#}:\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{D_{\mathrm{av}_{D}(a)}}}$, $(\mathrm{av}_{D})^{a}_{\\#}\colon\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{D_{\mathrm{av}_{D}(a)}}}$ are $(\mathrm{av}_{D})_{a}^{\\#}(Y^{\prime})=\\{p\in[{D}]_{\mathrm{av}_{D}(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\}$ $(\mathrm{av}_{D})^{a}_{\\#}(Y^{\prime})=\\{p\in[{D}]^{\mathrm{av}_{D}(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\},$ where $\mathit{supp}(p)=\\{y\in Y\mid p(y)>0\\}$ for $p\in\mathcal{D}(Y)$. ###### Proof 5.5. We only consider the primal cases, the dual ones are analogous. Let $a\in\mathbb{M}^{Y}$. * • _Constant:_ for all $0\sqsubset\theta\sqsubseteq\delta_{a}$ and $Y^{\prime}\subseteq[{Y}]_{a}$ we have $\displaystyle(c_{k})_{a,\theta}^{\\#}(Y^{\prime})$ $\displaystyle=\gamma_{c_{k}(a),\theta}\circ c_{k}\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=\\{z\in[{Z}]_{c_{k}(a)}\mid\theta\sqsubseteq c_{k}(a\oplus\theta_{Y^{\prime}})(z)\ominus c_{k}(a)(z)\\}$ $\displaystyle=\\{z\in[{Z}]_{c_{k}(a)}\mid\theta\sqsubseteq k\ominus k\\}=\\{z\in Z\mid\theta\sqsubseteq 0\\}=\emptyset$ Hence all values $\iota_{a}^{f}(Y^{\prime},z)$ are equal to $0$ and we have $\iota_{a}^{f}=\delta_{a}$. Replacing $\theta$ by $\iota_{a}^{f}$ we obtain $(c_{k})_{a}^{\\#}(Y^{\prime})=\emptyset$. * • _Reindexing:_ for all $0\sqsubset\theta\sqsubseteq\delta_{a}$ and $Y^{\prime}\subseteq[{Y}]_{a}$ we have $\displaystyle(u^{})_{a,\theta}^{\\#}(Y^{\prime})$ $\displaystyle=\gamma_{u^{}(a),\theta}\circ u^{}\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=\\{z\in[{Z}]_{u^{}(a)}\mid\theta\sqsubseteq(a\oplus\theta_{Y^{\prime}})(u(z))\ominus a(u(z))\\}.$ We show that this corresponds to $u^{-1}(Y^{\prime})=\\{z\in Z\mid u(z)\in Y^{\prime}\\}$. It is easy to see that for all $z\in u^{-1}(Y^{\prime})$, we have $(a\oplus\theta_{Y^{\prime}})(u(z))\ominus a(u(z))=\theta=a(u(z))\ominus(a\ominus\theta_{Y^{\prime}})(u(z))$ since $u(z)\in Y^{\prime}$ and $\theta\sqsubseteq\delta_{a}$. Since $u(z)\in Y^{\prime}\subseteq[{Y}]_{a}$, we have $u^{}(a)(z)=a(u(z))\neq 1$ and hence $z\in[{Z}]_{u^{}(a)}$. On the other hand, for all $z\not\in u^{-1}(Y^{\prime})$, we have $(a\oplus\theta_{Y^{\prime}})(u(z))=a(u(z))=(a\ominus\theta_{Y^{\prime}})(u(z))$ since $u(z)\notin Y^{\prime}$, and so $(a\oplus\theta_{Y^{\prime}})(u(z))\ominus a(u(z))=a(u(z))\ominus(a\ominus\theta_{Y^{\prime}})(u(z))=0\sqsubset\theta.$ Therefore $(u^{})_{a,\theta}^{\\#}(Y^{\prime})=u^{-1}(Y^{\prime})$. We observe that for $Y^{\prime}\subseteq[{Y}]_{a}$, $z\in[{Z}]_{u^{}(a)}$ either $u^{}(a\oplus\theta_{Y^{\prime}})(z)\ominus u^{}(a)(z)\sqsubset\theta$ for all $0\sqsubset\theta\sqsubseteq\delta_{a}$ – and in this case $\iota_{a}^{u^{}}(Y^{\prime},z)=0$ – or $u^{}(a\oplus\theta_{Y^{\prime}})(z)\ominus u^{}(a)(z)=\theta$ for all $0\sqsubset\theta\sqsubseteq\delta_{a}$ – and in this case $\iota_{a}^{u^{}}(Y^{\prime},z)=\delta_{a}$. By taking the minimum over all non-zero values, we get $\iota_{a}^{u^{}}=\delta_{a}$. And finally we observe that $(u^{})_{a}^{\\#}(Y^{\prime})=(u^{})_{a,\iota_{a}^{u^{}}}^{\\#}(Y^{\prime})=u^{-1}(Y^{\prime})$. * • _Minimum:_ let $0\sqsubset\theta\sqsubseteq\delta_{a}$. For all $Y^{\prime}\subseteq[{Y}]_{a}$ we have $\displaystyle(\min\nolimits_{\mathcal{R}})_{a,\theta}^{\\#}(Y^{\prime})$ $\displaystyle=\gamma_{\min\nolimits_{\mathcal{R}}(a),\theta}\circ\min\nolimits_{\mathcal{R}}\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=\\{z\in[{Z}]_{\min\nolimits_{\mathcal{R}}(a)}\mid\theta\sqsubseteq\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)\ominus\min_{y\mathcal{R}z}a(y)\\}$ We compute the value $V=\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)\ominus\min_{y\mathcal{R}z}a(y)$ and consider the following cases: * – Assume that there exists $\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$ where $\hat{y}\not\in Y^{\prime}$. Then $(a\oplus\theta_{Y^{\prime}})(\hat{y})=a(\hat{y})\sqsubseteq a(y)\sqsubseteq(a\oplus\theta_{Y^{\prime}})(y)$ for all $y\in\mathcal{R}^{-1}(z)$, which implies that $\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)=a(\hat{y})$. We also have $\min_{y\mathcal{R}z}a(y)=a(\hat{y})$ and hence $V=0$. * – Assume that $\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}$ and $\theta\sqsubseteq a(y)\ominus a(\hat{y})$ whenever $\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$, $y\not\in Y^{\prime}$ and $y\mathcal{R}z$. Since $\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}$ we observe that $\displaystyle\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)$ $\displaystyle=$ $\displaystyle\min\\{\min_{y\in\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}}(a(y)\oplus\theta),\min_{y\mathcal{R}z,y\not\in Y^{\prime}}a(y)\\}$ We can omit the values of all $y$ with $y\mathcal{R}z$, $y\not\in\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$, $y\in Y^{\prime}$, since we will never attain the minimum there. Now let $\hat{y}\in\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$ and $y$ with $y\mathcal{R}z$ and $y\not\in Y^{\prime}$. Then $\theta\sqsubseteq a(y)\ominus a(\hat{y})$ by assumption, which implies $a(\hat{y})\oplus\theta\sqsubseteq a(y)$, since $a(\hat{y})\sqsubseteq a(y)$ and 2.3(2) holds. From this we can deduce $\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)=a(\hat{y})\oplus\theta$. We also have $\min_{y\mathcal{R}z}a(y)=a(\hat{y})$ and hence – since $a(\hat{y})\sqsubseteq\overline{\theta}$ (due to $\theta\sqsubseteq\delta_{a}\sqsubseteq\overline{a(\hat{y})}$) and 2.3(9) holds – $V=(a(\hat{y})\oplus\theta)\ominus a(\hat{y})=\theta$. * – In the remaining case $\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}$ and there exist $\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$, $y\not\in Y^{\prime}$, $y\mathcal{R}z$ such that $a(y)\ominus a(\hat{y})\sqsubset\theta$. This implies $a(y)\sqsubseteq(a(y)\ominus a(\hat{y}))\oplus a(\hat{y})\sqsubset\theta\oplus a(\hat{y})$ since again $a(\hat{y})\sqsubseteq\overline{\theta}$ and 2.3(8) holds. Hence $\min_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)\sqsubseteq a(y)$, which means that $V\sqsubseteq a(y)\ominus a(\hat{y})\sqsubset\theta$. Summarizing, for $\theta\sqsubseteq\delta_{a}$ we observe that $V=\theta$ if and only if $\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}$ and $\theta\sqsubseteq a(y)\ominus a(\hat{y})$ whenever $\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$, $y\not\in Y^{\prime}$ and $y\mathcal{R}z$. Hence if $\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}$ we have $\iota_{a}^{\min\nolimits_{\mathcal{R}}}(Y^{\prime},z)=\min\\{a(y)\ominus a(\hat{y})\mid\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}},y\not\in Y^{\prime},y\mathcal{R}z\\}\cup\\{\delta_{a}\\}$ otherwise $\iota_{a}^{\min\nolimits_{\mathcal{R}}}(Y^{\prime},z)=0$. The values above are minimal whenever $Y^{\prime}=\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}$ and thus we have: $\iota_{a}^{\min\nolimits_{\mathcal{R}}}=\min_{z\in[{Z}]_{\min\nolimits_{\mathcal{R}}(a)}}\\{a(y)\ominus a(\hat{y})\mid y\mathcal{R}z,\hat{y}\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}},y\not\in\textit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\\}\cup\\{\delta_{a}\\}.$ Finally we deduce that $(\min\nolimits_{\mathcal{R}})_{a}^{\\#}(Y^{\prime})=(\min\nolimits_{\mathcal{R}})_{a,\iota_{a}^{\min\nolimits_{\mathcal{R}}}}^{\\#}(Y^{\prime})=\\{z\in[{Z}]_{\min\nolimits_{\mathcal{R}}(a)}\mid\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}\\}.$ * • _Maximum:_ let $0\sqsubset\theta\sqsubseteq\delta_{a}$. For all $Y^{\prime}\subseteq[{Y}]_{a}$ we have $\displaystyle(\max\nolimits_{\mathcal{R}})_{a,\theta}^{\\#}(Y^{\prime})$ $\displaystyle=\gamma_{\max\nolimits_{\mathcal{R}}(a),\theta}\circ\max\nolimits_{\mathcal{R}}\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=\\{z\in[{Z}]_{\max\nolimits_{\mathcal{R}}(a)}\mid\theta\sqsubseteq\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)\ominus\max_{y\mathcal{R}z}a(y)\\}$ We observe that $\displaystyle\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)$ $\displaystyle=$ $\displaystyle\max\\{\max_{y\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}}(a\oplus\theta_{Y^{\prime}})(y),\max_{\begin{subarray}{c}y\mathcal{R}z,y\in Y^{\prime}\\\ y\not\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\end{subarray}}(a(y)\oplus\theta)\\}$ We can omit the values of all $y$ with $y\mathcal{R}z$, $y\not\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}$, $y\not\in Y^{\prime}$, since we will never attain the maximum there. We now compute the value $V=\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)\ominus\max_{y\mathcal{R}z}a(y)$ and consider the following cases: * – Assume that there exists $\hat{y}\in\textit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}$ where $\hat{y}\in Y^{\prime}$. Then $(a\oplus\theta_{Y^{\prime}})(\hat{y})=a(\hat{y})\oplus\theta\sqsupseteq(a\oplus\theta_{Y^{\prime}})(y)\sqsupseteq a(y)$ for all $y\in\mathcal{R}^{-1}(z)$, which implies that $\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)=a(\hat{y})\oplus\theta$. We also have $\max_{y\mathcal{R}z}a(y)=a(\hat{y})$ and hence – since $a(\hat{y})\sqsubseteq\overline{\theta}$ and 2.3(9) holds – $V=(a(\hat{y})\oplus\theta)\ominus a(\hat{y})=\theta$. * – Assume that $\textit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}=\emptyset$. Now let $\hat{y}\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}$ and $y\not\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}$ with $y\mathcal{R}z$ and $y\in Y^{\prime}$. Then $\displaystyle\max_{y\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}}(a\oplus\theta_{Y^{\prime}})(y)$ $\displaystyle=$ $\displaystyle a(\hat{y})$ $\displaystyle\max_{\begin{subarray}{c}y\mathcal{R}z,y\in Y^{\prime}\\\ y\not\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\end{subarray}}(a(y)\oplus\theta)$ $\displaystyle=$ $\displaystyle a(y^{\prime})\oplus\theta$ for some value $y^{\prime}$ with $y^{\prime}\mathcal{R}z$, $y^{\prime}\in Y^{\prime}$, $y^{\prime}\not\in\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}$, that is $a(y^{\prime})\sqsubset a(\hat{y})$. So then either $\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)=a(\hat{y})$ and $V=a(\hat{y})\ominus a(\hat{y})=0$. Or $\max_{y\mathcal{R}z}(a\oplus\theta_{Y^{\prime}})(y)=a(y^{\prime})\oplus\theta$ and by 2.3(11) $V=(a(y^{\prime})\oplus\theta)\ominus a(\hat{y})\sqsubset\theta$. Summarizing, for $\theta\sqsubseteq\delta_{a}$ we observe that $V=\theta$ if and only if $\textit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset$, where the latter condition is independent of $\theta$. Hence, as in the case of reindexing, we have $\iota_{a}^{\max\nolimits_{\mathcal{R}}}=\delta_{a}$. Finally we have $(\max\nolimits_{\mathcal{R}})_{a}^{\\#}(Y^{\prime})=(\max\nolimits_{\mathcal{R}})_{a,\iota_{a}^{\max\nolimits_{\mathcal{R}}}}^{\\#}(Y^{\prime})=\\{z\in[{Z}]_{\max\nolimits_{\mathcal{R}}(a)}\mid\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset\\}.$ * • _Average:_ for all $0\sqsubset\theta\sqsubseteq\delta_{a}$ and $Y^{\prime}\subseteq[{Y}]_{a}$ by definition $\displaystyle(\mathrm{av}_{D})_{a,\theta}^{\\#}(Y^{\prime})$ $\displaystyle=\gamma_{\mathrm{av}_{D}(a),\theta}\circ\mathrm{av}_{D}\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=\\{p\in[{D}]_{\mathrm{av}_{D}(a)}\mid\theta\sqsubseteq\bigoplus_{y\in Y}p(y)\odot(a\oplus\theta_{Y^{\prime}})(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)\\}$ We show that this set corresponds to $\\{p\in[{D}]_{\mathrm{av}_{D}(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\}$. Consider $p\in[{D}]_{\mathrm{av}_{D}(a)}$ such that $\mathit{supp}(p)\subseteq Y^{\prime}$. Note that clearly $\bigoplus_{y\in Y^{\prime}}p(y)=1$. Now we have $\displaystyle\bigoplus_{y\in Y}p(y)\odot(a\oplus\theta_{Y^{\prime}})(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot(a(y)\oplus\theta)\oplus\bigoplus_{y\in Y\backslash Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle=\bigoplus_{y\in Y^{\prime}}(p(y)\odot a(y)\oplus p(y)\odot\theta)\oplus\bigoplus_{y\in Y\backslash Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ [by weak distributivity, since for $y\in Y^{\prime}\subseteq[{Y}]_{a}$, $a(y)\sqsubseteq\overline{\delta_{a}}$] $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\oplus\bigoplus_{y\in Y\backslash Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)$ [since, for $y\not\in Y^{\prime}\supseteq\mathit{supp}(p)$, $p(y)=0$ and thus $p(y)\odot a(y)=0$] $\displaystyle=(\bigoplus_{y\in Y^{\prime}}p(y))\odot\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)$ [by weak distributivity, since $p$ is a distribution] $\displaystyle=1\odot\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)$ [since $p$ is a distribution] $\displaystyle=\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)$ $\displaystyle=\theta$ In order to motivate the last passage, observe that for all $y\in Y^{\prime}\subseteq[{Y}]_{a}$, we have $a(y)\sqsubseteq\overline{\delta_{a}}$, and thus $\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\sqsubseteq\bigoplus_{y\in Y^{\prime}}p(y)\odot\overline{\delta_{a}}=(\bigoplus_{y\in Y^{\prime}}p(y))\odot\overline{\delta_{a}}=1\odot\overline{\delta_{a}}=\overline{\delta_{a}}$, where the third last passage is motivated by weak distributivity. Since $\theta\sqsubseteq\delta_{a}$, by 2.3(3), we have $\overline{\delta_{a}}\sqsubseteq\overline{\theta}$ and thus $\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\sqsubseteq\overline{\theta}$. In turn, using this fact, 2.3(9) motivates the last equality in the chain above, i.e., $\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)=\theta$. On the other hand, for all $p\in[{D}]_{\mathrm{av}_{D}(a)}$ such that $\mathit{supp}(p)\not\subseteq Y^{\prime}$, there exists $y^{\prime}\in Y\backslash Y^{\prime}$ such that $p(y^{\prime})\neq 0$. Then, we have $\displaystyle\bigoplus_{y\in Y}p(y)\odot(a\oplus\theta_{Y^{\prime}})(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot(a(y)\oplus\theta)\oplus\bigoplus_{y\in Y\backslash Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot\theta\oplus\bigoplus_{y\in Y^{\prime}}p(y)\odot a(y)\oplus\bigoplus_{y\in Y\backslash Y^{\prime}}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ [by weak distributivity, since for $y\in Y^{\prime}\subseteq[{Y}]_{a}$, $a(y)\sqsubseteq\overline{\delta_{a}}$] $\displaystyle=\bigoplus_{y\in Y^{\prime}}p(y)\odot\theta\oplus\bigoplus_{y\in Y}p(y)\odot a(y)\ominus\bigoplus_{y\in Y}p(y)\odot a(y)$ $\displaystyle\sqsubseteq\bigoplus_{y\in Y^{\prime}}p(y)\odot\theta$ [by 2.3(6)] $\displaystyle=\theta\odot\bigoplus_{y\in Y^{\prime}}p(y)$ [by weak distributivity, since $p$ is a distribution] $\displaystyle\sqsubset\theta$ In order to motivate the last inequality, we proceed as follows. We have that $\mathit{supp}(p)\not\subseteq Y^{\prime}$. Let $y_{0}\in\mathit{supp}(p)\backslash Y^{\prime}$. We know that $\overline{p(y_{0})}\sqsubseteq\bigoplus_{y\in Y\backslash\\{y\\}}p(y)\sqsubseteq\bigoplus_{y\in Y^{\prime}}p(y)$. Therefore $\overline{\bigoplus_{y\in Y^{\prime}}p(y)}\sqsubseteq p(y_{0})\neq 0$. Hence $\bigoplus_{y\in Y^{\prime}}p(y)\sqsubset 1$. The strict inequality above now follows, if we further show that given an $x\in\mathbb{M}$, $x\neq 1$ then $\theta\odot x\sqsubset\theta$. Note that $\overline{x}\neq 0$. Therefore $\theta=\theta\odot 1=\theta\odot(x\oplus\overline{x})=\theta\odot x\oplus\theta\odot\overline{x}$, where the last equality follows by weak distributivity. Now $\theta\odot\overline{x}\sqsubseteq\overline{x}\sqsubseteq\overline{\theta\odot x}$, and thus, by 2.3(9), we obtain $\theta\odot x=\theta\odot x\oplus\theta\odot\overline{x}\ominus\theta\odot\overline{x}=\theta\ominus\theta\odot\overline{x}\sqsubset\theta$, as desired. The last passage follows by the fact that $\theta,\overline{x}\neq 0$ and thus $\theta\odot\overline{x}\neq 0$. Since these results hold for all $\theta\sqsubseteq\delta_{a}$, we have $\iota_{a}^{\mathrm{av}_{D}}=\delta^{a}$. And finally $(\mathrm{av}_{D})_{a,\theta}^{\\#}(Y^{\prime})=(\mathrm{av}_{D})^{a,\theta}_{\\#}(Y^{\prime})=\\{p\in[{D}]_{\mathrm{av}_{D}(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\}$. When a non-expansive function arises as the composition of simpler ones (see 3.9) we can obtain the corresponding approximation by just composing the approximations of the simpler functions. ###### Proposition 5.6 (composing approximations). Let $g:\mathbb{M}^{Y}\to\mathbb{M}^{W}$ and $h:\mathbb{M}^{W}\to\mathbb{M}^{Z}$ be non-expansive functions. For all $a\in\mathbb{M}^{Y}$ we have that $(h\circ g)_{a}^{\\#}=h_{g(a)}^{\\#}\circ g_{a}^{\\#}$. Analogously $(h\circ g)^{a}_{\\#}=h^{g(a)}_{\\#}\circ g^{a}_{\\#}$ for the dual case. ###### Proof 5.7. Here we only consider the primal case, the dual case for $(h\circ g)_{\\#}^{a}$ is analogous. Let $0\sqsubset\theta\sqsubseteq\min\\{\iota_{a}^{g},\iota_{g(a)}^{h}\\}$. Then, by 3.26(2) we know that $g_{a}^{\\#}=g_{a,\theta}^{\\#}=\gamma_{g(a),\theta}\circ g\circ\alpha_{a,\theta}$ $h_{g(a)}^{\\#}=h_{g(a),\theta}^{\\#}=\gamma_{h(g(a)),\theta}\circ h\circ\alpha_{g(a),\theta}$ Now we will prove that $(h\circ g)_{a,\theta}^{\\#}=h_{g(a),\theta}^{\\#}\circ g_{a,\theta}^{\\#}$ First observe that $g(\alpha_{a,\theta}(Y^{\prime}))\in[{g(a)},{g(a\oplus\theta)}]\subseteq[{g(a)},{g(a)\oplus\theta}]$ for all $Y^{\prime}\subseteq[{Y}]_{a}$ by 3.7. Applying 3.26(2) on $h$ we obtain $\displaystyle(h\circ g)_{a,\theta}^{\\#}=\gamma_{h(g(a)),\theta}\circ h\circ g\circ\alpha_{a,\theta}(Y^{\prime})=h_{g(a),\theta}^{\\#}\circ\gamma_{g(a),\theta}\circ g\circ\alpha_{a,\theta}(Y^{\prime})$ $\displaystyle=$ $\displaystyle h_{g(a),\theta}^{\\#}\circ g_{a,\theta}^{\\#}(Y^{\prime})=h_{g(a)}^{\\#}\circ g_{a}^{\\#}(Y^{\prime})$ Hence all functions $(h\circ g)_{a,\theta}^{\\#}$ are equal and independent of $\theta$ and so it must hold that $(h\circ g)_{a,\theta}^{\\#}=(h\circ g)_{a}^{\\#}$. Then from 3.26 we can conclude $\min\\{\iota_{a}^{g},\iota_{g(a)}^{h}\\}\sqsubseteq\iota_{a}^{h\circ g}$. Furthermore functions can be combined via disjoint union, preserving non- expansiveness, as follows. ###### Proposition 5.8 (disjoint union of non-expansive functions). Let $f_{i}:\mathbb{M}^{Y_{i}}\to\mathbb{M}^{Z_{i}}$, for $i\in I$, be non- expansive and such that the sets $Z_{i}$ are pairwise disjoint. The function $\biguplus\limits_{i\in I}f_{i}:\mathbb{M}^{\bigcup_{i\in I}Y_{i}}\to\mathbb{M}^{\biguplus_{i\in I}Z_{i}}$ defined by $\biguplus_{i\in I}f_{i}(a)(z)=f_{i}(a\|_{Y_{i}})(z)\qquad\mbox{if $z\in Z_{i}$}$ is non-expansive. ###### Proof 5.9. For all $a,b\in\mathbb{M}^{\bigcup_{i\in I}Y_{i}}$ we have $\displaystyle\|\\!\|{\biguplus_{i\in I}f_{i}(b)\ominus\biguplus_{i\in I}f_{i}(a)}\|\\!\|$ $\displaystyle=\max_{z\in\biguplus_{i\in I}Z_{i}}(\biguplus_{i\in I}f_{i}(b)(z)\ominus\biguplus_{i\in I}f_{i}(a)(z))$ $\displaystyle=\max_{i\in I}\max_{z\in Z_{i}}(f_{i}(b\|_{Y_{i}})(z)\ominus f_{i}(a\|_{Y_{i}})(z))$ [since all $Z_{i}$ are disjoint] $\displaystyle=\max_{i\in I}\|\\!\|{f_{i}(b\|_{Y_{i}})\ominus f_{i}(a\|_{Y_{i}})}\|\\!\|$ [by def. of norm] $\displaystyle\sqsubseteq\max_{i\in I}\|\\!\|{b\|_{Y_{i}}\ominus a\|_{Y_{i}}}\|\\!\|$ [since all $f_{i}$ are non- expansive] $\displaystyle=\max_{i\in I}\max_{y\in Y_{i}}(b(y)\ominus a(y))$ $\displaystyle=\max_{y\in\bigcup_{i\in I}Y_{i}}(b(y)\ominus a(y))$ $\displaystyle=\|\\!\|{b\ominus a}\|\\!\|$ [by def. of norm] Also, the corresponding approximation of a disjoint union can be conveniently assembled from the approximations of its components. ###### Proposition 5.10 (disjoint union and approximations). The approximations for $\biguplus\limits_{i\in I}f_{i}$, where $f_{i}:\mathbb{M}^{Y_{i}}\to\mathbb{M}^{Z_{i}}$ are non-expansive and $Z_{i}$ are pairwise disjoint, have the following form. For all $a\colon{\bigcup_{i\in I}Y_{i}}\to\mathbb{M}$ and $Y^{\prime}\subseteq\bigcup_{i\in I}Y_{i}$: $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a}^{\\#}(Y^{\prime})=\biguplus_{i\in I}(f_{i})_{a\|_{Y_{i}}}^{\\#}(Y^{\prime}\cap Y_{i})\qquad\big{(}\biguplus_{i\in I}f_{i}\big{)}^{a}_{\\#}(Y^{\prime})=\biguplus_{i\in I}(f_{i})^{a\|_{Y_{i}}}_{\\#}(Y^{\prime}\cap Y_{i})$ ###### Proof 5.11. We just show the statement for the primal case, the dual case is analogous. We abbreviate $Y=\bigcup_{i\in I}Y_{i}$. Let $0\sqsubset\theta\sqsubseteq\delta_{a}$. According to the definition of $a$-approximation (3.25) we have for $Y^{\prime}\subseteq[{Y}]_{a}$: $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\theta}^{\\#}(Y^{\prime})=\gamma_{\biguplus\limits_{i\in I}f_{i}(a),\theta}\circ\biguplus\limits_{i\in I}f_{i}\circ\alpha_{a,\theta}$ $(f_{i})_{a\|_{Y_{i}},\theta}^{\\#}=\gamma_{f_{i}(a\|_{Y_{i}}),\theta}\circ f_{i}\circ\alpha_{a\|_{Y_{i}},\theta}$ for all $i\in I$. Our first step is prove that $\gamma_{\biguplus\limits_{i\in I}f_{i}(a),\theta}\circ\biguplus\limits_{i\in I}f_{i}\circ\alpha_{a,\theta}(Y^{\prime})=\biguplus_{i\in I}\gamma_{f_{i}(a\|_{Y_{i}}),\theta}\circ f_{i}\circ\alpha_{a\|_{Y_{i}},\theta}(Y^{\prime}\cap Y_{i})$ By simply expanding the functions we obtain $\gamma_{\biguplus\limits_{i\in I}f_{i}(a),\theta}\circ\biguplus\limits_{i\in I}f_{i}\circ\alpha_{a,\theta}(Y^{\prime})=\\{z\in Z_{i}\mid i\in I\ \land\ \theta\sqsubseteq f_{i}((a\oplus\theta_{Y^{\prime}})\|_{Y_{i}})(z)\ominus f_{i}(a\|_{Y_{i}})(z)\\}$ $\biguplus_{i\in I}\gamma_{f_{i}(a\|_{Y_{i}}),\theta}\circ f_{i}\circ\alpha_{a\|_{Y_{i}},\theta}(Y^{\prime}\cap Y_{i})=\biguplus_{i\in I}\\{z\in Z_{i}\mid\theta\sqsubseteq f_{i}(a\|_{Y_{i}}\oplus\theta_{Y^{\prime}\cap Y_{i}})(z)\ominus f_{i}(a\|_{Y_{i}})(z)\\}$ which are the same set, since for all $i\in I$ clearly $(a\oplus\theta_{Y^{\prime}})\|_{Y_{i}}=a\|_{Y_{i}}\oplus\theta_{Y^{\prime}\cap Y_{i}}$. This implies $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\theta}^{\\#}(Y^{\prime})=\biguplus_{i\in I}(f_{i})_{a\|_{Y_{i}},\theta}^{\\#}(Y^{\prime}\cap Y_{i}).$ Whenever $\theta\sqsubseteq\min\limits_{i\in I}\iota_{a}^{f_{i}}$, this can be rewritten to $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\theta}^{\\#}(Y^{\prime})=\biguplus_{i\in I}(f_{i})_{a\|_{Y_{i}}}^{\\#}(Y^{\prime}\cap Y_{i}).$ All functions $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\theta}^{\\#}$ are equal and independent of $\theta$ and so it must hold that $\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\theta}^{\\#}$ $=\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a}^{\\#}$. Then with 3.26 we can also conclude $\min\limits_{i\in I}\iota_{a}^{f_{i}}\sqsubseteq\iota_{a}^{\biguplus_{i\in I}f_{i}}$. Table 1. Basic functions $f\colon\mathbb{M}^{Y}\to\mathbb{M}^{Z}$ (constant, reindexing, minimum, maximum, average), function composition, disjoint union and the corresponding approximations $f_{a}^{\\#}\colon\mathbf{2}^{[{Y}]_{a}}\to\mathbf{2}^{[{Z}]_{f(a)}}$, $f^{a}_{\\#}\colon\mathbf{2}^{[{Y}]^{a}}\to\mathbf{2}^{[{Z}]^{f(a)}}$. _Notation:_ $\mathcal{R}^{-1}(z)=\\{y\in Y\mid y\mathcal{R}z\\}$, $\mathit{supp}(p)=\\{y\in Y\mid p(y)>0\\}$ for $p\in\mathcal{D}(Y)$, $\mathit{Min}_{a}=\\{y\in Y\mid a(y)\text{ minimal}\\}$, $\mathit{Max}_{a}=\\{y\in Y\mid a(y)\text{ maximal}\\}$, $a\colon Y\to\mathbb{M}$ function $f$ \| definition of $f$ \| $f_{a}^{\\#}(Y^{\prime})$ (above), $f_{\\#}^{a}(Y^{\prime})$ (below) ---\|---\|--- $c_{k}$ \| $f(a)=k$ \| $\emptyset$ ($k\in\mathbb{M}^{Z}$) \| \| $\emptyset$ $u^{}$ \| $f(a)=a\circ u$ \| $u^{-1}(Y^{\prime})$ ($u\colon Z\to Y$) \| \| $u^{-1}(Y^{\prime})$ $\min\nolimits_{\mathcal{R}}$ \| $f(a)(z)=\min\limits_{y\mathcal{R}z}a(y)$ \| $\\{z\in[{Z}]_{f(a)}\mid\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}\\}$ ($\mathcal{R}\subseteq Y\times Z$) \| \| $\\{z\in[{Z}]^{f(a)}\mid\mathit{Min}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset\\}$ $\max\nolimits_{\mathcal{R}}$ \| $f(a)(z)=\max\limits_{y\mathcal{R}z}a(y)$ \| $\\{z\in[{Z}]_{f(a)}\mid\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\cap Y^{\prime}\neq\emptyset\\}$ ($\mathcal{R}\subseteq Y\times Z$) \| \| $\\{z\in[{Z}]^{f(a)}\mid\mathit{Max}_{a\|_{\mathcal{R}^{-1}(z)}}\subseteq Y^{\prime}\\}$ $\mathrm{av}_{D}$ \| $f(a)(p)=\bigoplus\limits_{y\in Y}p(y)\odot a(y)$ \| $\\{p\in[{D}]_{f(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\}$ ($Z=D\subseteq\mathcal{D}(Y)$) \| \| $\\{p\in[{D}]^{f(a)}\mid\mathit{supp}(p)\subseteq Y^{\prime}\\}$ $h\circ g$ \| $f(a)=h(g(a))$ \| $h_{g(a)}^{\\#}\circ g_{a}^{\\#}(Y^{\prime})$ ($g\colon\mathbb{M}^{Y}\to\mathbb{M}^{W}$, \| \| $h^{g(a)}_{\\#}\circ g^{a}_{\\#}(Y^{\prime})$ $h\colon\mathbb{M}^{W}\to\mathbb{M}^{Z}$) \| \| $\biguplus\limits_{i\in I}f_{i}$ $I$ finite \| $f(a)(z)=f_{i}(a\|_{Y_{i}})(z)$ \| $\biguplus_{i\in I}(f_{i})_{a\|_{Y_{i}}}^{\\#}(Y^{\prime}\cap Y_{i})$ ($f_{i}\colon\mathbb{M}^{Y_{i}}\to\mathbb{M}^{Z_{i}}$, \| ($z\in Z_{i}$) \| $\biguplus_{i\in I}(f_{i})^{a\|_{Y_{i}}}_{\\#}(Y^{\prime}\cap Y_{i})$ $Y=\bigcup\limits_{i\in I}Y_{i}$, $Z=\biguplus\limits_{i\in I}Z_{i}$) \| \| We can then prove the desired results (non-expansiveness and approximation) for the basic building blocks and their composition (all schematically reported in Table 1). All basic functions in Def. 5.1 are non-expansive. Furthermore non-expansive functions are closed under composition and disjoint union. The approximations are the ones listed in the third column of Table 1. ###### Proof 5.12. Follows directly from Propositions 5.2, 5.4, 5.6, 5.8, 5.10 and 3.9. We can also specify the maximal decrease respectively increase that is propagated (here we are using the notation of 3.25). ###### Corollary 5.13. Let $f\colon\mathbb{M}^{Y}\to\mathbb{M}^{Z}$, $a\in\mathbb{M}^{Y}$ and $\iota_{a}^{f}$ be defined as in 3.25. In the dual view we have $\iota_{f}^{a}=\min\\{\iota_{f}^{a}(Y^{\prime},z)\mid Y^{\prime}\subseteq Y\ \land\ z\in Z\ \land\ \iota_{f}^{a}(Y^{\prime},z)\neq 0\\}\cup\\{\delta^{a}\\}$, where the set $\\{\theta\sqsubseteq\delta^{a}\mid z\in f^{a,\theta}_{\\#}(Y^{\prime})\\}$ has a maximum for each $z\in[{Z}]^{f(a)}$ and $Y^{\prime}\subseteq[{Y}]^{a}$, that we denote by $\iota_{f}^{a}(Y^{\prime},z)$. We consider the basic functions from Def. 5.1, function composition as in 3.9 and disjoint union as in 5.8 and give the corresponding values for $\iota_{a}^{f}$ and $\iota_{f}^{a}$. For greatest fixpoints (primal case) we obtain: • $\iota_{a}^{c_{k}}=\iota_{a}^{u^{}}=\iota_{a}^{\max_{\mathcal{R}}}=\iota_{a}^{\mathrm{av}_{D}}=\delta^{a}$ • $\iota_{a}^{\min_{\mathcal{R}}}=\min\limits_{z\in[{Z}]_{\min_{\mathcal{R}}(a)}}\\{a(y)\ominus a(\hat{y})\mid$ $y\mathcal{R}z,y\notin\mathit{Min}_{a\mid_{\mathcal{R}^{-1}(z)}},\hat{y}\in\mathit{Min}_{a\mid_{\mathcal{R}^{-1}(z)}}\\}\cup\\{\delta^{a}\\}$ * • $\iota_{a}^{g\circ f}\sqsupseteq\min\\{\iota_{a}^{f},\iota_{f(a)}^{g}\\}$ * • $\iota_{a}^{\biguplus_{i\in I}f_{i}}=\min_{i\in I}\iota_{a\|_{Y_{i}}}^{f_{i}}$ For least fixpoints (dual case) we obtain: * • $\iota_{c_{k}}^{a}=\iota_{u^{}}^{a}=\iota_{\min_{\mathcal{R}}}^{a}=\iota_{\mathrm{av}_{D}}^{a}=\delta_{a}$ • $\iota_{\max_{\mathcal{R}}}^{a}=\min\limits_{z\in[{Z}]^{\min_{\mathcal{R}}(a)}}\\{a(\hat{y})\ominus a(y)\mid$ $y\mathcal{R}z,\hat{y}\in\mathit{Max}_{a\mid_{\mathcal{R}^{-1}(z)}},y\notin\mathit{Max}_{a\mid_{\mathcal{R}^{-1}(z)}}\\}\cup\\{\delta_{a}\\}$ * • $\iota_{g\circ f}^{a}\sqsupseteq\min\\{\iota_{f}^{a},\iota_{g}^{f(a)}\\}$ * • $\iota_{\biguplus_{i\in I}f_{i}}^{a}=\min_{i\in I}\iota_{f_{i}}^{a\|_{Y_{i}}}$ ###### Proof 5.14. The values $\iota_{a}^{f}$ can be obtained by inspecting the proofs of Propositions 5.4, 5.6 and 5.8. It only remains to show that $\iota:=\iota_{a}^{\biguplus_{i\in I}f_{i}}\sqsubseteq\min_{i\in I}\iota_{a\|_{Y_{i}}}^{f_{i}}$ (cf. 5.8), which means showing $\iota\sqsubseteq\iota_{a\|_{Y_{i}}}^{f_{i}}$ for every $i\in I$. We abbreviate $\iota_{i}:=\iota_{a\|_{Y_{i}}}^{f_{i}}$. If $\iota\sqsupset\iota_{i}$ for some $i\in I$, we will find a $z\in[{Z_{i}}]_{f_{i}(a)}$ and $Y^{\prime}\subseteq[{Y}]_{a}$, such that $z\in(f_{i})^{\\#}_{a\|_{Y_{i}},\iota_{i}}(Y^{\prime}\cap Y_{i})=(f_{i})^{\\#}_{a\|_{Y_{i}}}(Y^{\prime}\cap Y_{i})$ but $z\notin(f_{i})_{a\|_{Y_{i}},\iota}^{\\#}(Y^{\prime}\cap Y_{i})$ by definition (cf. 3.22). This is a contradiction since $\displaystyle z\in\biguplus_{i\in I}(f_{i})^{\\#}_{a\|_{Y_{i}}}(Y^{\prime}\cap Y_{i})=\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a}^{\\#}(Y^{\prime})=\big{(}\biguplus_{i\in I}f_{i}\big{)}_{a,\iota}^{\\#}(Y^{\prime})=\biguplus_{i\in I}(f_{i})_{a\|_{Y_{i}},\iota}^{\\#}(Y^{\prime}\cap Y_{i})$ and since $z\in Z_{i}$, $z\not\in(f_{i})_{a\|_{Y_{i}},\iota}^{\\#}(Y^{\prime}\cap Y_{i})$ and cannot be contained in the union. The arguments for the values $\iota_{f}^{a}$ in the dual case are analogous. ## 6\. Applications ### 6.1. Termination probability We start by making the example from the introduction (Section 1) more formal. Consider a Markov chain $(S,T,\eta)$, as defined in the introduction (Fig. 1), where we restrict the codomain of $\eta\colon S\backslash T\to\mathcal{D}(S)$ to $D\subseteq\mathcal{D}(S)$, where $D$ is finite (to ensure that all involved sets are finite). Furthermore let $\mathcal{T}\colon[0,1]^{S}\to[0,1]^{S}$ be the function (Fig. 1) whose least fixpoint $\mu\mathcal{T}$ assigns to each state its termination probability. The function $\mathcal{T}$ can be written as $\mathcal{T}=(\eta^{}\circ\mathrm{av}_{D})\uplus c_{k}$ where $k\colon T\to[0,1]$ is the constant function $1$ defined only on terminal states. ###### Proof 6.1. Let $t\colon S\to[0,1]$. For $s\in T$ we have $\displaystyle((\eta^{}\circ\mathrm{av}_{D})\uplus c_{k})(t)(s)$ $\displaystyle=c_{k}(t)(s)$ [since $s\in T$] $\displaystyle=k(s)=1$ [by definition of $c_{k}$ and $k$] $\displaystyle=\mathcal{T}(t)(s)$ [since $s\in T$] For $s\notin T$ we have $\displaystyle((\eta^{}\circ\mathrm{av}_{D})\uplus c_{k})(t)(s)$ $\displaystyle=\eta^{}\circ\mathrm{av}_{D}(t)(s)$ [since $s\notin T$] $\displaystyle=\mathrm{av}_{D}(t)(\eta(s))$ [by definition of reindexing] $\displaystyle=\sum_{s^{\prime}\in S}\eta(s)(s^{\prime})\cdot t(s^{\prime})$ [by definition of $\mathrm{av}_{D}$] $\displaystyle=\mathcal{T}(t)(s)$ [since $s\notin T$] From this representation and section 5 it is obvious that $\mathcal{T}$ is non-expansive. Given a function $t\colon S\to[0,1]$, the $t$-approximation for $\mathcal{T}$ in the dual sense is $\mathcal{T}_{\\#}^{t}\colon\mathbf{2}^{[{S}]^{t}}\to\mathbf{2}^{[{S}]^{\mathcal{T}(t)}}$ with $\mathcal{T}_{\\#}^{t}(S^{\prime})=\\{s\in[{S}]^{\mathcal{T}(t)}\mid s\notin T\land\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}.$ ###### Proof 6.2. In the following let $t\colon S\to[0,1]$ and $S^{\prime}\subseteq[{S}]^{t}$. By subsection 6.1 we know that $\mathcal{T}=(\eta^{}\circ\mathrm{av}_{D})\uplus c_{k}$, then by Propositions 5.10, 5.6, and 5.4 we have $\displaystyle\mathcal{T}_{\\#}^{t}(S^{\prime})$ $\displaystyle=((\eta^{}\circ\mathrm{av}_{D})\uplus c_{k})_{\\#}^{t}(S^{\prime})$ $\displaystyle=(\eta^{}\circ\mathrm{av}_{D})_{\\#}^{t}(S^{\prime})\cup(c_{k})_{\\#}^{t}(S^{\prime})$ $\displaystyle=(\eta^{})_{\\#}^{\mathrm{av}_{D}(t)}\circ(\mathrm{av}_{D})_{\\#}^{t}(S^{\prime})\cup(c_{k})_{\\#}^{t}(S^{\prime})$ $\displaystyle=\\{s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}\mid\eta(s)\in\\{q\in[{D}]^{\mathrm{av}_{D}(t)}\mid\mathit{supp}(q)\subseteq S^{\prime}\\}\\}\cup\emptyset$ $\displaystyle=\\{s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}\mid\eta(s)\in[{D}]^{\mathrm{av}_{D}(t)}\land\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}$ Observe that actually for all $s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}$ it always holds that $\eta(s)\in[{D}]^{\mathrm{av}_{D}(t)}$. In fact, since $s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}$ we must have that $\eta^{}(\mathrm{av}_{D}(t))(s)=\mathrm{av}_{D}(t)(\eta(s))\neq 0$, and thus $\eta(s)\in\\{q\in D\mid\mathrm{av}_{D}(t)(q)\neq 0\\}=[{D}]^{\mathrm{av}_{D}(t)}$. Therefore, we have that $\displaystyle\\{s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}\mid\eta(s)\in[{D}]^{\mathrm{av}_{D}(t)}\land\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}$ $\displaystyle=\\{s\in[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}\mid\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}$ Finally, the set above is the same as $\\{s\in[{S}]^{\mathcal{T}(t)}\mid s\notin T\land\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}=\\{s\in[{S\backslash T}]^{\mathcal{T}(t)}\mid\mathit{supp}(\eta(s))\subseteq S^{\prime}\\}$ because, for all $s\in S\backslash T$, hence $s\notin T$, we have that $\mathcal{T}(t)(s)=\sum_{s^{\prime}\in S}\eta(s)(s^{\prime})\cdot t(s^{\prime})=\eta^{}(\mathrm{av}_{D}(t))(s)$, and so $[{S\backslash T}]^{\mathcal{T}(t)}=[{S\backslash T}]^{\eta^{}(\mathrm{av}_{D}(t))}$. At this point we have all the ingredients needed to formalise the application presented in the introduction. We refrain from repeating the same example, but rather present a new example that allows us to illustrate the question of the largest decrease for a fixpoint that still guarantees a pre-fixpoint (the dual problem is treated in Proposition 4.4). ###### Example 6.3. Consider the following Markov chain where $S=\\{x_{1},x_{2},x_{3}\\}$ are non- terminal states. The least fixpoint of the underlying fixpoint function $\mathcal{T}$ is clearly the constant $0$, since no state can reach a terminal state. $x_{1}$ --- $x_{2}$ --- $x_{3}$ --- 11$\frac{1}{2}$$\frac{1}{2}$ Now consider the function $t\colon S\to[0,1]$ defined by $t(x_{1})=0.1$, $t(x_{2})=0.5$ and $t(x_{3})=0.9$. This is also a fixpoint of $\mathcal{T}$. Observe that $\mathcal{T}^{a}_{\\#}(S)=S$ and thus, clearly, $\nu\mathcal{T}^{a}_{\\#}=S$. According to (the dual of) Def. 4.4 we have $\delta^{a,S}=0.1$ and thus, by (the dual of) Proposition 4.5, the function $t^{\prime}=t\ominus 0.1_{S}$, with $t^{\prime}(x_{1})=0$, $t^{\prime}(x_{2})=0.4$, and $t^{\prime}(x_{3})=0.8$, is a pre-fixpoint. Indeed, $\mathcal{T}(t^{\prime})(x_{1})=0$, $\mathcal{T}(t^{\prime})(x_{2})=0.4$ and $\mathcal{T}(t^{\prime})(x_{3})=0.8$. This is not the largest decrease producing a pre-fixpoint. In fact, we can choose $\theta=0.9$, greater than $\delta^{a,S}$ and we have that $a\ominus\theta_{S}$ is the constant $0$, i.e., the least fixpoint of $\mathcal{T}$. However, if we take $\theta^{\prime}=0.5\sqsubset\theta$, then $t\ominus\theta^{\prime}_{S}$ is not a pre-fixpoint. In fact $(t\ominus\theta^{\prime}_{S})(x_{2})=0$, while $\mathcal{T}(t\ominus\theta^{\prime}_{S})(x_{2})=0.2$. This means that the set of decreases (beyond $\delta^{a,S}$) producing a pre-fixpoint is not downward- closed and hence the largest decrease cannot be found by binary search, while, as already mentioned, a binary search will work for decreases below $\delta^{a,S}$. It is well-known that the function $\mathcal{T}$ can be tweaked in such a way that it has a unique fixpoint, coinciding with $\mu\mathcal{T}$, by determining all states which cannot reach a terminal state and setting their value to zero [bk:principles-mc]. Hence fixpoint iteration from above does not really bring us any added value here. It does however make sense to use the proof rule in order to guarantee lower bounds via post-fixpoints. Furthermore, termination probability is a special case of the considerably more complex stochastic games that will be studied in Section 7, where the trick of modifying the function is not applicable. ### 6.2. Branching Distances for Metric Transition Systems In this section we consider metric transition systems (MTS) and their (symmetrical) branching distances, as studied in [afs:linear-branching- metrics, bcdgr:algorithms-mean-payoff-games]. In a MTS each state has a set of successors and some given weight in $[0,1]$. The behavioural distance between two states is the maximum of the difference between their weights and the Hausdorff distance of the weights of their successors. We first consider the Hausdorff lifting and the corresponding approximation. #### Hausdorff lifting. Given a metric on a set $X$, the Hausdorff metric is obtained by lifting the original metric to $\mathbf{2}^{X}$. Here we define this for general distance functions on $\mathbb{M}$, not restricting to metrics. In particular the Hausdorff lifting is given by a function $\mathcal{H}:\mathbb{M}^{X\times X}\to\mathbb{M}^{\mathbf{2}^{X}\times\mathbf{2}^{X}}$ where $\mathcal{H}(d)(X_{1},X_{2})=\max\\{\max_{x_{1}\in X_{1}}\min_{x_{2}\in X_{2}}d(x_{1},x_{2}),\max_{x_{2}\in X_{2}}\min_{x_{1}\in X_{1}}d(x_{1},x_{2})\\}.$ An alternative characterisation of the Hausdorff lifting due to Mémoli [m:wasserstein], also observed in [bbkk:coalgebraic-behavioral-metrics], is more convenient for our purposes. Let $u:\mathbf{2}^{X\times X}\to\mathbf{2}^{X}\times\mathbf{2}^{X}$ be defined by $u(C)=(\pi_{1}[C],\pi_{2}[C])$, where $\pi_{1},\pi_{2}$ are the projections $\pi_{i}:X\times X\to X$ and $\pi_{i}[C]=\\{\pi_{i}(c)\mid c\in C\\}$. Then $\mathcal{H}(d)(X_{1},X_{2})=\min\\{\max_{(x_{1},x_{2})\in C}d(x_{1},x_{2})\mid C\subseteq X\times X\ \land\ u(C)=(X_{1},X_{2})\\}$. Relying on this characterisation, we can obtain the result below, from which we deduce that $\mathcal{H}$ is non-expansive and construct its approximation as the composition of the corresponding functions from Table 1. It holds that $\mathcal{H}=\min\nolimits_{u}\circ\max\nolimits_{\in}$ where $\max_{\in}\colon\mathbb{M}^{X\times X}\to\mathbb{M}^{\mathbf{2}^{X\times X}}$, with $\mathrel{\in}\ \subseteq(X\times X)\times\mathbf{2}^{X\times X}$ the “is-element-of”-relation on $X\times X$, and $\min_{u}\colon\mathbb{M}^{\mathbf{2}^{X\times X}}\to\mathbb{M}^{\mathbf{2}^{X}\times\mathbf{2}^{X}}$. ###### Proof 6.4. Let for $d:X\times X\to\mathbb{M}$, $X_{1},X_{2}\subseteq X$. Then we have $\displaystyle\min\nolimits_{u}(\max\nolimits_{\in}(d))(X_{1},X_{2})$ $\displaystyle=$ $\displaystyle\min_{u(C)=(X_{1},X_{2})}(\max\nolimits_{\in}(d))(C)=\min_{u(C)=(X_{1},X_{2})}\max_{(x_{1},x_{2})\in C}a(x_{1},x_{2})$ which is exactly the definition of the Hausdorff lifting $\mathcal{H}(d)(X_{1},X_{2})$ via couplings, due to Mémoli [m:wasserstein]. We next determine the approximation of the Hausdorff lifting in the dual sense. Intuitively, given a distance function $d$ and a relation $R$ on $X$, such function characterises those pairs $(X_{1},X_{2})$ ($X_{1},X_{2}\subseteq X$) whose distance in the Hausdorff metric decreases by a constant when we decrease the distance $d$ for all pairs in $R$ by the same constant. The approximation for the Hausdorff lifting $\mathcal{H}$ in the dual sense is as follows. Let $d\colon X\times X\to\mathbb{M}$, then $\mathcal{H}_{\\#}^{d}\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}}$ with $\displaystyle\mathcal{H}_{\\#}^{d}(R)$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}\mid$ $\displaystyle\qquad\forall x_{1}\in X_{1}\big{(}\min_{x_{2}^{\prime}\in X_{2}}d(x_{1},x_{2}^{\prime})=\mathcal{H}(d)(X_{1},X_{2})\,\Rightarrow\,\exists x_{2}\in X_{2}\colon$ $\displaystyle\qquad\qquad\qquad\qquad(x_{1},x_{2})\in R\land d(x_{1},x_{2})=\mathcal{H}(d)(X_{1},X_{2})\big{)}\mathop{\land}$ $\displaystyle\qquad\forall x_{2}\in X_{2}\big{(}\min_{x_{1}^{\prime}\in X_{1}}d(x_{1}^{\prime},x_{2})=\mathcal{H}(d)(X_{1},X_{2})\,\Rightarrow\,\exists x_{1}\in X_{1}\colon$ $\displaystyle\qquad\qquad\qquad\qquad(x_{1},x_{2})\in R\land d(x_{1},x_{2})=\mathcal{H}(d)(X_{1},X_{2})\big{)}\\}$ ###### Proof 6.5. Let $d\colon X\times X\to\mathbb{M}$ and $R\subseteq[{X\times X}]^{d}$. Then we have: $\displaystyle\mathcal{H}_{\\#}^{d}(R)$ $\displaystyle=$ $\displaystyle(\min\nolimits_{u})_{\\#}^{\max_{\in}(d)}((\max\nolimits_{\in})_{\\#}^{d}(R))$ where $\displaystyle(\max\nolimits_{\in})_{\\#}^{d}\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{\mathbf{2}^{X\times X}}]^{\max_{\in}(d)}}$ $\displaystyle(\min\nolimits_{u})_{\\#}^{\max_{\in}(d)}\colon\mathbf{2}^{[{\mathbf{2}^{X\times X}}]^{\max_{\in}(d)}}\to\mathbf{2}^{[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}}$ We are using the approximations associated to non-expansive functions, given in 5.4, and obtain: $\displaystyle\mathcal{H}_{\\#}^{d}(R)$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}\mid\mathit{Min}_{\max_{\in}(d)\|_{u^{-1}(X_{1},X_{1})}}\cap(\max\nolimits_{\in})_{\\#}^{d}(R)\neq\emptyset\\}$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}\mid\exists C\subseteq X\times X,u(C)=(X_{1},X_{2}),$ $\displaystyle\qquad C\in(\max\nolimits_{\in})_{\\#}^{d}(R),\mathrm{max}_{\in}(d)(C)=\min_{u(C^{\prime})=(X_{1},X_{2})}\mathrm{max}_{\in}(d)(C^{\prime})\\}$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}\mid\exists C\subseteq X\times X,u(C)=(X_{1},X_{2}),$ $\displaystyle\qquad C\in(\max\nolimits_{\in})_{\\#}^{d}(R),\max d[C]=\min_{u(C^{\prime})=(X_{1},X_{2})}\max d[C^{\prime}]\\}$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}\mid\exists C\subseteq X\times X,u(C)=(X_{1},X_{2}),$ $\displaystyle\qquad\mathit{Max}_{d\|_{C}}\subseteq R,\max d[C]=\mathcal{H}(d)(X_{1},X_{2})\\}$ We show that this is equivalent to the characterisation in the statement of the lemma. • Assume that for all $x_{1}\in X_{1}$ such that $\min_{x_{2}^{\prime}\in X_{2}}d(x_{1},x_{2}^{\prime})=\mathcal{H}(d)(X_{1},X_{2})$, there exists $x_{2}\in X_{2}$ such that $(x_{1},x_{2})\in R$ and $d(x_{1},x_{2})=\mathcal{H}(d)(X_{1},X_{2})$ (and vice versa). We define a set $C_{m}$ that contains all such pairs $(x_{1},x_{2})$, obtained from this guarantee. Now let $x_{1}\not\in\pi_{1}[C_{m}]$. Then necessarily $\min_{x_{2}^{\prime}\in X_{2}}d(x_{1},x_{2}^{\prime})<\mathcal{H}(d)(X_{1},X_{2})$ (because the minimal distance to an element of $X_{2}$ cannot exceed the Hausdorff distance of the two sets). Construct another set $C^{\prime}$ that contains all such $(x_{1},x_{2})$ where $x_{2}$ is an argument where the minimum is obtained. Also add elements $x_{2}\not\in\pi_{2}[C_{m}]$ and their corresponding partners to $C^{\prime}$. The $C=C_{m}\cup C^{\prime}$ is a coupling for $X_{1},X_{2}$, i.e., $u(C)=(X_{1},X_{2})$. Furthermore $\mathit{Max}_{d\|_{C}}=C_{m}\subseteq R$ and $\max d[C]=\max d[C_{m}]=\mathcal{H}(d)(X_{1},X_{2})$. * • Assume that there exists $C\subseteq X\times X$, $u(C)=(X_{1},X_{2})$, $\mathit{Max}_{d\|_{C}}\subseteq R$, $\max d[C]=\mathcal{H}(d)(X_{1},X_{2})$. Now let $x_{1}\in X_{1}$ such that $\min_{x_{2}^{\prime}\in X_{2}}d(x_{1},x_{2}^{\prime})=\mathcal{H}(d)(X_{1},X_{2})$. Since $C$ is a coupling of $X_{1},X_{2}$, there exists $x_{2}\in X_{2}$ such that $(x_{1},x_{2})\in C\subseteq R$. It is left to show that $d(x_{1},x_{2})=\mathcal{H}(d)(X_{1},X_{2})$, which can be done as follows: $\mathcal{H}(d)(X_{1},X_{2})=\min_{x_{2}^{\prime}\in X_{2}}d(x_{1},x_{2}^{\prime})\leq d(x_{1},x_{2})\leq\max d[C]=\mathcal{H}(d)(X_{1},X_{2}).$ For an $x_{2}\in X_{2}$ such that $\min_{x_{1}^{\prime}\in X_{1}}d(x_{1}^{\prime},x_{2})=\mathcal{H}(d)(X_{1},X_{2})$ the proof is analogous. #### Metric transition systems. A _metric transition system_ is a triple $(X,w,\eta)$ where $X$ is a finite set of states, and $w:X\to[0,1]$ and $\eta:X\to\mathbf{2}^{X}$ are functions that assign a weight $w(x)$ and a set of successors $\eta(x)$ to each $x\in X$. The _MTS pseudo-metric_ 222Different from a metric, for a pseudo-metric $d$ the fact that $d(x,y)=0$ does not necessarily imply $x=y$. is the least fixpoint of the function $\mathcal{J}\colon[0,1]^{X\times X}\to[0,1]^{X\times X}$ defined, for $d\colon X\times X\to[0,1]$ and $x_{1},x_{2}\in X$, as: $\mathcal{J}(d)(x_{1},x_{2})=\max\\{\mathcal{H}(d)(\eta(x_{1}),\eta(x_{2})),\|w(x_{1})-w(x_{2})\|\\}$ where $\mathcal{H}$ is the Hausdorff lifting (for $\mathbb{M}=[0,1]$) defined earlier. Now, let $\bar{w}\colon X\times X\to[0,1]$ be the weight distance function defined for $x_{1},x_{2}\in X$ via $\bar{w}(x_{1},x_{2})=\|w(x_{1})-w(x_{2})\|.$ The function $\mathcal{J}$ can be written as $\mathcal{J}=\max\nolimits_{p}\circ\left((\eta\times\eta)^{}\circ\mathcal{H}\uplus c_{\bar{w}}\right)$ where $p\colon(X\times X)\uplus(X\times X)\to(X\times X)$ with $p((x_{1},x_{2}),i)=(x_{1},x_{2})$. (We use $i\in\\{0,1\\}$ to distinguish the elements in the disjoint union.) ###### Proof 6.6. Let $d\colon X\times X$ and $x_{1},x_{2}\in X$, then we have $\displaystyle\mathcal{J}(d)(x_{1},x_{2})$ $\displaystyle=\max\nolimits_{p}\left(((\eta\times\eta)^{}\circ\mathcal{H}\uplus c_{\bar{w}})(d)\right)(x_{1},x_{2})$ $\displaystyle=\max\\{((\eta\times\eta)^{}\circ\mathcal{H}(d))(x_{1},x_{2}),\bar{w}(x_{1},x_{2})\\}$ $\displaystyle=\max\\{\mathcal{H}(d)(\eta(x_{1}),\eta(x_{2})),\bar{w}(x_{1},x_{2})\\}$ $\displaystyle=\max\\{\mathcal{H}(d)(\eta(x_{1}),\eta(x_{2}),\|w(x_{1})-w(x_{2})\|\\}.$ From this representation and section 5 it follows that $\mathcal{J}$ is non- expansive. Let $d\colon X\times X\to[0,1]$. The approximation for $\mathcal{J}$ in the dual sense is $\mathcal{J}_{\\#}^{d}\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{X\times X}]^{\mathcal{J}(d)}}$ with $\mathcal{J}_{\\#}^{d}(Z)=\\{(x_{1},x_{2})\in[{X\times X}]^{\mathcal{J}(d)}\mid\bar{w}(x_{1},x_{2})<\mathcal{H}(d)(\eta(x_{1}),\eta(x_{2}))\land(\eta(x_{1}),\eta(x_{2}))\in\mathcal{H}^{d}_{\\#}(Z)\\}$ ###### Proof 6.7. Let $d\colon X\times X\to[0,1]$ and $X^{\prime}\subseteq\mathbf{2}^{[{X\times X}]^{d}}$. We abbreviate $g=(\eta\times\eta)^{}\circ\mathcal{H}\colon[0,1]^{X\times X}\to[0,1]^{X\times X}$ and hence $\mathcal{J}=\max_{p}(g\uplus\bar{w})$. Thus we obtain $\mathcal{J}_{\\#}^{d}(X^{\prime})=(\max\nolimits_{p})_{\\#}^{g(d)\uplus\bar{w}}\left(g_{\\#}^{d}(X^{\prime})\uplus(c_{\bar{w}})_{\\#}^{d}(X^{\prime})\right).$ Since $c_{\bar{w}}\colon[0,1]^{\emptyset}\to[0,1]^{X\times X}$ is a constant function, we conclude $(c_{\bar{w}})_{\\#}^{d}(X^{\prime})=\emptyset.$ Now $g_{\\#}^{d}=((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(d)}\circ\mathcal{H}_{\\#}^{d}$ where $\displaystyle\mathcal{H}_{\\#}^{d}$ $\displaystyle\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}}$ $\displaystyle((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(d)}$ $\displaystyle\colon\mathbf{2}^{[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(d)}}\to\mathbf{2}^{[{X\times X}]^{g(d)}}.$ It holds that $((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(d)}=(\eta\times\eta)^{-1}$ and hence $(x_{1},x_{2})\in g_{\\#}^{d}(X^{\prime})\Leftrightarrow(\eta(x_{1}),\eta(x_{2}))\in\mathcal{H}_{\\#}^{d}(X^{\prime}).$ Lastly, we obtain $\displaystyle\mathcal{J}_{\\#}^{d}(X^{\prime})$ $\displaystyle=(\max\nolimits_{p})_{\\#}^{g(d)\uplus\bar{w}}((g\uplus\bar{w})^{d}_{\\#}(X^{\prime})\times\\{0\\})$ $\displaystyle=\\{(x_{1},x_{2})\in\mathbf{2}^{[{X\times X}]^{\mathcal{J}(d)}}\mid\mathit{Max}_{(g(d)\uplus\bar{w})\mid_{p^{-1}(\\{(x_{1},x_{2})\\})}}\subseteq g_{\\#}^{d}(X^{\prime})\times\\{0\\}\\}.$ We have that $p^{-1}(\\{(x_{1},x_{2})\\})=\\{((x_{1},x_{2}),0),((x_{1},x_{2}),1)\\}$. The inclusion $\mathit{Max}_{(g(d)\uplus\bar{w})\mid_{p^{-1}(\\{(x_{1},x_{2})\\})}}\subseteq g_{\\#}^{d}(X^{\prime})\times\\{0\\}$ can only hold if $g(d)(x_{1},x_{2})>\bar{w}(x_{1},x_{2})$ (and hence the maximum is achieved by $g(d)$ instead of $\bar{w}$) and additionally $((x_{1},x_{2}),0)\in g_{\\#}^{d}(X^{\prime})\times\\{0\\}$. Hence $\displaystyle\mathcal{J}_{\\#}^{d}(Z)$ $\displaystyle=\\{(x_{1},x_{2})\in[{X\times X}]^{\mathcal{J}(d)}\mid\bar{w}(x_{1},x_{2})<g(d)(x_{1},x_{2})\land(x_{1},x_{2})\in g^{d}_{\\#}(Z)\\}$ $\displaystyle=\\{(x_{1},x_{2})\in\mathbf{2}^{[{X\times X}]^{\mathcal{J}(d)}}\mid\bar{w}(x_{1},x_{2})<\mathcal{H}(d)(\eta(x_{1}),\eta(x_{2}))\land(\eta(x_{1}),\eta(x_{2}))\in\mathcal{H}^{d}_{\\#}(Z)\\}.$ ###### Example 6.8. We consider the MTS depicted below. $x:0.1$ --- $y:0.6$ --- $z:0.3$ --- Here, $\eta(x)=\\{x,z\\}$, $\eta(y)=\\{x,y,z\\}$ and $\eta(z)=\\{x\\}$. Additionally we have $w(x)=0.1$, $w(y)=0.6$ and $w(z)=0.3$ resulting in $\bar{w}(x,y)=0.5$, $\bar{w}(x,z)=0.2$ and $\bar{w}(y,z)=0.3$. The least fixpoint of $\mathcal{J}$ is a pseudo-metric $\mu\mathcal{J}$ given by $\mu\mathcal{J}(x,y)=\mu\mathcal{J}(y,z)=0.5$ and $\mu\mathcal{J}(x,z)=0.3$. (Since $\mu\mathcal{J}$ is a pseudo-metric, the remaining entries are fixed: $\mu\mathcal{J}(u,u)=0$ and $\mu\mathcal{J}(u,v)=\mu\mathcal{J}(v,u)$ for all $u,v\in\\{x,y,z\\}$.) Now consider the pseudo-metric $d$ with $d(x,y)=d(x,z)=d(y,z)=0.5$. This is also a fixpoint of $\mathcal{J}$. Note that $\mathcal{H}(d)(\eta(x),\eta(y))=\mathcal{H}(d)(\eta(x),\eta(z))=\mathcal{H}(d)(\eta(y),\eta(z))=0.5$. Let us use our technique in order to verify that $d$ is not the least fixpoint of $\mathcal{J}$, by showing that $\nu\mathcal{J}_{\\#}^{d}\neq\emptyset$. We start fixpoint iteration with the approximation $\mathcal{J}_{\\#}^{d}$ from the top element $[{X\times X}]^{d}$, which is given by the symmetric closure333We denote the symmetric closure of a relation $R$ by $S(R)$. of $\\{(x,y),(x,z),(y,z)\\}$ (since reflexive pairs do not contain slack). We first observe that $(x,y),(y,x)\notin\mathcal{J}_{\\#}^{d}(S(\\{(x,y),(x,z),(y,z)\\}))$ since $\bar{w}(x,y)=0.5\not<\mathcal{H}(d)(\eta(x),\eta(y))=0.5$. Next, $(y,z),(z,y)\notin\mathcal{J}_{\\#}^{d}(S(\\{(x,z),(y,z)\\}))$ since $(\eta(y),\eta(z))\not\in\mathcal{H}_{\\#}^{d}(S(\\{(x,z),(y,z)\\}))$. In order to see this, consider the approximation of the Hausdorff lifting in Lemma 6.2 and note that for $y\in\eta(y)$ we have $\min_{u\in\eta(z)}d(y,u)=0.5=\mathcal{H}(d)(\eta(y),\eta(z))$, but $(y,x)\notin S(\\{(x,z),(y,z)\\})$ (where $x$ is the only element in $\eta(z)$). The pairs $(x,z),(z,x)$ on the other hand satisfy all conditions and hence $\nu\mathcal{J}_{\\#}^{d}=S(\\{(x,z)\\})=\mathcal{J}_{\\#}^{d}(S(\\{(x,z)\\}))\neq\emptyset$ Thus we conclude that $d$ is not the least fixpoint, but, according to Proposition 4.5, we can decrease the value of $d$ in the positions $(x,z),(z,x)$ and obtain a pre-fixpoint from which we can continue fixpoint iteration. ### 6.3. Bisimilarity In order to define standard bisimilarity we use a variant $\mathcal{G}$ of the Hausdorff lifting $\mathcal{H}$ defined before, where $\max$ and $\min$ are swapped. More precisely, $\mathcal{G}:\mathbb{M}^{X\times X}\to\mathbb{M}^{\mathbf{2}^{X}\times\mathbf{2}^{X}}$ is defined, for $d\in\mathbb{M}^{X\times X}$, by $\mathcal{G}(d)(X_{1},X_{2})=\max\\{\min_{(x_{1},x_{2})\in C}d(x_{1},x_{2})\mid C\subseteq X\times X\ \land\ u(C)=(X_{1},X_{2})\\}$. ###### Lemma 6.9. The approximation for the adapted Hausdorff lifting $\mathcal{G}$ in the primal sense is as follows. Let $a\colon X\times X\to\\{0,1\\}$, then $\mathcal{G}^{\\#}_{a}\colon\mathbf{2}^{[{X\times X}]_{a}}\to\mathbf{2}^{[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{a}}$ with $\displaystyle\mathcal{G}_{a}^{\\#}(R)$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{\mathcal{H}(a)}\mid$ $\displaystyle\qquad\qquad\quad\forall x_{1}\in X_{1}\exists x_{2}\in X_{2}\colon\big{(}(x_{1},x_{2})\not\in[{X\times X}]_{a}\lor(x_{1},x_{2})\in R\big{)}$ $\displaystyle\qquad\qquad\mathop{\land}\forall x_{2}\in X_{2}\exists x_{1}\in X_{1}\colon\big{(}(x_{1},x_{2})\not\in[{X\times X}]_{a}\lor(x_{1},x_{2})\in R\big{)}\\}$ ###### Proof 6.10. We rely on the characterisation of $\mathcal{H}_{\\#}^{a}$ (dual case) of subsection 6.2 and we examine the case where $\mathbb{M}=\\{0,1\\}$. In this case, whenever we have $(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]^{\mathcal{H}(a)}$ it must necessarily hold that $\mathcal{H}(a)(X_{1},X_{2})=1$. Hence, the first part of the conjunction simplifies to: $\forall x_{1}\in X_{1}\big{(}\min_{x_{2}^{\prime}\in X_{2}}a(x_{1},x_{2}^{\prime})=1\,\Rightarrow\,\exists x_{2}\in X_{2}\colon(x_{1},x_{2})\in R\land a(x_{1},x_{2})=1\big{)},$ from which we can omit $a(x_{1},x_{2})=1$ from the conclusion, since this holds automatically. Furthermore $\min_{x^{\prime}_{2}\in X_{2}}a(x_{1},x^{\prime}_{2})=1$ can be rewritten to $\forall x_{2}\in X_{2}\colon a(x_{1},x_{2})=1$. This gives us: $\displaystyle\forall x_{1}\in X_{1}\big{(}\lnot\forall x_{2}\in X_{2}\colon a(x_{1},x_{2})=1\mathop{\lor}\exists x_{2}\in X_{2}\colon(x_{1},x_{2})\in R\big{)}$ $\displaystyle\equiv$ $\displaystyle\forall x_{1}\in X_{1}\big{(}\exists x_{2}\in X_{2}\colon a(x_{1},x_{2})=0\mathop{\lor}\exists x_{2}\in X_{2}\colon(x_{1},x_{2})\in R\big{)}$ $\displaystyle\equiv$ $\displaystyle\forall x_{1}\in X_{1}\exists x_{2}\in X_{2}\big{(}(x_{1},x_{2})\not\in[{X\times X}]^{a}\mathop{\lor}(x_{1},x_{2})\in R\big{)}.$ Since this characterisation is independent of the order, we can replace $[{X\times X}]^{a}$ by $[{X\times X}]_{a}$ and obtain a characterizing condition for $\mathcal{G}_{a}^{\\#}$ (primal case). Now we can define the fixpoint function for bisimilarity and its corresponding approximation. For simplicity we consider unlabelled transition systems, but it would be straightforward to handle labelled transitions. Let $X$ be a finite set of states and $\eta:X\to\mathbf{2}^{X}$ a function that assigns a set of successors $\eta(x)$ to a state $x\in X$. The fixpoint function for bisimilarity $\mathcal{B}:\\{0,1\\}^{X\times X}\to\\{0,1\\}^{X\times X}$ can be expressed by using the Hausdorff lifting $\mathcal{G}$ with $\mathbb{M}=\\{0,1\\}$. Bisimilarity on $\eta$ is the greatest fixpoint of $\mathcal{B}=(\eta\times\eta)^{}\circ\mathcal{G}$. ###### Proof 6.11. Let for $a:X\times X\to\\{0,1\\}$, $x,y\in X$. Then we have $\displaystyle(\eta\times\eta)^{}\circ\mathcal{G}(a)(x,y)$ $\displaystyle=$ $\displaystyle\mathcal{G}(a)(\eta(x),\eta(y))$ $\displaystyle=$ $\displaystyle\max\nolimits_{u}(\min\nolimits_{\in}(a))(\eta(x),\eta(y))$ $\displaystyle=$ $\displaystyle\max_{u(C)=(\eta(x),\eta(y))}(\min\nolimits_{\in}^{X\times X}(a))(C)$ $\displaystyle=$ $\displaystyle\max_{u(C)=(\eta(x),\eta(y))}\min_{(x^{\prime},y^{\prime})\in C}a(x^{\prime},y^{\prime})$ Now we prove that this, indeed, corresponds with the standard bisimulation function, i.e. $\max_{u(C)=(\eta(x),\eta(y))}\min_{(x^{\prime},y^{\prime})\in C}a(x^{\prime},y^{\prime})=1$ if and only if for all $x^{\prime}\in\eta(x)$ there exists $y^{\prime}\in\eta(y)$ such that $a(x^{\prime},y^{\prime})=1$ and vice versa. For the first implication, assume that $\max_{u(C)=(\eta(x),\eta(y))}\min_{(x^{\prime},y^{\prime})\in C}a(x^{\prime},y^{\prime})=1$. This means that there exists $C\subseteq X\times X$ such that $u(C)=(\pi_{1}(C),\pi_{2}(C))=(\eta(x),\eta(y))$ and $\min_{(x^{\prime},y^{\prime})\in C}a(x^{\prime},y^{\prime})=1$. Then we have two cases. Either $C=\emptyset$, which means that $\eta(x)=\eta(y)=\emptyset$, that is, $x$ and $y$ have no successors, and so the bisimulation property vacuously holds. Otherwise, $C\neq\emptyset$, and we must have $a(x^{\prime},y^{\prime})=1$ for all $(x^{\prime},y^{\prime})\in C$. Then, since $(\pi_{1}(C),\pi_{2}(C))=(\eta(x),\eta(y))$, for all $x^{\prime}\in\eta(x)$ there must exists $y^{\prime}\in\eta(y)$ such that $(x^{\prime},y^{\prime})\in C$, and thus $a(x^{\prime},y^{\prime})=1$. Vice versa, for all $y^{\prime}\in\eta(y)$ there must exists $x^{\prime}\in\eta(x)$ such that $(x^{\prime},y^{\prime})\in C$, and thus $a(x^{\prime},y^{\prime})=1$. So the bisimulation property holds. For the other implication, assume that for all $x^{\prime}\in\eta(x)$ there exists $y^{\prime}\in\eta(y)$ such that $a(x^{\prime},y^{\prime})=1$ and call $c_{1}(x^{\prime})$ such a $y^{\prime}$. Vice versa, assume also that for all $y^{\prime}\in\eta(y)$ there exists $x^{\prime}\in\eta(x)$ such that $a(x^{\prime},y^{\prime})=1$ and call $c_{2}(y^{\prime})$ such a $x^{\prime}$. This means that for all $x^{\prime}\in\eta(x)$ and $y^{\prime}\in\eta(y)$, we have $a(x^{\prime},c_{1}(x^{\prime}))=a(c_{2}(y^{\prime}),y^{\prime})=1$. Now let $C^{\prime}=\\{(x^{\prime},y^{\prime})\in\eta(x)\times\eta(y)\mid c_{1}(x^{\prime})=y^{\prime}\lor x^{\prime}=c_{2}(y^{\prime})\\}$. Since we assumed that for all $x^{\prime}\in\eta(x)$ there exists $y^{\prime}\in\eta(y)$ such that $c_{1}(x^{\prime})=y^{\prime}$, we must have that $\pi_{1}(C^{\prime})=\eta(x)$. The same holds for all $y^{\prime}\in\eta(y)$, thus $\pi_{2}(C^{\prime})=\eta(y)$. Therefore, we know that $u(C^{\prime})=(\eta(x),\eta(y))$, and we can conclude by showing that $a(x^{\prime},y^{\prime})=1$ for all $(x^{\prime},y^{\prime})\in C^{\prime}$, in which case also $\max_{u(C)=(\eta(x),\eta(y))}\min_{(x^{\prime},y^{\prime})\in C}a(x^{\prime},y^{\prime})=1$. By definition of $C^{\prime}$ either $c_{1}(x^{\prime})=y^{\prime}$ or $x^{\prime}=c_{2}(y^{\prime})$, or both, must hold. Assume the first one holds, the other case is similar. Then, we can immediately conclude since by hypothesis we know that $a(x^{\prime},c_{1}(x^{\prime}))=1$. Since we proved that the function $\mathcal{B}$ is the same of the standard bisimulation function, then its greatest fixpoint $\nu\mathcal{B}$ is the bisimilarity on $\eta$. Since we are interested in the greatest fixpoint, we are working in the primal sense. Bisimulation relations are represented by their characteristic functions $a\colon X\times X\to\\{0,1\\}$, in fact the corresponding relation can be obtained by taking the complement of $[{X\times X}]_{a}=\\{(x_{1},x_{2})\in X_{1}\times X_{2}\mid a(x_{1},x_{2})=0\\}$. Let $a\colon X\times X\to\\{0,1\\}$. The approximation for the bisimilarity function $\mathcal{B}$ in the primal sense is $\mathcal{B}_{a}^{\\#}\colon\mathbf{2}^{[{X\times X}]_{a}}\to\mathbf{2}^{[{X\times X}]_{\mathcal{B}(a)}}$ with $\displaystyle\mathcal{B}_{a}^{\\#}(R)$ $\displaystyle=$ $\displaystyle\\{(x_{1},x_{2})\in[{X\times X}]_{\mathcal{B}(a)}\mid$ $\displaystyle\quad\forall y_{1}\in\eta(x_{1})\exists y_{2}\in\eta(x_{2})\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R)\big{)}$ $\displaystyle\ \land\forall y_{2}\in\eta(x_{2})\exists y_{1}\in\eta(x_{1})\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R)\big{\\}}$ ###### Proof 6.12. From subsection 6.2 we know that $\displaystyle\mathcal{G}_{a}^{\\#}\colon[{X\times X}]_{a}$ $\displaystyle\to$ $\displaystyle[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{\mathcal{G}(a)}$ $\displaystyle\mathcal{G}_{a}^{\\#}(R)$ $\displaystyle=$ $\displaystyle\\{(X_{1},X_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{\mathcal{G}(a)}\mid$ $\displaystyle\quad\forall x_{1}\in X_{1}\exists x_{2}\in X_{2}\colon\big{(}(x_{1},x_{2})\not\in[{X\times X}]_{a}\lor(x_{1},x_{2})\in R\big{)}$ $\displaystyle\mathop{\land}\forall x_{2}\in X_{2}\exists x_{1}\in X_{1}\colon\big{(}(x_{1},x_{2})\not\in[{X\times X}]_{a}\lor(x_{1},x_{2})\in R\big{)}\\}.$ Furthermore $\displaystyle((\eta\times\eta)^{})_{\mathcal{G}(a)}^{\\#}\colon[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{\mathcal{G}(a)}$ $\displaystyle\to$ $\displaystyle[{X\times X}]_{\mathcal{B}(a)}$ $\displaystyle((\eta\times\eta)^{})_{\mathcal{G}(a)}^{\\#}(R)$ $\displaystyle=$ $\displaystyle(\eta\times\eta)^{-1}(R)$ Composing these functions we obtain: $\displaystyle\mathcal{B}_{a}^{\\#}\colon[{X\times X}]_{a}$ $\displaystyle\to$ $\displaystyle[{X\times X}]_{\mathcal{B}(a)}$ $\displaystyle\mathcal{B}_{a}^{\\#}(R)$ $\displaystyle=$ $\displaystyle(\eta\times\eta)^{-1}(\\{(Y_{1},Y_{2})\in[{\mathbf{2}^{X}\times\mathbf{2}^{X}}]_{\mathcal{G}(a)}\mid$ $\displaystyle\qquad\forall y_{1}\in Y_{1}\exists y_{2}\in Y_{2}\colon\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R\big{)}$ $\displaystyle\quad\,\mathop{\land}\forall y_{2}\in Y_{2}\exists y_{1}\in Y_{1}\colon\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R\big{)}\\})$ $\displaystyle=$ $\displaystyle\\{(x_{1},x_{2})\in[{X\times X}]_{\mathcal{B}(a)}\mid$ $\displaystyle\quad\forall y_{1}\in\eta(x_{1})\exists y_{2}\in\eta(x_{2})\colon\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R\big{)}$ $\displaystyle\mathop{\land}\forall y_{2}\in\eta(x_{2})\exists y_{1}\in\eta(x_{1})\colon\big{(}(y_{1},y_{2})\not\in[{X\times X}]_{a}\lor(y_{1},y_{2})\in R\big{)}\\}.$ We conclude this section by discussing how this view on bisimilarity can be useful: first, it again opens up the possibility to compute bisimilarity – a greatest fixpoint – by iterating from below, through smaller fixpoints. This could potentially be useful if it is easy to compute the least fixpoint of $\mathcal{B}$ inductively and continue from there. Furthermore, we obtain a technique for witnessing non-bisimilarity of states. While this can also be done by exhibiting a distinguishing modal formula [hm:hm-logic, c:automatically-explaining-bisim] or by a winning strategy for the spoiler in the bisimulation game [s:bisim-mc-other-games], to our knowledge there is no known method that does this directly, based on the definition of bisimilarity. With our technique we can witness non-bisimilarity of two states $x_{1},x_{2}\in X$ by presenting a pre-fixpoint $a$ (i.e., $\mathcal{B}(a)\leq a$) such that $a(x_{1},x_{2})=0$ (equivalent to $(x_{1},x_{2})\in[{X\times X}]_{a}$) and $\nu\mathcal{B}_{a}^{\\#}=\emptyset$, since this implies $\nu\mathcal{B}(x_{1},x_{2})\leq a(x_{1},x_{2})=0$ by our proof rule. There are two issues to discuss: first, how can we characterise a pre-fixpoint of $\mathcal{B}$ (which is quite unusual, since bisimulations are post- fixpoints)? In fact, the condition $\mathcal{B}(a)\leq a$ can be rewritten to: for all $(x_{1},x_{2})\in[{X\times X}]_{a}$ there exists $y_{1}\in\eta(x_{1})$ such that for all $y_{2}\in\eta(x_{2})$ we have $(y_{1},y_{2})\in[{X\times X}]_{a}$ (_or_ vice versa). Second, at first sight it does not seem as if we gained anything since we still have to do a fixpoint computation on relations. However, the carrier set is $[{X\times X}]_{a}$, i.e., a set of non- bisimilarity witnesses and this set can be small even though $X$ might be large. ###### Example 6.13. We consider the transition system depicted below. $x$$y$$u$ Our aim is to construct a witness showing that $x,u$ are not bisimilar. This witness is a function $a\colon X\times X\to\\{0,1\\}$ with $a(x,u)=0=a(y,u)$ and for all other pairs the value is $1$. Hence $[{X\times X}]_{a=\mathcal{B}(a)}=[{X\times X}]_{a}=\\{(x,u),(y,u)\\}$ and it is easy to check that $a$ is a pre-fixpoint of $\mathcal{B}$ and that $\nu\mathcal{B}_{a}^{}=\emptyset$: we iterate over $\\{(x,u),(y,u)\\}$ and first remove $(y,u)$ (since $y$ has no successors) and then $(x,u)$. This implies that $\nu\mathcal{B}\leq a$ and hence $\nu\mathcal{B}(x,u)=0$, which means that $x,u$ are not bisimilar. ###### Example 6.14. We modify 6.13 and consider a function $a$ where $a(x,u)=0$ and all other values are $1$. Again $a$ is a pre-fixpoint of $\mathcal{B}$ and $\nu\mathcal{B}\leq a$ (since only reflexive pairs are in the bisimilarity). However $\nu\mathcal{B}_{a}^{}\neq\emptyset$, since $\\{(x,u)\\}$ is a post- fixpoint. This is a counterexample to completeness discussed after subsection 4.2. Intuitively speaking, the states $y,u$ over-approximate and claim that they are bisimilar, although they are not. (This is permissible for a pre- fixpoint.) This tricks $x,u$ into thinking that there is some wiggle room and that one can increase the value of $(x,u)$. This is true, but only because of the limited, local view, since the “true” value of $(y,u)$ is $0$. ### 6.4. Behavioural metrics for probabilistic automata We now consider behavioural metrics for probabilistic automata, which involve both non-deterministic branching (as in Section 6.2) as well as probabilistic branching. Before we start, we first consider the Kantorovich lifting and the corresponding approximation. #### Kantorovich lifting. The Kantorovich (also known as Wasserstein) lifting converts a metric on $X$ to a metric on probability distributions over $X$. As for the Hausdorff lifting, we lift distance functions that are not necessarily metrics. Furthermore, in order to ensure finiteness of all the sets involved, we restrict to $D\subseteq\mathcal{D}(X)$, some finite set of probability distributions over $X$. A _coupling_ of $p,q\in D$ is a probability distribution $c\in\mathcal{D}(X\times X)$ whose left and right marginals are $p,q$, i.e., $p(x_{1})=m_{c}^{L}(x_{1}):=\sum_{x_{2}\in X}c(x_{1},x_{2})$ and $q(x_{2})=m_{c}^{R}(x_{2}):=\sum_{x_{1}\in X}c(x_{1},x_{2})$. The set of all couplings of $p,q$, denoted by $\Omega(p,q)$, forms a polytope with finitely many vertices [pc:computational-ot]. The set of all polytope vertices that are obtained by coupling any $p,q\in D$ is also finite and is denoted by $\mathit{VP}_{D}\subseteq\mathcal{D}(X\times X)$. The Kantorovich lifting is given by $\mathcal{K}:[0,1]^{X\times X}\to[0,1]^{D\times D}$ where $\mathcal{K}(d)(p,q)=\min_{c\in\Omega(p,q)}\sum_{(x_{1},x_{2})\in X\times X}c(x_{1},x_{2})\cdot d(x_{1},x_{2}).$ The coupling $c$ can be interpreted as the optimal transport plan to move goods from suppliers to customers [v:optimal-transport]. Again we provide an alternative characterisation, which shows non-expansiveness of $\mathcal{K}$ and allows one to derive its approximations. Let $u:\mathit{VP}_{D}\to D\times D$, $u(c)=(m_{c}^{L},m_{c}^{R})$. Then $\mathcal{K}=\min\nolimits_{u}\circ\mathrm{av}_{\mathit{VP}_{D}}$ where $\mathrm{av}_{\mathit{VP}_{D}}\colon[0,1]^{X\times X}\to[0,1]^{\mathit{VP}_{D}}$, $\min_{u}\colon[0,1]^{\mathit{VP}_{D}}\to[0,1]^{D\times D}$. ###### Proof 6.15. It holds that $u^{-1}(p,q)=\Omega(p,q)\cap\textit{VP}_{D}$ for $p,q\in D$. Furthermore note it is sufficient to consider as couplings the vertices, i.e., the elements of $\mathit{VP}_{D}$, since the minimum is always attained there [pc:computational-ot]. Hence we obtain for $d\colon X\times X\to[0,1]$, $p,q\in D$: $\displaystyle\min\nolimits_{u}(\mathrm{av}_{\mathit{VP}_{D}}(d))(p,q)$ $\displaystyle=$ $\displaystyle\min_{c\in\Omega(p,q)\cap\mathit{VP}_{D}}\mathrm{av}_{\mathit{VP}_{D}}(d)(c)$ $\displaystyle=$ $\displaystyle\min_{c\in\Omega(p,q)\cap\mathit{VP}_{D}}\sum_{x_{1},x_{2}\in X\times X}c(x_{1},x_{2})\cdot d(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle\min_{c\in\Omega(p,q)}\sum_{x_{1},x_{2}\in X\times X}c(x_{1},x_{2})\cdot d(x_{1},x_{2})$ $\displaystyle=$ $\displaystyle\mathcal{K}(d)(p,q)$ We now present the approximation of the Kantorovich lifting in the dual sense. Intuitively, given a distance function $d$ and a relation $M$ on $X$ it characterises those pairs $(p,q)$ of distributions whose distance in the Kantorovich metric decreases by a constant when we decrease the distance $d$ for all pairs in $M$ by the same constant. ###### Lemma 6.16. Let $d\colon X\times X\to[0,1]$. The approximation for the Kantorovich lifting $\mathcal{K}$ in the dual sense is $\mathcal{K}_{\\#}^{d}\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{D\times D}]^{\mathcal{K}(d)}}$ with $\displaystyle\mathcal{K}_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle\\{(p,q)\in[{D\times D}]^{\mathcal{K}(d)}\mid\exists c\in\Omega(p,q),\mathit{supp}(c)\subseteq M,$ $\displaystyle\qquad\sum_{u,v\in S}d(u,v)\cdot c(u,v)=\mathcal{K}(d)(p,q)\\}.$ ###### Proof 6.17. Let $d\colon X\times X\to[0,1]$ and $M\subseteq[{X\times X}]^{d}$. Then we have: $\displaystyle\mathcal{K}_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle(\min\nolimits_{u})_{\\#}^{\mathrm{av}_{\mathit{VP}_{D}}(d)}((\mathrm{av}_{\mathit{VP}_{D}})_{\\#}^{d}(M))$ where $\displaystyle(\mathrm{av}_{\mathit{VP}_{D}})_{\\#}^{d}\colon\mathbf{2}^{[{X\times X}]^{d}}\to\mathbf{2}^{[{\mathit{VP}_{D}}]^{\mathrm{av}_{\mathit{VP}_{D}}(d)}}$ $\displaystyle(\min\nolimits_{u})_{\\#}^{\mathrm{av}_{\mathit{VP}_{D}}^{X\times X}(d)}\colon\mathbf{2}^{[{\mathit{VP}_{D}}]^{\mathrm{av}_{\mathit{VP}_{D}}(d)}}\to\mathbf{2}^{[{D\times D}]^{\mathcal{K}(d)}}$ We are using the approximations associated to non-expansive functions, given in 5.4, and obtain: $\displaystyle\mathcal{K}_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle\\{(p,q)\in[{D\times D}]^{\mathcal{K}(d)}\mid\mathit{Min}_{\mathrm{av}_{\mathit{VP}_{D}}^{X\times X}(d)\|_{u^{-1}(p,q)}}\cap(\mathrm{av}_{\mathit{VP}_{D}}^{X\times X})_{\\#}^{d}(M)\neq\emptyset\\}$ $\displaystyle=$ $\displaystyle\\{(p,q)\in[{D\times D}]^{\mathcal{K}(d)}\mid\exists c\in\Omega(p,q),c\in(\mathrm{av}_{\mathit{VP}_{D}})_{\\#}^{d}(M),$ $\displaystyle\qquad\mathrm{av}_{\mathit{VP}_{D}}(d)(c)=\min_{c^{\prime}\in\Omega(p,q)}\mathrm{av}_{\mathit{VP}_{D}}^{X\times X}(d)(c^{\prime})\\}$ $\displaystyle=$ $\displaystyle\\{(p,q)\in[{D\times D}]^{\mathcal{K}(d)}\mid\exists c\in\Omega(p,q),c\in(\mathrm{av}_{\mathit{VP}_{D}})_{\\#}^{d}(M),$ $\displaystyle\qquad\mathrm{av}_{\mathit{VP}_{D}}(d)(c)=\mathcal{K}(d)(p,q)\\}$ $\displaystyle=$ $\displaystyle\\{(p,q)\in[{D\times D}]^{\mathcal{K}(d)}\mid\exists c\in\Omega(p,q),\mathit{supp}(c)\subseteq M,$ $\displaystyle\qquad\sum_{u,v\in S}d(u,v)\cdot c(u,v)=\mathcal{K}(d)(p,q)\\}$ #### Probabilistic automata. We now compare our approach with [bblmtv:prob-bisim-distance-automata], which describes the first method for computing behavioural distances for probabilistic automata. Although the behavioural distance arises as a least fixpoint, it is in fact better, even the only known method, to iterate from above, in order to reach this least fixpoint. This is done by guessing and improving couplings, similarly to what happens for strategy iteration discussed later in Section 7. A major complication, faced in [bblmtv:prob- bisim-distance-automata], is that the procedure can get stuck at a fixpoint which is not the least and one has to determine that this is the case and decrease the current candidate. In fact this paper was our inspiration to generalise this technique to a more general setting. A _probabilistic automaton_ is a tuple $\mathcal{A}=(S,L,\eta,\ell)$, where $S$ is a non-empty finite set of states, $L$ is a finite set of labels, $\eta\colon S\to\mathbf{2}^{\mathcal{D}(S)}$ assigns finite sets of probability distributions to states and $\ell\colon S\to L$ is a labelling function. (In the following we again replace $\mathcal{D}(S)$ by a finite subset $D$.) The _probabilistic bisimilarity pseudo-metrics_ is the least fixpoint of the function $\mathcal{M}\colon$ $[0,1]^{S\times S}\to[0,1]^{S\times S}$ where for $d\colon S\times S\to[0,1]$, $s,t\in S$: $\mathcal{M}(d)(s,t)=\begin{cases}1&\mbox{if $\ell(s)\neq\ell(t)$}\\\ \mathcal{H}(\mathcal{K}(d))(\eta(s),\eta(t))&\mbox{otherwise}\end{cases}$ where $\mathcal{H}$ is the Hausdorff lifting (for $\mathbb{M}=[0,1]$) and $\mathcal{K}$ is the Kantorovich lifting defined earlier. Now assume that $d$ is a fixpoint of $\mathcal{M}$, i.e., $d=\mathcal{M}(d)$. In order to check whether $d=\mu f$, [bblmtv:prob-bisim-distance-automata] adapts the notion of a self-closed relation from [f:game-metrics-markov-decision]. ###### Definition 6.18 ([bblmtv:prob-bisim-distance-automata]). A relation $M\subseteq S\times S$ is _self-closed_ with respect to $d=\mathcal{M}(d)$ if, whenever $s\,M\,t$, then • $\ell(s)=\ell(t)$ and $d(s,t)>0$, * • if $p\in\eta(s)$ and $d(s,t)=\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})$, then there exists $q\in\eta(t)$ and $c\in\Omega(p,q)$ such that $d(s,t)=\sum_{u,v\in S}d(u,v)\cdot c(u,v)$ and $\mathit{supp}(c)\subseteq M$, * • if $q\in\eta(t)$ and $d(s,t)=\min_{p^{\prime}\in\eta(s)}\mathcal{K}(d)(p^{\prime},q)$, then there exists $p\in\eta(s)$ and $c\in\Omega(p,q)$ such that $d(s,t)=\sum_{u,v\in S}d(u,v)\cdot c(u,v)$ and $\mathit{supp}(c)\subseteq M$. The largest self-closed relation, denoted by $\approx_{d}$ is empty if and only if $d=\mu f$ [bblmtv:prob-bisim-distance-automata]. We now investigate the relation between self-closed relations and post-fixpoints of approximations. For this we will first show that $\mathcal{M}$ can be obtained as the composition of non-expansive functions, which proves that it is indeed non-expansive. Furthermore, this decomposition will help in the comparison. The fixpoint function $\mathcal{M}$ characterizing probabilistic bisimilarity pseudo-metrics can be written as: $\mathcal{M}=\max\nolimits_{\rho}\circ(((\eta\times\eta)^{}\circ\mathcal{H}\circ\mathcal{K})\uplus c_{l})$ where $\rho\colon(S\times S)\uplus(S\times S)\to(S\times S)$ with $\rho((s,t),i)=(s,t)$. Furthermore $l\colon S\times S\to[0,1]$ is defined as $l(s,t)=0$ if $\ell(s)=\ell(t)$ and $l(s,t)=1$ if $\ell(s)\neq\ell(t)$. ###### Proof 6.19. In fact, given $d\colon S\times S\to[0,1]$, we have $\displaystyle\max\nolimits_{\rho}((((\eta\times\eta)^{}\circ\mathcal{H}\circ\mathcal{K})\uplus c_{l})(d))(s,t)$ $\displaystyle=$ $\displaystyle\max\\{(\eta\times\eta)^{}\circ\mathcal{H}\circ\mathcal{K})(d)(s,t),c_{l}(s,t)\\}$ $\displaystyle=$ $\displaystyle\max\\{\mathcal{H}(\mathcal{K}(d)(\eta(s),\eta(t)),c_{l}(s,t)\\}$ $\displaystyle=$ $\displaystyle\mathcal{M}(d)(s,t)$ Hence $\mathcal{M}$ is a composition of non-expansive functions and thus non- expansive itself. We do not spell out $\mathcal{M}_{\\#}^{d}$ explicitly, but instead show how it is related to self-closed relations. Let $d\colon S\times S\to[0,1]$ where $d=\mathcal{M}(d)$. Then $\mathcal{M}_{\\#}^{d}\colon\mathbf{2}^{[{S\times S}]^{d}}\to\mathbf{2}^{[{S\times S}]^{d}}$, where $[{S\times S}]^{d}=\\{(s,t)\in S\times S\mid d(s,t)>0\\}$. Then $M$ is a self-closed relation with respect to $d$ if and only if $M\subseteq[{S\times S}]^{d}$ and $M$ is a post-fixpoint of $\mathcal{M}_{\\#}^{d}$. ###### Proof 6.20. First note that whenever $M$ is self-closed, it holds that $d(s,t)>0$ for all $(s,t)\in M$ and hence $M\subseteq[{S\times S}]^{d}$. We abbreviate $g=(\eta\times\eta)^{}\circ\mathcal{H}\circ\mathcal{K}\colon[0,1]^{S\times S}\to[0,1]^{S\times S}$ and hence $\mathcal{M}=\max_{\rho}\circ(g\uplus c_{l})$. The approximation $g_{\\#}^{d}$ (yet to be determined) is of type $g_{\\#}^{d}\colon\mathbf{2}^{[{S\times S}]^{d}}\to\mathbf{2}^{[{S\times S}]^{g(d)}}$. In the following, we are using the approximations associated to non-expansive functions, given in 5.4. Since $c_{l}\colon[0,1]^{S\times S}\to[0,1]^{S\times S}$ is a constant function, we have $(c_{l})_{\\#}^{d}\colon\mathbf{2}^{[{S\times S}]^{d}}\to\mathbf{2}^{[{S\times S}]^{l}},\quad(c_{l})_{\\#}^{d}(M)=\emptyset.$ Hence $\displaystyle(g\uplus c_{l})_{\\#}^{d}\colon\mathbf{2}^{[{S\times S}]^{d}}$ $\displaystyle\to$ $\displaystyle\mathbf{2}^{[{S\times S}]^{g(d)}\uplus[{S\times S}]^{l}}$ $\displaystyle(g\uplus c_{l})_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle g_{\\#}^{d}(M)\times\\{0\\}\cup\emptyset\times\\{1\\}=g_{\\#}^{d}(M)\times\\{0\\}$ for $M\subseteq[{S\times S}]^{d}$. Furthermore we obtain $\displaystyle\mathcal{M}_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle(\max\nolimits_{\rho})_{\\#}^{g(d)\uplus l}((g\uplus c_{l})_{\\#}^{d}(M))$ $\displaystyle=$ $\displaystyle(\max\nolimits_{\rho})_{\\#}^{g(d)\uplus l}(g_{\\#}^{d}(M)\times\\{0\\})$ $\displaystyle=$ $\displaystyle\\{(s,t)\in[{S\times S}]^{\mathcal{M}(d)}\mid\mathit{Max}_{(g(d)\uplus c_{l})\|_{\rho^{-1}(\\{(s,t)\\})}}\subseteq g_{\\#}^{d}(M)\times\\{0\\}\\}$ In order to proceed, we examine $\rho^{-1}(\\{(s,t)\\}=\\{((s,t),0),((s,t),1)\\}$. Whenever $\ell(s)\neq\ell(t)$, we have $c_{l}(s,t)=1\geq g(d)(s,t)$, hence $\mathit{Max}_{(g(d)\uplus l)\|_{\rho^{-1}(\\{(s,t)\\})}}$ contains at least $((s,t),1)$, which is not contained in $g_{\\#}^{d}(M)\times\\{0\\}$, which means that the condition is not satisfied. Whenever $\ell(s)=\ell(t)$, we have $c_{l}(s,t)=0<g(d)(s,t)$ (note that $(s,t)\in[{S\times S}]^{\mathcal{M}(d)}$, hence $g(d)(s,t)>0$), so $\mathit{Max}_{(g(d)\uplus l)\|_{\rho^{-1}(\\{(s,t)\\})}}$ contains only $((s,t),0)$, which is contained in $g_{\\#}^{d}(M)\times\\{0\\}$ iff $(s,t)\in g_{\\#}^{d}(M)$. Summarizing, we obtain $\displaystyle\mathcal{M}_{\\#}^{d}(M)$ $\displaystyle=$ $\displaystyle\\{(s,t)\in[{S\times S}]^{\mathcal{M}(d)}\mid\ell(s)=\ell(t),(s,t)\in g_{\\#}^{d}(M)\\}$ $\displaystyle=$ $\displaystyle\\{(s,t)\in S\times S\mid d(s,t)>0,\ell(s)=\ell(t),(s,t)\in g_{\\#}^{d}(M)\\}$ For the last step observe that $d=\mathcal{M}(d)$. It is left to characterise $g_{\\#}^{d}$, where $g=(\eta\times\eta)^{}\circ\mathcal{H}\circ\mathcal{K}$. We have $g_{\\#}^{d}=((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(\mathcal{K}(d))}\circ\mathcal{H}_{\\#}^{\mathcal{K}(d)}\circ\mathcal{K}_{\\#}^{d}$ where $\displaystyle\mathcal{K}_{\\#}^{d}\colon\mathbf{2}^{[{S\times S}]^{d}}$ $\displaystyle\to$ $\displaystyle\mathbf{2}^{[{D\times D}]^{\mathcal{K}(d)}}$ $\displaystyle\mathcal{H}_{\\#}^{\mathcal{K}(d)}\colon\mathbf{2}^{[{D\times D}]^{\mathcal{K}(d)}}$ $\displaystyle\to$ $\displaystyle\mathbf{2}^{[{\mathbf{2}^{D}\times\mathbf{2}^{D}}]^{\mathcal{H}(\mathcal{K}(d))}}$ $\displaystyle((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(\mathcal{K}(d))}\colon\mathbf{2}^{[{\mathbf{2}^{D}\times\mathbf{2}^{D}}]^{\mathcal{H}(\mathcal{K}(d))}}$ $\displaystyle\to$ $\displaystyle\mathbf{2}^{[{S\times S}]^{g(d)}}.$ It holds that $((\eta\times\eta)^{})_{\\#}^{\mathcal{H}(\mathcal{K}(d))}=(\eta\times\eta)^{-1}$ and hence $(s,t)\in g_{\\#}^{d}(M)\iff(\eta(s),\eta(t))\in\mathcal{H}_{\\#}^{\mathcal{K}(d)}(\mathcal{K}_{\\#}^{d}(M)).$ Using the characterisation of the associated approximation of the Hausdorff lifting in subsection 6.2, we obtain that this is equivalent to for all $p\in\eta(s)$, whenever $\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})=\mathcal{H}(\mathcal{K}(d))(\eta(s),\eta(t))$, then there exists $q\in\eta(t)$ such that $(p,q)\in\mathcal{K}_{\\#}^{d}(M)$ and $\mathcal{K}(d)(p,q)=\mathcal{H}(\mathcal{K}(d))(\eta(s),\eta(t))$ (and vice versa), assuming that $\ell(s)=\ell(t)$ (this is a requirement in the definition of $\mathcal{M}_{\\#}^{d}(M)$), since then we have $\mathcal{H}(\mathcal{K}(d))(\eta(s),\eta(t))=d(s,t)>0$ and hence $(\eta(s),\eta(t))\in[{\mathbf{2}^{D}\times\mathbf{2}^{D}}]^{\mathcal{H}(\mathcal{K}(d))}$. Since also $d=\mathcal{M}(d)$, the condition above can be rewritten to for all $p\in\eta(s)$, whenever $\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})=d(s,t)$, then there exists $q\in\eta(t)$ such that $(p,q)\in\mathcal{K}_{\\#}^{d}(M)$ and $\mathcal{K}(d)(p,q)=d(s,t)$ (and vice versa). From 6.16 we know that $(p,q)\in\mathcal{K}_{\\#}^{d}(M)$ iff $\mathcal{K}(d)(p,q)>0$ and there exists $c\in\Omega(p,q)$ such that $\mathit{supp}(c)\subseteq M$ and $\sum_{u,v\in S}c(u,v)\cdot d(u,v)=\mathcal{K}(d)(p,q)$. We instantiate the condition above accordingly and obtain for all $p\in\eta(s)$, whenever $d(s,t)=\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})$, then there exists $q\in\eta(t)$ such that there exists $c\in\Omega(p,q)$ with $\mathit{supp}(c)\subseteq M$, $\mathcal{K}(d)(p,q)=\sum_{u,v\in S}c(u,v)\cdot d(u,v)$ and $\mathcal{K}(d)(p,q)=d(s,t)$ (and vice versa). The two last equalities can be simplified to $d(s,t)=\sum_{u,v\in S}c(u,v)\cdot d(u,v)$, since $\mathcal{K}(d)(p,q)\leq\sum_{u,v\in S}c(u,v)\cdot d(u,v)=d(s,t)=\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})\leq\mathcal{K}(d)(p,q)$ and hence $\mathcal{K}(d)(p,q)=d(s,t)$ can be inferred from the remaining conditions. We finally obtain the following equivalent characterisation: for all $p\in\eta(s)$, whenever $d(s,t)=\min_{q^{\prime}\in\eta(t)}\mathcal{K}(d)(p,q^{\prime})$, then there exists $q\in\eta(t)$ such that there exists $c\in\Omega(p,q)$ with $\mathit{supp}(c)\subseteq M$, $d(s,t)=\sum_{u,v\in S}c(u,v)\cdot d(u,v)$ (and vice versa). Hence we obtain that $(s,t)\in g_{\\#}^{d}(M)$ is equivalent to the the second and third item of Def. 6.18 (under the assumption that $\ell(s)=\ell(t)$), while the first item is covered by the other conditions ($d(s,t)>0$ and $\ell(s)=\ell(t)$) in the characterisation of $\mathcal{M}_{\\#}^{d}(M)$. ## 7\. Simple stochastic games In this section we show how our techniques can be applied to simple stochastic games [condon92, c:algorithms-ssg]. In particular, we present two novel algorithms based on strategy iteration and discuss some runtime results. ### 7.1. Introduction to simple stochastic games. A simple stochastic game is a state-based two-player game where the two players, Min and Max, each own a subset of states they control, for which they can choose the successor. The system also contains sink states with an assigned payoff and averaging states which randomly choose their successor based on a given probability distribution. The goal of Min is to minimise and the goal of Max to maximise the payoff. Simple stochastic games are an important type of games that subsume parity games and the computation of behavioural distances for probabilistic automata (cf. Section 6.4, [bblmtv:prob-bisim-distance-automata]). The associated decision problem (if both players use their best strategies, is the expected payoff of Max greater than $\frac{1}{2}$?) is known to lie in $\mathsf{NP}\cap\mathsf{coNP}$, but it is an open question whether it is contained in $\mathsf{P}$. There are known randomised subexponential algorithms [bv:randomized-algorithms-games]. It has been shown that it is sufficient to consider positional strategies, i.e., strategies where the choice of the player is only dependent on the current state. The expected payoffs for each state form a so-called value vector and can be obtained as the least solution of a fixpoint equation (see below). A _simple stochastic game_ is given by a finite set $V$ of nodes, partitioned into $\mathit{MIN}$, $\mathit{MAX}$, $\mathit{AV}$ (average) and $\mathit{SINK}$, and the following data: $\eta_{\min}:\mathit{MIN}\to\mathbf{2}^{V}$, $\eta_{\max}:\mathit{MAX}\to\mathbf{2}^{V}$ (successor functions for Min and Max nodes), $\eta_{\mathrm{av}}:\mathit{AV}\to D$ (probability distributions, where $D\subseteq\mathcal{D}(V)$ finite) and $w:\mathit{SINK}\to[0,1]$ (weights of sink nodes). The fixpoint function $\mathcal{V}\colon[0,1]^{V}\to[0,1]^{V}$ is defined below for $a\colon V\to[0,1]$ and $v\in V$: $\displaystyle\mathcal{V}(a)(v)=\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ \max_{v^{\prime}\in\eta_{\max}(v)}a(v^{\prime})&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}\eta_{\mathrm{av}}(v)(v^{\prime})\cdot a(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ The _least_ fixpoint of $\mathcal{V}$ specifies the average payoff for all nodes when Min and Max play optimally. In an infinite game the payoff is $0$. In order to avoid infinite games and guarantee uniqueness of the fixpoint, many authors [hk:nonterminating-stochastic-games, c:algorithms-ssg, tvk:strategy-improvement-ssg] restrict to stopping games, which are guaranteed to terminate for every pair of Min/Max-strategies. Here we deal with general games where more than one fixpoint may exist. Such a scenario has been studied in [kkkw:value-iteration-ssg], which considers value iteration to under- and over-approximate the value vector. The over-approximation faces challenges with cyclic dependencies, similar to the vicious cycles described earlier. Here we focus on strategy iteration, which is usually less efficient than value iteration, but yields a precise result instead of approximating it. ###### Example 7.1. We consider the game depicted below. Here $\min$ is a Min node with $\eta_{\min}(\min)=\\{\textbf{1},\mathrm{av}\\}$, $\max$ is a Max node with $\eta_{\max}(\max)=\\{\bm{\varepsilon},\mathrm{av}\\}$, 1 is a sink node with payoff 1, $\bm{\varepsilon}$ is a sink node with some small payoff $\varepsilon\in(0,1)$ and $\mathrm{av}$ is an average node which transitions to both $\min$ and $\max$ with probability $\frac{1}{2}$. Min should choose $\mathrm{av}$ as successor since a payoff of $1$ is bad for Min. Given this choice of Min, Max should not declare $\mathrm{av}$ as successor since this would create an infinite play and hence the payoff is $0$. Therefore Max has to choose $\bm{\varepsilon}$ and be content with a payoff of $\varepsilon$, which is achieved from all nodes different from $\bm{1}$. 1 --- $\min$ --- $\mathrm{av}$ --- $\bm{\varepsilon}$ --- $\max$ --- $\frac{1}{2}$$\frac{1}{2}$ In order to be able to determine the approximation of $\mathcal{V}$ and to apply our techniques, we consider the following equivalent definition. $\mathcal{V}=(\eta_{\min}^{}\circ\min\nolimits_{\in})\uplus(\eta_{\max}^{}\circ\max\nolimits_{\in})\uplus(\eta_{\mathrm{av}}^{}\circ\mathrm{av}_{D})\uplus c_{w}$, where $\mathrel{\in}\ \subseteq V\times\mathbf{2}^{V}$ is the “is- element-of”-relation on $V$. ###### Proof 7.2. Let $a\colon V\to[0,1]$. For $v\in\mathit{MAX}$ we have $\mathcal{V}(a)(v)=(\eta^{}_{\max}\circ\max\nolimits_{\in})(a)(v)=\max\nolimits_{\in}(a)(\eta_{\max}(v))=\max_{v^{\prime}\in\eta_{\max}(v)}a(v^{\prime}).$ For $v\in\mathit{MIN}$ we have $\mathcal{V}(a)(v)=(\eta^{}_{\min}\circ\min\nolimits_{\in})(a)(v)=\min\nolimits_{\in}(a)(\eta_{\min}(v))=\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime}).$ For $v\in\mathit{AV}$ we have $\mathcal{V}(a)(v)=(\eta^{}_{\mathrm{av}}\circ\mathrm{av}_{D})(a)(v)=\mathrm{av}_{D}(a)(\eta_{\mathrm{av}}(v))=\sum_{v^{\prime}\in V}\eta_{\mathrm{av}}(v)(v^{\prime})\cdot a(v^{\prime}).$ For $v\in\mathit{SINK}$ we have $\mathcal{V}(a)(v)=c_{w}(a)(v)=w(v)$. As a composition of non-expansive functions, $\mathcal{V}$ is non-expansive as well. Since we are interested in the least fixpoint we work in the dual sense and obtain the following approximation, which intuitively says: we can decrease a value at node $v$ by a constant only if, in the case of a Min node, we decrease the value of one successor where the minimum is reached, in the case of a Max node, we decrease the values of all successors where the maximum is reached, and in the case of an average node, we decrease the values of all successors. Let $a\colon V\to[0,1]$. The approximation for the value iteration function $\mathcal{V}$ in the dual sense is $\mathcal{V}_{\\#}^{a}\colon\mathbf{2}^{[{V}]^{a}}\to\mathbf{2}^{[{V}]^{\mathcal{V}(a)}}$ with $\displaystyle\mathcal{V}_{\\#}^{a}(V^{\prime})$ $\displaystyle=$ $\displaystyle\\{v\in[{V}]^{\mathcal{V}(a)}\mid\big{(}v\in\mathit{MIN}\land\mathit{Min}_{a_{\|\eta_{\min}(v)}}\cap V^{\prime}\not=\emptyset\big{)}\mathop{\lor}$ $\displaystyle\quad\big{(}v\in\mathit{MAX}\land\mathit{Max}_{a_{\|\eta_{\max}(v)}}\subseteq V^{\prime}\big{)}\lor\big{(}v\in\mathit{AV}\land\mathit{supp}(\eta_{\mathrm{av}}(v))\subseteq V^{\prime}\big{)}\\}$ ###### Proof 7.3. Let $a\colon V\to[0,1]$ and $V^{\prime}\subseteq[{V}]^{a}$. By Proposition 5.10 we have: $\displaystyle\mathcal{V}_{\\#}^{a}(V^{\prime})=$ $\displaystyle\big{(}\mathit{MIN}\cap(\eta^{}_{\min}\circ\min\nolimits_{\in})^{a}_{\\#}(V^{\prime})\big{)}\cup\big{(}\mathit{MAX}\cap(\eta^{}_{\max}\circ\max\nolimits_{\in})^{a}_{\\#}(V^{\prime})\big{)}\cup$ $\displaystyle\big{(}\mathit{AV}\cap(\eta^{}_{\mathrm{av}}\circ\mathrm{av}_{D})^{a}_{\\#}(V^{\prime})\big{)}\cup\big{(}\mathit{SINK}\cap(c_{w})_{\\#}^{a}(V^{\prime})\big{)}$ It holds that $({\eta^{}_{\min}})_{\\#}^{\min\nolimits_{\in}(v)}=\eta^{-1}_{\min}$, $({\eta^{}_{\max}})_{\\#}^{\max\nolimits_{\in}(v)}=\eta^{-1}_{\max}$ and $({\eta^{}_{\mathrm{av}}})_{\\#}^{\mathrm{av}_{D}(v)}=\eta^{-1}_{\mathrm{av}}$. Using previous results (5.4) we deduce $\displaystyle v\in(\eta^{}_{\min}\circ\min\nolimits_{\in})^{a}_{\\#}(V^{\prime})\Leftrightarrow\eta_{\min}(v)\in(\min\nolimits_{\in})^{a}_{\\#}(V^{\prime})\Leftrightarrow\mathit{Min}_{a_{\|\eta_{\min}(v)}}\cap V^{\prime}\not=\emptyset$ $\displaystyle v\in(\eta^{}_{\max}\circ\max\nolimits_{\in})^{a}_{\\#}(V^{\prime})\Leftrightarrow\eta_{\max}(v)\in(\max\nolimits_{\in})^{a}_{\\#}(V^{\prime})\Leftrightarrow\mathit{Max}_{a_{\|\eta_{\max}(v)}}\subseteq V^{\prime}$ $\displaystyle v\in(\eta^{}_{\mathrm{av}}\circ{\mathrm{av}^{V}_{D}})_{\\#}^{a}(V^{\prime})\Leftrightarrow\eta_{\mathrm{av}}(v)\in(\mathrm{av}^{V}_{D})^{a}_{\\#}(V^{\prime})\Leftrightarrow\mathit{supp}(\eta_{\mathrm{av}}(v))\subseteq V^{\prime}$ Lastly $(c_{w})^{a}_{\\#}(V^{\prime})=\emptyset$ for any $V^{\prime}\subseteq V$ since $c_{w}$ is a constant function which concludes the proof. ### 7.2. Strategy iteration from above and below. We describe two algorithms based on the idea of strategy iteration, first introduced by Hoffman and Karp in [hk:nonterminating-stochastic-games], that are novel, as far as we know. The first iterates to the least fixpoint from above and uses the techniques described in Section 4 in order not to get stuck at a larger fixpoint. The second iterates from below: the role of our results is not directly visible in the code of the algorithm, but its non-trivial correctness proof is based on the proof rule introduced earlier. We first recap the underlying notions. A Min-strategy is a mapping $\tau\colon\mathit{MIN}\to V$ such that $\tau(v)\in\eta_{\mathrm{min}}(v)$ for every $v\in\mathit{MIN}$. Following such a strategy, Min decides to always leave a node $v$ via $\tau(v)$. Analogously, a Max-strategy is a function $\sigma\colon\mathit{MAX}\to V$. Fixing a strategy for either player induces a modified value function. If $\tau$ is a Min-strategy, we obtain $\mathcal{V}_{\tau}$ which is defined exactly as $\mathcal{V}$ but for $v\in\mathit{MIN}$ where we set $\mathcal{V}_{\tau}(a)(v)=a(\tau(v))$. Analogously, for $\sigma$ a Max-strategy, $\mathcal{V}_{\sigma}$ is obtained by setting $\mathcal{V}_{\sigma}(a)(v)=a(\sigma(v))$ when $v\in\mathit{MAX}$. If both players fix their strategies, the game reduces to a Markov chain. ###### Lemma 7.4. For any pair of strategies $\sigma,\tau$ we have $\mathcal{V}_{\sigma}\leq\mathcal{V}\leq\mathcal{V}_{\tau}$. ###### Proof 7.5. Given any $a\colon V\to[0,1]$ and $v\in V$, we have $\displaystyle\mathcal{V}_{\sigma}(a)(v)$ $\displaystyle=\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ a(\sigma(v^{\prime}))&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ c_{w}(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle\leq\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ \max_{v^{\prime}\in\eta_{\max}(v)}a(v^{\prime})&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ c_{w}(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle=\mathcal{V}(a)(v)$ The same proof idea can be applied to show $\mathcal{V}\leq\mathcal{V}_{\tau}$. In order to describe our algorithms we also need the notion of a _switch_. Assume that $\tau$ is a Min-strategy and let $a$ be a (pre-)fixpoint of $\mathcal{V}_{\tau}$. Min can now potentially improve her strategy for nodes $v\in\mathit{MIN}$ where $\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})<a(\tau(v))$, called _switch nodes_. This results in a Min-strategy $\tau^{\prime}=\mathit{sw}_{\min}(\tau,a)$, where444If the minimum is achieved in several nodes, Min simply chooses one of them. However, she will only switch if this strictly improves the value. $\tau^{\prime}(v)=\text{arg}\min_{v^{\prime}\in\eta_{\min}(v)}a^{(i)}(v^{\prime})$ for a switch node $v$ and $\tau^{\prime}$, $\tau$ agree otherwise. Also, $\mathit{sw}_{\max}(\sigma,a)$ is defined analogously for Max strategies. Determine $\mu\mathcal{V}$ (from above) (1) Guess a Min-strategy $\tau^{(0)}$, $i:=0$ (2) $a^{(i)}:=\mu\mathcal{V}_{\tau^{(i)}}$ (3) $\tau^{(i+1)}:=\mathit{sw}_{\min}(\tau^{(i)},a^{(i)})$ (4) If $\tau^{(i+1)}\neq\tau^{(i)}$ $i:=i+1$ then goto 2. (5) Compute $V^{\prime}=\nu\mathcal{V}_{\\#}^{a}$, where $a=a^{(i)}$. (6) If $V^{\prime}=\emptyset$ then stop and return $a^{(i)}$. Otherwise set $a^{(i+1)}:=a-(\iota_{\mathcal{V}}^{a}(V^{\prime}))_{V^{\prime}}$, $\tau^{(i+2)}:=\mathit{sw}_{\min}(\tau^{(i)},a^{(i+1)})$, $i:=i+2$, goto 2 (a) Strategy iteration from above Determine $\mu\mathcal{V}$ (from below) (1) Guess a Max-strategy $\sigma^{(0)}$, $i:=0$ (2) $a^{(i)}:=\mu\mathcal{V}_{\sigma^{(i)}}$ (3) $\sigma^{(i+1)}:=\mathit{sw}_{\max}(\sigma^{(i)},a^{(i)})$ (4) If $\sigma^{(i+1)}\neq\sigma^{(i)}$ then set $i:=i+1$ and goto 2 Otherwise stop and return $a^{(i)}$. (b) Strategy iteration from below Figure 3. Strategy iteration from above and below Now strategy iteration from above works as described in Fig. 3(a). The computation of $\mu\mathcal{V}_{\tau^{(i)}}$ in the second step intuitively means that Max chooses his best answering strategy and we compute the least fixpoint based on this answering strategy. At some point no further switches are possible and we have reached a fixpoint $a$, which need not yet be the least fixpoint. Hence we use the techniques from Section 4 to decrease $a$ and obtain a new pre-fixpoint $a^{(i+1)}$, from which we can continue. The correctness of this procedure partially follows from 4.2 and 4.5, however we also need to show the following: first, we can compute $a^{(i)}=\mu\mathcal{V}_{\tau^{(i)}}$ efficiently by solving a linear program (cf. subsection 7.2) by adapting [condon92]. Second, the chain of the $a^{(i)}$ decreases, which means that the algorithm will eventually terminate (cf. subsection 7.2). Strategy iteration from below is given in Fig. 3(b). At first sight, the algorithm looks simpler than strategy iteration from above, since we do not have to check whether we have already reached $\nu\mathcal{V}$, reduce and continue from there. However, in this case the computation of $\mu\mathcal{V}_{\sigma^{(i)}}$ via a linear program is more involved (cf. subsection 7.2), since we have to pre-compute (via greatest fixpoint iteration over $\mathbf{2}^{V}$) the nodes where Min can force a cycle based on the current strategy of Max, thus obtaining payoff $0$. This algorithm does not directly use our technique but we can use our proof rules to prove the correctness of the algorithm (subsection 7.2). In particular, the proof that the sequence $a^{(i)}$ increases is quite involved: we have to show that $a^{(i)}=\mu\mathcal{V}_{\sigma^{(i)}}\leq\mu\mathcal{V}_{\sigma^{(i+1)}}=a^{(i+1)}$. This could be done by showing that $\mu\mathcal{V}_{\sigma^{(i+1)}}$ is a pre- fixpoint of $\mathcal{V}_{\sigma^{(i)}}$, but there is no straightforward way to do this. Instead, we prove this fact using our proof rules, by showing that $a^{(i)}$ is below the least fixpoint of $\mathcal{V}_{\sigma^{(i+1)}}$. The algorithm generalises strategy iteration by Hoffman and Karp [hk:nonterminating-stochastic-games]. Note that we cannot simply adapt their proof, since we do not assume that the game is stopping, which is a crucial ingredient. In the case where the game is stopping, the two algorithms coincide, meaning that we also provide an alternative correctness proof in this situation, while other correctness proofs [condon92] are based on linear algebra and inverse matrices. The least fixpoints of $\mathcal{V}_{\tau}$ and $\mathcal{V}_{\sigma}$ can be determined by solving linear programs. ###### Proof 7.6. We adapt the linear programs found in the literature on simple stochastic games (see e.g. [condon92]). The least fixpoint $a=\mu\mathcal{V}_{\tau}$ can be determined by solving the following linear program: $\displaystyle\min$ $\displaystyle\sum_{v\in V}a(v)$ $\displaystyle a(v)=a(\tau(v))$ $\displaystyle v\in\mathit{MIN}$ $\displaystyle a(v)\geq a(v^{\prime})$ $\displaystyle\forall v^{\prime}\in\eta_{\max}(v),v\in\mathit{MAX}$ $\displaystyle a(v)=\sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})$ $\displaystyle v\in\mathit{AV}$ $\displaystyle a(v)=w(v)$ $\displaystyle v\in\mathit{SINK}$ By having $a(v)\geq a(v^{\prime})$ for all $v^{\prime}\in\eta_{\max}(v)$ and $v\in\mathit{MAX}$ we guarantee $a(v)=\max_{v^{\prime}\in\eta_{\max}(v)}$ $a^{(i)}(v^{\prime})$ since we minimise. The minimisation also guarantees computation of the least fixpoint (in particular, nodes that lie on a cycle will get a value of $0$). Hence, the linear program correctly characterises $\mu\mathcal{V}_{\tau}$. Given a strategy $\sigma$ for Max, we can determine $a=\mu\mathcal{V}_{\sigma}$ by solving the following linear program: $\displaystyle\max$ $\displaystyle\sum_{v\in V}a(v)$ $\displaystyle a(v)=0$ $\displaystyle v\in C_{\sigma}$ $\displaystyle a(v)\leq a(v^{\prime})$ $\displaystyle\forall v^{\prime}\in\eta_{\min}(v),v\in\mathit{MIN},v\not\in C_{\sigma}$ $\displaystyle a(v)=a(\sigma(v))$ $\displaystyle v\in\mathit{MAX},v\not\in C_{\sigma}$ $\displaystyle a(v)=\sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})$ $\displaystyle v\in\mathit{AV},v\not\in C_{\sigma}$ $\displaystyle a(v)=w(v)$ $\displaystyle v\in\mathit{SINK}$ The set $C_{\sigma}$ contains those nodes which will guarantee a non- terminating play if Min plays optimally, given the fixed Max-strategy $\sigma$. The set $C_{\sigma}$ can again be computed via fixpoint-iteration by computing the greatest fixpoint of $c_{\sigma}$ via Kleene iteration on $\mathbf{2}^{V}$ from above: $\displaystyle c_{\sigma}\colon\mathbf{2}^{V}$ $\displaystyle\to$ $\displaystyle\mathbf{2}^{V}$ $\displaystyle c_{\sigma}(V^{\prime})$ $\displaystyle=$ $\displaystyle\\{v\in V\mid(v\in\mathit{MIN}\land\eta_{\min}(v)\cap V^{\prime}\neq\emptyset)\lor(v\in\mathit{MAX}\land\sigma(v)\in V^{\prime})$ $\displaystyle\qquad\mathop{\lor}(v\in\mathit{AV}\land\mathit{supp}(\eta_{\mathrm{av}}(v))\subseteq V^{\prime})\\}$ It is easy to see that $C_{\sigma}=\nu c_{\sigma}$ contains all those nodes from which Min can force a non-terminating play and hence achieve payoff $0$. (Note that there are further nodes that guarantee payoff $0$ – namely sinks with that payoff and nodes which can reach such sinks – but those will obtain value $0$ in any case.) We now show that this linear program computes $\mu\mathcal{V}_{\sigma}$: first, by requiring $a(v)\leq a(v^{\prime})$ for all $v\in\mathit{MIN}$, $v^{\prime}\in\eta_{\min}(v)$, we guarantee $a(v)=\min_{v^{\prime}\in\eta_{\min}}a(v^{\prime})$ since we maximise. Hence we obtain the greatest fixpoint of the following function $\mathcal{V}^{\prime}_{\sigma}\colon[0,1]^{V}\to[0,1]^{V}$: $\displaystyle\mathcal{V}^{\prime}_{\sigma}(a)(v)$ $\displaystyle=\begin{cases}0&v\in C_{\sigma}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV},v\not\in C_{\sigma}\\\ a(\sigma(v))&v\in\mathit{MAX},v\not\in C_{\sigma}\\\ \min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN},v\not\in C_{\sigma}\\\ w(v)&v\in\mathit{SINK}\\\ \end{cases}$ It is easy to show that the least fixpoints of $\mathcal{V}^{\prime}_{\sigma}$ and $\mathcal{V}_{\sigma}$ agree, i.e., $\mu\mathcal{V}^{\prime}_{\sigma}$ and $\mu\mathcal{V}_{\sigma}$: • $\mu\mathcal{V}^{\prime}_{\sigma}\leq\mu\mathcal{V}_{\sigma}$ can be shown by observing that $\mathcal{V}^{\prime}_{\sigma}\leq\mathcal{V}_{\sigma}$. * • $\mu\mathcal{V}_{\sigma}\leq\mu\mathcal{V}^{\prime}_{\sigma}$ can be shown by proving that $\mu\mathcal{V}^{\prime}_{\sigma}$ is a pre-fixpoint of $\mathcal{V}_{\sigma}$, which can be done via a straightforward case analysis. We have to show $\mathcal{V}_{\sigma}(\mu\mathcal{V}^{\prime}_{\sigma})(v)\leq\mu\mathcal{V}^{\prime}_{\sigma}(v)$ for all $v\in V$. We only spell out the case where $v\in\mathit{AV}$, the other cases are similar. In this case either $v\not\in C_{\sigma}$, which means that $\mathcal{V}_{\sigma}(\mu\mathcal{V}^{\prime}_{\sigma})(v)=\mathcal{V}^{\prime}_{\sigma}(\mu\mathcal{V}^{\prime}_{\sigma})(v)=\mu\mathcal{V}^{\prime}_{\sigma}(v).$ If instead $v\in C_{\sigma}$, we have that $\mathit{supp}(\eta_{\mathrm{av}}(v))\subseteq C_{\sigma}$ and so $\mu\mathcal{V}^{\prime}_{\sigma}(v^{\prime})=0$ for all $v^{\prime}\in\mathit{supp}(\eta_{\mathrm{av}}(v))$. Hence $\mathcal{V}_{\sigma}(\mu\mathcal{V}^{\prime}_{\sigma})(v)=\sum_{v^{\prime}\in V}\eta_{\mathrm{av}}(v)(v^{\prime})\cdot\mu\mathcal{V}^{\prime}_{\sigma}(v^{\prime})=0=\mu\mathcal{V}^{\prime}_{\sigma}(v)$ If we can now show that $\mathcal{V}^{\prime}_{\sigma}$ has a unique fixpoint, we are done. The argument for this goes as follows: assume that this function has another fixpoint $a^{\prime}$ different from $\mu\mathcal{V}^{\prime}_{\sigma}$. Clearly $[{V}]^{a^{\prime}}\cap C_{\sigma}=\emptyset$, where $[{V}]^{a^{\prime}}=\\{v\in V\mid a^{\prime}(v)\neq 0\\}$. Hence, if we compare $(\mathcal{V}^{\prime}_{\sigma})_{\\#}^{a}\colon\mathbf{2}^{[{V}]^{a}}\to\mathbf{2}^{[{V}]^{\mathcal{V}^{\prime}_{\sigma}(a)}}$ (defined analogously to subsection 7.1) and $c_{\sigma}$ above, we observe that $(\mathcal{V}^{\prime}_{\sigma})_{\\#}^{a^{\prime}}\subseteq c_{\sigma}\|_{\mathbf{2}^{[{V}]^{a^{\prime}}}}$. (Both functions coincide, apart from their treatment of nodes $v\in\mathit{MIN}$, where $c_{\sigma}(V^{\prime})$ contains $v$ whenever one of its successors is contained in $V^{\prime}$, whereas $(\mathcal{V}^{\prime}_{\sigma})_{\\#}^{a^{\prime}}(V^{\prime})$ additionally requires that the value of this successor is minimal.) Since $a^{\prime}$ is not the least fixpoint we have by 4.2 that $\emptyset\neq\nu(\mathcal{V}^{\prime}_{\sigma})_{\\#}^{a^{\prime}}\subseteq\nu(c_{\sigma}\|_{\mathbf{2}^{[{V}]^{a^{\prime}}}})\subseteq\nu c_{\sigma}=C_{\sigma}.$ This is a contradiction, since $[{V}]^{a^{\prime}}\cap C_{\sigma}=\emptyset$ as observed above. This shows that $\mathcal{V}^{\prime}_{\sigma}$ has a unique fixpoint and completes the proof. Note that if we do not explicitly require that the values of all nodes in $C_{\sigma}$ are $0$, $\mathcal{V}^{\prime}_{\sigma}$ will potentially have several fixpoints and the linear program would not characterise the least fixpoint. Strategy iteration from above and below both terminate and compute the least fixpoint of $\mathcal{V}$. ###### Proof 7.7. _Strategy iteration from above:_ We start by showing the following: Given any $a^{(i)}$ and a new switched Min- strategy $\tau^{(i+1)}$, i.e., $\tau^{(i+1)}=\mathit{sw}_{\min}(\tau^{(i)},a^{(i)})$, then $a^{(i)}$ is a pre-fixpoint of $\mathcal{V}_{\tau^{(i+1)}}$. By choice of $\tau^{(i+1)}$ we have $\displaystyle\mathcal{V}_{\tau^{(i+1)}}(a^{(i)})(v)$ $\displaystyle=\begin{cases}a^{(i)}(\tau^{(i+1)}(v))&v\in\mathit{MIN}\\\ \max_{v^{\prime}\in\eta_{\max}(v)}a^{(i)}(v^{\prime})&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a^{(i)}(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle=\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a^{(i)}(v^{\prime})&v\in\mathit{MIN}\\\ \max_{v^{\prime}\in\eta_{\max}(v)}a^{(i)}(v^{\prime})&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a^{(i)}(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle=\mathcal{V}(a^{(i)})(v)$ By 7.4 $\mathcal{V}\leq\mathcal{V}_{\tau^{(i)}}$ holds and since $a^{(i)}$ is a fixpoint of $\mathcal{V}_{\tau^{(i)}}$ we conclude $\displaystyle\mathcal{V}_{\tau^{(i+1)}}(a^{(i)})(v)=\mathcal{V}(a^{(i)})(v)\leq\mathcal{V}_{\tau^{(i)}}(a^{(i)})(v)=a^{(i)}(v)$ Thus we have $a^{(i+1)}\leq a^{(i)}$ (by Knaster-Tarski, since $a^{(i)}$ is a pre-fixpoint of $\mathcal{V}_{\tau^{(i+1)}}$ and $a^{(i+1)}$ is its least fixpoint). Furthermore we know that $a^{(i)}$ is not a fixpoint of $\mathcal{V}_{\tau^{(i+1)}}$ (otherwise we could not have performed a switch) and hence $a^{(i+1)}$ is strictly smaller than $a^{(i)}$ for at least one input. Since there are only finitely many strategies we will eventually stop switching and reach a fixpoint $a=a^{(j)}$ for an index $j$. Then, if $V^{\prime}=\nu\mathcal{V}_{\\#}^{a}=\emptyset$ then $a$ is the least fixpoint and we conclude. Otherwise, we determine $a^{(j+1)}=a-(\iota_{\mathcal{V}}^{a}(V^{\prime}))_{V^{\prime}}$. By 4.5 (dual version), $a^{(j+1)}$ is a pre-fixpoint of $\mathcal{V}$. Now Min will choose her best strategy $\tau=\tau^{(j+2)}=\mathit{sw}_{\min}(\tau^{(i)},a^{(i+1)})$ and we continue computing $a^{(j+2)}=\mu\mathcal{V}_{\tau^{(j+2)}}$. First, observe that since $a^{(j+1)}$ is a pre-fixpoint of $\mathcal{V}$, it is also a pre-fixpoint of $\mathcal{V}_{\tau^{(j+2)}}$. In fact, $\mathcal{V}$ and $\mathcal{V}_{\tau^{(j+1)}}$ coincide on all nodes $v\not\in\mathit{MIN}$. If $v\in\mathit{MIN}$, we have $\displaystyle\mathcal{V}_{\tau^{(j+2)}}(a^{(j+1)})(v)$ $\displaystyle=$ $\displaystyle a^{(j+1)}(\tau^{(j+2)}(v))$ $\displaystyle=$ $\displaystyle\min_{v^{\prime}\in\eta_{\min}(v)}a^{(j+1)}(v^{\prime})=\mathcal{V}(a^{(j+1)})(v)\leq a^{(j+1)}(v).$ Hence it follows by Knaster-Tarski that $a^{(j+2)}=\mu\mathcal{V}_{\tau^{(j+2)}}\leq a^{(j+1)}$. In turn, $a^{(j+1)}<a^{(j)}$ since $V^{\prime}$ is non-empty and hence also $a^{(j+2)}<a^{(j)}$ (where $<$ on tuples means means $\leq$ in all components and $<$ in at least one component.) This means that the chain $a^{(i)}$ is strictly descending. Hence, at each iteration we obtain a new strategy and, since the number of strategies is finite, the iteration will eventually stop. Hence the algorithm terminates and stops at the least fixpoint of $\mathcal{V}$. _Strategy iteration from below:_ We start as follows: Assume $a$ is the least fixpoint of $\mathcal{V}_{\sigma}$, i.e. $a=\mu\mathcal{V}_{\sigma}$ and $\sigma^{\prime}$ the new best strategy for Max obtained by switching with respect to $a$, i.e., $\sigma^{\prime}=\mathit{sw}_{\max}(\sigma,a)$. We have to show that $a^{\prime}=\mu\mathcal{V}_{\sigma^{\prime}}$ lies above $a$ ($a^{\prime}\geq a$). Here we use our proof rules (see subsection 4.2) and show the following: * • First, observe that $a$ is a post-fixpoint of $\mathcal{V}_{\sigma^{\prime}}$. For any $v\in V$ we have $\displaystyle a(v)=\mathcal{V}_{\sigma}(a)(v)$ $\displaystyle=\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ a(\sigma(v))&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle\leq\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ \max_{v^{\prime}\in\eta_{\max}(v)}a(v^{\prime})&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle=\begin{cases}\min_{v^{\prime}\in\eta_{\min}(v)}a(v^{\prime})&v\in\mathit{MIN}\\\ a(\sigma^{\prime}(v))&v\in\mathit{MAX}\\\ \sum_{v^{\prime}\in V}a(v^{\prime})\cdot\eta_{\mathrm{av}}(v)(v^{\prime})&v\in\mathit{AV}\\\ w(v)&v\in\mathit{SINK}\end{cases}$ $\displaystyle=\mathcal{V}_{\sigma^{\prime}_{\max}}(a)(v)$ * • Next we show that $\nu(\mathcal{V}_{\sigma^{\prime}})_{}^{a}=\emptyset$, thus proving that $a\leq\mu\mathcal{V}_{\sigma^{\prime}}=a^{\prime}$ by subsection 4.2. Note that $(\mathcal{V}_{\sigma^{\prime}})_{}^{a}\colon[{V}]^{a=\mathcal{V}_{\sigma^{\prime}}(a)}\to[{V}]^{a=\mathcal{V}_{\sigma^{\prime}}(a)}$, i.e., it restricts to those elements of $a$ where $a$ and $\mathcal{V}_{\sigma^{\prime}}(a)$ coincide. Whenever $v\in\mathit{MAX}$ is a node where the strategy has been “switched” with respect to $a$, we have $\mathcal{V}_{\sigma^{\prime}}(a)(v)=a(\sigma^{\prime}(v))>a(\sigma(v))=a(v).$ The first equality above is true by the definition of $\mathcal{V}_{\sigma^{\prime}}$ and the last equality holds since $a$ is a fixpoint of $\mathcal{V}_{\sigma}$. So if $v$ is a switch node, it holds that $v\not\in[{V}]^{a=\mathcal{V}_{\sigma}(a)}$. By contraposition if $v\in[{V}]^{a=\mathcal{V}_{\sigma}(a)}$, $v$ cannot be a switch node. We next show that $(\mathcal{V}_{\sigma})_{}^{a}$, $(\mathcal{V}_{\sigma^{\prime}})_{}^{a}$ agree on $[{V}]^{a=\mathcal{V}_{\sigma^{\prime}}(a)}\subseteq[{V}]^{a}=[{V}]^{a=\mathcal{V}_{\sigma}(a)}$ (remember that $a$ is a fixpoint of $\mathcal{V}_{\sigma}$). It holds that $\displaystyle(\mathcal{V}_{\sigma})_{}^{a}(V^{\prime})$ $\displaystyle=$ $\displaystyle\gamma_{\mathcal{V}_{\sigma}(a),\iota}(\mathcal{V}_{\sigma}(\alpha_{a,\iota}(V^{\prime})))$ $\displaystyle(\mathcal{V}_{\sigma^{\prime}})_{}^{a}(V^{\prime})$ $\displaystyle=$ $\displaystyle\gamma_{\mathcal{V}_{\sigma^{\prime}}(a),\iota}(\mathcal{V}_{\sigma^{\prime}}(\alpha_{a,\iota}(V^{\prime})))\cap[{V}]^{a=\mathcal{V}_{\sigma^{\prime}}(a)}$ for a suitable constant $\iota$ and if we choose $\iota$ small enough we can use the same constant in both cases. Now let $v\in[{V}]^{a=\mathcal{V}_{\sigma^{\prime}}(a)}$: by definition it holds that $v\in(\mathcal{V}_{\sigma})_{}^{a}(V^{\prime})=\gamma_{\mathcal{V}_{\sigma}(a),\iota}(\mathcal{V}_{\sigma}(\alpha_{a,\iota}(V^{\prime})))$ if and only if $\mathcal{V}_{\sigma}(\alpha_{a,\iota}(V^{\prime}))(v)\ominus\mathcal{V}_{\sigma}(a)(v)\geq\iota$. Since, by the considerations above, $v$ is not a switch node, $\mathcal{V}_{\sigma}(b)(v)=\mathcal{V}_{\sigma^{\prime}}(b)(v)$ for all $b$ and we can replace $\mathcal{V}_{\sigma}$ by $\mathcal{V}_{\sigma^{\prime}}$, resulting in the equivalent statement $v\in\gamma_{\mathcal{V}_{\sigma^{\prime}}(a),\iota}(\mathcal{V}_{\sigma^{\prime}}(\alpha_{a,\iota}(V^{\prime})))$, also equivalent to $v\in(\mathcal{V}_{\sigma^{\prime}})_{}^{a}(V^{\prime})$. Thus $\nu(\mathcal{V}_{\sigma^{\prime}})_{}^{a}\subseteq\nu(\mathcal{V}_{\sigma})_{}^{a}=\emptyset$. Hence we obtain an ascending sequence $a^{(i)}$. Furthermore, whenever we perform a switch, we know that $a^{(i)}$ is not a fixpoint of $\mathcal{V}_{\sigma^{(i+1)}}$ (otherwise we could not have performed a switch) and hence $a^{(i+1)}$ is strictly larger than $a^{(i)}$ for at least one input. Since there are only finitely many strategies we will eventually stop switching and reach the least fixpoint. ###### Example 7.8. Ex. 7.1 is well suited to explain our two algorithms. Starting with strategy iteration from above, we may guess $\tau^{(0)}(\min)=\textbf{1}$. In this case, Max would choose $\mathrm{av}$ as successor and we would reach a fixpoint, where each node except for $\bm{\varepsilon}$ is associated with a payoff of $1$. Next, our algorithm would detect the vicious cycle formed by $\min$, $\mathrm{av}$ and $\max$. We can reduce the values in this vicious cycle and reach the correct payoff values for each node. For strategy iteration from below assume that $\sigma^{(0)}(\max)=\mathrm{av}$. Given this strategy of Max, Min can force the play to stay in a cycle formed by $\min$, $\mathrm{av}$ and $\max$. Thus, the payoff achieved by the Max strategy $\sigma^{(0)}$ and an optimal play by Min would be $0$ for each of these nodes. In the next iteration Max switches and chooses $\bm{\varepsilon}$ as successor, i.e. $\sigma^{(1)}(\max)=\bm{\varepsilon}$, which results in the correct values. ### 7.3. Runtime results We implemented strategy iteration from above and from below – in the following abbreviated by SIA and SIB – and classical Kleene iteration (KI) in MATLAB. In Kleene iteration we terminate with a tolerance of $10^{-14}$, i.e., we stop if the change from one iteration to the next is below this value. In order to test the algorithms we created random stochastic games with $n$ nodes, where each Max, Min respectively average node has a maximal number of $m$ successors. For each node we choose randomly one of the four types of nodes. Sink nodes are given a random weight uniformly in $[0,1]$. Max and Min nodes are randomly assigned to successors and for an average nodes we assign a random number to each of its successors, followed by normalisation to obtain a probability distribution. We performed 1000 runs with different randomly created systems for each value of $n$ and $m=\frac{n}{2}$. The table below shows the runtimes in seconds and the number of iterations. Also, for SIB, we display the number of nodes with a payoff of $0$ (for an optimal play of Min) and the number of times SIA got stuck at any other fixpoint which is not $\mu\mathcal{V}$ (all numbers – runtime, iterations, etc. – are summed up over all $1000$ runs). \| runtime (seconds) \| number of iterations \| number nodes \| number of ---\|---\|---\|---\|--- $~{}~{}~{}n~{}~{}~{}$ \| KI \| SIA \| SIB \| KI \| SIA \| SIB \| payoff 0 \| other fp 10 \| 0.59 \| 20.28 \| 18.49 \| 47302 \| 2259 \| 2152 \| 2439 \| 508 20 \| 1.05 \| 31.71 \| 25.96 \| 30275 \| 3620 \| 3018 \| 4714 \| 743 30 \| 2.03 \| 35.98 \| 29.77 \| 27361 \| 3881 \| 3275 \| 7268 \| 771 40 \| 3.77 \| 38.84 \| 32.67 \| 26999 \| 3850 \| 3296 \| 9806 \| 756 50 \| 5.31 \| 38.09 \| 31.85 \| 26604 \| 3799 \| 3215 \| 12573 \| 734 60 \| 7.63 \| 40.33 \| 34.37 \| 26467 \| 3737 \| 3218 \| 15151 \| 727 70 \| 10.77 \| 45.00 \| 37.50 \| 26569 \| 3751 \| 3154 \| 17473 \| 751 80 \| 15.38 \| 54.89 \| 46.72 \| 26179 \| 3713 \| 3105 \| 20031 \| 752 90 \| 16.07 \| 52.21 \| 43.52 \| 26401 \| 3695 \| 3083 \| 22390 \| 777 100 \| 19.46 \| 60.29 \| 50.88 \| 26464 \| 3654 \| 3062 \| 25163 \| 751 Note that SIB always performs slightly better than SIA. Moreover KI neatly beats both of them. Here we need to remember that KI only converges to the solution and it is known that the rate of convergence can be exponentially slow [c:algorithms-ssg]. Note that the linear optimisation problems are quite costly to solve, especially for large systems. Thus additional iterations are substantially more costly compared to KI. Observe also that SIA has to perform more iterations than SIB, which explains the slightly higher runtime. The number of nodes with a payoff of 0 seems to grow linearly with the number of nodes in the system. The number of times SIA gets stuck at a fixpoint different from $\mu\mathcal{V}$ however seems to be independent of the system size and comparatively small. We performed a second comparison, where we assigned to sink nodes a value in $\\{0,1\\}$, which is often done for simple stochastic games. \| runtime \| number of iterations \| number nodes \| number of ---\|---\|---\|---\|--- $~{}~{}~{}n~{}~{}~{}$ \| KI \| SIA \| SIB \| KI \| SIB \| SIA \| payoff 0 \| other fp 10 \| 0.36 \| 14.51 \| 14.58 \| 42547 \| 1703 \| 1702 \| 5484 \| 219 20 \| 1.00 \| 19.85 \| 19.98 \| 29515 \| 2385 \| 2478 \| 8168 \| 137 30 \| 1.97 \| 20.45 \| 20.77 \| 27643 \| 2367 \| 2469 \| 11502 \| 33 40 \| 3.30 \| 20.13 \| 20.94 \| 26761 \| 2306 \| 2383 \| 14989 \| 12 50 \| 4.96 \| 20.24 \| 20.94 \| 26562 \| 2253 \| 2306 \| 18821 \| 2 60 \| 6.87 \| 20.57 \| 21.19 \| 26560 \| 2176 \| 2227 \| 22573 \| 0 70 \| 9.14 \| 21.95 \| 22.35 \| 26146 \| 2142 \| 2186 \| 26260 \| 0 80 \| 11.73 \| 24.69 \| 24.94 \| 26235 \| 2084 \| 2131 \| 30136 \| 0 90 \| 14.73 \| 28.90 \| 28.71 \| 26330 \| 2066 \| 2091 \| 33930 \| 0 100 \| 18.22 \| 34.75 \| 34.84 \| 26227 \| 2051 \| 2068 \| 37496 \| 0 Here, SIA performs very similar to SIB. The SIA approach seems to suffer, since Max can easily find himself in a situation where he can never reach a 1-sink, since only half of the sink nodes are of this kind. Additionally for these systems a significantly larger number of nodes have a payoff of 0 and SIA is less likely to get stuck at a fixpoint different from $\mu\mathcal{V}$. These factors seem to be correlated since it is now “harder” for Min to choose a bad successor (with a value greater than 0). ## 8\. Conclusion It is well-known that several computations in the context of system verification can be performed by various forms of fixpoint iteration and it is worthwhile to study such methods at a high level of abstraction, typically in the setting of complete lattices and monotone functions. Going beyond the classical results by Tarski [t:lattice-fixed-point], combination of fixpoint iteration with approximations [CC:TLA, bkp:abstraction-up-to-games-fixpoint] and with up-to techniques [p:complete-lattices-up-to] has proven to be successful. Here we treated a more specific setting, where the carrier set consists of functions from a finite set into an MV-chain and the fixpoint functions are non-expansive (and hence monotone), and introduced a novel technique to obtain upper bounds for greatest and lower bounds for least fixpoints, also providing associated algorithms. Such techniques are applicable to a wide range of examples and so far they have been studied only in quite specific scenarios, such as in [bblmtv:prob-bisim-distance-automata, f:game-metrics-markov-decision, kkkw:value-iteration-ssg]. In the future we plan to lift some of the restrictions of our approach. First, an extension to an infinite domain $Y$ would of course be desirable, but since several of our results currently depend on finiteness, such a generalisation does not seem to be easy. The restriction to total orders, instead, seems easier to lift: in particular, if the partially ordered MV-algebra $\bar{\mathbb{M}}$ is of the form $\mathbb{M}^{I}$ where $I$ is a finite index set and $\mathbb{M}$ an MV-chain. (E.g., finite Boolean algebras are of this type.) In this case, our function space is $\bar{\mathbb{M}}^{Y}=\big{(}\mathbb{M}^{I}\big{)}^{\raisebox{-2.0pt}{\scriptsize$Y$}}\cong\mathbb{M}^{Y\times I}$ and we have reduced to the setting presented in this paper. This will allow us to handle featured transition systems [ccpshl:simulation-product- line-mc] where transitions are equipped with boolean formulas. There are several other application examples that did not fit into this paper, but that can also be handled by our approach, for instance coalgebraic behavioural metrics [bbkk:coalgebraic-behavioral-metrics]. We also plan to investigate other types of games, such as energy games [bcdgr:algorithms-mean- payoff-games]. While here we introduced strategy iteration techniques for simple stochastic games, we also want to check whether we can provide an improvement to value iteration techniques, combining our approach with [kkkw:value-iteration-ssg]. We also plan to study whether some examples can be handled with other types of Galois connections: here we used an additive variant, but looking at multiplicative variants (multiplication by a constant factor) might also be fruitful. _Acknowledgements:_ We are grateful to Ichiro Hasuo for making us aware of stochastic games as an application domain. 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# Uncovering and displaying the coherent groups of rank data by exploratory riffle shuffling Vartan Choulakian and Jacques Allard Université de Moncton, Canada email: <EMAIL_ADDRESS><EMAIL_ADDRESS> (November 2020) ###### Abstract Let $n$ respondents rank order $d$ items, and suppose that $d<<n$. Our main task is to uncover and display the structure of the observed rank data by an exploratory riffle shuffling procedure which sequentially decomposes the n voters into a finite number of coherent groups plus a noisy group: where the noisy group represents the outlier voters and each coherent group is composed of a finite number of coherent clusters. We consider exploratory riffle shuffling of a set of items to be equivalent to optimal two blocks seriation of the items with crossing of some scores between the two blocks. A riffle shuffled coherent cluster of voters within its coherent group is essentially characterized by the following facts: a) Voters have identical first TCA factor score, where TCA designates taxicab correspondence analysis, an L1 variant of correspondence analysis; b) Any preference is easily interpreted as riffle shuffling of its items; c) The nature of different riffle shuffling of items can be seen in the structure of the contingency table of the first-order marginals constructed from the Borda scorings of the voters; d) The first TCA factor scores of the items of a coherent cluster are interpreted as Borda scale of the items. We also introduce a crossing index, which measures the extent of crossing of scores of voters between the two blocks seriation of the items. The novel approach is explained on the benchmarking SUSHI data set, where we show that this data set has a very simple structure, which can also be communicated in a tabular form. Key words: Borda score and scale; exploratory riffle shuffle; coherent group; coherent cluster; crossing index; taxicab correspondence analysis. AMS 2010 subject classifications: 62H25, 62H30 ## 1 Introduction Ordering the elements of a set is a common decision making activity, such as, voting for a political candidate, choosing a consumer product, etc. So there is a huge literature concerning the analysis and interpretation of preference data scattered in different disciplines. Often rank data is heterogenous: it is composed of a finite mixture of components. The traditional methods of finding mixture components of rank data are mostly based on parametric probability models, distance or latent class models, and are useful for sparse data and not for diffuse data. Rank data are sparse if there are at most a small finite number of permutations that capture the majority of the preferences; otherwise they are diffuse. As a running example in this paper, we will consider the famous benchmarking SUSHI data set enumerating $n=5000$ preferences of $d=10$ sushis, see $\left[1\right]$. The SUSHI data set is diffuse, because there are at most three counts for one observed permutation. It has been analyzed, among others by $\left[2,3,4\right]$. A second data set that we shall also analyze is the APA dataset of size $n=5738$ by $d=5$, see $\left[5\right]$. APA data set is also considered as non sparse by $\left[2\right]$, because all the 120 permutations exist with positive probability. For a general background on statistical methods for rank data, see the excellent monograph by $\left[6\right]$ and the book $\left[7\right]$. ### 1.1 Riffle shuffle The riffle shuffle, see $\left[8\right]$, is considered the most popular method of card shuffling, in which one cuts a deck of $d$ cards (aka items) into two piles of sizes $d_{1}$ and $d_{2}$, respectively, and successively drops the cards, one by one, so that the piles are interleaved into one deck again. Let $V,$ named a voting profile, be a set of $n$ preferences on $d$ items. Based on riffle shuffling ideas, $\left[2\right]$ proposed the notion of riffled independence to model the joint probability distribution of observed preferences of $V$. Independently, $\left[9\right]$ used it for visual exploration of $V$, naming it two blocks partition of the Borda scored items with crossing of some scores; this will be further developed here under the following important Assumption: $d<<n.$ This means that the sample size $n$ is quite large compared to the number of items $d$. SUSHI and APA data sets satisfy this assumption. The most important first step in the application of a riffle shuffling procedure is how to partition the $d$ items into two disjoint subsets. In the probabilistic riffle shuffling approach of $\left[2\right]$, the partitioning step is essentially done using some adhoc approach in the case of the SUSHI data set or based on background second order information of the items in the case of the APA data set. While in the exploratory riffle shuffling approach of this paper an optimal partition is obtained by maximizing the cut norm of row centered data, or equivalently by taxicab correspondence analysis of nega coded data. We compare the two formulations of riffle shuffle, probabilistic and exploratory, in section 10. ### 1.2 Aim Our aim is to explore and display a given voting profile $V$ by sequentially partitioning it into $G$ coherent groups plus a noisy group; that is, $V=\cup_{g=1}^{G}cohG(g)\cup noisyG,$ (1) where $G$ represents the number of coherent groups and $cohG(g)$ is the $g$th coherent group. Furthermore, each coherent group is partitioned into a finite number of coherent clusters; that is, $cohG(g)=\cup_{\alpha=1}^{c_{g}}cohC_{g}(\alpha)\text{ \ for }g=1,...,G,$ (2) where $c_{g}$ represents the number of coherent clusters in the $g$th coherent group. So the coherent clusters are the building blocks for the coherent groups. We note the following facts: Fact 1: The assumption $d<<n$ induces the new notion of coherency for the clusters and consequently for the groups; it is a stronger characterization than the notion of interpretability for groups as discussed in $\left[9\right]$. Fact 2: Each coherent group and its clusters have the same latent variable summarized by the Borda scale. Fact 3: Given that the proposed method sequentially peels the data like Occam’s razor, the number of groups $G$ is calculated automatically. Furthermore, outliers or uninformative voters belonging to the $noisyG$ are easily tagged. Fact 4: The approach is exploratory, visual, data analytic and is developed within the framework of taxicab correspondence analysis (TCA). TCA is an L1 variant of correspondence analysis developed by $\left[10\right]$. TCA is a dimension reduction technique similar to principal component analysis. In this paper, we shall use only the first TCA factor scores of the items and of the voters. Two major advantages of our method are: First, we can easily identify outliers. For the SUSHI data, our method tags 12.36% of the voters as outliers, which form the noisy group. While no outliers in the SUSHI data have been identified in $\left[3,\ 4\right]$. Second, it provides a tabular summary of the preferences which compose a coherent group. For instance, consider the first mixture component of the SUSHI data given in $\left[4\right]$, where the modal ordering is almost the same as the Borda scale ordering of the ten sushis in cohG(1) obtained by our method, see Table 14 in this paper. The sample size of their first mixture component is 27.56 %, which is much smaller than 48.36%, the sample size of our cohG(1), see Table 14. However, Table 13 of this paper provides a tabular-visual summary of the 2418 preferences which form cohG(1). The visual summary describes different kinds of equivalent similar riffle shufflings of the 2418 preferences, and it provides further insight into the structure of the data. Such interesting visual summaries are missing in $\left[3,\ 4\right]$. ### 1.3 Highlights of a coherent cluster A coherent cluster of voters has interesting mathematical properties and is essentially characterized by the following facts: a) Voters have identical unique first TCA factor score. b) Any voter preference is easily interpreted as a particular riffle shuffling of its items. c) The nature of riffle shuffling of the items can be observed in the structure of the contingency table of the first-order marginals constructed from the Borda scorings of the voters belonging to the coherent cluster. d) The first TCA factor scores of the items of a coherent cluster are interpreted as Borda scale of the items. e) We also introduce the crossing index, which measures the extent of interleaving or the crossing of scores of voters between two blocks seriation of the items in a coherent cluster. ### 1.4 Organization This paper has eleven sections and its contents are organized as follows: Section 2 presents an overview of TCA; section 3 presents some preliminaries on the Borda coding of the data and related tables and concepts; section 4 presents Theorem 1, which shows that the first principal dimension of TCA clusters the voters into a finite number of clusters; section 5 discusses coherent clusters and their mathematical properties; section 6 discusses riffle shuffling in a coherent cluster; section 7 introduces the crossing index; section 8 introduces the coherent groups; section 9 presents the analysis of APA data set; section 10 presents a comparison of the two formulations of riffle shuffle probabilistic and exploratory; and finally we conclude in section 11. All mathematical proofs are relegated to the appendix. Details of the computation are shown only for the first coherent group of SUSHI data set. ## 2 An overview of taxicab correspondence analysis Consider a $n\times p$ matrix $\mathbf{X}$ where $X_{ij}\geq 0.$ We have $\sum_{j=1}^{p}\sum_{i=1}^{n}\mathbf{X}_{ij}=X_{\ast\ast}.$ Let $\mathbf{P=X/}X_{\ast\ast}$ be the correspondence matrix associated to X; and as usual, we define $p_{i\ast}=\sum_{j=1}^{p}p_{ij}$, $p_{\ast j}=\sum_{i=1}^{n}p_{ij}$. Let $\mathbf{D}_{n}=Diag(p_{i\ast})$ a diagonal matrix with diagonal elements $p_{i\ast}$. Similarly $\mathbf{D}_{p}=Diag(p_{\ast j})$. Let $k=rank(\mathbf{P)}-1$. In TCA the calculation of the dispersion measures $(\delta_{\alpha})$, principal axes ($\mathbf{u}_{\alpha},\mathbf{v}_{\alpha}),$ principal basic vectors $(\mathbf{a}_{\alpha},\mathbf{b}_{\alpha}),$ and principal factor scores $(\mathbf{f}_{\alpha},\mathbf{g}_{\alpha})$ for $\alpha=1,...,k$ is done in a stepwise manner. We put $\mathbf{P}_{1}=(p_{ij}^{(1)}=p_{ij}-p_{i\ast}\ p_{\ast j})$. Let $\mathbf{P_{\alpha}}$ be the residual correspondence matrix at the $\alpha$-th iteration. The variational definitions of the TCA at the $\alpha$-th iteration are $\displaystyle\delta_{\alpha}$ $\displaystyle=$ $\displaystyle\max_{\mathbf{u\in\mathbb{R}}^{p}}\frac{\left\|\left\|\mathbf{P_{\alpha}u}\right\|\right\|_{1}}{\left\|\left\|\mathbf{u}\right\|\right\|_{\infty}}=\max_{\mathbf{v\in\mathbb{R}}^{n}}\ \frac{\left\|\left\|\mathbf{P_{\alpha}^{\prime}v}\right\|\right\|_{1}}{\left\|\left\|\mathbf{v}\right\|\right\|_{\infty}}=\max_{\mathbf{u\in\mathbb{R}}^{p},\mathbf{v\in\mathbb{R}}^{n}}\frac{\mathbf{v}^{\prime}\mathbf{P_{\alpha}u}}{\left\|\left\|\mathbf{u}\right\|\right\|_{\infty}\left\|\left\|\mathbf{v}\right\|\right\|_{\infty}},$ $\displaystyle=$ $\displaystyle\max\|\|\mathbf{P_{\alpha}u\|\|}_{1}\ \ \text{subject to }\mathbf{u}\in\left\\{-1,+1\right\\}^{p},$ $\displaystyle=$ $\displaystyle\max\|\|\mathbf{P_{\alpha}^{\prime}v\|\|}_{1}\ \ \text{subject to }\mathbf{v}\in\left\\{-1,+1\right\\}^{n},$ $\displaystyle=$ $\displaystyle\max\mathbf{v}^{\prime}\mathbf{P_{\alpha}u}\text{ \ subject to \ }\mathbf{u}\in\left\\{-1,+1\right\\}^{p},\mathbf{v}\in\left\\{-1,+1\right\\}^{n}.$ The $\alpha$-th principal axes are $\mathbf{u}_{\alpha}\ =\arg\max_{\mathbf{u}\in\left\\{-1,+1\right\\}^{p}}\left\|\left\|\mathbf{P_{\alpha}u}\right\|\right\|_{1}\text{ \ \ and \ \ }\mathbf{v}_{\alpha}\ =\arg\max_{\mathbf{v}\in\left\\{-1,+1\right\\}^{n}}\left\|\left\|\mathbf{P_{\alpha}^{\prime}v}\right\|\right\|_{1}\text{,}$ (3) and the $\alpha$-th basic principal vectors are $\mathbf{a}_{\alpha}=\mathbf{P_{\alpha}u}_{\alpha}\text{ \ and \ }\mathbf{b}_{\alpha}=\mathbf{P_{\alpha}^{\prime}v}_{\alpha},$ (4) and the $\alpha$-th principal factor scores are $\mathbf{f}_{\alpha}=\mathbf{D}_{n}^{-1}\mathbf{a}_{\alpha}\text{ \ and \ }\mathbf{g}_{\alpha}=\mathbf{D}_{p}^{-1}\mathbf{b}_{\alpha};$ (5) furthermore the following relations are also useful $\mathbf{u}_{\alpha}=sgn(\mathbf{b}_{\alpha})=sgn(\mathbf{g}_{\alpha})\text{ \ and \ }\mathbf{v}_{\alpha}=sgn(\mathbf{a}_{\alpha})=sgn(\mathbf{f}_{\alpha}),$ (6) where $sgn(.)$ is the coordinatewise sign function, $sgn(x)=1$ if $x>0,$ and $sgn(x)=-1$ if $x\leq 0.$ The $\alpha$-th taxicab dispersion measure $\delta_{\alpha}$ can be represented in many different ways $\begin{array}[]{cccc}\delta_{\alpha}&=&\left\|\left\|\mathbf{P_{\alpha}u}_{\alpha}\right\|\right\|_{1}=\left\|\left\|\mathbf{a}_{\alpha}\right\|\right\|_{1}=\mathbf{a}_{\alpha}^{\prime}\mathbf{v}_{\alpha}=\left\|\left\|\mathbf{D}_{n}\mathbf{f}_{\alpha}\right\|\right\|_{1}=\mathbf{u}_{\alpha}^{\prime}\mathbf{D}_{n}\mathbf{f}_{\alpha},&\\\ &=&\left\|\left\|\mathbf{P_{\alpha}^{\prime}v}_{\alpha}\right\|\right\|_{1}=\left\|\left\|\mathbf{b}_{\alpha}\right\|\right\|_{1}=\mathbf{b}_{\alpha}^{\prime}\mathbf{u}_{\alpha}=\left\|\left\|\mathbf{D}_{p}\mathbf{g}_{\alpha}\right\|\right\|_{1}=\mathbf{v}_{\alpha}^{\prime}\mathbf{D}_{p}\mathbf{g}_{\alpha}.&\end{array}$ (7) The $(\alpha+1)$-th residual correspondence matrix is $\mathbf{P_{\alpha+1}}=\mathbf{P_{\alpha}-D}_{n}\mathbf{f}_{\alpha}\mathbf{g}_{\alpha}^{{}^{\prime}}\mathbf{D}_{p}/\delta_{\alpha}.$ (8) An interpretation of the term $\mathbf{D}_{n}\mathbf{g}_{\alpha}\mathbf{f}_{\alpha}^{{}^{\prime}}\mathbf{D}_{p}/\delta_{\alpha}$ in (8) is that, it represents the best rank-1 approximation of the residual correspondence matrix $\mathbf{P_{\alpha}}$, in the sense of taxicab norm. In CA and TCA, the principal factor scores are centered; that is, $\sum_{i=1}^{n}f_{\alpha}(i)p_{i\ast}=0=\sum_{j=1}^{p}g_{\alpha}(j)p_{\ast j}\text{ \ \ \ for \ \ }\alpha=1,...,k.$ (9) The reconstitution formula in TCA and CA is $p_{ij}=p_{i.}p_{.j}\left[1+\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}\right].$ (10) In TCA, the calculation of the principal component weights, $\mathbf{u}_{\alpha}$ and $\mathbf{v}_{\alpha},$ and the principal factor scores, $\mathbf{g}_{\alpha}$ and $\mathbf{f}_{\alpha},$ can be accomplished by two algorithms. The first one is based on complete enumeration based on equation (3). The second one is based on iterating the transition formulae (4,5,6). This is an ascent algorithm; that is, it increases the value of the objective function at each iteration, see $\left[11\right]$. The iterative algorithm could converge to a local maximum; so it should be restarted from several initial configurations. The rows or the columns of the data can be used as starting values. The TCA map is obtained by plotting $(\mathbf{g}_{1},\mathbf{g}_{2})$ or $(\mathbf{f}_{1},\mathbf{f}_{2}).$ ## 3 Preliminaries In this section we review a) The Borda scoring of a voting profile V into R and the Borda scale; b) Contingency table of the first order marginals of R; c) The coded tables Rdouble and R${}_{nega}.$ ### 3.1 Borda scorings and Borda scale Let $A=\\{a_{1},a_{2},\ldots,a_{d}\\}$ denote a set of $d$ alternatives/candidates/items, and $V$ a set of $n$ voters/individuals/judges. In this paper we consider the linear orderings/rankings/preferences, in which all $d$ objects are rank-ordered according to their levels of desirability by the $n$ voters. We denote a linear order by a sequence $\mathbf{s}=(a_{k_{1}}\succ a_{k_{2}}\succ\ldots\succ a_{k_{d}})$, where $a_{k_{1}}\succ a_{k_{2}}$ means that the alternative $a_{k_{1}}$ is preferred to the alternative $a_{k_{2}}.$ The Borda scoring of $\mathbf{s}$, see $\left[12\right],$ is the vector $b(\mathbf{s)}$ where to the element $a_{k_{j}}$the score of $(d-j)$ is assigned, because $a_{k_{j}}$ is preferred to $(d-j)$ other alternatives; or equivalently it is the $j$th most preferred alternative. Let $\mathbf{R=(}r_{ij})$ be the matrix having $n$ rows and $d$ columns, where $r_{ij}$ designates the Borda score of the $i$th voter’s preference of the $j$th alternative. We note that the $i$th row of $\mathbf{R}$ will be an element of $S_{d}$ the set of permutations of the elements of the set $\left\\{0,1,2,...,d-1\right\\}.$ A toy example of $\mathbf{R}$ is presented in Table 1 for $n=4$ and $d=3$. The Borda scale of the elements of $A$ is $\mathbf{\beta}=\mathbf{1}_{n}^{\prime}\mathbf{R}/n,$ where $\mathbf{1}_{n}$ is a column vector of $1$’s having $n$ coordinates. The Borda scale seriates/orders the $d$ items of the set $A$ according to their average scores: $\mathbf{\beta}(j)>\mathbf{\beta}(i)$ means item $j$ is preferred to item $i$, and $\mathbf{\beta}(j)=\mathbf{\beta}(i)$ means both items $(a_{i},a_{j})$ are equally preferred. In the toy example of Table 1, the Borda scale seriates $\\{A,B\\}\succ C$. Similarly, we define the reverse Borda score of $\mathbf{s}$ to be the vector $\overline{b}$($\mathbf{s)}$, which assigns to the element $a_{k_{j}}$the score of $(j-1).$ We denote $\overline{\mathbf{R}}\mathbf{=(}\overline{r}_{ij})$ to be the matrix having $n$ rows and $d$ columns, where $\overline{r}_{ij}$ designates the reverse Borda score of the $i$th judge’s nonpreference of the $j$th alternative. The reverse Borda scale of the $d$ items is $\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}}/n.$ We note that $\mathbf{R+}\overline{\mathbf{R}}=(d-1)\mathbf{1}_{n}\mathbf{1}_{d}^{\prime}$ and $\mathbf{\beta+}\overline{\mathbf{\beta}}=(d-1)\mathbf{1}_{d}^{\prime}.$ Table 1: Toy example with $n=4$ preferences of $d=3$ items. --- \| $\mathbf{R}$ \| \| \| \| $\overline{\mathbf{R}}$ \| $A\succ B\succ C$ \| 0 \| 1 \| 2 \| 2 \| 1 \| 0 $A\succ C\succ B$ \| 1 \| 0 \| 2 \| 1 \| 2 \| 0 $B\succ A\succ C$ \| 0 \| 2 \| 1 \| 2 \| 0 \| 1 $B\succ C\succ A$ \| 1 \| 2 \| 0 \| 1 \| 0 \| 2 Borda scale $\mathbf{\beta}$ \| 0.5 \| 1.25 \| 1.25 \| \| \| $\text{reverse Borda scale}\overline{\text{ }\mathbf{\beta}}$ \| \| \| \| 1.5 \| 0.75 \| 0.75 nega $n\overline{\mathbf{\beta}}$ \| \| \| \| 6 \| 3 \| 3 ### 3.2 Contingency table of first-order marginals The contingency table of first order marginals of an observed voting profile $V$ on $d$ items is a square $d\times d$ matrix M, where $\mathbf{M(}i,j\mathbf{)}$ stores the number of times that item $j$ has Borda score $i$ for $i=0,...,d-1,$ see $\left[6,\ \text{p.17}\right]$. Table 2 displays the matrix M for the toy example $\mathbf{R}$ displayed in Table 1. We note the following facts: a) It has uniform row and column marginals equal to the sample size. b) We can compute the Borda scale $\mathbf{\beta}$ from it. c) It reveals the nature of crossing of scores attributed to the items for a given binary partition of the items. For the toy example, consider the partition $\left\\{C\right\\}$ and $\left\\{B,A\right\\}$ with attributed scores of $\left\\{0\right\\}$ and $\left\\{1,2\right\\}$ respectively (this is the first step in a riffle shuffle). Then the highlighted cells (marked in bold) in Table 2 show that there are two crossing of scores, permutation (transposition) of the scores 0 and 1, between the sets $\left\\{C\right\\}$ and $\left\\{B,A\right\\}$, (this is the second step in a random shuffle). Furthermore the third row of Table 2 shows that the score 2 is equally attributed to both items of the set $\left\\{B,A\right\\}$ and it never crossed to $\left\\{C\right\\}$. Table 2: The matrix of first-order marginals of R. --- \| $C$ \| $B$ \| $A$ \| row sum $0$ \| 2 \| 1 \| 1 \| 4 $1$ \| 2 \| 1 \| 1 \| 4 $2$ \| 0 \| 2 \| 2 \| 4 column sum \| 4 \| 4 \| 4 \| Borda scale $\mathbf{\beta}$ \| 0.5 \| 1.25 \| 1.25 \| ### 3.3 Coded tables $\mathbf{R}_{double}$ and $\mathbf{R}_{nega}$ Our methodological approach is based on Benzécri’s platform, see $\left[13,\ p.1113\right],$ that we quote: “ the main problem inductive statistics has to face is to build tables that, through appropriate coding and eventual supplementation, give to the available data such a shape that the analysis is able to extract from it the answer to any question that we are allowed to ask”. Italics are ours. There are three elements in Benzécri’s platform: a) coding, a kind of pre- processing of data, will be discussed in the following paragraph; b) eventual supplementation consists in applying TCA and not correspondence analysis (CA), because in the CA case we do not have a result similar to Theorem 1; c) question that we are allowed to ask is to explore and visualize rank data. Within the CA framework, there are two codings of rank data $\mathbf{R}_{double}$ and $\mathbf{R}_{nega.}$. #### 3.3.1 $\mathbf{R}_{double}$ The first one is the doubled table of size $(2n)\times d$ $\mathbf{R}_{double}=(_{\overline{\mathbf{R}}}^{\mathbf{R}})$ proposed independently by $\left[14,\ 15\right]$, where they showed that CA of $\mathbf{R}_{double}$ is equivalent to the dual scaling of Nishisato coding of rank data, see $\left[16\right]$. CA of $\mathbf{R}_{double}$ is equivalent to CA of its first residual correspondence matrix $\mathbf{P}_{double}^{1}=\frac{1}{t}(_{-(r_{ij}-\frac{d-1}{2})}^{(r_{ij}-\frac{d-1}{2})}),$ where $t=nd(d-1)$. The structure of $\mathbf{P}_{double}^{1}$ shows that each row is centered as in Carroll’s multidimensional preference analysis procedure, MDPREF, exposed in Alvo and Yu (2014, p.15). In TCA the objective function to maximize is a combinatorial problem, see equation (3); and the first iteration in TCA of $\mathbf{R}_{double}$ corresponds to computing $\begin{array}[]{cccc}\delta_{1}^{double}&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\|\|(\mathbf{v}^{t}\ \|\ \mathbf{-v}^{t})\mathbf{P}_{double}^{1}\|\|_{1}&\\\ &=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{2}{t}\sum_{j=1}^{d}\|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})v_{i}\|\text{.}&\end{array}$ #### 3.3.2 $\mathbf{R}_{nega}$ In the second approach, we summarize $\overline{\mathbf{R}}$ by its column total; that is, we create a row named $\mathbf{nega=}$ $n\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}},$ then we vertically concatenate $\mathbf{nega}$ to $\mathbf{R}$, thus obtaining $\mathbf{R}_{nega}=(_{\mathbf{nega}}^{\mathbf{R}})$ of size $(n+1)\times d.$ $\left[17\right]$ discussed the relationship between TCA of $\mathbf{R}_{double}$ and TCA of $\mathbf{R}_{nega}$: TCA of $\mathbf{R}_{nega}$ can be considered as constrained TCA of $\mathbf{R}_{double}$, because we are constraining the vector $\mathbf{-v}^{t}=\mathbf{-1}_{n}^{t}$ in (11); that is, the objective function to maximize corresponds to computing $\begin{array}[]{cccc}\delta_{1}&=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\|\|(\mathbf{v}^{t}\ \|\ \mathbf{-1}_{n}^{t})\mathbf{P}_{double}^{1}\|\|_{1}&\\\ &=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\|\|(\mathbf{v}^{t}\ \ \mathbf{-}1)\mathbf{P}_{nega}^{1}\|\|_{1}\\\ &=&\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{1}{t}\sum_{j=1}^{d}\|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})(v_{i}+1)\|\text{.}&\end{array}$ So, we see that if in (11) the optimal value of $\mathbf{v}=\mathbf{1}_{n}$, then $\delta_{1}^{double}=\delta_{1},$ otherwise $\delta_{1}^{double}>\delta_{1}$. Let $\mathbf{v}_{1}=\arg\max_{\mathbf{v\in}\left\\{-1,1\right\\}^{n}}\frac{1}{t}\sum_{j=1}^{d}\|\sum_{i=1}^{n}(r_{ij}-\frac{d-1}{2})(v_{i}+1)\|.$ Define the set of indices $I_{+}=\left\\{i\|v_{1i}=1\right\\}$ and $I_{-}=\left\\{i\|v_{1i}=-1\right\\},$ where $\mathbf{v}_{1}=(v_{1i}).$ Then $\delta_{1}=\frac{2}{t}\sum_{j=1}^{d}\|\sum_{i\in I_{+}}(r_{ij}-\frac{d-1}{2})\|$ (13) shows that the summation in (13) is restricted to the subset of assessors that belong to $I_{+}$. The subset $I_{+}$ indexes the voters having the same direction in their votes. Given that we are uniquely interested in the first TCA dimension, all the necessary information is encapsulated in $I_{+}$, as discussed in $\left[17,\ 9\right]$ using other arguments. Furthermore, $\delta_{1}$ in (13) equals four times the cut norm of $\mathbf{R}_{centered}(i,j)=\frac{1}{t}(r_{ij}-\frac{d-1}{2}),$ where the cut norm is defined to be $\displaystyle\left\|\left\|\mathbf{R}_{centered}\right\|\right\|_{cut}$ $\displaystyle=$ $\displaystyle\max_{S,T}\frac{1}{t}\sum_{j\in S}\sum_{i\in T}(r_{ij}-\frac{d-1}{2})$ $\displaystyle=$ $\displaystyle\frac{1}{t}\sum_{j\in S_{+}}\sum_{i\in I_{+}}(r_{ij}-\frac{d-1}{2})$ $\displaystyle=$ $\displaystyle\delta_{1}/4,$ where $S\subseteq\left\\{1,...,d\right\\}$ and $T\subseteq I;$ it shows that the subsets $I_{+}$ and $S_{+}$ are positively associated, for further details see for instance, $\left[18,\ 19\right]$. In the sequel, we will consider only the application of TCA to $\mathbf{R}_{nega}$. ## 4 First TCA voter factor scores of Rnega We show the results on the SUSHI data set enumerating $n=5000$ preferences of $d=10$ sushis, see $\left[1\right]$. Even though, our interest concerns only the first TCA voter factor scores of a voting profile $V_{1},$ it is a common practice in CA circles to present the principal map of the row and column projections. Figures 1 and 2 display the principal maps obtained from CA and TCA of $R_{nega}$ of the SUSHI data denoted by $V_{1}$. We observe that, TCA clusters the voters into a finite number of discrete patterns, while CA does not: This is the main reason that we prefer the use of TCA to the use of the classical well known dimension reduction technique CA. We have the following theorem concerning the first TCA principal factor scores of the voters belonging to a profile $V_{1}$, $f_{1}(i)$ for $i=1,...,n$, where the first principal axis partitions the $d$ items into $d_{1}$ and $d_{2}$ parts such that $d=d_{1}+d_{2}.$ (a) Figure 1:CA map of SUSHI rank data (b) Figure 2: TCA map of SUSHI rank data Theorem 1 a) The maximum number of distinct clusters of the $n$ voters belonging to $V_{1}$ on the first TCA principal axis (distinct $f_{1}(i\mathbf{)}$ values for $i\mathbf{\in}V_{1})$ is $d_{1}d_{2}+1.$ b) The maximum value that $f_{1}(i\mathbf{)}$ can attain is $2\frac{d_{1}d_{2}}{d(d-1)}.$ c) The minimum value that $f_{1}(i\mathbf{)}$ can attain is $-2\frac{d_{1}d_{2}}{d(d-1)}.$ d) If the number of distinct clusters is maximum, $d_{1}d_{2}+1$, then the gap between two contiguous $f_{1}(i\mathbf{)}$ values is $\frac{4}{d(d-1)}.\vskip 12.0pt plus 4.0pt minus 4.0pt$ Remark 1 a) We fix $f_{1}(nega)<0$ to eliminate the sign indeterminacy of the first bilinear term in (10). b) We partition $V_{1}$ into $d_{1}d_{2}+1$ clusters, $V_{1}=\cup_{\alpha=1}^{d_{1}d_{2}+1}V_{1,\alpha}$, where the voters of the $\alpha$th cluster are characterized by their first TCA factor score; that is, $V_{1,\alpha}=\left\\{i\in V_{1}\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)=}2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}\right\\}$ for $\alpha=1,...,d_{1}d_{2}+1$. Example 1: In Figure 2, $d_{1}=4$ and $d_{2}=6,$ and we observe Fact 1: by Theorem 1a, 5000 preferences are clustered into $d_{1}d_{2}+1=25$ clusters on the first TCA principal axis. Fact 2: by Theorem 1b, the maximum value of $f_{1}(i\mathbf{)=}$ $48/90\mathbf{.}$ Fact 3: by Theorem 1c, the minimum value of $f_{1}(i\mathbf{)=}$ $-48/90\mathbf{.}$ Fact 4: by Theorem 1d, the gap separating two contiguous clusters of voters on the first TCA principal axis is $4/90.\vskip 12.0pt plus 4.0pt minus 4.0pt$ A cluster of voters defined in Remark 1b, $V_{1,\alpha}$ for $\alpha=1,...,d_{1}d_{2}+1,$ can be classified as coherent or incoherent. And this will be discussed in the next section. ## 5 Coherent cluster The following definition characterizes a coherent cluster. Definition 1 (Coherency of a cluster of voters $V_{1,\alpha}$ for $\alpha=1,...,d_{1}d_{2}+1$) A cluster of voters $V_{1,\alpha}$ $\subseteq V_{1}$ is coherent if $f_{1}^{V_{1,\alpha}}(v\mathbf{)=}2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}$ for all $v\mathbf{\in}V_{1,\alpha},$ where $f_{1}^{V_{1,\alpha}}(i\mathbf{)}$ is the first TCA factor score of the voter $i\mathbf{\in}V_{1,\alpha}$ obtained from TCA of subprofile $V_{1,\alpha}.\vskip 12.0pt plus 4.0pt minus 4.0pt$ Remark 2: a) It is important to distinguish between $f_{1}^{V_{1}}(i\mathbf{)}$ for $i=1,...,\|V_{1}\|$ where $n=\|V_{1}\|,$ and $f_{1}^{V_{1,\alpha}}(i\mathbf{)}$ for $i=1,...,\|V_{1,\alpha}\|,$ where $\|V_{1,\alpha}\|$ represents the sample size of the cluster $\|V_{1,\alpha}\|.$ b) Definition 1 implies that a cluster $V_{1,\alpha}$ is coherent when for all voters $i\mathbf{\in}V_{1,\alpha}$ the first TCA factor score $f_{1}^{V_{1,\alpha}}(i\mathbf{)}$ does not depend on the voter $i$, but it depends on $(\alpha,d_{1},d_{2}).\vskip 12.0pt plus 4.0pt minus 4.0pt$ Corollary 1: It follows from Remark 1a and equation (13) that, a necessary condition, but not sufficient, for a cluster $V_{1,\alpha}$ to be coherent is that its first TCA factor score obtained from TCA of $V_{1}$ is strictly positive; that is, $0<f_{1}^{V_{1}}(i)$ for $i\in V_{1,\alpha}.$ (a) Figure 3 : TCA map of $\mbox{V}_{1,1}$. (b) Figure 4 : TCA map of $\mbox{V}_{1,2}$. (c) Figure 5 : TCA map of $\mbox{V}_{1,3}$. (d) Figure 6 : TCA map of $\mbox{V}_{1,4}$. (e) Figure 7 : TCA map of $\mbox{V}_{1,5}$. (f) Figure 8 : TCA map of $\mbox{V}_{1,6}$. (g) Figure 9 : TCA map of $\mbox{V}_{1,7}$. (h) Figure 10: TCA map of $\mbox{V}_{1,8}$. Example 2: Figures 3 through 9 show the coherency of the clusters of voters $V_{1,\alpha}$ for $\alpha=1,...,7,$ where dots represent clusters of voters; while Figure 10 shows the incoherence of the cluster $V_{1,8}.$ Further, the first three columns of Table 3 display the mathematical formulation of the 7 coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$ for $\alpha=1,...,7$ as defined in Remark 1b and their sample sizes $\|V_{1,\alpha}\|.$ Table 3: Characteristics of $cohC_{1}(\alpha)=$ $V_{1,\alpha}$ of SUSHI data. --- $\alpha$ \| $\|V_{1,\alpha}\|$ \| description of $V_{1,\alpha}$ \| $\delta_{1}(V_{1,\alpha})$ \| $T_{v\in V_{1,\alpha}}(\tau_{J_{1}}(S_{1}))$ \| $Cross(V_{1,\alpha})$ $1$ \| $314$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=48/90\right\\}$ \| $48/90$ \| $6$ \| $0$ $2$ \| $235$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=44/90\right\\}$ \| $44/90$ \| $7$ \| $1/12$ $3$ \| $326$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=40/90\right\\}$ \| $40/90$ \| $8$ \| $2/12$ $4$ \| $315$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=36/90\right\\}$ \| $36/90$ \| $9$ \| $3/12$ $5$ \| $452$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=32/90\right\\}$ \| $32/90$ \| $10$ \| $4/12$ $6$ \| $375$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=28/90\right\\}$ \| $28/90$ \| $11$ \| $5/12$ $7$ \| $401$ \| $\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=24/90\right\\}$ \| $24/90$ \| $12$ \| $6/12$ Proposition 1: For a voting profile $V$, $\delta_{1}(V)\geq\|f_{1}(nega)\|$, where $\delta_{1}(V)$ is the first TCA dispersion value obtained from TCA of $V,$ and $f_{1}(nega)$ is the first TCA factor score of the row $nega$. The equality in Proposition 1 is attained only for coherent clusters as shown in the following result. Proposition 2: The first TCA dispersion value of a coherent cluster $cohC_{1}(\alpha)$ satisfies $\displaystyle\delta_{1}(cohC_{1}(\alpha))$ $\displaystyle=$ $\displaystyle\|f_{1}^{V_{1,\alpha}}(nega)\|.$ $\displaystyle\mathbf{=}$ $\displaystyle 2\frac{d_{1}d_{2}}{d(d-1)}-(\alpha-1)\frac{4}{d(d-1)}$ Example 3: propostion 2 can be observed by looking at the columns 3 and 4 of Table 3 which concern the 7 coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$ for $\alpha=1,...,7$. While for the incoherent cluster $V_{1,8}$ with sample size of $\|V_{1,8}\|=335,$ we observe: $V_{1,8}=\left\\{i\mathbf{:}f_{1}^{V_{1}}(i\mathbf{)}=20/90=0.222\right\\},$ and by Proposition 1, $\delta_{1}(V_{1,8})=0.2354$ $>2/9.$ This means that the 335 voters belonging to $V_{1,8}$ form a cluster within the whole sample of 5000 voters, but separated as 335 voters they do not form a coherent cluster. ### 5.1 Interpretability of a coherent cluster The following result shows that for coherent clusters, the first TCA dimension can be interpreted as Borda scaled factor. Proposition 3: The first TCA column factor score of the item $j,$ $g_{1}(j),$ is an affine function of the Borda scale $\beta(j);$ that is, $g_{1}(j)=\frac{2}{d-1}\beta(j)-1$ for $j=1,...,d.$ Or $corr(\mathbf{g}_{1},\mathbf{\beta})=1.\vskip 12.0pt plus 4.0pt minus 4.0pt$ Remark 3: The first TCA principal factor score of item $j$ for $j=1,...,d$ is bounded: $-1\leq g_{1}(j)\leq 1,$ because $0\leq\beta(j)\leq d-1.\vskip 12.0pt plus 4.0pt minus 4.0pt$ Example 4: Table 4 displays the Borda scales of the items, sushis, in the seven coherent clusters $cohC_{1}(\alpha)=V_{1,\alpha}$ for $\alpha=1,...,7.$ To identify the sushi type, one has to refer to Figure 2; for instance, $j10$ corresponds to $10cucumber$ $roll$ in Figure 2. We observe the following main fact: For each of the seven coherent clusters, the first TCA principal axis produced the same binary partition of the items: $J_{1}=\left\\{j10,j7,j4,j9\right\\}$ characterized by $4.5>\beta(j_{1})$ for $j_{1}\in J_{1}$, and $J_{2}=\left\\{j3,j1,j2,j6,j5,j8\right\\}$ characterized by $\beta(j_{1})$ $>4.5$ for $j_{2}\in J_{2}.$ The six sushis in $J_{2}$ have Borda scales above average score of $4.5=(0+9)/2$; while the four sushis in $J_{1}$ have Borda scales below average score of $4.5.\vskip 12.0pt plus 4.0pt minus 4.0pt\ $ Table 4: Borda scales of the 10 sushis in the seven coherent clusters. --- Borda scale \| items \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 $\mathbf{\beta}(cohC_{1}(1))$ \| 0.66 \| 1.31 \| 1.87 \| 2.16 \| 5.55 \| 5.78 \| 6.03 \| 6.58 \| 7.31 \| 7.52 $\mathbf{\beta}(cohC_{1}(2))$ \| 0.69 \| 1.29 \| 2.44 \| 2.59 \| 5.47 \| 5.43 \| 5.50 \| 6.35 \| 7.38 \| 7.86 $\mathbf{\beta}(cohC_{1}(3))$ \| 0.65 \| 1.60 \| 3.04 \| 2.71 \| 5.25 \| 5.25 \| 5.39 \| 6.26 \| 7.17 \| 7.68 $\mathbf{\beta}(cohC_{1}(4))$ \| 0.83 \| 1.79 \| 3.10 \| 3.28 \| 5.30 \| 4.74 \| 5.22 \| 6.34 \| 6.76 \| 7.64 $\mathbf{\beta}(cohC_{1}(5))$ \| 1.12 \| 2.02 \| 3.26 \| 3.60 \| 5.70 \| 4.74 \| 5.27 \| 5.75 \| 5.99 \| 7.60 $\mathbf{\beta}(cohC_{1}(6))$ \| 1.12 \| 2.33 \| 3.62 \| 3.93 \| 5.68 \| 4.98 \| 5.21 \| 5.33 \| 5.25 \| 7.56 $\mathbf{\beta}(cohC_{1}(7))$ \| 1.42 \| 2.74 \| 3.84 \| 4.00 \| 5.45 \| 4.70 \| 5.02 \| 5.26 \| 5.20 \| 7.38 Now we ask the question what are the differences among the seven coherent clusters? The answer is riffle shuffling of the scores of the items, which we discuss next. ## 6 Exploratory riffle shuffling $\left[8\right]$ is the seminal reference on riffle shuffling of cards. $\left[2\right]$ generalized the notion of independence of two subsets of items to riffled independence to uncover the structure of rank data. Within the framework of data analysis of preferences, exploratory riffle shuffling can be described in the following way. We have two sets: $J$ a set of $d$ distinct items and $S$ a set of $d$ Borda scores. We partition both sets into two disjoint subsets of sizes $d_{1}$ and $d_{2}=d-d_{1};$ that is, $J=J_{1}\cup J_{2}$ with $J_{1}=\left\\{j_{1},j_{2},...,j_{d_{1}}\right\\}$ and $S=S_{1}\cup S_{2}$ with $S_{1}=\left\\{0,1,...,d_{1}-1\right\\}.$ Riffle shuffling consists of two steps. In the first step, we attribute the scores of $S_{1}$ to $J_{1}$ and the scores of $S_{2}$ to $J_{2}.$ In the second step, we permute some scores attributed to $J_{1}$ with the same number of scores attributed to $J_{2}.$ The second step can be mathematically described as an application of a permutation $\tau$, such that $\tau_{J}(S_{1},S_{2})=(\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2})).$ We interpret $\tau_{J_{1}}(S_{1})$ as the set of scores attributed to $J_{1},$ and $\tau_{J_{2}}(S_{2})$ as the set of scores attributed to $J_{2}.$ Example 5: Table 5 displays a toy example with $n=7$ voters’ Borda scorings of $d=10$ items with $J_{1}=\left\\{a,b,c,d\right\\}$ and $J_{2}=\left\\{e,f,g,h,i,j\right\\}.$ We observe the following: a) The first four voters have only done the first step in a riffle shuffle: each one of them has attributed the scores in $S_{1}=\left\\{0,1,2,3\right\\}$ to the items in $J_{1}$ and the scores in $S_{2}=\left\\{4,5,6,7,8,9\right\\}$ to the items in $J_{2}.$ This can be described as $\tau_{J}(S_{1},S_{2})=(S_{1},S_{2});$ that is the permutation $\tau$ is the identity permutation; so there is no crossing of scores between $J_{1}$ and $J_{2}$. b) Voters 5, 6 and 7 have done both steps in a riffle shuffle. Voters 5 and 6 have permuted score 3 with 5, so we have $\tau_{J_{1}}(S_{1})=\left\\{0,1,2,\mathbf{5}\right\\}$ and $\tau_{J_{2}}(S_{2})=\left\\{4,\mathbf{3},6,7\mathbf{,}8,9\right\\}$. Voter 7 has permuted the scores $\left\\{\mathbf{2,3}\right\\}$ with $\left\\{\mathbf{4,5}\right\\}$, so we have $\tau_{J_{1}}(S_{1})=\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}$ and $\tau_{J_{2}}(S_{2})=\left\\{\mathbf{2},\mathbf{3},6,7\mathbf{,}8,9\right\\}$. Further, we note by $\|\tau_{J_{1}}(S_{1})\|$ the number of voters who have done the riffle shuffle $(\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2}))$. So $\|\tau_{J_{1}}(S_{1})=\left\\{0,1,2,3\right\\}\|=4,$ $\|\left\\{0,1,2,\mathbf{5}\right\\}\|=2$ and $\|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}\|=1.$ The permuted scores between the two blocks of items is in bold in Table 5. Table 5: Borda scorings of 10 items by 7 voters. --- voter \| items \| a \| b \| c \| d \| e \| f \| g \| h \| i \| j 1 \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 2 \| 0 \| 2 \| 3 \| 1 \| 6 \| 4 \| 5 \| 8 \| 7 \| 9 3 \| 3 \| 2 \| 1 \| 0 \| 5 \| 6 \| 4 \| 9 \| 7 \| 8 4 \| 2 \| 1 \| 0 \| 3 \| 8 \| 7 \| 9 \| 4 \| 5 \| 6 5 \| 0 \| 1 \| 2 \| 5 \| 4 \| 3 \| 6 \| 7 \| 8 \| 9 6 \| 1 \| 2 \| 5 \| 0 \| 3 \| 6 \| 4 \| 9 \| 7 \| 8 7 \| 0 \| 4 \| 5 \| 1 \| 6 \| 8 \| 9 \| 2 \| 7 \| 3 Remark 4: A useful observation that we get from Example 5 is that we can concentrate our study either on $J_{1}$ or on $J_{2}:$ For if we know $\tau_{J_{1}}(S_{1})$, the scores attributed to $J_{1},$ we can deduce $\tau_{J_{2}}(S_{2})$, the scores attributed to $J_{2}$ because of mutual exclusivity constraints ensuring that any two items, say $a$ and $b,$ never map to the same rank by a voter. A simple measure of magnitude of $(d_{1},d_{2})$ riffle shuffling of a voter $i$ is the sum of its Borda scores attributed to the items in $J_{1};$ that is, $T_{i}(\tau_{J_{1}}(S_{1}))=\sum_{j\in J_{1}}r_{ij},$ where $r_{ij}$ is the Borda score attributed to item $j$ by voter $i$. In Table 5, for the first four voters, $T_{i}(\tau_{J_{1}}(S_{1}))=6$ for $i=1,...,4,$ which is the minimum attainable sum of scores; it implies that for these voters there is no crossing of scores between the two blocks $J_{1}$ and $J_{2}$. While for voters 5 and 6, $T_{i}(\tau_{J_{1}}(S_{1}))=8$ for $i=5,6;$ for voter 7, $T_{7}(\tau_{J_{1}}(S_{1}))=10$. These values show that the crossing of scores between the two blocks $J_{1}$ and $J_{2}$ of voters 5 and 6 are at a lower level than the crossing of scores for voter 7. For relatively small sample sizes, it is easy to enumerate the different types of $(d_{1},d_{2})$ riffle shuffles. For relatively large sample sizes, we use the contingency table of first-order marginals, that we discuss next. ### 6.1 Types of $(d_{1},d_{2})$ riffle shufflings in a coherent cluster The contingency table of first order marginals of an observed voting profile $V$ on $d$ items is a square $d\times d$ matrix M, where $\mathbf{M(}i,j\mathbf{)}$ stores the number of times that item $j$ has Borda score $i$ for $i=0,...,d-1,$ see subsection 3.2. It helps us to observe types of $(d_{1},d_{2})$ riffle shufflings in a coherent cluster as we explain in Example 6. Table 6: M${}_{1,1},$ contingency table of first-order marginals of $cohC_{1}(1).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 174 \| 92 \| 37 \| 11 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 314 1 \| 88 \| 88 \| 76 \| 62 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 314 2 \| 38 \| 78 \| 91 \| 107 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 314 3 \| 14 \| 56 \| 110 \| 134 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 314 4 \| 0 \| 0 \| 0 \| 0 \| 92 \| 78 \| 73 \| 38 \| 21 \| 12 \| 314 5 \| 0 \| 0 \| 0 \| 0 \| 95 \| 77 \| 59 \| 42 \| 23 \| 18 \| 314 6 \| 0 \| 0 \| 0 \| 0 \| 47 \| 63 \| 70 \| 65 \| 37 \| 32 \| 314 7 \| 0 \| 0 \| 0 \| 0 \| 35 \| 49 \| 45 \| 72 \| 68 \| 45 \| 314 8 \| 0 \| 0 \| 0 \| 0 \| 32 \| 27 \| 32 \| 62 \| 87 \| 74 \| 314 9 \| 0 \| 0 \| 0 \| 0 \| 13 \| 20 \| 35 \| 35 \| 78 \| 133 \| 314 $\beta$ \| 0.66 \| 1.31 \| 1.87 \| 2.16 \| 5.55 \| 5.78 \| 6.03 \| 6.58 \| 7.31 \| 7.52 \| Table 7: M${}_{1,2},$ contingency table of first-order marginals of $cohC_{1}(2).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 127 \| 70 \| 32 \| 6 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 235 1 \| 69 \| 82 \| 38 \| 46 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 235 2 \| 32 \| 56 \| 62 \| 85 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 235 3 \| 0 \| 0 \| 0 \| 0 \| 55 \| 59 \| 74 \| 29 \| 15 \| 3 \| 235 4 \| 7 \| 27 \| 103 \| 98 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 235 5 \| 0 \| 0 \| 0 \| 0 \| 68 \| 60 \| 42 \| 41 \| 11 \| 13 \| 235 6 \| 0 \| 0 \| 0 \| 0 \| 49 \| 53 \| 35 \| 48 \| 32 \| 18 \| 235 7 \| 0 \| 0 \| 0 \| 0 \| 26 \| 35 \| 42 \| 48 \| 40 \| 44 \| 235 8 \| 0 \| 0 \| 0 \| 0 \| 28 \| 15 \| 22 \| 44 \| 70 \| 56 \| 235 9 \| 0 \| 0 \| 0 \| 0 \| 9 \| 13 \| 20 \| 25 \| 67 \| 101 \| 235 $\beta$ \| 0.69 \| 1.29 \| 2.44 \| 2.59 \| 5.47 \| 5.43 \| 5.50 \| 6.35 \| 7.38 \| 7.86 \| Table 8: M${}_{1,3},$ contingency table of first-order marginals of $cohC_{1}(3).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 182 \| 97 \| 33 \| 14 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 326 1 \| 104 \| 100 \| 46 \| 76 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 326 2 \| 19 \| 41 \| 41 \| 70 \| 40 \| 37 \| 46 \| 17 \| 12 \| 3 \| 326 3 \| 16 \| 35 \| 53 \| 51 \| 39 \| 48 \| 43 \| 22 \| 13 \| 6 \| 326 4 \| 3 \| 29 \| 62 \| 61 \| 40 \| 41 \| 43 \| 32 \| 9 \| 6 \| 326 5 \| 2 \| 24 \| 91 \| 54 \| 39 \| 43 \| 23 \| 24 \| 16 \| 10 \| 326 6 \| 0 \| 0 \| 0 \| 0 \| 70 \| 65 \| 51 \| 60 \| 45 \| 35 \| 326 7 \| 0 \| 0 \| 0 \| 0 \| 53 \| 36 \| 52 \| 74 \| 56 \| 55 \| 326 8 \| 0 \| 0 \| 0 \| 0 \| 35 \| 33 \| 33 \| 57 \| 80 \| 88 \| 326 9 \| 0 \| 0 \| 0 \| 0 \| 10 \| 23 \| 35 \| 40 \| 95 \| 123 \| 326 $\beta$ \| 0.65 \| 1.60 \| 3.04 \| 2.71 \| 5.25 \| 5.25 \| 5.39 \| 6.26 \| 7.17 \| 7.68 \| Table 9: M${}_{1,4},$ contingency table of first-order marginals of $cohC_{1}(4).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 164 \| 93 \| 44 \| 14 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 315 1 \| 78 \| 71 \| 30 \| 36 \| 10 \| 31 \| 32 \| 9 \| 16 \| 2 \| 315 2 \| 44 \| 53 \| 49 \| 50 \| 32 \| 39 \| 27 \| 10 \| 8 \| 3 \| 315 3 \| 22 \| 52 \| 58 \| 87 \| 24 \| 20 \| 24 \| 15 \| 7 \| 6 \| 315 4 \| 5 \| 17 \| 35 \| 43 \| 51 \| 61 \| 41 \| 25 \| 23 \| 14 \| 315 5 \| 1 \| 11 \| 61 \| 46 \| 43 \| 42 \| 34 \| 35 \| 26 \| 16 \| 315 6 \| 1 \| 18 \| 38 \| 39 \| 52 \| 37 \| 37 \| 49 \| 28 \| 16 \| 315 7 \| 0 \| 0 \| 0 \| 0 \| 49 \| 44 \| 51 \| 61 \| 54 \| 56 \| 315 8 \| 0 \| 0 \| 0 \| 0 \| 37 \| 28 \| 47 \| 72 \| 69 \| 52 \| 315 9 \| 0 \| 0 \| 0 \| 0 \| 17 \| 13 \| 22 \| 39 \| 84 \| 140 \| 315 $\beta$ \| 0.83 \| 1.79 \| 3.10 \| 3.28 \| 5.30 \| 4.74 \| 5.22 \| 6.34 \| 6.76 \| 7.64 \| Table 10: M${}_{1,5},$ contingency table of first-order marginals of $cohC_{1}(5).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 188 \| 99 \| 36 \| 10 \| 6 \| 25 \| 30 \| 22 \| 34 \| 2 \| 452 1 \| 132 \| 109 \| 69 \| 57 \| 12 \| 30 \| 21 \| 13 \| 9 \| 0 \| 452 2 \| 69 \| 88 \| 59 \| 67 \| 28 \| 46 \| 40 \| 28 \| 20 \| 7 \| 452 3 \| 39 \| 72 \| 85 \| 92 \| 34 \| 44 \| 21 \| 31 \| 25 \| 9 \| 452 4 \| 12 \| 35 \| 76 \| 81 \| 50 \| 57 \| 53 \| 36 \| 38 \| 14 \| 452 5 \| 6 \| 29 \| 63 \| 72 \| 63 \| 64 \| 53 \| 41 \| 40 \| 21 \| 452 6 \| 3 \| 11 \| 34 \| 36 \| 71 \| 68 \| 64 \| 75 \| 45 \| 45 \| 452 7 \| 3 \| 9 \| 30 \| 37 \| 87 \| 45 \| 62 \| 73 \| 57 \| 49 \| 452 8 \| 0 \| 0 \| 0 \| 0 \| 71 \| 41 \| 47 \| 72 \| 95 \| 126 \| 452 9 \| 0 \| 0 \| 0 \| 0 \| 30 \| 32 \| 61 \| 61 \| 89 \| 179 \| 452 $\beta$ \| 1.12 \| 2.02 \| 3.26 \| 3.60 \| 5.70 \| 4.74 \| 5.27 \| 5.75 \| 5.99 \| 7.60 \| Table 11: M${}_{1,6},$ contingency table of first-order marginals of $cohC_{1}(6).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 151 \| 81 \| 31 \| 14 \| 8 \| 14 \| 19 \| 18 \| 39 \| 0 \| 375 1 \| 112 \| 79 \| 44 \| 33 \| 12 \| 21 \| 26 \| 25 \| 19 \| 4 \| 375 2 \| 66 \| 72 \| 52 \| 63 \| 16 \| 24 \| 29 \| 22 \| 28 \| 3 \| 375 3 \| 26 \| 52 \| 68 \| 68 \| 22 \| 45 \| 31 \| 29 \| 25 \| 9 \| 375 4 \| 8 \| 26 \| 42 \| 37 \| 52 \| 67 \| 41 \| 45 \| 45 \| 12 \| 375 5 \| 8 \| 27 \| 56 \| 61 \| 44 \| 49 \| 42 \| 36 \| 28 \| 24 \| 375 6 \| 3 \| 21 \| 36 \| 52 \| 64 \| 42 \| 49 \| 50 \| 29 \| 29 \| 375 7 \| 0 \| 7 \| 25 \| 31 \| 70 \| 43 \| 44 \| 59 \| 45 \| 51 \| 375 8 \| 1 \| 10 \| 21 \| 16 \| 66 \| 33 \| 46 \| 49 \| 45 \| 88 \| 375 9 \| 0 \| 0 \| 0 \| 0 \| 21 \| 37 \| 48 \| 42 \| 72 \| 155 \| 375 $\beta$ \| 1.12 \| 2.33 \| 3.62 \| 3.93 \| 5.68 \| 4.98 \| 5.21 \| 5.33 \| 5.25 \| 7.56 \| Table 12: M${}_{1,7},$ contingency table of first-order marginals of $cohC_{1}(7).$ --- Borda \| items scores \| j10 \| j7 \| j4 \| j9 \| j3 \| j1 \| j2 \| j6 \| j5 \| j8 \| sum 0 \| 129 \| 65 \| 46 \| 14 \| 11 \| 24 \| 35 \| 23 \| 52 \| 2 \| 401 1 \| 122 \| 77 \| 53 \| 35 \| 14 \| 28 \| 24 \| 19 \| 25 \| 4 \| 401 2 \| 74 \| 69 \| 50 \| 51 \| 24 \| 41 \| 36 \| 31 \| 19 \| 6 \| 401 3 \| 36 \| 51 \| 31 \| 66 \| 44 \| 48 \| 39 \| 46 \| 30 \| 10 \| 401 4 \| 24 \| 50 \| 40 \| 71 \| 51 \| 45 \| 38 \| 37 \| 27 \| 18 \| 401 5 \| 7 \| 45 \| 49 \| 56 \| 43 \| 53 \| 39 \| 48 \| 32 \| 29 \| 401 6 \| 5 \| 23 \| 73 \| 68 \| 42 \| 50 \| 42 \| 33 \| 40 \| 25 \| 401 7 \| 3 \| 10 \| 31 \| 28 \| 85 \| 39 \| 43 \| 54 \| 51 \| 57 \| 401 8 \| 1 \| 3 \| 17 \| 5 \| 58 \| 46 \| 47 \| 65 \| 57 \| 102 \| 401 9 \| 0 \| 8 \| 11 \| 7 \| 29 \| 27 \| 58 \| 45 \| 68 \| 148 \| 401 $\beta$ \| 1.42 \| 2.74 \| 3.84 \| 4.00 \| 5.45 \| 4.70 \| 5.02 \| 5.26 \| 5.20 \| 7.38 \| Example 6: Tables 6 to 12 display M1,α for $\alpha=1,...,7,$ the contingency tables of first order marginals of the seven coherent clusters of the SUSHI data, respectively. We observe the following: Each one of them reveals the nature of the riffle shuffles of its coherent cluster, which are summarized in Table 13. The number of observed $(4,6)$ blocks of scores for the seven coherent clusters, ($\tau_{J_{1}}(S_{1}),\tau_{J_{2}}(S_{2})),$ is only 27 in Table 13 out of the possible total number of $10!/(4!6!)=210$. The counts of the observed $(4,6)$ blocks do not seem to be uniformly distributed in Table 13. Furthermore, we observe that as $\alpha$ increases from 1 to 7, the magnitude of riffle shuffles, $T_{v}(\tau_{J_{1}}(S_{1})),$ increases in the coherent clusters from 6 to 12. Integers in bold in Table 13 are the shuffled-crossed scores. Table 13: Types of riffle shuffles in the 7 coherent clusters of SUSHI data. --- $cohC_{1}(\alpha)$ \| scores given to \| sum of \| \| $cohC_{1}(\alpha)$ \| scores given to \| sum of \| \| $\left\\{j10,j7,j4,j9\right\\}$ \| scores \| count \| \| $\left\\{j10,j7,j4,j9\right\\}$ \| scores \| count $cohC_{1}(1)$ \| $\left\\{0,1,2,3\right\\}$ \| 6 \| 314 \| $cohC_{1}(6)$ \| $\left\\{0,1,2,\mathbf{8}\right\\}$ \| 11 \| 48 $cohC_{1}(2)$ \| $\left\\{0,1,2,\mathbf{4}\right\\}$ \| 7 \| 235 \| \| $\left\\{0,1,\mathbf{7},3\right\\}$ \| 11 \| 63 $cohC_{1}(3)$ \| $\left\\{0,1,2,\mathbf{5}\right\\}$ \| 8 \| 171 \| \| $\left\\{0,\mathbf{6},2,3\right\\}$ \| 11 \| 53 \| $\left\\{0,1,\mathbf{4},3\right\\}$ \| 8 \| 155 \| \| $\left\\{\mathbf{5},1,2,3\right\\}$ \| 11 \| 98 $cohC_{1}(4)$ \| $\left\\{0,1,2,\mathbf{6}\right\\}$ \| 9 \| 96 \| \| $\left\\{0,1,\mathbf{4,6}\right\\}$ \| 11 \| 59 \| $\left\\{0,1,\mathbf{5},3\right\\}$ \| 9 \| 119 \| \| $\left\\{0,\mathbf{4},2,\mathbf{5}\right\\}$ \| 11 \| 54 \| $\left\\{0,\mathbf{4},2,3\right\\}$ \| 9 \| 100 \| $cohC_{1}(7)$ \| $\left\\{0,1,2,\mathbf{9}\right\\}$ \| 12 \| 26 $cohC_{1}(5)$ \| $\left\\{0,1,2,\mathbf{7}\right\\}$ \| 10 \| 79 \| \| $\left\\{0,1,\mathbf{8},3\right\\}$ \| 12 \| 26 \| $\left\\{0,1,\mathbf{6},3\right\\}$ \| 10 \| 84 \| \| $\left\\{0,\mathbf{7},2,3\right\\}$ \| 12 \| 33 \| $\left\\{0,\mathbf{5},2,3\right\\}$ \| 10 \| 85 \| \| $\left\\{\mathbf{6},1,2,3\right\\}$ \| 12 \| 43 \| $\left\\{\mathbf{4},1,2,3\right\\}$ \| 10 \| 119 \| \| $\left\\{0,\mathbf{4,5},3\right\\}$ \| 12 \| 38 \| $\left\\{0,1,\mathbf{4,5}\right\\}$ \| 10 \| 85 \| \| $\left\\{0,\mathbf{4},2,\mathbf{6}\right\\}$ \| 12 \| 39 \| \| \| \| \| $\left\\{0,1,\mathbf{4},\mathbf{7}\right\\}$ \| 12 \| 49 \| \| \| \| \| $\left\\{0,1,\mathbf{5,6}\right\\}$ \| 12 \| 82 \| \| \| \| \| $\left\\{\mathbf{4},1,2,\mathbf{5}\right\\}$ \| 12 \| 65 The counts in Table 13 are calculated from M1,α for $\alpha=1,...,7,$ by reasoning on the permutation of scores between the sets $S_{1}$ and $S_{2}$. Here are the details, where $J_{1}=\left\\{j10,j7,j4,j9\right\\}$. a)$\ cohC_{1}(1)$ $\|\left\\{0,1,2,3\right\\}\|=314,$ which is the number of $0$s attributed to $J_{1}$ in M${}_{1,1}.\ $Among the M1,α for $\alpha=1,...,7$, note that M1,1 is the only contingency table of first-order marginals which is block diagonal. b) $cohC_{1}(2)$ $\|\left\\{0,1,2,4\right\\}\|=235,$ which is the number of $4$s attributed to $J_{1}$ in M${}_{1,2}.$ c) $cohC_{1}(3)$ $\|\left\\{0,1,2,5\right\\}\|=171,$ which is the number of $5$s attributed to $J_{1}\ $in M${}_{1,3}.$ $\|\left\\{0,1,\mathbf{4},3\right\\}\|=155,$ which is the number of $4$s attributed to $J_{1}$ in M${}_{1,3}.$ d) $cohC_{1}(4)$ $\|\left\\{0,1,2,\mathbf{6}\right\\}\|=96,$ which is the number of $6$s attributed to $J_{1}$ in M${}_{1,4}.$ $\|\left\\{0,1,\mathbf{5},3\right\\}\|=119,$ which is the number of $5$s attributed to $J_{1}$ in M${}_{1,4}.$ $\|\left\\{0,\mathbf{4},2,3\right\\}\|=100,$ which is the number of $4$s attributed to $J_{1}$ in M${}_{1,4}.$ e) $cohC_{1}(5)$ $\|\left\\{0,1,2,\mathbf{7}\right\\}\|=79,$ which is the number of $7$s attributed to $J_{1}$ in M${}_{1,5}.$ $\|\left\\{0,1,\mathbf{6},3\right\\}\|=84,$ which is the number of $6$s attributed to $J_{1}$ in M${}_{1,5}.$ $\|\left\\{0,\mathbf{5},2,3\right\\}\|=85,$ which is the number of $1$s not attributed to $J_{1}$ in M${}_{1,5}.$ $\|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}\|+\|\left\\{0,\mathbf{5},2,3\right\\}\|=170,$ which is the total number of $5$s attributed to $J_{1}\ $in $\mathbf{M}_{1,5};$ so $\|\left\\{0,1,\mathbf{4},\mathbf{5}\right\\}\|=170-85=85.$ $\|\left\\{\mathbf{4},1,2,3\right\\}\|=119,$ which is the number of $0$s not attributed to $J_{1}\ $in $\mathbf{M}_{1,5}.$ f) $cohC_{1}(6)$ $\|\left\\{0,1,2,\mathbf{8}\right\\}\|=48,$ which is the number of $8$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ $\|\left\\{0,1,\mathbf{7},3\right\\}\|=63,$ which is the number of $7$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ $\|\left\\{\mathbf{5},1,2,3\right\\}\|=98,$ which is the number of $0$s not attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ $\|\left\\{0,\mathbf{4},2,\mathbf{5}\right\\}\|=152-98=54,$ where $152$ is the total number of $5$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ $\|\left\\{0,1,\mathbf{4},\mathbf{6}\right\\}\|=113-54=59,$ where $113$ is the total number of $4$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ $\|\left\\{0,\mathbf{6},2,3\right\\}\|=112-59=53,$ where $112$ is the total number of $6$s attributed to $J_{1}\ $in $\mathbf{M}_{1,6}.$ g) $cohC_{1}(7)$ $\|\left\\{0,1,2,\mathbf{9}\right\\}\|=26,$ which is the number of $9$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $\|\left\\{0,1,\mathbf{8},3\right\\}\|=26,$ which is the number of $8$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ For the remaining counts, we have to solve the following system of 7 linear equations, where, $u=\|\left\\{0,\mathbf{7},2,3\right\\}\|$, $t=\|\left\\{0,\mathbf{4},\mathbf{5},3\right\\}\|$, $s=\|\left\\{0,\mathbf{4},2,\mathbf{6}\right\\}\|$, $w=\|\left\\{0,1,\mathbf{4,7}\right\\}\|$, $z=\|\left\\{0,1,\mathbf{5,6}\right\\}\|$, $x=\|\left\\{\mathbf{6},1,2,3\right\\}\|$, and $y=\|\left\\{\mathbf{4},1,2,5\right\\}\|$. $x+y=147,\ $which is the number of $0$s not attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $u+w=72,\ $which is the number of $7$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $s+z+x=169,\ $which is the number of $6$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $t+z+y=157,\ $which is the number of $5$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $t+s+w+y=185,\ $which is the number of $4$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $u+t+x=158,\ $which is the number of $3$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ $u+s+x+y=218,\ $which is the number of $2$s attributed to $J_{1}\ $in $\mathbf{M}_{1,7}.$ ## 7 Crossing index The following $(d_{1},d_{2})$ crossing index is based on the internal dispersion of a voting profile. Definition 3: For a voting profile $V$ we define its crossing index to be $\displaystyle Cross(V)$ $\displaystyle=$ $\displaystyle 1-\frac{\delta_{1}(V_{d_{1},d_{2}})}{\max_{V}\delta_{1}(V_{d_{1},d_{2}})},$ $\displaystyle=$ $\displaystyle 1-\frac{\delta_{1}(V_{d_{1},d_{2}})}{2\frac{d_{1}d_{2}}{d(d-1)}}\ \ \text{by Proposition 2.}$ where $\delta_{1}(V_{d_{1},d_{2}})$ is the first taxicab dispersion obtained from TCA of $V$ and $(d_{1},d_{2})$ represents the optimal TCA binary partition of the $d$ items of $V$ such that $d=d_{1}+d_{2}.\vskip 12.0pt plus 4.0pt minus 4.0pt$ Proposition 4: The crossing index of a coherent cluster is $Cross(cohC(\alpha))=\frac{2(\alpha-1)}{d_{1}d_{2}}.$ Example 7: The last column in Table 3 contains the values of the crossing indices of the seven coherent clusters of the first iteration of SUSHI data. We observe: a) $Cross(cohC_{1}(1))=0$, because the structure of its matrix of first order marginals, M${}_{1,1},$ is block diagonal; which means that the permutation $\tau$ is the identical permutation, so there are no crossing of scores between the two subsets of items $J_{1}$ and $J_{2}$ in $cohC_{1}(1).$ b) $Cross(cohC_{1}(\alpha))$ for $\alpha=1,...,7$ is a uniformly increasing function of $\alpha,$ similar in spirit to the $T_{v}(\tau_{J_{1}}(S_{1}))$ statistic. c) For the incoherent cluster $V_{1,8}$, we have: $\delta_{1}(V_{1,8})=0.2354$ given in Example 3; and $d_{1}=d_{2}=5$ from Figure 10. So $Cross(V_{1,8})=1-\frac{0.2354}{2(5)(5)/(10(9))}=1-0.4237=0.5763.$ ## 8 Coherent group Our aim is to explore a given voting profile $V$ by uncovering its coherent mixture groups, see equation (1); that is, $V=\cup_{g=1}^{G}cohG(g)\cup noisyG$, where $G$ represents the number of coherent groups and $cohG(g)$ is the $g$th coherent group. The computation is done by an iterative procedure in $n_{G}$ steps for $n_{G}\geq G$ that we describe: For $g=1$; let $V_{1}=V;$ compute $cohG(1)$ from $V_{1},$ then partition $V_{1}=V_{2}\cup cohG(1);$ For $g=2$; compute $cohG(2)$ from $V_{2},$ then partition $V_{2}=V_{3}\cup cohG(2);$ By continuing the above procedure, after $n_{G}$ steps, we get $V=\cup_{g=1}^{n_{G}}cohG(g).\vskip 12.0pt plus 4.0pt minus 4.0pt$ However, some of the higher ordered coherent groups may have relatively small sample sizes; so by considering these as outliers, we lump them together thus forming the noisy group denoted by $noisyG$ in equation (1). Let us recall the definition of a coherent group given in equation 2 $cohG(g)=\cup_{\alpha=1}^{c_{g}}cohC_{g}(\alpha)\text{ \ for }g=1,...,G;$ that is, a coherent group is the union of its coherent clusters. This implies that the sample size of $cohG(g)$ equals the sum of the sample sizes of its coherent clusters $\|cohG(g)\|\ =\sum_{\alpha=1}^{c_{g}}\|cohC_{g}(\alpha)\|.$ As an example, for the SUSHI data, from the 2nd column of Table 3 we can compute the sample size of the first coherent group $\displaystyle\|cohG(1)\|$ $\displaystyle=$ $\displaystyle\sum_{\alpha=1}^{c_{g}=7}\|cohC_{1}(\alpha)\|$ $\displaystyle=$ $\displaystyle 2418.$ Furthermore, $cohG(1)$ is composed of 27 observed riffle shuffles summarized in Table 13, which provides quite a detailed view of its inner structure. The next result shows important characteristics of a coherent group inherited from its coherent clusters. Theorem 2: ( Properties of a coherent group $cohG(g)$) a) The first principal column factor score $\mathbf{g}_{1}$ of the $d$ items in a coherent group is the weighted average of the first principal column factor score $\mathbf{g}_{1}$ of the $d$ items of its coherent clusters; that is, $\displaystyle g_{1}(j$ $\displaystyle\in$ $\displaystyle cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{\|cohC_{g}(\alpha)\|}{\|cohG(g)\|}g_{1}(j\in cohC_{g}(\alpha))\text{\ \ for }j=1,...,d.$ $\displaystyle=$ $\displaystyle\frac{2}{d-1}\sum_{\alpha=1}^{c_{g}}\frac{\|cohC_{g}(\alpha)\|}{\|cohG(g)\|}\beta(j\in cohC_{g}(\alpha))-1\text{ \ \ by Proposition 3.}$ And $corr(\mathbf{g}_{1}(cohG(g),\mathbf{\beta}(cohG(g))=1.$ b) The first TCA dispersion value of a coherent group is the weighted average of the first TCA dispersion values of its coherent clusters; that is, $\delta_{1}(cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{\|cohC_{g}(\alpha)\|}{\|cohG(g)\|}\delta_{1}(cohC_{g}(\alpha)).$ c) The crossing index of a coherent group is the weighted average of the crossing indices of its coherent clusters; that is, $Cross(cohG(g))=\sum_{\alpha=1}^{c_{g}}\frac{\|cohC_{g}(\alpha)\|}{\|cohG(g)\|}Cross(cohC_{g}(\alpha)).$ Example 8: Table 14 summarizes the first four coherent groups of SUSHI data, which emerged after 5 iterations. For $g=1$, we get $cohG(1)=\cup_{\alpha=1}^{c_{1}=7}cohC_{1}(\alpha);$ that is, the first coherent group of voters, the majority, is composed of 48.36% of the sample with crossing index of 27.3%. Standard errors of the Borda scale of the items in $cohG(1)\ $in Table 14 are: $(0.046,0.051,0.042,0.042,0.053,0.047,0.037,0.034,0.037,0.025).$ We can discern the following grouped seriation (bucket ranking) of the items $j8\succ j5\succ j6\succ\left\\{j3,j2\right\\}\succ j1\succ\left\\{j9,j4\right\\}\succ\left\\{j7\right\\}\succ\left\\{j10\right\\}.$ The groupings are based on the standard 95% confidence intervals of the Borda scale of the items. The 2nd coherent group $cohG(2),$ summarized by its Borda scales in Table 14, is made up of eight coherent clusters; it is composed of 19.0% of the sample with crossing index of 35.38%. The voters in this coherent group disapprove $\left\\{uni(seaurchin),sake(salmonroe)\right\\},$ which are considered more ”daring sushis”. The third coherent group $cohG(3),$ summarized by its Borda scales in Table 14, is made up of eight coherent clusters; it is composed of 13.24% of the sample with crossing index of 27.3%. The voters in this coherent group prefer the three types of tuna sushis with sea urchin sushis. The fourth coherent group $cohG(4),$ summarized by its Borda scales in Table 14, is made up of eight coherent clusters; it is composed of 6.94% of the sample with crossing index of 35.27%. The voters disapprove the three types of tuna sushis. Remark 6: a) Note that the number of preferred sushis in $cohG(1)$ and $cohG(2)$ are six; that is $\|J_{2}\|=6.$ While the number of preferred sushis in $cohG(3)$ and $cohG(4)$ are four. b) The four coherent groups summarized in Table 14 can also be described as two bipolar latent factors: By noting that the only major difference between the first two coherent groups is that (5\. uni (sea urchin), 6. sake (salmon roe)) are swapped with (7\. tamago (egg), 4. ika (squid)). While the only major difference between the third and fourth coherent groups is that the three tunas are swapped with (4\. ika (squid), 5. uni (sea urchin), 1. ebi (shrimp)). c) We consider the fifth group as noisy (outliers not shown) composed of 12.36% of the remaining sample: it contains $cohG(5)=\cup_{\alpha=1}^{2}cohC_{5}(\alpha)$ whose sample size is $38$, a very small number. For the sake of completeness we also provide the sample sizes of its two coherent clusters $\|cohC_{5}(1)\|=22$ and $\|cohC_{5}(2)\|=16$. Table 14: The first four coherent groups of SUSHI data and related statistics. --- $\mathbf{cohG(1)=\cup}_{\alpha=1}^{7}\mathbf{cohC}_{1}\mathbf{(\alpha)}$ \| $\mathbf{\beta}$ \| $\mathbf{cohG(2)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{2}\mathbf{(\alpha)}$ \| $\mathbf{\beta}$ 8\. toro (fatty tuna) \| $\mathbf{7.62}$ \| 8\. toro (fatty tuna) \| $\mathbf{6.15}$ 5\. uni (sea urchin) \| $\mathbf{6.31}$ \| 2\. anago (sea eel) \| $\mathbf{5.97}$ 6\. sake (salmon roe) \| $\mathbf{5.92}$ \| 1\. ebi (shrimp) \| $\mathbf{5.92}$ 3\. maguro (tuna) \| $\mathbf{5.49}$ \| 7\. tamago (egg) \| $\mathbf{5.76}$ 2\. anago (sea eel) \| $\mathbf{5.35}$ \| 3\. maguro (tuna) \| $\mathbf{5.55}$ 1\. ebi (shrimp) \| $\mathbf{5.04}$ \| 4\. ika (squid) \| $\mathbf{5.41}$ 9\. tekka-maki (tuna roll) \| 3.27 \| 9\. tekka-maki (tuna roll) \| 3.80 4\. ika (squid) \| 3.10 \| 10\. kappa-maki (cucumber roll) \| 2.56 7\. tamago (egg) \| 1.94 \| 6\. sake (salmon roe) \| 2.45 10\. kappa-maki (cucumber roll) \| 0.97 \| 5\. uni (sea urchin) \| 1.44 $Cross(cohG(1))=27.3\%$ \| \| $Cross(cohG(2))=35.38\%$ \| $\|cohG(1)\|=2418\ (48.36\%)$ \| \| $\|cohG(2)\|=955\ (19.10\%)$ \| \| \| \| $\mathbf{cohG(3)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{3}\mathbf{(\alpha)}$ \| $\mathbf{\beta}$ \| $\mathbf{cohG(4)=\cup}_{\alpha=1}^{8}\mathbf{cohC}_{4}\mathbf{(\alpha)}$ \| $\mathbf{\beta}$ 8\. toro (fatty tuna) \| $\mathbf{7.31}$ \| 4\. ika (squid) \| $\mathbf{6.67}$ 6\. sake (salmon roe) \| $\mathbf{6.62}$ \| 5\. uni (sea urchin) \| $\mathbf{6.50}$ 3\. maguro (tuna) \| $\mathbf{6.30}$ \| 6\. sake (salmon roe) \| $\mathbf{6.43}$ 9\. tekka-maki (tuna roll) \| $\mathbf{6.00}$ \| 1\. ebi (shrimp) \| $\mathbf{6.16}$ 7\. tamago (egg) \| 3.76 \| 8\. toro (fatty tuna) \| 3.69 4\. ika (squid) \| 3.41 \| 7\. tamago (egg) \| 3.39 2\. anago (sea eel) \| 3.00 \| 2\. anago (sea eel) \| 3.21 1\. ebi (shrimp) \| 2.92 \| 9\. tekka-maki (tuna roll) \| 3.14 10\. kappa-maki (cucumber roll) \| 2.86 \| 10\. kappa-maki (cucumber roll) \| 2.99 5\. uni (sea urchin) \| 2.80 \| 3\. maguro (tuna) \| 2.80 $Cross(cohG(3))=31.37\%$ \| \| $Cross(cohG(4))=35.27\%$ \| $\|cohG(3)\|\ =662\ (13.24\%)$ \| \| $\|cohG(4)\|\ =347\ (6.94\%)$ \| ## 9 APA data set The 1980 American Psychological Association (APA) presidential election had five candidates: $\left\\{A,C\right\\}$ were research psychologists, $\left\\{D,E\right\\}$ were clinical psychologists and $B$ was a community psychologist. In this election, voters ranked the five candidates in order of preference. Among the 15449 votes, 5738 votes ranked all five candidates. We consider the data set which records the 5738 complete votes; it is available in $\left[20,\ p.96\right]$ and $\left[5,\ Table\ 1\right]$. The winner was candidate $C$. (a) Figure 11 : TCA map of $\mbox{Coh}_{1}C_{1}$ of APA data (b) Figure 12: TCA map of $\mbox{Coh}_{1}C_{2}$ of APA data Table 15 compares the results obtained by our method and the best distance- based mixture model given in $\left[21\right]$. Distance-based models have two parameters, a central modal ranking and a precision parameter. The precision parameter measures the peakedness of the distribution. $\left[21\right]$ found that the Cayley distance produced better results than the Kendall and Spearman distances using BIC (Bayesian information criterion) and ICL (integrated complete likelihood) criteria. Parts a and b of Table 15, are reproduced from $\left[21,\ \text{Tables 4 and 5}\right]$. Part c of Table 15 summarizes the results of our approach, where we only describe the first four coherent groups: We find only the first two coherent groups as meaningfully interpretable based on the a priori knowledge of the candidates. Voters in $cohG(1)$, with sample size of 31%, prefer the research oriented psychologists $\left\\{A,C\right\\}$ over the rest. Voters in $cohG(2)$, with sample size of 23.7%, prefer the clinical psychologists $\left\\{D,E\right\\}$ over the rest. We interpret $cohG(3)$ and $cohG(4)$ as mixed B with 14.23% and 12.% of the voters, respectively. Additionally, there is a $noisyG$ making up 19.1% of the sample, which comprises $cohG(5)$ displayed in Table 15. $\left[5\right]$ discussed this data set quite in detail; surprisingly, our results confirm his observations: a) There are two groups of candidates, $\left\\{A,C\right\\}$ and $\left\\{D,E\right\\}.$ The voters line up behind one group or the other ; b) The APA divides into academicians and clinicians who are on uneasy terms. Voters seem to choose one type or the other, and then choose within; but the group effect predominates ; c) Candidate $B$ seems to fall in the middle, perhaps closer to $D$ and $E$. The following important observation emerges from the comparison of results in Table 15. We have two distinct concepts of groups for rank data, categorical and latent variable based. To see this, consider groups 3 and 4 in part a of Table 15: Group 3 is based on the modal category $B\succ C\succ A\succ D\succ E$ and group 4 is based on the modal category $B\succ C\succ A\succ E\succ D.$ The only difference between these two modal categories is the permutation of the least ranked two clinical psychologist candidates $\left\\{D,E\right\\};$ this difference is not important and does not appear in our approach, which is a latent variable approach. Table 15: A summary of results derived from three methods of analysis of --- APA election data. Parts a) and b) are from Murphy and Martin (2003). a) Parameters of the best mixture model selected, Cayley-based, using BIC Group \| sample% \| modal orderings \| precision $1$ \| $42$ \| $D\succ B\succ E\succ C\succ A$ \| $0.16$ $2$ \| $31$ \| $C\succ D\succ E\succ A\succ B$ \| $0.79$ $3$ \| $12$ \| $B\succ C\succ A\succ D\succ E$ \| $1.52$ $4$ \| $8$ \| $B\succ C\succ A\succ E\succ D$ \| $1.81$ $5$ \| $7$ \| $B\succ D\succ A\succ E\succ C$ \| $1.72$ b) Parameters of the best mixture model selected, Cayley-based, using ICL Group \| sample% \| modal ordering \| precision $1$ \| $100$ \| $B\succ C\succ A\succ E\succ D$ \| $0.25$ c) The first five coherent groups, each composed of two coherent clusters. Group \| sample% \| $\beta(C)$ \| $\beta(A)$ \| $\beta(B)$ \| $\beta(E)$ \| $\beta(D)$ \| $Cross$ cohG(1) Research \| $31.0$ \| $\mathbf{3.55}$ \| $\mathbf{3.15}$ \| $1.31$ \| $1.15$ \| $0.85$ \| $10.22\%$ cohG(2) Clinical \| $23.7$ \| $0.83$ \| $1.28$ \| $1.28$ \| $\mathbf{3.31}$ \| $\mathbf{3.30}$ \| $12.90\%$ cohG(3) mixed B \| $14.2$ \| $0.66$ \| $\mathbf{2.70}$ \| $\mathbf{2.96}$ \| $0.71$ \| $\mathbf{2.97}$ \| $12.45\%$ cohG(4) mixed B \| $12.0$ \| $\mathbf{2.85}$ \| $0.77$ \| $\mathbf{2.86}$ \| $\mathbf{2.80}$ \| $0.72$ \| $10.22\%$ cohG(5) outlier \| $8.6$ \| $0.96$ \| $\mathbf{3.30}$ \| $1.31$ \| $\mathbf{3.40}$ \| $1.00$ \| $9.88\%$ ### 9.1 Description The eight coherent clusters of the first four coherent groups can simply be described as: $coh_{1}C(1):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{A,C\right\\}}\left\\{3,4\right\\}=\left\\{3,4\right\\})=7$ for $v=1,...,1233.$ $coh_{1}C(2):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{A,C\right\\}}\left\\{3,4\right\\}=\left\\{\mathbf{2},4\right\\})=6$ for $v=1,...,545.$ $coh_{2}C(1):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{D,E\right\\}}\left\\{3,4\right\\}=\left\\{3,4\right\\})=7$ for $v=1,...,834.$ $coh_{2}C(2):T_{v}(\tau_{J_{2}}(S_{2})=\tau_{\left\\{D,E\right\\}}\left\\{3,4\right\\}=\left\\{\mathbf{2},4\right\\})=6$ for $v=1,...,526.$ $coh_{3}C(1):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{C,E\right\\}}\left\\{0,1\right\\}=\left\\{0,1\right\\})=1$ for $v=1,...,512.$ $coh_{3}C(2):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{C,E\right\\}}\left\\{0,1\right\\}=\left\\{0,\mathbf{2}\right\\})=2$ for $v=1,...,305.$ $coh_{4}C(1):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{A,D\right\\}}\left\\{0,1\right\\}=\left\\{0,1\right\\})=1$ for $v=1,...,350.$ $coh_{4}C(2):T_{v}(\tau_{J_{1}}(S_{1})=\tau_{\left\\{A,D\right\\}}\left\\{0,1\right\\}=\left\\{0,\mathbf{2}\right\\})=2$ for $v=1,...,338.$ In this case, we can also visualize all the orderings belonging to a coherent group: Figures 11 and 12 display all the preferences belonging to the two coherent clusters of the first coherent group. The label $CAEBD162$ in Figure 11 should be interpreted as the preference $C\succ A\succ E\succ B\succ D$ repeated 162 times. ## 10 Riffle independence model Riffle independence is a nonparametric probabilistic modelling method of preferences developed by $\left[2\right]$, which generalizes the independence model. It can be described in the following way: (a) Partition the set $J$ of $d$ distinct items into two disjoint subsets $J_{1}$ of size $d_{1}$ and $J_{2}$ of size $d_{2}$. Then generate an ordering of items within each subset according to a certain ranking model. This implies that any ordering of the $d$ items can be written as a direct product of two disconnected orderings; which in its turn implies the independence of the two subsets $J_{1}$ and $J_{2}$. So the model complexity of this step is of order $d_{1}!+d_{2}!.$ (b) Interleave the two independent orderings for these two subsets using a riffle shuffle to form a combined ordering. An interleaving is a binary mapping from the set of orderings to $\left\\{J_{1},J_{2}\right\\}$. The model complexity of this step is of order $d!/(d_{1}!d_{2}!).$ The interleaving step generates the riffled independence of the two subsets $J_{1}$ and $J_{2}$. So the combined model complexity of both steps is $d_{1}!+d_{2}!+d!/(d_{1}!d_{2}!)$ which is much smaller than $d!=(d_{1}+d_{2})!$. For example, consider an ordering of the items in the set $J=\left\\{A,B,C,D,E,F\right\\}$ from its two subsets $J_{1}=\left\\{A,C\right\\}$ and $J_{2}=\left\\{B,D,E,F\right\\}.\ $In the first step, relative orderings of the items in $J_{1}$ and $J_{2}$ are drawn independently. Suppose we obtain the relative ordering $\varphi(J_{1})=(C\succ A)$ in $J_{1},$ and the relative ordering $\varphi(J_{2})=(B\succ D\succ F\succ E)$ in $J_{2}.$ Then, in the second step, the two relative orderings are combined by interleaving the items in the two subsets. For instance, if the interleaving process is $\omega(J_{1},J_{2})=(J_{1},J_{2},J_{2},J_{1},J_{2},J_{2})$, where the relative ordering of the items in each subset remains unchanged, the combined ordering is then determined by the composition $\displaystyle\omega(J_{1},J_{2})\ast(\varphi(J_{1}),\varphi(J_{2}))$ $\displaystyle=$ $\displaystyle(C\succ B\succ D\succ A\succ F\succ E)$ $\displaystyle=$ $\displaystyle\varphi(J).$ Given the two subsets $J_{1}$ and $J_{2}$ with their orderings $\varphi(J_{1})$ and $\varphi(J_{2})$ and interleaving $\omega(J_{1},J_{2})$ generated from models with probability distributions $f_{J_{1}},$ $g_{J_{2}}$ and $m_{\omega}$, respectively, the probability of observed ordering under the riffle independence model is $P(\varphi(J))=m_{\omega}(\omega(J_{1},J_{2}))f_{J_{1}}(\varphi(J_{1}))g_{J_{2}}(\varphi(J_{2}).$ There are two formulations of riffle shuffle for rank data in statistics: probabilistic and exploratory. In the riffled independence model, the set of items is partitioned recursively, while in the exploratory approach the set of voters is partitioned recursively. ## 11 Conclusion The main contribution of this paper is the introduction of an exploratory riffle shuffling procedure to reveal and display the structure of diffuse rank data for large sample sizes. The new notion of a coherent cluster, that we developed, is simply based on the geometric notion of taxicab projection of points on the first TCA axis globally and locally; furthermore, it has nice mathematical properties. Coherent clusters of a coherent group represent the same latent variable opposing preferred items to disliked items, and can easily be interpreted and displayed. Like Occam’s razor, step by step, our procedure peels the essential structural layers (coherent groups) of rank data. Our method was able to discover some other aspects of the rank data, such as outliers or small groups, which are eclipsed or masked by well established methods, such as distance or random utility based methods. The major reasons for this is that in random utility based methods the multivariate nature of a preference is reduced to binary preferences (paired comparisons), and in Mallows distance related methods distances between any two preferences are bounded. We presented a new index, $Cross$, that quantifies the extent of crossing of scores between the optimal binary partition of the items that resulted from TCA. The crossing index of a group is based on the first taxicab dispersion measure: it takes values between 0 and 100%, so it is easily interpretable. The proposed approach can easily be generalized to the analysis of rankings with ties and partial rankings. The package TaxicabCA written in R available on CRAN can be used to do the calculations. Acknowledgement: Choulakian’s research has been supported by NSERC grant (RGPIN-2017-05092) of Canada. References $\left[1\right]\ \ $Kamishima, T. (2003). Nantonac collaborative filtering: recommendation based on order responses. In: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’03, 583–588. ACM, New York. $\left[2\right]\ \ $Huang, J., Guestrin, C. (2012). Uncovering the riffled independence structure of ranked data. Electronic Journal of Statistics, 6, 199-230. $\left[3\right]\ \ $Lu, T., Boutilier, C. (2014). Effective sampling and learning for Mallows models with pairwise preference data. Journal of Machine Learning Research, 15, 3783-3829. $\left[4\right]\ \ $Vitelli, V., Sørenson, Ø., Crispino, M., Frigessi, A., Arjas, E. (2018). Probabilistic preference learning with the Mallows rank model. Journal of Machine Learning Research, 18, 1-49. $\left[5\right]\ \ $Diaconis, P. (1989). 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Appendix Let $\mathbf{R}=(r_{ij})$ for $i=1,...,n$ and $j=1,...,d$ represent the Borda scorings for preferences, where $r_{ij}$ takes values $0,...,d-1.$ Similarly, let $\overline{\mathbf{R}}$ represent the reverse Borda scorings, whose column sums are the cordinates of the row named $\mathbf{nega=}$ $n\overline{\mathbf{\beta}}=\mathbf{1}_{n}^{\prime}\overline{\mathbf{R}}.$ We consider the application of TCA to the data set $\mathbf{R}_{nega}=(_{\mathbf{nega}}^{\mathbf{R}})$ of size $(n+1)\times d.$ So let $\mathbf{P}=\mathbf{R}_{nega}/t$ be the correspondence table associated with $\mathbf{R}_{nega},$ where $t=2n\sum_{j=0}^{d-1}=nd(d-1).$ We have $\begin{array}[]{cccc}p_{i\ast}&=&\frac{1}{2n}\;\;\;\text{for}\quad i=1,...,n&\end{array}$ (14) $\begin{array}[]{cccc}&=&\frac{1}{2}\;\;\;\text{for}\quad i=n+1,&\end{array}$ (15) and $p_{\ast j}=\frac{1}{d}\text{\ \ \ \ \ for\ \ \ }j=1,...,d.$ (16) The first residuel correspondence matrix will be $\begin{array}[]{ccccc}p_{ij}^{(1)}&=&p_{ij}-p_{i\ast}p_{\ast j}&\qquad\qquad\qquad(17)\\\ &=&\frac{r_{ij}}{t}-\frac{1}{2n}.\frac{1}{d}\text{\ \ \ \ for\ \ \ \ }i=1,...,n&\qquad\qquad\qquad(18)\\\ &=&\frac{\mathbf{nega}_{j}}{t}-\frac{1}{2}.\frac{1}{d}\ \ \ \ \text{for\ \ \ \ }i=n+1.&\qquad\qquad\qquad(19)\end{array}$ Consider the nontrivial binary partition of the set $S=\left\\{0,1,...,d-1\right\\}$ into $S=S_{1}\cup S_{2},$ where $\|S_{1}\|=d_{1},$ $\|S_{2}\|=d_{2}$ and $d=d_{1}+d_{2}.$ To eliminate the sign indeterminacy in the first TCA principal axis, we fix $\mathbf{v}_{1}(nega)=\mathbf{v}_{1}(n+1)=-1;$ and we designate by $S_{1}$ the set of item indices such that the first TCA principal axis coordinates are negative, that is, $\mathbf{u}_{1}(j)=-1$ for $j\in S_{1}.$ It follows that $\mathbf{u}_{1}(j)=1$ for $j\in S_{2}$. Now we have by (4) for $i=1,...,n$ $\begin{array}[]{llll}a_{i1}&=&\sum_{j=1}^{d}\mathbf{u}_{1}(j)p_{ij}^{(1)}\\\ &=&\sum_{j\in S_{1}}\mathbf{u}_{1}(j)p_{ij}^{(1)}+\sum_{j\in S_{2}}\mathbf{u}_{1}(j)p_{ij}^{(1)}&\\\ &=&-\sum_{j\in S_{1}}p_{ij}^{(1)}+\sum_{j\in S_{2}}p_{ij}^{(1)}\\\ &=&-2\sum_{j\in S_{1}}p_{ij}^{(1)}\ \ \ \ \text{by\ \ }(17)&\\\ &=&-2\sum_{j\in S_{1}}(\frac{r_{ij}}{t}-\frac{1}{2n}.\frac{1}{d})\text{\ \ \ by\ \ (18)}&\\\ &=&\frac{d_{1}}{nd}-\frac{2}{t}\sum_{j\in S_{1}}r_{ij};&\end{array}$ (20) and from which we deduce by (5) for $i=1,...,n$ $\begin{array}[]{lll}f_{i1}&=&\frac{a_{i1}}{p_{i\ast}}\\\ &=&\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\sum_{j\in S_{1}}r_{ij}.\end{array}$ (21) We have the following Theorem concerning the first TCA principal factor scores of respondents $f_{i1}$ for $i=1,...,n.$ Theorem 1: a) The maximum number of distinct clusters of $n$ respondents on the first TCA principal axis (distinct $f_{i1}$ values$)$ is $d_{1}d_{2}+1.$ Proof: We consider the two extreme cases of $S_{1}$ and calculate the summation term in (21): For $S_{1}=\left\\{0,1,...,d_{1}-1\right\\}$,$\ \sum_{j\in S_{1}}r_{ij}=\sum_{j=0}^{d_{1}-1}j=\frac{d_{1}(d_{1}-1)}{2}.$ For $S_{1}=\left\\{d-d_{1},1,...,d-1\right\\},$ $\sum_{j\in S_{1}}r_{ij}=\sum_{j=d-d_{1}}^{d-1}j=\sum_{j=d_{2}}^{d-1}j=\frac{d_{1}(d_{2}+d-1)}{2}$. It follows that $\frac{d_{1}(d_{1}-1)}{2}\leq\sum_{j\in S_{1}}r_{ij}\leq\frac{d_{1}(d_{2}+d-1)}{2};$ so $\sum_{j\in S_{1}}r_{ij}$ can take at most $\frac{d_{1}(d_{2}+d-1)}{2}-\frac{d_{1}(d_{1}-1)}{2}+1=d_{1}d_{2}+1$ values. b) The maximum value that $f_{i1}$ can attain is $2\frac{d_{1}d_{2}}{d(d-1)}.$ Proof: From (21) and Part a, it follows that the maximum value that $f_{i1}$ can attain is $(\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\frac{d_{1}(d_{1}-1)}{2})=2\frac{d_{1}d_{2}}{d(d-1)}.$ c) The minimum value that $f_{i1}$ can attain is $-2\frac{d_{1}d_{2}}{d(d-1)}.$ Proof: From (21) and Part a, it follows that the minimum value that $f_{i1}$ can attain is $(\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\frac{d_{1}(d_{2}+d-1)}{2})=-2\frac{d_{1}d_{2}}{d(d-1)}.$ d) If the number of distinct clusters is maximum, $d_{1}d_{2}+1$, then the gap between two contiguous $f_{i1}$ values is $\frac{4}{d(d-1)}.$ Proof: Suppose that the number of distinct clusters is maximum, $d_{1}d_{2}+1$. We consider the first TCA factor score $f_{i1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}\sum_{j\in S_{1}}r_{ij}$ which is different in value from the two extreme values $\pm 2\frac{d_{1}d_{2}}{d(d-1)}.$ Then $f_{i_{1}1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}(-1+\sum_{j\in S_{1}}r_{ij})$ will be the contiguous higher value to $f_{i1};$ and similarly $f_{i_{2}1}=\frac{2d_{1}}{d}-\frac{4}{d(d-1)}(1+\sum_{j\in S_{1}}r_{ij})$ will be the contiguous lower value to $f_{i1};$ and the required result follows. Proposition 1: For a voting profile $V$, $\delta_{1}\geq\|f_{1}(\mathbf{nega})\|$. Proof: Let $\mathbf{a}_{1}=(_{a_{1}(nega)}^{\mathbf{a}_{11}}).$We need the following three observations. First, it is well known that $\mathbf{a}_{1}$ is centered by (5) and (9), $\displaystyle\mathbf{1}_{n+1}^{\prime}\mathbf{a}_{1}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle=$ $\displaystyle\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}+a_{1}(nega);$ from which we get, $\|\mathbf{1}_{n}^{\prime}\ \mathbf{a}_{11}\|=\|a_{1}(nega)\|.$ (22) Second, by triangle inequality of the L1 norm we have $\|\|\mathbf{a}_{11}\|\|_{1}\geq\|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}\|.$ (23) Third, the marginal relative frequency of the nega row is $p_{nega\ast}=1/2$ by (15) , and $f_{i1}=a_{i1}/p_{i\ast}$ for $i=1,...,n+1$ by (5); so we have $f_{1}(nega)=2a_{1}(nega).$ (24) Now we have by (7) $\begin{array}[]{llll}\delta_{1}&=&\|\|\mathbf{a}_{1}\mathbf{\|\|}_{1}\\\ &=&\|\|\mathbf{a}_{11}\|\|_{1}+\|a_{1}(nega)\|\\\ &\geq&\|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}\|+\|a_{1}(nega)\|\text{\ \ \ by \ \ (23)}\\\ &=&2\|a_{1}(nega)\|\ \ \ \ \text{by\ \ \ (22)}\\\ &=&\|f_{1}(nega)\|\text{\ \ \ \ by\ \ (24)}\end{array}$ Propostion 2: Let $cohC_{m}(\alpha)=V_{m,\alpha}$ be the $\alpha$th coherent cluster of the $m$th coherent group characterized by $f_{1}^{V_{m,\alpha}}(\mathbf{\sigma)=}f_{\alpha}^{V_{m}}$ for all $\mathbf{\sigma\in}cohC_{m}(\alpha)$. Then $\delta_{1}=f_{\alpha}^{V_{m}}=-f_{1}(\mathbf{nega}).$ Proof: By Definition 1 of the coherency of the cluster $V_{m,\alpha},$ we have $0<f_{1}^{V_{m,\alpha}}(i)=f_{\alpha}^{V_{m}}$ for $i=1,...,\|cohC_{m}(\alpha)\|$; by (5) it follows that $0<a_{i1}=$ $f_{\alpha}^{V_{m}}/n$ for $i=1,...,\|cohC_{m}(\alpha)\|$ ; so (25) becomes equality, $\|\|\mathbf{a}_{11}\|\|_{1}=\sum_{i=1}^{n}a_{i1}=\|\mathbf{1}_{n}^{\prime}\mathbf{a}_{11}\|$, and the required result follows. Proposition 3 is a corollary to the following general result Theorem 3: If the first TCA principal axis of the columns of $\mathbf{R}_{nega}$ is $\mathbf{v}_{1}=(_{-1}^{\mathbf{1}_{n}})$, then the first principal column factor score $\mathbf{g}_{1}$ of the $d$ items is an affine function of the Borda scale $\mathbf{\beta};$ that is, $g_{1}(j)=\frac{2}{d-1}\beta(j)-1$ or $corr(\mathbf{g}_{1},\mathbf{\beta})=1.$ Proof: Suppose that $\mathbf{v}_{1}=(_{-1}^{\mathbf{1}_{n}});$ then by (4) for $j=1,...,d$ $\displaystyle b_{1}(j)$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n+1}v_{1}(i)p_{ij}^{(1)}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}p_{ij}^{(1)}-p_{(n+1)j}^{(1)}$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n}p_{ij}^{(1)}\text{ \ \ by\ \ \ (17)}$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n}(p_{ij}-p_{i\ast}p_{\ast j})$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n}r_{ij}/t-p_{\ast j}\ \ \ \text{by\ \ (14)}$ $\displaystyle=$ $\displaystyle 2n\beta(j)/t-p_{\ast j}$ Thus by (5) for $j=1,...,d$ $\displaystyle g_{1}(j)$ $\displaystyle=$ $\displaystyle b_{1}(j)/p_{\ast j}$ $\displaystyle=$ $\displaystyle\frac{2n\beta(j)/t-p_{\ast j}}{p_{\ast j}}$ $\displaystyle=$ $\displaystyle\frac{2\beta(j)}{d-1}-1.$ Proposition 4: The crossing index of a coherent cluster is $Cross(cohC(\alpha))=\frac{2(\alpha-1)}{d_{1}d_{2}}.$ Proof: Easily shown by using Definition 3 and Proposition 2. The proof of Theorem 2a easily follows from Theorem 3. The proof of Theorem 2b is similar to the proof of Propostion 1. The proof of Theorem 2c is similar to the proof of Propostion 4.
# Existence of Primitive Normal Pairs with One Prescribed Trace over Finite Fields Hariom Sharma, R. K. Sharma ###### Abstract Given $m,n,q\in\mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in\mathbb{F}_{q}$, we establish a sufficient condition for the existence of primitive pair $(\alpha,f(\alpha))$ in $\mathbb{F}_{q^{m}}$ such that $\alpha$ is normal over $\mathbb{F}_{q}$ and $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=a$, where $f(x)\in\mathbb{F}_{q^{m}}(x)$ is a rational function of degree sum $n$. Further, when $n=2$ and $q=5^{k}$ for some $k\in\mathbb{N}$, such a pair definitely exists for all $(q,m)$ apart from at most $20$ choices. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India Keywords: Finite Fields, Characters, Primitive element, Normal element 2010 Math. Sub. Classification: 12E20, 11T23 111emails: <EMAIL_ADDRESS>(Hariom<EMAIL_ADDRESS>(Rajendra) ## 1 Introduction Given the positive integers $m$ and $q$ such that $q$ is a prime power, $\mathbb{F}_{q}$ denotes the finite field of order $q$ and $\mathbb{F}_{q^{m}}$ be the extension of $\mathbb{F}_{q}$ of degree $m$. A generator of the cyclic multiplicative group $\mathbb{F}_{q^{m}}^{}$ is known as a primitive element of $\mathbb{F}_{q^{m}}$. For a rational function $f(x)\in\mathbb{F}_{q^{m}}(x)$ and $\alpha\in\mathbb{F}_{q^{m}}$, we call a pair $(\alpha,f(\alpha))$ a primitive pair in $\mathbb{F}_{q^{m}}$ if both $\alpha$ and $f(\alpha)$ are primitive elements of $\mathbb{F}_{q^{m}}$. Further, $\alpha$ is normal over $\mathbb{F}_{q}$ if the set $\\{\alpha,\alpha^{q},\alpha^{q^{2}},\cdots,\alpha^{q^{m-1}}\\}$ forms a basis of $\mathbb{F}_{q^{m}}$ over $\mathbb{F}_{q}$. Also, the trace of $\alpha$ over $\mathbb{F}_{q}$, denoted by $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha)$ is given by $\alpha+\alpha^{q}+\alpha^{q^{2}}+\cdots+\alpha^{q^{m-1}}$. Primitive normal elements play a vital role in coding theory and cryptography [1]. Therefore, study of existence of such elements is an active area of research. We refer to [12] for the existence of primitive and normal elements in finite fields. Existence of both primitive and normal elements simultaneously was first established by Lenstra and Schoof in [11]. Later on, by using sieving techniques, Cohen and Huczynska [7] provided a computer-free proof of it. In 1985, Cohen studied the existence of primitive pair $(\alpha,f(\alpha))$ in $\mathbb{F}_{q}$ for the rational function $f(x)=x+a,a\in\mathbb{F}_{q}$. Many more researchers worked in this direction and proved the existence of primitive pair for more general rational function [8, 2, 14, 3]. Additionally, in the fields of even order, Cohen[5] established the existence of primitive pair $(\alpha,f(\alpha))$ in $\mathbb{F}_{q^{n}}$ such that $\alpha$ is normal over $\mathbb{F}_{q}$, where $f(x)=\frac{x^{2}+1}{x}$. Similar result has been obtained in [2] for the rational function $f(x)=\frac{ax^{2}+bx+c}{dx+e}$. Another interesting problem is to prove the existence of primitive pair with prescribed traces which have been discussed in [13, 10, 15]. In this article, we consider all the conditions simultaneously and prove the existence of primitive pair $(\alpha,f(\alpha))$ in $\mathbb{F}_{q^{m}}$ such that $\alpha$ is normal over $\mathbb{F}_{q}$ and for prescribed $a\in\mathbb{F}_{q}$, $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=a$, where $f(x)$ is more general rational function. To proceed further, we shall use some basic terminology and conventions used in [8]. To say that a non zero polynomial $f(x)\in\mathbb{F}_{q^{m}}[x]$ has degree $n\geq 0$ we mean that $f(x)=a_{n}x^{n}+\cdots+a_{0}$, where $a_{n}\neq 0$ and write it as $\deg(f)=n$. Next, for a rational function $f(x)=f_{1}(x)/f_{2}(x)\in\mathbb{F}_{q^{m}}(x)$, we always assume that $f_{1}$ and $f_{2}$ are coprime and degree sum of $f=\deg(f_{1})+\deg(f_{2})$. Also, we can divide each of $f_{1}$ and $f_{2}$ by the leading coefficient of $f_{2}$ and suppose that $f_{2}$ is monic. Further, we say that a rational function $f\in\mathbb{F}_{q^{m}}(x)$ is exceptional if $f=cx^{i}g^{d}$ for some $c\in\mathbb{F}_{q^{m}},i\in\mathbb{Z}$(set of integers) and $d>1$ divides $q^{m}-1$ or $f(x)=x^{i}$ for some $i\in\mathbb{Z}$ such that $\gcd(q^{m}-1,i)\neq 1.$ Finally, we introduce some sets which have an important role in this article. For $n_{1},n_{2}\in\mathbb{N}$, $S_{q,m}(n_{1},n_{2})$ will be used to denote the set of non exceptional rational functions $f=f_{1}/f_{2}\in\mathbb{F}_{q^{m}}(x)$ with $\deg(f_{1})\leq n_{1}$ and $\deg(f_{2})\leq n_{2}$, and $T_{n_{1},n_{2}}$ as the set of pairs $(q,m)\in\mathbb{N}\times\mathbb{N}$ such that for any given $f\in S_{q,m}(n_{1},n_{2})$ and prescribed $a\in\mathbb{F}_{q}$, $\mathbb{F}_{q^{m}}$ contains a normal element $\alpha$ with $(\alpha,f(\alpha))$ a primitive pair and $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=a$. Define $S_{q,m}(n)=\bigcup\limits_{n_{1}+n_{2}=n}S_{q,m}(n_{1},n_{2})$ and $T_{n}=\bigcap\limits_{n_{1}+n_{2}=n}T_{n_{1},n_{2}}$. By [4], for $m\leq 2$, there does not exist any primitive element $\alpha$ such that $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=0$. Therefore, we shall assume $m\geq 3$ throughout the article. In this paper, for $n\in\mathbb{N}$, we take $f(x)\in S_{q,m}(n)$ a general rational function of degree sum $n$ and $a\in\mathbb{F}_{q}$, and prove the existence of normal element $\alpha$ such that $(\alpha,f(\alpha))$ is a primitive pair in $\mathbb{F}_{q^{m}}$ and $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=a$. To be more precise, in section $3$, we obtain a sufficient condition for the existence of such elements in $\mathbb{F}_{q^{m}}$. In section $4$, we further improve the condition by proving a generalization of sieving technique due to Anju and Cohen[6]. In section $5$, we demonstrate the application of the results of section $3$ and section $4$ by working with the finite fields of characteristic $5$ and $n=2$. More precisely, we get a subset of $T_{2}$. ## 2 Preliminaries In this section, we provide some preliminary notations, definitions and results which are required further in this article. Throughout this article, $m\geq 3$ is an integer, $q$ is an arbitrary prime power and $\mathbb{F}_{q}$ is a finite field of order $q$. For each $k(>1)\in\mathbb{N}$, $\omega(k)$ denotes the number of prime divisors of $k$ and $W(k)$ denotes the number of square free divisors of $k$. Also for $g(x)\in\mathbb{F}_{q}[x]$, $\Omega_{q}(g)$ and $W(g)$ denote the number of monic irreducible(over $\mathbb{F}_{q})$ divisors of $g$ and number of square free divisors of $g$ respectively, i.e., $W(k)=2^{\omega(k)}$ and $W(g)=2^{\Omega_{q}(g)}$. For a finite abelian group $G$, a homomorphism $\chi$ from $G$ into the multiplicative group $S^{1}=\\{z\in\mathbb{C}:\|z\|=1\\}$ is known as a character of $G$. The set of all characters of $G$ forms a group under multiplication, which is isomorphic to $G$ and is denoted by $\widehat{G}$. Further, the character $\chi_{0}$, defined as $\chi_{0}(g)=1$ for all $g\in G$ is called the trivial character of $G$. The order of a character $\chi$ is the smallest positive integer $r$ such that $\chi^{r}=\chi_{0}$. For a finite field $\mathbb{F}_{q^{m}}$, the characters of the additive group $\mathbb{F}_{q^{m}}$ and the multiplicative group $\mathbb{F}^{}_{q^{m}}$ are called additive characters and multiplicative characters respectively. A multiplicative character $\chi\in\widehat{\mathbb{F}}_{q^{m}}^{}$ is extended from $\mathbb{F}^{}_{q^{m}}$ to $\mathbb{F}_{q^{m}}$ by the rule $\chi(0)=\begin{cases}0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ if }\chi\neq\chi_{0}\\\ 1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ if }\chi=\chi_{0}\end{cases}.$ For more fundamentals on characters, primitive elements and finite fields, we refer the reader to [12]. For a divisor $u$ of $q^{m}-1$, an element $w\in\mathbb{F}_{q^{m}}^{}$ is $\mathop{\mbox{$u$-$\mathit{free}$}}$, if $w=v^{d}$, where $v\in\mathbb{F}_{q^{m}}$ and $d\|u$ implies $d=1$. It is easy to observe that an element in $\mathbb{F}_{q^{m}}^{}$ is $\mathop{\mbox{$(q^{m}-1$)-$\mathit{free}$}}$ if and only if it is primitive. A special case of [16, Lemma 10], provides an interesting result. ###### Lemma 2.1. Let $u$ be a divisor of $q^{m}-1$, $\xi\in\mathbb{F}_{q^{m}}^{}$. Then $\sum_{d\|u}\frac{\mu(d)}{\phi(d)}\sum_{\chi_{d}}\chi_{d}(\xi)=\begin{cases}\frac{u}{\phi(u)}&\quad\text{ if }\xi\text{ is }\mathop{\mbox{$u$-$\mathit{free}$}},\\\ 0&\quad\text{otherwise.}\end{cases}$ where $\mu(\cdot)$ is the M$\ddot{\text{o}}$bius function and $\phi(\cdot)$ is the Euler function, $\chi_{d}$ runs through all the $\phi(d)$ multiplicative characters over $\mathbb{F}_{q^{m}}^{}$ with order $d$. Therefore, for each divisor $u$ of $q^{m}-1$, $\rho_{u}:\alpha\mapsto\theta(u)\sum_{d\|u}\frac{\mu(d)}{\phi(d)}\sum_{\chi_{d}}\chi_{d}(\alpha),$ (2.1) gives a characteristic function for the subset of $\mathop{\mbox{$u$-$\mathit{free}$}}$ elements of $\mathbb{F}_{q^{m}}^{}$, where $\theta(u)=\frac{\phi(u)}{u}$. Also, for each $a\in\mathbb{F}_{q}$, $\tau_{a}:\alpha\mapsto\frac{1}{q}\sum\limits_{\psi\in\widehat{\mathbb{F}}_{q}}\psi(\text{ Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha)-a)$ is a characterstic function for the subset of $\mathbb{F}_{q^{m}}$ consisting elements with $\text{ Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha)=a$. From [12, Theorem 5.7] every additive character $\psi$ of $\mathbb{F}_{q}$ can be obtained by $\psi(a)=\psi_{0}(ua)$, where $\psi_{0}$ is the canonical additive character of $\mathbb{F}_{q}$ and $u$ is an element of $\mathbb{F}_{q}$ corresponding to $\psi$. Thus $\tau_{a}(\alpha)=\frac{1}{q}\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(\text{ Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(u\alpha)-ua)$ $=\frac{1}{q}\sum\limits_{u\in\mathbb{F}_{q}}\hat{\psi_{0}}(u\alpha)\psi_{0}(-ua),$ (2.2) where $\hat{\psi_{0}}$ is the additive character of $\mathbb{F}_{q^{m}}$ defined by $\hat{\psi_{0}}(\alpha)=\psi_{0}(\text{ Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha))$. In particular, $\hat{\psi_{0}}$ is the canonical additive character of $\mathbb{F}_{q^{m}}$. The additive group of $\mathbb{F}_{q^{m}}$ is an $\mathbb{F}_{q}[x]$-module under the rule $f\leavevmode\nobreak\ o\leavevmode\nobreak\ \alpha=\sum\limits_{i=1}^{k}a_{i}\alpha^{q^{i}}$; for $\alpha\in\mathbb{F}_{q^{m}}$ and $f(x)=\sum\limits_{i=1}^{k}a_{i}x^{i}\in\mathbb{F}_{q}[x]$. For $\alpha\in\mathbb{F}_{q^{m}}$, the $\mathbb{F}_{q}$-order of $\alpha$ is the unique monic polynomial $g$ of least degree such that $g\leavevmode\nobreak\ o\leavevmode\nobreak\ \alpha=0$. Observe that $g$ is a factor of $x^{m}-1$. Similarly, by defining the action of $\mathbb{F}_{q}[x]$ over $\widehat{\mathbb{F}}_{q^{m}}$ by the rule $\psi\leavevmode\nobreak\ o\leavevmode\nobreak\ f(\alpha)=\psi(f\leavevmode\nobreak\ o\leavevmode\nobreak\ \alpha)$, where $\psi\in\widehat{\mathbb{F}}_{q^{m}},\alpha\in\mathbb{F}_{q^{m}}$ and $f\in\mathbb{F}_{q}[x]$, $\widehat{\mathbb{F}}_{q^{m}}$ becomes an $\mathbb{F}_{q}[x]$-module, and the unique monic polynomial $g$ of least degree such that $\psi\leavevmode\nobreak\ o\leavevmode\nobreak\ g=\chi_{0}$ is called the $\mathbb{F}_{q}$-order of $\psi$. Further there are $\Phi_{q}(g)$ characters of $\mathbb{F}_{q}$-order $g$, where $\Phi_{q}(g)$ is the analogue of Euler’s phi-function on $\mathbb{F}_{q}[x]$(see [12]). Similar to above, for $g\|x^{m}-1$ an element $\alpha\in\mathbb{F}_{q^{m}}$ is $g$-$free$, if $\alpha=h\leavevmode\nobreak\ o\leavevmode\nobreak\ \beta$, where $\beta\in\mathbb{F}_{q^{m}}$ and $h\|g$ implies $h=1.$ It is straightforward that an element in $\mathbb{F}_{q^{m}}$ is $(x^{m}-1)$-$free$ if and only if it is normal. Also, for $g\|x^{m}-1$ an expression for the characteristic function for $g$-$free$ elements is given by $\kappa_{g}:\alpha\mapsto\Theta(g)\sum_{h\|g}\frac{\mu^{\prime}(d)}{\Phi_{q}(h)}\sum_{\psi_{h}}\psi_{h}(\alpha),$ (2.3) where $\Theta(g)=\frac{\Phi_{q}(g)}{q^{deg(g)}}$, the internal sum runs over all characters $\psi_{h}$ of $\mathbb{F}_{q}$-order $h$ and $\mu^{\prime}$ is the analogue of the M$\ddot{\text{o}}$bius function defined as $\mu^{\prime}(g)=\begin{cases}(-1)^{s}&\text{if }$g$\text{ is a product of $s$ distinct monic irreducible polynomials},\\\ 0&\quad\text{otherwise.}\end{cases}$ Following results of D. Wang and L. Fu will play a vital role in our next section. ###### Lemma 2.2. [9, Theorem 4.5] Let $f(x)\in\mathbb{F}_{{q}^{d}}(x)$ be a rational function. Write $f(x)=\prod_{j=1}^{k}f_{j}(x)^{n_{j}}$, where $f_{j}(x)\in\mathbb{F}_{{q}^{d}}[x]$ are irreducible polynomials and $n_{j}$ are non zero integers. Let $\chi$ be a multiplicative character of $\mathbb{F}_{q^{d}}$. Suppose that the rational function $\prod_{i=0}^{d-1}f(x^{q^{i}})$ is not of the form $h(x)^{\text{ord}(\chi)}$ in $\mathbb{F}_{q^{d}}(x),$ where ord$(\chi)$ is the order of $\chi$, then we have $\big{\|}\sum_{\alpha\in\mathbb{F}_{q},f(\alpha)\neq 0,f(\alpha)\neq\infty}\chi(f(\alpha))\big{\|}\leq(d\sum_{j=1}^{k}\deg(f_{j})-1)q^{\frac{1}{2}}.$ ###### Lemma 2.3. [9, Theorem 4.6] Let $f(x),g(x)\in\mathbb{F}_{q^{m}}(x)$ be rational functions. Write $f(x)=\prod_{j=1}^{k}f_{j}(x)^{n_{j}}$, where $f_{j}(x)\in\mathbb{F}_{{q}^{m}}[x]$ are irreducible polynomials and $n_{j}$ are non zero integers. Let $D_{1}=\sum_{j=1}^{k}\deg(f_{j})$, let $D_{2}=max(\deg(g),0)$, let $D_{3}$ be the degree of denominator of $g(x)$, and let $D_{4}$ be the sum of degrees of those irreducible polynomials dividing denominator of $g$ but distinct from $f_{j}(x)(j=1,2,\cdots,k)$. Let $\chi$ be a multiplicative character of $\mathbb{F}_{q^{m}}$, and let $\psi$ be a non trivial additive character of $\mathbb{F}_{q^{m}}$. Suppose $g(x)$ is not of the form $r(x)^{q^{m}}-r(x)$ in $\mathbb{F}_{q^{m}}(x)$. Then we have the estimate $\big{\|}\sum_{\alpha\in\mathbb{F}_{q^{m}},f(\alpha)\neq 0,\infty g(\alpha)\neq\infty}\chi(f(\alpha))\psi(g(\alpha))\big{\|}\leq(D_{1}+D_{2}+D_{3}+D_{4}-1)q^{\frac{m}{2}}.$ ## 3 Sufficient condition Let $l_{1},l_{2}\in\mathbb{N}$ be such that $l_{1},l_{2}\|q^{m}-1$. Also, $a\in\mathbb{F}_{q}$, $f(x)\in S_{q,m}(n)$ and $g\|x^{m}-1$, then $N_{f,a,n}(l_{1},l_{2},g)$ denote the number of elements $\alpha\in\mathbb{F}_{q^{m}}$ such that $\alpha$ is both $\mathop{\mbox{$l_{1}$-$\mathit{free}$}}$ and $g$-$free$, $f(\alpha)$ is $\mathop{\mbox{$l_{2}$-$\mathit{free}$}}$ and $\text{Tr}_{\mathbb{F}_{q^{m}}/\mathbb{F}_{q}}(\alpha^{-1})=a$. We now prove one of the sufficient condition as follows. ###### Theorem 3.1. Let $m,n\text{ and }q\in\mathbb{N}$ such that $q$ is a prime power and $m\geq 3$. Suppose that $q^{\frac{m}{2}-1}>(n+2)W(q-1)^{2}W(x^{m}-1).$ (3.1) Then $(q,m)\in T_{n}$. ###### Proof. To prove the result, it is enough to show that $N_{f,a,n}(q^{m}-1,q^{m}-1,x^{m}-1)>0$ for every $f(x)\in S_{q,m}(n)$ and prescribed $a\in\mathbb{F}_{q}$. Let $f(x)\in S_{q,m}(n)$ be any rational function and $a\in\mathbb{F}_{q}$. Let $U_{1}$ be the set of zeros and poles of $f(x)$ in $\mathbb{F}_{q^{m}}$ and $U=U_{1}\cup\\{0\\}$. Assume $l_{1},l_{2}$ be divisors of $q^{m}-1$ and $g$ be a divisor of $x^{m}-1$. Then by definition $N_{f,a,n}(l_{1},l_{2},g)=\sum_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\rho_{l_{1}}(\alpha)\rho_{l_{2}}(f(\alpha))\tau_{a}(\alpha^{-1})\kappa_{g}(\alpha)$ now using (2.1), (2.2) and (2.3), $\displaystyle N_{f,a,n}(l_{1},l_{2},g)=\frac{\theta(l_{1})\theta(l_{2})\Theta(g)}{q}\sum\limits_{\begin{subarray}{c}d_{1}\|l_{1},d_{2}\|l_{2}\\\ h\|g\end{subarray}}\frac{\mu(d_{1})}{\phi(d_{1})}\frac{\mu(d_{2})}{\phi(d_{2})}\frac{\mu^{\prime}(h)}{\Phi_{q}(h)}\sum\limits_{\chi_{d_{1}},\chi_{d_{2}},\psi_{h}}\chi_{f,a}(d_{1},d_{2},h),$ (3.2) where $\chi_{f,a}(d_{1},d_{2},h)=\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{d_{1}}(\alpha)\chi_{d_{2}}(f(\alpha))\psi_{h}(\alpha)\hat{\psi_{0}}(u\alpha^{-1})$. Since $\psi_{h}$ is an additive character of $\mathbb{F}_{q^{m}}$ and $\hat{\psi_{0}}$ is canonical additive character of $\mathbb{F}_{q^{m}}$, therefore there exists $v\in\mathbb{F}_{q^{m}}$ such that $\psi_{h}(\alpha)=\hat{\psi_{0}}(v\alpha)$. Hence $\chi_{f,a}(d_{1},d_{2},h)=\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{d_{1}}(\alpha)\chi_{d_{2}}(f(\alpha))\hat{\psi_{0}}(v\alpha+u\alpha^{-1})$. At this point, we claim that if $(d_{1},d_{2},h)\neq(1,1,1)$, where third $1$ denotes the unity of $\mathbb{F}_{q}[x]$, then $\|\chi_{f,a}(d_{1},d_{2},h)\|\leq(n+2)q^{\frac{m}{2}+1}$. To see the claim, first suppose $d_{2}=1$, then $\chi_{f,a}(d_{1},d_{2},h)=\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{d_{1}}(\alpha)\hat{\psi_{0}}(v\alpha+u\alpha^{-1})$. Here, if $vx+ux^{-1}\neq r(x)^{q^{m}}-r(x)$ for any $r(x)\in\mathbb{F}_{q^{m}}(x)$ then by Lemma 2.3 $\|\chi_{f,a}(d_{1},d_{2},h)\|\leq 2q^{\frac{m}{2}+1}+(\|U\|-1)q\leq(n+2)q^{\frac{m}{2}+1}$. Also, if $vx+ux^{-1}=r(x)^{q^{m}}-r(x)$ for some $r(x)\in\mathbb{F}_{q^{m}}(x)$ then following [Comm. Anju], it is possible when $u=v=0$, which implies, $\|\chi_{f,a}(d_{1},d_{2},h)\|\leq\|U\|q<(n+2)q^{\frac{m}{2}+1}$. Now suppose $d_{2}>1$. Let $d$ be the least common multiple of $d_{1}$ and $d_{2}$. Then [12] suggests that there exists a character $\chi_{d}$ of order $d$ such that $\chi_{d_{2}}=\chi_{d}^{d/d_{2}}$. Also, there is an integer $0\leq k<q^{m}-1$ such that $\chi_{d_{1}}=\chi_{d}^{k}$. Consequently, $\chi_{f,a}(d_{1},d_{2},h)=\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{d}(\alpha^{k}f(\alpha)^{d/d_{2}})\hat{\psi_{0}}(v\alpha+u\alpha^{-1})$. At this moment, first suppose $vx+ux^{-1}\neq r(x)^{q^{m}}-r(x)$ for any $r(x)\in\mathbb{F}_{q^{m}}(x)$. Then Lemma 2.3 implies that $\|\chi_{f,a}(d_{1},d_{2},h)\|\leq(n+2)q^{\frac{m}{2}+1}$. Also, if $vx+ux^{-1}=r(x)^{q^{m}}-r(x)$ for some $r(x)\in\mathbb{F}_{q^{m}}(x)$, then following [15] we get $u=v=0$. Therefore, $\chi_{f,a}(d_{1},d_{2},h)=\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{d}(\alpha^{k}f(\alpha)^{d/d_{2}})$. Here, if $x^{k}f(x)^{d/d_{2}}\neq r(x)^{d}$ for any $r(x)\in\mathbb{F}_{q^{m}}(x)$, then using Lemma 2.2 we get $\|\chi_{f,a}(d_{1},d_{2},h)\|\leq nq^{\frac{m}{2}+1}<(n+2)q^{\frac{m}{2}+1}$. However, $x^{k}f(x)^{d/d_{2}}=r(x)^{d}$ for some $r(x)\in\mathbb{F}_{q^{m}}(x)$ gives that $f$ is exceptional(see [8]). Hence, from the above discussion along with (3.2), we get $\displaystyle N_{f,a,n}(l_{1},l_{2},g)\geq\frac{\theta(l_{1})\theta(l_{2})\Theta(g)}{q}(q^{m}-\|U\|-((n+2)q^{\frac{m}{2}+1})(W(l_{1})W(l_{2})W(g)-1))$ $\displaystyle\geq\frac{\theta(l_{1})\theta(l_{2})\Theta(g)}{q}(q^{m}-(n+1)-((n+2)q^{\frac{m}{2}+1})(W(l_{1})W(l_{2})W(g)-1))$ $\displaystyle\geq\frac{\theta(l_{1})\theta(l_{2})\Theta(g)}{q}(q^{m}-(n+2)q^{\frac{m}{2}+1}W(l_{1})W(l_{2})W(g))$ (3.3) Thus, if $q^{\frac{m}{2}-1}>(n+2)W(l_{1})W(l_{2})W(g)$, then $N_{f,a,n}(l_{1},l_{2},g)>0$ for all $f(x)\in S_{q}(n)$ and prescribed $a\in\mathbb{F}_{q}$. The result now follows by taking $l_{1}=l_{2}=q^{m}-1$ and $g=x^{m}-1$. ∎ ## 4 Sieving Results Here, we state some results, their proofs have been omitted as they follow on the lines of the results in [10] and have been used frequently in [13, 8, 10, 14, 2]. ###### Lemma 4.1. Let $k\text{ and }P$ be co-prime positive integers and $g,G\in\mathbb{F}_{q}[x]$ be co-prime polynomials. Also, let $\\{p_{1},p_{2},\cdots,p_{r}\\}$ be the collection of all prime divisors of $P$, and $\\{g_{1},g_{2},\cdots,g_{s}\\}$ contains all the irreducible factors of $G$. Then $\displaystyle N_{f,a,n}(kP,kP,gG)\geq\sum\limits_{i=1}^{r}N_{f,a,n}(kp_{i},k,g)+\sum\limits_{i=1}^{r}N_{f,a,n}(k,kp_{i},g)$ $\displaystyle+\sum\limits_{i=1}^{s}N_{f,a,n}(k,k,gg_{i})-(2r+s-1)N_{f,a,n}(k,k,g).$ ###### Lemma 4.2. Let $l,m,q\in\mathbb{N}$, $g\in\mathbb{F}_{q}[x]$ be such that $q$ is a prime power, $m\geq 3$ and $l\|q^{m}-1$, $g\|x^{m}-1$. Let $c$ be a prime number which divides $q^{m}-1$ but not $l$, and $e$ be irreducible polynomial dividing $x^{m}-1$ but not $g$. Then $\displaystyle\|N_{f,a,n}(cl,l,g)-\theta(c)N_{f,a,n}(l,l,g)\|\leq(n+2)\theta(c)\theta(l)^{2}\Theta(g)W(l)^{2}W(g)q^{\frac{m}{2}},$ $\displaystyle\|N_{f,a,n}(l,cl,g)-\theta(c)N_{f,a,n}(l,l,g)\|\leq(n+2)\theta(c)\theta(l)^{2}\Theta(g)W(l)^{2}W(g)q^{\frac{m}{2}}$ and $\displaystyle\|N_{f,a,n}(l,l,eg)-\Theta(e)N_{f,a,n}(l,l,g)\|\leq(n+2)\theta(l)^{2}\Theta(e)\Theta(g)W(l)^{2}W(g)q^{\frac{m}{2}}.$ ###### Theorem 4.1. Let $l,m,q\in\mathbb{N}$, $g\in\mathbb{F}_{q}[x]$ be such that $q$ is a prime power, $m\geq 3$ and $l\|q^{m}-1$, $g\|x^{m}-1$. Also, let $\\{p_{1},p_{2},\cdots p_{r}\\}$ be the collection of primes which divides $q^{m}-1$ but not $l$, and $\\{g_{1},g_{2},\cdots g_{s}\\}$ be the irreducible polynomials dividing $x^{m}-1$ but not $g$. Suppose $\delta=1-2\sum\limits_{i=1}^{r}\frac{1}{p_{i}}-\sum\limits_{i=1}^{s}\frac{1}{q^{\deg(g_{i})}},\delta>0$ and $\Delta=\frac{2r+s-1}{\delta}+2$. If $q^{\frac{m}{2}-1}>(n+2)\Delta W(l)^{2}W(g)$ then $(q,m)\in T_{n}.$ Now, we present a more effective sieving technique than Theorem 4.1, which is an extension of the result in [6]. For this, we adopt some notations and conventions from [6] as described. Let $\text{Rad}(q^{m}-1)=kPL$, where $k$ is the product of smallest prime divisors of $q^{m}-1$, $L$ is the product of large prime divisors of $q^{m}-1$ denoted by $L=l_{1}\cdot l_{2}\cdots l_{t}$, and rest of the prime divisors of $q^{m}-1$ lie in $P$ and denoted by $p_{1},p_{2},\cdots,p_{r}$. Similarly, $\text{Rad}(x^{m}-1)=gGH$, where $g$ is the product of irreducible factors of $x^{m}-1$ of least degree, and irreducible factors of large degree are factors of $H$ which are denoted by $h_{1},h_{2},\cdots,h_{u}$ and rest lie in $G$ and denoted by $g_{1},g_{2},\cdots,g_{s}$. ###### Theorem 4.2. Let $m,q\in\mathbb{N}$ such that $q$ is a prime power and $m\geq 3$. Using above notations, let $\text{Rad}(q^{m}-1)=kPL$, $\text{Rad}(x^{m}-1)=gGH$, $\delta=1-2\sum\limits_{i=1}^{r}\frac{1}{p_{i}}-\sum\limits_{i=1}^{s}\frac{1}{q^{\deg(g_{i})}},\epsilon_{1}=\sum\limits_{i=1}^{t}\frac{1}{l_{i}},\leavevmode\nobreak\ \epsilon_{2}=\sum\limits_{i=1}^{u}\frac{1}{q^{\deg(h_{i})}}\text{ and }\delta\theta(k)^{2}\Theta(g)-(2\epsilon_{1}+\epsilon_{2})>0$. Then $q^{\frac{m}{2}-1}>(n+2)[\theta(k)^{2}\Theta(g)W(k)^{2}W(g)(2r+s-1+2\delta)+(t-\epsilon_{1})+(2/(n+2))(u-\epsilon_{2})\\\ +(n/(n+2))(1/q^{m/2})(t+u-\epsilon_{1}-\epsilon_{2})]/[\delta\theta(k)^{2}\Theta(g)-(2\epsilon_{1}+\epsilon_{2})]$ (4.1) implies $(q,m)\in T_{n}$. ###### Proof. Clearly, $N_{f,a,n}(q^{m}-1,q^{m}-1,x^{m}-1)=N_{f,a,n}(kPL,kPL,gGH)\geq N_{f,a,n}(kP,kP,gG)\\\ +N_{f,a,n}(L,L,H)-N_{f,a,n}(1,1,1).$ (4.2) Further, by Lemma 4.1 $N_{f,a,n}(kP,kP,gG)\geq\delta N_{f,a,n}(k,k,g)+\sum\limits_{i=1}^{r}\\{N_{f,a,n}(kp_{i},k,g)-\theta(p_{i})N_{f,a,n}(k,k,g)\\}\\\ +\sum\limits_{i=1}^{r}\\{N_{f,a,n}(k,kp_{i},g)-\theta(p_{i})N_{f,a,n}(k,k,g)\\}+\sum\limits_{i=1}^{s}(N_{f,a,n}(k,k,gg_{i})-\Theta(g_{i})N_{f,a,n}(k,k,g))$ . Using (3.3) and Lemma 4.2, we get $\displaystyle N_{f,a,n}(kP,kP,gG)\geq\delta\theta(k)^{2}\Theta(g)\big{(}q^{m-1}-(n+2)W(k)^{2}W(g)q^{\frac{m}{2}}\big{)}$ $\displaystyle-(n+2)\theta(k)^{2}\Theta(g)W(k)^{2}W(g)\big{(}\sum\limits_{i=1}^{r}2\theta(p_{i})+\sum\limits_{i=1}^{s}\Theta(g_{i})\big{)}q^{\frac{m}{2}}$ $\displaystyle=\theta(k)^{2}\Theta(g)\big{(}\delta q^{m-1}-(n+2)(2r+s-1+2\delta)W(k)^{2}W(g)q^{\frac{m}{2}}\big{)}.$ (4.3) Again, by Lemma 4.1 $N_{f,a,n}(L,L,H)-N_{f,a,n}(1,1,1)\geq\sum\limits_{i=1}^{t}N_{f,a,n}(l_{i},1,1)+\sum\limits_{i=1}^{t}N_{f,a,n}(1,l_{i},1)\\\ +\sum\limits_{i=1}^{u}N_{f,a,n}(1,1,h_{i})-(2t+u)N_{f,a,n}(1,1,1)$ $=\sum\limits_{i=1}^{t}\\{N_{f,a,n}(l_{i},1,1)-\theta(l_{i})N_{f,a,n}(1,1,1)\\}+\sum\limits_{i=1}^{t}\\{N_{f,a,n}(1,l_{i},1)-\theta(l_{i})N_{f,a,n}(1,1,1)\\}\\\ +\sum\limits_{i=1}^{u}\\{N_{f,a,n}(1,1,h_{i})-\Theta(h_{i})N_{f,a,n}(1,1,1)\\}-(2\epsilon_{1}+\epsilon_{2})N_{f,a,n}(1,1,1)$ (4.4) By (3.2), for a prime divisor $l$ of $q^{m}-1$, $\|N_{f,a,n}(l,1,1)-\theta(l)N_{f,a,n}(1,1,1)\|=\frac{\theta(l)}{\phi(l)q}\|\sum\limits_{\chi_{l}}\chi_{f,a}(l,1,1)\|,$ where $\displaystyle\|\chi_{f,a}(l,1,1)\|=\|\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{l}(\alpha)\hat{\psi_{0}}(u\alpha^{-1}\|\leq q^{\frac{m}{2}+1}+nq.$ Hence, $\|N_{f,a,n}(l,1,1)-\theta(l)N_{f,a,n}(1,1,1)\|\leq\theta(l)(q^{\frac{m}{2}}+n).$ Similarly, $\displaystyle\|\chi_{f,a}(1,l,1)\|=\|\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\chi_{l}(f(\alpha))\hat{\psi_{0}}(u\alpha^{-1}\|\leq(n+1)q^{\frac{m}{2}+1},$ which further implies $\|N_{f,a,n}(1,l,1)-\theta(l)N_{f,a,n}(1,1,1)\|\leq(n+1)q^{\frac{m}{2}}$. Also, for an irreducible divisor $h$ of $x^{m}-1$, $\|\chi_{f,a}(1,1,h)\|=\|\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\psi_{h}(\alpha)\hat{\psi_{0}}(u\alpha^{-1}\|\\\ =\|\sum\limits_{u\in\mathbb{F}_{q}}\psi_{0}(-au)\sum\limits_{\alpha\in\mathbb{F}_{q^{m}}\setminus U}\hat{\psi_{0}}(v\alpha+u\alpha^{-1}\|\leq 2q^{\frac{m}{2}+1}+nq.$ Therefore, $\|N_{f,a,n}(1,1,h)-\Theta(h)N_{f,a,n}(1,1,1)\|\leq\Theta(h)(q^{\frac{m}{2}}+n)$. Using these bounds in (4.4), we have $N_{f,a,n}(L,L,H)-N_{f,a,n}(1,1,1)\geq-\sum\limits_{i=1}^{t}\theta(l_{i})(q^{\frac{m}{2}}+n)-\sum\limits_{i=1}^{t}\theta(l_{i})(n+1)q^{\frac{m}{2}}-\sum\limits_{i=1}^{u}\Theta(h_{i})(2q^{\frac{m}{2}}+n)-(2t+u)N_{f,a,n}(1,1,1)$. Now, $N_{f,a,n}(1,1,1)\leq q^{m-1}$ together with $\sum\limits_{i=1}^{t}\theta(l_{i})=(t-\epsilon_{1})$ and $\sum\limits_{i=1}^{u}=(u-\epsilon_{2})$ implies $N_{f,a,n}(L,L,H)-N_{f,a,n}(1,1,1)\geq-\\{(n+2)(t-\epsilon_{1})+2(u-\epsilon_{2})\\}q^{\frac{m}{2}}\\\ -n(t+u-\epsilon_{1}-\epsilon_{2})-(2\epsilon_{1}+\epsilon_{2})q^{m-1}.$ (4.5) Now using (4.3) and (4.5) in (4.2) we get, $N_{f,a,n}(q^{m}-1,q^{m}-1,x^{m}-1)\geq\\{\delta\theta(k)^{2}\Theta(g)-(2\epsilon_{1}+\epsilon_{2})\\}q^{m-1}-\theta(k)^{2}\Theta(g)(n+2)\\\ (2r+s-1+2\delta)W(k)^{2}W(g)q^{\frac{m}{2}}-\\{(n+2)(t-\epsilon_{1})+2(u-\epsilon_{2})\\}q^{\frac{m}{2}}-n(t+u-\epsilon_{1}-\epsilon_{2})\\\ \\\ =q^{\frac{m}{2}}\big{[}\big{(}\delta\theta(k)^{2}\Theta(g)-(2\epsilon_{1}+\epsilon_{2})\big{)}q^{\frac{m}{2}-1}-(n+2)\\{\theta(k)^{2}\Theta(g)(2r+s-1+2\delta)W(k)^{2}W(g)\\\ -\\{(t-\epsilon_{1})+(2/(n+2))(u-\epsilon_{2})\\}-(n/(n+2))(1/q^{m/2})(t+u-\epsilon_{1}-\epsilon_{2})\\}\big{]}$ Thus $q^{\frac{m}{2}-1}>(n+2)[\theta(k)^{2}\Theta(g)W(k)^{2}W(g)(2r+s-1+2\delta)+(t-\epsilon_{1})+(2/(n+2))(u-\epsilon_{2})\\\ +(n/(n+2))(1/q^{m/2})(t+u-\epsilon_{1}-\epsilon_{2})]/[\delta\theta(k)^{2}\Theta(g)-(2\epsilon_{1}+\epsilon_{2})]$ implies $N_{f,a,n}(q^{m}-1,q^{m}-1,x^{m}-1)>0$ i.e., $(q,m)\in T_{n}$. ∎ It is easy to observe that Theorem 4.1 is a special case of Theorem 4.2 and can be obtained by setting $t=u=\epsilon_{1}=\epsilon_{2}=0$. ## 5 Working Example However the results discussed above are applicable for arbitrary natural number $n$ and the finite field $\mathbb{F}_{q^{m}}$ of any prime characteristic. Though to demonstrate the application of above results and make the calculations uncomplicated we assume that $q=5^{k}$ for some $k\in\mathbb{N}$ and $n=2$, and work on the set $T_{2}$. Precisely, in this section, we prove the following result. ###### Theorem 5.1. Let $q=5^{k}$ for some $k\in\mathbb{N}$ and $m\geq 3$ is an integer. Then $(q,m)\in T_{2}$ unless one of the following holds: 1. 1. $q=5,5^{2},5^{3},5^{4},5^{5},5^{6},5^{8},5^{10}$ and $m=3$; 2. 2. $q=5,5^{2},5^{3},5^{4}$ and $m=4$; 3. 3. $q=5,5^{2}$ and $m=5,6;$ 4. 4. $q=5$ and $m=7,8,10,12.$ We shall divide it in two parts, in first part we shall work on $m\geq 5$ and in second we shall consider $m=3,4$. For further calculation work and to apply the previous results we shall need the following lemma which can also be developed from [5, Lemma 6.2]. ###### Lemma 5.1. Let $M$ be a positive integer, then $W(M)<4515\times M^{1/8}.$ ### 5.1 Part 1. In this part, we assume $m\geq 5$ and write $m=m^{\prime}5^{j}$, where $j\geq 1$ is an integer and $5\nmid m^{\prime}$. Then $\Omega_{q}(x^{m}-1)=\Omega_{q}(x^{m^{\prime}}-1)$ which further implies $W(x^{m}-1)=W(x^{m^{\prime}}-1)$. Further, we shall divide the discussion in two cases. $\bullet\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ m^{\prime}\|q-1$ $\bullet\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ m^{\prime}\nmid q-1$ Case 1. $m\|q-1$. Clearly [12, Theorem 2.47] implies that $\Omega_{q}(x^{m^{\prime}}-1)=m^{\prime}$. Let $l=q^{m}-1\text{ and }g=1$ in Theorem 4.1 then $\Delta=\frac{q^{2}+(a-3)q+2}{(a-1)q+1}$, where $a=\frac{q-1}{m^{\prime}},$ which further implies $\Delta<q^{2}$. Hence $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-3}>4W(q^{m}-1)^{2}.$ However, by Lemma 5.1, it is sufficient if $q^{\frac{m}{4}-3}>4\cdot(4515)^{2},$ which holds for $q\geq 125$ and for all $m\geq 28$. In particular, for $q\geq 125$ and for all $m^{\prime}\geq 28$. Next, we examine all the cases where $m^{\prime}\leq 27$. For this we set $l=q^{m}-1$ and $g=1$ in Theorem 4.1 unless mentioned. Then $\delta=1-\frac{m^{\prime}}{q}$ and $\Delta=2+\frac{(m^{\prime}-1)q}{q-m^{\prime}}$ 1\. $m^{\prime}=1.$ Here $m=5^{j}$ for some integer $j\geq 1$ and $\Delta=2$. Then by Theorem 4.1 it is sufficient if $q^{\frac{m}{2}-1}>4\cdot 2\cdot W(q^{m}-1)^{2}$. Again Lemma 5.1 implies $(q,m)\in T_{2}$ if $q^{\frac{m}{4}-1}>8\cdot(4515)^{2}$ i.e., $q^{\frac{5^{j}}{4}-1}>8\cdot(4515)^{2}$, which holds for all choices of $(q,m)$ except $(5,5),(5,5^{2}),(5^{2},5),(5^{2},5^{2}),(5^{3},5),(5^{4},5),\leavevmode\nobreak\ \cdots,(5^{46},5)$ which are $48$ in number. For these, we checked $q^{\frac{m}{2}-1}>4\cdot 2\cdot W(q^{m}-1)^{2}$ directly by factoring $q^{m}-1$ and got it verified except the pairs $(5,5),(5^{2},5),(5^{3},5),\\\ (5^{4},5)$ and $(5^{6},5)$. 2\. $m^{\prime}=2$. In this case, $m=2\cdot m^{j}$ for some $j\geq 1$ and $\Delta=2+\frac{q}{q-2}<4$. Similar to the above case, it is sufficient if $q^{\frac{2\cdot 5^{j}}{4}-1}>16\cdot(4515)^{2}$, which is true except the $9$ pairs $(5,10),(5,50),(5^{2},10),(5^{3},10),\cdots,(5^{8},10)$, and the verification of $q^{\frac{m}{2}-1}>4\cdot 4\cdot W(q^{m}-1)^{2}$ for these pairs yield the only possible exceptions as $(5,10)\text{ and }(5^{2},10)$. Following the similar steps for the rest of the values of $m^{\prime}\leq 27$ we get that there is no exception for many values of $m^{\prime}$. Values of $m^{\prime}$ with possible exceptional pairs is as below. 3\. $m^{\prime}=4.$ $(5,20)$. 4\. $m^{\prime}=6.$ $(5^{2},6),(5^{4},6)\text{ and }(5^{6},6).$ 5\. $m^{\prime}=8.$ $(5^{2},8)$. Furthermore, for the pairs $(5^{3},5),(5^{4},5),(5^{6},5),(5^{2},10),(5,20),(5^{4},6),(5^{6},6)$ and $(5^{2},8)$ Theorem 4.1 holds for some choice of $l$ and $g$ (see Table 1). Hence, only left possible exceptions in this case are $(5,5),(5^{2},5),(5,10)$ and $(5^{2},6)$. Table 1 Sr. No. $(q,m)$ $l$ $r$ $g$ $s$ $\delta>$ $\Delta<$ $4\Delta W(g)$ $W(l)^{2}<$ 1 $(5^{3},5)$ $2$ $5$ $1$ $1$ $0.705298$ $16.178405$ $518$ 2 $(5^{4},5)$ 6 6 $1$ 1 $0.581729$ $22.628164$ $2897$ 3 $(5^{6},5)$ 6 9 $1$ 1 0.390631 48.079201 6155 4 $(5^{2},10)$ 6 6 $1$ 2 0.503329 27.828038 3562 5 $(5,20)$ 6 6 $x^{2}+\beta^{3}x+\beta$ 2 0.183329 72.910743 18666 6 $(5^{4},6)$ 6 6 $1$ 6 0.476599 37.669274 4822 7 $(5^{6},6)$ 6 9 $1$ 6 0.330094 71.677019 9175 8 $(5^{2},8)$ 6 4 $1$ 8 0.401942 39.318735 5033 where $\beta$ is a primitive element of $\mathbb{F}_{5}$. Case 2. $m^{\prime}\nmid q-1$. Let the order of $q\mod m^{\prime}$ be denoted by $b$. Then $b\geq 2$ and degree of irreducible factors of $x^{m^{\prime}}-1$ over $\mathbb{F}_{q}$ is less than or equal to $b$. Let $M$ denotes the number of distinct irreducible factors of $x^{m}-1$ over $\mathbb{F}_{q}$ of degree less than $b$. Also let $\nu(q,m)$ denotes the ratio $\nu(q,m)=\frac{M}{m}$. Then, $m\nu(q,m)=m^{\prime}\nu(q,m^{\prime})$. For the further progress, we need the following two results which are the directly implied by Proposition $5.3$ of [7] and Lemma 7.2 of [5] respectively. ###### Lemma 5.2. Let $k,m,q\in\mathbb{N}$ be such that $q=5^{k}$ and $m^{\prime}\nmid q-1.$ In the notations of Theorem 4.1, let $l=q^{m}-1$ and $g$ is the product of irreducible factors of $x^{m}-1$ of degree less than $b$, then $\Delta<m^{\prime}$. ###### Lemma 5.3. Let $m^{\prime}>4$ and $m_{1}=\gcd(q-1,m^{\prime})$. Then following bounds hold. 1. 1. For $m^{\prime}=2m_{1}$, $\nu(q,m^{\prime})=\frac{1}{2};$ 2. 2. for $m^{\prime}=4m_{1},\nu(q,m^{\prime})=\frac{3}{8};$ 3. 3. for $m^{\prime}=6m_{1},\nu(q,m^{\prime})=\frac{13}{36};$ 4. 4. otherwise, $\nu(q,m^{\prime})\leq\frac{1}{3}$. At this point we note that $m^{\prime}=1,2$ and $4$ divide $q-1$ for any $q=5^{k}$ and have been discussed in above case, whereas $m^{\prime}=5$ is not possible. Therefore, in this case we need to discuss $m^{\prime}=3$ and $m^{\prime}\geq 6$. First consider $m^{\prime}=3$. Then $m=3\cdot 5^{j}$ for some integer $j\geq 1$. Also, $m^{\prime}\nmid q-1$ implies if $q=5^{k}$ then $k$ is odd and $x^{m^{\prime}}-1$ is the product of a linear factor and a quadratic factor. Thus, $W(x^{m}-1)=W(x^{m^{\prime}}-1)=2^{2}=4$ and $(\ref{main})$ implies $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-1}>16\cdot W(q^{m}-1)^{2}.$ By Lemma 5.1, it is sufficient if $q^{\frac{m}{4}-1}>16\cdot(4515)^{2}$, which hold for $q=5$ and $m\geq 53$, $q=125$ and $m\geq 21$, $q\geq 5^{5}$ and $m\geq 14$. Thus, only possible exceptions are $(5,15)$ and $(125,15)$. For these two possible exceptions we checked $q^{\frac{m}{4}-1}>16\cdot W(q^{m}-1)^{2}$ directly by factoring $q^{m}-1$ and got it verified for $(125,15)$. Hence only possible exception for $m^{\prime}=3$ is $(5,15)$. Now suppose $m^{\prime}\geq 6$. At this point, in Theorem 4.1 let $l=q^{m}-1$ and $g$ be the product of irreducible factors of $x^{m}-1$ of degree less than $b$. Therefore, Lemma 5.2 along with Theorem 4.1 implies $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-1}>4\cdot m^{\prime}\cdot W(q^{m}-1)^{2}\cdot 2^{m^{\prime}\nu(q,m^{\prime})}$. By Lemma 5.1, it is sufficient if $\displaystyle q^{\frac{m}{4}-1}>4\cdot m\cdot(4515)^{2}\cdot 2^{m\nu(q,m^{\prime})}.$ (5.1) Further, we shall discuss it in four cases as follows. 1. $m^{\prime}\neq 2m_{1},4m_{1},6m_{1}.$ Here, Lemma 5.3 implies $\nu(q,m^{\prime})=\frac{1}{3}$. Using this in (5.1) we get $(q,m)\in T_{2}$ if $q^{\frac{m}{4}-1}>4\cdot m\cdot(4515)^{2}\cdot 2^{\frac{m}{3}}$, which holds for $q^{m}\geq 5^{145}$. Next, for $q^{m}\leq 5^{144}$, we verified $q^{\frac{m}{2}-1}>4\cdot m\cdot W(q^{m}-1)^{2}\cdot 2^{\frac{m}{3}}$ by factoring $q^{m}-1$ and got a list of $20$ possible exception as follows. $(5,6),(5,7),(5,9),(5,11),(5,12),(5,13),(5,14),(5,17),(5,18),(5,19),(5,21),\\\ (5,22),(5,27),(5,30),(5,36),(5^{2},7),(5^{2},9),(5^{2},11),(5^{3},6),(5^{5},6)$. 2. $m^{\prime}=2m_{1}.$ In this case, $\nu(q,m)=\frac{1}{2}$. Therefore, (5.1) implies $(q,m)\in T_{2}$ if $q^{\frac{m}{4}-1}>4\cdot m\cdot(4515)^{2}\cdot 2^{\frac{m}{2}}$, which holds for $q=5$ and $m\geq 466$ while for $q\geq 25$ it is sufficient that $m\geq 56$. Here, for $q=5$, we have $m^{\prime}=8$ only. Thus possible exception for $q=5$ are $(5,8),(5,40)$ and $(5,200)$. On the other hand, for $q\geq 25$ and $q^{m}<25^{56}$ along with above three possible exceptions we checked $q^{\frac{m}{2}-1}>4\cdot m\cdot W(q^{m}-1)^{2}\cdot 2^{\frac{m}{2}}$ and got it verified except $(5,8),(5,40)\text{ and }(5^{3},8)$. 3. $m^{\prime}=4m_{1}.$ Here, $\nu(q,m)=\frac{3}{8}$. Again, (5.1) gives $(q,m)\in T_{2}$ if $q^{\frac{m}{4}-1}>4\cdot m\cdot(4515)^{2}\cdot 2^{\frac{3m}{8}}$, which is true for $q^{m}\geq 5^{176}$. On the other side, verification of $q^{\frac{m}{2}-1}>4\cdot m\cdot W(q^{m}-1)^{2}\cdot 2^{\frac{3m}{8}}$ for $q^{m}<5^{176}$ provides only possible exception as $(5,16)$. 4. $m^{\prime}=6m_{1}.$ Similar to the above case, we have $\nu(q,m)=\frac{13}{36}$ and $q^{\frac{m}{4}-1}>4\cdot m\cdot(4515)^{2}\cdot 2^{\frac{13m}{36}}$ holds for $q^{m}\geq 5^{164}$. Also, for $q^{m}<5^{164}$, $q^{\frac{m}{2}-1}>4\cdot m\cdot W(q^{m}-1)^{2}\cdot 2^{\frac{13m}{36}}$ holds for all $(q,m)$ except $(5,24)$. Table 2 Sr. No. $(q,m)$ $l$ $r$ $g$ $s$ $\delta>$ $\Delta<$ $4\Delta W(g)$ $W(l)^{2}<$ 1 $(5,11)$ 2 1 $1$ 3 0.799359 7.004009 225 2 $(5,13)$ 2 1 $1$ 4 0.795199 8.287731 266 3 $(5,14)$ 2 4 $x+1$ 3 0.059683 169.55170 5426 4 $(5,17)$ 2 2 $1$ 2 0.795110 8.288442 266 5 $(5,18)$ 6 5 $1$ 6 0.061578 245.59029 31436 6 $(5,19)$ 2 3 $1$ 3 0.789208 12.136745 389 7 $(5,21)$ 2 4 $1$ 5 0.689908 19.393614 621 8 $(5,22)$ 2 5 $x+1$ 5 0.014867 943.67119 30198 9 $(5,27)$ 2 7 $1$ 4 0.561470 32.277659 1033 10 $(5,30)$ 6 9 $x+1$ 3 0.110695 182.67531 23383 11 $(5,36)$ 6 9 $x^{4}-1$ 8 0.170222 148.86660 152440 12 $(5^{2},7)$ 2 4 1 3 0.219683 47.520125 1521 13 $(5^{2},9)$ 6 5 1 5 0.421578 35.208505 4507 14 $(5^{2},11)$ 2 5 1 3 0.176146 70.124930 2244 15 $(5^{3},6)$ 6 5 1 4 0.525578 26.734639 3423 16 $(5^{5},6)$ 6 9 10 4 0.390055 55.838482 7148 17 $(5,15)$ 2 5 1 2 0.473298 25.241167 808 18 $(5,40)$ 6 9 $x^{2}+\beta^{3}x+\beta$ 4 0.088640 238.91192 61162 19 $(5^{3},8)$ 6 6 1 6 0.454072 39.438940 5049 20 $(5,16)$ 6 4 $x+1$ 7 0.038742 363.35624 46510 21 $(5,24)$ 6 6 $x^{4}-1$ 10 0.086200 245.61740 251513 Next, we refer to Table 2 to note that Theorem 4.1 holds for the pairs $(5,11)$, $(5,13)$, $(5,14)$, $(5,15)$, $(5,16)$, $(5,17)$, $(5,18)$, $(5,19)$, $(5,21)$, $(5,22)$, $(5,24)$, $(5,27)$, $(5,30)$, $(5,36)$, $(5,40)$, $(5^{2},7)$, $(5^{2},9)$, $(5^{2},11)$, $(5^{3},6)$, $(5^{3},8)$, $(5^{5},6)$. Thus, only left possible exceptions in the case $m^{\prime}\nmid q-1$ are $(5,6)$,$(5,7)$, $(5,8)$, $(5,9)$, and $(5,12).$ ### 5.2 Part 2. In this part we shall consider $m=3,4.$ Following result will be required for further calculation, which follows on the lines of [6, Lemma 51]. ###### Lemma 5.4. Let $k\in\mathbb{N}$ such that $\omega(k)\geq 2828$. Then $W(k)<k^{\frac{1}{13}}.$ Also, $W(x^{m}-1)\leq 16$. Now, first assume $\omega(q^{m}-1)\geq 2828$, then (3.1) and Lemma 5.4 together implies $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-1}>64\cdot q^{\frac{2m}{13}}$ i.e., $q^{\frac{9m}{26}-1}>64$ or $q^{m}>64^{\frac{26m}{9m-26}}$, sufficient if $q^{m}>64^{78}$, which is true for $\omega(q^{m}-1)\geq 2828$. To make further progress we follow [13]. Next, assume $88\leq\omega(q^{m}-1)\leq 2827$. In Theorem 4.1, let $g=x^{m}-1$ and $l$ to be the product of least $88$ primes dividing $q^{m}-1$ i.e., $W(l)=2^{88}$. Then $r\leq 2739$ and $\delta$ will be at least its value when $\\{p_{1},p_{2},\cdots,p_{2739}\\}=\\{461,463,\cdots,25667\\}$. This gives $\delta>0.0041806$ and $\Delta<1.3101\times 10^{6}$, hence $4\Delta W(g)W(l)^{2}<8.0309\times 10^{60}=R$ (say). By Theorem 4.1 $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-1}>R$ or $q^{m}>R^{\frac{2m}{m-2}}$. But $m\geq 3$ implies $\frac{2m}{m-2}\leq 6$. Therefore, if $q^{m}>R^{6}$ or $q^{m}>2.6828\times 10^{365}$ then $(q,m)\in T_{2}$. Hence, $\omega(q^{m}-1)\geq 152$ gives $(q,m)\in T_{2}$. Repeating this process of Theorem 4.1 for the values in Table 3 implies $(q,m)\in T_{2}$ if $q^{\frac{m}{2}-1}>889903387$. Thus, for $m=3$ it is sufficient if $q>(889903387)^{2}$ and for $m=4$ we need $q>889903387$. Hence, only possible exceptions are $(5,3),(5^{2},3),\cdots,(5^{25},3)$ and $(5,4),(5^{2},4),\cdots,(5^{12},4)$. However, Table 4 implies that Theorem 4.1 holds for $(5^{9},3),(5^{11},3),(5^{12},3),(5^{13},3),\cdots,(5^{25},3)$ and $(5^{6},4),(5^{7},4),\cdots,(5^{12},4)$. Thus, only possible exceptions here are $(5,3),(5^{2},3),\cdots,(5^{8},3)$ and $(5^{10},3)$, and $(5,4),(5^{2},4),\cdots,(5^{5},4)$. Table 3 Sr. No. \| $a\leq\omega(q^{m}-1)\leq b$ \| $W(l)$ \| $\delta>$ \| $\Delta<$ \| $4\Delta W(g)$ $W(l)^{2}$ $<$ ---\|---\|---\|---\|---\|--- 1 \| $a=17,\leavevmode\nobreak\ \leavevmode\nobreak\ b=151$ \| $2^{17}$ \| $0.0347407$ \| $7687.5008$ \| $8.4526\times 10^{15}$ 2 \| $a=9,\leavevmode\nobreak\ \leavevmode\nobreak\ b=51$ \| $2^{9}$ \| $0.0550187$ \| $1510.5788$ \| $2.5344\times 10^{10}$ 3 \| $a=7,\leavevmode\nobreak\ \leavevmode\nobreak\ b=37$ \| $2^{7}$ \| $0.0064402$ \| $9163.1796$ \| $9608289244$ 4 \| $a=7,\leavevmode\nobreak\ \leavevmode\nobreak\ b=36$ \| $2^{7}$ \| $0.0191790$ \| $2973.9903$ \| $3118453847$ 5 \| $a=7,\leavevmode\nobreak\ \leavevmode\nobreak\ b=34$ \| $2^{7}$ \| $0.0458469$ \| $1158.0218$ \| $1214272852$ 6 \| $a=7,\leavevmode\nobreak\ \leavevmode\nobreak\ b=33$ \| $2^{7}$ \| $0.0602354$ \| $848.6790$ \| $889903387$ Table 4 Sr. No. $(q,m)$ $l$ $r$ $g$ $s$ $\delta>$ $\Delta<$ $4\Delta W(g)$ $W(l)^{2}<$ 1 $(5^{9},3)$ 2 7 1 2 0.801533 20.714128 663 2 $(5^{11},3)$ 2 4 1 2 0.925433 11.725177 376 3 $(5^{12},3)$ 6 9 1 3 0.330478 62.518314 8003 4 $(5^{13},3)$ 2 4 1 2 0.910167 11.888295 381 5 $(5^{14},3)$ 6 10 1 3 0.508443 45.269297 5795 6 $(5^{15},3)$ 2 10 1 2 0.603902 36.773815 1177 7 $(5^{16},3)$ 6 9 1 3 0.368379 56.291827 7206 8 $(5^{17},3)$ 2 6 1 2 0.930565 15.970005 512 9 $(5^{18},3)$ 6 12 1 3 0.499055 54.098369 6925 10 $(5^{19},3)$ 2 5 1 2 0.924693 13.895837 445 11 $(5^{20},3)$ 6 15 1 3 0.183646 176.24807 22560 12 $(5^{21},3)$ 2 9 1 2 0.822416 25.102645 804 13 $(5^{22},3)$ 6 10 1 3 0.522529 44.102865 5646 14 $(5^{23},3)$ 2 7 1 2 0.920550 18.294603 586 15 $(5^{24},3)$ 6 14 1 3 0.296682 103.11815 13200 16 $(5^{25},3)$ 2 14 1 2 0.666688 45.498589 1456 17 $(5^{6},4)$ 6 6 1 4 0.485944 32.867712 4208 18 $(5^{7},4)$ 2 6 1 4 0.105913 143.62473 4596 19 $(5^{8},4)$ 2 7 1 4 0.054494 313.95724 10047 20 $(5^{9},4)$ 6 9 1 4 0.330476 65.544620 8390 21 $(5^{1}0,4)$ 6 9 1 4 0.568640 38.930216 4984 22 $(5^{1}1,4)$ 2 8 1 4 0.039829 479.03888 15330 23 $(5^{1}2,4)$ 6 9 1 4 0.368379 59.006421 7553 Further, for all the left possible exceptions we checked Theorem 4.2 and got it verified in case of $(5^{7},3),(5^{5},4)$ and $(5,9)$ for the values in Table 5. Table 5 Sr. No. $(q,m)$ $k$ $P$ $L$ $f$ $G$ $H$ $R^{\prime}<$ 1 $(5,9)$ 2 589 829 $x-1$ $x^{2}+x+1$ $x^{6}+x^{3}+1$ 269 2 $(5^{7},3)$ 2 229469719 519499 $x-1$ 1 $x^{2}+x+1$ 262 3 $(5^{9},4)$ 6 216878233 9161 $x+1$ $x^{2}+x+\beta^{3}$ $x+\beta^{3}$ 2788 Where, $R^{\prime}$ represent the right hand side value of (4.1). Hence, all the results from part 1 and part 2 collectively implies Theorem 5.1. ## References [1] G. B. Agnew, R. C. Mullin, I. M. Onyszchuk, and S. A. Vanstone. An implementation for a fast public-key cryptosystem. J. Cryptology, 3(2):63–79, 1991. * [2] Anju and R. K. Sharma. Existence of some special primitive normal elements over finite fields. Finite Fields Appl., 46:280–303, 2017. * [3] A. Booker, S. D. Cohen, N. Sutherland, and T. Trudgian. Primitive values of quadratic polynomials in a finite field. Math. Comp., 88(318):1903–1912, 2019. * [4] W. S. Chou and S. D. Cohen. Primitive elements with zero traces. Finite Fields Appl., 7(1):125–141, 2001. * [5] S. D. Cohen. 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# Mathematical foundations of moral preferences Valerio Capraro<EMAIL_ADDRESS>Department of Economics, Middlesex University, The Burroughs, London NW4 4BT, U.K. Matjaž Perc <EMAIL_ADDRESS>Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan Alma Mater Europaea ECM, Slovenska ulica 17, 2000 Maribor, Slovenia Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria ###### Abstract One-shot anonymous unselfishness in economic games is commonly explained by social preferences, which assume that people care about the monetary payoffs of others. However, during the last ten years, research has shown that different types of unselfish behaviour, including cooperation, altruism, truth-telling, altruistic punishment, and trustworthiness are in fact better explained by preferences for following one’s own personal norms – internal standards about what is right or wrong in a given situation. Beyond better organising various forms of unselfish behaviour, this moral preference hypothesis has recently also been used to increase charitable donations, simply by means of interventions that make the morality of an action salient. Here we review experimental and theoretical work dedicated to this rapidly growing field of research, and in doing so we outline mathematical foundations for moral preferences that can be used in future models to better understand selfless human actions and to adjust policies accordingly. These foundations can also be used by artificial intelligence to better navigate the complex landscape of human morality. ## I Introduction Most people are not completely selfish. Given the right circumstances, they are happy to give up a part of their benefit to help other people or the society as a whole. Psychologists and economists have long observed that some people act unselfishly even in one-shot anonymous interactions, when there are no direct or indirect benefits for doing so rapoport1965prisoner ; engel_ee11 . The question is why? Understanding what motivates people to act unselfishly in one-shot, anonymous interactions is of great theoretical and practical importance. Theoretically, it may lead to a more complete and precise mathematical framework to formalise human decision-making, while practically, it may suggest policies and interventions to promote unselfish behaviour, with the ultimate goal of improving our societies. To study one-shot unselfishness, behavioural scientists usually turn to laboratory experiments using economic games, in which experimental subjects have to make monetary decisions that involve various forms of other-regarding behaviour. In this context, and throughout this review, selfishness and other- regarding behaviour is defined with respect to monetary payoffs. Clearly, a behaviour that is unselfish from the point of view of monetary outcomes may turn out to be selfish from a more general perspective that takes into account also psychological benefits and costs. For example, some people may engage in unselfish behaviour to decrease negative mood cialdini1973transgression or increase positive feelings andreoni1990impure . Therefore, in the last decades, behavioural scientists have been trying to mathematically explain unselfish behaviour by means of a utility function that depends on factors other than solely the monetary payoff of the decision maker. Based on empirical data scholars have initially advanced the explanation that human unselfishness in one-shot anonymous interactions is primarily driven by people not caring only about their own monetary payoff, but caring, at least to some extent, also about the monetary payoffs of the other people involved in the interaction ledyard1994public ; levine1998modeling ; fehr1999theory ; bolton2000erc ; andreoni2002giving ; charness2002understanding . However, about fifteen years ago, this _social preference hypothesis_ came under critique because some experiments showed that two particular forms of unselfish behaviour, altruistic punishment and altruism (see Table 1 for these definitions), could not be entirely explained by preferences defined solely over monetary outcomes. In 2010, building on work on the effect of social norms on people’s behaviour smith2010theory ; durkheim2017regles ; parsons1937structure ; geertz1973interpretation ; schwartz1977normative ; elster1989social ; cialdini1990focus ; bicchieri2005grammar ; bicchieri2016norms ; hawkins2019emergence , Bicchieri and Chavez bicchieri2010behaving proposed to explain altruistic punishment assuming that people have preferences for following their “personal norms” (what they personally believe to be the right thing to do) beyond the monetary consequences that this action brings about. Subsequently, Krupka and Weber krupka2013identifying proposed to explain altruism using “injunctive norms” (what one believes others would approve/disapprove); however, in their analysis, they did not consider a potential role of personal norms. In the last five years, numerous other experiments challenged social preference models in several behavioural domains, other than altruistic punishment and altruism biziou2015does ; schram2015inducing ; kimbrough2016norms ; eriksson2017costly ; capraro2018right ; tappin2018doing ; capraro2019power ; huang2019choosing ; moreover, the best interpretation of these results turns out to be in terms of personal norms, rather than other types of norms. Namely, the best way to organise these results is through the moral preference hypothesis, according to which people have preferences for following their personal norms, beyond the economic consequences that these actions bring about. This framework organises several forms of one-shot, anonymous unselfish behaviour, including cooperation, altruism, altruistic punishment, trustworthiness, honesty, and the equality-efficiency trade-off. We note at this stage that personal norms are not universally given. They certainly depend on the culture; for example, they can come from the internalisation of cultural values schwartz1977normative . But they can also depend on the individual; anecdotal evidence suggests that, even within the same family, there might be people with different beliefs about what is right or wrong in a given situations. We will discuss this in more details in Section VII.6. The moral preference hypothesis also holds promise of being very useful in practice. The idea is simple. If people care about doing the right thing, then just providing cues that make the rightness of an action salient should work just fine in promoting desirable behaviour. In fact, research has already demonstrated the applicability of this approach outside of the laboratory, showing in particular that nudges towards doing the right thing can increase charitable donations capraro2019increasing . In the light of ample empirical research supporting the moral preference hypothesis, theoretical research aiming to formalise human decision-making by means of a mathematical framework is also at a crossroads. On the one hand, the traditional approach involving monetary payoffs has worked well in explaining many challenging aspects of pro-social behaviour. But on the other, experiments indicate that there are likely hard boundaries to this simplistic approach, which will thus have to be amended by more avant-garde concepts, including formalising the intangibles of moral psychology and philosophy. Here we review this rapidly growing field of research within the following sections. Section II reviews the main economic games that have been developed to study one-shot unselfishness. Section III reviews social preference models, as earlier attempts to explain unselfishness in one-shot economic games within a unified theoretical framework. This section also describes a number of experiments that violate social preference models. Section IV shows how these experiments can be organised by general moral preferences for doing what one believes to be the right thing. Section V focuses on practical applications of the moral preference hypothesis. Section VI reviews the models of moral preferences that have been introduced so far and proposes a new model that explicitly takes into account the importance of personal norms. Lastly, Section VII outlines a number of key questions for future work, while Section VIII summarises the main conclusions. Taken together, this review thus outlines a mathematical formalism for morality, which shall inform future models aimed at better understanding selfless actions as well as artificial intelligence that strives to emulate counterintuitive human decision-making. ## II Measures of unselfish behaviour There are various forms of unselfish behaviour. For example, giving money to a homeless person on the street is, in principle, quite different from collaborating with a colleague on a common project, or from telling the truth when one is tempted to lie. To take this source of heterogeneity into account, scholars have developed a series of simple games and decision problems that are meant to prototypically represent different types of unselfish behaviour. These are simple scenarios in which experimental subjects can make decisions that have real consequences. To incentivise these decisions, behavioural scientists usually use monetary payoffs (at least among adult subjects, whereas other forms of remuneration, such as stickers, might be more effective among children). In this review, we will be mainly focused on one-shot decisions that are _purely_ unselfish, meaning that they bring no monetary benefit to the decision maker (and possibly bring a cost), no matter the beliefs of the decision maker regarding the behaviour of other people involved in the interaction. Specifically, we measure altruistic behaviour using the dictator game (see Table 1 for all the definitions), cooperative behaviour in pairwise interactions using the prisoner’s dilemma, truth-telling using the sender- receiver game, the tradeoff between equality and efficiency using the trade- off game, trustworthiness using player 2 in the trust game, and altruistic punishment using player 2 in the ultimatum game. In the last section we will also briefly consider decisions that are _strategically_ unselfish, such as trust (player 1 in the trust game) and strategic fairness (player 1 in the ultimatum game), which might actually maximise the payoff of the decision maker, depending on their beliefs about the behaviour of the second player. The distinction between pure unselfishness and strategic unselfishness generalises the distinction between pure cooperation and strategic cooperation, introduced by Rand in his meta-analysis rand2016cooperation , where “pure cooperation” was defined as paying a cost to benefit another person, regardless of the behaviour of the other person, as opposite of “strategic cooperation”, which might maximise the cooperator’s payoff, depending on the other person’s behaviour. ## III Social preferences and their limitations Behavioural scientists have long recognised that some people do act unselfishly even in one-shot anonymous interactions. For example, the first comprehensive empirical work on the one-shot prisoner’s dilemma dates back to 1965 rapoport1965prisoner . Formal frameworks to explain one-shot unselfishness have a more recent history, starting in 1994, when Ledyard observed that cooperation, altruism, and altruistic punishment could be explained by assuming that people maximise a utility function that depends not only on their own monetary payoff, but also on the total monetary payoff of the other people that are involved in the interaction ledyard1994public . See Table 2 for the exact mathematical definition. Since then, several models have been introduced. In 1998, Levine levine1998modeling proposed a utility function in which the level of altruism depends on the level of altruism of the other players. Subsequently, in 1999, Fehr and Schmidt fehr1999theory proposed a framework according to which players care about minimising inequities. In 2000, Bolton and Ockenfels bolton2000erc followed a similar idea and introduced a general inequity aversion model, in which the utility of an action depends negatively on the distance between the amount of money the decision maker gets if that action is implemented and the amount of money the decision maker would get if the equal allocation were implemented. The authors proposed an explicit mathematical formula only for the case of $n=2$ players. In 2002, Andreoni and Miller andreoni2002giving estimated the behaviour of experimental subjects in a number of dictator game choices using a specific utility function taking into account altruistic tendencies as well as potential convexity in the preferences. In the same year, Charness and Rabin charness2002understanding introduced a general utility function which, depending on the relative relationship between its two parameters, can cover several cases, including competitive preferences, inequity aversion preferences, and social efficiency preferences. We refer to Table 2 for the exact functional forms. (Besides these models, scholars have also studied models that can be applied to specific subsets of one-shot anonymous interactions (e.g., andreoni1990impure ). In this review, we focus on models that can be applied to any one-shot anonymous interaction involving unselfish behaviour). While differing in many details, all social preference models share one common property: they assume that the utility of a decision maker is a function of the monetary payoffs of the available actions. This assumption came under considerable criticism for the first time in 2003 when Falk, Fehr and Fischbacher falk2003nature showed that rejection rates in the ultimatum game depend on the choice set available to the proposer. Specifically, the split (8,2) — 8 to the proposer and 2 to the responder — is more likely to be accepted in ultimatum games in which the only other choice available to the proposer is (10,0), compared to ultimatum games in which the only other choice available to the proposer is (5,5). Therefore, responders prefer accepting (8,2) over rejecting it in the former case, but they prefer rejecting (8,2) over accepting it in the latter one, despite the fact that these choices have the same monetary consequences in the two cases. Clearly, this cannot be explained by any model of social preferences. See bicchieri2010behaving ; eriksson2017costly for conceptual replications. Shortly after, in 2005, Uri Gneezy introduced the sender-receiver game gneezy2005deception . In his experiments, decision makers were less likely to implement an allocation of money when implementing this allocation also required misreporting a private information. Also this finding cannot be explained by any model of social preferences and, more generally, also not by any utility function that depends only on the monetary payoffs that are associated with the available actions. This thus indicates that (some) people have an intrinsic cost of lying, which goes beyond their preferences toward monetary outcomes. To further support this interpretation, several scholars have independently studied the sender-receiver game in contexts in which lying would benefit both the sender and the receiver to the same extent. This case is particularly important because, when the benefit for the sender is equal to the benefit for the receiver, all social preference models predict that the totality of people would lie. However, this prediction turned out to be violated in experiments, which showed that a significant proportion of people tell the truth cappelen2013we ; erat2012white ; biziou2015does . Subsequently social preference models came under critique also in one of the behavioural domains in which they had been most successful, namely in research involving the dictator game. Dana, Cain and Dawes dana2006you and Lazear, Malmendier and Weber lazear2012sorting observed that some dictator game givers would prefer to altogether avoid the dictator game interaction if given the chance. These people thus preferred giving over keeping in a context in which they were forced to play the dictator game, but preferred keeping over giving in a context in which they could choose whether to play the dictator game or not. This finding, as in the preceding examples, cannot be explained by any utility function that is based solely on monetary outcomes. For the same game, and along similar lines, List list2007interpretation , Bardsley bardsley2008dictator , and Cappelen et al. cappelen2013give found that extending the choice set of the dictator by adding the possibility to take money from the recipient has the effect to make some dictators less likely to give. Therefore, these dictators preferred giving over keeping, when the taking option was not available, but preferred keeping over giving, when the taking option was available. This finding likewise cannot be explained by any preference over monetary payoffs. A conceptually similar point was also made by Krupka and Weber krupka2013identifying and Capraro and Vanzo capraro2019power , who found that even minor changes in the instructions of the dictator game can notably impact people’s behaviour. In the years after 2013, the inability of purely monetary-based models to explain empirically observed behaviour engulfed many other games and decision problems, whose experimental regularities had been previously thought to be explainable in terms of social preferences. Examples included the prisoner’s dilemma kimbrough2016norms ; capraro2018right , the trust game kimbrough2016norms , as well as different variants of the trade-off game capraro2018right ; tappin2018doing ; huang2019choosing , thus resulting in a crisis of the social preference hypothesis. ## IV The rise of the moral preference hypothesis To solve a crisis, one needs a paradigm shift. The shift started in 2010, when Bicchieri and Chavez bicchieri2010behaving proposed an elegant solution for one of the aforementioned empirical observations. This solution builds on classic work suggesting that, in everyday life, people’s behaviour is partly determined by what they believe to be the norms in a given context smith2010theory ; durkheim2017regles ; parsons1937structure ; geertz1973interpretation ; schwartz1977normative ; elster1989social ; cialdini1990focus ; bicchieri2005grammar ; bicchieri2016norms ; hawkins2019emergence . This observation led behavioural scientists to propose several classifications of norms. Particularly relevant for the thesis of this review is the distinction between personal and social norms schwartz1977normative . And moreover, among the social norms, the distinction between injunctive and descriptive norms cialdini1990focus . Personal norms refer to internal standards about what is right or wrong in a given situation; injunctive norms refer to what other people approve or disapprove of in that situation; descriptive norms refer to what other people actually do. In one- shot anonymous games, like the games considered in this review, the distinction among personal, descriptive, and injunctive norms roughly corresponds to Bicchieri’s personal normative beliefs, empirical expectations, and normative expectations bicchieri2005grammar . See Table 3 for precise definitions. The groundbreaking intuition of Bicchieri and Chavez bicchieri2010behaving was to apply the theory of norms to deviations from monetary-based social preferences in the ultimatum game. Specifically, Bicchieri and Chavez showed that the ultimatum game offer that is consider to be fair by responders depends on the choice set available to the proposer; moreover, responders tend to reject offers that they consider unfair. This suggests that altruistic punishment is driven by responders following their personal norms, beyond the monetary consequences that these actions bring about. In particular, this explains the aforementioned results of Falk, Fehr, and Fischbacher falk2003nature , that responders reject the same offer at different rates depending on the other offers available to the proposer. Shortly after, in 2013, Krupka and Weber krupka2013identifying applied a similar approach to several variants of the dictator game. However, instead of focusing on personal norms, they focused on injunctive norms. For each of the available actions, subjects were asked to declare whether they found the corresponding action to be “very socially inappropriate”, “somewhat socially inappropriate”, “somewhat socially appropriate”, or “very socially appropriate”. Subjects were given a monetary prize if they matched the modal choice made by other participants. Observe that, in this way, Krupka and Weber incentivised the elicitation of the injunctive norm. (The elicitation of personal norms cannot be incentivised.) In doing so, Krupka and Weber found that people believe that others think that avoiding a dictator game interaction is far less socially inappropriate than keeping the whole amount of money in a dictator game that one is obliged to play. Therefore, the empirical results summarised above regarding dictator games with an exit option dana2006you ; lazear2012sorting can be explained simply by a change in the perception of what is the injunctive norm in that context. Similarly, Krupka and Weber found that people believe that others think that keeping the money in a dictator game with a taking option is far less socially inappropriate than keeping the money in the dictator game without the taking option. In this way, they could explain also the results of List list2007interpretation , Bardsley bardsley2008dictator , and Cappelen et al. cappelen2013give in terms of a change in the perception of the injunctive norm. Finally, Krupka and Weber presented a novel experiment in which subjects played the dictator game in either of two variants: in the Standard variant, dictators started with $10 and had to decide how much of it, if any, to give to the recipient; in the Bully variant, the money was initially split equally among the dictator and the recipient, and the dictator could either give, take, or do nothing. The authors found that people were more altruistic in the Bully variant compared to the Standard variant, and this was driven by the fact that people rated “taking from the recipient” far less socially appropriate than “not giving to the recipient”. The work of Krupka and Weber suggests that taking into account injunctive norms is important to explain deviations from social preference models in the dictator game. But are the injunctive norms really the main force behind the observed behavioural changes, or are there also other norms playing more primary roles? In the last five years, a set of empirical studies tried to address this question. Schram and Charness schram2015inducing analysed the behaviour of dictators who were given an advice from third parties about the injunctive norm. They observed that dictators became more pro-social only when their choices were made public. By contrast, when their choices remained private, they found no significant increase in pro-sociality, compared to the case in which they did not receive any information about the injunctive norm. These results indicate that, although injunctive norms might correlationally explain behavioural changes in anonymous (and thus private) dictator game experiments, they are unlikely to be the primary motivation. In fact, being that these games were played anonymously, in front of the screen of a computer, the intuition suggests that the norms primarily at play are the personal norms. Two recent works provide evidence for this hypothesis. Capraro and Vanzo capraro2019power found that framing effects in the dictator game generated by morally loaded instructions can be explained by changes in the perception of what people “personally think to be the right thing” in the given context (i.e., their personal norms). Capraro et al. capraro2019increasing showed that making personal norms salient prior to playing the dictator game (by asking subjects to state what they personally think to be the morally right thing to do) has a strong effect on subsequent dictator game donations, even persisting to a second-stage prisoner’s dilemma interaction. This set of works thus suggests that dictator game giving is driven by personal norms. Putting this together with the results of Bicchieri and Chavez, we obtain that both altruism and altruistic punishment can be explained by people following their personal norms. More recently, this finding has been not only replicated, but, more importantly, also extended to explain several other forms of unselfish behaviour. In 2016, Kimbrough and Vostroknutov kimbrough2016norms introduced a task “that measures subjects’ preferences for following rules and norms, in a context that has nothing to do with social interaction or distributional concerns”. They found that this measure of norm-sensitivity predicts dictator game altruism, trust game trustworthiness (but not trust), and ultimatum game rejection thresholds (but not offers). Taken together, this indicates that altruism, trustworthiness, and altruistic punishment are driven by a common desire to adhere to a personal norm. In 2017, Eriksson et al. eriksson2017costly conducted an ultimatum game experiment under two different conditions. The difference, however, was only in the labels that were used to describe the action of refusing the proposer’s offer. In one treatment, this action was labeled “rejecting the proposer’s offer”, while in the other treatment, the same action was labeled “reducing the proposer’s payoff”. Since these two options are monetarily equivalent, any utility function depending only on the monetary payoffs of the available actions predict that responders should behave the same way in both cases. But contrary to this prediction, Eriksson et al. found that responders displayed higher rejection thresholds in the “rejection frame” than in the “reduction frame”. Moreover, they showed that the observed framing effect could be explained by a change in what people think to be the right thing to do. Specifically, subjects tended to rate the action of reducing the proposer’s offer to be morally worse than the action of rejecting the proposer’s offer, in spite of the fact that these two actions had the same monetary consequences. In 2018, Capraro and Rand capraro2018right showed that behaviour in the trade-off game is highly sensitive to the labels used to describe the available actions. In line with Eriksson et al. eriksson2017costly , Capraro and Rand also found that their framing effects could be explained by a change in what people think to be the right thing to do. Notably, framing effects in the trade-off game have been replicated several times tappin2018doing ; huang2019choosing ; capraro2019preferences ; capraro2020gender ; capraro2020does and a recent work has shown that these moral framings tap into relatively internalised moral preferences capraro2020does . Moreover, Capraro and Rand also considered a situation in which the personal norm conflicted with the descriptive norm, and found that people tend to follow the personal norm, rather than the descriptive norm. The same research also revealed a correlation between the framing effect in the trade-off game and giving in the dictator game and cooperation in the prisoner’s dilemma, thus indicating that not only trade-off decisions are driven by personal norms, but that altruism and cooperation are also subject to that same facilitator. Cooperative behaviour is also typically correlated to altruistic behaviour capraro2014heuristics ; peysakhovich2014humans ; reigstad2017extending , suggesting that they are driven by a common underlying motivation. To the best of our knowledge, there are no works directly exploring the role of personal norms on truth-telling in the sender-receiver game. However, Biziou-van-Pol et al. biziou2015does have shown that there is a positive correlation between truth-telling in the sender-receiver game (in the Pareto white lie condition), giving in the dictator game, and cooperation in the prisoner’s dilemma, suggesting that these types of behaviours are driven by a common motivation. Since the aforementioned research suggests that altruism and cooperation are driven by personal norms, this correlation suggests that lying aversion is so too. In sum, research accumulated in the last ten years suggests that several forms of one-shot, anonymous unselfishness, including altruism, altruistic punishment, truth-telling, cooperation, trustworthiness, and the equality- efficiency trade-off, can be explained using a unified theoretical framework, whereby people have moral preferences for following their personal norms, beyond the monetary payoff that these actions bring about. Of course, this is not meant to imply that monetary payoffs do not play any role in explaining one-shot unselfishness, but simply that something else, in addition to monetary payoffs, should be taken into account. The thesis is that this ‘something else’ are the personal norms, which gives rise to the moral preference hypothesis as described in Table 4. Also, this is not meant to imply that other types of norms play no role in these forms of one-shot selfless behaviour. For example, nudging the injunctive norm in the prisoner’s dilemma capraro2019increasing and in the trade-off game human2020effect has a similar effect as nudging the personal norm. Moreover, it is possible that social norms ultimately drive personal norms, because they allow to enhance or preserve one’s sense of self-worth and avoid self-concept distress, resulting in a self-reinforcing behaviour that eventually benefits one’s own self-image schwartz1977normative . However, the aforementioned literature suggests that, at a proximate level, personal norms have a greater explanatory power, in the sense that they consistently explain people’s behaviour also in games where injunctive norms have been shown to play a limited role (e.g., dictator game) or where descriptive norms play a limited role (e.g., the trade-off game). ## V Practical applications Behavioural scientists and policy makers have been using norm-based interventions to foster pro-sociality in real life for decades (bicchieri2009right, ; krupka2009focusing, ; zafar2011experimental, ; raihani2014dictator, ; d2017push, ; frey2004social, ; croson2010gendered, ; cialdini1991focus, ; ferraro2013using, ; agerstrom2016using, ; goldstein2008room, ; hallsworth2017behavioralist, ). Although these paternalistic interventions have been criticised because they subtly violate people’s freedom of choice hausman2010debate and can be exploited by malicious institutions glaeser2005paternalism (see sunstein2014nudge for a response to these critiques), they are well-studied because, compared to standard procedures to foster pro-sociality (punishment and rewards), they allow to save the monitoring cost that the institution needs to pay in order to know who to punish or reward. Norm-based interventions typically manipulate the descriptive or the injunctive norm in a given context, and show that this has an effect on people’s behaviour in that same context. The more recent works reviewed in the previous section, showing that unselfish behaviour in one-shot, anonymous economic games is primarily driven by a desire to follow the personal norms, suggest that a more effective mechanism to increase pro-sociality might be to use norm-based interventions that target personal norms, rather than social norms. The interest in targeting personal norms, compared to other mechanisms to promote pro-sociality, is also that targeting personal norms is potentially cheaper than other mechanisms. Clearly, it is cheaper than punishment and rewards because it avoids the monitoring cost. Additionally, it saves the cost of collecting information about the behaviour or the moral judgments of other people, which forms the basis of interventions targeting social norms. In recent years, there has been a growing body of research exploring the effect of nudging personal norms on various forms of unselfish behaviour. Some works using economic games found that making personal norms salient increases donations in the dictator game branas2007promoting ; capraro2019increasing , cooperation in the prisoner’s dilemma dal2014right ; capraro2019increasing , as well as decreases in-group favouritism, at least on average bilancini2019right . This suggests that nudging personal norms might be effective to increase pro-sociality in one-shot anonymous decisions that have consequences outside the laboratory. Along these lines, Capraro et al. capraro2019increasing found that asking people to report what they personally think is the morally right thing to do increases crowdsourced charitable donations by 44%. ## VI Models of moral preferences We have thus seen that several forms of unselfish behaviour can be organised by moral preferences for following the personal norms. The question is, can we model this using a formal utility function? There have been some attempts to formalise people’s tendency to follow a norm benabou2006incentives ; levitt2007laboratory ; lopez2008aversion ; andreoni2009social ; dellavigna2012testing ; kessler2012norms ; alger2013homo ; krupka2013identifying ; kimbrough2016norms ; kimbrough2020injunctive ; kimbrough2020theory . Most of these models, however, are either very specific in the sense that they can be applied only to certain games, or do not distinguish among different types of norms. Three models can be applied to every game of interest in this review (and, more generally, to every one-shot game) and distinguish among different types of norms. Levitt and List levitt2007laboratory introduced a model where the utility of an action $a$ depends on the monetary payoff associated to that action, $v_{i}(\pi_{i}(a))$, as well as on the moral cost (or benefit), $m(a)$, associated to that action: $u_{i}(a)=v_{i}(\pi_{i}(a))+m(a).$ Levitt and List assumed that the moral cost (or benefit) depends primarily on three factors: whether the action is recorded or performed in the presence of an observer, whether the action has negative consequences on other players, and whether the action is in line with “social norms or legal rules that govern behavior in a particular society”. Therefore, Levitt and List’s model, although useful in many circumstances, does only mention social norms, while ignoring the effect of personal norms. A similar model was considered by Krupka and Weber krupka2013identifying , with the key difference that they focused on injunctive norms specifically. Krupka and Weber introduced a function $N$ defined over the set of available actions that, given an action $a$, returns a number $N(a)$ representing the extent to which society views $a$ as socially appropriate. They also assumed that people are heterogeneous in the extent to which they care about doing what society considers to be appropriate. In doing so, they obtain the utility function: $u_{i}(a)=v_{i}(\pi_{i}(a))+\gamma_{i}N(a).$ As mentioned above, one of the main contributions of Krupka and Weber was to introduce an experimental technique to elicit the injunctive norm. To this end, they asked participants to rate each of the available actions in terms of their social appropriateness. Participants were incentivised to match the modal choice of the other participants. Very recently, in 2020, Kimbrough and Vostroknutov presented a different approach, but still based on injunctive norms kimbrough2020theory . Specifically, they introduced the utility function $u_{i}(a)=v_{i}(\pi_{i}(a))+\phi_{i}\eta(a),$ where $\phi_{i}$ represents the extent to which $i$ cares about following the injunctive norm, and $\eta(a)$ represents a measure of whether the society thinks that $a$ is socially appropriate. Although this utility function looks very similar to the one proposed by Krupka and Weber, it differs from it in one important dimension. While Krupka and Weber’s social appropriateness, $N(a)$, is computed by asking participants what they think others would approve or disapprove (and therefore it need not depend only on the monetary consequences of the available actions), Kimbrough and Vostroknutov’s injunctive norm, $\eta$, is built axiomatically from the game and it is assumed to be inversely proportional to the overall dissatisfaction of the players, defined as the difference between what they get in a given scenario and what they could have gotten in others. This implies that one limitation of this approach is that people always prefer Pareto dominant allocations over Pareto dominated ones. But, in experiments, this property is not always satisfied. For example, when lying is Pareto dominant, some people still tell the truth, and these people tend to cooperate in a subsequent prisoner’s dilemma and give in a subsequent dictator game biziou2015does . Moreover, in trade-off games framed in such a way that the Pareto dominant allocation is presented as morally wrong, people tend to choose the Pareto dominated option capraro2018right ; tappin2018doing . In sum, previous formal models consider only social norms or, more specifically, injunctive norms. But, as we have seen in the previous sections, unselfish behaviour in one-shot anonymous interactions is often driven by personal norms, rather than by social norms. Taking inspiration from the above models, one can formalise this using the utility function: $u_{i}(a)=v_{i}(\pi_{i}(a))+\mu_{i}P_{i}(a),$ where $\mu_{i}$ represents the extent to which player $i$ cares about doing what s/he personally thinks to be the morally right thing to do and $P_{i}(a)$ represents the extent to which $i$ personally thinks that $a$ is morally right. This functional form might superficially seem similar to the ones discussed earlier, but it differs from those in two important points. One point is that the personal norm $P_{i}(a)$ typically depends on the individual $i$, whereas the injunctive norm depends on the society and the culture in which the individual is embedded. The second point is the very fact that $P_{i}$ represents the extent to which $i$ thinks that $a$ is the morally right thing to do, whereas $m(a),N(a)$, and $\eta(a)$ represent social norms. In general, the personal norm might not be aligned with the social norms. In practice, $P_{i}(a)$ can be estimated using a suitable experiment, whereas $\mu_{i}$ and $v_{i}$ can be estimated, on average, using statistical techniques, following a similar method as the one developed by Krupka and Weber for injunctive norms krupka2013identifying . Specifically, one can estimate $P_{i}(a)$ by asking subjects to self-report the extent to which they personally think that action $a$ is the morally right thing to do. Then one can use these ratings to predict the behaviour, using a simple regression. The coefficient of this regression will give the average of the $\mu_{i}$’s. Also, putting the monetary payoffs in the regression, one can also get an estimation for the average of the $v_{i}$’s. This utility function based on personal norms has a greater predictive power than its counterparts based only on social norms, in the sense that it explains behaviour in a larger set of games, compared to their counterparts based on social norms. We have seen earlier that Schram and Charness schram2015inducing found that making the injunctive norm salient does _not_ increases altruistic behaviour in the anonymous dictator game. D’Adda et al. d2017push found that making the descriptive norm salient has only a marginally significant effect on anonymous dictator game giving; this effect also vanishes in a second interaction, played immediately after. Along the same lines, Dimant, van Kleef and Shalvi dimant2019requiem found that promoting the injunctive norm and promoting the descriptive norm does _not_ have any effect on people’s honesty in a deception game in which subjects can lie for their benefit. On the other hand, numerous works have shown that nudging personal norms impacts several forms of unselfish behaviour, ranging from altruism branas2007promoting ; capraro2019increasing , altruistic punishment eriksson2017costly , cooperation dal2014right ; capraro2019increasing , and the equality-efficiency trade-off capraro2018right . Moreover, the effect typically persists for at least another interaction and even spills across contexts capraro2019increasing . All these results are consistent with a utility function based on personal norms and are not consistent with a utility function based only on social norms. We present a summary of all above-discussed moral preference models in Table 5. ## VII Future work This is an exciting field of research, which provides a unified view of human choices in several contexts of decision-making, while having, at the same time, significant practical implications. Nonetheless, there are several questions that need to be explored in future research, as detailed in what follows and summarised in Table 6. ### VII.1 The utility function From a mechanistic perspective, the moral preference hypothesis raises the question of how can we express the utility function of a decision maker. Scholars have tried to give mathematical sense to people’s morality since the foundation of mathematical economics jevons1879theory ; bentham1996collected . About two centuries later, the question is still open, even in the simple setting of one-shot anonymous interactions. One simple way to do so is to assume that people are torn between maximising their monetary payoff and doing what they personally think to be the morally right thing. This can be done with a utility function of the shape $u_{i}(a)=v_{i}(\pi_{i}(a))+\mu_{i}P_{i}(a)$. Although this utility function outperforms their counterparts based on social norms, as well as social preferences, it undoubtedly represents just a first candidate. Future work should explore other ways to formalise moral preferences, through finer experiments with the power to detect small variations in how people weight their personal norm against monetary incentives. Future work should also find ways to estimate what people think to be the right thing in a given context, without asking it to the participants in a separate experiment. The literature reviewed above shows that, in many cases, it is enough to change only one word in the instructions of a decision problem to change people’s perception of what is the right thing to do in a given context. This suggests that $P_{i}(a)$ partly depends on the language in which the action $a$ is presented. Exploring this dependence can greatly improve the predictive power of the utility function. How can one do so? Recent work shows that emotional content in messages increases their diffusion in social media brady2017emotion ; brady2019ideological ; brady2019mad . Translating this finding in the context of one-shot games, it suggests that the emotions carried by the instructions of the decision problem might contribute to the computation of $P_{i}$. Along these lines, it is possible that one can use sentiment analysis to better estimate $P_{i}$. Sentiment analysis is a technique developed by computational linguists that allows to assign a polarity to any given piece of text pang2002thumbs . In principle, this polarity could enter the utility function of a decision maker and work as an additional motivation or obstacle for choosing an action, beyond its monetary consequences. In any case, mathematically describing or at least quantifying the seemingly intangible moral preferences, and in doing so building bridges between computational linguistics, behavioural economics, and moral psychology, is a fascinating direction for future work. ### VII.2 Evolution of norms Where do personal norms come from? One explanation is that they come from the internalisation of behaviours that, although not individually optimal in the short term, are optimal in the long run. It is therefore important to understand which unselfish behaviours can be selected in the long term, and under which conditions. A promising line of research uses evolutionary game theory and statistical physics to find the conditions that promote the evolution of cooperation on networks perc_pr17 . More recently, scholars have started applying similar techniques also to study the evolution of other forms of unselfish behaviour capraro_fp18 , such as truth-telling in the sender- receiver game capraro2019evolution ; capraro_pre20 and trustworthiness in the trust game kumar2020evolution . Some works along this line have also looked at the evolution of choices in the ultimatum game page_prsb00 ; killingback_prsb01 ; iranzo_pone12 ; szolnoki_prl12 . Future work should extend the same techniques to other forms of unselfish behaviour. ### VII.3 Personal norms versus social norms The experimental literature reviewed in the previous sections suggests that several forms of one-shot, anonymous unselfishness can be unified under a framework according to which people have preferences for following their personal norms. Moreover, preliminary evidence suggests that nudging personal norms can be an effective tool for fostering pro-sociality: making personal norms salient affects altruism, cooperation, altruistic punishment, and trade- off decisions between equality and efficiency branas2007promoting ; capraro2019increasing ; eriksson2017costly ; dal2014right . This, of course, does not mean that the social norms play no role at all. For example, nudging injunctive norms has a significant effect on the one-shot, anonymous, prisoner’s dilemma capraro2019increasing and the trade-off game human2020effect . One question that is still open, however, is whether these effects are fundamentally distinct from the effect of nudging personal norms. It is indeed possible that nudging injunctive norms in these games also nudge personal norms, and this is what makes people change their behaviour. A working paper suggests that people who follow injunctive norms in the trade- off game are different from those who follow personal norms human2020effect . Therefore, it is possible that a larger model taking into account both personal and injunctive norms might have an even greater predictive power, at least in some contexts, than a model based exclusively on personal norms. Further experiments comparing the effect of nudging different norms are needed to clarify this point. The evidence in this case is indeed still lacunar. One study compared the relative effect of the descriptive and the injunctive norms in the dictator game, and found that people tend to follow the descriptive norm bicchieri2009right . Another study compared the relative effect of nudging personal norms and the descriptive norms in the trade-off game, and found that people tend to follow the personal norms capraro2018right . The aforementioned working paper compared the effect of nudging the personal and the injunctive norm in the trade-off game and found that they have a similar effect; moreover, when the two norms are in conflict, some people follow the personal norm and other follow the injunctive norm human2020effect . This suggests that people’s behaviour depends on their focus of attention within an interconnected matrix of norms. Therefore, future work should explore norm salience, also in cases where more than one type of norm is simultaneously made salient. Research should also go beyond anonymous decisions and investigate what happens when choices are observable. The intuition suggests that when choices are observable, social norms may play a bigger role compared to when they remain private; in line with this intuition, Schram and Charness schram2015inducing showed that nudging the injunctive norms impacts public but not private dictator game giving. However, no studies compared the relative effectiveness of targeting different norms in public decisions. ### VII.4 Boundary conditions of interventions based on personal norms Having in mind potential practical applications, another important question concerns the boundary conditions of interventions based on personal norms. From a temporal perspective, previous research suggests that interventions targeting personal norms can last for several interactions within the same experiment dal2014right ; capraro2019increasing . However, it seems unrealistic to expect that their effect will last indefinitely. For example, a recent field experiment targeting injunctive norms found an effect that diminishes after repeated interventions, although it can be restored after waiting a sufficient amount of time between interventions ito2018moral . From the decisional context point of view, there will certainly be behavioural domains in which targeting personal norms might not be as effective. For example, a recent work suggests that risky cooperation in the stag-hunt game is primarily driven by preferences for efficiency, rather than by preferences for following personal norms capraro2019preferences . ### VII.5 External validity of interventions based on personal norms Given the potential relevance of this line of work for the society at large, future studies should explore the external validity of interventions based on personal norms. At the time of this writing, only one study investigated the effect of nudging personal norms in contexts in which decisions have consequences outside the laboratory. This study found that nudging personal norms increases crowdsourced charitable donations to real humanitarian organisations by 44% capraro2019increasing . ### VII.6 The moral phenotype and its topology We have seen that different forms of unselfish behaviour can be explained by a general tendency to do the right thing. We are tempted to call this tendency “moral phenotype”, extending the notion of “cooperative phenotype” introduced by Peysakhovich, Nowak, and Rand peysakhovich2014humans . See also reigstad2017extending . In their work, Peysakhovich and colleagues observed that pro-social behaviours in the dictator game, the public goods game (a variant of the prisoner’s dilemma with more than two players), and the trust game (both players) were all correlated; and they termed this general pro- social tendency cooperative phenotype. Therefore, the cooperative phenotype is uni-dimensional. On the other hand, the moral phenotype is likely to be multi- dimensional. For example, we have seen earlier that both altruistic punishment and altruistic giving are driven by preferences for doing the right thing. However, Peysakhovich, Nowak, and Rand peysakhovich2014humans found that they are not correlated. It is possible that they are not correlated because they come from different personal norms. The multi-dimensionality of morality is not a new idea, and several authors have come to suggest it in the last decades from different routes. For example, Haidt and colleagues argue that differences in people’s moral concerns can be explained by individual differences across six “foundations” haidt2004intuitive ; graham2009liberals ; haidt2012righteous . Kahane, Everett and colleagues have shown that (act) utilitarianism decomposes itself in at least two dimensions kahane2018beyond ; everett2020switching . Curry, Mullins, and Whitehouse curry2019good have reported that seven moral rules are universal across societies, but societies vary on how they rank them. However, we are not aware of any work exploring how different personal norms link to different forms of one-shot unselfishness. Another topological property of the moral phenotype that deserves further scrutiny is the boundary. Does, for example, the moral phenotype include decisions that are strategically unselfish, such as strategic fairness (ultimatum game offers) and trust (trust game transfers), both of which maximise the decision maker’s payoff depending on the decision maker’s beliefs about the behaviour of the other player? Previous evidence is limited and mixed. Bicchieri and Chavez bicchieri2010behaving showed that ultimatum game offers are partly driven by normative beliefs; Peysakhovich, Nowak, and Rand peysakhovich2014humans found that trustees’ decisions correlate with dictator game and public goods game decisions. By contrast, Kimbrough and Vostroknutov kimbrough2016norms found that trustees’ and proposers’ decisions are not correlated to their measure of norm-sensitivity. ### VII.7 A dual-process approach to personal norms Do personal norms come out automatically, or do they require deliberation? Research recently explored the cognitive basis of unselfish behaviour, by using cognitive process manipulation, such as time pressure and cognitive load, in order to favour instinctive responses rand_n12 ; andersen2018allowing ; bereby2018honesty ; bouwmeester2017registered ; capraro2019time ; capraro2017deliberation ; chen2019cognitive ; chuan2018field ; everett2017deliberation ; holbein2019insufficient . It has been shown that promoting intuition favours cooperation rand2016cooperation and altruistic punishment hallsson2018fairness . The evidence regarding altruism is instead more mixed rand2016social ; fromell2020altruism . Instead, a meta-analysis suggests that intuition decreases truth-telling, when lying harms abstract others, while leaving it unaffected when it harms concrete others kobis2019intuitive . Furthermore, results are inconclusive in the context of trustworthiness and the equality-efficiency trade-off (see capraro2019dual for a review). This line of work suggests that whether personal norms come out automatically or require deliberation may not have a general answer, but might depend on the specific behavioural context, and possibly also on the individual characteristics of the decision maker. More work is needed to understand which personal norms, in which context, and for which people, become internalised as automatic reactions. ## VIII Conclusions The moral preference hypothesis is emerging as a unified framework to explain a wide range of one-shot, anonymous unselfish behaviours, including cooperation, altruism, altruistic punishment, truth-telling, trustworthiness, and the equality-efficiency trade-off. This framework has promising practical implications, given that interventions making personal norms salient have been shown to be effective at increasing charitable donations. Future work should explore further mathematical formalisations of moral preferences in terms of a utility function, investigate the evolution and internalisation of personal norms, study the external validity and the boundary conditions of policy interventions based on personal norms, compare the relative effectiveness of targeting different types of norms, examine the topology of the moral phenotype, and analyse the cognitive foundations of morality, possibly using a dual-process perspective. Overall, the goal of this line of research should be to build bridges between different scientific disciplines to arrive at a better, perhaps more mechanistic, explanation of human decision-making. The outlined mathematical formalism for morality should be used to inform future models aimed at better understanding selfless actions, and it should also be used in artificial intelligence to better navigate the complex landscape of human morality and to better emulate human decision-making. Ultimately, the goal is to use the obtained insights to develop more efficient policies and interventions to increase good virtues and decrease bad ones, and to collectively strive towards better human societies. The past century has seen strict compartmentalisation of different scientific disciplines leading to groundbreaking and important discoveries that might had been impossible without it. But while technology and industry might fare well on idiosyncratic breakthroughs, human societies do not. The grandest challenges of today remind us that sustainable social welfare and organisation require a wholesome interdisciplinary and cross-disciplinary approach, and we hope this review will be an inspiration towards this goal. ###### Acknowledgements. This work was supported by the Slovenian Research Agency (Grant Nos. P1-0403, J1-2457, J4-9302, and J1-9112). ## References * (1) Rapoport, A., Chammah, A. M., and Orwant, C. J. 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Prisoner’s dilemma: We measure cooperative behaviour using the prisoner’s dilemma. Two players simultaneously decide whether to cooperate or to defect. Cooperating means paying a cost $c$ to give a benefit $b>c$ to the other player; defecting means doing nothing. Sender-Receiver game: We measure lying aversion using the sender-receiver game. The _sender_ is given a private information and has to report it to the _receiver_. In some experiments the receiver is passive gneezy2013measuring ; biziou2015does , in others is active gneezy2005deception ; erat2012white . Here we focus on the case in which the receiver is passive. In this case, if the sender reports the truthful information, then the sender and the receiver are paid according to Option A; if the sender reports an untruthful information, then the sender and the receiver are paid according to Option B. Only the sender knows the exact payoffs associated to the two options. Depending on these payoffs, one can classify lies into four main classes: black lies are those that benefit the sender at a cost to the receiver; altruistic white lies are those that benefit the receiver at a cost to the sender; Pareto white lies are those that benefit both the sender and the receiver; spiteful lies are those that harm both the sender and the receiver. Trade-Off game: We measure the trade-off between equality and efficiency using the trade-off game. A decision-maker has to decide between two possible allocations of money that affect people other than the decision-maker. One decision is equal (i.e., all people involved in the interaction receive the same monetary payoff), the other decision is efficient (i.e., the sum of the monetary payoffs of all people is greater than it is in the equal allocation). Trust game: We measure trustworthiness using the second player in the trust game. The _truster_ is given a certain amount of money and has to decide how much of it, if any, to transfer to the _trustee_. The amount sent to the trustee is multiplied by a constant (usually equal to 3) and given to the trustee. The trustee decides how much of the amount s/he received to return to the truster. Ultimatum game: We measure altruistic punishment using the second player in the ultimatum game. The _proposer_ makes an offer about how to split a sum of money between him/herself and the _responder_. The responder decides whether to accept or reject the offer. If the offer is accepted, the proposer and the responder get paid according to the agreed offer; if the offer is rejected neither the proposer nor the responder get any money. Rejecting a low offer is considered to be a measure of altruistic punishment. Table 2: Social preference models Let $x_{i}$ be the monetary payoff of player $i$. Social preference models assume that the utility function of player $i$, $u_{i}$, is defined over the monetary payoffs that are associated with the available actions. The main functional forms that have been proposed are the following. Ledyard (1994): $u_{i}(x_{1},\ldots,x_{n})=x_{i}+\alpha_{i}\sum_{j\neq i}x_{j}$, where $\alpha_{i}$ is an individual parameter representing $i$’s level of altruism. People with $\alpha_{i}=0$ maximise their monetary payoff; people with $\alpha_{i}>0$ are altruistic; people with $\alpha_{i}<0$ are spiteful. Levine (1998): $u_{i}(x_{1},\ldots,x_{n})=x_{i}+\sum_{j\neq i}\frac{\alpha_{i}+\lambda\alpha_{j}}{1+\lambda}x_{j}$, where $\alpha_{i}$ is an individual parameter representing $i$’s level of altruism, whereas $\lambda\in[0,1]$ is a parameter representing how sensitive players are to the level of altruism of the other players. Fehr and Schmidt (1999): $u_{i}(x_{1},\ldots,x_{n})=x_{i}-\frac{\alpha_{i}}{n-1}\sum_{j\neq i}\max(x_{j}-x_{i},0)-\frac{\beta_{i}}{n-1}\sum_{j\neq i}\max(x_{i}-x_{j},0)$, where $\alpha_{i},\beta_{i}$ are individual parameters representing the extent to which player $i$ cares about disadvantageous and advantageous inequities, respectively Bolton and Ockenfels (2000): $u_{i}(x_{1},x_{2})=\alpha_{i}x_{i}-\frac{\beta_{i}}{2}\left(\sigma_{i}-\frac{1}{2}\right)^{2}$, where $\sigma_{i}=\frac{x_{i}}{x_{1}+x_{2}}$, with $\sigma_{i}=\frac{1}{2}$ if $x_{1}+x_{2}=0$, $\alpha_{i}>0$ is an individual parameter representing the extent to which player $i$ cares about their own monetary payoff, and $\beta_{i}>0$ is an individual parameter representing the extent to which player $i$ cares about minimising the distance between their share and the fair share. Andreoni and Miller (2002): $u_{1}(x_{1},x_{2})=\left(\alpha_{1}x_{1}^{\rho_{1}}+(1-\alpha_{1})x_{2}^{\rho_{1}}\right)^{1/\rho_{1}}$, where $\alpha_{1}$ represents the extent to which the dictator cares about their own payoff, whereas $\rho_{1}$ takes into account a potential convexity in the preferences. Charness and Rabin (2002): $u_{2}(x_{1},x_{2})=(\rho_{2}r+\sigma_{2}s)x_{1}+(1-\rho_{2}r-\sigma_{2}s)x_{2}$. Depending on the relative relationship between $\rho_{2}$ and $\sigma_{2}$, this utility function can cover several cases, including competitive preferences, inequity aversion preferences, and social efficiency preferences. Table 3: The classification of norms Behavioural scientists have long been aware of the fact that people’s behaviour in a given context is influenced by what are perceived to be the norms in that context. In the same context, multiple norms might be at play. Scholars have proposed several norm classifications. In this review, we will be mainly concerned with the following three. Schwartz schwartz1977normative classified norms into two main categories, namely _personal norms_ and _social norms_. Personal norms refer to internal standards about what is right and what is wrong in a given context. Social norms refer to rules and standards of behaviour that affect the choices of individuals without the force of law. Social norms are typically externally motivated. Cialdini, Reno and Kallgren cialdini1990focus focused on social norms and classified them into two main categories, namely _injunctive norms_ and _descriptive norms_. Injunctive norms refer to what people think others would approve or disapprove. Descriptive norms refer to what others actually do. Bicchieri bicchieri2005grammar proposed a classification in three main categories, namely _personal normative beliefs_ , _empirical expectations_ , and _normative expectations_. Personal normative beliefs refer to personal beliefs about what should happen in a given situation. Empirical expectations refer to personal beliefs about how others would behave in a given situation. Normative expectations refer to personal beliefs about what others think one should do. Therefore, to the extent to which people believe that what should (or should not) happen in a given situation corresponds to their internal standards about what is right (or wrong), then Bicchieri’s personal normative beliefs correspond to Schwartz’s personal norms. In one-shot anonymous games (where decision makers receive no information about the behaviour of other people playing in the same role), descriptive norms correspond to empirical expectations (we replace the actual behaviour of others with the beliefs). Finally, normative expectations correspond to injunctive norms. Therefore, at least for the games and decision problems considered in this review, Bicchieri’s classification can be interpreted as a synthesis of the previous two classifications. Table 4: The moral preference hypothesis Previous work explained unselfish behaviour in one-shot, anonymous economic games using social preferences defined over monetary outcomes. According to this “social preference hypothesis”, some people act unselfishly because they do not only care about their own monetary payoff, but they also care about the monetary payoffs of other people. However, especially in the last five years, numerous experiments challenged social preference models. The best way to organise these results is through the moral preference hypothesis, according to which people have preferences for following their own personal norms – what they think to be the right thing to do – beyond the monetary consequences that these actions bring about. This framework outperforms the social preference hypothesis at organising cooperation in the prisoner’s dilemma, altruism in the dictator game, altruistic punishment in the ultimatum game, trustworthiness in the trust game, truth-telling in the sender-receiver game, and trade-off decisions between equality and efficiency in the trade-off game. Table 5: Moral preference models Let $a$ be an action for player $i$. Moral preference models assume that the utility function of player $i$, $u_{i}$, describes a tension between the material payoff associated to $a$, $v_{i}(\pi_{i}(a))$, and the moral utility. The main functional forms that have been proposed are the following. Levitt and List (2007): $u_{i}(a)=v_{i}(\pi_{i}(a))+m(a)$. The moral cost or benefit associated to $a$, $m(a)$, is assumed to depend on whether the action is observable, on the material consequences of that action, and on the set of _social norms_ and rules in place in the society where the decision maker lives. Krupka and Weber (2013): $u_{i}(a)=v_{i}(\pi_{i}(a))+\gamma_{i}N(a)$, where $\gamma_{i}$ is the extent to which $i$ cares about following the _injunctive norm_ and $N(a)$ represents the extent to which society views $a$ as socially appropriate. Kimbrough and Vostroknutov (2020): $u_{i}(a)=v_{i}(\pi_{i}(a))+\phi_{i}\eta(a)$, where $\phi_{i}$ is the extent to which $i$ cares about following the _injunctive norm_ and $\eta(a)$ represents the extent to which society views $a$ as socially appropriate. (The main difference between $\eta(a)$ and $N(a)$ regards the way they are computed.) Our proposal: $u_{i}(a)=v_{i}(\pi_{i}(a))+\mu_{i}P_{i}(a)$, where $\mu_{i}$ represents the extent to which $i$ cares about following their own _personal norms_ and $P_{i}(a)$ represents the extent to which $i$ personally thinks that $a$ is the right thing to do. Table 6: Outstanding challenges • Exploring in which contexts interventions targeting personal norms are more effective at promoting one-shot unselfish behaviour than interventions targeting social norms. • Finding the boundary conditions of interventions targeting personal norms. • Investigating the dimension and the boundary of the “moral phenotype”, to understand how different personal norms can drive different forms of unselfish behaviour and whether the moral phenotype includes behaviours that are strategically unselfish, such as strategic fairness and trust. • Building bridges between computational linguistics, moral psychology, and behavioural economics, with the goal of understanding how to express people’s utility function also in terms of the instructions of a decision problem. • Using techniques from evolutionary game theory, applied mathematics, network science, and statistical physics to explore which types of unselfish behaviour are more likely to evolve in order to understand which personal norms are more likely to be internalised. • Exploring the cognitive basis of personal norms using a dual-process perspective.
[table]capposition=top # Variational manifold learning from incomplete data: application to multislice dynamic MRI Qing Zou, Abdul Haseeb Ahmed, Prashant Nagpal, Sarv Priya, Rolf F Schulte, Mathews Jacob Qing Zou and Mathews Jacob are with the Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA, USA (e-mail: <EMAIL_ADDRESS>and [email protected]). Abdul Haseeb Ahmed is with Philips Healthcare, Rochester, MN, USA (e-mail: [email protected]). Prashant Nagpal is with the Department of Radiology, University of Wisconsin- Madison, Madison, WI, USA (email: [email protected]). Sarv Priya is with the Department of Radiology, The University of Iowa, Iowa City, IA, USA (e-mail: [email protected]). Rolf F Schulte is with General Electric Healthcare, Munich, Germany (email: [email protected]). This work is supported by NIH under Grants R01EB019961 and R01AG067078-01A1. This work was conducted on an MRI instrument funded by 1S10OD025025-01. ###### Abstract Current deep learning-based manifold learning algorithms such as the variational autoencoder (VAE) require fully sampled data to learn the probability density of real-world datasets. Once learned, the density can be used for a variety of tasks, including data imputation. However, fully sampled data is often unavailable in a variety of problems, including the recovery of dynamic and high-resolution MRI data considered in this work. To overcome this problem, we introduce a novel variational approach to learn a manifold from undersampled data. The VAE uses a decoder fed by latent vectors, drawn from a conditional density estimated from the fully sampled images using an encoder. Since fully sampled images are not available in our setting, we approximate the conditional density of the latent vectors by a parametric model whose parameters are estimated from the undersampled measurements using back- propagation. We use the framework for the joint alignment and recovery of multislice free breathing and ungated cardiac MRI data from highly undersampled measurements. Most of the current self-gating and manifold cardiac MRI approaches consider the independent recovery of images from each slice; these methods are not capable of exploiting the inter-slice redundancies in the datasets and require sophisticated post-processing or manual approaches to align the images from different slices. By contrast, the proposed scheme is able to align the multislice data and exploit the redundancies. Experimental results demonstrate the utility of the proposed scheme in dynamic imaging alignment and reconstructions. ###### Index Terms: Variational autoencoder; Generative model; CNN; Manifold approach; Unsupervised learning; Free-breathing cardiac MRI; Image reconstruction ## I Introduction Deep generative models [1] that rely on convolutional neural networks (CNNs) are now widely used to represent data living on nonlinear manifolds. For instance, the variational autoencoder (VAE) [2] represents the data points as CNN mappings of the latent vectors, whose parameters are learned using the maximum likelihood formulation. Since the exact log-likelihood of the data points is intractable, VAE relies on the maximization of a lower bound of the likelihood, involving an approximation for the conditional density of the latent variable represented by an encoder neural network. The VAE framework offers several benefits over the vanilla autoencoder [1], including improved generalization [3] and ability to disentangle the important latent factors [4, 5]. Unfortunately, most of the current generative models are learned from fully sampled datasets. Once learned, the probability density of the data can be used as a prior for various applications, including data imputation [6, 7]. Unfortunately, fully-sampled datasets are often not available in many high- resolution structural and dynamic imaging applications to train autoencoder networks. The main focus of this paper is to introduce a variational framework to learn a deep generative manifold directly from undersampled/incomplete measurements. The main application motivating this work is the multislice free-breathing and ungated cardiac MRI. Breath-held CINE imaging, which provides valuable indicators of abnormal structure and function, is an integral part of cardiac MRI exams. Compressed sensing [8, 9, 10, 11] and deep learning methods have emerged as powerful options to reduce the breath-hold duration, with excellent performance [12, 13, 14, 15, 16]. Despite these advances, breath-held CINE imaging is challenging for several subject groups, including pediatric and chronic obstructive pulmonary disease (COPD) subjects. Several authors have introduced self-gating [17, 18, 19, 20, 21, 22] and manifold approaches [23, 24, 25, 26, 27] to enable free-breathing and ungated single-slice cardiac MRI. For instance, the smoothness regularization on manifolds (SToRM) approach [28, 29, 30] models the images as points on a low-dimensional manifold whose structure is exploited using a kernel low-rank formulation [29, 30] to recover the images from highly undersampled measurements. Recently, deep learning- based manifold models were introduced [31, 32, 33] to further improve the performance; these schemes learn a deep generative network and its latent variables directly from the measured k-space data using a non-probabilistic formulation. All of the previously described free-breathing cardiac MRI reconstruction approaches (e.g., compressed sensing-based approaches, manifold approaches, and deep learning-based approaches) independently recover the data from each slice. Cardiac MRI often relies on slice-by-slice acquisition to preserve myocardium to blood pool contrast, resulting from the in-flow of blood from unexcited regions to the slice of interest; the improved contrast facilitates segmentation. The above-mentioned 2D self-gating and manifold methods are thus unable to exploit the extensive redundancies between adjacent slices, which could offer improved performance. Note that the respiratory and cardiac motion during the acquisition of the different slices is often very different; this makes the direct 3D extension of the 2D self-gating and manifold methods impossible. Another challenge with the approaches mentioned above is the need for post-processing methods to determine matching slices at specific cardiac/respiratory phases for estimation of cardiac parameters (e.g., ejection fraction, strain). Several post-processing methods have been introduced to align the data post reconstruction [24, 34, 35, 36, 37]. Because these methods require fully sampled data, they will not facilitate the exploitation of the inter-slice redundancies during image recovery. We introduce a novel variational framework for the joint recovery and alignment of multislice data from the entire heart. This approach combines the undersampled k-t space data from different slices, possibly acquired with multiple cardiac and respiratory motion patterns, to recover the 3D dynamic MRI dataset. We use a 3D CNN generative model, which takes in a latent vector and outputs a 3D image volume. The time-varying latent vectors capture the intrinsic variability in the dataset, including cardiac and respiratory motion. The latent variables and the parameters of the 3D CNN are jointly learned from the multislice k-t space data using a maximum likelihood formulation. Since the likelihood is not tractable, we maximize its variational lower bound involving a model for the conditional distribution of the latent variables, which is conceptually similar to the VAE approach [2]. The VAE scheme uses an encoder network to derive the conditional probabilities of the latent vectors from fully sampled data [2]. This approach is not directly applicable in our setting because each data sample is measured using a different measurement operator. We hence model the conditional densities as a Gaussian distribution whose parameters are learned from the undersampled data directly using back-propagation. We use a Gaussian prior on the latent variables while deriving the evidence-based lower bound (ELBO); the Gaussian prior ensures that the latent variables from different slices have similar distributions, facilitating the alignment of the slices. We note that the direct extension of our previous generative manifold model [31, 32] to the 3D setting does not have any constraint on the latent variables; this extension results in poor alignment of the slices and degradation in image quality in the 3D setting. We also use smoothness priors on the latent variables to further improve the performance. Once learned, the representation can be used to generate matching 3D volumes with any desired cardiac/respiratory phase by exciting the generator with appropriate latent vectors. This approach of learning a generative model of the entire heart may thus be viewed as a paradigm shift from conventional slice-by-slice image-recovery algorithms[17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. ## II Background on dynamic MRI ### II-A multislice free-breathing MRI: problem statement The main application considered in this paper is the recovery of 3D cardiac volumes of the heart from undersampled 2D multislice k-t space data acquired in the free-breathing and ungated setting. In particular, we consider the recovery of the time series $\mathbf{x}(\mathbf{r},t_{z})$, where $\mathbf{r}=(x,y,z)$ represents the spatial coordinates and $t_{z}$ denotes the time frame during the acquisition of the $z^{\rm th}$ slice. We model the acquisition of the data as $\mathbf{b}(t_{z})=\mathcal{A}_{t_{z}}\Big{(}\mathbf{x}(\mathbf{r},t_{z})\Big{)}+\mathbf{n}_{t_{z}},$ (1) where $\mathbf{b}(t_{z})$ is the k-t space data of the $z^{\rm th}$ slice at the $t^{\rm th}$ time frame. Here, $\mathcal{A}_{t_{z}}$ are the time- dependent measurement operators, which evaluate the multi-channel single-slice Fourier measurements of the 3D volume $\mathbf{x}(\mathbf{r},t_{z})$ on the trajectory $k_{t_{z}}$ corresponding to the time point $t$. Specifically, $\mathcal{A}_{t_{z}}$ extracts the $z^{\rm th}$ slice from the volume $\mathbf{x}(\mathbf{r},t_{z})$ and evaluates its single-slice measurements. $\mathbf{n}_{t_{z}}$ represents the noise in the measurements. ### II-B CNN-based generative manifold models in dynamic MRI CNN-based generative models were recently introduced for single-slice dynamic MRI [31]. This scheme models the 2-D images in the time series as the output of a CNN generator $\mathcal{D}_{\theta}$: $\mathbf{x}_{i}=\mathcal{D}_{\theta}(\mathbf{c}_{i}),\quad i=1,\cdots,M.$ The input $\mathbf{c}_{i}$ is the latent vector, which lives in a low- dimensional subspace. The recovery of the images in the time series involves the minimization of the criterion $\displaystyle\mathcal{C}(\mathbf{c},\theta)=$ $\displaystyle\underbrace{\sum_{i=1}^{N}\\|\mathcal{A}_{i}\left(\mathcal{D}_{\theta}(\mathbf{c}_{i})\right)-\mathbf{b}_{i}\\|^{2}}_{\scriptsize\mbox{data term}}$ $\displaystyle+\lambda_{1}\underbrace{\\|J_{\mathbf{c}}\mathcal{D}_{\theta}(\mathbf{c})\\|^{2}}_{\scriptsize\mbox{net reg.}}+\lambda_{2}\underbrace{\\|\nabla_{i}\mathbf{c}_{i}\\|^{2}}_{\scriptsize\mbox{latent reg.}}.$ (2) The first term in the cost function is a measure of data consistency, while the second term is a network regularization term that controls the smoothness of the generated manifold [31]. The last term is the temporal smoothness of the latent variables, which is used to further improve the performance. ## III Variational manifold learning We now introduce a novel variational formulation to learn a manifold from undersampled measurements, which is the generalization of the seminal VAE approach [2] to the undersampled setting. We will first present the proposed approach in a simple and general setting for simplicity and ease of understanding. The use of this variational manifold model for the joint alignment and recovery of 3D images from 2-D multislice MRI data will be described in Section IV. ### III-A General problem statement and intuition We assume that the images in the time series, indexed by $i$, live on a smooth manifold $\mathcal{M}$ and hence can be modeled as the output of a CNN-based generator: $\mathbf{x}_{i}=\mathcal{D}_{\theta}(\mathbf{c}_{i}),$ (3) where $\mathbf{c}_{i}$ is the low-dimensional latent variable corresponding to $\mathbf{x}_{i}$. Here, $\theta$ denotes the weights of the generator, which is shared for all the images. Most generative models consider the learning of the above model from fully sampled data. By contrast, we consider the recovery from incomplete measurements $\mathbf{b}_{i}=\mathcal{A}_{i}(\mathbf{x}_{i})+\mathbf{n}_{i}.$ (4) Here, $\mathcal{A}_{i}$ is an undersampled measurement operator corresponding to the $i^{\rm th}$ image frame. Here, $\mathbf{n}_{i}\in\mathcal{N}(\mathbf{0},\sigma^{2}\mathbf{I})$ are noise vectors. Note that the measurement operators for each $\mathbf{x}_{i}$ are different. If the same sampling operators are used for all the data points, it is impossible to recover the images without additional prior information. We assume that the sampling operators satisfy the following properties: 1. 1. We assume $\mathcal{A}_{i}$ to be a rectangular sub-matrix, obtained by picking specific rows of an orthonormal measurement operator (e.g., Fourier transform). 2. 2. We assume that the measurement operators $\mathcal{A}\sim\mathcal{S}$ are drawn from a distribution and satisfy $\mathbb{E}_{\mathcal{A}\sim S}[\mathcal{A}^{T}\mathcal{A}]=\mathcal{I},$ (5) which is the identity operator. The above condition guarantees diversity in the measurement operators. We now provide some intuition about why the learning of the model with the above settings will succeed under the restrictive assumptions on the measurement operators described above. In the noiseless setting, we consider the learning of the latent variables $\mathbf{c}_{i}$ and the weights $\theta$ by minimizing the empirical error: $\left\\{\theta^{},\mathbf{c}_{i}^{}\right\\}=\arg\min_{\theta,\mathbf{c}_{i}}\underbrace{\sum_{i}\\|\mathcal{A}_{i}\left(\mathbf{x}_{i}-\mathcal{D}_{\theta}(\mathbf{c}_{i})\right)\\|^{2}}_{\mathcal{L}}.$ (6) Here, $\mathbf{x}_{i}$ are the fully sampled data points. When $\mathcal{A}\sim\mathcal{S}$, this empirical sum approximates $\displaystyle\mathcal{L}$ $\displaystyle\approx$ $\displaystyle~{}\mathbb{E}_{\mathbf{x}\sim\mathcal{M}}~{}\mathbb{E}_{\mathcal{A}\sim S}\\|\mathcal{A}\left(\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c})\right)\\|^{2}$ $\displaystyle=$ $\displaystyle\mathbb{E}_{\mathbf{x}\sim\mathcal{M}}~{}\mathbb{E}_{\mathcal{A}\sim S}\left\langle\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c}),\mathcal{A}^{H}\mathcal{A}\left(\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c})\right)\right\rangle$ $\displaystyle=$ $\displaystyle\mathbb{E}_{\mathbf{x}\sim\mathcal{M}}~{}\left\langle\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c}),\underbrace{\mathbb{E}_{\mathcal{A}\sim S}[\mathcal{A}^{H}\mathcal{A}]}_{\mathcal{I}}\left(\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c})\right)\right\rangle$ $\displaystyle=$ $\displaystyle\arg\min_{\theta,\mathbf{c}}\mathbb{E}_{\mathbf{x}\sim\mathcal{M}}~{}\\|\mathbf{x}-\mathcal{D}_{\theta}(\mathbf{c})\\|^{2}.$ The above result follows from (5) and the orthonormality of the full measurement operator. This result shows that the recovery of the true manifold is feasible from undersampled data when the sampling operators satisfy the properties listed above. ### III-B Proposed algorithm We consider the recovery of the images $\mathbf{x}_{i}$ from their measurements (4) by maximizing their likelihood, specified by $p(\mathbf{b}_{i})=\frac{p(\mathbf{b}_{i},\mathbf{c}_{i})}{p(\mathbf{c}_{i}\|\mathbf{b}_{i})}$ (7) We note that the posterior $p(\mathbf{c}_{i}\|\mathbf{b}_{i})$ is not tractable. Following the VAE approach in [2], we use a surrogate distribution to approximate $p(\mathbf{c}_{i}\|\mathbf{b}_{i})$. The VAE formulation uses an encoder network to model $p(\mathbf{c}_{i}\|\mathbf{x}_{i})$ from the fully sampled data ($\mathbf{b}_{i}=\mathbf{x}_{i}$). Unfortunately, this approach is not directly applicable in our setting since $\mathbf{b}_{i}$ is the undersampled data, measured using $\mathcal{A}_{i}$ that vary with $i$. We propose to use a Gaussian model $q_{i}(\mathbf{c}_{i})\approx p(\mathbf{c}_{i}\|\mathbf{b}_{i})$, parameterized by its mean $\bm{\mu}_{i}$ and diagonal covariance matrix $\bm{\Sigma}_{i}$, and to estimate these parameters using back-propagation. Following a similar argument as in [2], we show in the Appendix that the likelihood term in (7) can be lower-bounded as $\displaystyle\log p(\mathbf{b}_{i})$ $\displaystyle\geq$ $\displaystyle\underbrace{-\frac{1}{2\sigma^{2}}\mathbb{E}_{\mathbf{c}_{i}\sim q_{i}(\mathbf{c}_{i})}\left[\\|\mathcal{A}_{i}\,\mathcal{D}_{\theta}(\mathbf{c}_{i})-\mathbf{b}_{i}\\|^{2}\right]}_{{\text{data term}}}$ (8) $\displaystyle\qquad-\qquad\underbrace{KL[q_{i}(\mathbf{c}_{i})\|\|p(\mathbf{c}_{i})]}_{L(q_{i}):\text{latent regularization}}.$ Here, $p(\mathbf{c}_{i})$ is a prior on the latent variables. In this work, we assume $p(\mathbf{c}_{i})=\mathcal{N}(\mathbf{0},\mathbf{I})$, where $\mathbf{I}$ is the identity matrix. In this case, the KL divergence can be explicitly evaluated as $L(\mathbf{c}_{i})=\frac{-\log[\det(\mathbf{\Sigma})]-n+{\mathrm{trace}}(\mathbf{\Sigma})+\bm{\mu}^{T}\bm{\mu}}{2},$ where we assume a latent space of dimension $n$. We hence solve for the unknown weights of the generator $\theta$ as well as the parameters of $q_{i}$ denoted by $\bm{\mu}_{i}$ and $\bm{\Sigma}_{i}$ by minimizing the negative of the lower bound in (8). Following [2], we use a Monte-Carlo approach to approximate the expectation in the data term. In particular, at each epoch of the training loop, we derive the samples $\mathbf{c}_{i}$ as $\mathbf{c}_{i}=\bm{\mu}_{i}+\bm{\Sigma}_{i}~{}\bm{\epsilon},$ (9) where $\bm{\epsilon}$ is a zero-mean unit variance Gaussian random variable. At each iteration, the estimation process thus involves the minimization of the criterion $\mathcal{C}\left(\theta,\\{\underbrace{\bm{\mu}_{i},\bm{\Sigma}_{i}}_{q_{i}}\\}\right)=\sum_{i=1}^{N_{\rm data}}\left(\\|\mathcal{A}_{i}\,\mathcal{D}_{\theta}(\mathbf{c}_{i})-\mathbf{b}_{i}\\|^{2}+\sigma^{2}~{}L(q_{i})\right),$ (10) with respect to the unknowns $\theta,\bm{\mu}_{i}$ and $\bm{\Sigma}_{i}$. Figure 1: Illustration of variational manifold learning in the context of learning the digit 1 from the MNIST dataset. We first trained the variational model from the fully sampled data. (a) shows several of the original images, and (b) shows the corresponding output of the generator (reconstructions). (c) illustrates the learned manifold; we sample the latent vectors on a uniform grid in the range $[-3,3]^{2}$ and show the corresponding reconstructions. Note that the latent vectors capture the intrinsic variability in the dataset. The second row shows the results from the variational model, which are trained with undersampled noisy measurements. In this setting, 70% of the pixel values are missing, and the remaining 30% of the pixel values are corrupted with Gaussian white noise with 0 mean and 0.05 standard deviation. The zero-filled images are shown in (d). In (e), we show the reconstructions from the undersampled measurements. Note that the reconstructions closely resemble the original digits in (a). (f) is the illustration of the manifold. Note that the variability in the manifold is captured in comparison to (c). ### III-C Illustration using MNIST data We provide a simple example for the illustration of the above variational model from undersampled data of the digit 1 in the MNIST dataset [38]. The images used are scaled to the range $[-1,1]$. The generator we used here is a simple CNN with three layers. ReLU activation function is used for the first two layers and tanh is used for the last layer. The dimension of the latent space is chosen as 2. In this example, all the trainable parameters are initialized as small random numbers, and the hyper- parameter for the latent regularization $L(\mathbf{c}_{i})$ is chosen as 1. We used 1,000 epoches to train the CNN generator. We first trained the model from the fully sampled data ($\mathcal{A}_{i}=\mathcal{I}$), whose results are shown in the first row of Fig. 1. Then we trained the model from undersampled noisy data. In the example, 70% of the pixel values in each image are missing, while Gaussian white noise with standard deviation 0.05 is added to the known 30% pixel values. The recovered images are shown in the second row of Fig. 1. We report the peak signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) for the results. ## IV Application to dynamic MRI We first describe the application of the algorithm in the single-slice free- breathing and ungated data, which is the setting considered in [31]. We then generalize the approach to the novel setting of the joint alignment and recovery of 3D MRI from multislice free-breathing data in Section IV-C. ### IV-A Acquisition scheme and pre-processing of data The datasets used in this work are acquired using a 2D (GRE) sequence with golden angle spiral readouts in the free-breathing and ungated setting on a MR750W scanner (GE Healthcare, Waukesha, WI, USA). The sequence parameters for the datasets are: FOV = 320 mm $\times$ 320 mm, flip angle = 18∘, slice thickness = 8 mm. The datasets were acquired using a cardiac multi-channel array with 34 channels. The Institutional Review Board at the University of Iowa approved the acquisition of the data, and written consents were obtained from the subjects. The number of slices acquired for different subjects varies. We used an algorithm developed in house to pre-select the coils that provide the best signal-to-noise ratio in the region of interest. A PCA-based coil combination scheme was then used such that the approximation error was less than $5\%$. We then estimated the coil sensitivity maps based on these virtual channels using ESPIRiT [39] and assumed them to be constant over time. A total of 3,192 spirals were acquired for each slice in the subjects with TR=8.4 ms, which corresponds to an acquisition time of 27 seconds. Among the 3,192 spirals, every sixth spiral was acquired with the same angle; these spirals were used for self-navigation in the reconstruction methods that require self-navigation. We binned the data from six spiral interleaves corresponding to 50 ms temporal resolution for each frame. (a) V-SToRM: SS (b) V-SToRM: MS Figure 2: Illustration of the proposed variational SToRM (V-SToRM) scheme. (a) single-slice setting: The 2D network $\mathcal{D}$ receives the latent vectors sampled from their respective latent distributions using (9). The measurements of the 2D-generated images obtained by the respective sampling operators $\mathcal{A}_{i}$ are compared to the acquired multi-channel measurements using the cost function specified by (11). (b) the multislice 3D setting: Similar to the single-slice setting, the inputs to the 3D network are samples from the respective latent distributions. The 3D volumes are sampled by the respective sampling operators $\mathcal{A}_{z,t}$, which extract the $z^{\rm th}$ slice and compare it to the measured data. The optimization criterion specified by (IV-C) is minimized in this case. ### IV-B Single-slice Variational SToRM algorithm Based on the analysis in the previous sections, we use the following scheme for the recovery of single-slice dynamic MRI. We use a re-parameterization layer to obtain the latent variables $\mathbf{c}(t)$ from the time-varying probability distributions $q(\mathbf{c}(t))$ with parameters $\bm{\mu}_{t}$ and $\bm{\Sigma}_{t}$. These latent variables are fed to the CNN generator $\mathcal{D}_{\theta}$, which generates the reconstructed volumes $\mathbf{x}(t)=\mathcal{D}_{\theta}(\mathbf{c}(t))$. The multi-channel, non- uniform, Fourier transform-based forward operators are applied on the reconstructed images, which are then compared to the actual noisy measurements $\mathbf{b}_{i}$. The illustration of this scheme is shown in Fig. 2 (a). The parameters in the generator and the $\bm{\mu}_{i}$ and the $\bm{\Sigma}_{i}$ are updated based on the loss function $\mathcal{L}(\theta,\\{\bm{\mu}_{t},\bm{\Sigma}_{t}\\})=\mathcal{C}(\theta,\\{\bm{\mu}_{t},\bm{\Sigma}_{t}\\})+\lambda_{1}\|\|\theta\|\|_{1}^{2}+\lambda_{2}\|\|\nabla\bm{\mu}_{t}\|\|^{2}.$ (11) Here, $\mathcal{C}(\theta,\\{\bm{\mu}_{t},\bm{\Sigma}_{t}\\})$ is defined in (10), which is the lower bound for maximum likelihood estimation. The second term in (11) is a regularization penalty on the generator weights. It has been shown in [31] that adding this term makes the training of the decoder more stable. The third term involves the temporal gradients of the latent vectors, which enforces the latent vectors to capture the smooth nature of motion patterns in the dynamic images. We use the ADAM optimization to determine the optimal parameters. We also adopt the progressive-in-time training strategy introduced in [31] to realize a computationally efficient reconstruction. We term this dynamic MRI reconstruction scheme as single-slice variational SToRM. ### IV-C Multislice Variational SToRM algorithm We now generalize the single-slice variational SToRM scheme for the joint alignment and recovery of multislice dynamic MRI. We assume that the image volume at the time point $t$ during the acquisition of the $z^{\rm th}$ slice, denoted by $\mathbf{x}(\mathbf{r},t_{z})$, as the output of the generator: $\mathbf{x}(\mathbf{r},t_{z})=\mathcal{D}_{\theta}\left(\mathbf{c}(t_{z})\right).$ Here, $\mathbf{c}(t_{z})$ are the low-dimensional latent vectors corresponding to slice $z$ at the time point $t$, which is formed by the re-parameterization layer. We note that the generator $\mathcal{D}_{\theta}$ is shared across all slices and time points; this approach facilitates the exploitation of the spatial redundancies between the slices and time points. We propose to jointly align and reconstruct the multislice MRI by jointly estimating the parameters $\theta$, $\bm{\mu}(t_{z})$ and $\bm{\Sigma}(t_{z})$ from the measured multislice data by minimizing the following cost function: $\displaystyle\mathcal{L}_{MS}(\theta,\bm{\mu}(t_{z}),\bm{\Sigma}(t_{z}))=$ $\displaystyle\mathcal{C}_{MS}(\theta,\bm{\mu}(t_{z}),\bm{\Sigma}(t_{z}))+\lambda_{1}\|\|\theta\|\|_{1}^{2}$ $\displaystyle+\lambda_{2}\sum_{z}\|\|\nabla_{t_{z}}\bm{\mu}(t_{z})\|\|^{2},$ (12) where $\mathcal{C}_{MS}=\displaystyle\sum_{z=1}^{N_{\rm slice}}\sum_{t=1}^{N_{\rm data}}\\|\mathcal{A}_{t_{z}}\left[\mathcal{D}_{\theta}(\mathbf{c}(t_{z}))\right]-\mathbf{b}_{t_{z}}\\|^{2}+\sigma^{2}~{}L(q(t_{z}))$ is the lower bound for maximum likelihood as the first term in (11). The illustration of this scheme is given in Fig. 2(b). The parameters of the shared 3D generator $\mathcal{D}_{\theta}$ are jointly learned in an unsupervised fashion from the measured k-t space data using the ADAM optimization algorithm. After the training process is complete, we will generate the image time series by feeding the generator with the latent variables of any specific slice. Following successful learning, we expect the volumes of the multislice reconstructions to have the same motion patterns characterized by the latent variables of that particular slice. We refer to this dynamic MRI reconstruction scheme as multislice variational SToRM, or V-SToRM. ### IV-D Comparison with state-of-the-art (SOTA) methods We compare the proposed V-SToRM approach with the following existing methods. * • Analysis SToRM [28]: The analysis SToRM model uses a kernel low-rank formulation, which involves the estimation of the manifold Laplacian matrix from the k-space navigators using kernel low-rank regularization. This Laplacian is then used to solve for the images. We note that the analysis SToRM approach has been demonstrated to yield improved performance over state- of-the-art self-gated methods, as shown in our prior work [28, 30]. We refer to this approach as A-SToRM. * • Single-slice generative SToRM [31]: The single-slice generative SToRM approach uses a CNN generator to generate the single-slice image series from the highly undersampled k-t space data. This scheme does not rely on a variational formulation. It performs the independent recovery of each slice and hence fails to exploit the inter-slice redundancies. We refer to this approach as G-SToRM:SS. * • Multislice generative SToRM: We extended the single-slice generative SToRM approach without the variational framework to the multislice setting. In particular, we use the CNN generator to produce the image volume; the generator parameters and the latent vectors for each slice are jointly learned. Finally, we feed the latent variables of a particular slice into the generator to obtain the aligned multislice reconstruction. We refer to this approach as G-SToRM:MS. For the quantitative comparisons, in addition to the SSIM metric, we also use the signal-to-error ratio (SER) defined as $\mathrm{SER}=20\cdot\log_{10}\frac{\|\|\mathbf{x}_{\rm ref}\|\|}{\|\|\mathbf{x}_{\rm ref}-\mathbf{x}_{\rm recon}\|\|}.$ Here, $\mathbf{x}_{\rm ref}$ and $\mathbf{x}_{\rm recon}$ represent the reference and the reconstructed images, respectively. The unit for SER is decibel (dB). In our free-breathing and ungated cardiac MRI setting, we usually do not have access to the ground truth. Therefore, in our work, we employ the analysis SToRM method using 25 seconds of data for the reconstruction as the simulated ground truth. ## V Experiments and results ### V-A Implementation details In this work, we use deep CNN to build the generator. The number of generator output channels is dependent on the specific datasets. For the experiments using the MNIST dataset, the channel is chosen as 1. By contrast, a two- channel output corresponding to the real and imaginary parts of the MR images is used for the rest of the experiments. In the MRI setting, we use a generator of 10 layers. The total number of trainable parameters is about 6 times the size of the image volume. For the convolutional layers in the generator, the activation function is chosen as leaky ReLU [40] except for the final layer, where $\tanh$ is used as the activation function. Random initialization is used to initialize the generator network. The algorithm has three free parameters, $\sigma^{2}$, $\lambda_{1}$, and $\lambda_{2}$. For each method, we optimize these parameters as well as the architecture of the generator on a single dataset such that the reconstructions closely match the 25-second A-SToRM reconstructions. Once the optimal parameters are determined, they are kept fixed for the remaining datasets. Our experiments showed that two latent vectors were sufficient for the good recovery of the single-slice datasets, which correspond to the cardiac and respiratory phases. In the multislice case, we required three to obtain good reconstructions. In this case, two of the three latent vectors captured cardiac and respiratory motion, respectively. The third latent vector seemed to capture a harmonic of the respiratory motion. Figure 3: Showcase of the single-slice V-SToRM. We trained the variational model using the data of one slice. We showed four different phases in the time series: diastole in End-Inspiration (E-I), diastole in End-Expiration (E-E), systole in End-Inspiration (E-I), and systole in End-Expiration (E-E), obtained from single-slice V-SToRM. The plot of the latent vectors are shown at the bottom of the figure, and the latent vectors corresponding to the four phases are indicated on the plot of the latent vectors. ### V-B Single-slice V-SToRM and comparisons In this section, we focus on single-slice V-SToRM; the reconstructions of a dataset and its latent vectors are shown in Fig. 3. We trained the variational model using the data of one slice. The latent vectors we obtained are shown at the bottom of Fig. 3. Four different phases in the time series are shown in the figure, and their corresponding latent vectors are indicated in the plot of the latent vectors. The comparisons between the single-slice V-SToRM and the state-of-the-art methods on a different dataset are shown in Fig. 4. In these experiments, we compare the region of interest for A-SToRM, G-SToRM, and V-SToRM reconstructions using the 7.5 seconds of data. We use A-SToRM reconstructions from 25 seconds of data as the reference. From Fig. 4, we see that G-SToRM (7.5 s) and V-SToRM (7.5) are able to reduce errors and noise in the images when compared to A-SToRM (7.5 s). The proposed V-SToRM (7.5 s) is able to provide sharper edges than G-SToRM (7.5 s). These observations are further confirmed by the quantitative results shown at the bottom of the figure. Figure 4: Comparisons with the state-of-the-art methods for single-slice results. The figure shows the visual comparison of three phases: the diastole phase (top row), the systole phase (third row), and the phase that is in between the diastole and systole phases (second row). The first three columns correspond to the reconstructions using the A-SToRM, G-SToRM, and V-SToRM approaches based on 7.5 seconds of data. The last column shows the reconstructions from A-SToRM based on 25 seconds of data; we use these reconstructions as references for quantitative comparisons. We also report the quantitative results at the bottom of the figure. (a) Alignment and recovery of eight slices using V-SToRM (b) Alignment and recovery of eight slices using G-SToRM (c) Latent vectors obtained by V-SToRM:MS (d) Latent vectors obtained by G-SToRM:MS Figure 5: Alignment and joint recovery of multislice data. In (a), we show the alignment and recovery of the eight slices obtained from the proposed multislice V-SToRM scheme. Four different phases in the time series for each slice are displayed. From (a), we see that all the slices have the same cardiac phase and respiratory phase, indicating that the multislice V-SToRM is able to align the slices. In (b), we show the alignment and recovery of the eight slices obtained from the generalization of single-slice G-SToRM to the multislice setting. We also use four different phases in the time series for each slice to illustrate the alignment of the multislice data. From (b), we see that some of the phases for some of the slices have poor image quality. In particular, the details in the cardiac regions are poorly captured, and in some cases the boundaries of the heart are not visible. These issues can be understood from the plot distributions of the latent vectors obtained by the multislice V-SToRM and G-SToRM:MS, shown in (c) and (d), respectively. We also plot the latent vectors for two of the slices for each method. Note that we generated the results in (a) and (b) by feeding the latent vectors corresponding to the second slice into the generators. The corresponding latent vectors used to generate the four different phases in (a) and (b) are indicated in the plot of the latent vectors in (c) and (d). From (c) and (d), we see that the latent vectors obtained from the proposed multislice V-SToRM scheme have similar distributions, whereas the distributions for the latent vectors obtained from G-SToRM:MS are very different. (a) Comparisons based on slice #3 (b) Comparisons based on slice #4 Figure 6: Comparisons of the image quality of the reconstructions. We compare the image quality of the multislice V-SToRM reconstructions with the image quality of the reconstructions from A-SToRM, G-SToRM:SS, and G-SToRM:MS. The multislice dataset used in this example has four slices, and we show two of the slices in the figure to do the comparisons. For each slice, we show three different phases: the diastole phase, the systole phase, and the phase that is in between the diastole and systole phases. For each sub-figure, the first four columns represent the reconstruction from A-SToRM, G-SToRM:SS, G-SToRM:MS, and the proposed multislice V-SToRM based on 6 seconds of data. The last column shows the reconstructions using A-SToRM based on 25 seconds of data; they are used as simulated references for the quantitative results, which are shown at the bottom of each sub-figure. From both the visual comparisons and the quantitative results, we see that the multislice V-SToRM scheme is able to provide comparable reconstructions when compared to the competing methods. We also highlighted some of the phases in the multislice G-SToRM reconstruction, from which we see that G-SToRM:MS has some issues in generating some of the image frames. ### V-C Joint alignment and recovery of multislice data In this section, we show the results of the joint alignment and recovery of multislice data using the proposed multislice V-SToRM scheme. We also compare the alignment results obtained from the straightforward multislice extension of the G-SToRM scheme. The results are shown in Fig. 5. More results are shown in the supplementary material. The dataset used in Fig. 5 was acquired with eight slices that covered the whole heart. We trained the variational model based on the undersampled k-t space data and fed the latent vectors corresponding to the second slice to the generator, which produces the aligned multislice reconstructions. Shown in the figures are four time points based on the different phases identified by the latent variables. The rows in Fig. 5 (a) correspond to diastole in End- Inspiration, diastole in End-Expiration, systole in End-Inspiration, and systole in End-Expiration for each slice obtained using the proposed multislice V-SToRM scheme. From Fig. 5 (a), we see that the proposed multislice V-SToRM scheme is able to jointly reconstruct and align the multislice free-breathing and ungated cardiac MRI. We note that all the slices in each row have the same cardiac phase and respiratory phase. In Fig. 5 (b), we show the corresponding results for the direct extension of the multislice G-SToRM approach. In particular, we trained the model using the undersampled k-t space data and fed the latent vectors corresponding to the second slice into the generator to produce the aligned multislice reconstructions. From Fig. 5 (b), we see that the multislice G-SToRM approach has some ability to align the multislice reconstructions. However, we find that the image quality for some of the frames (e.g., slices 5-8) is poor. For example, the diastole phases for the G-SToRM:MS reconstructions are blurred and the cardiac boundaries are missing. The reason for the poor reconstructions offered by multislice G-SToRM and the improved performance of V-SToRM can be easily appreciated from the distribution of the latent vectors shown in Fig. 5 (c) and Fig. 5 (d), respectively. The use of the variational formulation in V-SToRM encouraged the latent variables of the slices to approximate a Gaussian distribution. We also reported the KL divergence value compared to $\mathcal{N}(\mathbf{0},\mathbf{I})$ for each set of the latent vector in the figure. We note that the V-SToRM scheme offers low KL divergence values, indicating that the latent distribution of all the slices are roughly similar to a unit Gaussian. By contrast, the G-SToRM scheme cannot guarantee that the latent variables follow any distribution. We note from the top rows of (d) that the distribution of the latent variables of the second slice is very different from that of the other slices. When we feed the latent vectors of the second slice into the generator, the generator is only able to generate reasonable results for the second slice. ### V-D Comparison of image quality with state-of-the-art methods We compare the image quality of the multislice V-SToRM reconstructions with the image quality of the reconstructions from the state-of-the-art methods, including single-slice methods, in Fig. 6. Note that the motion patterns of the slices recovered by the single-slice methods may be very different. For comparison, we manually matched the images of the slices of the single-slice and multislice methods by their cardiac and respiratory phases. The quantitative comparisons of the slices are shown at the bottom of each sub- figure. We also show more results using another dataset in the supplementary material. The single-slice A-SToRM and G-SToRM:SS comparisons roughly match the observations in Fig. 4 and the results in [31]. The results show that the multislice V-SToRM approach is able to offer reconstructions that are less blurred and have fewer alias artifacts when compared to single-slice analysis methods (A-SToRM and G-SToRM:SS). The improved performance is also evidenced by the higher SER and SSIM values. We attribute the improved performance to the exploitation of the redundancies across slices, enabled by V-SToRM. We also note that the G-SToRM:MS method offers poor performance, evidenced by image blurring and missing details on the myocardium. The poor performance of G-SToRM:MS can be understood in terms of the differences in distribution of the latent vectors, shown in Fig. 5. ## VI Discussion and Conclusion In this work, we introduced an approach for the variational learning of a CNN manifold model from undersampled measurements. This work generalized the traditional VAE scheme to the undersampled setting. Unlike the traditional VAE scheme that uses an encoder to learn the conditional distribution from the images, we propose to learn the parameters of the distribution from the measurements using back-propagation. The application of the framework to multislice cardiac MRI data enabled the joint alignment and recovery from highly undersampled measurements. Unlike current single-slice methods that perform independent recovery of the slices, the proposed approach aligns the acquisitions and jointly recovers the images from the undersampled k-t space data. In addition to facilitating the exploitation of inter-slice redundancies, this approach also eliminates the need for post-processing schemes to match the phases of the slices. Our results show that the joint alignment and recovery of the slices offer reduced blurring and reduction of artifacts compared to the direct generalization of G-SToRM to the multislice setting. In particular, the variational framework encourages the latent variables of different slices to have the same distribution. By contrast, the G-SToRM framework cannot guarantee the similarity of the probability distributions; the improper alignment translates to image blurring and other artifacts. Similarly, the use of the CNN generator offers implicit spatial regularization, resulting in improved recovery over A-SToRM. A benefit with the proposed scheme is that it does not require fully sampled data to train the CNN. The subject-specific CNN parameters and the latent vectors are learned directly from the undersampled data. We note that the acquisition of fully sampled data to train neural networks is not always possible, especially in the high-resolution and dynamic MRI settings considered in this work. In this context, direct learning from undersampled data is desirable. However, a challenge of the proposed scheme when compared to pretrained deep learning methods that offer super-fast inference is the higher computational complexity. We will explore training strategies, including transfer learning and meta-learning, to reduce the run time in the future. ## VII Appendix In this appendix, we show that the likelihood term in (7) can be lower-bounded by (8). According to (7) and using the result of joint probability, we obtain $\displaystyle p(\mathbf{b}_{i})$ $\displaystyle=$ $\displaystyle\frac{p(\mathbf{b}_{i},\mathbf{c}_{i})}{q_{i}(\mathbf{c}_{i})}\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i}\|\mathbf{b}_{i})}$ (13) $\displaystyle=$ $\displaystyle\underbrace{\frac{p(\mathbf{b}_{i},\mathbf{c}_{i})}{p(\mathbf{c}_{i})}}_{p(\mathbf{b}_{i}\|\mathbf{c}_{i})}\frac{p(\mathbf{c}_{i})}{q_{i}(\mathbf{c}_{i})}\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i}\|\mathbf{b}_{i})}.$ Taking the logarithm on both sides of (13), we have $\log p(\mathbf{b}_{i})=\log p(\mathbf{b}_{i}\|\mathbf{c}_{i})-\log\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i})}+\log\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i}\|\mathbf{b}_{i})}.$ (14) Next, we take the expectation with respect to $\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})$ of both sides of (14), and realizing that $\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log p(\mathbf{b}_{i})=\log p(\mathbf{b}_{i})$, we obtain $\displaystyle\displaystyle\log p(\mathbf{b}_{i})=$ $\displaystyle\displaystyle\underbrace{\displaystyle\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log p(\mathbf{b}_{i}\|\mathbf{c}_{i})}_{{\text{data term}}}-\underbrace{\displaystyle\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i})}}_{KL[q_{i}(\mathbf{c}_{i})\|\|p(\mathbf{c}_{i})]}$ (15) $\displaystyle+\underbrace{\displaystyle\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log\frac{q_{i}(\mathbf{c}_{i})}{p(\mathbf{c}_{i}\|\mathbf{b}_{i})}}_{KL[q_{i}(\mathbf{c}_{i})\|\|p(\mathbf{c}_{i}\|\mathbf{b}_{i})]>0}.$ The last term is always greater than zero. The first term is the conditional density of the measurements $\mathbf{b}_{i}$ given the images $\mathbf{x}_{i}=\mathcal{D}_{\theta}(\mathbf{c}_{i})$. With the measurement model specified by (4), we obtain $\displaystyle\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log~{}p(\mathbf{b}_{i}\|\mathbf{c}_{i})=-\frac{1}{2\sigma^{2}}\displaystyle\mathop{\mathbb{E}}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\\|\mathcal{A}_{i}\,\mathcal{D}(\mathbf{c}_{i})-\mathbf{b}_{i}\\|^{2}+c,$ where $c$ is a constant independent of the parameters of interest. Ignoring the constant $c$ and plugging $\mathbb{E}_{\mathbf{c}_{i}\thicksim q_{i}(\mathbf{c}_{i})}\log~{}p(\mathbf{b}_{i}\|\mathbf{c}_{i})$ back into (15), we obtain the desired lower bound (8). ## Acknowledgments The authors would like to thank Ms. Melanie Laverman from the University of Iowa for making editorial corrections to refine this paper. 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# secureTF: A Secure TensorFlow Framework Do Le Quoc, Franz Gregor, Sergei Arnautov TU Dresden, Scontain UG , Roland Kunkel TU Dresden , Pramod Bhatotia TU Munich and Christof Fetzer TU Dresden, Scontain UG ###### Abstract. Data-driven intelligent applications in modern online services have become ubiquitous. These applications are usually hosted in the untrusted cloud computing infrastructure. This poses significant security risks since these applications rely on applying machine learning algorithms on large datasets which may contain private and sensitive information. To tackle this challenge, we designed secureTF, a distributed secure machine learning framework based on Tensorflow for the untrusted cloud infrastructure. secureTF is a generic platform to support unmodified TensorFlow applications, while providing end-to-end security for the input data, ML model, and application code. secureTF is built from ground-up based on the security properties provided by Trusted Execution Environments (TEEs). However, it extends the trust of a volatile memory region (or secure enclave) provided by the single node TEE to secure a distributed infrastructure required for supporting unmodified stateful machine learning applications running in the cloud. The paper reports on our experiences about the system design choices and the system deployment in production use-cases. We conclude with the lessons learned based on the limitations of our commercially available platform, and discuss open research problems for the future work. secure machine learning, confidential computing, intel software guard extensions (Intel SGX), tensorflow ## 1\. Introduction Machine learning has become an increasingly popular approach for solving various practical problems in data-driven online services (taigman2014deepface, ; bennett2007netflix, ; foster2014machine, ; deepmind_health, ). While these learning techniques based on private data arguably provide useful online services, they also pose serious security threats for the users. Especially, when these modern online services use the third-party untrusted cloud infrastructure for deploying these computations. In the untrusted computing infrastructure, an attacker can compromise the confidentiality and integrity of the computation. Therefore, the risk of security violations in untrusted infrastructure has increased significantly in the third-party cloud computing infrastructure (Santos2009, ). In fact, many studies show that software bugs, configuration errors, and security vulnerabilities pose a serious threat to computations in the cloud systems (Gunawi_bugs-in-the-cloud, ; Baumann2014, ; Santos2012, ). Furthermore, since the data is stored outside the control of the data owner, the third-party cloud platform provides an additional attack vector. The clients currently have limited support to verify whether the third-party operator, even with good intentions, can handle the data with the stated security guarantees (pesos, ; Vahldiek-Oberwagner2015, ). To overcome the security risks in the cloud, our work focuses on securing machine learning computations in the untrusted computing infrastructure. In this context, the existing techniques to secure machine learning applications are limiting in performance (graepel2012ml, ), trade accuracy for security (du2003using, ) or support only data classification (bost2015machine, ). Therefore, we want to build a secure machine learning framework that supports existing applications while retaining accuracy, supporting both training and classification, and without compromising the performance. Furthermore, our work strives to provide end-to-end security properties for the input data, ML models, and application code. To achieve our design goals, trusted execution environments (TEEs), such as Intel SGX (intel-sgx, ) or ARM TrustZone (arm-trustzone, ), provide an appealing way to build a secure machine learning system. In fact, given the importance of security threats in the cloud, there is a recent surge in leveraging TEEs for shielded execution of applications in the untrusted infrastructure (Baumann2014, ; arnautov2016scone, ; tsai2017graphene, ; shinde2017panoply, ; Orenbach2017, ). Shielded execution aims to provide strong confidentiality and integrity properties for applications using a hardware-protected secure memory region or enclave. While these shielded execution frameworks provide strong security guarantees against a powerful adversary, the TEEs have been designed to secure single- node in-memory (volatile) computations. Unfortunately, the trust of TEEs does not naturally extend to support distributed stateful applications running in the cloud. To build a secure machine learning framework that supports both training and classification phases, while providing all three important design properties: transparency, accuracy, and performance, we need to address several architectural challenges presented by TEEs, specifically Intel SGX, which acts as the root of trust. More specifically, in addition to the conventional architectural challenges posed by the SGX architecture in the single node setting, such as limited enclave memory and I/O bottlenecks, we need to address the following three important challenges in the context of distributed cloud computing: Firstly, we need to extend the trust of SGX to support the distributed TensorFlow framework, where the worker nodes are running in the remote distributed enclaves while ensuring that they execute correct code/computations and data. However, this is a challenging task since Intel SGX is designed for securing single machine computations. Secondly, we need to support practical features offered by the virtualized platforms in the public cloud service to enable elastic and fault-tolerant computing, i.e., scaling-up/down based on the workloads, and dealing with failures/migrations. To support these important requirements, we need to ensure the new worker node running in a container preserves the integrity, confidentiality of the data, ML models, and application code. However, the traditional remote attestation using the Intel Attestation Service (IAS) (costan2016intel, ) is impractical to support the elastic and fault-tolerant computing. Therefore, we need to redesign the mechanism to ensure an elastic trust establishment through a configuration and attestation service. Lastly, we need to support stateful machine learning applications that rely on reading the input data or write computation results from/to a file system storage as well as to the network. Unfortunately, Intel SGX is designed to protect only the data and computation residing in the volatile enclave memory. It does not provide any security guarantees for stateful machine learning computations across multiple machines. To overcome these design challenges, we present secureTF, a secure machine learning framework for the untrusted infrastructure. More specifically, we make the following contributions. * • We have designed and implemented secureTF as the end-to-end system based on TensorFlow that allows secure execution of the existing unmodified TensorFlow applications without compromising the accuracy. * • We optimized the performance to overcome the architectural limitation of Intel SGX in the context of machine learning workloads for distributed untrusted cloud computing environments. * • We report an extensive evaluation of secureTF based on microbenchmarks and production use-cases. Our evaluation shows that secureTF achieves reasonable performance overheads, while providing strong security with low TCB. secureTF is a commercially available platform, and it is currently used in production by four major customers. In this paper, we report on our experiences on building secureTF and deploying it in two production use-cases. We conclude the paper with the lessons learned based on the limitations of our system design, and a discussion on open research problems for the future work. ## 2\. Background and Threat Model ### 2.1. Machine Learning using TensorFlow Machine learning aims to automatically extract useful patterns in large-scale data by building probabilistic models (simeone2017brief, ). Machine learning approaches are often categorized into supervised, unsupervised and reinforcement learning. All forms have in common that they require datasets, a defined objective, a model and a mechanism to update the model according to new inputs. To generalize the machine learning approach for masses, Google proposed TensorFlow (abadi2016tensorflow, ) as a machine learning framework designed for heterogeneous distributed systems. TensorFlow requires the user first to define a directed graph consisting of nodes representing operations on incoming data. Nodes have zero or more inputs and outputs and perform operations on different levels of abstraction such as matrix multiplication, pooling or reading data from disk. Nodes can also have an internal state, depending on their type. Thus the whole graph can be stateful as well. After defining the graph, the user can perform calculations by starting a session and running the previously defined operations. TensorFlow uses a flow model for calculations. Through the division of the calculation in the graph into nodes, TensorFlow makes it easy to distribute the execution across different devices. Therefore, TensorFlow can be deployed on mobile devices, single personal computers, as well as computer clusters, by mapping the computation graph on available hardware. TensorFlow Lite (tensorflow-lite, ) is a feature-reduced version of TensorFlow, designed for mobile and embedded devices. Optimization for mobile devices is achieved by running a mobile-optimized interpreter that keeps the load at a lower level and having the overall binary size smaller when compared to full TensorFlow. The number of available operations for defining a graph is reduced to achieve a smaller memory footprint of the resulting binary. This comes at the cost of trainability of the graph, because TensorFlow Lite can only perform forward passes in graphs. Instead, a model must first be training with the full version of TensorFlow and then exported and converted to a special TensorFlow Lite model format. This format can then be used from the TensorFlow Lite API for inference. ### 2.2. Intel SGX and Shielded Execution Intel Software Guard Extension (SGX) is a set of x86 ISA extensions for Trusted Execution Environment (TEE) (costan2016intel, ). SGX provides an abstraction of a secure _enclave_ —a hardware-protected memory region for which the CPU guarantees the confidentiality and integrity of the data and code residing in the enclave memory. The enclave memory is located in the Enclave Page Cache (EPC)—a dedicated memory region protected by an on-chip Memory Encryption Engine (MEE). The MEE encrypts and decrypts cache lines that are written and read to EPC, respectively. Intel SGX supports a call-gate mechanism to control entry and exit into the TEE. Shielded execution based on Intel SGX aims to provide strong confidentiality and integrity guarantees for applications deployed on an untrusted computing infrastructure (Baumann2014, ; arnautov2016scone, ; tsai2017graphene, ; shinde2017panoply, ; Orenbach2017, ). Our work builds on the SCONE (arnautov2016scone, ) shielded execution framework. In the SCONE framework, the applications are linked against a modified standard C library (SCONE libc). In this model, the application’s address space is confined to the enclave memory, and interaction with the untrusted memory is performed via the system call interface. In particular, SCONE runtime provides an asynchronous system call mechanism (flexsc, ) in which threads outside the enclave asynchronously execute the system calls. Lastly, SCONE provides an integration to Docker for seamlessly deploying container images. ### 2.3. Threat Model We aim to protect against a very powerful adversary even in the presence of complex virtualization stacks in the cloud computing infrastructure (Baumann2014, ). In this setting, the adversary can control the entire system software stack, including the OS or the hypervisor, and is able to launch physical attacks, such as performing memory probes. In addition, we consider an untrusted network in the cloud environment, i.e., the adversary can drop, inject, replay, alter packages, or manipulate the routing of packages. This network model is consistent with the classic Dolev-Yao adversary model (DolevYao, ). Even under this extreme threat model, our goal is to guarantee the integrity, confidentiality, and freshness of data, code (e.g., Python code), and models of machine learning computation. We also provide bindings with Pesos (pesos, ), a secure storage system to protect against rollback attacks (Parno2011, ) on the data stored beyond the secure enclave memory. Our system is adaptable with SGXBounds (kuvaiskii2017sgxbounds, ); therefore, secureTF is resilient to memory safety vulnerabilities (intel-mpx, ). However, we do not protect against side-channel attacks based on cache timing and speculative execution (foreshadow, ), and memory access patterns (xu2015controlled, ; hahnel2017high, ). Mitigating side-channel attacks is an active area of research (varys, ). We do not consider denial of service attacks since these attacks are trivial for a third-party operator controlling the underlying infrastructure (Baumann2014, ), e.g., operating system (OS), and hypervisor. Lastly, we assume that the CPU hardware (including its implementation of SGX) are trusted and the adversary cannot physically open the processor packaging to extract secrets or corrupt the CPU system state. ## 3\. Design In this section, we present the design of secureTF. ### 3.1. System Overview secureTF is designed for secure distributed machine learning computations using the hardware-assisted trusted execution environment (TEE) technologies such as Intel SGX. Figure 1 depicts the high-level architecture of secureTF. Our system ensures not only the confidentiality, integrity and freshness of executions (e.g., training and classifying computations) but also the input data and machine learning models. At a high-level, the system works as follows: at the first step, when a user deploys a machine learning computation on a remote host (e.g., a public cloud), the user needs to establish trust into the secureTF instance running in the untrusted environment. To do so, the user performs the remote attestation mechanism provided by the TEE technology to ensure that the computation and the input data deployed in the remote environment are correct and not modified by anyone e.g., an attacker. After trusting the secureTF running in the remote environment, the user provides secrets including keys for encrypting/decrypting input and output data (e.g., input images and models, certificates for TLS connections), to the machine learning platform. After finishing the computation, secureTF returns the results back to the user via a TLS connection. Figure 1. System overview. Design goals. Our primary design goal is to achieve strong confidentiality and integrity properties. By confidentiality, we mean that all data including models handled by the machine learning framework and the machine learning framework code itself may not be disclosed to or obtainable by an unauthorized party. By integrity, we mean that modifications of the data handled by secureTF that were done by an unauthorized party must be detectable and should not compromise the internal state and functioning. In addition, while designing a practical system, we aim to achieve the following goals. * • Transparency: The secure framework must offer the same interface as the unprotected framework, and should run unmodified existing applications based on TensorFlow. * • Performance: We aim to impose as little overhead as possible when adding security to the machine learning framework. * • Accuracy: We do not aim to trade-off accuracy for security. Accuracy will be the same in the native TensorFlow framework as when using no security protection. ### 3.2. Design Challenges Building a practical secure distributed machine learning system using TEEs such as Intel SGX is not straightforward, in fact, we need to handle several challenges. ➊ Code modification. Intel SGX requires users to heavily modify the source code of their application to run inside enclaves. Thus, transparently supporting an unmodified machine learning framework to run inside enclaves is not a trivial task. ➋ Limited EPC size. Currently, Intel SGX supports only a limited memory space ($\sim 94$MB) for applications running inside enclaves. However, most machine learning computations, especially training, are extremely memory-intensive. ➌ Establishing the trust in a distributed system. Trust has to be established in the remote distributed enclaves to ensure that they execute correct code and data. However, this is a challenging task since Intel SGX is originally designed for a single machine. ➍ Elastic and fault tolerant computing support. Typically, public cloud services support elastic computing, i.e., when the input workload increases, the framework automatically spawns new service containers or instances to handle with the growth of requests. However, whenever spawning a new container, it requires to perform remote attestation to ensure the integrity, confidentiality of the machine learning application in that container before communicating with it. Unfortunately, the traditional attestation mechanism using the Intel Attestation Service (IAS) (costan2016intel, ) incurs significant overhead, thus it’s impractical in this setting. ➎ Stateful computing: security of network and file system. Machine learning applications running inside SGX enclaves need to read input data or write results from/to a file system, storage systems, or network. Unfortunately, Intel SGX is designed to protect only the stateless in-memory data and computation residing inside enclaves. It does not provide any security guarantees for state stateful machine learning computations across multiple machines. ### 3.3. System Design In this section, we present the detailed design of distributed secureTF that handles the aforementioned challenges in $\S$1. #### 3.3.1. System Components To overcome the challenge ➊ (see $\S$1), we built secureTF based on the SCONE shielded execution framework (arnautov2016scone, ). SCONE enables legacy applications to be executed in Intel SGX enclaves without source code changes. While there are other options available, we choose SCONE, because of the relatively small extra work required to run an application and comparatively small overhead compared to other available options. We leverage SCONE’s Docker container support to design secure distributed secureTF which allows users to perform machine learning computations in a secure manner on an untrusted environment such as a public cloud. Figure 2 shows the distributed architecture of secureTF. At the high-level, our systems consist of four core components: Configuration and Remote Attestation Service (CAS), secure machine learning containers including Tensorflow parameter servers and Tensorflow workers, network shield and file system shield, and adapted Tensorflow library. Figure 2. The distributed architecture of secureTF. We design the CAS component to handle the challenges ➌ and ➍. This component takes an important role in the distributed architecture of secureTF which transparently and automatically performs the remote attestation for secure machine learning containers before transferring secrets and configuration needed to run them. The CAS component is deployed inside an SGX enclave of an untrusted server in the cloud or on a trusted server under the control of the user. When a secure machine learning container is launched, it receives the necessary secrets and configuration from CAS after the remote attestation process, to run machine learning computations using the adapted Tensorflow library running inside an enclave. We design the network shield and the file system shield components to address the challenge ➎. All communications between secure machine learning containers and the CAS component are protected using the network shield component. Next, we provide the detailed design of each component. #### 3.3.2. Configuration and Remote Attestation Service CAS enhances the Intel attestation service (costan2016intel, ) to bootstrap and establish trust across the secureTF containers and maintain a secure configuration of the distributed secureTF framework. CAS itself is deployed in an Intel SGX enclave. In the case that CAS is deployed in an enclave on an untrusted server, the user of secureTF needs to establish trust into the CAS instance, i.e., he/she needs to perform remote attestation of CAS before transferring encryption keys and certificates to process the encrypted input data and machine learning models. By using CAS, we can maintain the original distributed architecture of Tensorflow machine learning framework. In addition, to guarantee the freshness of data during runtime, we design and implement an auditing service in CAS to keep track the data modification during machine learning computation. This mechanism allows secureTF to protect against rollback attacks. #### 3.3.3. Secure Machine Learning Containers To build secure machine learning containers, we make use of TensorFlow and TensorFlow Lite. TensorFlow Lite has the additional advantage of having a smaller memory footprint which helps us to handle the design challenge ➋. We use SCONE (arnautov2016scone, ) as an additional layer that allows access to SGX features with fewer changes to application code. Figure 3 presents the general architecture of a secure Tensorflow container using SCONE. Figure 3. The architecture of a secure machine learning container in secureTF. Since we build secure machine learning containers based on SCONE in secureTF, we use Docker (merkel2014docker, ) to conveniently deploy our system. No changes to the Docker engine is required. The design of a secure machine learning container in secureTF is composed of two components: (a) the secureTF controller that provides the necessary runtime environment for securing the TensorFlow library, and (b) secureTF TensorFlow library that allows deploying unmodified existing TensorFlow applications. We next describe these two components in detail. secureTF Controller. The secureTF controller is based on the SCONE runtime. Inside the SGX enclave, the controller provides a runtime environment for TensorFlow, which includes the network shield, the file system shield, and the user-level threading. Data, that is handled through file descriptors, is transparently encrypted and authenticated through the shields. The shields apply at each location where an application would usually trust the operating system, such as when using sockets or writing files to disk. The shields perform sanity checks on data passed from operating system to enclave to prevent Iago attacks (Checkoway2013, ). More specifically, these checks include bound checks and checking for manipulated pointers. This protection is required to fulfill the goal of not requiring the application to deal with untrusted systems (see challenge ➊ in $\S$1). File system shield. The file system shield protects confidentiality and integrity of data files. Whenever the application would write a file, the shield either encrypts and authenticates, simply authenticates or passes the file as is. The choice depends on user-defined path prefixes, which are part of the configuration of an enclave. The shield splits files into chunks that are then handled separately. Metadata for these chunks is kept inside the enclave, meaning it is protected from manipulation. The secrets used for these operations are different from the secrets used by the SGX implementation. They are instead configuration parameters at the startup time of the enclave. Network shield. TensorFlow applications do not inherently include end-to-end encryption for network traffic. Users who want to add security must apply other means to secure the traffic, such as a proxy for the Transport Layer Security (TLS) protocol. According to the threat model however, data may not leave the enclave unprotected, because the system software is not trusted. Network communication must therefore always be end-to-end protected. Our network shield wraps sockets, and all data passed to a socket will be processed by the network shield before given to system software. The shield then transparently wraps the communication channel in a TLS connection on behalf of the user application. The keys for TLS are saved in files and protected by the file system shield. User-level threading. Enclave transitions are costly and should therefore be avoided when possible. Many system calls require a thread to exit userspace and enter kernel space for processing. To avoid thread transitions out of enclaves as much as possible, the controller implements user space threading. When the OS assigns a thread to an enclave, it first executes an internal scheduler to decide, which application thread to execute. These application threads are then mapped to SGX thread control structures. When an application thread blocks, the controller is run again to assign the OS thread to a different application thread instead of passing control back to the operating system. In this way, the number of costly enclave transitions is reduced. When no application thread is ready for execution, the OS either backs off and waits inside the enclave, or outside, depending on the time required for an enclave transition. A side effect of this user-level threading scheme is that the controller does not require more OS threads than CPUs available to achieve full CPU utilization, which is usually the case for applications running under a conventional OS. #### 3.3.4. secureTF TensorFlow Library Machine learning applications consist of two major steps. In the first step, the model is trained, and thereafter, the model is employed for classification or inference tasks. Next, we explain the detailed design to run both training process and classification process with Intel SGX. Training process. For the training process, we use the full version of TensorFlow. Training in TensorFlow is usually performed on acceleration hardware such as GPUs and distributed across multiple machines. However, the secureTF controller requires SGX which is only available for CPUs. We therefore only support training on CPU. This limitation reduces the performance of the training process, but it is required to achieve the security goals. The secureTF controller allows easy distribution of the application in the form of Docker images. The training instances of secureTF can be distributed on multiple nodes, each running separate SGX hardware. The network shield applies transparent protection of the communication channel between instances. Scaling on the same instance, that is, on the same CPU is possible, but does decrease relative performance, because the limiting factor in our environment is EPC size, which is fixed for each CPU. Only horizontal scaling with more nodes can substantially increase performance. Classification process. The main reason for dividing the classification and training process in our design is that we can use different TensorFlow variants for each step. Running with Intel SGX imposes less overhead, if applications have a smaller memory footprint, because the limited EPC size is the major bottleneck (see challenge ➋ in $\S$1). TensorFlow Lite has a smaller memory footprint because it targets mobile devices. The drawback is however that it cannot perform training by design. Therefore, we can only use it for classification or inference. When protecting TensorFlow Lite with SCONE, the framework uses the SCONE C library instead of the common system library. The internals of TensorFlow Lite do otherwise not require change, as the interface of the SCONE C library is compatible. The interface for using the classification method of secureTF is the same as for TensorFlow Lite. Graph definitions created for TensorFlow Lite are compatible. ## 4\. Implementation We implement secureTF based on Tensorflow version $1.9.0$ and the SCONE framework (arnautov2016scone, ) to run machine learning computations within Intel SGX enclaves. We also consider other TEEs technologies such as ARM TrustZone (arm-trustzone, ) and AMD’s TEE, SME/SEV (amd_secure_technology, ). However, they have several limitations, e.g., ARM TrustZone supports only a single secure zone, and does not have any remote attestation mechanism, meanwhile, AMD’s TEE does not support integrity protection (mofrad2018comparison, ). We rely on SCONE to implement some features such as file system shield and network shield. However, it is not straightforward to use these features out- of-the-box to build secureTF. For example, SCONE does not support TLS connection via UDP which is required in Tensorflow. SCONE provides only confidentiality and integrity in network/storage shields, whereas, secureTF ensures also the freshness of data, code and models of machine learning computation. In addition, the memory management and user-level multithreading need to adapt/extend it to fit the custom scheduler and memory management of TensorFlow framework. Thus, we need to develop these missing parts of these features to implement the design of secureTF. In this section, we describe several challenges we faced during implementing secureTF and how we addressed them. We first present how to enable the security features in secureTF during the training process ($\S$4.1) and classifying process ($\S$4.2). Thereafter, we describe the implementation of the CAS component in $\S$4.3. ### 4.1. Training Process The typical user of TensorFlow uses the Python API for defining and training graphs, because it is the richest API. Using Python with SCONE would impose additional complexity because it requires the dynamic library open (dlopen) system call for imports. As the name implies, dlopen dynamically loads libraries during runtime of a program. However, SGX does not allow an enclave to be entered by a thread, unless it has been finalized according to the procedures of enclave creation. A library that is dynamically loaded would therefore not be represented in the enclave’s attestation hash. Consequently, dlopen is disabled by default for SCONE applications. To allow dlopen, we need to change the SCONE environment accordingly (i.e., SCONE_ALLOW_DLOPEN=yes). To ensure the security guarantee, we need to authenticate loaded libraries during runtime using the file system shield (see $\S$3.3). We support not only Python but also C++ API as native Tensorflow framework. In the previous version of secureTF, we did not support the Python API since, at that time, SCONE did not support fork system call which is required in the Python package (tensorscone-tech, ). The C++ version covers the low-level API of TensorFlow, meaning many convenience features such as estimators or monitored training are not available. However, implementation using C++ API provides much better performance compared to using Python API. There is one approach that let us use the convenience of the Python API for the definition of the computation graph. TensorFlow allows exporting graphs and parameters, such as learned biases that were created in the current session. Graph definitions and checkpoints containing the parameters can later be imported by another program. Importing and exporting are available in both the C++ and the Python API, and they use interchangeable formats. The user can therefore define a graph with the more high level Python API, including data inputs, and later import and run it with C++. If the application does not by default already export its model with a named interface, changes are required to the original program, so that either the name of operations in the graph can be known, or an interface is defined. For the training process, we used the full version of TensorFlow, not to be confused with TensorFlow Lite. A graph definition must be provided by the user in form of a graph frozen by a script packaged together with TensorFlow, when using either the Python or C++ API. If the user has used the C++ API for the definition, the full source definition of the graph can also be used. A frozen graph can be created from a graph definition exported from the Python script that defines the graph in the Protocol Buffers ((protobuf, )) exchange format. A checkpoint file containing all values of a graph that are not part of the graph definition, such as weights, biases and counters can be exported as well. Alternatively, the graph can also be exported as a blank slate without any initialized internal values. The initialization can then be done inside the secureTF environment, which is useful if a user wants to train the graph protected by SGX for the entire training process. The initialization operations are required when using the Python API and are therefore usually part of the exported graph. The user must also provide the inputs for training, such as a set of annotated images. secureTF protects the input data and code by activating the file system shield (see $\S$3.3). ### 4.2. Classification /Inference Process We implemented our design for inference/classifying computations in secureTF, by integrating the full Tensorflow with SCONE as we developed for the training computations. In addition, we provide a light-weight version for inference by adapting Tensorflow Lite (tensorflow-lite, ) framework to run with SCONE. We first ensured that Tensorflow and TensorFlow Lite compiles with the musl C library (alpine_faq, ) on Alpine Linux (alpine_linux, ), because SCONE enhanced the musl library to support legacy application running with Intel SGX. The musl libc is designed to be compatible with The GNU C Library (glibc) but more secure with a smaller code base. The issue we faced is that Tensorflow currently uses Identical code folding (ICF) (ICF, ), which is a compiler or linker feature, to eliminate identical function bodies at compile or link time in order to reduce the binary size. However, it is currently supported by gcc and the gold linker, but not by the musl linker or the compiler wrapper for musl. We therefore removed the ICF option for the binary targets in the TensorFlow source tree. Compiling the TensorFlow framework with and without ICF provides similar binary sizes. Therefore, the performance cost when deactivating ICF will also be minimal. The next issue is that TensorFlow also uses backtrace by default. This library is specific forglibc. We therefore could not use it directly with musl. To solve this issue, we either recompiled dependencies against the musl libc, or disabled backtrace in the building configuration of Tensorflow. After adapting the Tensorflow source code, compiling it with SCONE is quite straightforward by merely setting the environment variables CC and CXX to the SCONE C and C++ compilers (i.e., scone-gcc and scone-g++). Note that there is no standalone version of TensorFlow Lite available, meaning a user of TensorFlow Lite needs to build their application inside the TensorFlow source folder, with dependency targets set to TensorFlow Lite. Tensorflow uses Bazel as a build tool (bazel, ), however, Bazel also does not link library targets unless a binary target is created, which means TensorFlow Lite cannot be easily released from the source tree by compiling all libraries, and move them to the system’s include directories. Thus, we added compile targets that force linking as a workaround. The libraries could then be moved to other projects along with the header files, and used as third party dependencies. With this, we developed a classifier service from scratch. The service takes classification requests via network, and uses TensorFlow Lite for inference/classifying. For evaluation, we used an example available in the TensorFlow Lite source, which takes its inputs from the hard drive and prints the classification results to console. ### 4.3. Configuration and Remote Attestation Service For large-scale deployment secureTF, we design the Configuration and Remote Attestation Service component (CAS) to transparently perform the remote attestation and transfer keys to distributed secureTF containers (see $\S$3.3). We implement the CAS component using Rust (rust, ) programming language since it provides strong type safety. To run CAS with Intel SGX, we utilize the SCONE cross compiler to compile our implementation of CAS. We make use of an encrypted embedded SQLite (sqlite, ) to store encryption keys, certificates, and other secrets for Tensorflow computations (see $\S$3.3). This database itself also runs inside an enclave with the help of the SCONE framework. To allow a user of CAS can ensure that the code of CAS was not modified and indeed runs inside a valid SGX enclave, besides running CAS with SCONE, we implement CAS in the way that it has zero configuration parameters that can control its behavior. Thus, an attacker with root/privileged accesses cannot break the trust given by the user in CAS. A detail description of CAS regarding protection against rollback attacks and key management is provided in (palaemon, ). ## 5\. Evaluation In this section, we present the evaluation results of secureTF based on both microbenchmarks and macrobenchmarks with real world deployment. Figure 4. The attestation and keys transferring latency comparison between secureTF with the traditional way using IAS. ### 5.1. Experimental Setup Cluster setup. We used three servers with em SGXv1 support running Ubuntu Linux with a 4.4.0 Linux kernel, equipped with an Intel© Xeon© CPU E3-1280 v6 at 3.90GHz with 32 KB L1, 256 KB L2, and 8 MB L3 caches, and 64 GB main memory. These machines are connected with each other using a 1 Gb/s switched network. The CPUs update the latest microcode patch level. In addition, we used a Fujitsu ESPRIMO P957/E90+ desktop machine with an Intel© core i7-6700 CPU with 4 cores at 3.40GHz and 8 hyper-threads (2 per core). Each core has a private 32KB L1 cache and a 256KB L2 cache while all cores share an 8MB L3 cache. Datasets. We used two real world datasets: (i) Cifar-10 image dataset (krizhevsky2009learning, ) and (ii) MNIST handwritten digit dataset (mnist- dataset, ). Figure 5. Comparison between secureTF, native versions and the state-of-the- art Graphene system in terms of latency with different model sizes, (a) Densenet (42MB), (b) Inception_v3 (91MB), and (c) Inception_v4 (163MB). #1: Cifar-10. This dataset contains a labeled subset of a much larger set of small pictures of size 32x32 pixels collected from the Internet. It contains a total of 60,000 pictures. Each picture belongs to one of ten classes, which are evenly distributed, making a total of 6,000 images per class. All labels were manually set by human labelers. Cifar-10 has the distinct advantage that a reasonable good model can be trained in a relatively short time. The set is freely available for research purposes and has been extensively used for benchmarking machine learning techniques (xu2015empirical, ; he2016deep, ). #2: MNIST. The MNIST handwritten digit dataset(mnist-dataset, ) consists of $60000$ $28$ pixel images for training, and $10000$ examples for testing. Methodology. Before the actual measurements, we warmed up the machine by running at full load with IO heavy operations that require swapping of EPC pages. We performed measurements for classification and training both with and without the file system shield. For full end-to-end protection, the file system shield was required. We evaluate secureTF with the two modes: (i) hardware mode (HW) which runs with activated TEE hardware and (ii) simulation mode (SIM) which runs with simulation without Intel SGX hardware activated. We make use of this SIM mode during the evaluation to evaluate the performance overhead of the Intel SGX and to evaluate secureTF when the EPC size is getting large enough in the future CPU hardware devices. ### 5.2. Micro-benchmark: Remote Attestation and Keys Management In secureTF, we need to securely transfer certificates and keys to encrypt/decrypt the input data, models and the communication between worker nodes (in distributed training process). To achieve the security goal, we make use of the CAS component (see $\S$3.3) which attests Tensorflow processes running inside enclaves, before transparently provides the keys and certificates to encrypt/decrypt input data, models, and TLS communications. Note that the advantage of using CAS over the traditional way using IAS to perform attestation is that the CAS component is deployed on the local cluster where we deploy secureTF. Figure 4 shows the break-down latency in attestation and keys transferring of our component CAS and the method using IAS. The quote verification process in our CAS takes less than $1$ms, whereas in the IAS based method is $\sim 280$ms. In total, our attestation using CAS ($\sim 17$ms) is roughly $19\times$ faster than the traditional attestation using IAS ($\sim 325$ms). This is because the attestation using IAS requires providing and verifying the measured information contained in the quotes (costan2016intel, ) which needs several WAN communications to the IAS service. Figure 6. The effect of file system shield on the classification latency with different model sizes, (a) Densenet (42MB), (b) Inception_v3 (91MB), and (c) Inception_v4 (163MB). ### 5.3. Macrobenchmark: Classifying Process We evaluate the performance of secureTF in real-world deployments. First, we present the evaluation results of secureTF in detecting objects in images and classifying images using pre-trained deep learning models. Thereafter, in the next section, we report the performance results of secureTF in training deep learning models (see $\S$5.4). In the first experiment, we analyze the latency of secureTF in Sim mode and HW mode, and make a comparison with native versions using glibc and musl libc (i.e., running Tensorflow Lite with Ubuntu and Alpine linux) and a system (graphene-tesorflow-lite, ) provided by Intel using Graphene (tsai2017graphene, ). Graphene is an open-source SGX implementation of the original Graphene library OS. It follows a similar principle to Haven (Baumann2014, ), by running a complete library OS inside of SGX enclaves. Similar to SCONE (arnautov2016scone, ), Graphene offers developers the option to run their applications with Intel SGX without requiring code modifications. All evaluated systems except the Graphene-based system run inside a Docker container. To conduct this experiment, we use the Desktop machine (see $\S$ 5.1) to install Ubuntu 16.04 since the Graphene based system does not work with Ubuntu 18.04. The evaluated systems run with single thread because of the current version of the Graphene-based system does not support multiple threads, i.e., to run the classification process, we use the same input arguments for the classification command line: $\$\ label\\_image-m\ model.tflite-i\ input.bmp-t\ 1$. For the latency measurement, we calculate the average over $1,000$ runs. We use a single bitmap image from the Cifar-10 dataset as an input of evaluated systems. Models. For classifying images, we use several pre-trained deep learning models with different sizes including Inception-v3 (szegedy2016rethinking, ) with the size of $91$MB, Inception-v4 (szegedy2017inception, ) with the size of $163$MB and Densenet (huang2017densely, ) with the size of $42$MB. We manually checked the correctness of a single classification by classifying the image with the TensorFlow label_image application involving no self-written code and running directly on the host without containerization. We later compared the results to the ones provided by secureTF and other evaluated systems, we could confirm that indeed the same classifying result was produced by the evaluated systems. #1: Effect of input model sizes. Figure 5 shows the latency comparison between secureTF with Sim and HW mode, native Tensorflow Lite with glibc, native Tensorflow Lite with musl libc, and Graphene-based system. secureTF with Sim mode incurs only $\sim 5\%$ overhead compared to the native versions with different model sizes. In addition, secureTF with Sim mode achieves a latency $1.39\times$, $1.14\times$, and $1.12\times$ lower than secureTF with HW mode with the model size of $42$MB, $91$MB, and $162$MB, respectively. This means that operations in the libc of secureTF introduce a lightweight overhead. This is because secureTF handles certain system calls inside the enclave and does not need to exit to the kernel. In the Sim mode, the execution is not performed inside hardware SGX enclaves, but secureTF still handles some system calls in userspace, which can positively affect performance. We perform an analysis using strace tool to confirm that some of the most costly system calls of secureTF are indeed system calls that are handled internally by the SCONE runtime. Interestingly, the native Tensorflow Lite running with glibc is the same or slightly faster compared to the version with musl libc. The reason for this is that both C libraries excel in different areas, but glibc has the edge over musl in most areas, according to microbenchmarks (clib_compare, ), because glibc is tailored for performance, whereas musl is geared towards small size. Because of this difference in goals, an application may be faster with musl or glibc, depending on the performance bottlenecks that limit the application. Differences in performance of both C libraries must therefore be expected. In comparison to Graphene-based system, secureTF with HW mode is faster and faster than Graphene-based system when we increase the size of input models, specially when it exceeds the limit of the Intel SGX EPC size ($\sim 94$MB). In particular, with the model size of $42$MB, secureTF with HW mode is only $1.03\times$ faster compared to Graphene-based system, however, with the model size of $163$MB, secureTF with HW mode is $\sim 1.4\times$ compared to Graphene-based system. The reason for this is that when the application allocates memory size larger than the EPC size limit, the performance of reads and writes severely degraded because it performs encrypting data and paging operations which are very costly. To reduce this overhead, we reduce the size of our libraries loaded into SGX enclaves. Instead of adding the whole OS libc into SGX enclaves as Graphene did, we make use of SCONE libc (arnautov2016scone, ) which is a modification of musl libc having much smaller size. In this library, system calls are not executed directly but instead are forwarded to the outside of an enclave via the asynchronous system call interface (see $\S$3.3). This interface together with the user level scheduling allows secureTF to mask system call latency by switching to other application threads. Thus, we expect this speedup factor of secureTF compared to Graphene-based system will increase more when the size of the input model size is increased and when the application runs with multiple threads. #2: Effect of file system shield. One of real world usecases of secureTF is that a user not only wants to acquire classifying results but also wants to ensure the confidentiality of the input images since they may contain sensitive information, e.g., handwritten document images. At the same time, the user wants to protect her/his machine learning models since he/she had to spend a lot of time and cost to train the models. To achieve this level of security, the user activates the file system shield of secureTF which allows he/she to encrypt the input including images and models and decrypt and process them within an SGX enclave (see $\S$3.3). Figure 7. The latency comparison in classifying cifar-10 images with different numbers of CPU cores and nodes. In this experiment, we evaluate the effect of this file system shield on the overall performance of secureTF. As previous experiments, we use the same input Cifar-10 images. Figure 6 shows the latency of secureTF when running with/without activating the file system shield with different models. The file system shield incurs significantly small overhead on the performance of the classification process. secureTF with Sim mode running with the file system shield is $0.12\%$ slower than secureTF with Sim mode running without the file system shield. Whereas in the secureTF with HW mode, the overhead is $0.9\%$. The lightweight overhead comes from the fact that our file system shield uses Intel-CPU-specific hardware instructions to perform cryptographic operations and these instructions can reach a throughput of up to 4 GB/s, while the model is about 163 MB in size. This leads to a negligible overhead on the startup of the application only. Figure 8. The training latency comparison between secureTF with different modes and native Tensorflow. #3: Scalability. To evaluate the scalability of secureTF, we measure the latency of secureTF in classifying $800$ cifar-10 images, with different number of CPU cores (scale-up), and different number of physical nodes (scale- out). Figure 7 shows that secureTF both in Sim and HW mode scale quite well from $1$ CPU core to $4$ CPU cores. However, secureTF in HW mode does not scale from $4$ CPU cores to $8$ CPU cores. The reason for this is that the EPC size is limited to ~$94MB$. When secureTF runs with $8$ cores it requires more than the capacity of the current version of Intel SGX. Thus, it requires to perform the paging mechanism which is very expensive. For scale-out evaluating, we keep each node to run with $4$ CPU cores. As we expected, secureTF in both Sim and HW mode scale well with different numbers of physical nodes. The latency of secureTF in HW mode with $1$ node is $1180$s whereas with $3$ nodes the latency is $403$s. #4: TensorFlow and TensorFlow Lite comparison. To show the advantage of using TensorFlow Lite in secureTF instead of TensorFlow for inference or classification, we make a comparison between them. In this experiment, we use the same input model (i.e., Inception_v3 model) and input image to evaluate the performance of secureTF using TensorFlow and TensorFlow Lite in HW mode. secureTF with TensorFlow Lite achieves a $\sim 71\times$ lower latency ($0.697$s) compared to secureTF with TensorFlow ($49.782$s). The reason for this is that, the binary size of secureTF with TensorFlow Lite is only $1.9$MB, meanwhile the binary size of secureTF with TensorFlow is $87.4$MB; and note that the Intel SGX enclave EPC size is limited to $\sim 94$MB. ### 5.4. Macrobenchmark: Distributed Training In this section, we evaluate the performance of secureTF in training distributed deep learning models at scale. In these experiments, we use MNIST handwritten digit dataset (see $\S$5.1) and three physical servers having the same configuration described in $\S$5.1. We keep the same batch size of $100$ and learning rate as $0.0005$, then measure the end-to-end latency of secureTF with different modes including HW mode, Sim mode, with and without activating the network shield, and a native version of Tensorflow. Figure 8 shows that secureTF with different modes scales almost linearly with the number of workers. secureTF, with full features running in HW mode, achieves a speedup of $1.96\times$ and $2.57\times$ when it runs with 2 and 3 workers, respectively. Unsurprisingly, this mode of secureTF is roughly $14\times$ slower compared to the native version due to the fact that the training process requires memory-intensive computations and the enclave EPC size is limited to $\sim 94$MB. However, we believe that Intel will release new generation of its hardware which supports much large EPC sizes, thus we performed the experiments to evaluate secureTF in the SIM mode, to see the overhead of secureTF in the case the EPC size is enough for the training process. The slowdown factor in comparison to the native version, is reduced to $6\times$ and $2.3\times$ with secureTF in the SIM mode with and without activating the network shield, respectively. This indicates that the main overhead of the current implementation is the network shield. In addition, note that the slowdown in SIM mode is because of a scheduling issue in SCONE. We have reported this issue, it’s now fixed in the current version of SCONE. From the results of experiments, we can learn that with the current Intel SGX hardware capacity, performing securely inference/classification inside Intel SGX is practical, but it is not feasible for securely training deep learning (see $\S$7.1). ## 6\. Real-World Deployments secureTF is a commercial platform, and it is actively used by four customers (names omitted) in production. We next present the secureTF deployment for two use cases. Figure 9. Deployment #1: secure document digitization. ### 6.1. Secure Handwritten Documents Analysis The first use case of secureTF is to perform secure handwritten documents analytics (see Figure 9). A company (name omitted) is using a public cloud to automatically translate handwritten documents into digital format using machine learning. Customers of this company not only want to acquire the inference results, but also want to ensure the confidentiality of the input since the handwritten document images contain sensitive information. At the same time, the company wants to protect its Python code for the inference engine as well as its machine learning models. To achieve this level of security, the company has deployed our framework—secureTF. The company uses the file system shield to encrypt Python code and models used for the inference. Meanwhile, the customers make use of the attestation mechanism of secureTF to attest the enclave running the service, and then send their handwritten document images via the TLS connections to this service to convert them into digital text documents. Figure 10. Deployment #2: secure federated learning. ### 6.2. Secure Federated Learning: Medical Use-case The second use case of secureTF is secure federated learning (FL) (federated- learning, ) (see Figure 10). FL is proposed to allow multiple parties to jointly train a model that takes benefits from diverse datasets from the parties. In our second use-case, several hospitals are actively collaborating to train a model for diagnosing brain tumors. However, at the same time, they want to protect patients’ data regarding their privacy. Thus, each hospital performs the training locally using its local data and thereafter shares the model parameters with the global training computation without sharing its actual data. Unfortunately, the local model may reveal private information (deeplearning-DP, ). These local models have been demonstrated to be vulnerable to several privacy attacks (federated-learning-privacy, ). In addition, there is empirical evidence to the risks presented by machine learning models, e.g., the work by Matt et al (model-inversion-attacks, ) demonstrates that extracted images from a face recognition system look similar to images from the underlying training dataset. To handle this issue, these hospitals make use of secureTF to run the global training inside Intel SGX enclaves. They only share their local model after performing the attestation over the enclaves. The communication with the global training enclaves are performed via TLS connections. ## 7\. Discussion and Lessons Learned In this section, we discuss the lessons learned based on the limitations of our commercially available platform, and also, present open research problems for the future work. ### 7.1. Training Vs Classification The limited EPC size has different implications for training vs classifications. As shown in $\S$5, training deep learning with the larger datasets inside the enclave is performance-wise limiting due to EPC paging. However, the EPC size is quite practical for classifying/inference processes since the size of the deployed ML model is usually much smaller than the original training data. As discussed in $\S$6, we are effectively using secureTF for image classification ($\S$6.1), and federated machine learning use case (see $\S$6.2). To improve the performance of the training phase in the limited enclave memory regions, we are exploring two avenues: (1) data normalization: we can further improve the training performance, by normalizing input data, e.g., in image recognition services, all input images can be normalized to the size of $32\times 32$; and (2) Ice lake CPUs Intel announced the next generation processors called ice lake which supports larger EPC size (icelake, ). ### 7.2. ML Model Optimizations To further improve the performance, we are exploring perform optimizations for the ML models leveraging pruning and quantization tools, such as Intel OpenVINO Toolkit (openvino, ). Since TensorFlow models are typically abstracted as directed computation graphs (see $\S$2), where nodes are operations and edges present the communication between operations. By performing optimization on the model graphs such as pruning unnecessary edges and nodes, we can significantly improve the performance of classification/inference computations. The optimization also provides an opportunity to deploy ML inference service at edge devices supporting SGX (nuc, ) in edge computing. In fact, we have been working with an IoT-based company to use secureTF for securely deploying the latest trained models at the edge, while achieving high-performance. ### 7.3. Security Analysis and Properties secureTF protects machine learning computations against attackers with privileged access by executing securely these computations inside Intel SGX enclaves. All data (input training/inference data, model, and Python code) and communications outside enclaves are always encrypted. The encrypted data is only decrypted inside enclaves. The keys or secrets to decrypt the data are protected inside the CAS component which is also running inside an enclave. The CAS component only provides these secrets via TLS connections to the machine learning enclaves after attesting these enclaves. A detailed security analysis of CAS is provided in (palaemon, ). Intel SGX is typically vulnerable to side-channel attacks (sidechannel- attack1, ; sidechannel-attack2, ; gotzfried2017cache, ; vanbulck2018foreshadow, ; weisse2018foreshadowNG, ; spectre1, ; spectre2, ). Although this type of attacks are out-of-scope of our work, it is worth to mention that the version of SCONE, which was integrated in secureTF, can not only protect against L1-based side channels attacks (varys, ) but also Iago attacks (checkoway2013iago, ). We can also make use of LLVM-extensions, e.g., speculative load hardening (SpecLH2019, ) to prevent exploitable speculation which helps us to present the variants of Spectre attacks (spectre1, ; spectre2, ). In addition, the next generation of Intel CPUs (icelake, ) seems to provide hardware-based solutions to handle several types of side-channel attacks. secureTF supports only TLS-based communications to protect against eavesdropping on any communication between the CAS and computation nodes in a distributed setting. In secureTF, the TLS certificates are generated inside the SGX enclave running CAS, and thus they cannot be seen by any human. This mechanism allows secureTF to handle man-in-the-middle attacks. However, TLS and its predecessor are also vulnerable to side-channel attacks, e.g., attacks on RSA (drown-attack, ; robot-attack, ). Thus, in secureTF, we recommend to completely disable RSA encryption and replace it by forward-secret key exchanges e.g., Elliptic-curve Diffie–Hellman (ECDHE) encryption (tls-book, ). ### 7.4. GPUs Support Graphics Processing Units (GPUs) have become popular and essential accelerators for machine learning (bekkerman2011scaling, ). Unfortunately, trusted computing in GPUs is not commercially available, except research prototypes, such as Graviton (graviton, ). Therefore, secureTF provides security properties by relying on Intel SGX which is supported only for CPUs. Technically, secureTF can also offer the GPU support, however, it requires weakening the threat model, i.e., we need to assume that the GPU computations and the communication between GPU and CPU are secure. The relaxation of the threat model may be acceptable in practice for several use cases, e.g., when users just want to protect their Python code and models for machine learning computations. secureTF can ensure the code and models are encrypted. However, this extension may not practical for many other use cases (graviton, ). Therefore, we are currently investigating GPU enclave research proposals, e.g., Graviton (graviton, ) and HIX (hix, ) which proposed hardware extensions to provide a secure environment on GPUs. ## 8\. Related Work In this section, we summarize the related work about secure machine learning and shielded execution using Intel SGX. Early works on preserving-privacy data mining techniques have relied on randomizing user data (du2003using, ; bost2015machine, ; PrivApprox2017, ). These approaches trade accuracy for privacy. They include a parameter that allows making a trade-off between privacy and accuracy. The proposed algorithms aim to provide privacy of computation, but they do not protect the results themselves in the cloud, nor do they secure the classification phase. While this can protect the users privacy, it does not cover training as in secureTF. Further, we target to provide the same accuracy level as the native execution. Graepel et al. (graepel2012ml, ) developed machine learning algorithms to perform both training and classification on encrypted data. The solution is based on the properties of homomorphic encryption. However, homomorphic encryption schemes provide restrictive computing operations, and incur high performance overheads. There have been a series of recent works (secureml, ; gazelle, ; cryptflow, ; delphi, ) aimed to provide secure machine learning platforms with Secure multiparty computation (MPC). Especially, Delphi (delphi, ) and CrypTFlow (cryptflow, ) demonstrated that they outperform previous works. However, these systems also were designed only for securing inferences. secureTF is instead based on a hardware-based encryption approach (i.e., Intel SGX) and it supports both training and inference computations. Shielded execution provides strong security guarantees for legacy applications running on untrusted platforms (Baumann2014, ). Prominent examples include Haven (Baumann2014, ), SCONE (arnautov2016scone, ), and Graphene-SGX (tsai2017graphene, ). Our work builds on the SCONE framework. Intel SGX has become available in clouds (IBMCloudSGX, ; AzureSGX, ), unleashing a plethora of services to be ported, including Web search (sgx-websearch, ), actor framework (eactors, ), storage (pesos, ; speicher, ), leases (t-lease, ), monitoring and profilers (tee-perf, ; teemon, ), software update (TSR, ), FaaS (clemmys, ), networking (shieldbox, ; slick, ), and data analytics systems (sgx-pyspark, ; opaque, ; Schuster2015, ). Recently, several secure machine learning systems have been proposed, which rely on Intel SGX to support secure machine learning (privado, ; slalom, ; chiron, ; ohrimenko, ). Privado (privado, ) proposes a mechanism to obtain oblivious neural networks. Then, it executes the oblivious neural network inside SGX enclaves for secure inferencing. Slalom (slalom, ) makes use of a combination of Intel SGX and untrusted GPUs to secure Deep Neural Networks (DNNs) computations. The idea of Slalom is that it splits the DNN computations into linear operations (e.g., matrix multiplications) on GPUs, whereas performing the non-linear operations (eg. ReLUs operations) inside Intel SGX enclaves. This approach allows achieving much better performance since the intensive computation is performed with GPUs. Unfortunately, Slalom still has several limitations. First, as Privado, it focuses only on secure inferences. It refers to secure training computations as a research challenge. Second, it requires Tensorflow users to heavily modify or redevelop their existing code. Third, it does not support distributed settings, i.e., it does not support secure connections between SGX enclaves. Finally, Slalom is not production ready, in fact, it indicates that it can be used only for testing. Chiron (chiron, ) is the most relevant for secureTF, where they leveraged Intel SGX for privacy-preserving machine learning services. Unfortunately, Chiron is a single-threaded system within an enclave. In addition, Chiron requires adding an interpreter and model compiler into enclaves which introduce significant runtime overhead since the limited EPC size. The work from Ohrimenko et al. (ohrimenko, ) also used Intel SGX to secure machine learning computations, however, it supports only a limited number of operators. In contrast, we propose secureTF — a practical distributed machine learning framework for securing both training and inference computations. ## 9\. Conclusion In this paper, we report on our experience with building and deploying secureTF, a secure TensorFlow-based machine learning framework leveraging the hardware-assisted TEEs, specifically Intel SGX. secureTF extends the security properties of a secure stateless enclave in a single node to secure unmodified distributed stateful machine learning applications. Thereby, it provides a generic platform for end-to-end security for the input data, ML model, and application code. Moreover, it supports both training and classification phases while providing all three important design properties for the secure machine learning workflow: transparency, accuracy, and performance. secureTF is a commercially available platform, and is currently being used in production by four major customers. While there are several open challenges and limitations of our system, our experience shows that secureTF strives for a promising approach: it incurs reasonable performance overheads, especially in the classification/inference process, while providing strong security properties against a powerful adversary. Lastly, we also discussed several open challenges and on-going extensions to the system. Acknowledgements. We thank our shepherd Professor Sara Bouchenak and the anonymous reviewers for their insightful comments and suggestions. This work has received funding from the Cloud-KRITIS Project and the European Union’s Horizon 2020 research and innovation programme under the LEGaTO Project (legato-project.eu), grant agreement No 780681. ## References * [1] Alpine Linux. https://alpinelinux.org/. Accessed: May, 2020. * [2] Alpine Linux FAQ. https://wiki.musl-libc.org/faq.html. Accessed: May, 2020. * [3] AMD Secure Technology. https://www.amd.com/en/technologies/security. 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11institutetext: Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland 11email<EMAIL_ADDRESS> http://szajowski.wordpress.com/ 22institutetext: Wrocław University of Science and Technology 22email<EMAIL_ADDRESS> # Operation comfort vs. the importance of system components Krzysztof J. Szajowski 11 Małgorzata Średnicka 22 ###### Abstract The paper focuses on portraying importance measures that are reasonably helpful in analyzing system reliability and its development. The presented measures concern coherent binary and multistate systems and help to distinguish the most influential elements of the system, which require more attention. Importance measures are presented for systems with known structure (e.g. parallel, series) and for repairable or nonrepairable components Subject Classification:MSC 90B25 $\cdot$ (62N05; 60K10) ###### Keywords: component importance coherent system binary system multistate system Barlow- Proschan measure Birnbaum measure Natvig measure universal generating function. Table of Contents section.1.1 subsection.1.1.1subsection.1.1.2subsection.1.1.3subsection.1.1.4subsection.1.1.5subsection.1.1.6section.1.2 subsection.1.2.1subsection.1.2.2subsection.1.2.3subsection.1.2.4subsection.1.2.5subsection.1.2.6subsection.1.2.7section.1.3subsection.1.3.1subsection.1.3.2subsection.1.3.3section.Asection.A.1section.A.2section.A.3section.A ## 1 Introduction ### 1.1 Preliminaries Let’s consider a system111System (in Ancient Greek: ςυςτηµα –romanized: systema – a complex thing) – a set of interrelated elements realizing the assumed goals as a whole., a complex structure with specific functionality. Contemporary systems are characterized by their structural complexity. In the process of designing the system, the most important thing is its preparation for the implementation of the assumed goals. The mathematical model of the system is based on the set theory as the family of subsets of given set $\mathbb{C}=\\{c_{1},\ldots,c_{n}\\}$ having some properties. An example is technical devices whose design is dictated by the need to perform specific functions. The constructed system should function in a planned and predictable manner. This property is a requirement that should also be considered in the design and construction (fabrication) process. The goal is therefore to reduce the risk222It is difficult to define _risk_ in general. In short, when we think about risk, we mean the possibility of an unexpected loss caused by an unpredictable event or harmful behavior (human, machine, animal, nature). One can think about possibility of loss or injury. From the other side, the risk is the chance or probability that a person (a system) will be harmed or experience an adverse health (functioning) effect if exposed to a hazard. It may also apply to situations with property or equipment loss, or harmful effects on the environment. Therefore, we are talking about reducing ownership and loss as a result of a random event. Risk reduction means minimizing the chance of a loss occurring or limiting its size. In order to better understand the risk and possibilities of risk management, the task of measuring risk has been set. The task is not formulated so that its solution is universal. This allowed to determine the desired properties of such measures [3]. of a break in the planned operation of the system. Therefore, ensuring reliability and proper operation is of great importance in system analysis and management of its operation. One of the measures to assess the quality of a solution is system performance. Correct and expected operation gives the expected results - both in terms of size, time of achievement and costs (outlays) of receiving them. These expectations are achieved by ensuring reliable system operation. The performance of the system is therefore affected by the reliability of its components and its structure. At the same time, not only the reliability of the system depends on these factors. In the event of a failure, it is important to be able to _localize the damage_ more easily and to remove it (repair it). Therefore, it is obvious that not all elements have the same effect on the functioning of the system. To improve system reliability and readiness, as well as streamline maintenance and repair activities, the importance of system components should be explored for both reliability and maintenance - including diagnostics, maintenance and repairs. Without proper analysis, it is impossible to predict the significance of individual elements for these features. Individual elements may affect each of them to a different degree. There are known results on the evaluation of the weight of components on the reliability of the system. The introduced measures of significance of elements on reliability will be the basis for the introduction of diagnostic algorithms, about the possibility of which they wrote at the end of his seminal paper by Birnbaum(1968, 1969) (v. Barlow et al. (1975)). The indication of these algorithms is the subject of this study. In order to determine the significance of the reliability of individual system components to the reliability of the whole system, measures are constructed that are sensitive to modifications in the system. This allows the rationalization of the structure and subsequent planning for optimal maintenance. The issues are complex due to the fact that it is necessary to take into account both the effective reliability of the constructed system and the cost of maintaining it in readiness in a given period. Profitability analysis is of great importance. It is natural to formulate the problem by defining the primary goal of minimizing the cost while guaranteeing the expected levels of reliability. With this approach, it is possible to define weights for the cost of individual elements in a given time horizon, while ensuring a certain level of security or readiness. This approach can be found at paper by Wu and Coolen(2013). At the same time, one should not forget about the other key goals and parameters in system analysis. Their inclusion in the balanced model is possible with the use of natural methods of analysis in the formulation of many criteria, based on elements of game theory. We are trying to present the issue comprehensively, although there is currently no consistent approach on the way of determining the importance of elements in the system. This is because the loss of functionality of an element often does not clearly affect the system’s ability to perform tasks. This aspect is highlighted by numerous examples presented in the literature, which show a significant impact of the state of the environment in which the systems are operated (time of day, weather conditions, environmental pollution). In addition, attention should be paid to the cause-effect relationships of the work of the elements. We often deal with a sequential progression of damage and degradation of elements, which means that it is possible to propose a modeling method without the possibility of creating a universal model, the calibration of which allows for a proper description of the analyzed problem. The methodological limitations mentioned here mean that the proposed methods are a development of the problems that we mention, but we do not exclude that the approach may also enrich other analyzes based on other premises and conditions. In order to organize the methodology, we will use the systems classification, which will allow us to formulate assumptions. Wherever possible, we provide the source of inspiration (a description of an issue in which the proposed approach can be modeled, or we cite sources in the literature that use an analogous model), although we realize that getting to the original formulation of a concept or approach does not have to be the best justification and motivation for the proposed approach. ### 1.2 Availability for planned tasks determines reliability. We analyze the system (layout, structure) as one whole, carrying out a specific simple task. We consider systems that are made up of elements. The system is operational if it can accomplish the task for which it was created. With this formulation, we assume that the task execution time is infinitely short, so the possibility of failure in the course of the task can be neglected. The analysis of the role of the components in such a system comes down to the assessment of the impact of the reliability of a specific element on the reliability of the whole. For this category of tests, measures of the importance of elements will be helpful, which allow for the assessment (measure) of the improvement in the system reliability resulting from the improvement of the reliability of a given component. Such measures are useful in determining the components whose reliability should be improved to obtain the maximum improvement in system reliability. Examples of such measures can be found in the works Birnbaum et al. (1961), Birnbaum (1969), Natvig (1985), Boland et al. (1988). We want, at this level of generality, to measure the weight of an element related to its place in the structure, and structure has a role when the system is intentionally designed. This analysis is also performed when the reliability of the components is unknown. Hence we say that we are looking for a measure of the significance of the structural element (structural importance measure). The factor that we want to include in the analysis is not only the position, but also the reliability of the element. While still maintaining the assumption that the system takes an infinitely short time to complete a task, we introduce a measure of the element’s significance for reliability reasons (reliability importance measure). If the time needed to perform the task cannot be omitted, or the tasks are repeated, and we know the reliability functions of the elements, the element significance measure should also take into account the changes in the reliability of elements over time. The inclusion of the reliability function in the element significance analysis can be performed in various ways: global, local or for a fixed time period ( various lifetime importance measure). The aspects presented relate to the readiness to perform the task, excluding the need for maintenance and repair, including the costs of these activities (cost of parts, repair and maintenance time, penalties for non-availability). In system maintenance tasks, in determining component importance, issues such as detecting damaged components at system shutdown are important. The element that should be checked in the first place (because it is most suspected of causing a failure) can be treated as important for the efficient conduct of maintenance or repair (v. e.g. Ping (2004)). ### 1.3 Raised the role of the element in failure. At the time of failure (and not analysis during construction), the system analyzes the maintenance team. It can monitor the state of the system. He wants to find out what the elements meant for the observed state. To facilitate this analysis, we determine the posterior weights of the elements. Otherwise, in these considerations, the measure of importance of a component (group of components) in a given system is based on the quantification of the "role" of that component (group of components) in the failure of that system. Examples of such measures can be found in Fussell and Vesely (1972), Barlow and Proschan (1975), El-Neweihi et al. (1978), El-Neweihi and Sethuraman (1991) and Abouammoh et al. (1994). Defined measures (indices) of significance allow us to identify the components (groups) that are probably responsible for "causing" a system failure. Establishing these indexes, in turn, leads to an effective control and maintenance principle, as well as optimizing the storage of spare parts and optimal allocation of repairs to the appropriate maintenance technicians of the relevant system components. The purpose of such research is to propose new importance measures for degrading components (v. Cao et al. (2019)). The motivation is based on Shapley value, which can provide answers about how important players are to the whole cooperative game and what payoff each player can reasonably expect. The proposed importance measure characterizes how a specific degrading component contributes to the degradation of system reliability by using Shapley value. Degradation models are also introduced to assess the reliability of degrading components. The reliability of system consisting independent degrading components is obtained by using structure functions, while reliability of system comprising correlated degrading components is evaluated with a multivariate distribution. The ranking of degrading components according to this importance measure depends on the degradation parameters of components, system structure and parameters characterizing the association of components. A reliability degradation of engineering systems and equipment are often attributed to the degradation of a particular or set of components that are characterized by degrading features. This approach reflects the responsibility of each degrading component for the deterioration of system reliability. The results are also able to give timely feedback of the expected contribution of each degrading component to system reliability degradation. ### 1.4 General systems classification The systems can be split into two categories: 1. (i) Binary systems (BS) 2. (ii) Multistate systems (MSS) A binary system (i) is a system comprised of $n$ elements. It has precisely two states: $0$ \- when the system is failed and $1$ \- when the system is functioning. However, term "binary" pertains to the components of the system that define it. In this case components may be in only one of two states $1$ \- when the component is functioning perfectly and $0$ \- when the element is absolutely damaged. Nevertheless, binary systems not always meet the real life problems. Frequently we have to reckon with elements that undergo only partial failure, but do not cease to perform their operation and do not cause the entire system to cease function. This is the case of the multistate systems (ii) with the same properties as (i) beside states of components. Binary systems are discussed in chapter 2, while the discussion of multistate systems are moved to next paper. There are three main classes of importance measures (v. Birnbaum(1969), Amrutkar and Kamalja(2017)) 1. (i) Reliability importance measure 2. (ii) Structural importance measure 3. (iii) Lifetime importance measure 4. (iv) Failure and its recovery costs importance measure Reliability importance measure (i) focuses on the change in the reliability of the system due to reliability change of the particular component. The measure is evaluated with respect to the specific finite period of time and depends on the components reliability and on the system structure. Nonetheless, if reliability of the components are unknown, then we consider the case of the structural importance measure (ii). To apply it we are obligated to know the structure of the system. Hence, this measure indicates importance of the system by checking significance of the positions occupied by individual components. The lifetime importance measure (iii) depends on the lifetime distribution of component and also on component position in the system. This measure can be divided into two categories with respect to being the function of time: Time Independent Lifetime importance and Time Dependent Lifetime importance. Lastly but not least, the cost of failure and its recovery importance measure (iv) depends on the lifetime distribution of component, its position in the system and loss related to non-availability of the system, diagnosis and repair. It is a new look at the importance of the components of a complex system. The analysis and significance measure proposed in this paper is based on the possibility of observing the components and a rational system maintenance policy, which consists in stopping the system for maintenance and repairs at a time when it pays off to a sufficient number of components. The details are based on a cooperative analysis of costs and losses in the operation of such a system (v. the section 2.7, Szajowski and Yasuda (1997)). ### 1.5 Review of importance measure concepts Since Birnbaum(1968, 1969) the importance measures were investigated and extended in various directions (v. Amrutkar and Kamalja(2017)). Ramamurthy(1990) shows the relation of these idea to the research on the cooperative games. These relationships can be helpful in determining the importance of elements for the reliability of the system and at the same time a role in the possibility of efficient diagnosis in the event of a failure, as well as in determining the rules of procedure for removing a failure. Removing the failure causes that the features of the element and the repaired module are restored. However, it should be remembered that the method of repair and the quality of the elements used reproduce the original features to varying degrees (v. e.g. Navarro et al.(2019)). This has an impact on further operation, diagnosis and maintenance (uplift). Rules are easier to set when they are associated with objective measures of the features of components, modules and the system. Analysis of significance measures in the context of repairs helps to understand such relationships. Let us therefore establish these relationships (v Do and Bérenguer (2020)). ###### Definition 1 (The structure) For a non-empty and finite set $\mathbf{N}$333The list of symbols and abbreviations used in the work has been collected in the section abbreviation on page LABEL:skroty., we denote by $\mathcal{P}$ the family of subsets $\mathbf{N}$ having the following properties 1. (1) $\emptyset\in{\mathcal{P}}$; 2. (2) ${\mathbf{N}}\in{\mathcal{P}}$; 3. (3) $S\subseteq T\subseteq{\mathbf{N}}$ and $S\in{\mathcal{P}}$ imply $T\in{\mathcal{P}}$. The family $\mathcal{P}$ is called structure. This basic structure has been studied in many areas under a variety of names. The monograph by Ramamurthy(1990) unified the definitions and concepts in two main fields of application, that is cooperative game theory (simple games) (v. Appendix 0.B, Chapt. 10 in Tijs(2003)) and reliability theory (semi-coherent and coherent structures, v. Esary and Proschan (1963), Barlow and Wu (1978), Ohi (2010)). In reliability theory, consider the set ${\mathbf{N}}=\\{1,2,\ldots,n\\}$ of components with which a system $g$ has been built. The state of the system as well as any component can either be $0$ (a failed state) or $1$ (a functioning state). The knowledge of the system is represented by the knowledge of the structure function of the system which is defined as a switching function (boolean) $g:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ of $n$ variables (or $n$ dimensional vector $\vec{x}$)444With the same symbol, we denote the system and the analytical description of the system using the structure function wherever it does not lead to misunderstandings.. The structure function $g$ (simply the structure $g$) is called semi-coherent if 1. (1) $g$ is monotone, i.e. $\overrightarrow{x}\preceq\overrightarrow{y}$ implies $g(\overrightarrow{x})\leq g(\overrightarrow{y})$; 2. (2) $g(\vec{0})=0$ and $g(\vec{1})=1$. The semi-coherent structure can be called coherent when all its elements are significant. A subset $A\subset{\mathbf{N}}$ is called a path set of $g$, if $g(\vec{1}^{A},\vec{0}^{N\setminus A})=1$, i.e. the system is working if the items forming the set $A$ [resp. $N\setminus A$] are working [resp. failed]. Similarly, $A\subset{\mathbf{N}}$ is called a cut set of $g$, if $g(\vec{0}^{A},\vec{1}^{N\setminus A})=0$. Obviously, the assemblage of path [cut] sets of a semi-coherent structure $g$ satisfies the three properties of the basic structure mentioned at the beginning. ### 1.6 Cooperative games vs. semi-coherent systems [30, Sec. 2] indicates the correspondence between the terminology of cooperative game theory and reliability by means of a list of equivalent notions: player or component; simple game or semi-coherent structure; characteristic function or structure function; winning [blocking] coalition or path [cut] set; minimal winning [blocking] coalition or minimal path [cut] set. The review of the various types of simple games and semi-coherent structures encountered in the literature are mentioned there. The most interesting is [30, Ch. 3], where detailed study of the problem of assessing the importance [power] of components [players] comprising the system [game] is described. The emphasis is on the probabilistic approach to the quantification of relative importance. ## 2 Binary systems ### 2.1 Preliminary remarks Importance measures are helpful in deciding on the development of which element to emphasize in order to improve the functioning of the system, through indicating those more meaningful. A system yield function was a concept of a general measure of importance, firstly introduced by Birnbaum(1968). His idea take into account the structure of the system only. Further, the research on the topic went in various direction (cf. Xie(1987)). New variants of importance measures can be found in Dui et al.(2017), Wu and Coolen(2013), Dutuit and Rauzy(2015). Importance measures have been widely used as important decision-aiding indicators in various purposes such as reliability studies, risk analysis and maintenance optimization. A novel time- dependent importance measure for systems composed of multiple non repairable components is proposed by Do and Bérenguer (2020). The proposed importance measure of a component (module of components) is defined as its ability to improve the system reliability during a mission given the current conditions (states or degradation levels) of its components. To take into account economic aspects, like e.g. maintenance costs, economic dependence between components and the cost benefit thanks to maintenance operations, an extension of the proposed importance measure is then investigated. Thanks to these proposed importance measures, the component (group of components) can be _rationally_ selected for preventive maintenance regarding to the reliability criteria or the financial issues. The new treatment of the mentioned topic is the subject of the section 2.7. ### 2.2 Coherence and system structure In this paper, we will consider coherent structures, i.e. that are nondecreasing functions. We call these structures monotonic. We will not consider structures whose state does not depend on the states of their elements. ###### Definition 2 The structure $\phi$ is called semi-coherent if for states $\overrightarrow{x}$ and $\overrightarrow{y}$, such that $\overrightarrow{x}\preceq\overrightarrow{y}$ implies $\phi(\overrightarrow{x})\leq\phi(\overrightarrow{y}),$ and coherent if additionally it complies with $\phi(\vec{1})=1$ and $\phi(\vec{0})=0$. In multi-component system to classify a structure as coherent, we have to introduce more notation and some properties [11], [10]. Thus, at the very beginning we assume that $n$ components comprise the system, denoted by $\vec{c}=(c_{1},c_{2},\dots,c_{n})$. Of the two available states (S)- failed (F) or functioning (working W) - each component can only have one, what can be defined by a binary indicator variable $x_{i}=\mathbb{I}_{\textbf{W}}(c_{i})$, $c_{i}\in\\{\textbf{F},\textbf{W}\\}$ for every $i=1,2,\ldots,n$. In other words, it is a state vector (vector of component states) $\overrightarrow{x}=(x_{1},x_{2},\ldots,x_{n})$. The comparison of the state vectors can be described with following notation [10] based on the component states for $i=1,\ldots,n$: $\displaystyle\overrightarrow{x}=\overrightarrow{y}$ if $y_{i}=x_{i}$, $\displaystyle\overrightarrow{x}\preceq\overrightarrow{y}$ if $x_{i}\geq y_{i}$, $\displaystyle\overrightarrow{x}\prec\overrightarrow{y}$ if $\overrightarrow{x}\preceq\overrightarrow{y}$ and $\overrightarrow{x}\neq\overrightarrow{y}$ Moreover, we assume that a system composed of $n$ elements whose states are binary also has only two states possible - failed or functioning. Let $\phi:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ be the structure function. If inequality $x_{i}\leq y_{i}$ for $i=1,\ldots,n$ fulfills conditions from the definition 2 and the structure is monotonic and irreducible, then the structure function $\phi$ is called coherent. The structure function $\phi$ for every $j=1,2,\ldots,n$ may be presented in the manner of $\displaystyle\phi(\overrightarrow{x})$ $\displaystyle=x_{j}\cdot\delta_{j}(\overrightarrow{x})+\mu_{j}(\overrightarrow{x})$ (1) where $\displaystyle\delta_{j}(\overrightarrow{x})$ $\displaystyle=\phi(1,\overrightarrow{x}_{-j})-\phi(0,\overrightarrow{x}_{-j})$ (2) $\displaystyle\mu_{j}(\overrightarrow{x})$ $\displaystyle=\phi(0,\overrightarrow{x}_{-j}).$ (3) Hence, the component $c_{j}$ with the state $x_{j}$ does not influence $\delta_{j}(\overrightarrow{x})$ and $\mu_{j}(\overrightarrow{x})$. ### 2.3 Reliability importance. If for $i=1,\ldots,n$ we consider independent elements $X_{i}$, then the system reliability is defined as a function of reliability of its components, which is equal to the probability that the whole system will keep functioning. Let assume the coherent system with a vector $\overrightarrow{\textbf{p}}=(p_{1},\dots,p_{n})$ of the components reliabilities, and in that case the reliability function is expressed as $h(\overrightarrow{\textbf{p}})=\textbf{P}\\{\omega:\phi(\overrightarrow{X}(\omega))=1\|\overrightarrow{\textbf{p}}\\}=\textbf{E}[\phi(\overrightarrow{X})\|\overrightarrow{\textbf{p}}],$ (4) where $h(\overrightarrow{\textbf{p}})$ is the reliability of the structure $\phi$ as the function of the reliability of their components. From equations (1) and (4) we have $h(\overrightarrow{\textbf{p}})=p_{i}\cdot\textbf{E}[\delta_{i}(X)]+\textbf{E}[\mu_{i}(X)]$ (5) for every $i=1,\ldots,n$ and from (1) and (5) we obtain the _reliability importance_ of the component $c_{i}$ in the system $\phi$ $I_{\phi}(i;\overrightarrow{\textbf{p}})=I_{h}(i;\overrightarrow{\textbf{p}})\stackrel{{\scriptstyle\text{\cite[cite]{[\@@bibref{Number}{spivak1965calculus}{}{}]}}}}{{=}}\textbf{D}_{p_{i}}h(\overrightarrow{\textbf{p}})=\frac{\partial}{\partial p_{i}}h(\overrightarrow{\textbf{p}})=\frac{\partial h(\overrightarrow{\textbf{p}})}{\partial p_{i}}\stackrel{{\scriptstyle\eqref{hp}}}{{=}}\textbf{E}[\delta_{i}(\overrightarrow{X})],$ (6) which was first introduced by Birnbaum (1969). These _importance measures_ are known as a vector $\vec{B}(\overrightarrow{\textbf{p}})$ having coordinates $B(i\|\overrightarrow{\textbf{p}})=\textbf{D}_{p_{i}}h(\overrightarrow{\textbf{p}})=\textbf{D}_{1-p_{i}}(1-h(\overrightarrow{\textbf{p}})),\qquad i=1,2,...,n,$ (7) where $B(i\|\overrightarrow{\textbf{p}})$ is $\overrightarrow{\textbf{p}}$ dependent. If reliabilities $\overrightarrow{\textbf{p}}$ are unknown, we obtain the _structural importance_ , defined as $B(i\|\overrightarrow{\textbf{p}})=I_{\phi}(i;\overrightarrow{\textbf{p}})=\textbf{D}_{p_{i}}h(\overrightarrow{\textbf{p}})\Bigg{\rvert}_{p_{1}=\ldots=p_{n}=\frac{1}{2}},\qquad i=1,2,...,n,$ (8) what will be discussed in section 2.6. The _reliability importance_ (v. [10]) of a component $c_{i}$ is defined as $\displaystyle I_{\phi}(i,r;\overrightarrow{\textbf{p}})$ $\displaystyle=\textbf{P}\\{\phi(\overrightarrow{X})=r\|X_{i}=r;\overrightarrow{\textbf{p}}\\}-\textbf{P}\\{\phi(\overrightarrow{X})=r\|\overrightarrow{\textbf{p}}\\},$ $\displaystyle=\textbf{P}\\{\phi(\overrightarrow{X})=r\|(r,\overrightarrow{\textbf{p}}_{-i})\\}-\textbf{P}\\{\phi(\overrightarrow{X})=r\|\overrightarrow{\textbf{p}}\\},$ for the functioning of the structure $\phi$ with $r=1$, while for the failure of the structure $\phi$ with $r=0$. Hence, the _compound reliability importance_ of the component $c_{i}$ for the structure $\phi$ is $\displaystyle I_{\phi}(i;\overrightarrow{\textbf{p}})$ $\displaystyle=I_{\phi}(i,1;\overrightarrow{\textbf{p}})+I_{\phi}(i,0;\overrightarrow{\textbf{p}}),$ (9) what is exactly equal to $\displaystyle I_{\phi}(i;\overrightarrow{\textbf{p}})$ $\displaystyle=\frac{\partial h(\overrightarrow{\textbf{p}})}{\partial p_{i}}=\textbf{E}[\delta_{i}(\overrightarrow{X})]$ (10) We can easily get (10) from (9) $\displaystyle I_{\phi}(i;\overrightarrow{\textbf{p}})$ $\displaystyle=I_{\phi}(i,1;\overrightarrow{\textbf{p}})+I_{\phi}(i,0;\overrightarrow{\textbf{p}})$ $\displaystyle=\textbf{P}\\{\phi(\overrightarrow{X})=1\|X_{i}=1;\overrightarrow{\textbf{p}}\\}-\textbf{P}\\{\phi(\overrightarrow{X})=1;\overrightarrow{\textbf{p}}\\}$ $\displaystyle\quad+\textbf{P}\\{\phi(\overrightarrow{X})=0\|X_{i}=0;\overrightarrow{\textbf{p}}\\}-\textbf{P}\\{\phi(\overrightarrow{X})=0;\overrightarrow{\textbf{p}}\\}$ $\displaystyle=\textbf{P}\\{\phi(\overrightarrow{X})=1\|X_{i}=1;\overrightarrow{\textbf{p}}\\}-(1-\textbf{P}\\{\phi(\overrightarrow{X})=0\|X_{i}=0;\overrightarrow{\textbf{p}}\\})$ $\displaystyle=\textbf{P}\\{\phi(\overrightarrow{X})=1\|X_{i}=1;\overrightarrow{\textbf{p}}\\}-\textbf{P}\\{\phi(\overrightarrow{X})=1\|X_{i}=0;\overrightarrow{\textbf{p}}\\}.$ Hence, the equivalent definition of the reliability importance [43] is $I_{h}(i;\overrightarrow{\textbf{p}})=h(1,\overrightarrow{\textbf{p}}_{-i})-h(0,\overrightarrow{\textbf{p}}_{-i})=\textbf{E}\big{[}\phi(1_{i},\overrightarrow{X})-\phi(0_{i},\overrightarrow{X})\big{]}=\textbf{E}\delta_{i}(\overrightarrow{X}).$ (11) For the coherent system, the reliability of each element and the reliability importance belongs to the interval $(0,1)$. From (11) we obtain $I_{h}(i;\overrightarrow{\textbf{p}})=\textbf{P}\\{\phi(1_{i},\overrightarrow{X})-\phi(0_{i},\overrightarrow{X})=1\\}.$ (12) From equations (11) and (12) we conclude that $I_{h}(i)$ can be interpreted as the probability that a system has a state, in which it is spoiled due to the $i$-th element being out of order. ###### Example 1 (Birnbaum reliability importance - series) Let us consider the series structure presented in Fig. 1 composed of three independent components, where each component $c_{1},c_{2},c_{3}$ has a corresponding reliability $\overrightarrow{\textbf{p}}=(0.95,0.99,0.96)$. $c_{1}$$c_{2}$$c_{3}$ Figure 1: Series structure Then, at any time $t$ the system reliability is equal to $h(\overrightarrow{\textbf{p}})=\prod^{3}_{i=1}p_{i}=0.90288$ and the Birnbaum reliability importance (7) for components $c_{1},c_{2},c_{3}$ is $(B(1\|\overrightarrow{\textbf{p}}),B(2\|\overrightarrow{\textbf{p}}),B(3\|\overrightarrow{\textbf{p}}))=(\prod_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 1}}^{3}p_{i},\prod^{3}_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 2}}p_{i},\prod^{3}_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 3}}p_{i})=(0.9504,0.912,0.9405).$ In the series system we may see that the component having the smallest reliability is the most meaningful for the system. ###### Example 2 (Birnbaum reliability importance - parallel) Let us consider the parallel structure presented in Fig. 2 composed of three independent components $c_{1}$$c_{2}$$c_{3}$ Figure 2: Parallel structure. where components $c_{1},c_{2},c_{3}$ have the same reliabilities like in Example 2. Then, the system reliability at time $t$ is equal to $h(\overrightarrow{\textbf{p}})=\coprod^{3}_{i=1}p_{i}=1-\prod^{3}_{i=1}(1-p_{i})=0.99998$ and the Birnbaum reliability importance (7) for components $c_{1},c_{2},c_{3}$ is $\displaystyle(B(1\|\overrightarrow{\textbf{p}}),B(2\|\overrightarrow{\textbf{p}}),B(3\|\overrightarrow{\textbf{p}}))$ $\displaystyle=(\prod_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 1}}^{3}(1-p_{i}),\prod^{3}_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 2}}(1-p_{i}),\prod^{3}_{\genfrac{}{}{0.0pt}{}{i=1}{i\neq 3}}(1-p_{i}))$ $\displaystyle=(0.0004,0.002,0.0005).$ In the parallel system we may see that the component having the greatest reliability is the most relevant for the system. ### 2.4 Lifetime importance measure. If $n$ components comprise the system, then we assume that for $t\geq 0$ and $i=1,2,...,n$ a stochastic process $X_{i}(\omega,t)$ defines the $i$-th component’s state with $X_{i}(\omega,t)$ equal to $0$ or $1$, reliant on failure or functioning of the system at moment $t$, respectively. Let $\xi_{i}(\omega)=\inf\\{t\in\Re^{+}:X_{i}(\omega,t)=0\\}$–the life time of $i$th element and denote $Q_{i}(s)=\textbf{P}\\{\omega;\xi_{i}(\omega)\geq s\\}$. Assuming continuous life distribution of the $i$-th component $Q_{i}(t)=\textbf{P}\\{\omega:X_{i}(\omega,t)=1\\}$ and the structure $\phi$, at each moment $t$ there is defined the reliability of the structure by the adequate function $h(\overrightarrow{Q}(t))$ (see (4)). Based on these denotation we have the process of system states $X(\omega,t)=\phi(\overrightarrow{X}(\omega,t))$ and the system’s reliability function $Q(t)$ can be derived. Hence, $\displaystyle Q(t)=h(\overrightarrow{Q}(t))$ $\displaystyle=h\big{(}{Q}_{1}(t),{Q}_{2}(t),\ldots,{Q}_{n}(t)\big{)}$ (13) $\displaystyle=\textbf{P}\\{\omega:\phi(\overrightarrow{X}(\omega,t))=1\\}=\textbf{E}\big{[}\phi(X(\omega,t)\big{]}.$ (14) Let us calculate the density function $f(t)=-Q^{\prime}(t)=-q(t)$ of the survival time distribution of the structure with the structure function $h$. $\displaystyle f(t)=-\frac{d}{dt}Q(t)$ $\displaystyle\stackrel{{\scriptstyle\eqref{RelFunSys}}}{{=}}-\left\langle\nabla h(\overrightarrow{Q}(t)),\overrightarrow{\textbf{q}}(t)\right\rangle$ (15) $\displaystyle\stackrel{{\scriptstyle\eqref{Gradh}}}{{=}}-\left\langle\overrightarrow{\textbf{I}}_{h}(\overrightarrow{Q}(t)),\overrightarrow{\textbf{q}}(t)\right\rangle=-\left\langle\textbf{E}[\overrightarrow{\delta}(\overrightarrow{X})],\overrightarrow{\textbf{q}}(t)\right\rangle.$ First, Birnbaum in 1968 introduced importance measures for fixed function of time $t$, while Barlow and Proschan in 1975 freed the measure from the time- dependence. They proposed probability that the system and $i$-th component failures coincide, which means that the $i$-th component impaired the system. ###### Fact 2.1 For $i=1,2,...,n$ let the $i$-th component have a distribution $F_{i}$, reliability $Q_{i}$ and density $f_{i}$. Then, the probability that system failure occurred at time t and was caused by a component $i$ is defined as $\frac{f_{i}(t)\cdot[h(1,\overrightarrow{Q}_{-i}(t))-h(0,\overrightarrow{Q}_{-i}(t))]}{\left\langle\nabla h(\overrightarrow{Q}(t)),\overrightarrow{\textbf{f}}(t)\right\rangle}=\frac{f_{i}(t)\cdot I_{h}(i;\overrightarrow{Q}(t))}{\sum_{k=1}^{n}f_{k}(t)\cdot I_{h}(k;\overrightarrow{Q}(t))}.$ (16) ###### Proof The probability of the system functioning at time $t$ if the $i$-th element is functioning and that the system is not functioning at time $t$ if the $i$-th element is not functioning is $\displaystyle P\big{[}\phi(1,\overrightarrow{X}_{-i}(t))-\phi(0,\overrightarrow{X}_{-i}(t))=1\big{]}$ $\displaystyle=h(1,\overrightarrow{Q}_{-i}(t))-h(0,\overrightarrow{Q}_{-i}(t))$ $\displaystyle=I_{h}(i;\overrightarrow{Q}(t)).$ (17) Therefore, the numerator in (16) multiplied by $dt$ represents the probability that in the interval $[t,t+dt]$ the $i$-th component led to the failure of the system and the denominator multiplied by $dt$ stands for the probability that the system failed in the given interval [5]. ###### Fact 2.2 In consequence of equation (17), the probability of $i$ causing the system failure in time interval $[0,t]$, while the system failure occurs in the same period of time $[0,t]$ is $\frac{\int_{0}^{t}\cdot I_{h}(i;\overrightarrow{Q}(u))dF_{i}(u)}{\sum_{k=1}^{n}\int_{0}^{t}\cdot I_{h}(k;\overrightarrow{Q}(u))dF_{k}(t)}=\frac{\int_{0}^{t}[h(1,\overrightarrow{Q}_{-i}(u))-h(0,\overrightarrow{Q}_{-i}(u))]dF_{i}(u)}{\int_{0}^{t}\sum_{k=1}^{n}[h(1,\overrightarrow{Q}_{-k}(u))-h(0,\overrightarrow{Q}_{-k}(u))]dF_{k}(u)}.$ (18) When in equation (18) $t\to\infty$, then it is a probability of $i$ leading the system to the total failure. In this regard, the denominator is equal to $1$. We assume that this limit is a definition of component importance. ###### Definition 3 As a consequence of (18), the probability of $i$ causing the system failure is denoted as $I_{h}(i;\overrightarrow{Q})=\int\limits_{0}^{\infty}[h(1,\overrightarrow{Q}(t))-h(0,\overrightarrow{Q}(t)(t))]dF_{i}(t)$ (19) where $I_{h}(i;\overrightarrow{Q})$ is precisely the lifetime importance measure of the $i$-th component. ###### Fact 2.3 Importance measure properties 1. 1. $I_{h}(i;\overrightarrow{Q})\in[0,1]$ 2. 2. $I_{h}(i;\overrightarrow{Q})\in(0,1)$ if $n\geq 2$ 3. 3. $\sum_{i=1}^{n}I_{h}(i;\overrightarrow{Q})=1$ However, Birnbaum in 1969 extended reliability importance of components, he was not able to free measure from time dependence though. Probability distribution $F_{i}(t)=P\\{\xi_{i}\leq t\\}$ was considered with assumption of each $i$-th component having a life length $\xi_{i}$ [10]. Therefore, using this assumption and those from the beginning of this section, we have the lifetime importance measure given by $\displaystyle I_{h}^{i}(t)=\textbf{P}\Big{[}\phi\big{(}1,\overrightarrow{X}_{-i}(t)\big{)}-\phi\big{(}0,\overrightarrow{X}_{-i}(t)\big{)}=1\Big{]}=$ (20) $\displaystyle=h(1,\overrightarrow{Q}_{-i}(t))-h(0,\overrightarrow{Q}_{-i}(t)),$ (21) what describes probability at time $t$ that the system is in the state $t$ in which the $i$-th component is crucial for the system. If the $i$-th component is series or parallel to the system, then it has a corresponding formula to the structural importance case [41]. ### 2.5 Module importance Multi-component and coherent system may be partitioned into modules, which in other words are sub-systems consisting of different components. As Birnbaum [10] proposed, a module importance for fixed time with coherent structure $\phi$ is expressed by $\phi(x)=\phi(x_{1},x_{2},...,x_{n})=x_{1}\cdot\delta_{x_{1}}(\phi;x)+\mu_{x_{1}}\cdot(\phi;x)$ (22) and coherent structure $\Psi(y)$ denoted as $\Psi(y)=\Psi(y_{1},y_{2},...,y_{m}),$ (23) we may achieve the structure $\chi$, if in $\phi(x)$ an element $x_{1}$ is substituted by the coherent module $\Psi(y)$: $\displaystyle\chi(y_{1},...,y_{m},x_{2},...,x_{n})=\phi[\Psi(y_{1},...,y_{m}),x_{2},...,x_{n}]=\phi[\Psi_{1}(y),x]=$ (24) $\displaystyle=\Psi(y)\cdot\delta_{x_{1}}\cdot[\phi;x]+\mu_{x_{1}}\cdot[\phi;x].$ (25) From (25) we deduce that $\begin{gathered}\delta_{x_{1}}(\chi;y_{1},...,y_{m},x_{2},...,x_{n})=\\\ =\chi(1,y_{2},...,y_{m},x_{2},...,x_{n})-\chi(0,y_{2},...,y_{m},x_{2},...,x_{n})=\\\ =\delta_{x_{1}}(\phi;x)\cdot[\Psi(1,y)-\Psi(0_{1},y)]=\delta_{y_{1}}(\Psi;y)\cdot\delta_{x_{1}}(\phi;x).\end{gathered}$ (26) If we base on equations (10) and (26), then we obtain the importance of a module defined as $I_{y_{i}}(\chi;y_{1},...,y_{m},x_{2},...,x_{n})=I_{x_{i}}(\phi;x)\cdot I_{y_{i}}(\Psi;y).$ (27) For the system $\chi$ we may derive the importance of every component comprising the module $\Psi$ by repeating the procedure of substituting modules for components till none is left [10]. Different definition of module importance of the coherent system was proposed by Barlow and Proschan in [5] (cf. [8]). ###### Definition 4 For $n$ components let introduce a structure $\phi$ that is coherent, subset of 1,2,…,n given by $M$ and its complement $M^{C}$, and coherent system $\chi$ comprised of components in $M$. Then, the module $(M,\chi)$ of the coherent system $\phi$ is defined as $\phi(x)=\Psi[\chi(x^{M}),x^{M^{C}}],$ (28) where $x^{M^{C}}$ is a complement of a subset $M$. The module importance $I_{h}(M)$ is the probability of the module causing system failure. ###### Theorem 2.4 If $i\in M$ and $f$ denotes module’s reliability function, then $I_{h}(i)=\int\limits_{0}^{\infty}\big{[}h(1^{M},\bar{F}(t))-h(0^{M},\bar{F}(t))\big{]}\cdot\big{[}f(1_{i},\bar{F}(t))-f(0_{i},\bar{F}(t))\big{]}d\bar{F}_{i}(t)$ (29) and $I_{h}(M)=\sum_{i\in M}I_{h}(i)$ (30) ###### Proof (29) Probability of functioning of the system at time $t$, if and only if the module functions firmly, is represented by $h(1^{M},\bar{F}(t))-h(0^{M},\bar{F}(t))=P\big{[}\phi(1^{M},X(t))-\phi(0^{M},X(t))=1\big{]},$ while the probability of module functioning at time $t$, if and only if the component $i$ functions, is denoted as $f(1_{i},\bar{F}(t))-f(0_{i},\bar{F}(t))=P\big{[}\chi(1_{i},X(t))-\chi(0_{i},X(t))=1\big{]},$ In the system with modules, component $i$ may only cause system failure through module failure, hence $\displaystyle\sum_{i\in M}I_{h}(i)=\int\limits_{0}^{\infty}\big{[}h(1^{M},\bar{F}(t))-h(0^{M},\bar{F}(t))\big{]}\cdot\sum_{i\in M}\big{[}f(1_{i},\bar{F}(t))-f(0_{i},\bar{F}(t))\big{]}dF_{i}(t)=$ $\displaystyle=-\int\limits_{0}^{\infty}\big{[}h(1^{M},\bar{F}(t))-h(0^{M},\bar{F}(t))\big{]}\frac{d}{dt}f(\bar{F}(t))dt=I_{h}(M)$ ###### Note 1 Definition of the module importance proposed by Birnbaum is slightly different from the one introduced by Barlow and Proschan. In Birnbaum’s definition importance of the module’s component is equal to the component’s importance for the module multiplied by the importance of the module for the system. This is not consistent with Barlow and Proschan definition due to the fact that for each $x$ expression $r(x)=s(x)\cdot u(x)$ doesn’t imply $\int_{a}^{b}r(x)dx=\int_{a}^{b}s(x)dx\cdot\int_{a}^{b}u(x)dx.$ ###### Lemma 1 If a component $i$ is serial to the system, then the importance $I_{h}(i)$ increases in $F_{i}(t)$ and $\bar{F}_{j}(t)$ when $i\neq j$. Otherwise, if component $i$ is parallel to the system, then the importance $I_{h}(i)$ decreases. ###### Proof With assumption that component $i$ is serial to the system, we obtain $I_{h}(i)=\int_{0}^{\infty}h(1_{i},\bar{F}(t))dF_{i}(t),$ while $h(0_{i},\bar{F}(t))=0$ due to the hypothesis. Since $h(1_{i},p)$ increases in each $p$, $I_{h}(i)$ increases in $\bar{F}_{j}(t)$, if $i\neq j$. Moreover, $h(1_{i},\bar{F}(t)$ decreases in $t$, therefore $I_{h}(i)$ increases in $F_{i}(t)$. ###### Lemma 2 If component $i$ is serial or parallel to the system and all components have the same distribution $F$, then for $i\neq j$ we obtain $I_{h}(i)\geq I_{h}(j)$. ###### Proof If we assume that $i$ is serial to the system and use the fact that components are stochastically alike, $I_{h}(k)$ may be treated as the permutations’ proportion from $1$ to $n$ corresponding to the failure of the system by cause of $k$. Hence, computation of $I_{h}(k)$ proceeds with the interchange of $j$ and $i$ in each permutation. This calculation method shows that the number of permutations, in which the failure is caused by $i$, is not smaller than the number of permutations in which the failure is caused by $j$. By using lemmas 1 and 2 we may introduce the following theorem 2.5. ###### Theorem 2.5 If we assume that the $i$-th component is serial or parallel to the system and $t\geq 0$, $j\neq i$, then the true is $F_{j}(t)\leq F_{i}(t)$ and $I_{h}(j)\leq I_{h}(i)$. ### 2.6 Structural importance At times we have to face the situation when information about component reliabilities are missing. In that case we have to consider the impact of various components to the system. And so, we define the structural importance. Measure introduced by Birnbaum [10] requires specifying structure function equalities (1), (2), (3). ###### Definition 5 1. a) A component $c_{i}$ is indispensable for the $\phi$ structure at the vector of states $x$ when $\phi(1_{j},x)-\phi(0_{j},x)=\delta_{j}(x)=1$ (31) 2. b) A component $c_{i}$ is indispensable at the vector of states $x$ for the functioning of the structure $\phi$ when $(1-x_{j})\cdot\delta_{j}(x)=1$ (32) 3. c) A component $c_{i}$ is indispensable at the vector of states $x$ for the failure of the structure $\phi$ when $x_{j}\cdot\delta_{j}(x)=1$ (33) To clarify, if $c_{i}$ is indispensable at the state vector $x$, then it is equally indispensable for both functioning or failure, when coordinates of the state vector $x$ equal to $0$ or $1$. Hence, the structural importance of a component $c_{i}$ for the functioning of $\phi$ is defined as $I_{\phi}(j,1)=2^{-n}\sum_{(x)}(1-x_{j})\cdot\delta_{j}(x),$ (34) where the sum covers all $2^{n}$ unit cube’s vertices. The structural importance of a component $c_{i}$ for failure of structure $\phi$ is defined as $I_{\phi}(j,0)=2^{-n}\sum_{(x)}x_{j}\cdot\delta_{j}(x)$ (35) and the structural importance of a component $c_{i}$ for the structure $\phi$ is defined as $I_{\phi}(j)=I_{\phi}(j,1)+I_{\phi}(j,0)=\sum_{(x)}\delta_{j}(x).$ (36) To conclude, if a component $c_{i}$ is indispensable at the state vector $\overrightarrow{x}$ for functioning of structure $\phi$, then the component $c_{i}$ is indispensable at $(1,\overrightarrow{x}_{-j})$ for failure, meanwhile, if a component $c_{i}$ is indispensable at the state vector $x$ for failure of structure $\phi$, then the component $c_{i}$ is indispensable at $(0,\overrightarrow{x}_{-j})$ for functioning. Due to communication between vertices at which $c_{i}$ is responsible for failure or functioning, the number of each type of vertices is the same. Hence, from equalities (34), (35) and (36) follows $I_{\phi}(j,1)=I_{\phi}(j,0)=\frac{1}{2}I_{\phi}(j).$ (37) From (37) we deduce that there is no purpose dividing the structural importance into the one for failure and the one for functioning, unlike the reliability importance. When we consider continuous life distribution of components, we shall use the structural importance measure introduced by Barlow and Proschan. The importance of component $c_{i}$ proposed in fact 2.1 with assumption that all components have the same life distribution $F_{1}=F_{2}=...=F_{n}$, then in structure $\phi$, such an importance becomes the structural importance of component $c_{i}$ denoted as $I_{\phi}(i)$. By substituting $p$ for $\bar{F}_{i}(t)$ for $i=1,...,n$, we obtain $I_{\phi}(i)=\int\limits_{0}^{1}[h(1_{i},p)-h(0_{i},p)]dp,$ (38) where vector $(1,\overrightarrow{\textbf{p}}_{-i})$ has $1$ in the $i$-th position and $p$ everywhere else i.e. $\overrightarrow{\textbf{p}}_{-i}=\vec{p}_{-i}$. To compute the structural importance presented by Barlow and Proschan (1975), first we need to introduce some definitions. ###### Definition 6 1. a) A set of elements that allow proper operating of the system is called a path set. If the path set is irreducible, then it is called a minimal path set. 2. b) At the same time, a set of elements that can by their own effect failure of the system is called a cut set. If the cut set is irreducible, then it is called a minimal cut set. 3. c) A vector $(1_{i},x)$, which fulfills conditions of $\phi(0_{i},x)=0$ and $\phi(1_{i},x)=1$, is called a critical path vector for the $i$-th component. Hence, for the $i$-th component a critical path set is denoted as $\\{i\\}\cup\\{j\|x_{j}=1,i\neq j\\}.$ It means that functioning of the system or its failure is determined by a component $c_{i}$. A critical path vector for a component $c_{i}$ with size $r$ can be presented as $1+\sum_{i\neq j}x_{j}=r,\qquad r=1,2,...,n.$ Therefrom, we may introduce a number of critical path vector $n_{r}(i)$ for $i$-th component of size $r$ specified as $n_{r}(i)=\sum_{\sum_{i\neq j}x_{j}=r-1}[\phi(1,\overrightarrow{x}_{-j})-\phi(0,\overrightarrow{x}_{-j})]$ Hence, the structural importance $I_{\phi}(i)$ may be expressed as the number of critical path vectors $n_{r}(i)$. ###### Theorem 2.6 $I_{\phi}(i)=\sum_{r=1}^{n}n_{r}(i)\cdot\frac{(n-r)!(r-1)!}{n!}$ (39) ###### Proof If we merge equations (38) and (39), we obtain $\displaystyle I_{\phi}(i)=\int\limits_{0}^{1}[h(1_{i},p)-h(0_{i},p)]dp=$ (40) $\displaystyle=\int\limits_{0}^{1}\Big{[}\sum_{x}[\phi(1_{i},x)-\phi(0_{i},x)]\cdot p^{\sum_{j\neq i}x_{j}}\cdot(1-p)^{n-1-\sum_{j\neq i}x_{j}}\Big{]}dp=$ (41) $\displaystyle=\int\limits_{0}^{1}\sum_{r=1}^{n}n_{r}(i)\cdot p^{r-1}(1-p)^{n-r}dp=\sum_{r=1}^{n}n_{r}(i)\cdot\frac{(n-r)!(r-1)!}{n!}.$ (42) Equation (39) can be expressed as $I_{\phi}(i)=\frac{1}{n}\sum_{r=1}^{n}n_{r}(i)\tbinom{n-1}{r-1}^{-1},$ (43) where the numerator $n_{r}(i)$ stands for the critical path vectors with size $r$ and the denominator stands for the number of results with precisely $r-1$ components functioning among the exactly $n-1$ components, without component $i$. It means that the $i$-th component’s structural importance is in other words the average probability that for the $i$-th component the vector is the critical path vector. Expression (42) can be also translated into $I_{\phi}(i)=\int\limits_{0}^{1}\Big{[}\sum_{r=1}^{n}n_{r}(i)\cdot\tbinom{n-1}{r-1}^{-1}\tbinom{n-1}{r-1}\cdot(1-p)^{n-r}\cdot p^{r-1}\Big{]}dp$ (44) where $\tbinom{n-1}{r-1}\cdot(1-p)^{n-r}\cdot p^{r-1}$ is the probability that $r-1$ among $n-1$ elements, omitting the $i$-th one, function. Furthermore, $n_{r}(i)\cdot\tbinom{n-1}{r-1}^{-1}$ is the probability that functioning components $i$ and $r-1$ comprise the critical path set for the component $i$. Hence, equation (44) stands for the probability of $i$ causing the system failure. Integrating it over $p$ means that the component reliability $p$ has a uniform distribution $p\sim\mathcal{U}(0,1)$. If we compare Barlow and Proschan structural importance $I_{\phi}(i;\overrightarrow{\textbf{p}})=\int\limits_{0}^{1}[h(1,\vec{p}_{-i})-h(0,\vec{p}_{-i})]dp,$ (45) with Birnbaum structural importance $B(i;\overrightarrow{.5})=I_{\phi}(i;\overrightarrow{.5})=\frac{\partial h(p)}{\partial p_{i}}\Bigg{\rvert}_{p_{1}=...=p_{n}=\frac{1}{2}}=h\big{(}1,\overrightarrow{.5}_{-i}\big{)}-h\big{(}0,\overrightarrow{.5}_{-i}\big{)}$ (46) we see that Birnbaum sets $p=\tfrac{1}{2}$ in order to compute the difference $h(1,\overrightarrow{\textbf{p}}_{-i})-h(0,\overrightarrow{\textbf{p}}_{-i})$, while Barlow and Proschan compute this difference for $p\in[0,1]$. Moreover, from equation (46) we can deduce that $B(i;\overrightarrow{\textbf{p}})=I_{\phi}(i;\overrightarrow{\textbf{p}})=\sum_{x}\frac{1}{2^{n-1}}\cdot[\phi(1,\overrightarrow{.5}_{-i})-\phi(0,\overrightarrow{.5}_{-i})].$ Hence, Birnbaum structural importance can be given as $I_{\phi}(i;\overrightarrow{.5})=\sum_{r=1}^{n}\frac{n_{r}(i)}{2^{n-1}}.$ (47) If we compare expressions (39) and (47), we can see that in $I_{\phi}(i)$ the number of critical vector path $n_{r}(i)$ has a weight $(n-r)!(r-1)!/n!$, meanwhile Birnbaum uses the same weight $1/(2^{n-1})$ everywhere. Due to behavior of $(n-r)!(r-1)!/n!$ for different $n$ we deduce that only very large or very small critical paths may reach the greatest weight [5]. ###### Example 3 (Structural importance - series / parallel structure) Let consider a structure of $n$ elements, which $k$ are in series and $n-k$ are parallel. This example concerns a structure of five components $c_{1},c_{2},c_{3},c_{4},c_{5}$, $n=5$ and $k=2$ presented in figure 3. $c_{1}$$c_{2}$$c_{3}$$c_{4}$$c_{5}$ Figure 3: Series and parallel structure Reliability of each component $c_{i}$ is unknown, however we may derive the system reliability for different $p_{i}$ $h(\overrightarrow{\textbf{p}})=p_{1}\cdot p_{2}\cdot\big{[}1-(1-p_{3})\cdot(1-p_{4})\cdot(1-p_{5})\big{]}.$ The structural importance’s assumption is that the reliabilities $p_{1}=p_{2}=...=p_{n}=p$ are identical. Since $I_{B}(j;\overrightarrow{\textbf{p}})=\textbf{D}_{p_{j}}h(\overrightarrow{\textbf{p}}),$ we may derive formulas for the structural importance for each component $\displaystyle I_{B}(j)=\prod^{k}_{\begin{subarray}{c}i=1\\\ i\neq j\end{subarray}}p_{i}\big{[}1-\prod^{n}_{m=k+1}(1-p_{m})\big{]}\text{\quad for $j=1,...,k$,}$ (48) $\displaystyle I_{B}(j)=\prod^{k}_{i=1}p_{i}\prod^{n}_{\begin{subarray}{c}m=k+1\\\ m\neq j\end{subarray}}(1-p_{m})\text{\quad for $j=k+1,...,n$.}$ (49) For Birnbaum case, each reliability $p_{i}=\frac{1}{2}$, hence from (48) and (49) we obtain $\displaystyle I_{B}(j;\overrightarrow{.5})=2^{-(k-1)}-2^{-(n-1)}\text{\quad for $i=1,\ldots,k$,}$ $\displaystyle I_{B}(j;\overrightarrow{.5})=2^{-(n-1)}\text{\quad for $i=k+1,\ldots,n$.}$ Therefore, for structure in figure 3 with $k=2$ and $n=5$, we have $\displaystyle I_{B}(1;\overrightarrow{.5})=I_{B}(2;\overrightarrow{.5})=2^{-1}-2^{-4}=0.4375$ $\displaystyle I_{B}(3;\overrightarrow{.5})=I_{B}(4;\overrightarrow{.5})=I_{B}(5;\overrightarrow{.5})=2^{-4}=0.0625$ We can see that the components in series have much greater importance than the components in parallel. ###### Example 4 (Minimal path and cut sets) $A$$B$123487569 Figure 4: Graph $G_{A,B}$ Let’s consider the structure of order $10$ with edges of graph $G_{A,B}$ as elements of the system. Every set of edges connecting vertices $A$ and $B$ is a path and every set of edges, when removed, disconnecting vertices $A$ and $B$ is a cut [22]. Directed graph $G_{A,B}$ represents simple example of the network connecting two nodes (A and B) that is often used in examining reliability of computer networks. In order to define structure function $\phi(x)$, determining the minimal path and cut sets is obligatory. Table 1: Minimal path set Path \| Elements ---\|--- 1 \| 1 3 8 2 \| 1 4 7 8 3 \| 2 5 7 8 5 \| 2 6 9 The minimal path and cut sets presented in tables 1 and 2 respectively, allow to determine the structure function. Graph $G_{A,B}$ is described by four minimal path series structures $\displaystyle\rho_{1}(\overrightarrow{x})$ $\displaystyle=\prod_{i\in\\{1,3,8\\}}x_{i}$ $\displaystyle\rho_{2}(\overrightarrow{x})$ $\displaystyle=\prod_{i\in\\{1,4,7,8\\}}x_{i}$ $\displaystyle\rho_{3}(\overrightarrow{x})$ $\displaystyle=\prod_{i\in\\{2,5,7,8\\}}x_{i}$ $\displaystyle\rho_{4}(\overrightarrow{x})$ $\displaystyle=\prod_{i\in\\{2,6,9\\}}x_{i}$ Table 2: Minimal cut set Cut \| Elements ---\|--- 1 \| 1 2 2 \| 1 5 6 3 \| 1 5 9 4 \| 1 6 7 5 \| 1 6 8 6 \| 1 7 9 7 \| 2 3 4 8 \| 2 3 7 9 \| 2 8 10 \| 3 4 5 6 11 \| 3 4 5 9 12 \| 3 6 7 13 \| 3 7 9 14 \| 6 8 15 \| 8 9 and by fifteen minimal cut parallel structures $\displaystyle\kappa_{1}(\overrightarrow{x})$ $\displaystyle=x_{1}\amalg x_{2}$ $\displaystyle\kappa_{2}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{1,5,6\\}}x_{i}$ $\displaystyle\kappa_{3}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{1,5,9\\}}x_{i}$ $\displaystyle\kappa_{4}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{1,6,7\\}}x_{i}$ $\displaystyle\kappa_{5}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{1,6,8\\}}x_{i}$ $\displaystyle\kappa_{6}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{1,7,9\\}}x_{i}$ $\displaystyle\kappa_{7}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{2,3,4\\}}x_{i}$ $\displaystyle\kappa_{8}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{2,3,7\\}}x_{i}$ $\displaystyle\kappa_{9}(\overrightarrow{x})$ $\displaystyle=x_{2}\amalg x_{8}$ $\displaystyle\kappa_{10}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{3,4,5,6\\}}x_{i}$ $\displaystyle\kappa_{11}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{3,4,5,9\\}}x_{i}$ $\displaystyle\kappa_{12}(\overrightarrow{x})$ $\displaystyle=\coprod_{i\in\\{3,6,7\\}}x_{i}$ $\displaystyle\kappa_{13}(\overrightarrow{x})$ $\displaystyle=x_{3}\amalg x_{7}\amalg x_{9}$ $\displaystyle\kappa_{14}(\overrightarrow{x})$ $\displaystyle=x_{6}\amalg x_{8}$ $\displaystyle\kappa_{15}(\overrightarrow{x})$ $\displaystyle=x_{8}\amalg x_{9}$ The structure functions, if at least one of the minimal paths functions, can be presented as a parallel structure of minimal path series structure $\phi(\overrightarrow{x})=\coprod_{i=1}^{r}\rho_{i}(\overrightarrow{x})=1-\prod_{i=1}^{r}\big{[}1-\rho_{i}(\overrightarrow{x})\big{]},$ where $r$ is a number of minimal paths of graph $G_{A,B}$. Hence, the structure function of the graph $G_{A,B}$ can be written in the form of $\displaystyle\phi(\overrightarrow{x})$ $\displaystyle=\coprod_{i=1}^{4}\rho_{i}(\overrightarrow{x})=1-\prod_{j=1}^{4}(1-\rho_{j}(\overrightarrow{x}))$ $\displaystyle=1-(1-\prod_{i\in\\{1,3,8\\}}x_{i})(1-\prod_{i\in\\{1,4,7,8\\}}x_{i})(1-\prod_{i\in\\{2,5,7,8\\}}x_{i})(1-\prod_{i\in\\{2,6,9\\}}x_{i})$ and the structure function $\phi(\overrightarrow{x})$ is exactly equal to $\begin{split}\phi(\overrightarrow{x})&=x_{1}^{2}x_{2}x_{3}x_{4}x_{5}x_{7}^{2}x_{8}^{3}-x_{1}^{2}x_{2}^{2}x_{3}x_{4}x_{5}x_{6}x_{7}^{2}x_{8}^{3}x_{9}-x_{1}x_{2}x_{4}x_{5}x_{7}^{2}x_{8}^{2}\\\ &\quad- x_{1}^{2}x_{3}x_{4}x_{7}x_{8}^{2}-x_{1}x_{2}x_{3}x_{5}x_{7}x_{8}^{2}+x_{1}x_{2}^{2}x_{4}x_{5}\cdot x_{6}x_{7}^{2}x_{8}^{2}x_{9}\\\ &\quad+x_{1}^{2}x_{2}x_{3}x_{4}x_{6}x_{7}x_{8}^{2}x_{9}+x_{1}x_{2}^{2}x_{3}x_{5}x_{6}x_{7}x_{8}^{2}x_{9}+x_{1}x_{3}x_{8}+x_{1}x_{4}x_{7}x_{8}\\\ &\quad+x_{2}x_{5}\cdot x_{7}x_{8}-x_{1}x_{2}x_{3}x_{6}x_{8}x_{9}-x_{1}x_{2}x_{4}x_{6}x_{7}x_{8}x_{9}+x_{2}x_{6}x_{9}-x_{2}^{2}x_{5}x_{6}x_{7}x_{8}x_{9}.\end{split}$ Moreover, the structure function can be also presented in the form of the series structure of the minimal cut parallel structures $\phi(\overrightarrow{x})=\prod_{i=1}^{c}\kappa_{i}(\overrightarrow{x}),$ where $c$ is a number of minimal cuts of the graph $G_{A,B}$. If at least one of the minimal cuts fails, the structure fails as well. $G_{A,B}$ graph’s structure function can be written in the short form as $\phi(\overrightarrow{x})=\prod_{i=1}^{14}\kappa_{i}(\overrightarrow{x}).$ The structural importance measure presented by Birnbaum in 1968 and then by Barlow and Proschan in 1975 was once independently developed in the field of game theory (v. Appendix 0.C) by Shapley and Shubik in 1954(v. Shapley (1953)) and Banzhaf (1965), respectively (v. Ramamurthy (1990). ### 2.7 Importance measure based on multilateral stopping problem. The basis for the description of binary systems is the structure function described in appendix xxx. We consider semi-coherent structures, which means that the structure function has properties identical to the function aggregating players’ decisions in multi-person decision problems considered in the work of Szajowski and Yasuda (1997). Multi-player decision problems assume that each game participant has a preference function based on a scalar function defined on the states of a certain process. If the elements of the structure are assigned to conservators (hypothetical players) who take care of the condition of these elements so that they fulfill their functions properly, the mentioned function can estimate profits and losses resulting from the state of the element. In principle, this condition should be good, allowing the function of the element, or bad - excluding the element from functioning. However, in reality, it is the diagnostician who decides when to perform maintenance or replacement (and bear the cost of it), and only sometimes a failure introduces a forced repair. An element in a system usually lowers its efficiency (e.g., mating components in a driveline may need lubrication to reduce friction, which results in increased energy expenditure and lower system efficiency), but the maintenance downtime is wasted and cannot always be managed. The operating conditions of the system make it possible to determine the correct payment function (cost) for each maintenance technician. Each of the n conservators, observing the states on which its payment depends, decides whether to order a maintenance break or to carry out uninterrupted operation. For safety reasons and the structure of the system, it is clear whether such a decision of a single observer is effective - it can start work when the system is stopped, and the stoppage requires the consensus of conservators from some critical path. To analyze the effects of action, we will use the model of the following antagonistic game with elements of cooperation, which are defined by the function of the structure. Following the results of the author and Yasuda [37] the multilateral stopping of a Markov chain problem can be described in the terms of the notation used in the non-cooperative game theory (see [24], [15], [23], [28]). To this end the process and utilities of its states should be specified. ###### Definition 7 (ISS-Individual Stopping Strategies) Let $(\overrightarrow{X}_{n},{\mathcal{F}}_{n},{\textbf{P}}_{x})$, $n=0,1,2,\ldots,N$, be a homogeneous Markov chain with the state space $(\mathbb{E},{\mathcal{B}})$. * • The players are able to observe the Markov chain sequentially. The horizon can be finite or infinite: $N\in\mathbb{N}\cup\\{\infty\\}$. * • Each player has their utility function $f_{i}:\mathbb{E}\rightarrow\Re$, $i=1,2,\ldots,p$, such that ${\textbf{E}}_{x}\|f_{i}(\overrightarrow{X}_{1})\|<\infty$ and the cost function $c_{i}:\mathbb{E}\rightarrow\Re$, $i=1,2,\ldots,p$. * • If the process is not stopped at moment $n$, then each player, based on ${\mathcal{F}}_{n},$ can declare independently their willingness to stop the observation of the process. ###### Definition 8 (see [42]) An individual stopping strategy of the player $i$ (ISS) is the sequence of random variables $\\{\sigma_{n}^{i}\\}_{n=1}^{N}$, where $\sigma_{n}^{i}:\Omega\rightarrow\\{0,1\\}$, such that $\sigma_{n}^{i}$ is ${\mathcal{F}}_{n}$-measurable. The interpretation of the strategy is following. If $\sigma_{n}^{i}=1$, then player $i$ declares that they would like to stop the process and accept the realization of $X_{n}$. ###### Definition 9 (SS–Stopping Strategy (the aggregate function).) Denote $\sigma^{i}=(\sigma_{1}^{i},\sigma_{2}^{i},\ldots,\sigma_{N}^{i})$ and let ${\mathscr{S}}^{i}$ be the set of ISSs of player $i$, $i=1,2,\ldots,p$. Define ${\mathscr{S}}={\mathscr{S}}^{1}\times{\mathscr{S}}^{2}\times\ldots\times{\mathscr{S}}^{p}$. The element $\sigma=(\sigma^{1},\sigma^{2},\ldots,\sigma^{p})^{T}\in{\mathscr{S}}$ will be called the stopping strategy (SS). The stopping strategy $\sigma\in{\mathscr{S}}$ is a random matrix. The rows of the matrix are the ISSs. The columns are the decisions of the players at successive moments. The factual stopping of the observation process, and the players realization of the payoffs is defined by the stopping strategy exploiting $p$-variate logical function. Let $\delta:\\{0,1\\}^{p}\rightarrow\\{0,1\\}$ be the aggregation function. In this stopping game model the stopping strategy is the list of declarations of the individual players. The aggregate function $\delta$ converts the declarations to an effective stopping time. ###### Definition 10 (An aggregated SS) A stopping time $\tau_{\delta}(\sigma)$ generated by the SS $\sigma\in{\mathscr{S}}$ and the aggregate function $\delta$ is defined by $\tau_{\delta}(\sigma)=\inf\\{1\leq n\leq N:\delta(\sigma_{n}^{1},\sigma_{n}^{2},\ldots,\sigma_{n}^{p})=1\\}$ $(\inf(\emptyset)=\infty)$. Since $\delta$ is fixed during the analysis we skip index $\delta$ and write $\tau(\sigma)=\tau_{\delta}(\sigma)$. ###### Definition 11 (Process and utilities of its states) * • $\\{\omega\in\Omega:\tau_{\delta}(\sigma)=n\\}=\bigcap\nolimits_{k=1}^{n-1}\\{\omega\in\Omega:\delta(\sigma_{k}^{1},\sigma_{k}^{2},\ldots,\sigma_{k}^{p})=0\\}\cap\\{\omega\in\Omega:\delta(\sigma_{n}^{1},\sigma_{n}^{2},\ldots,\sigma_{n}^{p})=1\\}\in{\mathcal{F}}_{n}$; * • $\tau_{\delta}(\sigma)$ is a stopping time with respect to $\\{{\mathcal{F}}_{n}\\}_{n=1}^{N}$. * • For any stopping time $\tau_{\delta}(\sigma)$ and $\mathfrak{i}\in\\{1,2,\ldots,p\\}$ the payoff of player $\mathfrak{i}$ is defined as follows (cf. [35]): $f_{i}(X_{\tau_{\delta}(\sigma)})=f_{i}(X_{n})\mathbb{I}_{\\{\tau_{\delta}(\sigma)=n\\}}+\limsup_{n\rightarrow\infty}f_{i}(X_{n})\mathbb{I}_{\\{\tau_{\delta}(\sigma)=\infty\\}}.$ ###### Definition 12 [An equilibrium strategy (cf. [37])] Let the aggregate rule $\delta$ be fixed. The strategy ${}^{}\\!\sigma=({}^{}\\!\sigma^{1},{}^{}\\!\sigma^{2},\ldots,{}^{}\\!\sigma^{p})^{T}\in{\mathscr{S}}$ is an equilibrium strategy with respect to $\delta$ if for each $\mathfrak{i}\in\\{1,2,\ldots,p\\}$ and any $\sigma^{i}\in{\mathscr{S}}^{i}$ we have $v_{i}(\overrightarrow{x})={\textbf{E}}_{x}[f_{i}(\overrightarrow{X}_{\tau_{\delta}({}^{}\\!\sigma)})+\sum_{k=1}^{\tau_{\delta}({}^{}\\!\sigma)}c_{i}(\overrightarrow{X}_{k-1})]\leq{\textbf{E}}_{x}[f_{i}(\overrightarrow{X}_{\tau_{\delta}({}^{}\\!\sigma(i))})+\sum_{k=1}^{\tau_{\delta}({}^{}\\!\sigma(i))}c_{i}(\overrightarrow{X}_{k-1})].$ ###### Definition 13 [Voting Game Importance] Let the aggregate rule $\delta=h$ be fixed and the strategy ${}^{}\\!\sigma=({}^{}\\!\sigma^{1},{}^{}\\!\sigma^{2},\ldots,{}^{}\\!\sigma^{p})^{T}\in{\mathscr{S}}$ be an equilibrium strategy with respect to $\delta$. The voting game importance of the elements is the component of $\textbf{VGI}=\frac{\textbf{E}_{\overrightarrow{Q}^{0}}\overrightarrow{\textbf{v}}(\overrightarrow{X})}{\textbf{E}<\overrightarrow{\textbf{v}}(\overrightarrow{X}),\overrightarrow{Q}^{0}>}.$ The measure of significance of a structure element introduced in this way takes into account its role in the structure by the aggregation function $h$, it is normalized in the sense that the measures of all elements sum up to $1$. It takes into account the external loads of elements, the cost of maintenance and repairs. Its use requires in-depth knowledge of the system and its components, which is a significant obstacle in its introduction into diagnostic practice. The hardest part is figuring out the payout functions (cost, risk, profit). The simplified version of the method may include in the payout functions only the operating risk with components in a condition requiring maintenance or repair, which is usually associated with less safety. ## 3 Concluding remarks ### 3.1 Summary Ensuring the reliability and secure performance of the simple as well as complex systems has an indisputable significance in system analysis. Wherefore, the aim of the research was to answer the question how to recognize the most influential elements of the system so as to improve its reliability. This paper has demonstrated several approaches to the concept of importance measure depending on parameters and assumptions characterizing the system. The new approach is proposed in section 2.7. In this paper we have considered binary systems. Their extension in the form of multistate systems is subject of another paper. In addition, the assumption was their coherence. Limitations and assumptions of coherent system for binary systems have been presented. For two-state systems various importance measures have been introduced and also the concept of the module importance, that can be applied to any more complex system. We have looked into case when only structure of the system was known (structural importance measure), case when the system was dependent on both reliability of components and structure of the system (reliability importance measure), and case, when the measure was dependent on the lifetime distribution of the components and the system structure (lifetime importance measure). The measures of importance have been based on Barlow and Proschan and Birnbaum’s studies. The problem of choosing the proper importance measure was shown, e.g. due to the inconsistent behavior for different structures of the system. In addition, the relationship between importance measures in the reliability theory and power indices in the game theory have been discussed in the paper. This analysis showed that the importance measures first introduced by Birnbaum in 1968 became the foundation for further search of more convenient and versatile definitions of the importance of components in the system reliability. Since then, research has expanded in different directions but till nowadays importance evaluation of highly complex structures such as networks may cause many computational problems. Besides, restrictions regarding coherence may exclude examination of certain systems. Therefore, this subject is under constant exploration. ### 3.2 Exploratory importance measure research. There are many quantities estimated in probabilistic risk assessments (PRAs) to index the level of plant safety. If the PRA is to be used as a risk management tool to assist in the safe operation of the plant, it is essential that those elements of the plant design and its mode of operation that have the greatest impact on plant safety be identified. These elements may be identified by performing importance calculations. There are certain decisions that must be made before the importance calculation is carried out. The first is the definition of the events for which importance is to be evaluated; that is, to what level of resolution the analysis is to be performed. The second decision that must be made–and the major subject of this paper–is the choice of importance measure. Many measures of importance have been proposed; this discussion is restricted to three: the risk achievement (or degradation) worth, the risk reduction worth, and criticality importance. In Schmidt et al. (1985) these measures of importance are defined, their interrelationships are discussed, and a generalized importance measure is introduced. The use of these three measures is compared and their advantages and disadvantages are discussed. ### 3.3 Important direction of further investigations. When interpreting component importance (v. Wu and Coolen(2013)), concluded that the importance of a component should depend on the following factors: 1. 1. The location of the component in the system. 2. 2. The reliability of the component. 3. 3. The uncertainty in the estimate of the component reliability and related cost. 4. 4. The costs of maintaining this component in a given time interval $(0,t)$. (v. also Rausand et al. (2021)). The factor (3) highly depends on the statistical method implemented in the analyzes of exploratory data analyzes. Due to source of the data, the role of structure of the system to the reliability of it, the importance measure should take these elements into accounts. We are not observing the hidden state of the system directly and the information taken from the sensors should by interpreted and evaluated to infer on the hidden state of the elements and the system. The details of the construction needed, based on the results by Szajowski (2020), are subject of a paper under editorial process. Author Contributions: Both authors equally contributed to the conceptualization, methodology, formal analysis, investigation and writing–original draft preparation. Małgorzata Średnicka is responsible for the description of the importance measure concepts, examples, visualisation (v. [32]) and Krzysztof J. Szajowski is responsible for the project conceptualization and its administration. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. Appendices ## Appendix 0.A Structure functions. To study the relationship between the reliability of the components of a structure and the reliability of the structure itself, one has to know how the performance or failure of various components affect the performance or failure of the structure. We do this with the help of Boolean functions. In the reliability literature, Boolean functions are called structure functions. Structure functions serve as a conceptual model on which the theory of reliability is largely based. The state of the system is assumed to depend only on the states of the components. We shall distinguish between only two states – a functioning state and a failed state. This dichotomy applies to the structure as well as to each component. The assumption that the state of the system is completely determined by the states of its components implies the existence of a Boolean function $\varphi:B^{n}\rightarrow B$. A Boolean function of $n$ variables is a function on $B^{n}$ taking values in $B=\\{0,1\\}$. A system or structure is assumed to consist of an element of $\textbf{N}=\\{1,2,\ldots,n\\}$ \- the set of $n$ components. Let us consider the state of the system at a fixed moment of time. ###### Definition 14 The structure function of a system consisting of $n$ components is a Boolean function of $n$ variables. Let $\varphi$ be a structure on N and $i\in\textbf{N}$. The component $i$ is _irrelevant to the structure_ $\varphi$ if $\varphi(1,\overrightarrow{x}_{-i})=\varphi(0,\overrightarrow{x}_{-i})$ for all $\overrightarrow{x}\in B^{n}$ and relevant otherwise. The number of relevant components is called the order of the structure $\varphi$. The structure with no relevant components is a _degenerate structure_ , i.e., $\varphi(\overrightarrow{x})=1$ or $\varphi(\overrightarrow{x})=0$ for all $\overrightarrow{x}\in B^{n}$. Let $\varphi_{i}$, $i=1,2$ be two structures on $\textbf{N}=\\{1,2,\ldots,n\\}$. The linear composition of these two structures is a structure $h(\overrightarrow{x},x_{n+1})=x_{n+1}\varphi(\overrightarrow{x})+(1-x_{n+1})\varphi_{2}(\overrightarrow{x})$ on $\textbf{B}^{n+1}$. ###### Corollary 1 Any structure $\varphi$ of order $n$ is a linear composition of two structures of at most order $n-1$: $\varphi(\overrightarrow{x})=x_{i}\varphi(1,\overrightarrow{x}_{-i})+(1-x_{i})\varphi(0,\overrightarrow{x}_{-i}),\text{for every $\overrightarrow{x}\in B^{n}$, $i\in\textbf{N}$.}$ (50) ###### Definition 15 Let $\varphi$ be a structure on N, $A\subset\textbf{N}$ and $J=\textbf{N}\setminus A$. The collection of $A$ of components form a path (cut) set of $\varphi$ if $\varphi(\vec{1}^{A},\vec{0}^{J})=1$ ($\varphi(\vec{0}^{A},\vec{1}^{J})=0$). ###### Definition 16 Let $\varphi$ be a structure on N. Its dual $\varphi^{\mathcal{D}}$ is another structure on N defined by $\varphi^{\mathcal{D}}(\overrightarrow{x})=1-\varphi(\vec{1}-\overrightarrow{x})$ for every $\overrightarrow{x}\in\textbf{B}^{n}$. ###### Definition 17 Let $\varphi$ be a structure on N. A path (cut) set $S$ of $\varphi$ is called a minimal path (cut) set of $\varphi$ if $T\subset S$ implies that $T$ is not a path (cut) set of a structure $\varphi$. The family $\alpha(\varphi)$ ($\beta(\varphi)$) denotes collection of minimal path (cut) sets of the structure $\varphi$. ###### Proposition 1 For every semicoherent structure, $\varphi$ on N $\varphi(\overrightarrow{x})=1-\prod_{S\in\alpha(\varphi)}(1-\prod_{i\in S}x_{i})=\prod_{S\in\beta(\varphi)}(1-\prod_{i\in S}(1-x_{i}))\text{ for all $\overrightarrow{x}\in\textbf{B}^{n}$.}$ (51) ###### Remark 1 (A simple form of $\varphi$) Expanding either one of the two terms on the right hand side of the expression of Proposition 1 (putting $x_{i}^{r}=x_{i}$ for $r\geq 1$) we get a structure function in the form $\varphi(\overrightarrow{x})=\sum_{T\subseteq\textbf{N}}b_{T}\prod_{j\in T}x_{j}\text{for all $\overrightarrow{x}\in\textbf{B}^{n}$}$ (52) with $b_{T}$–some integer constants ($\prod_{j\in T}x_{j}=1$ for $T=\emptyset$). For any structure, there always exist at least one simple form and the simple form of a structure is unique. ## Appendix 0.B The simple game In game theory considers the set $N=\\{1,2,\ldots,n\\}$ of players and the power set $2^{N}$ of coalitions. A function $\lambda:2^{N}\rightarrow\\{0,1\\}$ is called a simple game on $N$ in characteristic function form if 1. (1) $\lambda(\emptyset)=0$; 2. (2) $\lambda(N)=1$; 3. (3) $S\subseteq T\subseteq N$ implies $\lambda(S)\leq\lambda(T)$. A coalition $S\subset N$ is called winning if $\lambda(S)=1$ and it is called blocking if $\lambda(N\setminus S)=0$. Indeed, the collection of winning (or blocking) coalitions in a simple game satisfies the three properties of the basic structure mentioned at the beginning. ## Appendix 0.C Power indexes In the field of game theory Shapley (1953) and Banzhaf (1965) consider the role of the players in a cooperative game to provide an idea of division of the gain (v. Ramamurthy (1990). First, Shapley and Shubik examined $n$-player games what let them formulate a characteristic value applicable to simple games. Hence, originally the Shapley-Shubik index measured a power of players in voting games. Their measure is a natural consequence of the influence of a given voter on the result. ###### Definition 18 The Banzhaf index of the $i$-th player [component] denoted as $\psi_{i}(g)$ is applicable for a semi-coherent structure $g$ on $N$, and is defined by $\psi_{i}(g)=\frac{\eta_{i}(g)}{2^{n-1}},$ (53) where $i\in N$, $r$ is a size of $\eta_{i}(g)$, which stands for the aggregated sum of critical path vectors of g, and $\eta_{i}(g)=\sum_{r=1}^{n}\eta_{i}(r,g)$. Definition 18 is identical to the structural importance (47) presented by Birnbaum. ###### Definition 19 For a semicoherent structure $g$ the Shapley-Shubik index is denoted as $\phi_{i}(g)$, given by $\phi_{i}=\sum_{r=1}^{n}\eta_{i}(r,g)\cdot\frac{(n-r)!(r-1)!}{n!},$ (54) where $i\in N$, $r$ is a size of $\eta_{i}(r,g)$, which stands for the number of critical path vectors of g. Definition 19 is identical to the structural importance (39) presented by Barlow and Proschan. ###### Fact 0.C.1 If we compare the expressions (53) and (54), we can see that the index introduced by Shapley-Shubik has a weight $(n-r)!(r-1)!/n!$ attached to $\eta_{i}(r,g)$, meanwhile the Banzhaf index is independent on $r$ and has always weight $1/(2^{n-1})$ attached to $\eta_{i}(r,g)$. Due to the behavior of $(n-r)!(r-1)!/n!$ for different $n$, we deduce that only very large or very small critical paths may reach the greatest weight. 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# : a Multi-modal Dataset of Election Fraud Claims on Twitter Anton Abilov1,2, Yiqing Hua1,2, Hana Matatov3, Ofra Amir3, Mor Naaman1,2 ###### Abstract The wide spread of unfounded election fraud claims surrounding the U.S. 2020 election had resulted in undermining of trust in the election, culminating in violence inside the U.S. capitol. Under these circumstances, it is critical to understand discussions surrounding these claims on Twitter, a major platform where the claims disseminate. To this end, we collected and release the dataset, a multi-modal dataset with 7.6M tweets and 25.6M retweets from 2.6M users related to voter fraud claims. To make this data immediately useful for a wide area of researchers, we further enhance the data with cluster labels computed from the retweet graph, user suspension status, and perceptual hashes of tweeted images. We also include in the dataset aggregated information for all external links and YouTube videos that appear in the tweets. Preliminary analyses of the data show that Twitter’s ban actions mostly affected a specific community of voter fraud claim promoters, and exposes the most common URLs, images and YouTube videos shared in the data. ## 1 Introduction Free and fair elections are the foundation of every democracy. The 2020 presidential election in the United States was probably one of the most consequential and contentious such events. Two-thirds of the voting-eligible population voted, resulting in the highest turnout in the past 120 years (Schaul, Rabinowitz, and Mellnik 2020). The Democratic Party candidate Joe Biden was elected as the president. Unfortunately, efforts to deligitimize the election process and its results were carried out before, throughout and after the election. Mostly unfounded claims of voter fraud (Frenkel 2020) were spread both through public statements by politicians, and on social media platforms. As a result, 34% of Americans say that they do not trust the election results as of December, 2020 (NPR 2020). Voter fraud claims without credible evidence have great ramifications on both the integrity of the election and the stability of the U.S. democracy. On January 6th, 2021, believing that the election was ‘stolen’, mobs breached U.S. capitol while the Congress voted to certify Biden as the winner of the election. Social media platforms like Facebook, Twitter, YouTube and Reddit play a significant role in political events (Vitak et al. 2011; Allcott and Gentzkow 2017), and the 2020 election was no exception (Ferrara et al. 2020). In particular, Twitter has been the focus of public and media attention as a prominent public square where ideas are adopted and claims – false or true – are spread (Vosoughi, Roy, and Aral 2018; Grinberg et al. 2019). It is thus important to understand the participants, discussions, narratives, and allegations around voter fraud claims on this specific platform. In this work, we release , a multi-modal Twitter dataset of 7.6M tweets and 25.6M retweets that are related to voter fraud claims. Using a manually curated set of keywords (e.g., “voter fraud” and “#stopthesteal”) that was further expanded using a data-driven approach, we streamed Twitter activities between October 23rd and December 16th, 2020. We performed various validations on the limits of our stream, given Twitter’s API constraints (Morstatter et al. 2013), and estimate that we were able to retrieve around 60% of the data containing our crawled keywords. We further enhanced the dataset in order to make it accessible for a broader set of researchers and future research: (1) We cluster users according to their retweeting dynamics and release the cluster labels; (2) Given Twitter’s widespread post-election suspension action, we crawl and include the user status as of January 10th, 2021; (3) We compute and share the perceptual hashes of 168K images that appeared in the data; (4) We aggregate and share metadata about 138K external links that appeared in the tweets, including 12K unique YouTube videos. Our dataset also allows researchers to calculate the amount of Twitter interactions with the collected tweets, users, and media items, including number of retweets and quotes from various clusters, or from suspended users. A preliminary analysis finds a significant cluster of users who were promoting the election fraud related claims, with nearly 7.8% of them suspended in January. The suspensions focused on a specific community within the cluster. A simple analysis of the distribution of images, based on visual similarity, exposes that the most broadly shared (by number of tweets) and the most retweeted images are different. Although recent research has shown that voter fraud claims are pushed mainly by mass media (Benkler et al. 2020), we also find that external links referenced by promoters of the claims point mostly to low-quality news websites, streaming services, and YouTube videos. Some of the widespread videos claiming ‘evidence’ of voter fraud were published by surprisingly small channels. Most strikingly, all of the top ten channels and videos spreading voter fraud claims were still available on YouTube as of January 11th, 2021. We believe that the release of , the largest public dataset of Twitter discussions around the voter fraud claims, with the enhanced labels and data, will help the broad research community better understand this important topic at a critical time. ## 2 Data Collection Our data collection process involved streaming Twitter data using a data- driven manually curated set of keywords and hashtags. We report on the span and volume of the collected data, as well as on analyses estimating its coverage. ### 2.1 Streaming Twitter data We used a data-driven approach to generate a list of keywords and hashtags related to election fraud claims in an iterative manner. We started with a single phrase and two derived keywords: voter fraud and #voterfraud. We first used a convenience sample of 11M political tweets consisting of the tweets of 2,262 U.S. political candidates and the replies to those tweets, collected between July 21st and Oct 22nd, 2020 using the Twitter Streaming API (Twitter 2019). We then identified hashtags that co-occur with our meta-seed keywords, voter fraud and voterfraud. We selected all hashtags that appeared in at least 10 tweets and co-occurred with either of the meta-seed keywords at least 50% of the time. From the resulting set, we manually filtered out those that were not directly relevant to voter fraud. To this end, two members of the research team reviewed the hashtags, including, if needed, searching for them on Twitter to see whether they produce relevant results. Only the hashtags that were agreed on by both evaluators were added, resulting in an initial set of hashtags that was added to the two original keywords. We computed the Jaccard coefficient between each of our seed hashtags and all other hashtags that appeared in the new stream. We added to our set all hashtags that had a Jaccard coefficient greater than 0.001 with any of the seed hashtags. Three members of the team again reviewed this list by 1) excluding hashtags that were not related to voter fraud, 2) adding corresponding keywords of the hashtags (e.g. #discardedballots corresponds to discarded ballots), and 3) adding relevant hashtags or keywords that the researchers observed while searching for hashtags from the generated list. Both the seed list and the final list of keywords and hashtags we used for streaming are included in Appendix A (Table 3). We collected data using the Twitter streaming API (Twitter 2019). The dataset includes tweets from 17:00, October 23rd, 2020 to 13:00 December 16th, 2020. We expanded the keywords list on Oct. 31st with additional keywords, and added #stopthesteal as it started trending on November 3rd. While streaming, we stored each tweet’s metadata (e.g., user ID, text, timestamp). We also downloaded all image media items included in the tweets. In total, we collected 3,781,524 original tweets, 25,566,698 retweets, and 3,821,579 quote tweets (i.e. tweets that include a reference to another tweet) discussing election fraud claims. Note that quote tweets are included in the Twitter stream when either the new tweet or the referenced (quoted) tweet include one of the keywords or hashtags on the list. In total, we collected tweets from 2,559,018 users who posted, shared or quoted one or more tweets with these keywords. ### 2.2 Coverage Analysis Since the Twitter streaming API provides only a sample of the tweets, especially for large-volume keywords (Morstatter et al. 2013), we performed multiple tests to evaluate and estimate the coverage of the dataset. This analysis suggests that the dataset covers over 60% of the content shared on Twitter using the keywords we tracked. ##### Retweet and quote coverage. We evaluated the coverage of retweet and quote tweets by comparing the counts of these objects in the stream to Twitter’s metadata. When a new retweet for an original tweet appears in the stream, the API returns the tweet’s metadata including the current retweet count and quote count of the original tweet. In other words, if an original tweet $t_{i}$ is retweeted, it will appear in the stream as a retweet $rt_{j}$, and the metadata for $rt_{j}$ will include the total number of retweets of $t_{i}$ so far. From this metadata, it is easy to define the retweet coverage as the ratio of the total number of retweets ($rt$ objects) streamed and stored in our dataset, over the sum of all retweet counts of the original $t$ tweets, returned by the API in the latest $rt$ retweet of each original tweet. The quote coverage is defined analogously. According to this analysis, the dataset captured 63.2% of the retweets and 62.6% of the quote tweets. These findings compare favorably with previous work that shows a single API client captures only 37.6% of the retweets through the Streaming API (Morstatter et al. 2013). ##### Comparison with #Election2020. To further evaluate the coverage on the voter fraud tweets, we compared our dataset with a previously published Twitter dataset of the U.S. 2020 election (Chen, Deb, and Ferrara 2020). The creators of the #Election2020 dataset used the streaming API to track 168 keywords that are broadly related to the election and 57 accounts that are tied to candidates running for president. As in , the keyword ‘voter fraud’ was also used to collect data for #Election2020. We used this overlap to estimate our coverage. First, we can directly compare the relative volume and overlap between the ‘voter fraud’ tweets in both datasets. We expect to have a higher volume of such tweets because of its more focused set of keywords. Second, if we assume sampling for both streams is independent and random, we could estimate the coverage of by looking at the proportion of #Election2020 tweets that also appear in our data. To this end, we extracted all tweets and retweets that contain this keyword from both datasets posted on two days following the November 3rd election data: November 6th and November 13th. The analysis, performed on December 17th, was limited to two days as we had to obtain the content of the tweets of the #Election2020 dataset by “hydrating” them (i.e. using the tweet IDs to get the full tweet text using the Twitter API). We were unable to hydrate the full data, presumably due to inactive accounts and deleted tweets. The hydration yielded 92.4% of the #Election2020 data on November 6th (a total of 1.4M tweets/3.5M retweets), and 91.1% of the data on November 13th (1.3M tweets/3M retweets). In total, our data includes 45,322 ‘voter fraud’ related tweets on November 6th, 2.3 times as much as recorded in #Election2020. The ratio is even higher on November 13th, when we obtained 47,313 tweets, 3.1 times as much as in #Election2020. Figure 1 breaks down the coverage by dates (separated by rows), in the two datasets (by different colors). From left to right, the bars show the percentages of tweets that are available only in our dataset (dark blue), that are available in both datasets (light blue), and that are available only in #Election2020 (yellow). On any given day, the dataset contains substantially more tweets related to voter fraud, as compared to #Election2020, especially when the estimated total volume is lower. On November 13th (second row), contained 95.7% of the combined data (left two bars) while #Election2020 only collected 30.7% (right two bars) of the tweets. These numbers also indicate that ’s sample includes 32.1% of the related samples in #Election2020 on November 6th and 85.9% on November 13th. We acknowledge that these two numbers are not consistent, presumably because of November 6th’s much higher volume of activity. If these samples are indeed independent, though, it means that our lower bound of coverage is November 6th’s 32.1%. Figure 1: Coverage comparison between our dataset and #Election2020 for tweets containing ‘voter fraud’. Based on these coverage analyses, we conclude that is, at the time of submission, the largest known public Twitter dataset of voter fraud claims and discussions. ## 3 Data Enhancement To ensure the reusability of our data, we took the following steps to enhance the dataset. In addition to raw streaming data, we clustered users according to the retweet dynamics and release the cluster labels. We also queried Twitter for the user status on 10th of January, and share the user status as active/not-found/suspended. Furthermore, to enable research on visual misinformation, we encode all images shared in the tweets with perceptual hash. Finally, we release the URLs, and the metadata of the YouTube videos that appeared in our dataset. Community \| Users \| Relative size \| % of users ---\|---\|---\|--- \| \| \| suspended 0 \| 860,976 \| 45.6% \| 1% 1 \| 437,783 \| 23.2% \| 4.6% 2 \| 342,184 \| 18.1% \| 14.1% 3 \| 33,857 \| 1.8% \| 1.5% 4 \| 23,414 \| 1.2% \| 1.6% (a) (a) (b) Figure 2: (a) Community statistics. (b) Retweet graph colored by communities. (c) Suspension status (orange: suspended users). ##### Retweet Graph Communities. Retweet networks have been frequently analyzed in previous works in order to understand political conversations on Twitter (Arif, Stewart, and Starbird 2018; Cherepnalkoski and Mozetič 2016). Using community detection algorithms, researchers are able to study key players, sharing patterns and content on different sides of a discussion surrounding a heated political topic. We first constructed a retweet graph of the dataset, where nodes represent users and directed edges correspond to retweets between the users. The direction of an edge corresponds to the direction of the information spreading in the retweet relation. Edges are weighted according to the number of times the corresponding source user has been retweeted. The resulting network consists of 1,887,736 nodes and 16,718,884 edges. To detect communities within the graph, we used the Infomap community detection algorithm (Bohlin et al. 2014), which captures the flow of information in directed networks. Using the default parameters, the algorithm produces thousands of clusters. By excluding all clusters that contain fewer than 1% of the nodes we are left with 90% of all nodes111Since the graph only includes retweeting and retweeted users, this number corresponds to 73.8% of all users in our dataset. which are clustered into five communities (see Figure 1(a)). In Figure 2(a), we visualize the retweet network using the Force Atlas 2 layout in Gephi (Bastian, Heymann, and Jacomy 2009), using a random sample of 44,474 nodes and 456,372 edges. The nodes are colored according to their computed community as described in Figure 1(a). Edges are colored by their source node. The visualization indicates that the nodes are split between two distinct clusters - community 0 (blue) on the left and communities 1, 2, 3 and 4 on the right. By examining the top users in each community, we conclude that community 0 mostly consists of accounts that detract the voter fraud claims, while the communities on the right consist of accounts that promote the voter fraud claims. Most of the tweets from these users are written in English, except for users in Community 3 who mainly post tweets in Japanese and users in Community 4 who write in Spanish. Community 2 is more deeply embedded in the promoter cluster compared to Community 1, as we observe tweets from Community 1 being retweeted by Community 0 on the left, but not from Community 2. We include the list of top 5 Twitter accounts in each community by the number of community retweets in the Appendix. For brevity, in the following analyses, we refer to the cluster on the left as the detractor cluster, and the cluster with community 1,2,3,4 on the right as the promoter cluster. Note that due to the partisan nature of the U.S. politics, most of the promoter users are likely aligned with right-leaning politics, and detractor users align with left-leaning politics. To identify users that are prominent within each of these two cluster, we calculate the closeness centrality of the user nodes in each cluster. In a retweet network this metric can be interpreted as a user’s ability to spread information to other users in the network (Okamoto, Chen, and Li 2008). We compute the top-k closeness centrality to find the 10,000 most central nodes within the detractor and promoter clusters (Bisenius et al. 2017). We release the author’s community label of each tweet, the community label of each user, and a user’s closeness centrality in the detractor and promoter clusters. We also include two additional metrics - retweet count by community $X$ and quote count by community $X$. For a tweet $t_{i}$, the retweet count by community $X$ is the total number of retweets $rt_{i}$ it received from each user $u_{X}$ in community $X$. The metric is computed analogously for quotes. ##### Labeling Suspended and Deleted Users When the electoral college were set to confirm the election results on January 6th, 2021, the allegations of voter fraud took a dramatic turn, which culminated in the storming of the US Capitol. Subsequently, Twitter took a harder stance on moderating content on their platform and suspended at least 70,000 accounts that were engaged in propagating conspiracy theories and sharing QAnon-content (Twitter 2021). This ban has substantial implications for researchers seeking to understand the spread of voter fraud allegations on Twitter, since the Twitter API does not allow the “hydration” of Tweets from suspended users. In order to understand the distribution of suspensions within our dataset, we queried the updated user status of all users in our dataset on January 10th, a few days following the ban. The Twitter API returns a user status that indicates if the user is active, suspended or not found (presumably deleted). In total, 3.9% of the accounts (99,884 accounts) in our data were suspended. We enhance the dataset by labeling tweets and users that were suspended. This metadata will enable both research and ease hydration by allowing hydraters to skip content that is no longer available. We also include two additional metrics for each tweet: retweet count by suspended users and quote count by suspended users. Due to its immense public interest, we have retained the full data we retrieved from the 99,884 suspended users including 1,240,405 tweets and 6,246,245 retweets. This detailed data is not part of . However, we will distribute an anonymized version of this data to published academic researchers upon request. ##### Images. Because of their persuasive power and ease of spread, there is a growing interest in analyzing how visual misinformation spreads both within a platform or across platforms (Zannettou et al. 2018; Highfield and Leaver 2016; Paris and Donovan 2019; Moreira et al. 2018; Zannettou et al. 2020). However, visual information such as images or videos is difficult for many researchers to study due to computational and storage costs. Here, we make the information about image content shared in easier to use by sharing perceptual hash values for these images. With these numeric hash values, researchers can easily find duplicates and near-duplicate images in tweets, without working directly with cumbersome image content. To this end, we download all image media items that were posted in the tweets in the streaming data, and encode them with three different types of perceptual hashes. Common perceptual hashes are binary strings designed such that the Hamming distance (Zauner, Steinebach, and Hermann 2011) between two hashes is close if and only if the two corresponding images are perceptually similar. In other words, an image that is only slightly transformed, for example, by re-sizing, cropping, or rotation, will have a similar hash value to the original image. However, as the definition of perceptual similarity is often subjective and the underlying algorithms are often different, various hashing functions have different performance characteristics dealing with various types of image transformations. Therefore, we encode the images in our dataset with three perceptual hash functions: the Perceptive Hash (pHash), the Average Hash (aHash), and the Wavelet Hash (wHash) (Petrov 2017; Zauner, Steinebach, and Hermann 2011). In total, our streamed tweets included 201,259 image URLs, 167,696 of them were retrieved during streaming. We provide some more details about the distribution of these images in Section 5. ##### External links. Misinformation campaigns are known to use broad cross-platform information, often via links to other sites (Wilson and Starbird 2020; Golovchenko et al. 2020). Hence, we extracted and publish the set of external (non-Twitter) URLs that were referenced in the tweets. For ease of use, we resolved URLs that point to a redirected location (e.g. bit.ly URLs) to their final destination URL. Our streamed tweets included references to a total of 138,411 unique URLs, appearing in 609,513 tweets. Since a large portion (over 12%) of all URLs in the data were YouTube links, we further enhanced the data with YouTube-specific metadata. A key motivation for this specific focus was the known role YouTube plays generally in spreading misinformation (Hussein, Juneja, and Mitra 2020; Papadamou et al. 2020) and specifically its role in the 2020 election and voter fraud claims (Kaplan 2020; Wakabayashi 2020). For each YouTube video that was shared in the collected tweets, we used YouTube’s Data API (YouTube 2021), to retrieve the video’s title, description, as well as the id and title of the channel that posted it. We retrieved the YouTube metadata on Jan 1st, 2021. On that data, out of the 13,611 unique video ids that we have queried, 1,608 were no longer available resulting in 12,003 YouTube URLs with full additional metadata. ## 4 Data Sharing and Format Our dataset is available for download under FAIR principles (Wilkinson et al. 2016) in CSV format222https://figshare.com/account/projects/96518/articles/13571084. The data includes “item data” tables for tweets, retweets, and users keyed by Twitter assigned IDs and augmented with additional metadata as described below. The data also includes the images that appear in the dataset, indexed by randomly genenerated unique IDs. Finally, the data includes aggregated tables for URLs and for YouTube videos including the information described in Section 3. The dataset tables and the fields they contain are summarized on Github333https://github.com/sTechLab/VoterFraud2020. The dataset conforms with FAIR principles. The dataset is _Findable_ as it is publicly available on Figshare, with a digital object identifier (DOI): 10.6084/m9.figshare.13571084. It is also _Accessible_ since it can be accessed by anyone in the world through the link. The datset is in csv format, hence it is _Interoperable_. We release the full dataset with descriptions detailed in this paper, as well as an online tool to explore the dataset at http://voterfraud2020.io, making the dataset _Reusable_ to the research community. The tables for Tweets and Retweets contain the full set of items that were collected, including from suspended users. These tables do not include raw tweet data beyond the ID, according to Twitter’s ToS. However, to support use of the data without being required to download (“hydrate”) the full set of tweets, we augment the Tweets table with several key properties. For each tweet we provide the number of total retweets as computed by Twitter (retweet/quote_count_metadata), as well as the number of retweets and quotes we streamed for this tweet from users in each of the five main communities (retweet/quote_count_community_X, X ranging from 0 to 4). Note that the latter do not add up to the Twitter metadata due to the coverage issues listed in Section 2.2. The Tweet table properties also include the user_community (0–4) for the user who posted the tweet, computed using methods listed in Section 3. Some of the Twitter accounts were not clustered into one of the five main communities. In this case, the user_community label is null. With this augmentation, researchers using this dataset could very quickly, for example, select and then hydrate a subset of the most retweeted tweets from non- suspended users in Community 2. As the tweet itself and the ID of the user who tweeted it is not available until hydration, Twitter’s users’ privacy is preserved. The Users table is similarly augmented with aggregate information about the importance of the user in the dataset, including the community that they belong to, their centrality in the two meta-clusters, detractor and promoter (closeness_centrality_detractor_cluster and closeness_centrality_promoter_cluster), and the amount of attention (retweets and quotes) they received from other users in the different communities. We also note whether, according to the data we collected, the user had been suspended. With this data, interested researchers can quickly focus their attention and research on the main actors in each community. The Images table includes all the image media items retrieved in the stream, their unique media ID, and the ID of the tweet in which the image was shared. We augment this table with the image hash using three types of perceptual hash functions – aHash, pHash and wHash, as detailed in Section 3. This augmentation, together with the link to the Tweet ID, will allow researchers to quickly identify and hydrate popular images using the tweet metadata. They can also quickly identify and get information for images that are similar to any other arbitrary image, by computing and comparting the perceptual hash values. The two aggregate tables, the URLs table and the YouTube Videos table again provide information about the popularity of the object in the dataset: aggregate retweet and quote counts, both using the Twitter metadata and the count of objects in our stream from the various communities. In addition, these tables are augmented with metadata about the item (URL or YouTube video) as noted in Section 3. ## 5 Data Analysis We performed a preliminary analysis of our dataset and its different modalities – tweets and users, images, external links – to demonstrate its potential interest and provide some initial guiding insights about the data. Figure 3: Temporal overview of the dataset showing number of streamed tweets, quotes and retweets per day. The shaded regions mark the expansions of the keyword set. (a) \| (b) \| (c) ---\|---\|--- \| \| Tweets \| Retweets (total) \| Retweets (in cluster) \| Tweets \| Retweets (total) \| Retweets (in cluster) \| Tweets \| Retweets (total) \| Retweets (in cluster) 15 \| 24,399 \| 18,020 \| 11 \| 20,104 \| 10,424 \| 34 \| 28,833 \| 10,250 Figure 4: Top three most retweeted images in the promoter cluster: (a)–(c), with the number of tweets, retweets as in metadata, and retweets by users in the cluster. Image (c) was cropped to fit the figure. ##### Tweets and users. Figure 3 shows the amount of retweets (green), original tweets (blue) and quote tweets (yellow) in the dataset over the time (X-axis) of the data collection. Three shaded regions, from left to right, mark the expansion of our set of keywords on October 31st (light blue, region b) and November 3rd (light green, region c). The Y-axis specifies the daily count. In general, except for the large increase after the election date (November 3rd, dotted vertical line), the volume of the stream remains roughly the same. On average, there are 170,938 tweets, 576,136 retweets, and 85,488 quote tweets per day after the election. Our manual inspection shows that top tweets retweeted by the detractor cluster often condemn the alleged voter fraud claims, while top tweets on the promoter cluster indeed make voter fraud claims. Not surprisingly, among the top ten most retweeted tweets in the promoter cluster, nine were tweeted by President Trump. We refer readers to our project website for more details about popular tweets. While the promoter clusters seems rather homogeneous (Figure 2(a)), users in Community 2 (yellow) stand out in both their level of activity and the rate in which they were suspended. Community 2 was highly active in our dataset. For example, Community 2 comprises 18.1% of the users, but contributed 68% of the tweets, and 74% of the retweets. Moreover, 14% of Community 2’s users were suspended by Twitter by the time we collected the account status data as described above, a much higher rate than the other communities, as shown in Figure 1(a). In total, Community 2 was responsible for 46.1% of all suspensions amongst the users we associated with the top five communities. The suspension effect, and its focus on Community 2, can also be observed in Figure 2(b). A full analysis of the suspended accounts and their network communities, and the potential impact of the suspension is out of scope for this dataset paper, but can be easily performed using the data we share in . For example, the data shows that 35% of the promoter cluster users that were retweeted more than 1,000 times (1,596 in total) were suspended. To conclude, our preliminary analysis shows that alleged election fraud claims mostly circulate in the promoter cluster, and in particular in Community 2 within the cluster. The most popular tweets (by retweet counts) supporting such claims often come from prominent accounts. The recent moderation efforts from Twitter seem to have effected the most active community that engaged in fraud related misinformation, and did not broadly target all accounts involved in promoting such claims. #### Images. We conducted a preliminary examination of matching and repeated images in to analyze the distribution of images related to voter fraud claims. Our data, using the perceptual hash functions described in Section 3, allows tracking of duplicate and near-duplicate images that were posted in multiple tweets. In this analysis, we experimented with three perceptual hash functions and refer to two images as matching if they have an identical perceptual hash value. In Figure 5(a), we show the cumulative distribution of the number of unique perceptual hashes in (Y-axis), with hash values sorted based on the number of unique tweets in which they appear, from the highest to the lowest (X-axis). For example, according to pHash, the 1,000 images shared in the largest number of unique tweets appeared together in 25,019 different tweets (not including retweets). Although in general the results are similar when using different hash functions, pHash is the most “conservative” in terms of assigning matches. Overall, our results are similar when using different hash functions. For example, there are 109,312 (out of 167,696) images with the same pHash value. Of these, 17,831 were shared in more than one tweet, an average of 4.27 times. In other words, 34% of the images instances in tweets appear in more than one tweet. Figure 5(b) presents the image that appeared in most number of unique tweets: the same perceptual hash value appeared in over 1,000 tweets, according to all three hash functions. We further investigate the popularity of images, defined by number of retweets, in particular, within the promoter and detractor clusters. After grouping images by the same pHash value, we present in Figure 4 the top three images that have been retweeted in the promoter cluster. Also note that despite the high values of metadata retweets and cluster retweets, all these “popular” images appeared in only a few original tweets in our data. For example, image (a) appeared in 15 tweets, whose metadata retweet (as returned from the API) counts add up to 24,399 in total, and was retweeted (as recorded in our dataset) from users in the promoter cluster 18,020 times. We note that images (a) and (b) were also the top two images retweeted by users in the suspended users set, with 5,547 and 3,122 retweets in that set, respectively (recall that as almost all suspended users belong to the promoter cluster). As expected, the most retweeted images in the two clusters are quite different. The three most retweeted images in the detractor cluster (not included for lack of space) have somewhat lower spread, appearing in tweets that were retweeted 10743, 6425, 3411 times (based on metadata). The top image is a screenshot of the NY Times front page of Nov 11th, 2020 reporting that top election officials across the country have not identified any fraud. The analysis presented above can be easily extended with less-strict image similarity matching by calculating the Hamming distance between a pair of perceptual hash values. In this initial analysis, we used a strict sense of similarity, treating images as similar only when they share the same perceptual hash values. (a) (b) Figure 5: (a) The cumulative number of repeated images by hash matches. (b) The most tweeted image. promoter cluster \| detractor cluster ---\|--- Domain \| Retweets \| Domain \| Retweets pscp.tv \| 51,822 \| washingtonpost.com \| 11,220 youtube.com \| 44,031 \| rawstory.com \| 9,267 thegatewaypundit.com \| 35,967 \| cnn.com \| 4,139 davidharrisjr.com \| 18,793 \| independent.co.uk \| 3,882 foxnews.com \| 17,332 \| nytimes.com \| 3,746 theepochtimes.com \| 15,297 \| newsweek.com \| 3,496 thedcpatriot.com \| 14,958 \| news.yahoo.com \| 2,899 thefederalist.com \| 13,288 \| deadstate.org \| 2,409 djhjmedia.com \| 11,816 \| theguardian.com \| 2,232 justthenews.com \| 11,149 \| politicususa.com \| 2,032 Table 1: Top 10 domains being retweeted in the promoter and the detractor clusters respectively, as well as the number of retweets by users in these clusters. ##### URLs. We conduct preliminary analyses of the external links that have been included in the tweets. Table 1 lists the top 10 domains that have been shared inside the detractor and promoter clusters respectively. Most of the links shared by users in the detractor clusters are to mainstream news media, such as the Washington Post, CNN, and the New York Times. The rest are other news-related websites. The links shared by users in the promoter cluster mostly point to low-quality news-related websites. The most shared domain in the promoter cluster is pscp.tv, a live video streaming app that’s owned by Twitter. YouTube stands out as the second most retweeted domain among the promoter users. This trend is reflected in multiple news reports, warning of the significant role that YouTube plays in spreading false information related to voter fraud claims (Frenkel 2020). The majority of the top 10 most retweeted videos by the promoter users falsely claim evidence of widespread election fraud. The users spreading these videos had significant overlap with the January (or earlier) suspension action by Twitter. For eight of these videos, around $30\%$ of the retweets of tweets sharing those videos were by accounts later suspended by Twitter. A scan of the top 10 YouTube channels retweeted in the promoter cluster shows that they were relatively large (millions of subscribers), though there are also several smaller channels. For example, the most retweeted channel, Precinct 13, has only 3.67K subscribers, has a video that appeared in 88 tweets and have been retweeted over 9K times. Despite YouTube’s announcement that it will take actions against content creators who falsely claim the existence of widespread voter fraud444see: https://twitter.com/YouTubeInsider/status/1347231471212371970, as of Jan 11th, the top 10 channels and videos listed in our tables are still available on YouTube. ## 6 Related Work and Datasets We review prior work using Twitter data analysing politically related events, with an emphasis on those that have released a public dataset. In particular, previous works had used and published Twitter data to study U.S. elections. Using tweets collected during the 2016 U.S. election, researchers have analysed information operations run by social bots (Rizoiu et al. 2018), characterized the dissemination of misinformation (Vosoughi, Roy, and Aral 2018) and its exposure to American voters (Grinberg et al. 2019). Work in Hua, Naaman, and Ristenpart (2020); Hua, Ristenpart, and Naaman (2020) characterized adversarial interaction against political candidates during the 2018 U.S. general election and shared 1.7M tweets interacting with political candidates. Focusing on the U.S. 2020 election, research studied false claims regarding mail-in ballots (Benkler et al. 2020) before the election as the COVID-19 pandemic made it hard to vote in person. Closest to our work is the #Election2020 dataset (Chen, Deb, and Ferrara 2020), which streamed a broad set of Twitter data for both political candidates’ tweets and related keywords. As discussed above, although some of the voter fraud related keywords were included in their data collection process, our dataset contains more than 2.3 times as much of the related data in #Election2020, for overlapping streaming keywords, presumably because of our more focused stream. Our stream also included a broader set of fraud-claim related keywords. In order to help understand the dissemination of misinformation cross platforms, Brena et al. (2019); Hui et al. (2018) used news articles as queries and released the tweets pointing to these articles. In 2018, Twitter published a list of accounts that the platform suspects to be related with Russia’s government controlled Internet Research Agency (Twitter 2018). This release enabled a number of studies that deepened our understanding of foreign information manipulation in the U.S. (Arif, Stewart, and Starbird 2018; Im et al. 2020; Badawy, Ferrara, and Lerman 2018). Most of the previous works that released Twitter datasets only included the tweet IDs, in accordance with Twitter’s Terms of Service. We keep to that practice, and augment the data without sharing tweet content, as detailed above, making our multi-modal dataset more accessible and useful to the research community. ## 7 Discussion and Conclusions The voter fraud allegations to discredit the U.S. 2020 presidential elections are likely to form one of the most consequential misinformation campaigns in modern history. It is critical to allow a diverse set of researchers to provide a deeper understanding of this effort, which will continue to have national and global impact for years to come. To enable that contribution, it is important to provide a public and accessible archive of this campaign on various social media platforms, including Twitter as we do in . The dataset has the potential to benefit the research community, and to further inform the public regarding both Twitter activities around the voter fraud claims, as well as Twitter’s response. Yet, the data has some limitations. We could not possibly capture the full extent of the voter fraud claims on Twitter, as our dataset was constructed by using matching keywords. Further, as discussed above, we do not have full coverage even for the keywords we tracked, though we estimate that we have a majority of the tweets with those keywords. Nevertheless, the breadth of the data enables various types of investigation using both the tweet data, as well as the aggregated data of URLs, videos and images used in the campaign. We propose three major categories of such investigation. First, researchers can use the dataset to study the spread, reach, and dynamics of the voter fraud campaign on Twitter. Researchers can describe and analyze the participants, including the activities of political candidates using information from orthogonal data sets of candidate accounts 555https://github.com/vegetable68/Midterm-2020-candidates, and the interaction between public figures and other accounts spreading claims and promoting certain narratives. Further, the data can help expose how different public figures spread different claims, for example the claims regarding the Dominion voting machines, what kind of engagement such narratives received. The data can also be used to understand the role of bots and other coordinated activities and campaigns in spreading this information. In general, the dataset can provide for analysis of the distribution of attention to these claims and how it spreads – via images, tweets, URLs – including comparison among different pre-computed communities and clusters. Second, we include auxiliary data – URLs including YouTube links, and image hashes – that can help researchers examine other sources of information and their roles in spreading these claims. For example, using the image hash values that were encoded using publicly available algorithms, researchers can easily map images not just within the Twitter data, but also within the larger web ecosystem. Researchers may combine our dataset with datasets that are collected from other social media platforms to examine how visual misinformation spread cross platforms (e.g., (Zannettou et al. 2018; Moreira et al. 2018)). A third potential area of investigation is Twitter’s response to the voter fraud claims. A specific question is the characterization of the suspended users, who are primarily part of a specific community even within the the group promoting voter fraud claims as shown above. Researchers can use the data to both understand Twitter’s non-public response, and its potential effectiveness, or even simulate the effectiveness of hypothetical earlier bans of the same population. As noted above, while Twitter’s terms forbid us from publicly sharing full data for these suspended users – the tweets for these users are no longer available on Twitter by their ID – we will make these tweets available privately to published academic researchers, as we believe these tweets are of immense and justified public interest. The publicly released data was collected and made available according to Twitter’s Terms of Service for academic researchers, following established guidelines for ethical Twitter data use (Rivers and Lewis 2014). By limiting to the Tweet IDs as the main data element, the dataset does not expose information about users whose data had been removed from the service. 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International Society for Optics and Photonics. ## Appendix A Appendix Community 0 --- ID \| Handle \| Active Status \| Retweets 32871086 \| kylegriffin1 \| active \| 76,302 1640929196 \| mmpadellan \| active \| 74,393 255812611 \| donwinslow \| active \| 69,796 216776631 \| BernieSanders \| active \| 60,961 15952856 \| AriBerman \| active \| 58,222 a) Community 1 25073877 \| realDonaldTrump \| suspended \| 1,560,373 187680645 \| LLinWood \| suspended \| 1,057,805 586707638 \| SidneyPowell1 \| suspended \| 633,273 240454812 \| GenFlynn \| suspended \| 334,197 1812055789 \| CodeMonkeyZ \| suspended \| 274,210 b) Community 2 2922345639 \| DonnaWR8 \| suspended \| 38,388 259260816 \| zeusFanHouse \| suspended \| 36,347 393190233 \| LeahR77 \| suspended \| 33,352 951302891708583936 \| TheRISEofROD \| suspended \| 32,992 32804484 \| Bubblebathgirl \| active \| 27,787 c) Commmunity 3 835040085573689346 \| ganaha_masako \| active \| 12,480 1128981340848656384 \| KadotaRyusho \| active \| 6,890 796450109986902016 \| yamatogokorous \| active \| 5,716 1166577240601239552 \| mei98862477 \| active \| 5,347 109458204 \| kohyu1952 \| active \| 5,244 d) Community 4 3393186119 \| FernandoAmandi \| active \| 4,217 1126414392080232449 \| POTUS_Trump_ESP \| active \| 2,981 1195348350620622850 \| TDN_NOTICIAS \| active \| 2,459 98294131 \| 1VAFI \| active \| 1,802 1068238181282267137 \| Gamusina77 \| active \| 1,638 Table 2: Top 5 Users in each community sorted by retweets from other users. Seed list \| #abolishdemocratparty #ballotharvasting #ballotvoterfraud #cheatingdemocrats #democratvoterfraud #gopvoterfraud #ilhanballotharvesting #ilhanomarballotharvesting #ilhanomarvoterfraud #mailinvoterfraud #stopvoterfraud #voterfraud #voterfraudbymail #voterfraudisreal ---\|--- Filtered \| #abolishdemocratparty Generated from the seed list \| #ballotharvesting #voterid #ilhanomarforprison #stopgopvoterfraud #ilhanomar #nancypelosiabusingpower #nancypelosimustresign #junkmailballots #traresforcongress #immigrationfraud #votebymailfraud #ballotfraud #exposed #votersuppression #ilhanresign #voteinperson #votebymail #video #lockherup #nomailinvoting #ilhanomarelectionfraud #taxfraud #ballotharvesting #massivemailinballots #arrestilhanomar #obamagate #ilhanomarlockherup #buyingvotes #2020election #campaignfraud #homewrecker #voteinperson #minneapolis #absenteeballots #trump2020 #arrestilhanomar #absenteeballot #darktolight #wwg1wga #terrorist #daveygravyspirualsavage #trump #fraud #liar #pizzagate #republicans #qproof #theawakening #voteatthepolls #marriedherbrother #glasshouses #sheepnomore #voteyouout #cheater #georgesoros #georgia #vote #walkaway #thegreatawakening #qanon #evil #savethechildren Keywords list 10/24 \| #ballotfraud #ballotharvesting #ballotvoterfraud #cheatingdemocrats #democratvoterfraud #ilhanomarballotharvesting #ilhanomarvoterfraud #mailinvoterfraud #nomailinvoting #stopgopvoterfraud #stopvoterfraud #votebymailfraud #voterfraud #voterfraudisreal Added on 10/31 \| #discardedballots #electionfraud #electioninterference #electiontampering #gopvoterfraud #hackedvotingmachines ‘destroyed ballots’ ‘discarded ballots’ ‘election fraud’ ‘election interference’ ‘election tampering’ ‘hacked voting machine’ ‘pre-filled ballot’ ‘stolen ballots’ ‘ballot fraud’ ‘ballot harvesting’ ‘cheating democrats’ ‘democrats cheat’ ‘harvest ballot’ ‘vote by mail fraud’ ‘voter fraud’ Added on 11/03 \| #stopthesteal Table 3: Hashtags and keywords related to election fraud.
# PiChu: Accelerating Block Broadcasting in Blockchain Networks with Pipelining and Chunking Kaushik Ayinala Baek-Young Choi Sejun Song University of Missouri-Kansas City, Kansas City, MO, USA Email: {kapnb, choiby<EMAIL_ADDRESS> ###### Abstract Blockchain technologies have been rapidly enhanced in recent years. However, its scalability still has limitations in terms of throughput and broadcast delay as the network and the amount of transaction data increase. To improve scalability of blockchain networks, we propose a novel approach named PiChu that accelerates block propagation in blockchain networks by pipelining and verifying chunks of a block in parallel. Accelerating block propagation reduces the mining interval and chance of fork occurring, which in turn increases throughput. Our approach can be applied to the blockchain networks either directly or with a minor modification to the consensus. Through an extensive and large scale simulations, we validate that the proposed PiChu scheme significantly enhances the scalability of blockchain networks. For instance, a 64 MB block can be broadcasted in just 80 seconds in a blockchain network with a million nodes. The efficiency of PiChu broadcasting increases with bigger block sizes and a larger number of nodes in the network. ###### Index Terms: blockchain, block propagation, chunking, pipelining, simulator, P2P network, scalability ©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. ## I Introduction Blockchain maintains a distributed ledger of the completed transactions as blocks and chains them sequentially using the previous block hash to maintain the order of completed transactions. Nodes in a blockchain are connected to each other on a peer-to-peer (P2P) network. A consensus protocol running at every node follows the agreement of a policy to add a block to the chain. There are several consensus schemes such as Proof of Work (PoW), Proof of Stake (PoS), Delegated Proof of Stake (DPoS), Practical Byzantine Fault Tolerance (PBFT), and Hybrid Consensus. For instance, Proof of Work (PoW) is one of the commonly used consensus algorithms introduced in bitcoin [1]. In the PoW consensus algorithm, each block contains a timestamp, nonce, hash of the block, difficulty target. Proof of Stake (PoS) is another well-known consensus algorithm introduced in PPCoin [2]. After validating a block, the node broadcasts or propagates it to the rest of the network. The time it takes to propagate a block depends on many factors, such as the size of a block, the average bandwidth of the nodes, and the maximum hop count or diameter of a network. Those factors have intricate relationships. When the number of nodes in the network increases, the network diameter increases along with a block broadcast time. Also, when throughput is increased via larger block size, a block broadcast time increases, causing the chance of undesirable forks. The blockchain network becomes unstable when there are too many forks, or forks do not resolve. Therefore, if we increase the throughput or capacity of the blockchain network, then it may become unstable. This causes the scalability problem [3, 4, 5] in the blockchain. In this paper, we propose a Pipelining and Chunking scheme for blockchain networks, named PiChu that is to expedite a block propagation by verifying consensus with the block header and incrementally forwarding the body of a block by small chunks over the P2P network, instead of a whole block at once. After receiving a chunk, a node will verify and forward the chunk. Accelerating block broadcast time improves the scalability of the blockchain, as the block interval can be reduced, the block size can be increased, and forks in the chain would be reduced. Since PiChu takes advantage of network pipelining, the efficiency is far better than the traditional approach. Our experimental results showed, on average, a magnitude ($\approx$ 13.6 times) less block broadcast time than the traditional method in a blockchain network with 65,536 nodes. PiChu technique can be applied directly to the existing consensus protocols with minimal change to the blockchain network. PiChu approach can be directly used with the existing consensus algorithms such as PoS, DPoS and PBFT that use a header only to verify a block. As for PoW that uses an entire block for a verification, PiChu approach can be employed with a minor modification in the consensus. Our contributions in this paper include i) proposing PiChu, a novel block broadcasting technique, ii) development of a versatile blockchain simulator, and iii) analysis and extensive evaluations of the efficiency of the proposed scheme. The rest of the paper is organized as follows. We discuss the existing works on blockchain scalability in Section II. Section III describes the proposed scheme in detail. The efficiency and the pseudo-code of our scheme are given in Section IV. Section V discusses the potential attacks and proposes countermeasures. Section VI explains the experiment environment and results. We conclude the paper in Section VII. ## II Related Work There are a number of studies to improve the scalability of the blockchain networks. They follow approaches like using multiple chains, sharding or exploiting network topology information. Monoxide [6] uses multiple chains to linearly scales the blockchain. It proposes Chu-ko-nu mining to maintain the same difficulty across all the chains, and proposes a protocol to handle inter-chain transactions. A node can mine a block in multiple or all chains by solving a single problem. Miners can choose the chains they want to work on. This may cause a chain to be abandoned if there are too many chains. Elastico [3] also linearly scales the blockchain by sharding but uses a single chain. Sharding involves dividing the network into groups or shards for a given amount of time. Each group will work on a different set of transactions. The size of the block increases with the number of nodes, which in turn increases the broadcast time. The size of the block is limited by bandwidth and latency. A scheme to speed up block propagation by choosing the closest neighbors as peers was proposed in [7], where the closest neighbor is determined by transmission latency. Another study [8] also improves the scalability by maintaining the network topology using a tree structure for a broadcast routing. Tree cluster routing is proposed to do routing during node failures. However, it does not address adapting to dynamic network conditions such as a new node’s join and handling a node or cluster failure. Velocity [9] improves block broadcasting by downloading the parts of a block from multiple neighbors. In the scheme, a block is converted into so-called fountain codes. The node that wants to receive a block sends a request message to all of its neighbors. The neighbors having the block sends a fountain code continuously. After receiving sufficient codes, the node rebuilds the block. Graphene [10] improves block propagation by reducing the transmission delay between the nodes. Graphene uses Bloom filters and Invertible Bloom Lookup Table (IBLT) to synchronize the block between peers. Bitcoin-NG [11] indirectly selects a leader for a given time frame, and the leader transmits the micro blocks throughout the time frame. The chain contains two types of blocks. They are key and micro blocks. The node that mines the key block becomes the leader. The consensus protocol for the key block is PoW. However, it is for a specific type of a consensus protocol and can not be used on other existing consensus protocols. To the best of our knowledge, this paper is the first work that uses the unique approach of pipelining and chunking for accelerating block propagation blockchain networks. The proposed scheme can be used along with existing scaling and acceleration techniques in a complementary manner. ## III PiChu: The Proposed Pipelining and Chunking Approach This section explains the proposed PiChu scheme. PiChu scheme involves first, sending a header as an invitation, then dividing the body of the block into chunks, and finally forwarding the chunks in pipeline. ### III-A Verification of a Block for a Consensus PiChu does a block verification for a consensus using only a header rather than a whole block. Most consensus algorithms including Proof Of Stake, Delegated Proof of Stake, Proof of Activity, Proof of Burn, Proof of Elapsed Time, and Leased Proof of Stake ([2, 12, 13, 14, 15]) need only the header for the consensus verification. Those consensus protocols require only the header to verify a block for the consensus, as shown in Equation (1). Equation (1) is the consensus between the nodes in PoS. Thus PiChu can be readily used on those blockchains. $Hash(Header)<C_{w}\textit{DifficulyTarget}$ (1) where, $C_{w}$ is a coin day weight. On the other hand, Proof of Work is a consensus protocol that requires an entire block for its verification. However, it can be made PiChu-capable with minor modifications. Note that nodes in a PoW blockchain follows Equation (2) to add a block to their chain. $Hash(Block)<\textit{DifficulyTarget}$ (2) Hash of all the transactions should be included in the header. The reward transaction should be included in the header. In Bitcoin, the size of nonce is 32 bit, but the difficulty target can be greater than $2^{32}$. Miners iterate nonce, but they may not find the nonce that satisfies the consensus, and then miners shuffle the transactions and iterate the nonce again. As consensus has to be verified with the header, there are no transactions for the miners to shuffle. So the size of the nonce has to be increased. After modifying the header, the PoW consensus can be verified by using the Equation (3). The PoW consensus is modified to use only the header. PiChu can now be used on modified PoW consensus. $Hash(Header^{\prime})<\textit{DifficulyTarget}$ (3) ### III-B Chunking and Propagation Scheme Chunking involves dividing the body of the block into multiple chunks of the same size. Each chunk should contain only complete transactions, and the remaining space in a chunk is padded. A miner appends some information about chunking to the header and signs it. The miner can not use any key to sign the header with metadata. He has to sign with the key that is used to claim the reward of the block. All the blockchains give rewards to the miners. The reward is included in the block. The reward should contain the public key of the miner. In consensus algorithms that need a whole block, the reward is included in the body of the block. We have to modify a whole-block consensus protocol in such a way that the reward for mining the block should be included in the header. Thus, a miner signs the header with metadata by using its reward private key. Miner sends the signature along with the header to its connection. Receiving nodes retrieves the header from the invitation and verifies the consensus. If consensus is correct, then nodes retrieve the miner reward public key and verify the signature of the invitation. If it is correct, then the node retrieves the information about the chunks from the invitation and uses it receives the chunks. After dividing the block into chunks, the miner appends the chunk number at the starting of the chunk to identify the order of the chunks. The miner then signs the chunk with metadata by using the reward private key, and also signs each chunk to prevent an intermediate node from tampering the data. When a node receives a chunk, it checks the integrity by using the reward public key. We discuss about the optimal chunk size in Section IV. Figure 1: Block structure in PiChu TABLE I: PiChu field types description Field Name \| Size \| Description ---\|---\|--- # of chunks \| 2 Bytes \| Number of chunks in the body of block, varies with number of transactions in the body $C_{i}$ \| 128 KBytes \| $i^{th}$ chunk in the body of the block $S(C_{i},K^{Pr})$ \| 64 Bytes \| signature of $C_{i}$ with ECDSA private key in the block header For every chunk, we send an additional 64 bytes as a signature, which increases the amount of data to be transferred for a block. Even though the data transmission size is increasing, the blocks are transmitting much faster. While storing the block, nodes can remove the metadata about the chunks. Figure 2: Block broadcast sequence in traditional blockchains Figure 3: Block broadcast sequence in PiChu blockchain ### III-C Pipelining In the general broadcast approach, when a node mines or receives a new block, it sends a block invitation to all of its neighbors. If a node receives a block invitation, it checks whether block exists or not. If the block does not exist, then the node replies with the block request message. After receiving the block request message, the node forwards the block. The receiving node verifies the block, and if the block is valid, then it sends the block invitation to all of its neighbors. The traditional block broadcast protocol is illustrated in Figure 3. As illustrated in Figure 3, when a node mines or receives a block, it sends an invitation to all the connected nodes with the PiChu header. The node that received the invitation message verifies whether the header achieves consensus or not. If the header achieves consensus and the node does not have that block, it sends a chunk request message back to the original node. Besides, it sends an invitation message to its neighbor nodes by using the PiChu header. After sending the header invitations to all the neighbors, the miner starts sending the chunks to the neighbors who sent the chunk request message. When a node receives chunks, it verifies the signature of each chunk by using the public key of the miner. Although an additional 64 bytes as a signature is required for each chunk, the overhead is trivial. As long as a chunk is verified, it forwards the chunk to its neighbor nodes, which sent a chunk request. The verification of chunk includes checking the integrity and validity of transactions in it. ## IV Analysis of PiChu Efficiency The broadcast time is proportional to the radius of the network. If the network radius increases, then broadcast time increases. The broadcast time is also proportional to the delay at each node. If the delay at each node increases, then block broadcast time increases and vice versa. On average, the broadcast time in a traditional blockchain network is equal to the radius of the network in hop counts multiplied by the nodal delay of a network, and the nodal delay is the sum of the transmission delay, propagation delay, and verification delay. The transmission delay depends on the block size, bandwidth, and the number of neighbors. The transmission delay is proportional to block size and the average degree of the nodes. The transmission delay is inversely proportional to the bandwidth. The verification delay also depends on the size of the block and the diameter of the blockchain network. The notations of the symbols used in this section are summarized in Table II. $\displaystyle T_{B}$ $\displaystyle=$ $\displaystyle R\times\left(T_{LinkTrans}+T_{LnkPrp}+T_{ver}\right)$ (4) $\displaystyle=$ $\displaystyle R\times\left(D_{conn}\times\frac{L_{B}}{B_{w}}+T_{LnkPrp}+T_{ver}\right)$ (5) In the PiChu scheme, the header is broadcasted first, then chunks are pipelined in parallel. So the time it takes to propagate the block is equal to the sum of the time it takes to broadcast the header and the time to transmit all the chunks from a node to another. The time it takes to transmits chunks from one node to another depends on the degree of the nodes, the number of chunks, metadata, and bandwidth. The size of metadata for each is 520 bits. $\displaystyle T_{PiChu}$ $\displaystyle=$ $\displaystyle T_{PH}+T_{DC}$ (7) $\displaystyle=$ $\displaystyle R\times\left(D_{conn}\frac{L_{H}}{B_{w}}+T_{LnkPrp}+T_{ver}\right)+T_{DC}$ $\displaystyle=$ $\displaystyle R\times\left(D_{conn}\times\frac{L_{H}}{B_{w}}+T_{LnkPrp}+T_{ver}\right)+$ $\displaystyle\frac{D_{conn}\times\left(N_{c}+520\right)\times L_{C}}{B_{w}}$ As seen in Equation (5), the block broadcast time depends on the product of the network radius and block size. If the block size is increased in Equation (5), then broadcast time increases by at least $R$ times. In Equation (7), we can observe that the block broadcast time depends on the product of the network radius and header size. If the block size is increased in Equation (7) then broadcast time increases by only block transmission delay between two nodes. So the PiChu block broadcast approach is very efficient than the general broadcast approach. The efficiency of the PiChu broadcast approach over the traditional broadcast approach increases with an increase in block size and number of nodes in the network. Algorithm 1 gives the pseudo-code of the PiChu scheme. It shows that when a node receives an invitation from a peer, it checks whether that header or block exists in the chain. If it does not exist, then the node requests and receives the chunks from the peer. When a chunk is received, it immediately forwards it to other peers. It also indicates that only one block is received and forwarded at a time. List HeaderConnections; Object CurrentHeader; while _True_ do Receive block header H as an invitation from node N; if _CurrentHeader == H_ then HeaderConnections.add(N); Continue; else if _CurrentHeader == null_ then CurrentHeader = H; if _If adding H makes a chain longest_ then if _Hash(Header) <D_ then sendToOthers(H); Retrieve $Pu_{k}$, $N_{c}$ from H; while _$N_{C}-- >0$_ do Receive a chunk; if _Chunk is valid_ then sendToOthers(Chunk); else Choose a node X from HeaderConnections; $N_{C}++$ ; Request X to pipeline last $N_{C}$ chunks; end while if _Block is valid_ then Add block to the chain; else Discard the block; end while Procedure _sendToOthers(_Data_)_ Send Data to other nodes in parallel Algorithm 1 Psuedo code of Chunking and Pipelining Block Broadcast Scheme The size of the chunk is bounded by the block size but should be large enough to overcome the metadata processing overhead. The transmission delay of a chunk should be less than the sum of propagation delay and protocol overhead so that there will be no extra delay at each forwarding node. A node has to receive the chunk before it receives the chunk request message from its neighbors so that it can immediately forward the chunk after receiving the message. The chunk size can be decided by Equation (10) below. $\displaystyle T_{tc}$ $\displaystyle<$ $\displaystyle T_{LnkPrp}+T_{proc}$ (8) $\displaystyle\frac{L_{C}D_{conn}}{B_{w}}$ $\displaystyle<$ $\displaystyle T_{LnkPrp}+T_{proc}$ (9) $\displaystyle L_{C}$ $\displaystyle<$ $\displaystyle\frac{(T_{LnkPrp}+T_{proc})\times B_{w}}{C_{m}}$ (10) TABLE II: Explanation of Notations Notation \| Explanation ---\|--- $L_{H}$ \| header size in bits $L_{B}$ \| block size in bits $N_{C}$ \| the number of chunks in a block $L_{C}$ \| chunk size in bits $B_{w}$ \| average bandwidth of a node $R$ \| radius of a network $T_{P}$ \| average broadcast time in a traditional blockchain network $D_{conn}$ \| average degree of connections of a node $T_{LinkTrans}$ \| average transmission delay between the nodes $T_{LnkPrp}$ \| average propagation delay between two nodes $T_{ver}$ \| average verification delay of a block $T_{PiChu}$ \| average delay to propagate block in PiChu scheme $T_{PH}$ \| average delay to propagate a header $T_{DC}$ \| average delay in transmitting all chunks from one node to another $T_{tc}$ \| transmission delay of a chunk $T_{proc}$ \| PiChu processing overhead delay ## V Defense against Potential Attacks In this section, we discuss the potential attacks and mitigation strategies in a PiChu enabled blockchain network. ### V-A Forwarding node tampers data An intermediate node can modify the data in the block before forwarding it to other nodes. If a malicious node modifies the data in the chunk and forwards it, then the receiving nodes can not verify the integrity of the chunk. Receiving nodes validates the integrity of the chunk by checking the signature of the chunk. A node that received a tampered chunk discards the chunk and disconnects from the node that sent it. The node has to receive the remaining chunks from other neighbors. A node can receive the header $H$ invitation from multiple neighbors. The node keeps a record $R$ of neighbors that sent the header $H$ invitation. When a tampered chunk is received for the block with header $H$, node disconnects from the sender and requests to pipeline remaining chunks from an optimal neighbor in the record $R$. The optimal neighbor is decided based on latency and transmission delay. ### V-B Miner includes invalid transactions in a block A miner can include invalid transactions in a block. This causes one or more chunks to contain invalid transactions. The header will be accepted by all the nodes, as it was valid. When receiving the chunks, nodes validate the chunks before forwarding them. If the chunk contains invalid transactions, then nodes can not validate that chunk. If it contains invalid transactions and integration is correct, then nodes can safely assume that the miner is malicious. After detecting that the miner is malicious, the node forwards the chunk with invalid transactions to neighbors so that other nodes can detect that miner is malicious. If the miner includes invalid transactions in the last chunk, then he can perform a denial of service on the network for the time it takes to broadcast the block. The time to propagate the block through PiChu is small compared to regular broadcast. So the time that the miner can perform a denial of service on the network is small. PiChu broadcasting is used on a block if adding that block to the chain makes it longer. We do this to reduce the forks in the chain and prioritize the miner that finds the block first. After detecting the invalid transactions in a chunk, nodes blacklist the header of that block. Nodes will not add a block with a blacklisted header to their blockchains. This causes the miner to lose the reward for a mined block. We can also revoke all the rewards that the miner accumulated. Miner has to lose his reward if he wants to perform a denial of service on the network. ### V-C Intermediate node delays the sending of the chunks A forwarding or intermediate node in the blockchain network can intentionally delay the forwarding of chunks. The nodes connected to the attacking node receives the chunks slower than their peers. If there are many attackers in the network, then the block broadcast time will increase. The mitigation for this attack is similar to the data tampering mitigation. The node keeps a record $R$ of neighbors that sent the header $H$ invitation. When a node detects or suspects that an intermediate is delaying the forwarding of chunks, then node disconnects from the forwarding node and requests to pipeline remaining chunks from a neighbor in the record $R$. ### V-D Miner dies while sending the block or sends only partial block Miner node might fail while sending the block or intentionally sends the partial block to perform an attack on the network. It might not be possible to differentiate whether the miner node died or intentionally sent the partial block, so the approach for the two cases is the same. When a node does not receive the chunk $X$ after receiving the chunk $X-1$ , it has to first decide whether miner died or the forward node died. A forwarding node might intentionally stop forwarding the chunks. To decide either miner died or the forwarding node died , the node requests the chunk $X$ from all of its neighbors. If any of the neighbors send the chunk $X$, then the forwarding node is failed. If no neighbor sends the chunk $X$ within a time frame, it is safe to assume that the miner died. If a forwarding node is failed, then the node terminates the connection with the forwarding node and receives the chunks from another neighbor. Assume that the miner died while sending the chunk $X$, and all the previous chunks are valid. All the nodes in the network will have chunks till $X-1$. Some blockchains can tolerate partial block in the chain, and other blockchains can not tolerate it. If it is tolerable, then nodes append a special chunk to the chunk $X-1$ that represents only partial block is received. After appending the special chunk, nodes will not receive any further chunks for that block. If partial blocks are not acceptable, nodes discard the chunks received for that block and keep the header in the blacklist. Nodes will not accept the block with a blacklisted header. As the header is not accepted in the blockchain, the miner will not get the reward for that block. Miner loses the reward for sending partial block. This gives the incentive to not send partial blocks. In another approach, nodes will not use the PiChu scheme for the blacklisted header and propagate through the general approach. In an aggressive approach, nodes can take away all the rewards that the miner accumulated. ## VI Experiment Results (a) Traditional approach (b) PiChu approach Figure 4: Block broadcast delay in a blockchain network [Number of nodes: 1K $\sim$ 1M nodes; Block sizes: 8KB $\sim$ 64MB]; (more than 15 times faster with PiChu in a million node network and 64MB blocks) Figure 5: Block broadcast time comparison: Traditional vs. PiChu (in a 65536 node network) Figure 6: Percentage of forks: Traditional vs. PiChu (in a 65536 node network) In order to validate the effectiveness of the PiChu scheme in a very large network with varied parameters, we have developed our own blockchain simulator. While there is an existing blockchain simulator called Simblock [16], it is not well-suited to simulate the block broadcasting in a network with a large number of nodes. Our simulator is developed in Java, and we made the source code publicly available through github [17]. It can simulate block broadcasting in a network with millions of nodes and supports a large block size. Our blockchain simulator can simulate block broadcast in traditional and PiChu approach. It takes the average bandwidth of nodes, average latency between nodes, the block size, chain length, number of nodes, and average degree of a node ($D_{n}$) as input. For a given number of nodes, the simulator generates a random graph topology based on the average degree per node. We first match general propagation results with real measurements data as well as other existing simulators for comparable settings of the experiments. Table III shows the simulation settings used in our study that is similar to [4] and [16]. The output of our simulation is showed and compared in Table IV. Our results are close to the real measurements. TABLE III: Simulation Settings Parameter \| Bitcoin \| Litecoin \| DodgeCoin ---\|---\|---\|--- # of the nodes \| 6000 \| 800 \| 600 Block Interval \| 10 min \| 2.5 min \| 1 min Block Size \| 534 KB \| 6.11 KB \| 8 KB # of the connections \| based on Miller.A[18] Bandwidth \| testmy.net [19] Propagation delay \| verizon [20] TABLE IV: Various simulators output \| Bitcoin \| Litecoin \| DodgeCoin ---\|---\|---\|--- (Block Interval) \| (10 m) \| (2 m 30 s) \| (1 m) $t_{MBP}$ of Real Measurement [4] \| 8.7 s \| 1.02 s \| 0.98 s $t_{MBP}$ from Gervais et. al. [4] \| 9.42 s \| 0.86 s \| 0.83 s $t_{MBP}$ from SimBlock [16] \| 8.94 s \| 0.85 s \| 0.82 s $t_{MBP}$ from our Simulator \| 9.55 s \| 1.04 s \| 1.07 s Measured $r_{f}$ \| 0.41% \| 0.27% \| 0.62 Gervais et al. $r_{f}$ \| 1.85% \| 0.24% \| 0.79% SimBlock $r_{f}$ \| 0.58% \| 0.30% \| 0.80% Our Simulator $r_{f}$ \| 0.55% \| 0.40% \| 0.70% First, we assess how the block broadcast delay varies with the number of nodes and the block size in the traditional broadcast scheme. The average bandwidth of the nodes and average latency between nodes for this experiment are taken from [19] and [20]. In this experiment, the degree of each node is varied between 8 and 12. The maximum number of connections for a node in bitcoin is 125 [18], and the maximum number of connections for a node in Ethereum is 50 [21]. Coinscope[18] found that the majority of nodes in the Bitcoin network have a degree between 8 and 12, even though the maximum number of connections is set to 125. The verification delay is set to 0.25 ms for a transaction[22]. After setting the parameters for the experiment, the number of nodes in the network and block size are varied. The results for this experiment are represented in Figure 4 (a). We can observe that when the number of nodes is constant, broadcast time increases linearly with an increase in block size. The broadcast time is proportional to the product of the network radius and block size. The broadcast time also increases with increasing the number of nodes. We then assess how the block broadcast delay vary with the number of nodes and the block size in the blockchain networks using the PiChu propagation technique. The parameters for this experiment are the same as the previous experiment except the $D_{n}$. PiChu technique works better if the degree of the nodes is small. In Equation (7), we can observe that when the block size is high, the broadcast delay majorly depends on the transmission delay of the block between two nodes. The efficiency of the PiChu depends on how fast a chunk can be transmitted from one node to another. If the degree of a node is high, then the time it takes to transmit a chunk from one node to another increases, and the efficiency of the PiChu decreases. The average degree of the node is set to 5. The reason is explained in the latter part of this section. After setting the parameters for this experiment, the number of nodes and block size are varied. Figure 4 (b) shows the output of this experiment. We can observe that for a given number of nodes, the block broadcast time increases linearly with increasing the block size, but the slope is less than the traditional approach broadcast time. When block size is large, the propagation delay mainly depended only on block size instead of the product of the network radius and block size. The broadcast time increases a little with an increase in the network radius. Figure 6 shows how the broadcast delay change with respect to block size for 65536 nodes in PiChu and the general approach. The block propagation with PiChu for 65536 nodes is 13.6 times less than the traditional approach. The block propagation with PiChu for million nodes is 16.3 times less than the traditional approach. By the experiment results, we can say that the PiChu propagation method is efficient than traditional propagation, and the efficiency of the PiChu increases with an increase in the number of nodes or block size. Figure 6 shows the percentage of forks occurring with respect to block size for 65536 nodes in PiChu and traditional approach. The block interval for this experiment is 10 minutes. The percentage of forks occurring in the PiChu approach is ten times less than the traditional approach. As the forks are reducing, we can increase the size of the block. Throughput increases with an increase in block size. The maximum possible block size for a given block interval is measured for a traditional blockchain and the PiChu blockchain. Both blockchain networks contain 65536 nodes. The maximum possible block size is for a given block interval is determined by increasing the block size until the forks are greater than or equal to 100 percent. When forks are greater than 100 percent, the blockchain becomes obsolete. Figure 8 shows the results. The maximum block size for a given interval is ten times higher in PiChu than traditional. In both approaches, the maximum block size increases with an increase in block interval. The broadcast time of a block in PiChu depends on the degree of nodes. In PiChu, the broadcast time increases with respect to the degree of nodes as the time takes to send the chunk to all the connections increases. In the traditional approach, the broadcast time might increase or decrease with respect to node degree. To confirm that the broadcast time increases with an increase in the node degree, we varied the node degree between 3 and 25. The degree can not be two as the topology of the network will be become linear or circular. The simulator settings are the same as the experimental settings. The number of nodes is 65536, and the block size is set to 64 MB. Figure 8 shows how the broadcast time increases with an increase in degree. The lowest broadcast time is recorded when the degree of the nodes is 3. If we choose the degree as three, then nodes are susceptible to Sybil attacks, and new nodes might find it difficult to discover the nodes. The degree should be as high as possible with reasonable broadcast time. We suggest the degree to be 5 as the block can be broadcast in under 80 seconds. It is to be noted that if we used degree 3 in our previous experiments, then the experimental results will be 1.6 times better. Figure 7: Block broadcast time with respect to maximum number of connections per node Figure 8: Maximum block size for a given interval in 65536 nodes network ## VII Conclusions We have proposed a block acceleration scheme via pipelining and chunking of a block, named PiChu, to address the issue of scalability and performance of a blockchain network. To the best of our knowledge, this is the first kind of approach for blockchain scalability. The approach can be employed with minimal modification of existing blockchain networks. We have shown the efficiency of the PiChu approach, both theoretically and extensive evaluation using our blockchain simulator. 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, Electronic address<EMAIL_ADDRESS>, # PRINCIPAL COMPONENT ANALYSIS FOR ESTIMATING PARAMETERS OF THE L1287 DENSE CORE BY FITTING MODEL SPECTRAL MAPS INTO OBSERVED ONES L. E. Pirogov Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia P. M. Zemlyanukha Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia ###### Abstract An algorithm has been developed for finding the global minimum of a multidimensional error function by fitting model spectral maps into observed ones. Principal component analysis is applied to reduce the dimensionality of the model and the coupling degree between the parameters, and to determine the region of the minimum. The $k$–nearest neighbors method is used to calculate the optimal parameter values. The algorithm is used to estimate the physical parameters of the contracting dense star-forming core of L1287. Maps in the HCO+(1–0), H13CO+(1–0), HCN(1–0), and H13CN(1–0) lines, calculated within a 1D microturbulent model, are fitted into the observed ones. Estimates are obtained for the physical parameters of the core, including the radial profiles of density ($\propto r^{-1.7}$), turbulent velocity ($\propto r^{-0.4}$), and contraction velocity ($\propto r^{-0.1}$). Confidence intervals are calculated for the parameter values. The power-law index of the contraction-velocity radial profile, considering the determination error, is lower in absolute terms than the expected one in the case of gas collapse onto the protostar in free fall. This result can serve as an argument in favor of a global contraction model for the L1287 core. ## 1 Introduction Studies on the structure and kinematics of the dense cores of molecular clouds provide information on the initial conditions and early stages of the star- formation process to be utilized in theoretical models. This is especially important when studying the regions of massive star and star cluster formation, which evolutionary scenarios are now only beginning to develop (see, e.g., [1, 2]). According to observational data, massive stars ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}8$ M⊙) and star clusters are formed in near-virial equilibrium dense cores that are located in filament-shaped massive gas-dust complexes and clumps (see, e.g., [3–7]). The existing theoretical models of star formation employ different assumptions about the initial core state. Thus, the isothermal sphere model, which is applied for describing the formation of isolated low- mass stars [8, 9], assumes that the quasi-equilibrium spherical core with a Bonnor-Ebert-type density profile (a flat segment near the center and a near-$\propto r^{-2}$ dependence in the envelope) evolves towards a singularity at the center (protostar), after which a collapse begins, which propagates ‘‘inside-out’’. The turbulent-core model [10], proposed for describing the formation of massive stars and star clusters, also considers, as the initial state, a hydrostatic-equilibrium sphere characterized by supersonic turbulence and a $\propto r^{-3/2}$ density profile [10, 11]. Both the isothermal sphere model and the turbulent core model use density and velocity profiles in the region where gas collapses onto the star of the form $\propto r^{-3/2}$ and $\propto r^{-1/2}$, respectively. As shown in [12], these profiles do not depend on the state of gas in the core. An alternative model of global hierarchical collapse [13] proceeds from the fact that the cores, like the parent clouds, are nonequilibrium objects that are in an ongoing process of global collapse even before the protostar formation, and their observed closeness to virial equilibrium is due, specifically, to the closeness of the free fall velocity to the virial one. In this model, which is based on the classical works of Larson and Penston [14, 15], after the formation of the protostar, the density profile in the envelope becomes $\propto r^{-2}$ and the contraction velocity is independent of the radial distance (see, e.g., [16, 13]). Near the protostar, where the collapse occurs, the radial profiles of density and contraction velocity are the same as in the isothermal-sphere and turbulent-core models. Thus, the information about the density profile is insufficient for us to make a choice between the above models; firstly, we need to know the velocity profile in the outer regions of the cores. The kinematics of the cores is estimated mainly from observations in molecular lines. The presence of systematic velocities along the line of sight leads to a shift in the centers of optically thick and thin lines (see, e.g., [17]) and to the appearance of asymmetry in the spectra of optically thick lines due to the absorption of the emission from the inner layers by outer ones and due to the Doppler effect (see, e.g., [18, 19]). The average contraction velocity of the core can be estimated within more or less simple models from asymmetric line observations at one point (see, e.g., [20–23]). To estimate the radial profile of systematic velocity, it is necessary to fit the model spectral maps into the observed ones, while simultaneously calculating or setting the profiles of the other physical parameters. Automatic fitting methods of model spectral maps into observed ones in the case of several free parameters are rarely used today. Researchers usually compare the spectra observed at individual points with simulated ones; less often, they use spectral maps, varying one or two parameters and considering the remaining ones to be specified from independent observations, theoretical model predictions, or preliminary calculation results (see, e.g., [24–29]). In this case, researchers either consider the systematic contraction velocity to be constant or use a radial profile $\propto r^{-1/2}$. Finding the optimal values while varying several parameters simultaneously may be difficult because the error function may have more than one local minimum and the parameters themselves may correlate with one another, leading to dependence on the initial conditions and to poor convergence. The use of special methods to search for the global minimum of the error function (e.g., the method of differential evolution [30]) in the case of a model with several free parameters may lead to considerable computational costs. In recent years, $Principal\leavevmode\nobreak\ Component\leavevmode\nobreak\ Analysis$ (PCA) has been successfully applied to studying experimental data [31]. Within this method, data are transformed to such an optimal basis in which linear relations between the basic vectors are excluded. This approach allows one to reduce the dimensionality of the data. This method is quite often used to reduce the dimensionality of astronomical data (see, e.g., [32] and references therein), but it can also be applied to the results of model calculations by reducing the dimensionality of the model and determining the range of parameter values near the minimum of the error function. The exact values of the model parameters, which correspond to the minimum of the error function, can be calculated by the regression method. For instance, the $k$–nearest neighbors ($k$NN) method [33] appears to fit this purpose. It is an analogue of the least-squares method, but unlike the latter, it allows one to choose, from a set of models, only ones that correspond to observational data by the least-squared error criterion. This work aims to develop an algorithm that uses PCA and $k$NN to fit model spectral maps into observed ones and to apply this algorithm for estimating the radial profile of systematic velocity and other physical parameters of the L1287 dense core. In this object, a cluster of low- and intermediate-mass stars is being formed, and the observed profiles of optically thick lines show an asymmetry pattern which indicates contraction (see, e.g., [34]). The analysis used observational data in the lines of HCO+(1–0) and HCN(1–0), which are indicators of high-density gas ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}10^{5}$ cm-3 [18]) and the isotope lines of these molecules. Observations in different lines of the HCO+ and HCN molecules are often used to search for massive cores with systematic motions in the line of sight (see, e.g., [35–39]). The HCN(1–0) line is, however, less often used for these purposes. It has three hyperfine components with different optical depths and intensity ratios that differ from the case of local thermodynamic equilibrium (LTE). The observed profiles of these components may overlap. To determine parameters of the physical and kinematic structure of the cores from the HCN(1–0) data it is necessarily to use non-LTE models (see, e.g., [40, 41]). In this work, we calculated the excitation of HCO+, HCN, and their isotopes using a 1D microturbulent spherically symmetric non-LTE model, the physical parameters of which, including the systematic velocity, were functions of the distance from the center. This paper consists of five sections and Appendix. Section 2 presents the algorithm for fitting model spectral maps into observed ones using PCA and $k$NN. Section 3.1 summarizes the observational data and physical properties of the L1287 core. Section 3.2 describes the application of the algorithm to the observational data on L1287 and the results of estimating the physical parameters. Sections 4 and 5 present the results and discussion and the conclusions, respectively. A description of the model is given in Appendix. ## 2 PCA-BASED ALGORITHM FOR FITTING MODEL SPECTRAL MAPS INTO OBSERVED ONES The process of fitting model spectral maps into observed ones by means of conventional iterative methods for estimating physical parameters is complicated by the fact that the multidimensional error function (the total discrepancy between the observed and model spectra) may have several local minima, which creates a dependence on the initial values. In this case, a correlation between the parameters may seriously worsen the convergence. Another approach is to calculate a set of model maps in advance for a grid of model parameters and select those that are close to the observed parameters. This is also a complicated approach because calculations for a discrete $n$-dimensional grid (where $n$ is the number of parameters) that is densely enough to cover the space of probable values may be beyond the computational capabilities. However, such a grid would obviously be redundant. If we specify the parameter values randomly, then, with calculated model maps for them, we can roughly determine a region within which lies the minimum of the error function. If we apply a certain transformation to the resulting region that minimizes the relations between the parameters and transform it to a new space of orthogonal vectors, we can reduce the dimensionality by discarding the vectors carrying minimum information about the model parameters. If we then fill the remaining vector space with a sufficiently dense grid and make the inverse transformation, we obtain a filled-in space of model parameters near the minimum, the exact value of which can be found by the regression method. One such transformation can be PCA, a classical method of dimensionality reduction [31]. It involves finding such a linear transformation where the initial set of parameters is represented by a vector basis (with principal components as the vectors), the correlations between the vectors being minimized. The described general principles enabled the development of an algorithm for finding the physical parameters of dense cores of molecular clouds by fitting model maps of the molecular lines into the observed ones. The algorithm involved a preliminary analysis of observational data and determination of the to-be-free parameters, PCA-based dimensionality reduction and determination of the region of model parameters near the minimum, and finding the optimal values of free parameters by the $k$NN method [33] and determination of the confidence region boundaries for each of them. The diagram of the algorithm is shown in Fig. 1. The optimal parameters were determined minimizing the error function: $\chi^{2}=\frac{1}{N_{p}-n}\sum_{j=1}^{N}\sum_{i=1}^{m}\frac{(I_{ij}^{obs}-I_{ij}^{mod})^{2}}{\sigma_{j}^{2}}\hskip 5.69054pt,$ (1) where $N$ is the number of spatial points in the map; $m$ is the number of channels in the spectrum; $I_{ij}^{obs}$ and $I_{ij}^{mod}$ are the observed and model intensities in the $i$th spectral channel for the $j$th point in the map, respectively; $\sigma_{j}$ is the standard deviation of the observed spectrum at point $j$, calculated from intensity fluctuations outside the line range; $N_{p}=m\times N$; and $n$ is the number of parameters in the model. 5mm normal Figure 1: Parameter determination diagram for fitting model spectral maps into observed ones. In the course of preliminary analysis, we determined the coordinates of the map’s central point (the center of the core) and the source velocity from the observational data in the optically thin line. Using a random number generator, we then created a set of model parameters with sufficiently wide ranges and calculated the model spectral maps and values of the error function. Within this set, we selected those parameters that satisfy the inequality $\chi^{2}\leq\chi_{min}^{2}(3\sigma_{obs})$, where $\chi_{min}^{2}(3\sigma_{obs})$ is the value of the error function at those model parameters that yield the minimum value $\chi^{2}$ for a given set when noise with a standard deviation of $3\sigma_{obs}$ is added to the observed intensities. For the resulting parameter sample, we calculated matrices of the direct and inverse transformation into the PC-space, reduced the number of components, and filled the remaining space with a regular grid. The grid nodes were transformed using the inverse transformation into the values of the physical parameters. Choosing a sufficient number of PCs is not a simple problem since any dimensionality reduction method causes information loss. An overview of possible options for solving this problem is given in Appendix to [42] and in the references to that paper. In linear PCA, which we applied, the loss of information can manifest itself in biased values of the physical parameters after the direct and inverse transformations. The number of remaining PCs, which determines the extent of the loss, was chosen in such a way that the ratio of the eigenvector sum for the PC covariance matrix to the eigenvector sum for the covariance matrix of the physical parameters differed from unity (the value in the case of the identity transformation) by no more than 10% and the bias in the parameter estimates did not create systematic errors [42, 43]. The final step was to calculate the physical parameter values corresponding to the exact minimum of the error function and estimate the errors. To this end, we used the $k$NN method [33], which we previously applied to estimate the physical parameters of the dense core of S255N [44]. The $k$NN method is similar to the least-squares method, but unlike the latter, it does not adjust the model parameters to the observed spectra but calculates the optimal parameter values from the previously obtained spectra by the same criterion. This method enables regression analysis between a set of model maps with different parameters and the observed map. Thus, among all the model maps, we found $k$ nearest ones to the observed map by the criterion of the minimum of the mean $\chi^{2}$ value. When there was no such minimum ($\chi^{2}$ increases with increasing $k$, and averaging across the models increases the error function), a denser grid was needed for PCs in the region near the supposed minimum. The optimal value of the $p$ parameter was the $\chi^{2}$-weighed mean over $k$ selected instances: $p=\frac{\sum_{i=1}^{\rm k}p_{i}/\chi_{i}^{2}}{\sum_{i=1}^{\rm k}1/\chi_{i}^{2}}\hskip 5.69054pt,$ (2) where $p_{i}$ and $\chi_{i}^{2}$ are the values of the parameter and error function for the $i$th point in the parameter space, respectively. Using maps of the object in several spectral lines of different optical depths, we can narrow down the range of confidence values by fitting the model spectra into these maps simultaneously. In this case, additional model parameters are the relative abundances of the molecules. The total error for the maps in several lines ($n_{lines}$) is written as $\chi^{2}=\frac{1}{N_{p}-n}\sum_{k=1}^{n_{lines}}\sum_{j=1}^{N_{k}}\sum_{i=1}^{m_{k}}\frac{(I_{ijk}^{obs}-I_{ijk}^{mod})^{2}}{\sigma_{jk}^{2}}\hskip 5.69054pt,$ (3) where $N_{p}=\sum_{k=1}^{n_{lines}}N_{k}\times m_{k}$; $N_{k}$ is the number of spatial points in the map in the $k$th line; $m_{k}$ is the number of channels in the spectrum of the $k$th line; and $\sigma_{jk}$ is the standard deviation of the observed spectrum in the $k$th line at point $j$. Since the parameter space is curvilinear, the confidence regions for the probable parameter values were determined by applying a cross-section of the multidimensional error function by the hyperplane $\chi^{2}=\chi^{2}_{\sigma}$. The calculation of $\chi^{2}_{\sigma}$ does not depend on the choice of the basis; it is convenient to perform it in the PC space. The threshold value was $\chi^{2}_{\sigma}=\chi_{min}^{2}(pc_{l}^{opt}\pm\sigma_{pc_{l}})$, i.e., the value of the error function in the case where one of the PCs ($pc_{l}$) takes a value displaced from the optimal one by $\sigma_{pc_{l}}$ and the other components vary in such a way that the error function takes the minimum value. As $\sigma_{pc_{l}}$, we took a symmetric estimate for the error of $pc_{l}$, which is a diagonal element of the matrix inverse to the Hesse matrix, (see, e.g., [45–47]), which element was calculated as $\beta_{lm}=\sum_{k=1}^{n_{lines}}\sum_{j=1}^{N_{k}}\sum_{i=1}^{m_{k}}\frac{1}{\sigma_{jk}^{2}}\frac{\partial I_{ijk}^{mod}}{\partial pc_{l}}\frac{\partial I_{ijk}^{mod}}{\partial pc_{m}}\hskip 5.69054pt,$ (4) where $pc_{l},pc_{m}$ are different PCs. The derivatives were calculated numerically over the entire set of model maps. After estimating the threshold value of $\chi^{2}_{\sigma}$, we constructed two-dimensional projections of the error function and its hyperplane cross-section in the plane of different pairs of model parameters and determined the confidence regions. In the general case, these regions are not symmetrical relative to the optimal parameter values. An example of using two-dimensional projections of the error function for estimating the confidence ranges of model parameters in the analysis of the L1287 molecular line maps is presented in Section 3.2. ## 3 ESTIMATING THE PHYSICAL PARAMETERS OF THE L1287 CORE ### 3.1 Observational Manifestations of L1287 The dark cloud L1287 is located at a distance of $0.93\pm 0.03$ kpc [48] and is shaped as a filament of $\sim 10$ pc in length. A dust emission map in continuum at a wavelength of 500 $\mu$m, which was acquired using the Herschel telescope towards L1287 (observation ID: 1342249229 [49]), is shown in Fig. 2 (different shades). In the central part of the cloud, there is a high-density core, which contains the source IRAS 00338+6312 [34]. In the core, two objects of type FU Ori (RNO 1B/1C) were also detected [51–53], as well as a cluster of IR and radio sources, likely associated with young stellar objects of low and intermediate mass [54, 53]. Maser lines of water [55] and methanol molecules [56] were also detected there. Molecular line observations [34, 57, 58] revealed a bipolar outflow in the northeastern and southwestern directions. Based on observations in the H13CO+(1–0) line, it was concluded [59] that the central part of the core contains a rotating disk of radius $\sim 7800$ AU, with the bipolar outflow oriented along the disk axis. Based on interferometry observations, the inner part of the core ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}0.1$ pc) is highly fragmented [60, 61]. In [61], a kinematic model was proposed for the central part of the L1287 core. In this model, gas motions towards the center from the core’s outer regions become nonisotropic near the center and transform into the disk rotation. 5mm normal Figure 2: Map of the L1287 dark cloud at wavelength 500 $\mu$m from the Herschel telescope observational data. The integrated intensity isolines in the HCO+(1–0) line correspond to 20% and 50% of the maximum (38.6 K km/s) [50]. The star symbol indicates the source IRAS 00338+6312. The emission region sizes of the L1287 core in the different molecular lines and in continuum vary from a few tenths to one parsec [34, 62–65]; the shape of the emission regions is roughly close to a spherically symmetric one. The profiles of optically thick lines in the L1287 core are asymmetric and have two peaks separated by a dip, with the amplitude of the blue peak in most of the maps being higher than that of the red one [25, 34, 62]. We observed the L1287 core in 2015 with the OSO-20m telescope in different lines in the frequency range of $\sim 85-89$ GHz [50]. The angular resolution of the observations was $\sim 42^{\prime\prime}$, which corresponds to a linear resolution of $\sim 0.19$ pc. The integrated intensity isolines of the HCO+(1–0) line, which were superimposed onto the Herschel map, are shown in Fig. 2. The asymmetric profiles of HCO+(1–0) and HCN(1–0), observed virtually throughout the entire region ($\sim 0.9$ pc), and the symmetric profiles of optically thin lines, which intensity peaks are close to the dips in the profiles of optically thick lines, are likely to be indicative of gas contraction. ### 3.2 Map Analysis of the L1287 Core in Different Molecular Lines The algorithm presented in Section 2 was applied for estimating the physical parameters of the L1287 core. To this end, we performed the fitting of the maps in the lines of HCO+(1–0), H13CO+(1–0), HCN(1–0), and H13CN(1–0), calculated within the 1D microturbulent model (see Appendix), into the central part of the observed region with an angular size of 80′′ ($\sim 0.4$ pc). The physical parameters (density, turbulent and systematic velocities, and kinetic temperature) were dependent on the distance to the center, $r$, by the law $P=P_{0}/(1+(r/R_{0})^{\alpha_{p}}$), where $R_{0}$ is the radius of the central layer. The free model parameters were the values $P_{0}$ for the radial profiles of density and turbulent and systematic velocities ($n_{0}$, $V_{turb}$, $V_{sys}$, respectively); the power-law indices $\alpha_{p}$ ($\alpha_{n}$, $\alpha_{turb}$, $\alpha_{sys}$), the relative abundances of the molecules ($X$), independent of the radial distance; and the outer radius ($R_{max}$) of the core. The kinetic temperature profile was set at $T=80$ K$/(1+(r/R_{0})^{0.3})$ and kept unchanged during the calculations. Meanwhile, the kinetic temperature varied from 40 K in the central layer to $\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}20$ K on the periphery, which is generally consistent with estimates based on observational data (see, e.g., [62, 63, 65, 50]). Although the dust temperatures for L1287 from the Herschel data are $\sim 15-24$ K (http://www.astro.cardiff.ac.uk/research/ViaLactea/) [66], the data of interferometric observations suggest that in the inner regions of the L1287 core ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}0.1$ pc), where the fragmentation effects are strong, the kinetic temperature of fragments may be as high as $\sim 80-100$ K (see [60]). Thus, in the calculations, 40 K was taken as an average value of kinetic temperature in the central layer, the radius of which ($R_{0}$) was set at $2\times 10^{16}$ cm ($\sim 1300$ AU). The radial velocity and the core center coordinates were estimated from the H13CO+(1–0) line. Then, we used a map in the HCO+(1–0) line to search for the minimum of the error function. Using a random number generator, we formed an array of 6000 parameter values, which were randomly and equiprobably distributed in the following ranges of eight parameters: $n_{0}$=[$10^{6.5}...10^{9}$] cm-3, $\alpha_{n}$=[1.3…2.5], $V_{turb}$=[1.4…7.5] km/s, $\alpha_{turb}$=[0.1…0.7], $V_{sys}$=[–1.3…–0.2] km/s, $\alpha_{sys}$=[–0.2…0.4], $X$(HCO+)=[$10^{-10.5}...10^{-8}$], $R_{max}$=[$10^{17.7}-10^{19.2}$] cm. Although we assumed that these ranges were certain to include the optimal parameter values for the L1287 core, their boundaries were not rigid and could be expanded by means of the inverse transformation from the PC space. For each value in the parameter array, we calculated a map in the HCO+(1–0) line and the error function. Based on the accepted criterion, $\chi^{2}\leq\chi_{min}^{2}(3\sigma_{obs})$, we selected 246 values from the initial set. This number was enough to construct the statistics in the PC space. For these values, we calculated a set of PCs using a procedure from the $scikit$-$learn$ library [67]. Using the dependence of $R$, the ratio of the sum of the diagonal components in the PC covariance matrix to the sum of the diagonal components in the covariance matrix of the physical parameters, on the number of components, we estimated the minimum number of PCs required to represent the physical parameters (Fig. 3). Figure 3 shows that the five PCs represent to a sufficient extent the eight physical parameters at a level of $R=0.9$. For the five PCs, the difference after the inverse transformation did not exceed 5% of the grid step for all the parameters, suggesting no distortions in subsequent calculations and no error accumulation. Figure 3 also shows the contribution of each component to the relative covariance matrix of the PCs. 5mm normal Figure 3: Dependence of the ratio of the sum of the diagonal components in the PC covariance matrix and the sum of the diagonal components in the covariance matrix of the physical parameters on the number of components (red crosses). The green circles show the contribution of an individual component to the relative PC covariance matrix. The dashed horizontal line indicates a cutoff level of 0.9. In the space of the five remaining PCs, we constructed a uniform five- dimensional grid with a center at the point of minimum of the error function; the grid size was consistent with $6\Delta(pc_{i})$, where $\Delta(pc_{i})$ is a standard deviation of the $i$th PC values, which was calculated from the selected set of points. The PC array was recalculated to the array of the physical parameter values, for which we calculated the spectral maps and estimated the error functions. From the calculated model maps, we estimated the optimal physical parameters from the HCO+(1–0) data by the $k$NN method. Varying by the least squares method, the relative abundances of H13CO+, HCN, and H13CN, we fitted the corresponding model spectral maps into the observed ones. In so doing, we also slightly adjusted the parameters within the error ranges calculated from the HCO+(1–0) data. By comparing the set of model spectral maps with the observed ones, we estimated the global error function in the four lines by equation (3). The resulting model spectra proved to be close to the observed ones up to a scale of $\sim 0.8$ pc, which confirmed the relevance of the applied model. 5mm normal Figure 4: Projections of the eight-dimensional error function $\chi^{2}$ onto the planes of the different parameter pairs calculated from the fitting of the model spectral maps in the lines HCO+(1–0), H13CO+(1–0), HCN(1–0) and H13CN(1–0) into the observed maps in the L1287 core. The dependencies of the error function on the individual parameters are given over each projection column. The red dots in the diagrams and the red vertical lines in the upper plots indicate the global minimum of the error function, which was obtained by $k$NN. The confidence regions for the optimal parameter values, which were calculated from the hyperplane $\chi_{\sigma}^{2}$ cross-sections of the error function, are shown with blue contours and horizontal lines in the two- dimensional projections and one-dimensional plots, respectively. Figure 4 presents a set of projections of the eight-dimensional error function onto the planes of the different parameter pairs and the error function projection dependencies on each of the parameters. It follows from the two- dimensional projections and the dependencies, that the model has a global minimum, and a confidence level can be determined for each of the parameters. Correlations are observed between some of the parameters. A clear correlation is observed between $R_{max}$ and the relative abundance of HCO+ ($X_{0}$), between $\alpha_{n}$ and $X_{0}$, and between the turbulent and systematic velocities in the central layer and the corresponding power-law indices of the radial profiles of these parameters. A weaker correlation exists between $\alpha_{n}$ and $n_{0}$ and between $R_{max}$ and $\alpha_{n}$. The exact position of the minimum was estimated by the $k$NN method from all the lines; it is marked by a red cross in the two-dimensional projections and by red vertical lines in the $\chi^{2}$ projection dependencies on individual parameters. The confidence regions were calculated using a cross-section of the error function by the hyperplane $\chi_{\sigma}$. The projections of the error function cross-sections by the $\chi_{\sigma}$ hyperplane are in fact contours in the planes of parameter pairs; they correspond to horizontal lines on upper plots (see Fig. 4). The confidence regions are not symmetric with respect to the optimal values. The distortions in the shape of the contours are due to observational noise and the discrete filling of the parameter space. When analyzing the dependencies of $\chi^{2}$ on individual parameters in broader ranges than those shown in Fig. 4, we found additional local minima, which values are, however, greater than the one corresponding to the global minimum and the corresponding parameter values considerably deviate from independent estimates. The estimates for the physical parameters of the L1287 core, which correspond to the global minimum of the error function, and the uncertainties of these estimates, which correspond to the boundaries of the confidence regions (Fig. 4), are given in Table 1. It should be noted that in accordance with the specified form of the radial profiles, the physical parameter values in the central layer are twice as low as the corresponding values of $n_{0}$, $V_{turb}$, and $V_{sys}$. 0mm Table 1: Resulting values of the physical parameters Parameter \| Value ---\|--- $n_{0}$(cm-3), 107 \| 2.6${}_{-1.3}^{+1.7}$ $\alpha_{n}$ \| 1.7${}_{-0.3}^{+0.1}$ Vturb (km/s) \| 5.6${}_{-1.4}^{+0.7}$ $\alpha_{turb}$ \| 0.44${}_{-0.13}^{0.05}$ Vsys (km/s) \| –0.66${}_{-0.24}^{+0.21}$ $\alpha_{sys}$ \| 0.1${}_{-0.13}^{+0.08}$ Rmax(pc) \| 0.8${}_{-0.25}^{+0.2}$ X(HCO+), 10-9 \| 1.0${}_{-0.4}^{+0.5}$ X(H13CO+), 10-11 \| 3.7${}_{-2.0}^{+2.4}$ X(HCN), 10-9 \| 2.5${}_{-1.1}^{+1.4}$ X(H13CN), 10-11 \| 8.5${}_{-4.8}^{+5.3}$ ## 4 RESULTS AND DISCUSSION Figures 5 and 6 show the spectral maps for the central part of the L1287 core ($\sim 0.4$ pc) in the HCO+(1–0), HCN(1–0), H13CO+(1–0), and H13CN(1–0) lines with the fitted model spectra corresponding to the global minimum of the error function. The asymmetry and dip in the observed profiles of the optically thick lines of HCO+(1–0) and HCN(1–0) are well reproduced by the model. In the central and southwestern parts of the analyzed region, the spectra of the optically thick lines exhibit high-velocity gas emission, which was ignored in the model calculations. The slight discrepancy between the model and observed spectra at the edges of the observed region may be due to a difference from spherical symmetry. The diameter (1.6 pc) of the model cloud exceeds the observed sizes of the emission regions in the different molecular lines, the dense gas indicators, HCO+(1–0), HCN(1–0), and NH3(1,1) ($\sim 0.3-0.5$ pc) [62, 63, 50], since it comprises the low-density outer layers, which cause the dip in the profiles of the optically thick lines as they absorb the emission from the central layers. 5mm normal Figure 5: Results of fitting the model spectra of HCO+(1–0) (left) and HCN(1–0) (right) (smooth red curves) into the observed ones (histograms, black lines) in the central part of the L1287 core. The horizontal axis plots the velocities in the range from –33 to –5 km/s. 5mm normal Figure 6: Results of fitting the model spectra of H13CO+(1–0) (left) and H13CN(1–0) (right) (smooth red curves) into the observed ones (histograms, black lines) in the central part of the L1287 core. The horizontal axis plots the velocities in the range from –28.5 to –7 km/s. The calculated physical parameters of the core are consistent, considering the errors (see Table 1), with estimates obtained from the data of independent observations. Thus, the model column density of molecular hydrogen for a region of radius $\sim 20^{\prime\prime}$ agrees with the value calculated from the data of dust observations with the Herschel telescope [66] ($4.6_{-2.3}^{+3.0}$ $10^{23}$ cm-2 and $(1.8\pm 1.2)\,10^{23}$ cm-2, respectively). The core mass calculated from the model for a region of radius $\sim 0.6$ pc is $\sim 1200$ M0; considering the error ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}50$%), associated primarily with the error of $n_{0}$, this mass is consistent with the value of 810 M0, obtained from the observations of dust for a region of similar radius [65]. Neither does the power-law index of the radial density profile $1.7_{-0.3}^{+0.1}$ contradict the value of $1.25\pm 0.2$, obtained from the observations of dust in continuum [65]. Both of these estimates lie in the value range for the power-law index of the density profile obtained for samples of dense cores associated with regions of massive star and star cluster formation (see, e.g., [65, 68, 69]) but are lower than 2, the value predicted by the isothermal sphere [8] and global collapse models [13]. The model abundance ratios of the main and rarer isotopes ([HCO+]/[H13CO+] and [HCN]/[H13CN]) are lower by a factor of $\sim 2$ than the isotope ratio [12C]/[13C]$\sim 58$, calculated from the heliocentric dependence of this ratio [70] for $R_{G}\sim 9$ kpc (L1287). However, the uncertainties of the model abundance ratios are rather high ($\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}80$%) to make further conclusions from this discrepancy. To verify the results obtained, they should be compared with the chemical modeling results. The turbulent velocity falls rather sharply with the distance from the center (from 2.8 km/s in the central layer to $\sim 0.6$ km/s in the outer layer), which is necessary for reproducing the shape of the dip in the HCO+(1–0) and HCN(1–0) spectra (solid curve $1$ in Fig. 7, right panel). The contraction velocity decreases weakly in absolute terms with the distance from the center ($\sim 0.33$ km/s in the central layer and $\sim 0.25$ km/s in the outer layer) (dashed curve $1$ in Fig. 7, right panel). Its average value across the model cloud is $0.26\pm 0.09$ km/s, which does not contradict the value of $\sim 0.22$ km/s, calculated from the HCO+(1–0) line parameters for the point (60′′,40′′) by the formula of the two-layer model [20] (the value given in [50] is underestimated). 5mm normal Figure 7: Model radial profiles of density and kinetic temperature (left panel) and contraction and turbulent velocities (right panel). Profiles were obtained in this work ($1$) and those from [25] ($2$). The cloud radius in the latter model was set to 1.4 pc, consistent with a distance of 0.93 kpc to L1287. The $1A$ mark in the right panel shows the contraction velocity profile obtained in our model with a fixed value of the power index, 0.5. The power-law index of the radial profile for contraction velocity obtained in the model calculations considering errors proved to be lower than 0.5 for the case of gas collapse onto the protostar in free fall [8, 10, 13]. In [25], the observational data for the core 0038 + 6312 (L1287) in the HCO+(1–0), CS(2–1), and CS(5–4) lines were compared with the calculated results for the model with the density and contraction velocity profiles $\propto r^{-3/2}$ and $\propto r^{-1/2}$, respectively. Although the intensities and widths of the model spectra proved to match the observed ones in the direction of individual positions quite well, considering the sensitivity and spectral resolution of the data, the dip in the HCO+(1–0) line profiles was not reproduced (see [25]). This is due to the difference of the radial profiles for velocity, which were assumed in the model [25], from the profiles obtained in our calculations (Fig. 7, right panel). The left panel in Fig. 7 shows the radial profiles of density and kinetic temperature for our model and for the model [25]. As shown in the model of the global hierarchical collapse (see, e.g., [13, 16]), if the core is globally out-of-equilibrium, it experiences contraction with a constant velocity and this contraction continues in the outer layers even after the protostar formation. For comparison, we fitted the model maps into the observed ones for the case where the power-law index of the radial profile of systematic velocity was fixed at 0.5. The corresponding velocity profile is marked $1A$ in Fig. 7 (right panel). Figure 8 shows the observed HCO+(1–0) and HCN(1–0) lines for the point (60′′,40′′) near the core center and the model spectra for the power-law index of systematic velocity of 0.1, which corresponds to the global minimum of the error function, and for the case when the index is 0.5, respectively. Upon comparison of the spectra, the model with the index 0.1 more accurately reproduces the intensities and widths of the asymmetric HCO+(1–0) profiles and, specifically, the profiles of three hyperfine HCN(1–0) components than the model with the index 0.5. A similar conclusion is true for the other points. Although in the southwestern part, the high-velocity emission associated with bipolar outflow more strongly distorts the spectra shape (Fig. 5), which makes it more difficult to compare the models. The fact that the value obtained in the model with the power-law index of the velocity profile as a free parameter turned out to be lower than 0.5 may suggest a likelihood of nonuniform gas contraction in the core – with constant velocity in the extended envelope and with the $\propto r^{-1/2}$ profile in the region near the center. The use of the model with a single power-law index gives a weighted average for the entire core in this case. 5mm normal Figure 8: Observed and model profiles of the lines HCO+(1–0) (left) and HCN(1–0) (right) towards the (60′′,40′′) position for models with different values of the power-law index in the radial profile of contraction velocity. Although we used a rather simplified 1D model with uniform power-law indices for the radial profiles of the physical parameters, it allows using the developed algorithm for fitting the model spectral maps into the observed ones with PCA and $k$NN to reproduce accurately the observed HCO+(1–0), HCN(1–0), H13CO+(1–0), and H13CN(1–0) line maps and estimate the radial profiles of the parameters in outer regions of the L1287 core ($r\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr>\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr>\cr\sim\cr}}}}0.1$ pc). Some of the differences between the model and observed spectra can be eliminated, apparently, within more complex models with composite radial profiles of the parameters and also within 2D or 3D models considering the possible spatial inhomogeneity of the velocity field, rotation, and high- velocity outflows. To reduce errors in the calculated parameters and to confirm the conclusion about the possible global contraction of the L1287 core, further observations are required in molecular lines of different optical depths with better spatial and spectral resolution. A distinctive feature of the developed algorithm is its lack of ties to a specific model and its capability of simultaneous analysis of spectral maps with different spatial resolutions and sizes, as well as maps in continuum. The main constraint is only the computational time required to construct the necessary statistics. ## 5 CONCLUSIONS The results of this work can be summarized as follows: (1) An algorithm was developed for finding the global minimum of the multidimensional error function and calculating the optimal parameter values when fitting model spectral maps into observed ones. The algorithm is based on applying principal component analysis to a given range of parameters, as a result of which a reduction is achieved in the model’s dimensionality and in the coupling degree between the parameters. Thus, the region of the minimum is determined. The optimal parameter values are calculated using the $k$ nearest neighbors method. Confidence regions for the optimal parameter values are determined using a cross-section of the error function by a hyperplane calculated in the PC space and its projections onto the various pairs of the parameters. The algorithm is not tied to a specific model. (2) The algorithm was used to perform the fitting of the model maps in the HCO+(1–0), H13CO+(1–0), HCN(1–0), and H13CN(1–0) lines into the observed maps of the protostellar core of L1287, in which the formation of a young stellar cluster is underway, and the asymmetry of the profiles of optically thick lines indicates contraction. The maps were calculated within a spherically symmetric 1D model in which the physical parameters (density and turbulent and systematic velocities) were functions of the distance from the center. Optimal values were calculated for the model parameters, and their errors were determined. It was found that density in the L1287 core decreases with the distance from the center as $\propto r^{-1.7}$ while turbulent and contraction velocities decrease as $\propto r^{-0.4}$ and $\propto r^{-0.1}$, respectively. The absolute value of the power-law index for the radial profile of contraction velocity, considering the probable error, is less than 0.5, a value expected in the case of gas collapse onto the protostar in free fall. This result may indicate global contraction in the L1287 core, which was predicted in several theoretical works. APPENDIX. MODEL DESCRIPTION Excitation of rotational levels of the HCO+ and HCN molecules and their isotopes and the profiles of the (1–0) transitions were calculated within a spherically symmetric microturbulent model. The model cloud is a set of concentric layers in which a certain physical parameter (density, kinetic temperature, turbulent and systematic velocity) was set constant, changing from one layer to another by the relationship $P=P_{0}/(1+(r/R_{0})^{\alpha_{p}}$), where $r$ is the distance from the center and $R_{0}$ is the radius of the central layer. This functional dependence, which is a simplified form of the Plummer function, is used quite often as a model density profile (see, e.g., [22]) to avoid singularity at the center. In our model, this form of the dependence was used for all the parameters. The values of $P_{0}$ and $\alpha_{p}$ for each parameter were varied while fitting the model profiles into the observed ones. The kinetic temperature profile was taken as $T=80$ K$/(1+(r/R_{0})^{0.3})$ and remained unchanged during calculations. It should be noted that kinetic temperature affects the intensities of the calculated HCO+(1–0) and HCN(1–0) lines to a lesser degree than density and concentration. Turbulent velocity was a parameter that gives an additional contribution - aside from the thermal one - to the local width of the lines. The relative molecular abundance was independent of radial distance. When calculating the excitation of HCN and H13CN, the hyperfine structure of the rotational spectrum and the overlapping of closely located hyperfine components [40, 71] was taken into account. The description of the model and calculation techniques for radiation transfer in the case of HCN is given in the Appendix to [40]. In our version of this model, the layer width increases by the power law with the distance from the center, and the radial profile of systematic velocity, which gives a Doppler shift to the local profile of the line, is taken into account. The calculations were conducted for 14 layers. The calculations used collisional probabilities of HCO+–H2 [72] and HCN–H2 taking into account hyperfine structure [73]. Excitation of rotational levels of a certain molecule was calculated by an iterative method, sequentially for one point in each layer, the radial distance of which is equal to the geometric mean of the inner and outer radii of the layer. To this end, a system of population balance equations was solved, while the populations in other layers were considered unchanged. After reaching the outer layer, the populations in each layer were compared with the values obtained in the previous iteration, and the process was repeated [40]. To increase the accuracy of calculating the radiation transfer in a moving medium, each layer was additionally divided into ten sublayers, with different systematic velocities. A test comparison of the calculated results for this model with the calculated results in [74] for a molecule with two energy levels showed that the calculated populations differ by no more than 0.4% in the case of line optical depth of $\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}60$. The model code, written in Fortran, was controlled by means of a module written in Python. Model spectra were calculated for the different impact parameters. Using the astropy.convolve_fft procedure [75], the resulting maps were convoluted channel by channel with a two-dimensional Gaussian of width 40′′ (the width of the main beam of the OSO-20m radio telescope at a frequency of $\sim 90$ GHz). 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# On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in ${\mathbb{R}}^{4}$ Ciprian Demeter Department of Mathematics, Indiana University, 831 East 3rd St., Bloomington IN 47405<EMAIL_ADDRESS> ###### Abstract. We prove sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb{R}}^{4}$. We place Bourgain’s seminal result [2] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves. ###### Key words and phrases: decoupling, Weyl sums, square root cancellation The author is partially supported by the NSF grant DMS-1800305 AMS subject classification: Primary 42A45, Secondary 11L07 ## 1\. Introduction Throughout this paper, $\phi_{3},\phi_{4}$ will be real analytic functions defined on some open interval containing $[\frac{1}{2},1]$, and satisfying $\\|\phi_{k}^{\prime}\\|_{C^{3}}=\sum_{1\leq n\leq 4}\max_{\frac{1}{2}\leq t\leq 1}\|\phi_{k}^{(n)}(t)\|\leq A_{1},\;\;k\in\\{3,4\\},$ (1) $A_{2}\leq\left\|\det\begin{bmatrix}\phi_{3}^{(3)}(t)&\phi_{3}^{(4)}(t)\\\ \phi_{4}^{(3)}(s)&\phi_{4}^{(4)}(s)\end{bmatrix}\right\|\leq A_{3},\;\;t,s\in[\frac{1}{2},1],$ (2) $\|\phi_{3}^{(3)}(t)\|\geq A_{4},\;\;t\in[\frac{1}{2},1].$ (3) $A_{1},\ldots,A_{4}$ are positive numbers that will determine the implicit constants in various inequalities. While $\phi_{3},\phi_{4}$ being $C^{4}$ would suffice for our purposes, we choose to work with real analytic functions for purely aesthetic reasons. The examples of most immediate interest are power functions $\phi_{3}(t)=t^{a}$, $\phi_{4}(t)=t^{b}$, with the real numbers $a$ and $b$ satisfying the restrictions $a\not=b$ and $a,b\not\in\\{0,1,2\\}$. For a finite interval $I\subset{\mathbb{Z}}$, we write (ignoring the dependence on $\phi_{k}$) ${\mathcal{E}}_{I,N}(x)=\sum_{n\in I}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}+\phi_{4}(\frac{n}{N})x_{4}).$ We make the following conjecture. ###### Conjecture 1.1. Assume $\alpha\geq\beta\geq 0$ and $\alpha+\beta=3$. Let $\omega_{3}=[0,N^{\alpha}]$ and $\omega_{4}=[0,N^{\beta}]$. Assume $\phi_{3},\phi_{4}$ are real analytic on $(0,3)$ and satisfy (1), (2) and (3). Then $\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)\|^{12}dx\lesssim_{\epsilon}N^{9+\epsilon}.$ (4) For virtually all conceivable applications of (4), both $\phi_{3}$ and $\phi_{4}$ will satisfy (3). Because of this symmetry, our restriction $\alpha\geq\beta$ is essentially meaningless. Let us write $A\lesssim B$ or $A=O(B)$ if $\|A\|\leq CB$ for some, possibly large, but universal constant $C$. The notation $A\sim B$ will mean that $A\lesssim B$ and $B\lesssim A$ at the same time. We will write $A\ll B$ or $A=o(B)$ if $\|A\|\leq cB$, for some small enough, universal constant $c$. The notation $A{\;\lessapprox}\;B$ will mean $A\lesssim(\log N)^{O(1)}B$, where $N$ will be the key scale parameter. Since $\|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)\|\sim N$ if $x\in[0,o(\frac{1}{N})]\times[0,o(\frac{1}{N^{2}})]\times[0,o(1)]^{2}$, the exponent 9 is optimal in (4). While the large value $N$ is attained by $\|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)\|$ for a small subset of $x$, estimate (4) implies that the average value of the exponential sum on the given domain is $O(N^{\frac{1}{2}+\epsilon})$, when measured in $L^{12}$. We call this $L^{12}$ square root cancellation. The relevance of the requirement $\alpha+\beta=3$ is that it guarantees $\|[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}\|(\sqrt{N})^{12}=N^{9}.$ The case $(\alpha,\beta)=(2,1)$ of the conjecture has been settled in [2]. This inequality was proposed by Huxley [8], with $\phi_{3}(t)=t^{3/2}$, $\phi_{4}(t)=t^{1/2}$. In [2], it serves as the main ingredient in sharpening the record on the Lindelöf hypothesis. The only other known case prior to our work was $(\alpha,\beta)=(3,0)$. The variable $x_{4}$ and the function $\phi_{4}$ play no role in this case, as $\phi_{4}(\frac{n}{N})x_{4}=O(1)$. We treat $e(\phi_{4}(\frac{n}{N})x_{4})$ as a coefficient $c_{n}=O(1)$. In this case, (4) follows from the inequality $\int_{[0,1]^{2}\times[0,N^{3}]}\|\sum_{n\in I}c_{n}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}\|^{12}dx_{1}dx_{2}dx_{3}\lesssim_{\epsilon}N^{9+\epsilon}\\|c_{n}\\|_{l^{\infty}}.$ Assuming $\phi_{3}$ satisfies (1) and (3), this was known as the Main Conjecture in Vinogradov’s Mean Value Theorem, and was first solved in [10], then in [5]. This reduction fails for all other values $\alpha<3$, since the length of the interval $\omega_{3}$ becomes too small. The proofs in [2] and [5] for $\alpha=2$ and $\alpha=3$ are fairly different, in spite of both relying entirely on abstract decoupling. Our main result here verifies Conjecture 1.1 in the range $\frac{3}{2}\leq\alpha<2$. ###### Theorem 1.2. Assume that $\frac{3}{2}\leq\alpha\leq 2$. Assume $\phi_{3},\phi_{4}$ are real analytic on $(0,3)$ and satisfy (1), (2) and (3). Then $\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{[\frac{N}{2},N],N}(x)\|^{12}dx\lesssim_{\epsilon}N^{9+\epsilon}.$ (5) If $\phi_{3}(t)=t^{a}$, $\phi_{4}(t)=t^{b}$ with $\alpha\leq a$, $\beta\leq b$, $a+b\leq 9$, $a\not=b$ and $a,b\not\in\\{0,1,2\\}$, a very simple rescaling argument for each member of the sum ${\mathcal{E}}_{[1,N],N}(x)=\sum_{1\leq M\leq N\atop{M\text{ dyadic}}}{\mathcal{E}}_{[\frac{M}{2},M],N}(x)$ shows that (5) holds with ${\mathcal{E}}_{[\frac{N}{2},N],N}$ replaced with ${\mathcal{E}}_{[1,N],N}(x)$. In particular, this is always the case for the moment curve $\phi_{3}(t)=t^{3}$, $\phi_{4}(t)=t^{4}$. Other cases such as $(\alpha,\beta)=(2,1)$, $(a,b)=(\frac{3}{2},\frac{1}{2})$ require slightly more sophisticated arguments similar to the one in [2], but will not be pursued here. Theorem 1.2 will follow from its bilinear analog. We prove this reduction in Section 7. ###### Theorem 1.3. Let $I_{1},I_{2}$ be intervals of length $\sim N$ in $[\frac{N}{2},N]$, with ${\operatorname{dist}\,}(I_{1},I_{2})\sim{N}$. Assume $\phi_{3},\phi_{4}$ are real analytic on $(0,2)$ and satisfy (1), (2) and (3). Assume that $\frac{3}{2}\leq\alpha\leq 2$. Then $\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{I_{1},N}(x){\mathcal{E}}_{I_{2},N}(x)\|^{6}dx\lesssim_{\epsilon}N^{9+\epsilon}.$ The implicit constant in this inequality is uniform over $A_{1},\ldots,A_{4}\sim 1$. The reason we prove Theorem 1.2 using bilinear, as opposed to linear or trilinear methods, is rather delicate. As explained earlier, our results are sharp, the exponent 9 in (5) cannot be lowered. Each of the decoupling inequalities relevant to this paper has a certain critical exponent $p_{c}>2$. Experience shows that to achieve sharp results via decoupling, this tool must be used at the critical exponent $p_{c}$. When we apply decoupling on spatial balls of radius $M$, we decouple the curve into arcs of length $M^{-1/2}$. It seems very likely that the critical exponent for such a decoupling in the linear setting is larger than 12. See [7] for a detailed discussion on this. Because of this, using linear $L^{12}$ decoupling turns out to be inefficient. Bilinearizing instead, gives us access to the $L^{6}$ decoupling of the parabola. This is an ideal scenario, since 6 is precisely the critical exponent in this setting. The fact that $12=6\times 2$ turns out to be crucial to our argument. The other factorization $12=4\times 3$ is also inefficient. An approach based on trilinearity would use $L^{4}$ estimates for hypersurfaces in ${\mathbb{R}}^{4}$. But the critical exponent here is $10/3$, not $4$. Most of the paper is devoted to proving Theorem 1.3. The proof will combine abstract decoupling methods with quadratic Weyl sum estimates. The decoupling techniques are introduced in Section 2. While these results are by now standard, the observation that the superficially stronger $l^{2}(L^{12})$ decoupling holds true for nondegenerate curves in the bilinear setting appears to be new. One of the key features in our argument is the use of this inequality in places where the $l^{12}(L^{12})$ decoupling used in [2] becomes inefficient. We combine this finer decoupling with estimates from number theory. In short, here is how our approach works. The initial integral involves quartic Weyl sums, for which sharp estimates are totally out of reach at the moment. Decoupling is applied once or twice in order to lower the complexity of the sums, to the level of manageable quadratic Weyl sums. These sums will appear in various combinations, and need to be tackled with extreme care, using various counting arguments such as Lemma 4.1 and Lemma 4.2. In Section 3, we start with a careful examination of Bourgain’s argument from [2], for $\alpha=2$. In many ways, this case turns out to be the easiest, as it works via just $l^{12}L^{12}$ decoupling, and without any input from number theory. In Section 4 we introduce our new methodology, addressing the symmetric case $\alpha=\beta=\frac{3}{2}.$ This ends up being the most delicate case, since it captures the biggest region near the origin where constructive (and near constructive) interference occurs. Also, it is in this case that the curve looks most genuinely four dimensional, as both $\omega_{3}$ and $\omega_{4}$ are large. For comparison, recall that when $\alpha=3$ the curve degenerates to a three dimensional one. Sections 5 and 6 extend our method to the remaining cases, by successively building on each other. The case $\frac{9}{5}\leq\alpha<2$ combines elements of both approaches. To the best of our knowledge, this paper represents the first systematic effort to combine abstract decoupling with Weyl sum estimates. The results proved here are part of the vast program initiated in [6], concerned with proving sharp $L^{p}$ estimates for the moment curve on spatial domains smaller than the torus. In [6] only the moment curve in ${\mathbb{R}}^{3}$ is considered, and all estimates there rely solely on decoupling techniques. There remain a lot of interesting related questions. One of them has to do with proving Conjecture 1.1 in the range $2<\alpha<3$. We may speculate that the solution would combine some of the tools from our paper with those used to solve the case $\alpha=3$, see also Remark 3.1. Second, $L^{p}$ moments are also worth investigating for smaller values $p<12$, in particular for $p=10$. See for example [1] for some recent progress and some interesting applications. Section 8 of our paper contains an example that describes some of the enemies and limitations of square root cancellation in this setting. It seems plausible that small cap decoupling for the parabola (see [6]) will be the right tool to attack this problem. We hope to address some of these questions in future work. ###### Acknowledgment. The author is grateful to Hongki Jung and Zane Li for pointing out a few typos in the earlier version of the manuscript. ## 2\. Decoupling for nondegenerate curves in ${\mathbb{R}}^{4}$ Let us start by recalling the decoupling for nondegenerate curves in ${\mathbb{R}}^{2}$. ###### Theorem 2.1 ([3]). Let $\phi:[0,1]\to{\mathbb{R}}$ be a $C^{3}$ function, with $\\|\phi^{\prime}\\|_{C^{2}}=A_{5}<\infty$ and $\min_{0\leq t\leq 1}\|\phi^{\prime\prime}(t)\|=A_{6}>0.$ Then for each $f:[0,1]\to{\mathbb{C}}$ and each ball $B_{N}\subset{\mathbb{R}}^{2}$ with radius $N$ we have $\\|\int_{[0,1]}f(t)e(tu+\phi(t)w)dt\\|_{L_{u,w}^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J\subset[0,1]\atop{\|J\|=N^{-1/2}}}\\|\int_{J}f(t)e(tu+\phi(t)w)dt\\|_{L_{u,w}^{6}(B_{N})}^{2})^{1/2}.$ The implicit constant is uniform over $A_{5}\sim 1$, $A_{6}\sim 1$. We use this result to illustrate in the simplest terms the reduction of cubic terms used in the next section for $\alpha=2$. Namely, let us show that for $1\leq\alpha<3$ $\int_{[0,1]^{2}}\|\sum_{1\leq m\leq M}e(mu+(m^{2}+\frac{m^{3}}{M^{\alpha}})w)\|^{6}dudw\lesssim_{\epsilon}M^{3+\epsilon}.$ (6) The term $\frac{m^{3}}{M^{\alpha}}w$ is not negligible (in the sense of Lemma 9.2), as it is not $O(1)$. However, after a change of variables and using periodicity in $u$, we may rewrite the integral as $\frac{1}{M^{4}}\int_{[0,M^{2}]^{2}}\|\sum_{1\leq m\leq M}e(\frac{m}{M}u+\phi(\frac{m}{M})w)\|^{6}dudw,$ where $\phi(t)=t^{2}+t^{3}M^{1-\alpha}$. Note that $A_{5},A_{6}\sim 1$, uniformly over $M$. Inequality (6) is now a standard consequence of Theorem 2.1 with $N=M^{2}$. The cubic term becomes a perturbation of the quadratic term, and does not significantly affect the constant $A_{6}$. Theorem 2.4 will formalize this approach in four dimensions. Throughout this section, $\phi_{3},\phi_{4}$ are arbitrary functions satisfying (1) and (2). We denote by $E$ the extension operator associated with the curve $\Phi$ $\Phi(t)=(t,t^{2},\phi_{3}(t),\phi_{4}(t)),\;t\in[\frac{1}{2},1].$ (7) More precisely, for $f:[\frac{1}{2},1]\to{\mathbb{C}}$ and $I\subset[\frac{1}{2},1]$ we write $E_{I}f(x)=\int_{I}f(t)e(tx_{1}+t^{2}x_{2}+\phi_{3}(t)x_{3}+\phi_{4}(t)x_{4})dt.$ The following $l^{6}(L^{6})$ decoupling was proved in [2], see also [4]. It is in fact a bilinear version of the $l^{12}L^{12}$ decoupling for the curve (7). ###### Theorem 2.2. Let $I_{1},I_{2}$ be two intervals of length $\sim 1$ in $[\frac{1}{2},1]$, with $dist(I_{1},I_{2})\sim 1$. Let also $f_{i}:[\frac{1}{2},1]\to{\mathbb{C}}$. Then for each ball $B_{N}$ of radius $N$ in ${\mathbb{R}}^{4}$ we have $\\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\frac{1}{3}+\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{6})^{1/6}.$ The sum on the right is over intervals $J_{i}$ of length $N^{-1/2}$. In [2], this result is used in conjunction with the following estimate, an easy consequence of transversality. $\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{6}\lesssim N^{-4}\\|E_{J_{1}}f_{1}\\|_{L^{6}(B_{N})}^{6}\\|E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{6}.$ (8) This is inequality (13) in [4], and there is a detailed proof there. For reader’s convenience, we sketch a somewhat informal argument below. The Fourier transform of $\|E_{J_{i}}f_{i}\|^{6}$ is supported inside a rectangular box with dimensions $\sim N^{-1}\times N^{-1}\times N^{-1}\times N^{-1/2}$. We have the following wavepacket representation on $B_{N}$, slightly simplified for exposition purposes $\|E_{J_{i}}f_{i}(x)\|^{6}\approx\sum_{P_{i}\in{\mathcal{P}}_{i}}a_{P_{i}}1_{P_{i}}(x).$ The coefficients $a_{P_{i}}$ are nonnegative reals. The rectangular boxes $P_{i}$ have dimensions $\sim N\times N\times N\times N^{1/2}$ and tile $B_{N}$. They can be thought of as $N^{1/2}$-neighborhoods of cubes with diameter $\sim N$ inside hyperplanes ${\mathcal{H}}_{i}$. Since ${\operatorname{dist}\,}(J_{1},J_{2})\sim 1$, the angle between the normal vectors of ${\mathcal{H}}_{i}$ is $\sim 1$. Thus, we have that $\|P_{1}\cap P_{2}\|\lesssim N^{3}$. We conclude by writing $\displaystyle\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{6}$ $\displaystyle\sim\sum_{P_{1}\in{\mathcal{P}}_{1}}\sum_{P_{2}\in{\mathcal{P}}_{2}}a_{P_{1}}a_{P_{2}}\|P_{1}\cap P_{2}\|$ $\displaystyle\lesssim N^{-4}\sum_{P_{1}\in{\mathcal{P}}_{1}}\sum_{P_{2}\in{\mathcal{P}}_{2}}a_{P_{1}}a_{P_{2}}\|P_{1}\|\|P_{2}\|$ $\displaystyle\approx N^{-4}\\|E_{J_{1}}f_{1}\\|_{L^{6}(B_{N})}^{6}\\|E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{6}.$ It is worth observing that $\lesssim$ in inequality (8) is essentially an (approximate) similarity $\approx$, making (8) extremely efficient. Indeed, since $P_{1}$ and $P_{2}$ intersect $B_{N}$, we have that $\|P_{1}\cap P_{2}\|\sim N^{3}$. To address new values of $\alpha$ in this paper, we will need the following $l^{2}(L^{6})$ decoupling. This implies the previous $l^{6}(L^{6})$ decoupling, and provides a critical improvement in the cases when the terms in the sum are of significantly different sizes. ###### Theorem 2.3. Let $I_{1},I_{2}$ be two intervals of length $\sim 1$ in $[\frac{1}{2},1]$, with $dist(I_{1},I_{2})\sim 1$. Let also $f_{i}:[\frac{1}{2},1]\to{\mathbb{C}}$. Then for each ball $B_{N}$ of radius $N$ in ${\mathbb{R}}^{4}$ we have $\\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{2})^{1/2}.$ The sum on the right is over intervals $J$ of length $N^{-1/2}$. ###### Proof. We briefly sketch the argument, that follows closely the one in [2]. Let $b(N)$ be the best constant such that $\\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\\|_{L^{6}(B_{N})}\leq b(N)(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|_{L^{6}(B_{N})}^{2})^{1/2}$ holds for all functions and balls as above. Fix $B_{N}$ and let ${\mathcal{B}}$ be a finitely overlapping cover of $B_{N}$ with balls $\Delta$ of radius $N^{2/3}$. It will soon become clear that the exponent $2/3$ is chosen in order to make cubic terms negligible. Applying this inequality on each $\Delta$, then summing up using Minkowski’s inequality shows that $\\|E_{I_{1}}f_{1}E_{I_{2}}f_{2}\\|_{L^{6}(B_{N})}\leq b(N^{2/3})(\sum_{H_{1}\subset I_{1}}\sum_{H_{2}\subset I_{2}}\\|E_{H_{1}}f_{1}E_{H_{2}}f_{2}\\|_{L^{6}(B_{N})}^{2})^{1/2}.$ The intervals $H$ have length $N^{-1/3}$. We next analyze each term in the sum. Let $l_{i}$ be the left endpoint of $H_{i}$, and write a generic point $t_{i}\in H_{i}$ as $t_{i}=l_{i}+s_{i}$, with $s_{i}\in[0,N^{-1/3}]$. We use Taylor’s formula for $k\in\\{3,4\\}$ $\phi_{k}(t_{i})=\phi_{k}(l_{i})+\phi_{k}^{\prime}(l_{i})s_{i}+\frac{\phi_{k}^{\prime\prime}(l_{i})}{2}s_{i}^{2}+\psi_{k,i}(s_{i}),$ where, due to (1), $\\|\psi_{k,i}\\|_{L^{\infty}([0,N^{-1/3}])}=O(\frac{1}{N})$. Let us write $\begin{cases}y_{1}=x_{1}+2l_{1}x_{2}+\phi_{3}^{\prime}(l_{1})x_{3}+\phi_{4}^{\prime}(l_{1})x_{4}\\\ y_{2}=x_{1}+2l_{2}x_{2}+\phi_{3}^{\prime}(l_{2})x_{3}+\phi_{4}^{\prime}(l_{2})x_{4}\\\ y_{3}=x_{2}+\frac{\phi_{3}^{\prime\prime}(l_{1})}{2}x_{3}+\frac{\phi_{4}^{\prime\prime}(l_{1})}{2}x_{4}\\\ y_{4}=x_{2}+\frac{\phi_{3}^{\prime\prime}(l_{2})}{2}x_{3}+\frac{\phi_{4}^{\prime\prime}(l_{2})}{2}x_{4}\end{cases}.$ It follows that $\|E_{H_{1}}f_{1}(x)E_{H_{2}}f_{2}(x)\|=$ $\|\int_{[0,N^{-1/3}]^{2}}f_{1}(l_{1}+s_{1})e(s_{1}y_{1}+s_{1}^{2}y_{3})f_{2}(l_{2}+s_{2})e(s_{2}y_{2}+s_{2}^{2}y_{4})e(L(y,s_{1},s_{2}))ds_{1}ds_{2}\|.$ Here $L(y,s_{1},s_{2})=x_{3}(\psi_{3,1}(s_{1})+\psi_{3,2}(s_{2}))+x_{4}(\psi_{4,1}(s_{1})+\psi_{4,2}(s_{2})).$ Lemma 9.1 shows that $x_{3},x_{4}$ depend linearly on $y_{1},y_{2},y_{3},y_{4}$ with coefficients $O(1)$. This allows us to write $e(L(y,s_{1},s_{2}))=e(\sum_{i=1}^{4}y_{i}(g_{i}(s_{1})+h_{i}(s_{2})))$ with $\\|g_{i}\\|_{\infty},\\|h_{i}\\|_{\infty}=O(\frac{1}{N})$. Letting $\bar{f}_{i}(s_{i})=f_{i}(l_{i}+s_{i})$ and $\begin{cases}\eta_{1}(s_{1},s_{2})=s_{1}+g_{1}(s_{1})+h_{1}(s_{2})\\\ \eta_{3}(s_{1},s_{2})=s_{1}^{2}+g_{3}(s_{1})+h_{3}(s_{2})\\\ \eta_{2}(s_{1},s_{2})=s_{2}+g_{2}(s_{1})+h_{2}(s_{2})\\\ \eta_{4}(s_{1},s_{2})=s_{2}^{2}+g_{4}(s_{1})+h_{4}(s_{2})\end{cases}$ we write $\|E_{H_{1}}f_{1}(x)E_{H_{2}}f_{2}(x)\|=$ $\|\int_{[0,N^{-1/3}]^{2}}\bar{f}_{1}(s_{1})e(y_{1}\eta_{1}(s_{1},s_{2})+y_{3}\eta_{3}(s_{1},s_{2}))\bar{f}_{2}(s_{2})e(y_{2}\eta_{2}(s_{1},s_{2})+y_{4}\eta_{4}(s_{1},s_{2}))ds_{1}ds_{2}\|.$ For $\bar{J}_{i}\subset[0,N^{-1/3}]$ we write ${\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}(y)=$ $\|\int_{\bar{J}_{1}\times\bar{J}_{2}}\bar{f}_{1}(s_{1})e(y_{1}\eta_{1}(s_{1},s_{2})+y_{3}\eta_{3}(s_{1},s_{2}))\bar{f}_{2}(s_{2})e(y_{2}\eta_{2}(s_{1},s_{2})+y_{4}\eta_{4}(s_{1},s_{2}))ds_{1}ds_{2}\|.$ This is the extension operator associated with the surface $(\eta_{1},\ldots,\eta_{4})$, applied to the function $f_{1}\otimes f_{2}$. We use Lemma 9.1 to write $\int_{B_{N}}\|E_{J_{1}}f_{1}(x)E_{J_{2}}f_{2}(x)\|^{6}dx=\int_{\bar{B}_{N}}{\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}(y)^{6}dy.$ Here $\bar{B}_{N}$ is a ball of radius $\sim N$ and $J_{i}=\bar{J}_{i}+l_{i}$. Note that the surface $(\eta_{1},\ldots,\eta_{4})$ is within $O(N^{-1})$ from the surface $(s_{1},s_{1}^{2},s_{2},s_{2}^{2}),\;\;s_{i}\in[0,N^{-1/3}],$ so -for decoupling purposes- the two surfaces are indistinguishable when paired with spatial variables $y$ ranging through a ball of radius $N$. The latter surface admits an $l^{2}(L^{6})$ decoupling, as can be easily seen by using Theorem 2.1 twice. The same remains true for the surface $(\eta_{1},\ldots,\eta_{4})$, and thus $\\|{\mathcal{I}}_{[0,N^{-1/3}],[0,N^{-1/3}]}\\|_{L^{6}({\bar{B}_{N}})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{{\bar{J}_{1},\bar{J}_{2}}\subset[0,N^{-1/3}]}\\|{\mathcal{I}}_{\bar{J}_{1},\bar{J}_{2}}\\|^{2}_{L^{6}({\bar{B}_{N}})})^{1/2}.$ The sum on the right is over intervals of length $N^{-1/2}$. If we undo the change of variables we find $\\|E_{H_{1}}f_{1}E_{H_{2}}f_{2}\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset H_{1}}\sum_{J_{2}\subset H_{2}}\\|E_{J_{1}}f_{1}E_{J_{2}}f_{2}\\|^{2}_{L^{6}(B_{N})})^{1/2}.$ Putting things together, we have proved the bootstrapping inequality $b(N)\lesssim_{\epsilon}N^{\epsilon}b(N^{2/3}).$ We conclude that $b(N)\lesssim_{\epsilon}N^{\epsilon}$, as desired. ∎ We will also record the following close relative of Theorem 2.3, that will be needed in the next sections. ###### Theorem 2.4. Assume $\psi_{1},\ldots,\psi_{4}:[-1,1]\to{\mathbb{R}}$ have $C^{3}$ norm $O(1)$, and in addition satisfy $\|\psi_{2}^{\prime\prime}(t)\|,\|\psi_{3}^{\prime\prime}(t)\|\ll 1,\;\forall\;\|t\|\leq 1$ and $\|\psi_{1}^{\prime\prime}(t)\|,\|\psi_{4}^{\prime\prime}(t)\|\sim 1,\;\forall\;\|t\|\leq 1.$ Let $E$ be the extension operator associated with the surface $\Psi(\xi_{1},\xi_{2})=(\xi_{1},\xi_{2},\psi_{1}(\xi_{1})+\psi_{2}(\xi_{2}),\psi_{3}(\xi_{1})+\psi_{4}(\xi_{2})),\;\;\|\xi_{1}\|,\|\xi_{2}\|\leq 1.$ More precisely, for $F:[-1,1]^{2}\to{\mathbb{C}}$, $R\subset[-1,1]^{2}$ and $x\in{\mathbb{R}}^{4}$ we write $E_{R}F(x)=\int_{R}F(\xi_{1},\xi_{2})e(x\cdot\Psi(\xi_{1},\xi_{2}))d\xi_{1}d\xi_{2}.$ Then for each ball $B_{N}\subset{\mathbb{R}}^{4}$ with radius $N$ we have $\\|E_{[-1,1]^{2}}F\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{H_{1},H_{2}\subset[-1,1]}\\|E_{H_{1}\times H_{2}}F\\|^{2}_{L^{6}(B_{N})})^{1/2},$ (9) where the sum is taken over intervals of length $N^{-1/2}$. In particular, for each constant coefficients $c_{m_{1},m_{2}}\in{\mathbb{C}}$ we have $\\|\sum_{m_{1}\leq N^{1/2}}\sum_{m_{2}\leq N^{1/2}}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{N^{1/2}},\frac{m_{2}}{N^{1/2}}))\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}\\|c_{m_{1},m_{2}}\\|_{l^{2}}\|B_{N}\|^{1/6},$ (10) while if $M\geq N^{1/2}$ we have $\\|\sum_{m_{1}\leq M}\sum_{m_{2}\leq M}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{M},\frac{m_{2}}{M}))\\|_{L^{6}(B_{N})}$ (11) $\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1},J_{2}}\\|\sum_{m_{1}\in J_{1}}\sum_{m_{2}\in J_{2}}c_{m_{1},m_{2}}e(x\cdot\Psi(\frac{m_{1}}{M},\frac{m_{2}}{M}))\\|^{2}_{L^{6}(B_{N})})^{1/2},$ with $J_{i}$ intervals of length $\sim MN^{-1/2}$ partitioning $[1,M]$. The implicit constants in both inequalities are independent of $N$, $M$ and of $\psi_{i}$. ###### Proof. The exponential sum estimates (10), (11) are standard consequences of (9), so we will focus on proving the latter. When $\Psi(\xi_{1},\xi_{2})=(\xi_{1},\xi_{2},C_{1}\xi_{1}^{2}+C_{2}\xi_{2}^{2},C_{3}\xi_{1}^{2}+C_{4}\xi_{2}^{2})$ with $\|C_{1}\|,\|C_{4}\|\sim 1$ and $\|C_{2}\|,\|C_{3}\|\ll 1$, the result follows by applying $l^{2}(L^{6})$ Theorem 2.1 twice (after an initial affine change of variables that reduces it to the case $C_{1}=C_{4}=1$, $C_{2}=C_{3}=0$). We use a bootstrapping argument similar to the one in Theorem 2.3. Let $d(N)$ be the smallest constant in (9). We need to prove that $d(N)\lesssim_{\epsilon}N^{\epsilon}$. We first note that $\\|E_{[-1,1]^{2}}F\\|_{L^{6}(B_{N})}\leq d(N^{2/3})(\sum_{U_{1},U_{2}\subset[-1,1]}\\|E_{U_{1}\times U_{2}}F\\|^{2}_{L^{6}(B_{N})})^{1/2},$ where $U_{i}$ are intervals of length $N^{-1/3}$ centered at $l_{i}$. When $\|t\|<\frac{1}{2}N^{-1/3}$ we have $\psi_{i}(t)=\psi_{i}(l_{i})+\psi_{i}^{\prime}(l_{i})t+C_{i}t^{2}+O(\frac{1}{N}),$ where $\|C_{1}\|,\|C_{4}\|\sim 1$ and $\|C_{2}\|,\|C_{3}\|\ll 1$. It follows that, after an affine change of variables, the restriction of $\Psi$ to $U_{1}\times U_{2}$ may be parametrized as $(\xi_{1},\xi_{2},C_{1}\xi_{1}^{2}+C_{2}\xi_{2}^{2}+O(\frac{1}{N}),C_{3}\xi_{1}^{2}+C_{4}\xi_{2}^{2}+O(\frac{1}{N})),\;\;\|\xi_{1}\|,\|\xi_{2}\|=O(N^{-1/3}).$ We decouple this on $B_{N}$, using the observation at the beginning of the proof. We find $\\|E_{U_{1}\times U_{2}}F\\|_{L^{6}(B_{N})}\lesssim_{\epsilon}N^{\epsilon}(\sum_{H_{1}\subset U_{1},H_{2}\subset U_{2}}\\|E_{H_{1}\times H_{2}}F\\|^{2}_{L^{6}(B_{N})})^{1/2}.$ It follows that $d(N)\lesssim_{\epsilon}N^{\epsilon}d(N^{2/3})$, which forces $d(N)\lesssim_{\epsilon}N^{\epsilon}$. ∎ ## 3\. Bourgain’s argument for the case $\alpha=2$ For the remainder of the paper, we will use the following notation ${\mathcal{E}}_{I}(x)=\sum_{n\in I}e(N\Phi(\frac{n}{N})x)=\sum_{n\in I}e(nx_{1}+\frac{n^{2}}{N}x_{2}+\phi_{3}(\frac{n}{N})Nx_{3}+\phi_{4}(\frac{n}{N})Nx_{4}).$ Note that, compared to ${\mathcal{E}}_{I,N}$, we dropped the subscript $N$ and renormalized the variables $x_{2}$, $x_{3}$ and $x_{4}$. Letting $\Omega=[0,N]^{3}\times[0,1]$ and using periodicity in $x_{1}$, we need to prove that $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{9+\epsilon}.$ Bourgain’s argument from [2] for the case $\alpha=2$ of Theorem 1.3 involves three successive decouplings. We simplify it slightly it and reduce it to only two decouplings. Step 1. Cover $\Omega$ with cubes $B$ of side length $1$, apply $l^{6}(L^{6})$ decoupling (Theorem 2.2) on each $B$ (or rather $NB$, after rescaling), then sum these estimates to get $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{2+\epsilon}\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}.$ Here $J_{1},J_{2}$ are intervals of length $N^{1/2}$. The remaining part of the argument will show the uniform estimate $O(N^{6+\epsilon})$ for each term $\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}.$ Fix $J_{i}=[h_{i},h_{i}+N^{1/2}]$. Step 2. Note that when $x\in\Omega$ $\|{\mathcal{E}}_{J_{i}}(x)\|=\|\sum_{m\leq N^{1/2}}c_{m}(x_{4})e(mu_{i}+m^{2}w_{i}+\eta_{i}(m)x_{3})\|$ where $\begin{cases}u_{i}=x_{1}+\frac{2h_{i}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{i}}{N})x_{3}\\\ w_{i}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{i}}{N})\frac{x_{3}}{2N}\\\ \eta_{i}(m)={m^{3}}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h_{i}}{N})}{3!N^{2}}+{m^{4}}\frac{\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{i}}{N})}{4!N^{3}}+\ldots\end{cases}.$ (12) We hide the whole contribution from $x_{4}$ into the coefficient $c_{m}(x_{4})$. Indeed, since $\phi_{4}(\frac{h_{i}+m}{N})Nx_{4}=\phi_{4}(\frac{h_{i}}{N})Nx_{4}+m\phi_{4}^{\prime}(\frac{h_{i}}{N})x_{4}+O(1),$ $x_{4}$ does not contribute significantly with quadratic or higher order terms, so it produces no cancellations. We will only use that $c_{m}(x_{4})=O(1)$. At this point, we seek a change of variables. We want the new domain of integration to be a rectangular box, to allow us to separate the four-variable integral $\int_{\Omega}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$ into the product of two-variable integrals. Note that the ranges of $x_{1},x_{2},x_{3}$ are the same, $[0,N]$, but $x_{4}$ is restricted to the smaller interval $[0,1]$. We cannot use periodicity to extend the range of $x_{4}$ to $[0,N]$, because the individual waves $e(m\phi_{4}^{\prime}(\frac{h_{i}}{N})x_{4})$ have different periods with respect to the variable $x_{4}$. Because of this, the variable $x_{4}$ is practically useless from this point on, it will not generate oscillations. To generate a fourth variable with range $[0,N]$ for the purpose of a change of coordinates, Bourgain produces a piece of magic. First, he applies (8) on each cube $NB$ $\displaystyle\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$ $\displaystyle=N^{-4}\int_{NB}\|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N}){\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})\|^{6}$ $\displaystyle\lesssim N^{-8}\int_{NB}\|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N})\|^{6}\int_{NB}\|{\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})\|^{6}=\int_{B}\|{\mathcal{E}}_{J_{1}}\|^{6}\int_{B}\|{\mathcal{E}}_{J_{2}}\|^{6}.$ Second, he uses the following abstract inequality, that only relies on the positivity of $\|{\mathcal{E}}_{J_{i}}\|^{6}$ $\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{J_{1}}\|^{6}\int_{B}\|{\mathcal{E}}_{J_{2}}\|^{6}\lesssim\int_{\Omega}dx\int_{(y,z)\in[-1,1]^{4}\times[-1,1]^{4}}\|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)\|^{6}dydz.$ Using periodicity in the $y_{1},z_{1}$ variables, this is $\lesssim\frac{1}{N^{2}}\int_{x_{1}\in[0,N]\atop{}_{x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}\in[-1,1]}}dx_{1}\ldots dz_{4}\int_{y_{1},z_{1},x_{2},x_{3}\in[0,N]}\|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)\|^{6}dy_{1}dz_{1}dx_{2}dx_{3}.$ In short, the variable $x_{1}$ is now replaced with the new variables $y_{1}$ and $z_{1}$. It remains to prove that $\int_{y_{1},z_{1},x_{2},x_{3}\in[0,N]}\|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)\|^{6}dy_{1}dz_{1}dx_{2}dx_{3}\lesssim_{\epsilon}N^{7+\epsilon},$ (13) uniformly over $x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$. With these variables fixed, we make the affine change of variables $(y_{1},z_{1},x_{2},x_{3})\mapsto(u_{1},u_{2},w_{1},w_{2})$ $\begin{cases}u_{1}=(y_{1}+x_{1})+\frac{2h_{1}}{N}(x_{2}+y_{2})+\phi_{3}^{\prime}(\frac{h_{1}}{N})(x_{3}+y_{3})\\\ u_{2}=(z_{1}+x_{1})+\frac{2h_{2}}{N}(x_{2}+z_{2})+\phi_{3}^{\prime}(\frac{h_{2}}{N})(x_{3}+z_{3})\\\ w_{1}=\frac{x_{2}+y_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}+y_{3}}{2N}\\\ w_{2}=\frac{x_{2}+z_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}+z_{3}}{2N}\end{cases}.$ (14) The Jacobian is $\sim\frac{1}{N^{2}}$, due to (3). Note that $x_{3}+y_{3}=A(w_{1}-w_{2})$, where $A$ depends just on $h_{1},h_{2}$, and $\|A\|\sim N$. Using (12) we may write the last integral as $N^{2}\times$ $\int_{\|u_{i}\|\lesssim N\atop{}_{\|w_{i}\|\lesssim 1}}\|\sum_{m_{i}\leq N^{\frac{1}{2}}}c_{m_{1},m_{2}}e(m_{1}u_{1}+m_{1}^{2}w_{1}+m_{2}u_{2}+m_{2}^{2}w_{2}+(\eta_{1}(m_{1})+\eta_{2}(m_{2}))A(w_{1}-w_{2}))\|^{6}.$ (15) The coefficient $c_{m_{1},m_{2}}$ depends only on $m_{1},m_{2},x_{4},y_{4},z_{4}$, but not on the variables of integration $u_{i},w_{i}$. The argument of each exponential may be rewritten as $\frac{m_{1}}{N^{1/2}}u_{1}N^{1/2}+(\psi_{1}(\frac{m_{1}}{N^{1/2}})+\psi_{2}(\frac{m_{2}}{N^{1/2}}))w_{1}N+$ $\frac{m_{1}}{N^{1/2}}u_{2}N^{1/2}+(\psi_{3}(\frac{m_{1}}{N^{1/2}})+\psi_{4}(\frac{m_{2}}{N^{1/2}}))w_{2}N$ where $\begin{cases}\psi_{1}(\xi)=\xi^{2}+&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{1}}{N})}{3!N^{3/2}}+{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{1}}{N})}{4!N^{2}}+\ldots\\\ \psi_{2}(\xi)=&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{2}}{N})}{3!N^{3/2}}+{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{2}}{N})}{4!N^{2}}+\ldots\\\ \psi_{3}(\xi)=&{-\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{1}}{N})}{3!N^{3/2}}-{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{1}}{N})}{4!N^{2}}-\ldots\\\ \psi_{4}(\xi)=\xi^{2}-&{\xi^{3}}\frac{A\phi_{3}^{\prime\prime\prime}(\frac{h_{2}}{N})}{3!N^{3/2}}-{\xi^{4}}\frac{A\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{2}}{N})}{4!N^{2}}-\ldots\end{cases}.$ These functions satisfy the requirements in Theorem 2.4. The integral (15) is the same as $N^{-3}\int_{u_{i}=O(N^{3/2})\atop{}_{w_{i}=O(N)}}\|\sum_{m_{1}\leq N^{1/2}}\sum_{m_{2}\leq N^{1/2}}c_{m_{1},m_{2}}e((u_{1},u_{2},w_{1},w_{2})\cdot\Psi(\frac{m_{1}}{N^{1/2}},\frac{m_{2}}{N^{1/2}}))\|^{6}du_{1}du_{2}dw_{1}dw_{2}.$ If we cover the domain of integration with $\sim N$ balls $B_{N}$ and apply (10) on each of them, we may dominate the above expression by $N^{-3}N(N^{\frac{1}{2}+\epsilon}N^{4/6})^{6}=N^{5+\epsilon}.$ This proves (13) and ends the proof. Note that this argument treats the cubic and higher order terms as perturbations of quadratic factors, as explained in the proof of (6). In summary, what is special about the case $\alpha=2$ is that the range of $x_{3}$ in our initial integral over $\Omega$ is $[0,N]$. This was needed in producing the large spatial range $w_{i}=O(N)$ for our final variables, crucial for the application of (10). This inequality provides decoupling into point masses, reducing the initial exponential sum to individual waves. In Section 6 we will see that when $\alpha$ is slightly smaller than 2, inequality (11) will have to replace (10), leading to quadratic Weyl sums whose handling demands number theory. ###### Remark 3.1. It is not clear whether a version of Bourgain’s method could be made to work in the range $2<\alpha<3$. If successful, this would potentially provide a new argument for Vinogradov’s Mean Value Theorem in ${\mathbb{R}}^{3}$. Decoupling on cubes with size $N^{\beta-1}$ and using (8) on balls $B_{N^{\beta}}$ leads to variables $y_{1},z_{1}$ with associated period equal to 1, much bigger than their range $N^{\beta-1}$. The change of variables (14) is no longer efficient in this case. ## 4\. Proof of Theorem 1.3 in the case $\alpha=\frac{3}{2}$ This time we let $\Omega=[0,1]\times[0,N]\times[0,N^{1/2}]\times[0,N^{1/2}]$. Recall that ${\mathcal{E}}_{I}(x)=\sum_{n\in I}e(nx_{1}+\frac{n^{2}}{N}x_{2}+\phi_{3}(\frac{n}{N})Nx_{3}+\phi_{4}(\frac{n}{N})Nx_{4}).$ We need to prove $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}dx\lesssim_{\epsilon}N^{8+\epsilon}.$ (16) In this case, we will only need assumptions (1) and (2), but not (3). We start by presenting a general principle that will explain the subtleties of our argument. See also Remark 4.3. Consider two partitions of $\Omega$, one into cubes $B$ with side length $l$ and another one into cubes $\Delta$ with side length $L\geq l$. The intervals $J_{i}$ have length $\sqrt{\frac{N}{l}}$ and partition $I_{i}$. The intervals $U_{i}$ have length $\sqrt{\frac{N}{L}}$ and partition $I_{i}$. The following holds, via two applications of Theorem 2.3 (on cubes $B$ and $\Delta$, combined with Minkowski’s inequality) $\\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\\|_{L^{6}(\Omega)}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1},J_{2}}\\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\\|_{L^{6}(\Omega)}^{2})^{1/2}\lesssim_{\epsilon}N^{\epsilon}(\sum_{U_{1},U_{2}}\\|{\mathcal{E}}_{U_{1}}{\mathcal{E}}_{U_{2}}\\|_{L^{6}(\Omega)}^{2})^{1/2}.$ (17) Also, combining the above inequalities with Hölder shows that $\\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\\|_{L^{6}(\Omega)}\lesssim_{\epsilon}$ $N^{\epsilon}(\sharp(J_{1},J_{2}))^{\frac{1}{3}}(\sum_{J_{1},J_{2}}\\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\\|_{L^{6}(\Omega)}^{6})^{1/6}\lesssim_{\epsilon}N^{\epsilon}(\sharp(U_{1},U_{2}))^{\frac{1}{3}}(\sum_{U_{1},U_{2}}\\|{\mathcal{E}}_{U_{1}}{\mathcal{E}}_{U_{2}}\\|_{L^{6}(\Omega)}^{6})^{1/6}.$ (18) Invoking periodicity in $x_{1},x_{2}$ and the invariance of (1), (2), (3) under the change of sign $\phi_{k}\mapsto-\phi_{k}$, (16) is equivalent with proving that $\int_{[-N^{1/2},N^{1/2}]\times[-N,N]\times[-N^{1/2},N^{1/2}]\times[-N^{1/2},N^{1/2}]}\|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)\|^{6}dx\lesssim_{\epsilon}N^{8+\frac{1}{2}+\epsilon}.$ (19) We first demonstrate the inefficiency of $l^{6}L^{6}$ decoupling for this case, by working with the smaller domain $S=[-o(N^{1/2}),o(N^{1/2})]^{4}.$ We cover $S$ with unit cubes and apply decoupling into intervals $J_{1},J_{2}$ of length $N^{1/2}$ as in the previous section, to dominate $\int_{S}\|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)\|^{6}dx\lesssim_{\epsilon}N^{\epsilon}N^{6(\frac{1}{2}-\frac{1}{6})}\sum_{J_{1},J_{2}}\int_{S}\|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)\|^{6}dx.$ (20) We will next show that the right hand side is too big, thus leading to an overestimate for our initial integral. When $J=[h,h+N^{1/2}]$ and $n=h+m\in J$ we write $\phi_{k}(\frac{n}{N})=\phi_{k}(\frac{h}{N})+\phi_{k}^{\prime}(\frac{h}{N})\frac{m}{N}+\frac{1}{2}\phi_{k}^{\prime\prime}(\frac{h}{N})(\frac{m}{N})^{2}+O(\frac{m}{N})^{3}.$ If $\|x_{3}\|,\|x_{4}\|\ll N^{1/2}$ and $m\leq N^{1/2}$, we guarantee that the contribution from higher order terms is small $O(\frac{m}{N})^{3}N(\|x_{3}\|+\|x_{4}\|)\ll 1.$ If we collect the contributions from linear and quadratic terms we find $\|{\mathcal{E}}_{J}(x)\|=\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+o(1))\|$ where $\begin{cases}u=x_{1}+\frac{h}{N}x_{2}+\phi_{3}^{\prime}(\frac{h}{N})x_{3}+\phi_{4}^{\prime}(\frac{h}{N})x_{4}\\\ w=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h}{N})\frac{x_{4}}{N}\end{cases}.$ Using Lemma 9.1 we write $\int_{S}\|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)\|^{6}dx\gtrsim$ $N^{2}\int_{(u_{1},u_{2})\in[0,o(N^{1/2})]^{2}\atop{(w_{1},w_{2})\in[0,o(N^{-1/2})]^{2}}}\|\sum_{m\leq N^{1/2}}e(mu_{1}+m^{2}w_{1}+o(1))\|^{6}\|\sum_{m\leq N^{1/2}}e(mu_{2}+m^{2}w_{2}+o(1))\|^{6}.$ We now use the fact that we have constructive interference $\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+o(1))\|\sim N^{1/2}$ on the set of measure $\sim\frac{1}{N}$ $(u,w)\in(\bigcup_{l\in\\{0,1,\ldots,o(\sqrt{N})\\}}[l,l+\frac{1}{\sqrt{N}}])\times[0,\frac{1}{N}].$ It follows that $\int_{S}\|{\mathcal{E}}_{J_{1}}(x){\mathcal{E}}_{J_{2}}(x)\|^{6}dx\gtrsim N^{2}N^{6}N^{-2}=N^{6}.$ It is not hard to prove that this lower bound is sharp, but this has no relevance to us here. The point of working with the symmetric domain $S$ was to make sure that $w_{1},w_{2}\sim\frac{1}{N}$ are in the new domain of integration. Going back to (20), the $l^{6}(L^{6})$ decoupling method leads to the upper bound $\int_{S}\|{\mathcal{E}}_{I_{1}}(x){\mathcal{E}}_{I_{2}}(x)\|^{6}dx\lesssim_{\epsilon}N^{9+\epsilon}.$ This falls short by the factor $N^{1/2}$ from proving (19). The second inequality in (4) shows that using $l^{6}(L^{6})$ decoupling on cubes $\Delta$ that are larger than $N$ will only worsen the upper bounds we get. On the other hand, working with smaller cubes will render decoupling inefficient. The resulting exponential sums will be very difficult to handle using number theory, since the cubic terms are no longer $O(1)$ in this case. Let us now describe the correct approach, that will critically rely on $l^{2}$, rather than $l^{6}$ decoupling. The following level set estimate will play a key role in various counting arguments. The main strength of the lemma is in the case when $\|l_{1}\|\sim\|l_{2}\|$. Throughout the remainder of the paper, the letter $l$ will be used to denote integers, and their relative proximity to powers of $2$ will be denoted using the symbol $\sim$. We make the harmless convention to write $0\sim 2^{0}$. ###### Lemma 4.1. Assume $\phi_{3}$, $\phi_{4}$ satisfy (1) and (2). Let $l_{1},l_{2}$ with $\max\\{\|l_{1}\|,\|l_{2}\|\\}\sim 2^{j}$, $j\geq 0$, and let $f(t)=l_{1}\phi_{3}^{\prime\prime}(t)+l_{2}\phi_{4}^{\prime\prime}(t).$ Then we can partition the range of $f$ into sets $R_{s}$ with $0\leq s\leq j$, each of which is the union of at most two intervals of length $\sim 2^{s}$, such that for each $v\in R_{s}$ we have $\|f^{-1}(v+[-O(1),O(1)])\cap[\frac{1}{2},1]\|\lesssim\frac{1}{\sqrt{2^{j+s}}}.$ All implicit constants are universal over all pairs of such $\phi_{3}$, $\phi_{4}$ and over $l_{1},l_{2},s$. ###### Proof. The result is trivial if $l_{1}=l_{2}=0$, so we will next assume that $\max\\{\|l_{1}\|,\|l_{2}\|\\}\geq 1$. We restrict $f$ to the interval $[\frac{1}{2},1]$. Since $\begin{bmatrix}f^{\prime}(t)\\\ f^{\prime\prime}(t)\end{bmatrix}=\begin{bmatrix}\phi_{3}^{(3)}(t)&\phi_{4}^{(3)}(t)\\\ \phi_{3}^{(4)}(t)&\phi_{4}^{(4)}(t)\end{bmatrix}\begin{bmatrix}l_{1}\\\ l_{2}\end{bmatrix},$ (2) implies that for each $t\in[\frac{1}{2},1]$ we have $\max\\{\|f^{\prime}(t)\|,\|f^{\prime\prime}(t)\|\\}\sim 2^{j}.$ (21) We let $t_{0}$ be a point in $[\frac{1}{2},1]$ where $\|f^{\prime}\|$ attains its minimum. If $\|f^{\prime}(t_{0})\|\sim 2^{j}$, then we may take $R_{j}$ to be the whole range of $f$, and all other $R_{s}$ to be empty. Indeed, the Mean Value Theorem shows that $\|f(t_{1})-f(t_{2})\|\gtrsim 1$ whenever $\|t_{1}-t_{2}\|\gtrsim 2^{-j}$. It is worth observing that if $\|l_{1}\|\gg\|l_{2}\|$, then (3) would immediately guarantee that $\|f^{\prime}(t_{0})\|\sim 2^{j}$. We now assume that $\|f^{\prime}(t_{0})\|\ll 2^{j}$. Due to (21), we must have that $\|f^{\prime\prime}(t_{0})\|\sim 2^{j}$. We write for $t\in[\frac{1}{2},1]$ $f(t)=f(t_{0})+f^{\prime}(t_{0})(t-t_{0})+\frac{f^{\prime\prime}(t_{0})(t-t_{0})^{2}}{2}+O(2^{j}(t-t_{0})^{3}).$ (22) Case 1. Consider $s$ with $2^{j}\geq 2^{s}>C\max\\{\frac{\|f^{\prime}(t_{0})\|^{2}}{2^{j}},1\\}$, for some large enough $C$ independent of $j$. Using this and (22), we see that $\|f(t)-f(t_{0})\|\ll 2^{s}\;\text{ whenever }\|t-t_{0}\|\ll 2^{\frac{s-j}{2}}.$ (23) Define $R_{s}=\\{v:\;\|v-f(t_{0})\|\sim 2^{s}\\}.$ Let $v\in R_{s}$ and let $w=v+O(1)$. Thus, we also have $\|w-f(t_{0})\|\sim 2^{s}$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$, $f(t_{2})=w$. Using (23) it follows that $\|t_{1}-t_{0}\|,\|t_{2}-t_{0}\|\gtrsim 2^{\frac{s-j}{2}}$. Our assumption shows that $2^{\frac{s-j}{2}}\gg\frac{\|f^{\prime}(t_{0})\|}{2^{j}}$. Thus, $\|t_{1}-t_{0}\|,\|t_{2}-t_{0}\|\gg\frac{\|f^{\prime}(t_{0})\|}{2^{j}}$, and using (22) again we conclude that $\|f(t_{i})-f(t_{0})\|\sim 2^{j}\|t_{i}-t_{0}\|^{2}.$ Thus, $\|t_{i}-t_{0}\|\sim 2^{\frac{s-j}{2}}$. Using again (22) we find that if $t_{1},t_{2}$ are on the same side of $t_{0}$ then $\|f(t_{1})-f(t_{2})\|\sim 2^{\frac{s+j}{2}}\|t_{1}-t_{2}\|.$ We conclude that $\|t_{1}-t_{2}\|\lesssim\frac{1}{\sqrt{2^{j+s}}}$, as desired. Next, we define $R_{s}$ for smaller values of $s$. We distinguish two cases. Case 2a. Assume now that $\|f^{\prime}(t_{0})\|\leq 2^{j/2}$. For $s$ such that $2^{s}$ is the largest dyadic power $\leq C\max\\{\frac{\|f^{\prime}(t_{0})\|^{2}}{2^{j}},1\\}=C$ we define $R_{s}=\\{v:\;\|v-f(t_{0})\|\lesssim 2^{s}\\}.$ We also let $R_{s^{\prime}}=\emptyset$ for smaller values of $s^{\prime}$. Let $v\in R_{s}$ and $w=v+O(1)$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$, $f(t_{2})=w$. Since in fact $\|f(t_{i})-f(t_{0})\|\lesssim 1$, (22) forces $\|t_{i}-t_{0}\|\lesssim 2^{-j/2}\sim\frac{1}{\sqrt{2^{j+s}}}$, as desired. Case 2b. Assume now that $\|f^{\prime}(t_{0})\|>2^{j/2}$. For $s$ such that $2^{s}$ is the largest dyadic power $\leq C\max\\{\frac{\|f^{\prime}(t_{0})\|^{2}}{2^{j}},1\\}=C\frac{\|f^{\prime}(t_{0})\|^{2}}{2^{j}}$ we define $R_{s}=\\{v:\;\|v-f(t_{0})\|\lesssim 2^{s}\\}.$ We also let $R_{s^{\prime}}=\emptyset$ for smaller values of $s^{\prime}$. Let $v\in R_{s}$ and $w=v+O(1)$. Let $t_{1},t_{2}$ be such that $f(t_{1})=v$, $f(t_{2})=w$. Using that $\|f^{\prime}(t)\|\geq\|f^{\prime}(t_{0})\|$ for all $t$, we find that $\|f(t_{1})-f(t_{2})\|\geq\|t_{1}-t_{2}\|\|f^{\prime}(t_{0})\|.$ We conclude that $\|t_{1}-t_{2}\|\lesssim\frac{1}{\|f^{\prime}(t_{0})\|}\sim\frac{1}{\sqrt{2^{j+s}}},$ as desired. ∎ From now on, we will implicitly assume that all Weyl sums are smooth, as in Lemma 9.2. This can be easily arranged using partitions of unity, namely working with smooth $\gamma$ satisfying $\sum_{l\in{\mathbb{Z}}}\gamma(\cdot+l)=1_{\mathbb{R}}.$ To simplify notation, these weights will be ignored. Cover $\Omega$ with unit cubes $B=B_{p,l_{1},l_{2}}=[0,1]\times[p,p+1]\times[l_{1},l_{1}+1]\times[l_{2},l_{2}+1]$ with $p\leq N,\;\;l_{1},l_{2}\leq N^{1/2}.$ We first write $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\sim\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}.$ We use $l^{2}$ decoupling (Theorem 2.3) on each $B$ $\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3}$ where $J_{i}$ is of the form $[h_{i},h_{i}+N^{1/2}]$. When $x\in B$ and $J=[h,h+N^{1/2}]$ $\|{\mathcal{E}}_{J}(x)\|=\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+{m^{3}}v+O(N^{-1/4}))\|$ where $\begin{cases}u=x_{1}+\frac{2h}{N}x_{2}+\phi_{3}^{\prime}(\frac{h}{N})x_{3}+\phi_{4}^{\prime}(\frac{h}{N})x_{4}\\\ w=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h}{N})\frac{x_{4}}{N}\\\ v=\frac{\phi_{3}^{\prime\prime\prime}(\frac{h}{N}){x_{3}}+\phi_{4}^{\prime\prime\prime}(\frac{h}{N}){x_{4}}}{6N^{2}}\end{cases}.$ (24) The term $O(N^{-1/4})$ can be dismissed as it produces tiny errors consistent with square root cancellation. Note that since $v=O(N^{-3/2})$, we have $\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w+{m^{3}}v)\|\approx\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w)\|.$ See Lemma 9.2 for a rigorous argument. The key point is that we may dismiss the cubic terms. Write $I(h_{1},h_{2},B)=$ $\int_{(u_{1},u_{2},w_{1},w_{2})\in[0,1]^{2}\times[\frac{a_{1}-O(1)}{N},\frac{a_{1}+O(1)}{N}]\times[\frac{a_{2}-O(1)}{N},\frac{a_{2}+O(1)}{N}]}\|\prod_{i=1}^{2}\sum_{m_{i}\leq N^{1/2}}e(m_{i}u_{i}+m_{i}^{2}w_{i})\|^{6}du_{1}du_{2}dw_{1}dw_{2},$ where $\begin{cases}a_{1}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\\\ a_{2}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\end{cases}.$ (25) Via the change of variables with Jacobian $\sim\frac{1}{N^{2}}$ (Lemma 9.1) $\begin{cases}u_{1}=x_{1}+\frac{2h_{1}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{1}}{N})x_{3}+\phi_{4}^{\prime}(\frac{h_{1}}{N})x_{4}\\\ w_{1}=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{4}}{N}\\\ u_{2}=x_{1}+\frac{2h_{2}}{N}x_{2}+\phi_{3}^{\prime}(\frac{h_{2}}{N})x_{3}+\phi_{4}^{\prime}(\frac{h_{2}}{N})x_{4}\\\ w_{2}=\frac{x_{2}}{N}+\frac{1}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}}{N}+\frac{1}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{4}}{N}\end{cases}$ we see that $\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}\lesssim N^{2}I(h_{1},h_{2},B).$ Writing $I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{N},\frac{a+O(1)}{N}]}\|\sum_{m\leq N^{1/2}}e(mu+m^{2}w)\|^{6}dudw$ we find that $\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}\lesssim N^{2}I_{a_{1}}I_{a_{2}}.$ Let us analyze (25). The question is, for fixed $B$, what are the values of $a_{1},a_{2}$ that arise (modulo $O(1)$ error terms), and what is their multiplicity, when $h_{1},h_{2}$ range through the multiples of $N^{1/2}$ in $[1,N]$. Assume $l_{1}\sim 2^{j_{1}}$, $l_{2}\sim 2^{j_{2}}$, with $2^{j_{1}},2^{j_{2}}\leq N^{1/2}$. We may assume $j_{1}\leq j_{2}$, the other case is completely similar. We apply Lemma 4.1 to $f(t)=\frac{1}{2}(l_{1}\phi_{3}^{\prime\prime}(t)+l_{2}\phi_{4}^{\prime\prime}(t))$. For each $0\leq s_{1},s_{2}\leq j_{2}$ and each $p$ we have $O(2^{s_{1}+s_{2}})$ pairs $(a_{1},a_{2})$ of integers with $a_{1}-p\in R_{s_{1}}(l_{1},l_{2})$ and $a_{2}-p\in R_{s_{2}}(l_{1},l_{2})$. Note that we index the intervals $R_{s_{i}}$ from Lemma 4.1 by $l_{1},l_{2}$. For each such pair $(a_{1},a_{2})$, (25) has $O(\frac{N}{2^{j_{2}}2^{\frac{s_{1}+s_{2}}{2}}})$ solutions $(h_{1},h_{2})$. When we count solutions, we tolerate error terms of size $O(1)$. Thus $\displaystyle\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}$ $\displaystyle\lesssim N^{2}\sum_{p\leq N}\sum_{2^{j_{2}}\lesssim N^{1/2}}\sum_{2^{j_{1}}\lesssim 2^{j_{2}}}\sum_{l_{1}\sim 2^{j_{1}}}\sum_{l_{2}\sim 2^{j_{2}}}\sum_{s_{1},s_{2}\leq j_{2}}(\frac{N}{2^{j_{2}+\frac{s_{1}+s_{2}}{2}}})^{3}(\sum_{a_{1}\in p+R_{s_{1}}(l_{1},l_{2})}\sum_{a_{2}\in p+R_{s_{2}}(l_{1},l_{2})}I_{a_{1}}^{1/3}I_{a_{2}}^{1/3})^{3}$ $\displaystyle\lesssim N^{2}\sum_{p\leq N}\sum_{2^{j_{2}}\lesssim N^{1/2}}\sum_{2^{j_{1}}\lesssim 2^{j_{2}}}2^{j_{1}+j_{2}}(\frac{N}{2^{j_{2}}})^{3}\sum_{s_{1},s_{2}\leq j_{2}}(\sum_{a_{1}\in p+R_{s_{1}}(l_{1},l_{2})}I_{a_{1}}^{2/3})^{3/2}(\sum_{a_{2}\in p+R_{s_{2}}(l_{1},l_{2})}I_{a_{2}}^{2/3})^{3/2}.$ The last inequality follows from Cauchy–Schwarz. Next, we observe that $p+R_{s_{i}}(l_{1},l_{2})\subset[p-O(2^{j_{2}}),p+O(2^{j_{2}})]$. These intervals are roughly the same for roughly $2^{j_{2}}$ values of $p$. We can thus dominate the above by $\displaystyle\begin{split}&{\;\lessapprox}\;N^{2}\sum_{2^{j_{2}}\lesssim N^{1/2}}\sum_{2^{j_{1}}\lesssim 2^{j_{2}}}2^{j_{1}+j_{2}}(\frac{N}{2^{j_{2}}})^{3}2^{j_{2}}\sum_{H\subset[0,N]\atop{\|H\|={2^{j_{2}}}}}(\sum_{a\in H}I_{a}^{2/3})^{3}\\\ &\sim N^{5}\sum_{2^{j}\lesssim N^{1/2}}\sum_{H\subset[0,N]\atop{\|H\|={2^{j}}}}(\sum_{a\in H}I_{a}^{2/3})^{3}.\end{split}$ (26) The sum runs over pairwise disjoint intervals $H$. It is easily seen to be $O(N^{8})$, by using the following lemma with $M=N^{1/2}$. ###### Lemma 4.2. Let $I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M^{2}},\frac{a+O(1)}{M^{2}}]}\|\sum_{m\leq M}e(mu+m^{2}w)\|^{6}dudw$ For each $2^{j}\leq M^{2}$ we have $\sum_{H\subset[0,M^{2}]\atop{\|H\|={2^{j}}}}(\sum_{a\in H}I_{a}^{2/3})^{3}{\;\lessapprox}\;M^{4}2^{2j}+M^{6}.$ ###### Proof. The arcs $\\{x\in[0,1):\;{\operatorname{dist}\,}(x-\frac{b}{q},{\mathbb{Z}})\leq\frac{1}{qM}\\}$, with $1\leq b\leq q\leq M$ and $(b,q)=1$, cover $[0,1)$. They may overlap, which leads to double counting in our argument, but this will be harmless. We consider the contribution from those $I_{a}$ with $\frac{a}{M^{2}}$ in some arc with $q\sim Q$. Here $Q$ is dyadic and $Q\lesssim M$. We separate the proof into two cases. Note that $H/M^{2}\subset[0,1]$ and has length $2^{j}/M^{2}$. Also, $\|b/q-b^{\prime}/q^{\prime}\|\geq 1/qq^{\prime}$. Case 1. $Q^{2}>\frac{M^{2}}{2^{j}}$. Each $H/M^{2}$ intersects $\lesssim\frac{2^{j}Q^{2}}{M^{2}}$ arcs with $q\sim Q$. For each such $b/q$, and each $1\leq 2^{k}\leq\frac{M}{q}$ there are $\sim\frac{M}{2^{k}Q}$ values of $a$ with $\|\frac{a}{M^{2}}-\frac{b}{q}\|\sim\frac{1}{qM2^{k}}.$ Call ${\mathbb{A}}(Q,k)$ the collection of all these $a$. For each $a\in{\mathbb{A}}(Q,k)$, Lemma 9.2 gives $I_{a}\lesssim\frac{1}{2^{k}M^{2}}(M^{1/2}2^{k/2})^{6}=2^{2k}M.$ The contribution from $a\in{\mathbb{A}}(Q,k)$ is $\sum_{H\subset[0,M^{2}]\atop{\|H\|={2^{j}}}}(\sum_{a\in H\cap{\mathbb{A}}(Q,k)}I_{a}^{2/3})^{3}\lesssim\frac{M^{2}}{2^{j}}(\frac{2^{j}Q^{2}}{M^{2}}\frac{M}{2^{k}Q})^{3}(M2^{2k})^{2}=M2^{k}2^{2j}Q^{3}.$ This is easily seen to be $O(2^{2j}M^{4})$, since $2^{k}=O(MQ^{-1})$ and $Q=O(M)$. The contribution to the full sum is acceptable, since there are ${\;\lessapprox}\;1$ values of $Q$ and $k$. Case 2. $Q^{2}<\frac{M^{2}}{2^{j}}$. There are $\lesssim Q^{2}$ arcs with $q\sim Q$. Essentially, each $H$ is either disjoint from all these (so not contributing at this stage) or (essentially) contained inside one of them. We distinguish two subcases. (a) If $\frac{2^{j}}{M^{2}}<\frac{1}{QM2^{k}}$ (this is stronger than $Q^{2}<\frac{M^{2}}{2^{j}}$), there are $\frac{1}{QM2^{k}}\frac{M^{2}}{2^{j}}$ intervals $H/M^{2}$ contained in $[\frac{b}{q}-\frac{1}{QM2^{k}},\frac{b}{q}+\frac{1}{QM2^{k}}]$. Their contribution is $\displaystyle\sum_{b<q\sim Q}\sum_{H/M^{2}\subset[\frac{b}{q}-\frac{1}{QM2^{k}},\frac{b}{q}+\frac{1}{QM2^{k}}]\atop{\|H\|={2^{j}}}}(\sum_{a\in H}I_{a}^{2/3})^{3}$ $\displaystyle\lesssim Q^{2}\frac{M}{Q2^{k}2^{j}}2^{3j}(2^{2k}M)^{2}$ $\displaystyle=2^{2j}M^{3}Q2^{3k}.$ Using our assumption, this is $O(2^{-j}M^{6})$. (b) If $\frac{2^{j}}{M^{2}}>\frac{1}{QM2^{k}}$, for each $b/q$ with $q\sim Q$ there is only one $H/M^{2}$ that intersects $\|t-\frac{b}{q}\|\sim\frac{1}{qM2^{k}}$ with at most $M^{2}\frac{1}{qM2^{k}}$ values of $a$ contributing from $H$. The contribution from the $O(Q^{2})$ arcs with denominator $\sim Q$ is $\lesssim Q^{2}(\frac{M}{Q2^{k}})^{3}(2^{2k}M)^{2}=\frac{2^{k}M^{5}}{Q}.$ Since $2^{k}\lesssim M$, this term is $O(M^{6})$. ∎ ###### Remark 4.3. One may wonder whether there is a clever way to estimate the sum $\sum_{B\subset\Omega}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3},$ without using number theory. To this end, the most natural thing to try is to use Minkowski’s inequality and to bound this expression by $(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\int_{\Omega}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3}.$ (27) However, a change of variables as before shows that for each $J_{1},J_{2}$ $\displaystyle\int_{\Omega}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$ $\displaystyle\geq N^{-1/2}\int_{[0,N^{1/2}]^{4}}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$ $\displaystyle\sim N^{-1/2}N^{2}[\int_{(u,w)\in[0,N^{1/2}]\times[0,N^{-1/2}]}\|\sum_{m=1}^{N^{1/2}}e(mu+m^{2}w)\|^{6}dudw]^{2}$ $\displaystyle\sim N^{11/2}.$ Using this approach, the upper bound we get for (27) is $N^{8+\frac{1}{2}}$. As in our earlier attempt to use $l^{6}$ rather than $l^{2}$ decoupling, this estimate falls short by a factor of $N^{1/2}$ from the sharp upper bound $N^{8}$. Also, due to (17), the expression (27) is only getting larger if $J_{i}$ are replaced with smaller intervals. Thus, decoupling on cubes larger than $B$ (such as $N^{1/2}$-cubes) only worsens our upper bound. A similar computation shows that the only case of Conjecture 1.1 that can be approached with $l^{6}L^{6}$ decoupling is the case $\alpha=2$ discussed in the previous section. ## 5\. The case $\frac{3}{2}<\alpha\leq\frac{9}{5}$ Let $\Omega=[0,N^{\frac{2\alpha}{3}-1}]\times[0,N]\times[N^{\frac{1}{2}},N^{\alpha-1}]\times[0,N^{\beta-1}]$. Using 1-periodicity in $x_{1}$, we need to prove that $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{7+\frac{2\alpha}{3}+\epsilon}.$ We cover $\Omega$ with cubes $B$ of side length $N^{\frac{2\alpha}{3}-1}$ and write $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\sim\sum_{B\subset\Omega}\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}.$ The size of these cubes is the smallest that will make cubic terms negligible after the decoupling. Since we need $\frac{2\alpha}{3}-1\leq\beta-1$ in order to not exceed $\Omega$, this leads to the restriction $\alpha\leq\frac{9}{5}$. Note also that the $x_{3}$ coordinate is $\geq N^{1/2}$. We can afford this omission because of the $\alpha=\frac{3}{2}$ case discussed in the previous section. Since we are about to decouple on cubes $B$ with size larger than 1, Remark 4.3 tells us that applying this method for $x$ near the origin leads to losses. Our next argument will make explicit use of the fact that $x_{3}$ is away from the origin. We use $l^{2}$ decoupling (Theorem 2.3) on each $B$ (or rather $NB$, after rescaling) $\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3}$ where $J_{i}$ is of the form $[h_{i},h_{i}+M]$, with $M=N^{1-\frac{\alpha}{3}}$. We write $B$ as $[0,N^{\frac{2\alpha}{3}-1}]\times[pN^{\frac{2\alpha}{3}-1},(p+1)N^{\frac{2\alpha}{3}-1}]\times[l_{1}N^{\frac{2\alpha}{3}-1},(l_{1}+1)N^{\frac{2\alpha}{3}-1}]\times[l_{2}N^{\frac{2\alpha}{3}-1},(l_{2}+1)N^{\frac{2\alpha}{3}-1}]$ with the integers $0\leq p\leq M^{2}$, $N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq l_{1}\leq N^{\frac{\alpha}{3}}$ and $0\leq l_{2}\leq N^{3-\frac{5\alpha}{3}}$. Since $\alpha>\frac{3}{2}$, we have that $l_{1}\gg l_{2}$. We let as before $I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M^{2}},\frac{a+O(1)}{M^{2}}]}\|\sum_{m\leq M}e(mu+m^{2}w)\|^{6}dudw$ With a change of variables as in the previous section, we have $\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}\sim N^{2}(N^{\frac{2\alpha}{3}-1})^{2}I_{a_{1}}I_{a_{2}}$ where $\begin{cases}a_{1}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{1}}{N})\\\ a_{2}=p+\frac{l_{1}}{2}\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})+\frac{l_{2}}{2}\phi_{4}^{\prime\prime}(\frac{h_{2}}{N})\end{cases}.$ (28) It is crucial that the cubic (and also the higher order) term is $O(1)$, cf. (24) $m^{3}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h}{N}){x_{3}}+\phi_{4}^{\prime\prime\prime}(\frac{h}{N}){x_{4}}}{N^{2}}=O(1),\;\;\forall x\in\Omega,$ so it may be neglected according to Lemma 9.2. If $l_{1}\sim 2^{j_{1}}$, it is immediate that $\|a_{1}-p\|\lesssim 2^{j_{1}}$, $\|a_{2}-p\|\lesssim 2^{j_{1}}$. Also, for fixed $a_{1},a_{2},p,l_{1},l_{2}$, (28) has $O((\frac{N}{M2^{j_{1}}})^{2})$ solutions $(h_{1},h_{2})$, modulo $O(1)$. We do not need Lemma 4.1 here, since this time $l_{1}$ is much larger than $l_{2}$. We now dominate $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}$ as before by $\lesssim_{\epsilon}N^{\epsilon}\sum_{j_{1}:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq 2^{j_{1}}\leq N^{\frac{\alpha}{3}}}\sum_{l_{1}\sim 2^{j_{1}}}\sum_{j_{2}:\;2^{j_{2}}\leq N^{3-\frac{5\alpha}{3}}}\sum_{l_{2}\sim 2^{j_{2}}}(\frac{N}{M2^{j_{1}}})^{6}N^{2}(N^{\frac{2\alpha}{3}-1})^{2}\sum_{p\leq M^{2}}(\sum_{a:\;\|a-p\|\lesssim 2^{j_{1}}}I_{a}^{1/3})^{6}.$ We use Cauchy–Schwarz for the last expression to dominate the above by $N^{\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq 2^{j}\leq N^{\frac{\alpha}{3}}}2^{j}N^{3-\frac{5\alpha}{3}}2^{-6j}N^{2\alpha}N^{2}(N^{\frac{2\alpha}{3}-1})^{2}2^{j}2^{3j}\sum_{\|H\|\sim 2^{j}}(\sum_{a\in H}I_{a}^{2/3})^{3}$ $=N^{3+\frac{5\alpha}{3}+\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq 2^{j}\leq N^{\frac{\alpha}{3}}}2^{-j}\sum_{\|H\|\sim 2^{j}}(\sum_{a\in H}I_{a}^{2/3})^{3}.$ Using Lemma 4.2, this is dominated by $N^{3+\frac{5\alpha}{3}+\epsilon}\sum_{j:\;N^{\frac{3}{2}-\frac{2\alpha}{3}}\leq 2^{j}\leq N^{\frac{\alpha}{3}}}(M^{4}2^{j}+M^{6}2^{-j})\lesssim N^{\epsilon}(N^{7+\frac{2\alpha}{3}}+N^{\frac{15}{2}+\frac{\alpha}{3}}).$ This is $O(N^{7+\frac{2\alpha}{3}+\epsilon})$, as desired, since $\alpha>\frac{3}{2}$. ## 6\. The case $\frac{9}{5}\leq\alpha<2$ Let $\Omega=[0,N^{\delta}]\times[0,N]\times[N^{\frac{4}{5}},N^{\alpha-1}]\times[0,N^{\beta-1}].$ Because of the case addressed in the previous section, we may assume $x_{3}\geq N^{\frac{4}{5}}$. This will buy us some extra flexibility in choosing $\delta$. In fact, we can work with any $\delta$ satisfying $2-\frac{3\beta}{2}\leq\delta\leq\frac{9}{5}-\beta.$ (29) We need to prove that $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{8+\delta+\epsilon}.$ We will first decouple on cubes $B$ with side length $N^{\beta-1}$. This is the largest size that is available to us, due to the range in the $x_{4}$ variable. Unlike the case from the previous section, the resulting intervals are not small enough to make the cubic terms negligible, to allow us to use estimates for quadratic Weyl sums. We will accomplish that by means of a further decoupling, on cubes of side length $N^{\delta}$, similar to the case $\alpha=2$ described earlier. To get started, we use $l^{2}$ decoupling (Theorem 2.3) on each cube $B$ of side length $N^{\beta-1}$ $\int_{B}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3},$ where $J_{i}=[h_{i},h_{i}+M]$ has length $M=N^{1-\frac{\beta}{2}}$. Next, we cover $\Omega$ with boxes $\Delta=[0,N^{\delta}]\times[pN^{\delta},(p+1)N^{\delta}]\times[lN^{\delta},(l+1)N^{\delta}]\times[0,N^{\beta-1}],$ with $p\leq N^{1-\delta}$, $N^{\frac{4}{5}-\delta}\leq l\leq N^{\alpha-1-\delta}$. If we sum up the above inequality over cubes $B\subset\Delta$ and use Minkowski’s inequality, we find $\int_{\Delta}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}N^{\epsilon}(\sum_{J_{1}\subset I_{1}}\sum_{J_{2}\subset I_{2}}(\sum_{B\subset\Delta}\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6})^{1/3})^{3}.$ (30) Next, we fix $J_{1},J_{2}$ and perform a second decoupling for the term $\int_{\Delta}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$. We proceed as in Section 3 $\displaystyle\int_{B}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}$ $\displaystyle=N^{-4}\int_{NB}\|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N}){\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})\|^{6}$ $\displaystyle\lesssim N^{-4-4\beta}\int_{NB}\|{\mathcal{E}}_{J_{1}}(\frac{\cdot}{N})\|^{6}\int_{NB}\|{\mathcal{E}}_{J_{2}}(\frac{\cdot}{N})\|^{6}=N^{4-4\beta}\int_{B}\|{\mathcal{E}}_{J_{1}}\|^{6}\int_{B}\|{\mathcal{E}}_{J_{2}}\|^{6}.$ Then $\sum_{B\subset\Delta}\int_{B}\|{\mathcal{E}}_{J_{1}}\|^{6}\int_{B}\|{\mathcal{E}}_{J_{2}}\|^{6}\lesssim N^{4-4\beta}\int_{\Delta}dx\int_{(y,z)\in[0,N^{\beta-1}]^{4}\times[0,N^{\beta-1}]^{4}}\|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)\|^{6}dydz.$ Combining these two and using periodicity in the $y_{1},z_{1}$ variables we get $\int_{\Delta}\|{\mathcal{E}}_{J_{1}}{\mathcal{E}}_{J_{2}}\|^{6}\lesssim N^{6-6\beta-2\delta}\times$ $\int_{(x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4})\in S}dx_{1}\ldots dz_{4}\int_{y_{1},z_{1}\in[0,N^{\delta}]\atop{x_{2}\in[pN^{\delta},(p+1)N^{\delta}]\atop{x_{3}\in[lN^{\delta},(l+1)N^{\delta}]}}}\|{\mathcal{E}}_{J_{1}}(x+y){\mathcal{E}}_{J_{2}}(x+z)\|^{6}dy_{1}dz_{1}dx_{2}dx_{3},$ where $S$ is characterized by $0\leq x_{1}\in[0,N^{\delta}],\;x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}\in[0,N^{\beta-1}].$ We seek to estimate the second integral uniformly over $x_{1},x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$. With these variables fixed, we make the affine change of variables $(y_{1},z_{1},x_{2},x_{3})\mapsto(u_{1},u_{2},w_{1},w_{2})$ $\begin{cases}u_{1}=(y_{1}+x_{1})+\frac{2h_{1}}{N}(x_{2}+y_{2})+\phi_{3}^{\prime}(\frac{h_{1}}{N})(x_{3}+y_{3})+\phi_{4}^{\prime}(\frac{h_{1}}{N})(x_{4}+y_{4})\\\ u_{2}=(z_{1}+x_{1})+\frac{2h_{2}}{N}(x_{2}+z_{2})+\phi_{3}^{\prime}(\frac{h_{2}}{N})(x_{3}+z_{3})+\phi_{4}^{\prime}(\frac{h_{2}}{N})(x_{4}+y_{4})\\\ w_{1}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})\frac{x_{3}}{2N}\\\ w_{2}=\frac{x_{2}}{N}+\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})\frac{x_{3}}{2N}\end{cases}.$ The Jacobian is $\sim\frac{1}{N^{2}}$, due to (3). The second integral is comparable to $N^{2}\int_{(u_{i},w_{i})\in[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{}^{2}},\frac{a_{i}+O(1)}{M_{}^{2}}]}\prod_{i=1}^{2}\|\sum_{m_{i}\leq M}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})\|^{6}du_{1}dw_{1}du_{2}dw_{2}.$ (31) Here $M_{}=N^{\frac{1}{2}-\frac{\delta}{2}}$, $a_{i}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{i}}{N})}{2}$ and $\eta_{i}(m)={m^{3}}\frac{\phi_{3}^{\prime\prime\prime}(\frac{h_{i}}{N})}{3!N^{2}}+{m^{4}}\frac{\phi_{3}^{\prime\prime\prime\prime}(\frac{h_{i}}{N})}{4!N^{3}}+\ldots$. Note that since $\frac{v}{N}=O(M^{-2})$ for $v$ equal to any of the variables $x_{4},y_{2},y_{3},y_{4},z_{2},z_{3},z_{4}$, we have dismissed the contribution of these variables associated with quadratic (as well as the higher order) terms. See Lemma 9.2. We may apply again Theorem 2.4, using that $x_{3}=A(w_{1}-w_{2})$ with $A=O(N)$. Note however that this time we cannot decouple into point masses (as in (10)), since $M_{}$ is significantly larger than 1. Instead, applying (11) with $N=(\frac{M}{M_{}})^{2}$ we dominate (31) by $N^{2+\epsilon}\times$ (32) $(\sum_{J_{1}^{\prime},J_{2}^{\prime}}[\int_{(u_{i},w_{i})\in[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{}^{2}},\frac{a_{i}+O(1)}{M_{}^{2}}]}\prod_{i=1}^{2}\|\sum_{m_{i}\in J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})\|^{6}du_{1}dw_{1}du_{2}dw_{2}]^{\frac{1}{3}})^{3}.$ The intervals $J_{i}^{\prime}$ partitioning $[1,M]$ have length $M_{}$. What we have gained by doing this decoupling is that, when $m_{i}$ is confined to a small interval $J_{i}^{\prime}=[h_{i}^{\prime},h_{i}^{\prime}+M_{}]$, the contribution of the term $\eta_{i}(m_{i})=\eta_{i}(h_{i}^{\prime}+m_{i}^{\prime})=\eta_{i}(h_{i}^{\prime})+\eta_{i}^{\prime}(h_{i}^{\prime})m_{i}^{\prime}+\eta_{i}^{\prime\prime}(h_{i}^{\prime})\frac{(m_{i}^{\prime})^{2}}{2}+O(\frac{(m_{i}^{\prime})^{3}}{N^{2}})$ (33) can be neglected. To see this, note first that $\eta_{i}^{\prime\prime}(h_{i}^{\prime})=\sum_{n\geq 3}\phi_{3}^{(n)}(\frac{h_{i}}{N})\frac{(h_{i}^{\prime})^{n-2}}{N^{n-1}(n-2)!}.$ Making another linear change of variables such that $w_{i}^{\prime}=w_{i}+\frac{\eta_{i}^{\prime\prime}(h_{i}^{\prime})}{2}A(w_{1}-w_{2}),$ we write, using that $\|x_{3}\|\leq N^{\alpha-1}$ $\prod_{i=1}^{2}\|\sum_{m_{i}\in J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i}+\eta_{i}(m_{i})x_{3})\|=\prod_{i=1}^{2}\|\sum_{m_{i}^{\prime}\in[1,M_{}]}e(m_{i}^{\prime}u_{i}^{\prime}+(m_{i}^{\prime})^{2}w_{i}^{\prime}+O((m_{i}^{\prime})^{3}\frac{N^{\alpha-1}}{N^{2}}))\|.$ The range of $w_{i}^{\prime}$ is (a subset of) $[\frac{a_{i}^{\prime}-O(1)}{M_{}^{2}},\frac{a_{i}^{\prime}+O(1)}{M_{}^{2}}]$, where $a_{i}^{\prime}=p+\frac{l}{2}\sum_{n\geq 2}\phi_{3}^{(n)}(\frac{h_{i}}{N})\frac{(h_{i}^{\prime})^{n-2}}{N^{n-2}(n-2)!}=p+l\frac{\phi_{3}^{{}^{\prime\prime}}(\frac{h_{i}+h_{i}^{\prime}}{N})}{2}.$ Since $\alpha-1-\frac{\beta}{2}\leq\delta$ by (29), we have that $a_{i}-a_{i}^{\prime}=O(1)$. Thus, the quadratic term in (33) will not affect the domain of integration. Moreover, the contribution of the higher order terms in (33) is negligible (cf. Lemma 9.2), as long as we can guarantee that we have $\frac{N^{\alpha-1}}{N^{2}}=O(M_{}^{-3})$. This is equivalent to $\delta\geq 1-\frac{2\beta}{3}$, and follows from (29) and the fact that $\beta\leq\frac{6}{5}$. Under this assumption, we dominate (32) by $N^{2+\epsilon}\left(\sum_{J_{1}^{\prime},J_{2}^{\prime}}\left[\prod_{i=1}^{2}\int_{[0,N^{\delta}]\times[\frac{a_{i}-O(1)}{M_{}^{2}},\frac{a_{i}+O(1)}{M_{}^{2}}]}\|\sum_{m_{i}\in J_{i}^{\prime}}e(m_{i}u_{i}+m_{i}^{2}w_{i})\|^{6}du_{i}dw_{i}\right]^{\frac{1}{3}}\right)^{3}.$ This is $\sim N^{2+2\delta+\epsilon}(\frac{M}{M_{}})^{6}I_{a_{1}}I_{a_{2}}$, where $I_{a}=\int_{[0,1]\times[\frac{a-O(1)}{M_{}^{2}},\frac{a+O(1)}{M_{}^{2}}]}\|\sum_{m\leq M_{}}e(mu+m^{2}w)\|^{6}dudw$ is independent of $J_{1}^{\prime},J_{2}^{\prime}$. Recall that $\begin{cases}a_{1}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{1}}{N})}{2}\\\ a_{2}=p+l\frac{\phi_{3}^{\prime\prime}(\frac{h_{2}}{N})}{2}\end{cases}.$ (34) Assume now that $l\sim 2^{j}$, with $N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}.$ For fixed $p,l$, and fixed $(a_{1},a_{2})$ (within a factor of $O(1)$), the system (34) has $\lesssim(\frac{N}{2^{j}M})^{2}$ solutions $(h_{1},h_{2})$. Getting back to (30), summing over $\Delta\subset\Omega$ we find that $\int_{\Omega}\|{\mathcal{E}}_{I_{1}}{\mathcal{E}}_{I_{2}}\|^{6}\lesssim_{\epsilon}$ $N^{6-6\beta-2\delta}\|S\|N^{2+2\delta+\epsilon}N^{3+3\delta}\sum_{p\leq M_{}^{2}}\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}}2^{-6j}\sum_{l\sim 2^{j}}(\sum_{\|a-p\|\lesssim 2^{j}}I_{a}^{1/3})^{6}.$ We use Cauchy–Schwarz to dominate this by $N^{4+4\delta+\beta+\epsilon}\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}}\sum_{H\subset[1,M_{}^{2}]\atop{\|H\|=2^{j}}}2^{-j}(\sum_{a\in H}I_{a}^{2/3})^{3}.$ Using Lemma 4.2, it remains to check that $\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}}M_{}^{4}2^{j}+\sum_{N^{\frac{4}{5}-\delta}\lesssim 2^{j}\lesssim N^{\alpha-1-\delta}}M_{}^{6}2^{-j}\lesssim N^{4-\beta-3\delta}.$ The first sum is in order, since $\alpha+\beta=3$. So is the second sum, as long as $\delta\leq\frac{9}{5}-\beta$, which is guaranteed by (29). ## 7\. Proof of Theorem 1.2 This section shows that Theorem 1.3 implies Theorem 1.2. The argument is inspired by [2]. The parameter $K$ will be very large and universal, independent of $N$, $\phi_{k}$. The larger the $K$ we choose to work with, the smaller the $\epsilon$ from the $N^{\epsilon}$ loss will be at the end of the section. ###### Proposition 7.1. Assume $\alpha+\beta=3$ and $\frac{3}{2}\leq\alpha\leq 2.$ Assume $\phi_{3},\phi_{4}:(0,3)\to{\mathbb{R}}$ are real analytic and satisfy (1), (2) and (3). Let as before $\omega_{3}=[0,N^{\alpha}]$, $\omega_{4}=[0,N^{\beta}]$ and ${\mathcal{E}}_{I,N}(x)=\sum_{n\in I}e(nx_{1}+n^{2}x_{2}+\phi_{3}(\frac{n}{N})x_{3}+\phi_{4}(\frac{n}{N})x_{4}).$ We consider arbitrary integers $N_{0},M$ satisfying $1\leq M\leq\frac{N_{0}}{K}$ and $N_{0}+[M,2M]\subset[\frac{N}{2},N]$. Let $H_{1},H_{2}$ be intervals of length $\frac{M}{K}$ inside $N_{0}+[M,2M]$ such that ${\operatorname{dist}\,}(H_{1},H_{2})\geq\frac{M}{K}$. Then $\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{6}\lesssim_{\epsilon}N^{7}M^{2+\epsilon}.$ ###### Proof. Write $H_{1}=N_{0}+I_{1}$, $H_{2}=N_{0}+I_{2}$ with $I_{1},I_{2}$ intervals of length $\frac{M}{K}$ inside $[M,2M]$ and with separation $\geq\frac{M}{K}$. We use the following expansion, certainly valid for all $m$ in $I_{i}$. $\displaystyle\phi_{3}(\frac{N_{0}+m}{N})$ $\displaystyle=Q_{3}(m)+\sum_{n\geq 3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})}{n!}(\frac{m}{N})^{n}$ $\displaystyle=Q_{3}(m)+(\frac{M}{N})^{3}\sum_{n\geq 3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-3}}{n!}(\frac{m}{M})^{n}.$ Here $Q_{3}(m)=A+Bm+Cm^{2}$ with $B=O(\frac{1}{N})$, $C=O(\frac{1}{N^{2}})$. We introduce the analogue $\tilde{\phi_{3}}$ of $\phi_{3}$ at scale $M$ $\tilde{\phi_{3}}(t)=\sum_{n\geq 3}\frac{\phi_{3}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-3}}{n!}t^{n}.$ This series is convergent as long as $\frac{N_{0}}{N}+t\in(0,3)$, so the new function is certainly real analytic on $(0,2)$, since $N_{0}\leq N$. Let $\delta>0$ be conveniently small. By choosing $K$ large enough we can make $\frac{M}{N}$ as small as we wish, so we may guarantee that for each $t\in[\frac{1}{2},1]$ we have $\|\tilde{\phi}_{3}^{(3)}(t)-{\phi}_{3}^{(3)}(\frac{N_{0}}{N})\|\leq\delta.$ (35) Thus, we can guarantee (3) for $\tilde{\phi}_{3}$, with a slightly smaller, but uniform value of $A_{4}$. The same will work with (1) and (2), as it will soon become clear. To this end, we may also enforce $\|\tilde{\phi}_{3}^{(4)}(t)\|\leq\delta.$ (36) We also define, with $Q_{4}(m)=D+Em+Fm^{2}$ satisfying $E=O(\frac{1}{N})$, $F=O(\frac{1}{N}^{2})$ $\displaystyle\phi_{4}(\frac{N_{0}+m}{N})$ $\displaystyle=Q_{4}(m)+\sum_{n\geq 3}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})}{n!}(\frac{m}{N})^{n}$ $\displaystyle=Q_{4}(m)+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{3!}(\frac{m}{N})^{3}+(\frac{M}{N})^{4}\sum_{n\geq 4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})(\frac{M}{N})^{n-4}}{n!}(\frac{m}{M})^{n}.$ The last two terms are equal to $\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}(\frac{M}{N})^{3}\tilde{\phi}_{3}(\frac{m}{M})+(\frac{M}{N})^{4}\sum_{n\geq 4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(n)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})n!}(\frac{M}{N})^{n-4}(\frac{m}{M})^{n}.$ Let $\tilde{\phi}_{4}$ be the analogue of $\phi_{4}$ at scale $M$ defined by $\tilde{\phi}_{4}(t)=\sum_{n\geq 4}\frac{\phi_{4}^{(n)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(n)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})n!}(\frac{M}{N})^{n-4}t^{n}.$ As before, by choosing $K$ large enough, we can arrange that for all $t\in[\frac{1}{2},1]$ $\|\tilde{\phi}_{4}^{(4)}(t)-\frac{\phi_{4}^{(4)}(\frac{N_{0}}{N})\phi_{3}^{(3)}(\frac{N_{0}}{N})-\phi_{4}^{(3)}(\frac{N_{0}}{N})\phi_{3}^{(4)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}\|\leq\delta.$ Combining this with (35) and (36) we may arrange that $\det\begin{bmatrix}\tilde{\phi}_{3}^{(3)}(t)&\tilde{\phi}_{3}^{(4)}(t)\\\ \tilde{\phi}_{4}^{(3)}(s)&\tilde{\phi}_{4}^{(4)}(s)\end{bmatrix}-\det\begin{bmatrix}\phi_{3}^{(3)}(\frac{N_{0}}{N})&\phi_{3}^{(4)}(\frac{N_{0}}{N})\\\ \phi_{4}^{(3)}(\frac{N_{0}}{N})&\phi_{4}^{(4)}(\frac{N_{0}}{N})\end{bmatrix}$ is as small in absolute value as we wish, uniformly over $t,s\in[\frac{1}{2},1]$. In particular, we can guarantee (2) for the pair $(\tilde{\phi}_{3},\tilde{\phi}_{4})$, with slightly modified, but uniform values of $A_{2},A_{3}$. Similar comments apply regarding (1). Now $\displaystyle\|{\mathcal{E}}_{H_{k},N}(x)\|$ $\displaystyle=\|\sum_{m\in I_{k}}e(mx_{1}+(m^{2}+2N_{0}m)x_{2}+\phi_{3}(\frac{N_{0}+m}{N})x_{3}+\phi_{4}(\frac{N_{0}+m}{N})x_{4})\|$ $\displaystyle=\|\sum_{m\in I_{i}}e(m(x_{1}+2N_{0}x_{2}+Bx_{3}+Ex_{4})+m^{2}(x_{2}+Cx_{3}+Fx_{4})$ $\displaystyle+(\frac{M}{N})^{3}\tilde{\phi}_{3}(\frac{m}{M})(x_{3}+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}x_{4})$ $\displaystyle+(\frac{M}{N})^{4}\tilde{\phi}_{4}(\frac{m}{M})x_{4})\|.$ Recall $N_{0}\sim N$, $B,E=O(1/N)$, $C,F=O(1/N^{2})$. We make the change of variables $\begin{cases}y_{1}=x_{1}+2N_{0}x_{2}+Bx_{3}+Ex_{4}\\\ y_{2}=x_{2}+Cx_{3}+Fx_{4}\\\ y_{3}=(\frac{M}{N})^{3}(x_{3}+\frac{\phi_{4}^{(3)}(\frac{N_{0}}{N})}{\phi_{3}^{(3)}(\frac{N_{0}}{N})}x_{4})\\\ y_{4}=(\frac{M}{N})^{4}x_{4}\end{cases}.$ Due to periodicity, we may extend the range of $x_{1}$ to $[0,N_{0}]$. This linear transformation maps $[0,N_{0}]\times[0,1]\times\omega_{3}\times\omega_{4}$ to a subset of a box $\tilde{\omega}_{1}\times\tilde{\omega}_{2}\times\tilde{\omega}_{3}\times\tilde{\omega_{4}}$ centered at the origin, with dimensions roughly $N_{0},1,M^{3}N^{\alpha-3},M^{4}N^{\beta-4}$. Thus $\|{\mathcal{E}}_{H_{k},N}(x)\|=\|{\mathcal{E}}_{I_{k},M}(y)\|$ where ${\mathcal{E}}_{I_{k},M}(y)=\sum_{m\in I_{k}}e(my_{1}+m^{2}y_{2}+\tilde{\phi}_{3}(\frac{m}{M})y_{3}+\tilde{\phi}_{4}(\frac{m}{M})y_{4}).$ We may write, using again periodicity in $y_{1}$ and $y_{2}$ $\displaystyle\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{6}$ $\displaystyle=\frac{1}{N_{0}}\int_{[0,N_{0}]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{6}$ $\displaystyle\leq(\frac{N}{M})^{7}\int_{[0,1]\times[0,1]\times\tilde{\omega}_{3}\times\tilde{\omega_{4}}}\|{\mathcal{E}}_{I_{1},M}(y){\mathcal{E}}_{I_{2},M}(y)\|^{6}.$ Finally, we use Theorem 1.3 with $N=M$, noting that $\tilde{\omega}_{3}\subset[-M^{\alpha},M^{\alpha}]$ and $\tilde{\omega}_{4}\subset[-M^{\beta},M^{\beta}]$, to estimate the last expression by $(\frac{N}{M})^{7}M^{9+\epsilon}=N^{7}M^{2+\epsilon}.$ ∎ We can now prove Theorem 1.2. Choose $K$ large enough, depending on $\epsilon$. Write ${\mathcal{H}}_{n}(I)$ for the collection of dyadic intervals in $I$ with length $\frac{N}{2K^{n}}$. We write $H_{1}\not\sim H_{2}$ to imply that $H_{1},H_{2}$ are not neighbors. Then $\|{\mathcal{E}}_{I,N}(x)\|\leq 3\max_{H\in{\mathcal{H}}_{1}(I)}\|{\mathcal{E}}_{H,N}(x)\|+K^{10}\max_{H_{1}\not\sim H_{2}\in{\mathcal{H}}_{1}(I)}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{1/2}.$ We repeat this inequality until we reach intervals in ${\mathcal{H}}_{l}$ of length $\sim 1$, that is $K^{l}\sim N$. We have $\displaystyle\|{\mathcal{E}}_{I,N}(x)\|$ $\displaystyle\lesssim 3^{l}+l3^{l}K^{10}\max_{1\leq n\leq l}\max_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim H_{2}\in{\mathcal{H}}_{n+1}(H)}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{1/2}$ $\displaystyle\lesssim(\log N)N^{\log_{K}3}\max_{1\leq n\leq l}\max_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim H_{2}\in{\mathcal{H}}_{n+1}(H)}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{1/2}.$ Using Corollary 7.1 we finish the proof $\displaystyle\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}$ $\displaystyle\|{\mathcal{E}}_{I,N}\|^{12}$ $\displaystyle\lesssim_{K}N^{\log_{K}3}\sum_{n}\sum_{H\in{\mathcal{H}}_{n}(I)}\max_{H_{1}\not\sim H_{2}\in{\mathcal{H}}_{n+1}(H)}\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{H_{1},N}(x){\mathcal{E}}_{H_{2},N}(x)\|^{6}$ $\displaystyle\lesssim_{K,\epsilon}N^{\epsilon+\log_{K}3}\sum_{n}{K^{n}}N^{7}(\frac{N}{K^{n}})^{2}$ $\displaystyle\lesssim_{K,\epsilon}N^{\epsilon+\log_{K}3}N^{9}.$ Choosing $K$ large enough, we may make $\log_{K}3$ as small as we wish. ## 8\. Other values of $p$ The reason Theorem 1.2 was accessible via the bilinear result in Theorem 1.3 has to do with the fact that $6$ is the critical exponent for the decoupling for the parabola (at the canonical scale). Thus, our arguments rely fundamentally on this dimensional reduction. In [6], the small cap decoupling for the parabola is settled, and the associated critical exponents lie between 4 and 6. In principle, this new tool can be used to determine $L^{p}$ moments for curves in ${\mathbb{R}}^{4}$, in the range $8\leq p\leq 12$. There are many possible things to consider in this direction. One is the following extension of Conjecture 1.1, that we use to illustrate a different type of obstruction that appears in this regime. This was observed in [1], in a related context. ###### Conjecture 8.1 (Square root cancellation in $L^{p}$). Let $11\leq p\leq 12$. Assume $\alpha\geq\beta\geq 0$ satisfy $\alpha+\beta=\frac{p}{2}-3$. Let $\phi_{3}$, $\phi_{4}$, $\omega_{3}$, $\omega_{4}$ be as in Theorem 1.2. Then $\int_{[0,1]\times[0,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{[\frac{N}{2},N],N}\|^{p}\lesssim_{\epsilon}N^{p-3+\epsilon}.$ The case $\beta=0$ was proved in [6] in the larger range $9\leq p<12$. As mentioned earlier, when $\beta=0$, the curve collapses to a three dimensional curve. However, the next result shows that the restriction $p\geq 11$ is needed if $\beta>0.$ The new obstruction can be described as constructive interference on spatially disjoint blocks. ###### Theorem 8.2. Assume $p<11$. Let $\omega_{3}=[-N^{\alpha},N^{\alpha}]$, $\omega_{4}=[-N^{\beta},N^{\beta}]$, $\alpha\geq\beta$ and $\alpha+\beta=\frac{p}{2}-3$. Assume also that $\beta>0$. Then, for some $\delta>0$ and $\phi_{3}(t)=t^{3}$, $\phi_{4}(t)=t^{4}$ we have $\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{[\frac{N}{2},N],N}\|^{p}\gtrsim N^{p-3+\delta}.$ ###### Proof. Lemma 8.3 shows that the integral is greater than $\sum_{J\subset I}\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{J,N}\|^{p},$ where the sum runs over intervals $J$ of length $M<N$, partitioning $[\frac{N}{2},N]$. The parameter $M$ will be determined later. In some sense, the components ${\mathcal{E}}_{J,N}$ behave as if they were spatially supported on pairwise disjoint sets. By periodicity $\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{J,N}\|^{p}=N^{-3}\int_{[-N^{2},N^{2}]\times[-N,N]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{J,N}\|^{p}.$ Write $H=[h+1,h+M]$. Note that $\|{\mathcal{E}}_{J,N}(x)\|=\|\sum_{1\leq m\leq M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})\|$ where $\begin{cases}y_{1}=x_{1}+2h{x_{2}}+\frac{3h^{2}}{N^{3}}x_{3}+\frac{4h^{3}}{N^{4}}x_{4}\\\ y_{2}=x_{2}+\frac{3h}{N^{3}}x_{3}+\frac{6h^{2}}{N^{4}}x_{4}\\\ y_{3}=\frac{x_{3}}{N^{3}}+\frac{4h}{N^{4}}x_{4}\\\ y_{4}=\frac{x_{4}}{N^{4}}\end{cases}.$ This change of variables maps $[-N^{2},N^{2}]\times[-N,N]\times\omega_{3}\times\omega_{4}$ to a set containing $S=[-o(N^{2}),o(N^{2})]\times[-o(N),o(N)]\times[-o(N^{\alpha-3}),o(N^{\alpha-3})]\times[-o(N^{\beta-4}),o(N^{\beta-4})].$ We have used that $3\geq\alpha\geq\beta-1$. Thus $\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{J,N}(x)\|^{p}dx$ $\geq N^{4}\int_{S}\|\sum_{1\leq m\leq M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})\|^{p}dy.$ Let $M=\max\\{N^{1-\frac{\alpha}{3}},N^{1-\frac{\beta}{4}}\\}.$ Note that since $\beta>0$ we have that $M=N^{1-\epsilon}$ for some $\epsilon>0$. Note also that $[0,o(M^{-3})]\times[0,o(M^{-4})]\subset[0,o(N^{\alpha-3})]\times[0,o(N^{\beta-4})].$ Using constructive interference we get $\int_{S}\|\sum_{1\leq m\leq M}e(my_{1}+m^{2}y_{2}+m^{3}y_{3}+m^{4}y_{4})\|^{p}dy\gtrsim M^{p-10}N^{3}.$ Putting things together we conclude that $\int_{[-1,1]\times[-1,1]\times\omega_{3}\times\omega_{4}}\|{\mathcal{E}}_{[\frac{N}{2},N],N}\|^{p}\gtrsim\frac{N}{M}N^{4}M^{p-10}N^{3}.$ Note that this is $\geq N^{p-3+\delta}$, for some $\delta>0$. ∎ ###### Lemma 8.3. Let ${\mathcal{R}}$ be a collections of rectangular boxes $R$ in ${\mathbb{R}}^{n}$, with pairwise disjoint doubles $2R$. Let $F$ be a Schwartz function in ${\mathbb{R}}^{n}$ that can be written as $F=\sum_{R\in{\mathcal{R}}}F_{R},$ with the spectrum of $F_{R}$ inside $R$. Then for each $2\leq p\leq\infty$ we have $(\\|F_{R}\\|_{p})_{l^{p}({\mathcal{R}})}\lesssim\\|F\\|_{p}.$ The implicit constant is independent of $F$ and ${\mathcal{R}}$. ###### Proof. Interpolate between $2$ and $\infty$. ∎ ## 9\. Auxiliary results This section records two auxiliary results that are used repeatedly throughout the paper. ###### Lemma 9.1. Assume $t,s\in[\frac{1}{2},1]$ satisfy $\|t-s\|\sim 1$. The Jacobian of the transformation $y=\psi(x)$ $\begin{cases}y_{1}=x_{1}+2tx_{2}+\phi_{3}^{\prime}(t)x_{3}+\phi_{4}^{\prime}(t)x_{4}\\\ y_{2}=x_{1}+2sx_{2}+\phi_{3}^{\prime}(s)x_{3}+\phi_{4}^{\prime}(s)x_{4}\\\ y_{3}=\frac{2x_{2}}{N}+\phi_{3}^{\prime\prime}(t)\frac{x_{3}}{N}+\phi_{4}^{\prime\prime}(t)\frac{x_{4}}{N}\\\ y_{4}=\frac{2x_{2}}{N}+\phi_{3}^{\prime\prime}(s)\frac{x_{3}}{N}+\phi_{4}^{\prime\prime}(s)\frac{x_{4}}{N}\end{cases}$ is $\sim\frac{1}{N^{2}}.$ Moreover, $\psi$ maps cubes $Q$ with side length $L$ to subsets of rectangular boxes of dimensions roughly $L\times L\times\frac{L}{N}\times\frac{L}{N}$. If the cube $Q$ is centered at the origin, $\psi(Q)$ contains the rectangular box $[-o(L),o(L)]^{2}\times[-o(\frac{L}{N}),o(\frac{L}{N})]^{2}$. ###### Proof. Let $\phi_{1}(u)=u$, $\phi_{2}(u)={u^{2}}$. Then the Jacobian is $\frac{1}{N^{2}}\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\ \phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\ \phi_{1}^{\prime\prime}(t)&\phi_{2}^{\prime\prime}(t)&\phi_{3}^{\prime\prime}(t)&\phi_{4}^{\prime\prime}(t)\\\ \phi_{1}^{\prime\prime}(s)&\phi_{2}^{\prime\prime}(s)&\phi_{3}^{\prime\prime}(s)&\phi_{4}^{\prime\prime}(s)\end{array}\right].$ (37) Note that $\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\ \phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\ \phi_{1}^{\prime\prime}(t)&\phi_{2}^{\prime\prime}(t)&\phi_{3}^{\prime\prime}(t)&\phi_{4}^{\prime\prime}(t)\\\ \phi_{1}^{\prime\prime}(s)&\phi_{2}^{\prime\prime}(s)&\phi_{3}^{\prime\prime}(s)&\phi_{4}^{\prime\prime}(s)\end{array}\right]=\lim_{\epsilon\to 0}\frac{1}{\epsilon^{2}}\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\ \phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\ \phi_{1}^{\prime}(t+\epsilon)&\phi_{2}^{\prime}(t+\epsilon)&\phi_{3}^{\prime}(t+\epsilon)&\phi_{4}^{\prime}(t+\epsilon)\\\ \phi_{1}^{\prime}(s+\epsilon)&\phi_{2}^{\prime}(s+\epsilon)&\phi_{3}^{\prime}(s+\epsilon)&\phi_{4}^{\prime}(s+\epsilon)\end{array}\right].$ A generalization of the Mean-Value Theorem (see [9], Voll II, part V, Chap 1, No. 95) guarantees that $\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(t)&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\ \phi_{1}^{\prime}(s)&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\ \phi_{1}^{\prime}(t+\epsilon)&\phi_{2}^{\prime}(t+\epsilon)&\phi_{3}^{\prime}(t+\epsilon)&\phi_{4}^{\prime}(t+\epsilon)\\\ \phi_{1}^{\prime}(s+\epsilon)&\phi_{2}^{\prime}(s+\epsilon)&\phi_{3}^{\prime}(s+\epsilon)&\phi_{4}^{\prime}(s+\epsilon)\end{array}\right]=$ $\displaystyle=\epsilon^{2}(t-s)^{2}(t+\epsilon-s)(s+\epsilon-t)\operatorname{det}\left[\begin{array}[]{cccc}\phi_{1}^{\prime}(\tau_{1})&\phi_{2}^{\prime}(\tau_{1})&\phi_{3}^{\prime}(\tau_{1})&\phi_{4}^{\prime}(\tau_{1})\\\ \phi_{1}^{\prime\prime}(\tau_{2})&\phi_{2}^{\prime\prime}(\tau_{2})&\phi_{3}^{\prime\prime}(\tau_{2})&\phi_{4}^{\prime\prime}(\tau_{2})\\\ \phi_{1}^{\prime\prime\prime}(\tau_{3})&\phi_{2}^{\prime\prime\prime}(\tau_{3})&\phi_{3}^{\prime\prime\prime}(\tau_{3})&\phi_{4}^{\prime\prime\prime}(\tau_{3})\\\ \phi_{1}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{2}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{3}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{4}^{\prime\prime\prime\prime}(\tau_{4})\end{array}\right]$ $\displaystyle=\epsilon^{2}(t-s)^{2}(t+\epsilon-s)(s+\epsilon-t)\operatorname{det}\left[\begin{array}[]{cccc}\phi_{3}^{\prime\prime\prime}(\tau_{3})&\phi_{4}^{\prime\prime\prime}(\tau_{3})\\\ \phi_{3}^{\prime\prime\prime\prime}(\tau_{4})&\phi_{4}^{\prime\prime\prime\prime}(\tau_{4})\end{array}\right]$ for some $\tau_{i}\in[\frac{1}{2},1]$ depending on $t,s,\epsilon$. The conclusion follows by letting $\epsilon\to 0$ and using (2). The second statement is immediate. To prove the last one, assume $y\in[-cL,cL]^{2}\times[-c\frac{L}{N},c\frac{L}{N}]^{2},$ for some small enough $c$, independent of $N$. We need to prove that $y=\psi(x)$ for some $x\in Q$. This can be seen by solving for $x$. For example, $x_{1}\sim N^{2}\operatorname{det}\left[\begin{array}[]{cccc}y_{1}&\phi_{2}^{\prime}(t)&\phi_{3}^{\prime}(t)&\phi_{4}^{\prime}(t)\\\ y_{2}&\phi_{2}^{\prime}(s)&\phi_{3}^{\prime}(s)&\phi_{4}^{\prime}(s)\\\ y_{3}&\frac{\phi_{2}^{\prime\prime}(t)}{N}&\frac{\phi_{3}^{\prime\prime}(t)}{N}&\frac{\phi_{4}^{\prime\prime}(t)}{N}\\\ y_{4}&\frac{\phi_{2}^{\prime\prime}(s)}{N}&\frac{\phi_{3}^{\prime\prime}(s)}{N}&\frac{\phi_{4}^{\prime\prime}(s)}{N}\end{array}\right].$ This and (1) show that $\|x_{1}\|\lesssim N^{2}(\frac{\|y_{1}\|+\|y_{2}\|}{N^{2}}+\frac{\|y_{3}\|+\|y_{4}\|}{N}).$ The same inequality holds for all $x_{i}$, which proves the desired statement. ∎ ###### Lemma 9.2. Let $\gamma$ be a Schwartz function supported on $[-2,2]$. Define the smooth Weyl sums for $u,w,v\in{\mathbb{R}}$ $G(u,w,v)=\sum_{k\in{\mathbb{Z}}}\gamma(k/M)e(ku+k^{2}w+k^{3}v).$ Let $1\leq b\leq q\leq M$ with $(b,q)=1$. Assume that ${\operatorname{dist}\,}(w-\frac{b}{q},{\mathbb{Z}}):=\varphi\leq\frac{1}{qM}$ and that $\|v\|\lesssim\frac{1}{M^{3}}$. Then for each $\epsilon>0$ we have $\|G(u,w,v)\|\lesssim_{\epsilon}\frac{M^{\epsilon}}{q^{1/2}}\min\\{M,\frac{1}{\varphi^{1/2}}\\}$ if $u\in{\mathcal{M}}=\bigcup_{m\in{\mathbb{Z}}}[\frac{m}{q}-\varphi M^{1+\epsilon},\frac{m}{q}+\varphi M^{1+\epsilon}]$ and $\|G(u,w,v)\|\lesssim_{\epsilon}M^{-100}$ if $u\not\in{\mathcal{M}}.$ ###### Proof. Invoking periodicity, we may assume that $w=\frac{b}{q}+\varphi$ with $\|\varphi\|\leq\frac{1}{Mq}$. Using the representation $k=rq+k_{1}$, $0\leq k_{1}\leq q-1$ and the Poisson summation formula we get $G(u,w,v)=\sum_{k_{1}=0}^{q-1}e(k_{1}^{2}b/q)\sum_{r\in{\mathbb{Z}}}\gamma(\frac{k_{1}+rq}{M})e((rq+k_{1})u+(rq+k_{1})^{2}\varphi+(rq+k_{1})^{3}v)$ $=\sum_{m\in{\mathbb{Z}}}\left[\frac{1}{q}\sum_{k_{1}=0}^{q-1}e(k_{1}^{2}b/q-k_{1}m/q)\right]\left[\int_{\mathbb{R}}\gamma(y/M)e((u+\frac{m}{q})y+\varphi y^{2}+vy^{3})dy\right]$ $=\sum_{m\in{\mathbb{Z}}}S(b,m,q)J(u,v,\varphi,m,q)$ (38) where $S(b,m,q)=\frac{1}{q}\sum_{k=0}^{q-1}e(k^{2}b/q-km/q)$ $\displaystyle J(u,v,\varphi,m,q)$ $\displaystyle=\int_{{\mathbb{R}}}\gamma(y/M)e((u+\frac{m}{q})y+\varphi y^{2}+vy^{3})dy$ $\displaystyle=M\int_{{\mathbb{R}}}\gamma(z)e(M(u+\frac{m}{q})z+\varphi M^{2}z^{2}+vM^{3}z^{3})dz.$ If $\|\varphi\|\lesssim\frac{1}{M^{2}}$, we are content with the bound $\|J(u,v,\varphi,m,q)\|\lesssim M$. Assume now that $\|\varphi\|\gg\frac{1}{M^{2}}$. The classical van der Corput estimate (second derivative test) reads $\|\int_{{\mathbb{R}}}\gamma(z)e(Az+Bz^{2}+Cz^{3})dz\|\lesssim\|B\|^{-1/2},$ if $\|B\|\gg\|C\|$. In our case $\|B\|=\|\varphi\|M^{2}\gg 1\gtrsim\|vM^{3}\|=\|C\|$. In either case we get $\|J(u,v,\varphi,m,q)\|\lesssim\min\\{M,\|\varphi\|^{-1/2}\\}.$ On the other hand, repeated integration by parts (first derivative test) shows that for each $\alpha>0$ $\|J(u,v,\varphi,m,q)\|\lesssim_{\alpha}\frac{1}{A^{\alpha}}$ when $\|A\|=M\|u+\frac{m}{q}\|\geq M^{\epsilon}\varphi M^{2}$. Thus, when $u\in{\mathcal{M}}$, only $O(M^{\epsilon})$ values of $m$ will have a non- negligible contribution to the sum, while if $u\not\in{\mathcal{M}}$ then the contribution from all $m$ will be negligible. Combining these with the classical estimate $\|S(b,m,q)\|\lesssim\frac{1}{\sqrt{q}}$ finishes the argument. ∎ ## References [1] Bourgain, J. Decoupling inequalities and some mean-value theorems, J. Anal. 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The Clarendon Press, Oxford University Press, New York, 1996 * [9] Polya, G. and Szegö, G., Problems and Theorems in Analysis, Springer-Verlag, New York, 1976. * [10] Wooley, Trevor D. The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math. 294 (2016), 532-561
# Minimal instance with no weakly stable matching for three-sided problem with cyclic incomplete preferences E.Yu. Lerner, R.E. Lerner ###### Abstract Given $n$ men, $n$ women, and $n$ dogs, each man has an incomplete preference list of women, each woman does an incomplete preference list of dogs, and each dog does an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that each of them enters in the preference list of the corresponding agent. We do a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog which do not live together currently but each of them would become “happier” if they do. Otherwise the matching is said to be stable (a weakly stable matching in 3-DSMI-CYC problem). We give an example of this problem for $n=3$ where no stable matching exists. Moreover, we prove the absence of such an example for $n<3$. Such an example was known earlier only for $n=6$ (Biro, McDermid, 2010). The constructed examples also allows one to decrease (in two times) the dimension of the recently constructed analogous example for complete preference lists (Lam, Plaxton, 2019). ## 1 Introduction Assume that there are $n$ men and $n$ women, and each one among them has a preference list of representatives of the opposite sex. A partition into heterogeneous families with no tuple of man and woman, who prefer each other rather than their partners (if they have ones), is called stable matching. The initial case of complete preference lists was studied by D. Gale and L.S. Shaple: a stable matching necessarily exists, an $O(n^{2})$-hard algorithm for forming it was proposed in [4]. Note that the algorithm is also applicable for the case of incomplete preference lists, but some men and women, possibly, remain single. A certain modification of the Gale–Shaple algorithm (see, for example, [5]) allows one to find, if possible, a matching without single men and women or to prove its absence, otherwise. In the case of random complete preference lists (when for each man and each woman the distribution of all permutations of representatives of the opposite sex is independent and uniform) the time necessary for finding a stable matching is $\Theta(n\ln(n))$ [5]. In this case, the mean value of the number of stable matchings also has the asymptote $n\ln(n)$ [9]. In [5] D. Knuth states the question whether it is possible to generalize the theory of stable matchings to the case of three genders. The most interesting variant in the $k$-gender case occurs when preferences are cyclic: representatives of the 1st gender rank representatives of the 2nd one, the latter do representatives of the 3rd gender, etc., and each representative of the $k$th gender has a preference list of representatives of the 1st gender (see [7, Chapter 5.6] for the non-cyclic variants of the $k$-gender case). A tuple containing exactly one representative of each gender is called a family, and the set of disjoint families is called a matching. A matching is said to be weakly stable, if there is no tuple outside this matching, each member of which would become “happier”, if they live together. In what follows, for brevity, we use the term “a stable matching” instead of the term “a weakly stable matching”. Let the number of representatives of each gender equal $n$. In [2], one proves that with complete preference lists a stable matching always exists, provided that $n\leqslant k$ (where $k$ is the number of genders). In [3], Eriksson et al. generalize this result for the case when $k=3$ and $n=k+1=4$. Ibid, one states the conjecture that the problem of finding a stable matching in 3-gender case with complete preference lists (problem 3-DSM-CYC or just 3DSM) has a solution for any $n$. Using a satisfiability problem formulation and an extensive computer-assisted search, the authors of [8] prove the validity of the conjecture stated by Eriksson et al. for $n=5$. In [10], one proves that with random preference lists the mean value of stable matchings in problem 3DSM grows as $\Omega(n^{2}\ln^{2}(n))$. The 3DSMI-problem (3-dimensional stable matching with incomplete preference lists) was studied by P. Biró and E. McDermid [1]. According to results a solution of 3DSMI does not necessarily exists in contrast to the two- dimensional case; they give an explicit example of problem 3DSMI for $n=6$ with no stable matching. Moreover, they prove that the problem of establishing the solvability of 3DSMI is NP-complete. Ibid, they state the problem of constructing an instance with no weakly stable matching for $n<6$. Finally, contrary to expectations, the conjecture stated by Eriksson et al. was recently refuted in [6]. Lam and Paxton associate problem 3DSMI with a certain problem 3DSM, where $n$ is 15 times greater than the initial dimension; this problem is solvable if and only if so is the initial problem 3DSMI. Therefore, the problem of establishing the solvability of problem 3DSM is NP-complete. The example described in the paper [1] allows one to construct an instance of problem 3DSM for $n=90=6\times 15$ with no stable matching. For this reason, the problem of finding an instance of 3DSMI with no weakly stable matching for $n<6$ becomes more actual. The construction of such instances for the least possible values of $n$ is the goal of this paper. First we constructed an instance of 3DSMI problem for $n=4$ and proved the absence of such instances for $n<3$. But after failing to prove the absence of such instances for $n=3$, we have proposed an algorithm for the computer search of all possible instances of 3DSMI problems for $n=3$. Unexpectedly, the algorithm has succeeded in constructing instances of rather simple 3DSMI problems without weakly stable matching for $n=3$. The rest part of the paper has the following structure. In Sect. 2, we present the formal definitions of 3DSMI-CYC in terms of the graph theory. In Sect. 4, we study some properties of graphs of problem 3DSMI-CYC, prove the absence of counterexamples for $n<3$. In Sect. 3 we describe various cases of problem 3DSMI for $n=3$ and consider the result of their computer enumeration. We consider several instances and explicitly prove the absence of a stable matching for each of them. In Sect. 4, we conclude by mentioning some potential future work. In Appendix, we describe our example for $n=4$ and prove that in this case no stable matching exists. ## 2 The statement of 3DSMI-CYC in terms of the graph theory Let $G$ be some directed graph. Denote the set of its edges by $E$; assume that no edges are multiple. Let the vertex set $V$ of the graph $G$ consist of 3 subsets, namely, the set of men $M$, women $F$, and dogs $D$. Any vertex $v\in M$ has outgoing edges directed to (certain) vertices in $F$, any vertex $v\in F$ has outgoing edges directed to (certain) vertices in $D$, and any vertex $v\in D$ has outgoing edges directed to (certain) vertices in $D$. Assume that $\|M\|=\|F\|=\|D\|$ (otherwise we supplement the corresponding subgraph with vertices that are not connected with the rest part of the graph). The number $n=\|M\|=\|F\|=\|D\|$ is called the problem dimension. Evidently, the length of all cycles in the graph $G$ is a multiple of $3$. Note also that this condition ensures the possibility to divide the vertex set of any orgraph $G$ into 3 subsets $M$, $F$, $D$ so that all its edges are directed as indicated above. Each edge $(v,v^{\prime})$, $v,v^{\prime}\in V$, corresponds to some positive integer $r(v,v^{\prime})$ which is called the rank of this edge. For fixed $v\in V$, all possible ranks $r(v,v_{1}),\ldots,r(v,v_{k})$ coincide with $\\{1,\ldots,k\\}$, where $k$ is the outgoing vertex degree $v$ (if $r(v,v^{\prime})=1$, then $v^{\prime}$ is the best preference for $v$, and so on). We understand a three-sided matching as a subgraph $H(V)$ of the graph $G$, where each vertex $v\in V$ has at most one outgoing edge and the following condition is fulfilled: if a vertex $v$ has an outgoing edge, then this edge belongs to a cycle of length 3 in the graph $H$. Cycles of length 3 in the graph $H$ are called families. Evidently, each family, accurate to a cyclic shift, takes the form $(m,f,d)$, where $m\in M$, $f\in F$, and $d\in D$. Note that in what follows, for convenience of denotations of families, we do not fix the order of genders in a family, i. e., we treat denotations of families as triples derived from an initial one by a cyclic shift as equivalent. In what follows, we sometimes use the notion of a family in a wider sense, namely, as any cycle of length 3 in the graph $G$. However, if some three- sided matching $H$ is fixed, then we describe other cycles of length 3 explicitly, applying the term “a family” only to cycles that enter in a three- sided matching. A matching ${\mathcal{M}}$ is a collection of all families of a three-sided matching $H$. For a vertex $v$, $v\in V$, in the matching ${\mathcal{M}}$, the rank $R_{\mathcal{M}}(v)$ is defined as the rank of the edge that goes out of this vertex in the subgraph $H$. If some vertex $v$ in the subgraph $H$ has no outgoing edge, then $R_{\mathcal{M}}(v)$ is set to $+\infty$. A triple $(v,v^{\prime},v^{\prime\prime})$ is said to be blocking for some matching ${\mathcal{M}}$, if it is a cycle in the graph $G$, and $r(v,v^{\prime})<R_{\mathcal{M}}(v),\quad r(v^{\prime},v^{\prime\prime})<R_{\mathcal{M}}(v^{\prime}),\quad r(v^{\prime\prime},v)<R_{\mathcal{M}}(v^{\prime\prime}).$ A matching ${\mathcal{M}}$ is said to be stable, if no blocking triple exists for it. Problem 3DSMI (3-dimensional stable matching with incomplete preference lists) consists in finding a stable matching for a given graph $G$. It is well known that it does not necessarily exists. Moreover, the problem of establishing its existence for a given graph $G$ is NP-complete. As was mentioned in the Introduction, this fact was proved by Biro and McDermid. They have constructed an explicit example of the graph $G$ of dimension 6, for which no stable matching exists. Moreover, the question of constructing similar examples for lesser dimensions was also stated by the mentioned authors. ## 3 The absence of examples of problem 3DSMI with no stable matching for $n<3$ Let $G$ and $G^{\prime}$ be two directed graphs defined on one and the same vertex set $V$ but, generally speaking, having distinct edge sets. Assume that rank functions $r_{G}$ and $r_{G^{\prime}}$ are defined on $E$ and $E^{\prime}$, correspondingly. Let $L\subseteq E\cap E^{\prime}$. We say that ranking orders $r_{G}$ and $r_{G^{\prime}}$ coincide on $L$, if for any two edges $(v,v^{\prime}),(v,v^{\prime\prime})$ in $L$, $r_{G}(v,v^{\prime})<r_{G}(v,v^{\prime\prime})\quad\iff\quad r_{G^{\prime}}(v,v^{\prime})<r_{G^{\prime}}(v,v^{\prime\prime}).$ ###### Lemma 1 For any graph $G$ of problem 3DSMI of dimension $n$ there exists a graph $G^{\prime}$ of the same dimension such that the outgoing degree of each its vertex is nonzero and there is the following correspondence between graphs $G$ and $G^{\prime}$: 1) the set of all possible families of graphs $G$ and $G^{\prime}$ coincide; 2) the ranking order of all edges that enter in these families also coincide. Proof: Let $v$ be a vertex in the graph $G$ having no outgoing edges. Then $v$ enters in no family of the graph $G$. Let us delete this vertex together with all edges that enter in it. Repeating this procedure several times, we get a graph $\widehat{G}$ such that each its vertex has at least one outgoing edge and its set of families coincides with that of the initial graph $G$. Let the symbol $\widehat{V}$ stand for the vertex set of the graph $\widehat{G}$, denote the set of its edges by $\widehat{E}$. Without loss of generality, we assume that the set of families of graphs $G$ and $\widehat{G}$ is nonempty. In this case, the set $\widehat{V}$ contains at least one vertex for each gender. Let us now restore the initial vertices belonging to the set $V\setminus\widehat{V}$ and for each of them arbitrarily construct at least one edge directed to some vertex in $\widehat{V}$ that corresponds to a proper gender. Since the incoming degree of restored vertices equals zero, they, as earlier, can enter in no family. Note that $\widehat{E}\subseteq E$ and, consequently, one can construct a rank function for the obtained graph $G^{\prime}$ preserving the ranking order of the graph $G$ on $\widehat{E}$. The obtained graph $G^{\prime}$ with the rank function defined in the indicated way is the desired one. $\square$ Lemma 1 allows one, when studying problems 3DSMI of dimension $n$, to restrict oneself to considering the corresponding graphs $G$ with nonzero outgoing degrees of all vertices, which we do in what follows. Let the symbol $G^{\prime\prime}$ stands for a subgraph of the graph $G$ consisting of its edges of rank 1. We call $G^{\prime\prime}$ the basic subgraph of the graph $G$. Since each vertex in the basic subgraph has exactly one outgoing edge, $G^{\prime\prime}$ represents a collection of cycles, whose lengths are multiples of 3, and trees of edges that lead to these cycles. ###### Theorem 1 Problem 3DSMI of dimension $n\leqslant 2$ always has a stable matching. Proof: Note that with $n=1$ the assertion of the lemma is trivial. In what follows, we restrict ourselves to considering the case of $n=2$. Note also that in this case a nonstable matching can contain only one family. The basic subgraph of the graph $G$ contains cycles either of length 3 or of length 6. Let us consider both cases sequentially. In the first case, the exist vertices $v_{0},v_{1},v_{2}$ such that $r(v_{0},v_{1})=r(v_{1},v_{2})=r(v_{2},v_{0})=1$. Therefore, if the family $(v_{0},v_{1},v_{2})$ is a matching, then these vertices enter in no blocking triple. But then the consideration is reduced to the case of $n=1$, which, as was mentioned above, is trivial. It remains to consider the case when the basic subgraph of the graph $G$ is a cycle of length 6, i.e., $C=(v_{0},v_{1},\ldots,v_{5})$. Without loss of generality, we assume that the graph $G$, which represents a counterexample to Theorem 1, along with the cycle $C$ contains the edge $(v_{2},v_{0})$ of rank 2. Then the only possible blocking triple to the matching of one family $(v_{0},v_{1},v_{2})$ is $(v_{2},v_{3},v_{4})$. Consequently, the graph $G$ also contains the edge $(v_{4},v_{2})$. But then the only possible blocking triple for the matching consisting of one family $(v_{2},v_{3},v_{4})$ is $(v_{4},v_{5},v_{0})$. In turn, the graph $G$ that consists of only a basic cycle $C$ and edges $(v_{0},v_{4})$, $(v_{4},v_{2})$, $(v_{2},v_{0})$ of rank 2 has a stable matching consisting of one family $(v_{0},v_{4},v_{2})$. Therefore, the graph $G$, along with the cycle $C$, contains at least 4 edges. Consequently, the graph $G$ of dimension $n=2$ has a matching of two families, and it is stable by definition. $\square$ ## 4 The examples of graphs $G$ of dimension $n=3$ with no stable matching In this section, we consider the case of $n=3$. Let us first classify all graphs of the problem of this dimension; this will facilitate their computer search. If the basic subgraph of the graph $G$ contains cycles of length 3, then there exist vertices $v_{0},v_{1},v_{2}$ such that $r(v_{0},v_{1})=r(v_{1},v_{2})=r(v_{2},v_{0})=1$. Therefore, if the family $(v_{0},v_{1},v_{2})$ enters in a matching, then these vertices can enter in no blocking triple. But then we again get the case of $n=2$ which is mentioned in the statement of Theorem 1 (compare this paragraph with the beginning of the proof of Theorem 1). Therefore, the basic subgraph of the graph $G$ represents either a cycle of length 9, or a cycle of length 6 with three edges that lead to this cycle. Altogether, accurate to the cyclic symmetry, there are 6 such subgraphs; they are shown in Fig. 1 Figure 1: 6 variants of the basic subgraph of the graph $G$. Each of 9 vertices of these subgraphs in the graph $G$ can have outgoing edges that lead to two remaining vertices of the corresponding gender (here we understand remaining vertices as those that differ from the vertex, to which the edge of the basic graph $G^{\prime\prime}$ is already directed). Generally speaking, the total number of possible cases is 5, namely, 1) the considered vertex has no more outgoing edges; 2)-3) the considered vertex has one more outgoing edge that leads to some vertex among two ones; 4)-5) the considered vertex has two edges, their ranks are equal to 2 and 3, we can associate ranks with these edges in two ways. Therefore, it suffices to consider $6\times 5^{9}$ problems 3DSMI. Evidently, for each of these problems there exist at most 27 families (27 blocking triples). The number of possible three-sided matching, as one can easily calculate, also is not so large. Namely, there exist, evidently, at most 27 matchings consisting of one family. In addition, there exist at most 108 matchings consisting of two families, namely, there are 27 ways to form a triple consisting of representatives of genders that enter in no matching, and 4 ways to choose partners among two women and two dogs entering the matching for a fixed man that also enters this matching. Finally, there is at least 36=108/3 three-sided matchings of 3 triples, in the previous enumeration each of them can be taken into account three times because each of 3 considered triples does not necessarily enters in the matching. Therefore, the total amount of three-sided matchings does not exceed 36+108+27=171. For each of them we need to find the first triple among 27 potential blocking ones that really is blocking. Therefore, the total amount of considered cases does not exceed $6\times 5^{9}\times 171\times 27\approx 54\times 10^{9}$. For generating these cases, we have written a program in Python. See the version of this program that calculates the number of counterexamples for each of basic graphs shown in Fig. 1 at https://github.com/reginalerner/3dsm/. For the first basic graph shown in Fig. 1 (a cycle formed by 9 vertices), no such graph for problem 3SDMI with no stable matching was obtained. As is mentioned in the Introduction, we even did not expect to find such instances for $n=3$. To our surprise, the computer search has found such counterexamples for each of the rest basic graphs. One of them is shown in Fig. 2. For convenience, we enumerate vertices of the graph by numbers $v$, $v=0,1,\ldots,8$. The value $v\bmod 3$ defines the gender that corresponds to the vertex $v$ . 501234678 Figure 2: The graph of problem 3DSMI of dimension 3 with no stable matching consisting of 16 edges. The rank of all edges indicated by solid bold lines equals 1. The dashed lines represent the edges with the rank 2. The rank of the “dotted” edge equals 3. ###### Theorem 2 The graph shown in Fig. 2 has no stable matching. Proof: Fig. 2 evidently demonstrates that each possible cycle of length 3 takes one of the following forms: $(0,1,5)$, $(0,7,8)$, $(1,2,3)$, $(1,5,3)$, $(2,3,4)$, $(3,4,5)$, and $(4,8,6)$. These cycles form families, while collections of disjoint families form matchings ${\mathcal{M}}$ in the problem. Evidently, if one can add a cycle $(v,v^{\prime},v^{\prime\prime})$ to a matching ${\mathcal{M}}$ (vertices $v,v^{\prime},v^{\prime\prime}$ do not enter in ${\mathcal{M}}$), then ${\mathcal{M}}$ is unstable, i.e., the triple $(v,v^{\prime},v^{\prime\prime})$ is blocking for ${\mathcal{M}}$. Therefore, candidates for stable matchings should be supplemented with possible cycles. We call such matchings uncompletable and consider only such ones. The union of vertices of three cycles that are listed above does not coincide with the set of all vertices of the graph shown in Fig. 2. On the other hand, by using the direct search method we can prove that any set consisting of one triple is completable. Therefore, each uncompletable matching consists of two families. Below we give their complete list together with blocking triples: 1) $\\{(0,1,5),(2,3,4)\\}$, the blocking triple is $(4,8,6)$; 2) $\\{(0,1,5),(4,8,6)\\}$, the blocking triple is $(1,2,3)$; 3) $\\{(0,7,8),(1,2,3)\\}$, the blocking triple is $(3,4,5)$; 4) $\\{(0,7,8),(1,5,3)\\}$, the blocking triple is $(2,3,4)$; 5) $\\{(0,7,8),(2,3,4)\\}$, the blocking triple is $(0,1,5)$; 6) $\\{(0,7,8),(3,4,5)\\}$, the blocking triple is $(0,1,5)$; 7) $\\{(1,2,3),(4,8,6)\\}$, the blocking triple is $(0,7,8)$ or $(3,4,5)$; 8) $\\{(1,5,3),(4,8,6)\\}$, the blocking triple is $(0,7,8)$. $\square$ One can easily give other examples of graphs with the same set of cycles, uncompletable matchings, and blocking triples. In particular, this property is characteristic for the graph that differs from that shown in Fig. 2 by the presence of the additional edge $(7,2)$ of rank 2 or the additional edge $(6,1)$ of rank 2, or both of these edges. Moreover, one can find other graphs consisting of 16 edges that have no stable matching. One of them is shown in Fig. 3 (any other graph with this property differs from the indicated one only in the fact that ranks of edges $(0,4)$ and $(0,7)$ have interchanged). Note that the graph shown in Fig. 3 is similar to that in our example for $n=4$ (see, for comparison, Fig. 4 in Appendix). 501234678 Figure 3: One more 3D graph of problem 3DSMI with no stable matching consisting of 16 edges. Denotations are the same as in Fig. 2. These graphs define the following families of forming matchings in problem 3DSMI: $(0,1,8)$, $(0,4,5)$, $(0,7,5)$, $(1,2,6)$, $(2,3,7)$, $(3,4,5)$, and $(3,7,5)$. The list of matchings with blocking triples looks as follows: 1) $\\{(0,1,8),(2,3,7)\\}$, the blocking triple is $(3,4,5)$; 2) $\\{(0,1,8),(3,4,5)\\}$, the blocking triple is $(1,2,6)$; 3) $\\{(0,1,8),(3,7,5)\\}$, the blocking triple is $(1,2,6)$; 4) $\\{(0,4,5),(1,2,6)\\}$, the blocking triple is $(2,3,7)$; 5) $\\{(0,4,5),(2,3,7)\\}$, the blocking triple is $(0,1,8)$; 6) $\\{(0,7,5),(1,2,6)\\}$, the blocking triple is $(2,3,7)$; 7) $\\{(1,2,6),(3,4,5)\\}$, the blocking triple is $(0,7,5)$. 8) $\\{(1,2,6),(3,7,5)\\}$, the blocking triple is $(0,4,5)$. Since the counterexamples considered above are diverse, they have some common properties. We are going to describe them in a separate paper. ## 5 Concluding remarks In this paper, we study the problem stated by Biro and McDermid in [1], namely, we seek for instances with no weakly stable matching for 3-DSM-CYC with $n<6$. In particular, we find the minimal value of $n$, with which such instances exist, and describe some of them. For $n=3$, all counterexamples seem to have one and the same structure. We are going to consider this structure in a separate paper. The idea of this study is due to the work of Lam and Paxton [6], who give an example of problem 3DSM-CYC for $n=90$ with no stable matching. This example is based on an analogous example proposed by Biro and McDermid for problem 3DSMI-CYC with $n=6$. Our example constructed for problem 3DSMI with $n=3$ allows one to make the dimension of an example for problem 3DSM with no stable matching as low as $n=45$. According to results obtained in Sect. 3, the further decrease of $n$ for problem 3DSMI is impossible. However, it seems possible to find problem 3DSM with no stable matching with $n<45$ using some other methods. Actually, Lam and Paxton studied not only 3-DSM-CYC, but also its $k$-gender analog, $k$-DSM-CYC, for arbitrary $k\geqslant 3$. First they have represented problem 3-DSMI-CYC as a particular case of $k$-DSMI-CYC with $n^{2}$ representatives of each gender. Then by the reduction from $k$-DSMI-CYC they have proved that $k$-DSM-CYC is NP-complete. Note that some development of ideas proposed in the paper [6] allows one to rather easily construct a counterexample of dimension $n=5$ for $k$-DSMI-CYC, $k>3$, basing on the graph shown in Fig. 2 via subdivision of outcoming edges of woman vertexes. Any of subdivided edges is converted to the chain with $k-3$ vertexes inside, one for each new gender. A $k$-gender family should contain the new vertexes from subdivided edge, so there is a biunique correspondence between new $k$-gender families and old 3-gender ones. If no stable matching exists for 3-gender families, then neither one exists for the new $k$-gender graph. Therefore, for any $k>2$, we have constructed an instance of problem $k$-DSMI- CYC with $n=5$ with no stable matching (where lists of preferences of two women, two men, and two dogs that are not shown in Fig. 2 can be arbitrary). The question about the existence of such counterexamples for $n=3$ and $n=4$ still remains open. We hope that this work can be useful in studying other questions related to other aspects of the generalization of the theory of stable matchings to the $k$-dimensional case, $k>2$. In our opinion, this study is far from completion. ## References * [1] P. Biró, E. McDermid, Three-sided stable matchings with cyclic preferences, Algorithmica 58 (2010) 5–18. * [2] E. Boros, V. Gurvich, S. Jaslar, D. Krasner, Stable matchings in three-sided systems with cyclic preferences. Discrete Math. 289(1–3) (2004) 1–10. * [3] K. Eriksson, J. Söstrand, P. Strimling, Three-dimensional stable matching with cyclic preferences, Math. Soc. Sci. 52 (2006) 77–87. * [4] D. Gale, L.S. Shapley, College admissions and the stability of marriage, Am. Math. Mon. 69 (1962), 9–15. * [5] D.E. Knuth, Stable marriage and its relation to other combinatorial problems: an introduction to the mathematical analysis of algorithms, in: CRM Proceedings and Lecture Notes, 1996. * [6] C.-K. Lam, C.G. Plaxton, On the Existence of Three-Dimensional Stable Matchings with Cyclic Preferences, in: Lecture Notes in Computer Science, 11801, 2019. Algorithmic Game Theory, 329–342. * [7] D.F. Manlove, Algorithmics of matching under preferences, Theor. Comput. Sci. World Scientific, 2013. * [8] K. Pashkovich, L. Poirrier, Three-dimensional stable matching with cyclic preferences, Optimization Letters (2020). * [9] B. Pittel, The average number of stable matchings, SIAM J. Discrete Math. 2 (1989) 530–549. * [10] B. Pittel, On random stable matchings: Cyclic ones with strict preferences and two-sided ones with partially ordered preferences, Advances in Appl. Math. 120 (2020) 1–27. ## Appendix. An example of a graph $G$ of dimension $n=4$ with no stable matching $v_{2}$$v_{1}$$v_{0}$$v_{8}$$v_{7}$$v_{6}$$v_{5}$$v_{4}$$v_{3}$$w_{2}$$w_{3}$$w_{4}$ Figure 4: An example of a graph with no stable matching for $n=4$ Consider the graph $G$ shown in Fig. 4. Here bold lines indicate edges of rank 1, dashed ones do edges of rank 2, and dotted ones do edges of rank 3. Therefore, the vertex set of the graph is $V=\\{v_{0},\ldots,v_{8},w_{2},w_{3},w_{4}\\},$ the remainder $i\bmod 3$ of the division of the vertex index $i$ by 3 defines the gender that corresponds to this vertex. Edges of the graph $G$ take one of the following forms: 1) $(v_{i},v_{(i+1)\bmod 9})$, $i=0,\ldots,8$; 2) $(w_{i},v_{i-2})$, $i=2,3,4$; 3) $(v_{i},w_{i+1})$, $i=1,2,3$; 4) $(v_{i},v_{(i+4)\bmod 9})$, $i=0,\ldots,8$. The rank of edges of the 1st and 2nd kinds equals 1; the rank of edges of the 3d kind equals 2; the rank of edges of the 4th kind equals 2 for $i=0,4,5,\ldots,8$ and 3 for $i=1,2,3$. In what follows in this section, we consider indices of all vertices modulo 9, for brevity of notations, we omit the symbol $\bmod\,9$ in subscripts. ###### Theorem 3 The graph $G$ shown in Fig. 4 has no stable matching. Proof: Analogously to the proof of Theorem 2, we consider only uncompletable matchings. One can easily see that all possible families of the graph $G$ take one of the following forms: $(v_{i},v_{i+1},v_{i+5}),i=0,\ldots,8,\quad\text{or}\quad(v_{i},v_{i+1},w_{i+2}),i=0,1,2.$ All uncompletable matchings for the graph $G$ consist of two or three such families. In particular, uncompletable matchings of two families take the form $\\{(v_{i},v_{i+1},v_{i+5}),(v_{i+2},v_{i+3},v_{i+7})\\},\quad i=0,\ldots,8,$ where one or two vertices among $v_{i+5},v_{i+7}$ can be replaced with those $w_{i+2}$ and $w_{i+4}$, correspondingly, (certainly, this replacement is possible, only if a vertex $w$ with the corresponding index exists). In any case, for such a matching ${\mathcal{M}}$ the blocking triple takes the form $(v_{i+3},v_{i+4},v_{i+8})$. Really, by definition, $R_{\mathcal{M}}(v_{i+4})=R_{\mathcal{M}}(v_{i+8})=\infty,\quad\text{and}\quad R_{\mathcal{M}}(v_{i+3})>1.$ Uncompletable matchings of three families for the graph $G$ take the form: ${\mathcal{M}}_{1}(i)=\\{(v_{i},v_{i+1},v_{i+5}),(v_{i+2},v_{i+6},v_{i+7}),(v_{i+3},v_{i+4},v_{i+8})\\},\quad i=0,\ldots,8,$ or ${\mathcal{M}}_{2}(i)=\\{(v_{i},v_{i+1},w_{i+2}),(v_{i+2},v_{i+6},v_{i+7}),(v_{i+3},v_{i+4},v_{i+8})\\},\quad i=0,1,2.$ Note that ${\mathcal{M}}_{2}(i)\equiv{\mathcal{M}}_{3}((i+3)\bmod 9)\equiv{\mathcal{M}}_{4}((i+6)\bmod 9)$, where ${\mathcal{M}}_{3}(i)=\\{(v_{i},v_{i+1},v_{i+5}),(w_{i+8},v_{i+6},v_{i+7}),(v_{i+3},v_{i+4},v_{i+8})\\},\quad i=3,4,5;$ ${\mathcal{M}}_{4}(i)=\\{(v_{i},v_{i+1},v_{i+5}),(v_{i+2},v_{i+6},v_{i+7}),(v_{i+3},v_{i+4},w_{i+5})\\},\quad i=6,7,8.$ In addition, ${\mathcal{M}}_{1}(i)\equiv{\mathcal{M}}_{1}((i+3)\bmod 9)\equiv{\mathcal{M}}_{1}((i+6)\bmod 9)$. Therefore, there exist, as a total, 6 uncompletable matchings of three families: ${\mathcal{M}}_{1}(0),{\mathcal{M}}_{2}(0),{\mathcal{M}}_{1}(1),{\mathcal{M}}_{2}(1),{\mathcal{M}}_{1}(2),{\mathcal{M}}_{2}(2).$ For matchings ${\mathcal{M}}_{1}(0),{\mathcal{M}}_{2}(0)$ the blocking triple is $(v_{1},v_{2},w_{3})$: $R_{{\mathcal{M}}_{1}(0)}(v_{1})=3>1$, $R_{{\mathcal{M}}_{2}(0)}(v_{1})=2>1$, $R_{{\mathcal{M}}_{1}(0)}(v_{2})=R_{{\mathcal{M}}_{2}(0)}(v_{2})=3>2$. Analogously, for matchings ${\mathcal{M}}_{1}(1),{\mathcal{M}}_{2}(1)$ the blocking triple is $(v_{2},v_{3},w_{4})$; for matchings ${\mathcal{M}}_{1}(2),{\mathcal{M}}_{2}(2)$ the blocking triple is $(v_{0},v_{1},w_{2})$. $\square$
# Distributional Anchor Regression Lucas Kook1,2111Corresponding author, email<EMAIL_ADDRESS>, Beate Sick1,2, Peter Bühlmann3 (1University of Zurich, Switzerland 2Zurich University of Applied Sciences, Switzerland 3ETH Zurich, Switzerland) ###### Abstract Prediction models often fail if train and test data do not stem from the same distribution. Out-of-distribution (OOD) generalization to unseen, perturbed test data is a desirable but difficult-to-achieve property for prediction models and in general requires strong assumptions on the data generating process (DGP). In a causally inspired perspective on OOD generalization, the test data arise from a specific class of interventions on exogenous random variables of the DGP, called anchors. Anchor regression models, introduced by Rothenhäusler et al. (2018), protect against distributional shifts in the test data by employing causal regularization. However, so far anchor regression has only been used with a squared-error loss which is inapplicable to common responses such as censored continuous or ordinal data. Here, we propose a distributional version of anchor regression which generalizes the method to potentially censored responses with at least an ordered sample space. To this end, we combine a flexible class of parametric transformation models for distributional regression with an appropriate causal regularizer under a more general notion of residuals. In an exemplary application and several simulation scenarios we demonstrate the extent to which OOD generalization is possible. ##### Keywords anchor regression, covariate shift, diluted causality, distributional regression, transformation models, out-of-distribution generalization ## 1 Introduction Common methods in supervised statistical learning assume the test data to follow the same distribution as the training data. This is implicitly exploited in e.g., cross-validation or by randomly splitting a dataset into a training and a test set, which has been demonstrated to be potentially flawed (Efron, 2020) due to concept drift or domain shift where new (test) data do not follow the same distribution as the training data. More recently, the problem has been referred to as out-of-distribution (OOD) generalization (Sun et al., 2019). The desire to achieve reliable test predictions under distributional shifts is ubiquitous in many fields of machine learning and statistics, such as transfer learning (Pan and Yang, 2009; Rojas-Carulla et al., 2018), domain adaptation (Magliacane et al., 2018; Redko et al., 2020), multi-task learning (Caruana, 1997), representation learning (Mitrovic et al., 2020) or prediction models in medical statistics (Subbaswamy and Saria, 2019). Accordingly, many different formulations of the problem of OOD generalization exist in the literature (a detailed overview can be found in Chen and Bühlmann, 2020). We will frame OOD generalization as the problem of robustly predicting an outcome in novel, unseen environments, based on data from a few observed environments and extend on the idea of anchor regression and causal regularization (Rothenhäusler et al., 2018; Bühlmann, 2020; Bühlmann and Ćevid, 2020) to develop distributional anchor regression. In such a framework, training a model on heterogeneous data is not a disadvantage but rather a necessity. ### 1.1 Related work It has been known for decades that a causal model is robust towards arbitrarily strong perturbations on components other than the response (Haavelmo, 1943). However, identifying causal structures is not only difficult but often leads to sub-par prediction performance when the test data contain perturbations of bounded strength (Rothenhäusler et al., 2018). Rothenhäusler et al. introduce linear anchor regression, which allows a trade-off between prediction performance and robustness against shift perturbations of a certain size. The framework of linear anchor regression was extended to deal with nonlinear regression between the response and covariates (Bühlmann, 2020). Furthermore, Christiansen et al. (2020) provide a causal framework to decide which assumptions are needed for and to what extent OOD generalization is possible. Anchor regression is related to Instrumental Variables (IV) regression. However, the main IV assumption that the instrument $A$ does not directly affect some hidden confounding variables $H$ is dropped, at the price of non- identifiability of the causal parameter (Angrist et al., 1996). A graphical description of the issue is given in Figure 1. $X$$A$$Y$$H$ $X$$A$$Y$$H$ Figure 1: Graphical models for the response variable $Y$, covariates $X$ and hidden confounders $H$: IV regression with instruments $A$ (left) and anchor regression with anchor $A$ (right). In anchor regression, $A$ is only required to be a source node but is allowed to directly influence response, covariates and hidden confounders. ### 1.2 Our Contribution In this work we develop a framework for distributional anchor regression in the broad class of transformation models (TMs, Hothorn et al., 2014). The resulting class of anchor TMs generalizes (non-) linear anchor regression to potentially censored responses and characterizes the full conditional distribution of $Y\|\text{\boldmath$X$}=\text{\boldmath$x$}$ instead of estimating solely the conditional mean function. While the $L_{2}$ anchor loss can be decomposed into a squared error and causal regularization term penalizing correlation between anchors and residuals, we propose a distributional anchor loss based on the negative log-likelihood and replacing the least-squares residuals by the more general score residuals. The proposed causal regularizer induces uncorrelatedness between the anchors and these score residuals. The resulting procedure is tailored towards protecting against distributional shifts induced by the anchors and naturally interpolates between the unpenalized maximum-likelihood estimate and a solution for which anchors and residuals are strictly uncorrelated. The latter may be thought of as a distributional IV-like objective but it generally does not estimate the causal model due to the fact that the anchor $A$ can also directly influence $H$ and $Y$ (see Figure 1). It leads to some invariance of the score residuals across the values of the anchors $A$, and such an invariance property has been referred to as “diluted causality” (Bühlmann, 2020). We implement all methods and algorithms in the R language for statistical computing (R Core Team, 2020) and the code is available on GitHub. In the appendix we present further details on notation, computation, and score residuals. ## 2 Background First, we introduce structural equation models (SEMs) before recapping linear anchor regression. In Section 2.3, we switch perspectives from modelling the conditional expectation to transformation models which enable to capture entire conditional distributions. The notation used in this work is described in Appendix A. ### 2.1 Structural Equation Models Let $Y$ be a response which takes values in $\mathbb{R}$, $X$ be a random vector of covariates taking values in $\mathbb{R}^{p}$, $H$ denotes hidden confounders with sample space $\mathbb{R}^{d}$, and $A$ are exogenous variables (called anchors, due to exogeneity; source node in the graph in Figure 1) taking values in $\mathbb{R}^{q}$. The SEM governing linear anchor regression is given by $\displaystyle\begin{pmatrix}Y\\\ \text{\boldmath$X$}\\\ \text{\boldmath$H$}\end{pmatrix}\leftarrow\text{$\mathbf{B}$}\begin{pmatrix}Y\\\ \text{\boldmath$X$}\\\ \text{\boldmath$H$}\end{pmatrix}+\text{$\mathbf{M}$}\text{\boldmath$A$}+\text{\boldmath$\varepsilon$},$ (1) with $(1+p+d)\times(1+p+d)$-matrix $\mathbf{B}$ which corresponds to the structure of the SEM in terms of a directed acyclic graph (DAG), the effect of $A$ enters linearly via the $(1+p+d)\times q$-matrix $\mathbf{M}$, and $\varepsilon$ denotes the error term with mutually independent components. The “$\leftarrow$” symbol is algebraically a distributional equality sign. It emphasizes the structural character of the SEM, saying that, e.g., $Y$ is only a function of the parents of the node $Y$ in the structural DAG and the additive component $(\text{\boldmath$M$}\text{\boldmath$A$}+\text{\boldmath$\varepsilon$})_{1}$. The anchors $A$ may be continuous or discrete. In the special case of discrete anchors each level can be viewed as an “environment”. We define perturbations as intervening on $A$, e.g., by $\operatorname{do}(\text{\boldmath$A$}=\text{\boldmath$a$})$, which replaces $A$ by $a$ in the SEM while leaving the underlying mechanism, i.e., the coefficients in the SEM, unchanged. In this work we restrict ourselves to $\operatorname{do}$\- (Pearl, 2009) and $\operatorname{push}$-interventions (Markowetz et al., 2005) on $A$, which in turn lead to shifts in the distribution of $X$. Since $A$ is exogenous and a source node in the graph, the specific type of intervention does not play a major role. Christiansen et al. (2020) show that under the above conditions OOD generalization is possible in linear models. ### 2.2 Linear Anchor Regression Linear anchor regression with its corresponding causal regularization estimates the linear regression parameter $\beta$ as $\displaystyle\hat{\text{\boldmath$\beta$}}=\operatorname{{arg\,min}}_{\text{\boldmath$\beta$}}\bigg{\\{}\left\lVert(\operatorname{Id}-\boldsymbol{\Pi}_{\text{$\mathbf{A}$}})(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2}/n+\gamma\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2}/n\bigg{\\}},$ where $0\leq\gamma\leq\infty$ is a regularization parameter and $\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}=\text{$\mathbf{A}$}(\text{$\mathbf{A}$}^{\top}\text{$\mathbf{A}$})^{-1}\text{$\mathbf{A}$}^{\top}$ denotes the orthogonal projection onto the column space of the anchors (Rothenhäusler et al., 2018). For $\gamma=1$ one obtains ordinary least squares, $\gamma\to\infty$ corresponds to two-stage least squares as in Instrumental Variables regression and $\gamma=0$ is adjusting for the anchor variables $A$ (being equivalent to ordinary least squares when regressing $Y$ on $X$ and $A$). Causal regularization encourages, for large values of $\gamma$, uncorrelatedness of the anchors $A$ and the residuals. As a procedure, causal regularization does not depend at all on the SEM in eq. (1). However, as described below, the method inherits a distributional robustness property, whose formulation depends on the SEM in eq. (1). Rothenhäusler et al. (2018) establish the duality between the $L_{2}$ loss in linear anchor regression and optimizing a worst case risk over specific shift perturbations. The authors consider shift perturbations $\nu$, which are confined to be in the set $\displaystyle C_{\gamma}:=\big{\\{}\text{\boldmath$\nu$}:\text{\boldmath$\nu$}=\text{$\mathbf{M}$}\text{\boldmath$\delta$},\;\text{\boldmath$\delta$}\mbox{ independent of }\text{\boldmath$\varepsilon$},\;\mathbb{E}[\text{\boldmath$\delta$}\text{\boldmath$\delta$}^{\top}]\preceq\gamma\mathbb{E}[\text{\boldmath$A$}\text{\boldmath$A$}^{\top}]\big{\\}},$ and which generate the perturbed response $Y^{\text{\boldmath$\nu$}}$, and covariates $\text{\boldmath$X$}^{\text{\boldmath$\nu$}}$ via $\displaystyle\begin{pmatrix}Y^{\text{\boldmath$\nu$}}\\\ \text{\boldmath$X$}^{\text{\boldmath$\nu$}}\\\ \text{\boldmath$H$}^{\text{\boldmath$\nu$}}\end{pmatrix}\leftarrow\text{$\mathbf{B}$}\begin{pmatrix}Y^{\text{\boldmath$\nu$}}\\\ \text{\boldmath$X$}^{\text{\boldmath$\nu$}}\\\ \text{\boldmath$H$}^{\text{\boldmath$\nu$}}\end{pmatrix}+\text{\boldmath$\nu$}+\text{\boldmath$\varepsilon$}.$ The set $C_{\gamma}$ contains all vectors which lie in the span of the columns of $M$ and thus in the same direction as the exogenous contribution $M$$A$ of the anchor variables. The average squared size of $\delta$ is limited to $\gamma$ times the smallest eigenvalue of the centered anchor’s variance- covariance matrix. Now, the explicit duality between the worst case risk over all shift perturbations of limited size and the $L_{2}$ anchor loss is given by $\displaystyle\sup_{\text{\boldmath$\nu$}\in C_{\gamma}}\mathbb{E}\left[(Y^{\text{\boldmath$\nu$}}-(\text{\boldmath$X$}^{\text{\boldmath$\nu$}})^{\top}\text{\boldmath$\beta$})^{2}\right]=\mathbb{E}\left[((\operatorname{Id}-P_{\text{\boldmath$A$}})(Y-\text{\boldmath$X$}^{\top}\text{\boldmath$\beta$}))^{2}\right]+\gamma\mathbb{E}\left[(P_{\text{\boldmath$A$}}(Y-\text{\boldmath$X$}^{\top}\text{\boldmath$\beta$}))^{2}\right],$ (2) where $P_{\text{\boldmath$A$}}=\mathbb{E}[\cdot\|\text{\boldmath$A$}]$ is the population analogue of $\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}$. We note that the right-hand side is the population analogue of the objective function in anchor regression. Hence, causal regularization in anchor regression provides guarantees for optimizing worst-case risk across a class of shift perturbations. The details are provided in Rothenhäusler et al. (2018). ### 2.3 Transformation Models We now switch perspective from models for the conditional mean to the conditional distributions. Specifically, we consider transformation models (Hothorn et al., 2014). TMs decompose the conditional distribution of $Y\|\text{\boldmath$x$}$ into a pre-defined simple distribution function $F_{Z}$, with log-concave density $f_{Z}$, and a (semi-) parametric transformation function $h(y\|\text{\boldmath$x$})$, which is monotone non- decreasing in $y$ $\displaystyle F_{Y\|\text{\boldmath$x$}}(y\|\text{\boldmath$x$})=F_{Z}(h(y\|\text{\boldmath$x$})).$ This way, the problem of estimating a conditional distribution simplifies to estimating the parameters of the transformation function $h=F_{Z}^{-1}\circ F_{Y\|\text{\boldmath$x$}}$ (since $F_{Z}$ is pre-specified and parameter- free). Depending on the complexity of $h$, very flexible conditional distributions can be modelled. Hothorn et al. (2018) give theoretical guarantees for the existence and uniqueness of the transformation function $h$ for absolute continuous, countably infinite and ordered discrete random variables. For the sake of generality, $h$ is parametrized in terms of a basis expansion in the argument $y$ which can be as simple as a linear function in $y$ or as complex as a basis of splines to approximate a smooth function in $y$. In this work, we assume the transformation function for a continuous responses can be additively decomposed into a linear predictor in $x$ and a smooth function in $y$ which is modelled as a Bernstein polynomial of order $P$ with parameters $\text{\boldmath$\theta$}\in\mathbb{R}^{P+1}$ (Hothorn et al., 2018), such that $h(y\|\text{\boldmath$x$})=\text{\boldmath$b$}_{\text{Bs},P}(y)^{\top}\text{\boldmath$\theta$}+\text{\boldmath$x$}^{\top}\beta$. Monotonicity of $\text{\boldmath$b$}_{\text{Bs},P}(y)^{\top}\text{\boldmath$\theta$}$ and thereby of $h(y\|\text{\boldmath$x$})$ can then be enforced via the $P$ linear constraints $\theta_{1}\leq\theta_{2}\leq\dots\theta_{P+1}$. In case of an ordinal response taking values in $\\{y_{1},y_{2},\dots,y_{K}\\}$, the transformation function is a monotone increasing step function, $h(y_{k}\|\text{\boldmath$x$})=\theta_{k}+\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$}$, for $k=1,\dots,K-1$ and the additional constraint $\theta_{K}=+\infty$. We summarize a transformation model based on its simple distribution function $F_{Z}$, basis $b$, which may include covariates, and parameters $\vartheta$, such that $F_{Y\|\text{\boldmath$x$}}(y\|\text{\boldmath$x$})=F_{Z}(\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$})$. For instance, for a transformation model with continuous response and explanatory variables $x$ we thus use $\text{\boldmath$b$}(y,\text{\boldmath$x$})=(\text{\boldmath$b$}_{\text{Bs},P}(y)^{\top},\text{\boldmath$x$}^{\top})^{\top}$ and $\text{\boldmath$\vartheta$}=(\text{\boldmath$\theta$}^{\top},\text{\boldmath$\beta$}^{\top})^{\top}$, yielding $h(y\|\text{\boldmath$x$})=\text{\boldmath$b$}_{\text{Bs},P}(y)^{\top}\text{\boldmath$\theta$}+\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$}$. For a TM with ordinal response we substitute the Bernstein basis with a dummy encoding of the response, which we denote by $\tilde{\text{\boldmath$y$}}$ (e.g., Kook et al., 2020). Also note that the unconditional case is covered by the above formulation as well, by omitting all explanatory variables from the TM’s basis. Figure 2: Illustration of an unconditional transformation model $(1-\exp(-\exp(\cdot)),\text{\boldmath$b$}_{\text{Bs},6},\text{\boldmath$\vartheta$})$ for the Old Faithful Geyser data (Azzalini and Bowman, 1990) using a Bernstein polynomial basis expansion of order 6 for the transformation function, $h(y)=\text{\boldmath$b$}_{\text{Bs},6}(y)$. The colored regions indicate the transport of probability mass from $\mathbb{P}_{Y}$ (lower right) to $\mathbb{P}_{Z}$ (upper left) via the transformation function $h(y)=\text{\boldmath$b$}(y)^{\top}\text{\boldmath$\vartheta$}$ (upper right). If $h$ is continuously differentiable, the density of $Y$ is given by $f_{Y}(y)=f_{Z}(h(y))h^{\prime}(y)$. Figure 2 illustrates the intuition behind transformation models. The transformation function (upper right panel) transforms the complex, bimodal distribution of $Y$ (lower panel) to $F_{Z}=F_{\operatorname{MEV}}$, the standard minimum extreme value distribution (upper left panel). ###### Definition 1 (Transformation model, Definition 4 in Hothorn et al. (2018)) The triple ($F_{Z}$, $b$, $\vartheta$) is called transformation model. ###### Example 1 (Linear regression) The normal linear regression model (Lm) is commonly formulated as $Y=\beta_{0}+\text{\boldmath$x$}^{\top}\tilde{\text{\boldmath$\beta$}}+\varepsilon$, $\varepsilon\sim\mathcal{N}(0,\sigma^{2})$, or $Y\|\text{\boldmath$x$}\sim\mathcal{N}(\beta_{0}+\text{\boldmath$x$}^{\top}\tilde{\text{\boldmath$\beta$}},\sigma^{2}).$ For a distributional treatment we write the above expression as $\displaystyle F_{Y\|\text{\boldmath$x$}}(y\|\text{\boldmath$x$})=\Phi\left(\frac{y-\beta_{0}-\text{\boldmath$x$}^{\top}\tilde{\text{\boldmath$\beta$}}}{\sigma}\right)=\Phi(\vartheta_{1}+\vartheta_{2}y-\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$}),$ (3) which can be understood as a transformation model by letting $\vartheta_{1}=-\beta_{0}/\sigma$, $\vartheta_{2}=1/\sigma$ and $\text{\boldmath$\beta$}=\tilde{\text{\boldmath$\beta$}}/\sigma$. Formally, it corresponds to the model $\displaystyle(F_{Z},\text{\boldmath$b$},\text{\boldmath$\vartheta$})=(\Phi,(1,y,\text{\boldmath$x$}^{\top})^{\top},(\vartheta_{1},\vartheta_{2},-\text{\boldmath$\beta$}^{\top})^{\top}).$ Note that the baseline transformation, $h(y\|\text{\boldmath$X$}=0)$, is constrained to be linear with constant slope $\vartheta_{2}$. Due to the linearity of $h$ and the choice $F_{Z}=\Phi$, the modeled distribution of $Y\|\text{\boldmath$x$}$ will always be normal with constant variance. By parametrizing $h$ in a smooth way, we arrive at much more flexible conditional distributions for $Y\|\text{\boldmath$x$}$. The parameters of a TM can be jointly estimated using maximum-likelihood. The likelihood can be written in terms of the simple distribution function $F_{Z}$, which makes its evaluation computationally more convenient. For a single datum $\\{(\underaccent{\bar}{\ry},\bar{y}],\text{\boldmath$x$}\\}$ with potentially censored response the log-likelihood contribution is given by (Lindsey et al., 1996) $\displaystyle\ell(\text{\boldmath$\vartheta$};y,\text{\boldmath$x$})=\begin{cases}\log f_{Z}(\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$})+\log\left(\text{\boldmath$b$}^{\prime}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$}\right),&y=(\underaccent{\bar}{\ry}+\bar{y})/2,\;\text{exact,}\\\ \log F_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$}),&y\in(-\infty,\bar{y}],\;\text{left,}\\\ \log\left(1-F_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$})\right),&y\in(\underaccent{\bar}{\ry},+\infty),\;\text{right,}\\\ \log\left(F_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$})-F_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$})\right),&y\in(\underaccent{\bar}{\ry},\bar{y}],\;\text{interval.}\end{cases}$ The likelihood is always understood as conditional on $X$ when viewing the covariables as random. Allowing for censored observations is of practical importance, because in many applications the response of interest is not continuous or suffers from inaccuracies, which can be taken into account via uninformative censoring. ###### Example 2 (Lm, cont’d) For an exact datum $\\{y,\text{\boldmath$x$}\\}$ the log-likelihood in the normal linear regression model is given by $\displaystyle\ell(\vartheta_{1},\vartheta_{2},\text{\boldmath$\beta$};y,\text{\boldmath$x$})=\log\phi\big{(}\vartheta_{1}+\vartheta_{2}y-\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$}\big{)}+\log(\vartheta_{2}),$ using the density approximation to the likelihood (Lindsey et al., 1996). Here, $\phi$ denotes the standard normal density, and $\text{\boldmath$b$}^{\prime}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$}=\frac{\partial\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$}}{\partial y}=\vartheta_{2}$. Now that we have established TMs and the log-likelihood function to estimate their parameters, we also need a more general notion of the residuals to formulate a causal regularizer for a distributional anchor loss. Most importantly, these residuals have to fulfill the same requirements as least squares residuals in the $L_{2}$ anchor loss. That is, they have to have zero expectation and a positive definite covariance matrix (e.g., Theorem 3 in Rothenhäusler et al., 2018). In the survival analysis literature, score residuals have received considerable attention, and fulfill the above requirements at least asymptotically (Lagakos, 1981; Barlow and Prentice, 1988; Therneau et al., 1990; Farrington, 2000). We now define score residuals for the general class of transformation models. ###### Definition 2 (Score residuals) Let $(F_{Z},\text{\boldmath$b$},\text{\boldmath$\vartheta$})$ be a fully specified TM. On the scale of the transformation function, add an additional parameter, $\alpha$, to arrive at the TM $(F_{Z},(\text{\boldmath$b$}^{\top},1)^{\top},(\text{\boldmath$\vartheta$}^{\top},-\alpha)^{\top})$ with distribution function $F_{Y\|\text{\boldmath$x$}}(y\|\text{\boldmath$x$})=F_{Z}(\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\text{\boldmath$\vartheta$}-\alpha)$. Because the model is fully specified, $\alpha$ is constrained to $0$. The score residual for a single datum $y\in(\underaccent{\bar}{y},\bar{y}]$ is now defined as $\displaystyle r:=\frac{\partial}{\partial\alpha}\ell(\text{\boldmath$\vartheta$},\alpha;y,\text{\boldmath$x$})\bigg{\rvert}_{\hat{\text{\boldmath$\vartheta$}},\alpha\equiv 0},$ (4) which can be understood as the score contribution of a single observation to test $\alpha=0$ for a covariate which is not included in the model. When viewed as a random variable, the vector of score residuals has mean zero asymptotically and its components are asymptotically uncorrelated (Farrington, 2000). The score residuals can be derived in closed form for a transformation model and observations under any form of uninformative censoring $\displaystyle r=\begin{cases}-f_{Z}^{\prime}(\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})/f_{Z}(\text{\boldmath$b$}(y,\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}}),&y=(\underaccent{\bar}{\ry}+\bar{y})/2,\;\text{exact,}\\\ -f_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})/F_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}}),&y=(-\infty,\bar{y}],\;\text{left,}\\\ f_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})/(1-F_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})),&y=(\underaccent{\bar}{\ry},+\infty),\;\text{right,}\\\ (f_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})-f_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})\big{/}&y=(\underaccent{\bar}{\ry},\bar{y}],\;\text{interval.}\\\ \quad(F_{Z}(\text{\boldmath$b$}(\bar{y},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}})-F_{Z}(\text{\boldmath$b$}(\underaccent{\bar}{\ry},\text{\boldmath$x$})^{\top}\hat{\text{\boldmath$\vartheta$}}))\end{cases}$ ###### Example 3 (Lm, cont’d) By including the addtitional intercept parameter in the normal linear model in eq. (3), the score residuals are given by $\displaystyle\frac{\partial}{\partial\alpha}\ell(\vartheta_{1},\vartheta_{2},\text{\boldmath$\beta$},\alpha;y,\text{\boldmath$x$})\bigg{\rvert}_{\hat{\vartheta}_{1},\hat{\vartheta}_{2},\hat{\text{\boldmath$\beta$}},\alpha\equiv 0}$ $\displaystyle=\frac{\partial}{\partial\alpha}\big{\\{}\log\phi\big{(}\vartheta_{1}+\vartheta_{2}y-\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$}-\alpha)+\log(\vartheta_{2})\big{\\}}\bigg{\rvert}_{\hat{\vartheta}_{1},\hat{\vartheta}_{2},\hat{\text{\boldmath$\beta$}},\alpha\equiv 0}$ $\displaystyle=\hat{\vartheta}_{1}+\hat{\vartheta}_{2}y-\text{\boldmath$x$}^{\top}\hat{\text{\boldmath$\beta$}}=\frac{y-\hat{\beta}_{0}-\text{\boldmath$x$}^{\top}\hat{\text{\boldmath$\beta$}}}{\hat{\sigma}}.$ In this simple case the score residuals are equivalent to scaled least-square residuals, which underlines the more general nature of score residuals. We are now ready to cast transformation models into the framework of SEMs. In the definition below, we define linear SEMs for TMs in terms of the transformed random variable $Z:=h(Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$})$ and transform it back to the scale of $Y$. This strategy is motivated by SEMs for generalized linear models (GLMs), which parametrize the natural parameter, instead of modelling the response directly (e.g., Skrondal and Rabe-Hesketh, 2004). ###### Definition 3 (Linear structural equation transformation model) Let the conditional distribution of $Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}$ be given by the transformation model cdf $F_{Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}}=F_{Z}\circ h$. We set up a SEM for the transformed response $\displaystyle h(Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$})=F_{Z}^{-1}(F_{Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}}(Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$})),$ which is distributed according to $F_{Z}$. By Corollary 1 in Hothorn et al. (2018), the transformation function exists, is unique and monotone non- decreasing in its argument. The following distributional equalities and structural equations summarize the data generating process $\displaystyle F_{Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}}(y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$})$ $\displaystyle=F_{Z}(h(y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}))$ $\displaystyle h(Y)$ $\displaystyle\leftarrow\text{\boldmath$b$}(Y)^{\top}\text{\boldmath$\theta$}+\text{$\mathbf{B}$}_{Y\text{\boldmath$X$}}\text{\boldmath$X$}+\text{$\mathbf{B}$}_{Y\text{\boldmath$H$}}\text{\boldmath$H$}+\text{$\mathbf{M}$}_{Y}\text{\boldmath$A$}$ $X$ $\displaystyle\leftarrow\text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$X$}}\text{\boldmath$X$}+\text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$H$}}\text{\boldmath$H$}+\text{$\mathbf{M}$}_{\text{\boldmath$X$}}\text{\boldmath$A$}+\varepsilon_{\text{\boldmath$X$}}$ $H$ $\displaystyle\leftarrow\text{$\mathbf{B}$}_{\text{\boldmath$H$}\text{\boldmath$H$}}\text{\boldmath$H$}+\text{$\mathbf{M}$}_{\text{\boldmath$H$}}\text{\boldmath$A$}+\varepsilon_{\text{\boldmath$H$}}$ $A$ $\displaystyle\leftarrow\varepsilon_{\text{\boldmath$A$}}.$ In contrast to the linear SEM in eq. (1), the SEM is set up for the transformed response. The basis expansion term $\text{\boldmath$b$}(y)^{\top}\text{\boldmath$\theta$}$ can be viewed as an intercept function, which fixes the overall shape of the transformation function. The remaining additive components of the transformation function, in turn, solely shift the transformation up- or downwards with the covariates. This may seem restrictive at first, however, all covariates influence not only the conditional mean, but all higher conditional moments of $F_{Y\|\text{\boldmath$x$},\text{\boldmath$h$},\text{\boldmath$a$}}$. A graphical representation of the SEM from Definition 3 is given in Figure 3. However, we do not display the possibility that some components of $X$ directly influence each other, and likewise for $H$. In fact, in the simulations in Section 4, the coefficients $\text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$X$}}=\text{$\mathbf{B}$}_{\text{\boldmath$H$}\text{\boldmath$H$}}=0$. $X$$A$$h(Y\|\text{\boldmath$X$},\text{\boldmath$H$},\text{\boldmath$A$})$$H$ $\text{$\mathbf{M}$}_{\text{\boldmath$X$}}$ $\text{$\mathbf{B}$}_{Y\text{\boldmath$X$}}$ $\text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$H$}}$ $\text{$\mathbf{B}$}_{Y\text{\boldmath$H$}}$ $\text{$\mathbf{M}$}_{\text{\boldmath$H$}}$ $\text{$\mathbf{M}$}_{Y}$ Figure 3: Linear structural equation model for a transformation model. Instead of setting up the SEM on the scale of $Y$, it is set up on the scale of the transformation function $h$. The conditional distribution of $Y$ is still fully determined by $h$ and $F_{Z}$. Next, we will present our main proposal on distributional anchor regression to achieve robust TMs with respect to perturbations on the anchor variables $A$. ## 3 Distributional Anchor Regression We now formulate distributional anchor regression, for which we consider a distributional loss function, i.e., the negative log-likelihood, which can take into account censored observations and captures the conditional distribution of $Y\|\text{\boldmath$X$}$, and a causal regularizer involving score residuals. We first give some intuition how to arrive at the distributional anchor loss, starting from the $L_{2}$ anchor loss. One can decompose the $L_{2}$ anchor loss $\displaystyle L_{2}(\text{\boldmath$\beta$};\text{\boldmath$y$},\text{$\mathbf{X}$},\text{$\mathbf{A}$})=\frac{1}{2}\bigg{(}\left\lVert(\operatorname{Id}-\boldsymbol{\Pi}_{\text{$\mathbf{A}$}})(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2}/n+\gamma\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2}/n\bigg{)}$ into a squared error and a pure penalty term. We rewrite $\displaystyle L_{2}(\text{\boldmath$\beta$};\text{\boldmath$y$},\text{$\mathbf{X}$},\text{$\mathbf{A}$})$ $\displaystyle\propto\left\lVert(\operatorname{Id}-\boldsymbol{\Pi}_{\text{$\mathbf{A}$}})(y-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2}+\gamma\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\text{\boldmath$y$}-\text{$\mathbf{X}$}\beta)\right\rVert_{2}^{2}$ $\displaystyle=\left\lVert\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$}\right\rVert_{2}^{2}+(\gamma-1)\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\right\rVert_{2}^{2},$ which is a sum of the squared-error loss and a causal regularizer involving the $L_{2}$ norm of the residuals $(\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})$ projected linearly onto the space spanned by the columns of $\mathbf{A}$ to encourage uncorrelatedness between the residuals and the anchor variables. The cross- terms when expanding the $L_{2}$ norm vanish because $\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}$ is an orthogonal projection. Now an intuitive choice for the distributional anchor regression loss would be $\displaystyle L(\text{\boldmath$\beta$};\text{\boldmath$y$},\text{$\mathbf{X}$},\text{$\mathbf{A}$})\propto-\ell(\text{\boldmath$\beta$};\text{\boldmath$y$},\text{$\mathbf{X}$})+(\gamma-1)\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}\text{\boldmath$r$}\right\rVert_{2}^{2},$ where the negative log-likelihood induced by a transformation model, $\ell(\cdot)$, replaces the squared error loss and, most importantly, $r$ denotes the vector of score residuals as defined in Section 2.3. Thus, the loss encourages uncorrelatedness between the anchor variables and the score residuals, particularly for large values of $\gamma$. The origin and importance of score residuals is outlined in Appendix B. We now give a definition for the distributional anchor loss. ###### Definition 4 (Distributional anchor loss) Consider a linear TM and its SEM, as in Definition 3. Then, the distributional anchor loss is defined as $\displaystyle L(\text{\boldmath$\vartheta$};\text{\boldmath$y$},\text{$\mathbf{X}$},\text{$\mathbf{A}$},\xi)=-\ell(\text{\boldmath$\vartheta$};\text{\boldmath$y$},\text{$\mathbf{X}$})/n+\xi\left\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}\text{\boldmath$r$}\right\rVert_{2}^{2}/n,$ where $\ell(\cdot)$ denotes the log-likelihood induced by a TM, $r$ denotes the vector of score residuals and $\xi\in[0,+\infty)$ controls the extent of causal regularization. As mentioned earlier, the log-likelihood is conditional on $X$. ###### Example 4 (Lm, cont’d) For normal linear regression with constant variance, the $L_{2}$ anchor loss and the distributional anchor loss are equivalent. This is because $\displaystyle L(\vartheta_{1},\vartheta_{2},\text{\boldmath$\beta$};\text{\boldmath$y$},\text{$\mathbf{X}$},\xi)$ $\displaystyle=-\log\phi(\vartheta_{1}+\vartheta_{2}\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})/n-\log(\vartheta_{2})+\xi\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\vartheta_{1}+\vartheta_{2}\text{\boldmath$y$}-\text{$\mathbf{X}$}\text{\boldmath$\beta$})\rVert_{2}^{2}/n$ $\displaystyle=\lVert\text{\boldmath$y$}-\beta_{0}-\text{$\mathbf{X}$}\tilde{\text{\boldmath$\beta$}}\rVert_{2}^{2}/(2\sigma^{2}n)+\xi\lVert\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}(\text{\boldmath$y$}-\beta_{0}-\text{$\mathbf{X}$}\tilde{\text{\boldmath$\beta$}})\rVert_{2}^{2}/(\sigma^{2}n)+C.$ Absorbing all additive constants into $C$ and factoring out the variance renders the above objective equivalent to the $L_{2}$ anchor loss for $\xi=(\gamma-1)/2$. In the following Section, we will empirically evaluate the prediction performance of transformation models estimated under the distributional anchor loss. Computational details for fitting TMs using the distributional anchor loss, are given in Appendix C. ## 4 Empirical Results We begin the section by describing the experimental setup in the application and simulation studies and then present the results in Sections 4.1 and 4.2. We consider median housing prices in the BostonHousing2 dataset (Harrison and Rubinfeld, 1978) to illustrate an application of anchor TMs in normal linear regression (Lm), which assumes normality and equal variances. To lift these assumptions, a continuous probit (c-probit) regression is used to model more complex, skewed distributions, which are typical for housing prices. Lastly we use a continuous logit (c-logit) model, which now takes into account the censored observations in the BostonHousing2 dataset and enables more easily interpretable shift effects on the log-odds scale. Furthermore, the proposed distributional framework for anchor regression is evaluated in simulation studies for Lm, c-probit and ordered logistic regression (o-logit). A summary of the models used to empirically evaluate anchor TMs is given in Table 1. Table 1: Transformation models used to illustrate the distributional anchor loss. $F_{\operatorname{SL}}=\operatorname{expit}$ denotes the standard logistic distribution. By $\tilde{\text{\boldmath$y$}}$ we denote the dummy encoded response, $\tilde{\text{\boldmath$y$}}=\text{\boldmath$e$}(k)$, for $Y$ taking class $y_{k}$, $k=1,\dots,K$. Here, $\text{\boldmath$e$}(k)$ denotes the $k$th unit vector. In the experiments, the basis functions for $y$ are Bernstein polynomials with maximum order $P$, $\text{\boldmath$b$}_{\text{Bs},P}(y)\in\mathbb{R}^{P+1}$. Because the transformation function $h(y)=\text{\boldmath$b$}(y)^{\top}\text{\boldmath$\vartheta$}$ must be monotone non-decreasing, we require some constraints on the parameters of the transformation function. Name \| Transformation Model \| Constraints \| Type Response ---\|---\|---\|--- Lm \| $\left(\Phi,(1,y,\text{\boldmath$x$}^{\top})^{\top},(\theta_{1},\theta_{2},-\text{\boldmath$\beta$}^{\top})^{\top}\right)$ \| $\theta_{2}>0$ \| Continuous c-probit \| $\left(\Phi,(\text{\boldmath$b$}_{\text{Bs},P}^{\top},\text{\boldmath$x$}^{\top})^{\top},(\text{\boldmath$\theta$}^{\top},-\text{\boldmath$\beta$}^{\top})^{\top}\right)$ \| $\theta_{1}\leq\dots\leq\theta_{P+1}$ \| Continuous c-logit \| $\left(F_{\operatorname{SL}},(\text{\boldmath$b$}_{\text{Bs},P}^{\top},\text{\boldmath$x$}^{\top})^{\top},(\text{\boldmath$\theta$}^{\top},-\text{\boldmath$\beta$}^{\top})^{\top}\right)$ \| $\theta_{1}\leq\dots\leq\theta_{P+1}$ \| Continuous o-logit \| $\left(F_{\operatorname{SL}},(\tilde{\text{\boldmath$y$}}^{\top},\text{\boldmath$x$}^{\top})^{\top},(\text{\boldmath$\theta$}^{\top},-\text{\boldmath$\beta$}^{\top})^{\top}\right)$ \| $\theta_{1}<\dots<\theta_{K-1},$ \| Ordinal \| \| $\theta_{K}=+\infty$ \| ### 4.1 Application: BostonHousing2 For the BostonHousing2 data we wish to predict corrected median housing prices (cmedv) from several socio-economical and environmental factors. These include per capita crime rates (crim), average number of rooms (rm), and nitric oxide concentration (nox) among others. Each observation corresponds to a single census tract in Boston. Individual cities will serve as anchors in this example because they are plausibly exogenous factors that induce heterogeneity in the observed covariates and housing prices. “Leave-one-environment-out” (LOEO) cross validation is used to demonstrate the change in estimated regression coefficients and NLL comparing a plain model without causal regularization ($\xi=0$) to three different anchor TMs over a large range of causal regularization (Figure 4). For some of the left-out towns the conditional distribution of cmedv will differ from the training distribution and contain unseen perturbations. In this case, the town will be harder to predict, leading to a worse cross-validated NLL compared to the environments which are not perturbed. We hypothesize, an anchor TM should improve prediction performance for the census tracts in these hard-to-predict towns, in analogy to the distributional robustness results described in eq. (2), whereas it should perform worse than a plain TM for environments with only mild perturbations. Figure 4: Leave-one-environment-out cross validation under increasing causal regularization for the BostonHousing2 data, with town as anchors. A linear (Lm), continuous probit (c-probit) and continuous logit (c-logit) model is fitted on 91 towns and used to predict the left out town. A: Out-of-sample NLL for the left-out census tracts. Beacon Hill, Back Bay and North End are consistently hardest to predict. Consequently, for these towns the cross- validated NLL improves with increasing causal regularization up to a certain point. For the majority of the remaining towns prediction performance decreases. We thus indeed improve worst-case prediction, in analogy to eq. (2). Note that $\log_{10}\xi=-\infty$ corresponds to the unpenalized model. B: Estimated regression coefficients, which are interpretable as difference in means (Lm), difference in transformed means (c-probit) and log odds-ratios (c-logit). Solely the c-logit model accounts for right-censored observations. With increasing causal regularization the estimates shrink towards zero, indicating that town may be a weak instrument (see Appendix D). First, a linear model assuming homoscedasticity and conditional normality is used to estimate the conditional distribution of cmdev depending on the socio- economic factors described above. A notable reduction in the observed worst- case loss is already observed for mild causal regularization ($\xi\in\\{1,10\\}$) without loosing too much predictive performance for the other environments (Figure 4 A). For stronger causal regularization, the cross-validated NLL becomes gradually worse. However, assuming a symmetric distribution for prices ignores the typical skewness of these outcomes. The c-probit model allows a non-linear basis expansion in the response and thus relaxes the homoscedasticity and conditional normality assumption. Essentially the same gain in terms of worst-case CV NLL is observed for the c-probit model compared to Lm. Figure 5 shows the predicted conditional densities for the three observations in Boston Beacon Hill and emphasizes the importance of modelling cmedv using a right-skewed distribution. A disadvantage of switching from a linear probit to a non-linear probit model is the loss of interpretability of the individual regression coefficients (e.g., Fahrmeir et al., 2007). Figure 5: Density estimates for the three census tracts (Loc 1, Loc 2, Loc 3) in Boston Beacon Hill, the hardest to predict town in terms of LOEO cross- validated NLL for $\xi=10$ (cf. Figure 4). Lm assumes equal variances and conditional normality, whereas c-probit loosens this assumption leading to more accurate, skewed distributions. Only c-logit takes into account right censoring in the data. However, also this disadvantage can be overcome in the framework of anchor TMs by specifying a different distribution function, $F_{Z}$, while keeping the basis expansion in the outcome equally flexible. In addition, the housing prices above $\$50^{\prime}000$ (cmedv = 50) are right-censored in the BostonHousing2 data, which is commonly ignored in analyses, but crucial to capture the uncertainty in predicting the skewed outcome (Gilley and Kelley Pace, 1996). The c-logit model now takes into account right censoring of the observations and yields regression coefficients that are interpretable as log odds-ratios (Lohse et al., 2017). Indeed, for census tract Loc 1 the c-logit model puts more probability mass on prices of cmedv beyond $\$50^{\prime}000$ relative to the density at its mode compared to the c-probit model, which treats the censored observations as exact (Figure 5). Taking into right censoring apparently made out-of-sample prediction for Boston Beacon Hill, Back Bay and North End easier, but the improvement through causal regularization diminished slightly (Figure 4 A). Coefficient estimates for all three models are shown in Figure 4 B. With increasing amount of causal regularization, all estimates shrink towards 0, which indicates town may be a weak instrument (Imbens and Rosenbaum, 2005), for more details see Appendix D. However, intermediate amounts of causal regularization yield estimates for which anchors and score residuals are somewhat de-correlated and still lead to the desired robustness against perturbations in unseen environments. ### 4.2 Simulations In this the section, the results of the simulation scenarios are presented. The parameters for the SEMs used to simulate from the models in Table 1 are summarized in Table 2. #### 4.2.1 Experimental Setup We begin with a comparison of linear anchor regression and the distributional version of linear anchor regression in scenario la, which was first used to study anchor regression in Bühlmann (2020). The non-linear scenario nla also stems from Bühlmann (2020), which we use to show how shift transformation models can estimate non-linear conditional expectation function, albeit for their linear model formulation in the covariates. For the last two scenarios iv1 and iv2, the IV assumptions hold, i.e., the anchors influence only $X$. Scenario iv1 showcases discrete anchors and a continuous response and a non- linear transformation function, which we model by a continuous probit regression. Scenario iv2 features an ordinal response and a more detailed simulation, including various shift strengths. In scenarios la, iv1 and iv2, the effect from $X$ to $Y$ is denoted by $\beta$, whereas the non-linear $f$ is used in scenario nla. For the data generating processes that involve transformation models, the transformation function $h$ is specified. For ordinal responses the number of classes, $K$, and for continuous outcomes, the maximum order of the Bernstein basis, $P$, determines the number of parameters for the baseline transformation. The parameters of the Bernstein basis are fixed by applying the transformation function $h$ to a $(P+1)$-vector of evenly spaced values in the desired support of $Y$. In turn, such a basis approximation leads to a distribution approximation for the true distribution of $Y$ which improves as $P$ increases. However, the transformation function is constrained to be monotone non-decreasing, which makes a parsimonious parametrization sufficient. Table 2: Simulation scenarios used to empirically evaluate distributional anchor regression. Scenarios la and nla are adapted from Bühlmann (2020) and will be used to evaluate linear and continuous probit anchor TMs. The seven covariates omitted in the table in both scenarios are noise covariates, i.e., $\beta_{j}=0,\;j\neq 2,3$. In nla, $f(\text{\boldmath$X$})=X_{2}+X_{3}+\mathds{1}(X_{2}\leq 0)+\mathds{1}(X_{2}\leq 0.5)\mathds{1}(X_{3}\leq 1)$. Both use $n_{\text{train}}=300$ and $n_{\text{test}}=2000$ observations. In the iv scenarios the instrumental variables assumptions hold, because the anchor neither influences the hidden confounders nor the response. The scenarios generalize Example 1 in Rothenhäusler et al. (2018) to anchor TMs with a continuous outcome (iv1) and an ordinal outcome (iv2). Both use $n_{\text{train}}=1000$ and $n_{\text{test}}=2000$ observations. Scenario \| Model \| Distributions \| Parameters \| Perturbation ---\|---\|---\|---\|--- la \| Lm \| $\begin{array}[]{rl}\varepsilon_{Y}&\sim\mathcal{N}(0,0.25^{2})\\\ \varepsilon_{\text{\boldmath$X$}}&\sim\mathcal{N}_{10}(0,\operatorname{Id})\\\ \varepsilon_{H}&\sim\mathcal{N}(0,1)\\\ \varepsilon_{\text{\boldmath$A$}}&\sim\mathcal{N}_{2}(0,\operatorname{Id})\\\ \end{array}$ \| $\begin{array}[]{rl}\beta_{j}&=3,\;j=2,3\\\ \text{$\mathbf{M}$}_{Y}&=(-2,0)\\\ \text{$\mathbf{M}$}_{X_{j}}&\sim\mathcal{N}_{2}(0,\operatorname{Id}),\;j=1,\dots,10\\\ \text{$\mathbf{M}$}_{\text{\boldmath$H$}}&=0\\\ \text{$\mathbf{B}$}_{Y\text{\boldmath$H$}}&=1\\\ \text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$H$}}&=(1,\dots,1)\\\ \end{array}$ \| $\text{\boldmath$A$}\sim\mathcal{N}_{2}(0,10\operatorname{Id}))$ nla \| c-probit \| $\begin{array}[]{rl}\varepsilon_{Y}&\sim\mathcal{N}(0,0.25^{2})\\\ \varepsilon_{\text{\boldmath$X$}}&\sim\mathcal{N}_{10}(0,0.5^{2}\operatorname{Id})\\\ \varepsilon_{H}&\sim\mathcal{N}(0,1)\\\ \varepsilon_{\text{\boldmath$A$}}&\sim\mathcal{N}_{2}(0,\operatorname{Id})\end{array}$ \| $\begin{array}[]{rl}f&=f(X_{2},X_{3})\\\ \text{$\mathbf{M}$}_{Y}&=\text{$\mathbf{M}$}_{\text{\boldmath$H$}}=0\\\ \text{$\mathbf{M}$}_{X_{j}}&=(1,1),\;j=1,\dots,10\\\ \text{$\mathbf{B}$}_{Y\text{\boldmath$H$}}&=3\\\ \text{$\mathbf{B}$}_{\text{\boldmath$X$}\text{\boldmath$H$}}&=(2,\dots,2)\\\ \end{array}$ \| $\begin{array}[]{l}\text{\boldmath$A$}\sim\mathcal{N}_{2}(\text{\boldmath$\mu$},\operatorname{Id}))\\\ \text{\boldmath$\mu$}\sim\mathcal{N}_{2}(1,2^{2}\operatorname{Id})\end{array}$ iv1 \| c-probit \| $\begin{array}[]{rl}\varepsilon_{X}&\sim\mathcal{N}(0,0.75^{2})\\\ \varepsilon_{H}&\sim\mathcal{N}(0,0.75^{2})\\\ \varepsilon_{A}&\sim\operatorname{Rademacher}\end{array}$ \| $\begin{array}[]{rl}\beta&=0.3\\\ \text{$\mathbf{M}$}_{Y}&=\text{$\mathbf{M}$}_{H}=0\\\ \text{$\mathbf{M}$}_{X}&=0.3\\\ \text{$\mathbf{B}$}_{YH}&=\text{$\mathbf{B}$}_{XH}=0.6\\\ h&=\Phi^{-1}\circ F_{\chi^{2}_{3}}\\\ P&=6\\\ \end{array}$ \| $\begin{array}[]{l}\operatorname{do}(A=3.6)\end{array}$ iv2 \| o-logit \| $\begin{array}[]{rl}\varepsilon_{X}&\sim\mathcal{N}(0,0.75^{2})\\\ \varepsilon_{H}&\sim\mathcal{N}(0,0.75^{2})\\\ \varepsilon_{A}&\sim\operatorname{Rademacher}\end{array}$ \| $\begin{array}[]{rl}\beta&=0.5\\\ \text{$\mathbf{M}$}_{Y}&=\text{$\mathbf{M}$}_{H}=0\\\ \text{$\mathbf{M}$}_{X}&\in\\{-1,0.5,1\\}\\\ \text{$\mathbf{B}$}_{YH}&=\text{$\mathbf{B}$}_{XH}=1.5\\\ h&=F_{\operatorname{SL}}^{-1}\circ\operatorname{Id}\\\ K&\in\\{4,6,10\\}\\\ \end{array}$ \| $\begin{array}[]{l}\operatorname{do}(A=a)\\\ a\in\\{0,1,1.8,3\\}\end{array}$ #### 4.2.2 Scenario la The linear anchor scenario la was first presented in Bühlmann (2020) for the $L_{2}$ anchor loss. The performance gain of using anchor regression compared to a plain linear model is shown in Figure 6 for the $L_{2}$ anchor loss (left) and the distributional anchor loss (right). Figure 6: Test performance averaged over 100 simulations for scenario la for $n_{\text{train}}=300$ and $n_{\text{test}}=2000$. The $\alpha$-quantiles of test absolute prediction error $\text{APE}:=\lvert y-\hat{y}\rvert$, where $\hat{y}$ denotes the conditional median, is shown for linear anchor regression (A) and the negative log-likelihood contributions for distributional (conditionally Gaussian) linear $L_{2}$ anchor regression (B). The two models are equivalent up to estimating the residual variance via maximum likelihood in the distributional anchor TM. The change in perspective from an $L_{2}$ to a distributional loss requires different evaluation metrics, of which the log-likelihood, being a proper scoring rule (Good, 1952), is the most natural choice. A performance gain across all quantiles of the log-likelihood contributions can be observed. However, the larger the quantile, the higher the performance gain. The extent of causal regularization was chosen based on the theoretical insight that, in a multivariate normal model, $\gamma$ can be interpreted as the quantile of a $\chi^{2}_{1}$ distribution, which relates the expected size of the unobserved perturbations to the conditional mean squared error given the anchors (Lemma 1 in Rothenhäusler et al., 2018). #### 4.2.3 Scenario nla In scenario nla, which features non-linear anchor regression, a continuous probit model is fitted. Figure 7 shows a vast gain in performance over all quantiles, comparable to what was observed in Bühlmann (2020) with a different nonlinear anchor boosting method. Figure 7: Test performance over 100 simulations for scenario nla with $n_{\text{train}}=300$ and $n_{\text{test}}=2000$. $\operatorname{NLL}$ (A) and $\alpha$-quantiles of the negative log-likelihood contributions (B) for the c-probit anchor TM. The test data are generated under strong push- interventions on the distribution of the anchors (cf. Table 2). This gain in performance can be explained in the causal generalization framework of Christiansen et al., because the causal function linearly extends outside the support of $\text{\boldmath$X$}_{\text{train}}$. #### 4.2.4 Scenario iv1 In scenario la the anchors influence the response and hidden confounders, violating the instrumental variable assumptions. Scenario iv1 explores binary anchors as valid instruments, while the baseline transformation $\text{\boldmath$b$}_{\text{Bs},6}(y)^{\top}\text{\boldmath$\theta$}$ is non- linear. Note that although the model formulation is linear in $\beta$, the conditional expectation function may be non-linear, because of the non-linear transformation function. This is inspired by Example 1 in Rothenhäusler et al. (2018) and translates it into a transformation model SEM from Definition 3 for continuous but non-normal outcomes. Figure 8: Test performance over 100 simulations for scenario iv1 with $n_{\text{train}}=1000$ and $n_{\text{test}}=2000$. Quantiles of the individual log-likelihood contributions (A) and estimates of $\beta$ (B) for increasingly strong causal regularization. The ground truth is indicated by a dashed line. The test data are generated under the intervention $\operatorname{do}(\text{\boldmath$A$}=3.6)$. The test data were generated using $\operatorname{do}(A=3.6)$, which leads to better predictive performance under stronger causal regularization (Figure 8 A). Additionally, although $A$ is a valid instrument, the causal parameter seems to be slightly biased in the finite sample case for larger $\xi$ (Figure 8 B). #### 4.2.5 Scenario iv2 The instrumental variable assumptions hold also in the last scenario iv2. However, the response’s distribution is now ordered categorical and more varying parameters are considered, including the number of classes of the response, the strength of the instruments and the perturbations in the test data (cf. Table 2). Figure 9: Test performance and coefficient estimates over 200 simulations for scenario iv2. Because the results are comparable for differing sample sizes and numbers of classes, solely the results for $n_{\text{train}}=1000$ and $K=10$ are displayed. A: Test log-likelihood contributions for varyingly strong instruments (columns) and perturbation sizes (rows). B: Parameter estimates $\hat{\beta}$ for all intervention scenarios together, since they do not influence estimation. The simulated ground truth $\beta=0.5$ is indicated with a dashed line. Figure 9 depicts test NLL alongside the estimated shift parameter, $\hat{\beta}$. Also here, in case of strong perturbations anchor regression protects against unseen perturbations for larger $\xi$ (e.g., $\operatorname{do}(A=1.8)$ and $\operatorname{do}(A=3)$ for $\text{$\mathbf{M}$}_{\text{\boldmath$A$}}=-0.5$) resulting in improved test predictions. However, if the shift perturbations are not innovative, test prediction suffers with increasing amounts of causal regularization (e.g., $\operatorname{do}(A=1)$ for $\text{$\mathbf{M}$}_{\text{\boldmath$X$}}=2$). Note the interplay between the strength of the anchors, $\text{$\mathbf{M}$}_{\text{\boldmath$X$}}$, and the strength of the shift interventions. For larger $\text{$\mathbf{M}$}_{\text{\boldmath$X$}}$, the training data becomes more heterogeneous and the larger shifts are not as innovative, resulting in a weaker performance of anchor TMs for increasing $\xi$. Again, the estimated shift parameter seems to be slightly biased (Figure 9 B). ## 5 Discussion and Outlook The proposed method of distributional anchor regression generalizes (non-) linear anchor regression beyond the assumptions of normality and homoscedasticity and beyond estimating solely a conditional mean. In an exemplary analysis of the BostonHousing2 data we have illustrated the flexibility of anchor TMs and demonstrated a gain in prediction performance in terms of worst-case cross validated log-likelihood, while preserving interpretability and appropriately accounting for censored observations. The simulations show comparable results to established linear and non-linear anchor regression models under both IV and invalid IV scenarios and extend the notion of invariance between residuals and environments to other than continuous responses. Although anchor TMs are generally unable to recover the causal parameter, we argue that the “diluted causal” (Bühlmann, 2020) parameter, $\hat{\text{\boldmath$\beta$}}^{\to\infty}:=\hat{\text{\boldmath$\beta$}}(\xi)$ as $\xi\to\infty$, is interesting in its own right, for it induces invariance between anchors and score residuals and allows robust test predictions in the presence of distributional shifts. Much like the causal parameter, the diluted causal parameter leads to (aspects of) invariant conditional distributions across environments. Even though the powerful causal interpretation is lost, distributional anchor regression yields models that allow causally flavored interpretations in terms of stabilization and robustification across environments. Anchor TMs estimate the whole conditional distribution and thus enable robust prediction of a multitude of responses, which we demonstrated for (censored) continuous and ordered categorical responses. Possible extensions of anchor TMs are numerous. For instance, other types of responses include count and time-to-event data. The framework of anchor TMs contains a fully parametric version of the Cox proportional hazard model (Hothorn et al., 2018), although an extension to classical survival models is also possible. For instance, the Cox proportional hazard model (Cox, 1972) can be fitted by substituting the likelihood for the partial likelihood (Cox, 1975) in the distributional anchor loss, while the score residuals are equivalent to martingale residuals (cf. Appendix B; Barlow and Prentice, 1988; Therneau et al., 1990). As in high- dimensional linear and non-linear anchor regression, anchor TMs could be fitted under a lasso penalty (Tibshirani, 1996). The idea of using a different class of residuals can also be translated to other model classes, such as deviance residuals for GLMs, as long as the theoretical requirements discussed in Section 3 are met. In terms of future work, further theoretical investigation of the distributional anchor loss, such as bounds on the generalization error, is warranted. So far we restricted distributional regression to linear (in $x$) TMs because of their already highly flexible nature and simplicity in the considered DGPs. However, more complex experimental designs require, for instance, random effects or time-varying effects of covariates in time-to- event data. Taken together, anchor TMs lay the foundation for future work on distributional extensions of anchor regression. ##### Acknowledgements. The research of L. Kook and B. Sick was supported by the Novartis Research Foundation (FreeNovation 2019). The research of P. Bühlmann was supported by the European Research Council under the Grant Agreement No 786461 (CausalStats - ERC-2017-ADG). We thank Torsten Hothorn for fruitful discussions and input on the exemplary application. We also thank Malgorzata Roos for her helpful comments on the manuscript. ## References Abadi et al. [2015] Martín Abadi et al. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, 2015\. URL https://tensorflow.org/. Software available from tensorflow.org. * Angrist et al. [1996] Joshua D. Angrist, Guido W. Imbens, and Donald B. Rubin. Identification of Causal Effects Using Instrumental Variables. _Journal of the American Statistical Association_ , 91(434):444–455, 1996. doi: 10.1080/01621459.1996.10476902. URL https://www.tandfonline.com/doi/abs/10.1080/01621459.1996.10476902. * Azzalini and Bowman [1990] A. Azzalini and A. W. Bowman. 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Random vectors are written like random variables, but in bold, $X$, and realizations thereof in lowercase bold, $x$. Stacked to a matrix for $n$ observations, we write $\text{$\mathbf{X}$}=(\text{\boldmath$x$}_{1}^{\top},\dots,\text{\boldmath$x$}_{n}^{\top})^{\top}\in\mathbb{R}^{n\times p}$. Matrices are written in bold, normal uppercase, $\mathbf{A}$, vectors in bold italic lowercase, $a$. The probability measure of a random variable $Y$ is denoted by $\mathbb{P}_{Y}$. Coefficient matrices in the SEMs are denoted by $\mathbf{M}$ for the anchors $A$ and by $\mathbf{B}$ for all other components. When restricting the coefficient matrix $\mathbf{B}$ to a single component, e.g., the effect of $X$ on $Y$, we write $\text{$\mathbf{B}$}_{Y\text{\boldmath$X$}}$. Because for $\mathbf{M}$ it is clear from context, we omit the $A$ in the index. ## Appendix B Background on Score Residuals Stigler [2016] considers residuals the seventh “pillar of statistical wisdom”, which highlights their conceptual importance. Here, we will briefly justify the use of the score residual in transformation models based on the work of Lagakos [1981], who introduced martingale residuals for survival analysis under interval censoring. Farrington [2000] presents a summary of the different types of residuals used in survival analysis. The scope of score residuals in transformation models is to take the notion of a residual and generalize it to a wider class of models, namely TMs, and allow for all kinds of uninformative censoring. In Definition 2, we note that the score residual can be interpreted as the score contribution of a newly introduced intercept parameter, which is constrained to zero. This is equivalent to formulating a score test for $\alpha=0$ for a covariate that is not yet included in the model. Since $\alpha$ is constrained to zero in the whole procedure, we do not need to evaluate the model under the alternative hypothesis. Note also, that we restrict $\alpha$ to an intercept term on the scale of the transformation function. Farrington gives a compelling argument why one should do so. This connection is drawn next. As an adjustment of Cox-Snell residuals [Cox and Snell, 1968], Lagakos proposes a centered version of Cox-Snell residuals. Barlow and Prentice [1988] later drew the connection to stochastic calculus and coined the term “martingale residual” [see also Therneau et al., 1990]. For interval and right censored, as well as exact responses, Lagakos residuals have a direct connection to score statistics [Farrington, 2000]. The setting is based in survival analysis, but the connection to transformation models will become apparent. Lagakos starts from a general proportional hazards model where $\displaystyle\lambda(t\|\text{\boldmath$x$})=\lambda(t)\exp(\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$})$ denotes the hazard function depending on time $t$ and covariates $x$, which can be decomposed into a product of a baseline hazard function $\lambda(t)$ and the exponential function of a linear predictor in the covariates $\beta$. The cumulative hazard function is then defined as $\displaystyle\Lambda(t\|\text{\boldmath$x$})=\int_{0}^{t}\lambda(u\|\text{\boldmath$x$})du.$ From there, the cdf can be recovered as $\displaystyle F_{T}(t\|\text{\boldmath$x$})=1-\exp(-\Lambda(t)\exp(\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$})).$ By definition $\Lambda(t)>0\;\forall t$, hence $\displaystyle F_{T}(t\|\text{\boldmath$x$})=1-\exp(-\exp(\log\Lambda(t)+\text{\boldmath$x$}^{\top}\text{\boldmath$\beta$})),$ which is a transformation model with Minimum-Extreme-Value inverse-link, $F_{Z}=F_{\operatorname{MEV}}$, and the transformation function, $h(t)=\log\Lambda(t)$, is the log cumulative baseline hazard. Farrington now assumes that the baseline hazard function belongs to a family $\mathcal{F}$ that is closed under scaling by a factor $\gamma\in\mathbb{R^{++}}$, i.e., $\displaystyle\Lambda(t\rvert x)\in\mathcal{F}\Rightarrow\gamma\Lambda(t\|\text{\boldmath$x$})\in\mathcal{F}.$ The Lagakos residual is now defined as $\displaystyle\hat{r}^{L}=\frac{\partial}{\partial\alpha}\ell\bigg{\rvert}_{\hat{\text{\boldmath$\beta$}},\hat{S}_{0}}$ for $\alpha=\log\gamma$ and $\hat{S}_{0}$ being the estimated survivor curve under $\text{\boldmath$x$}=0$, i.e., $\hat{S}_{0}=\exp(-\hat{\Lambda}(t\|\text{\boldmath$x$}=0))$. This translates to the family of log (cumulative) hazard functions being closed under addition of $\alpha=\log\gamma\in\mathbb{R}$, which is exactly the case for transformation models. This step corresponds to adding and constraining a new intercept term to zero on the scale of the transformation function. Lagakos residuals behave like the usual score statistic, i.e., they have mean zero asymptotically and are asymptotically uncorrelated. ## Appendix C Computational Details Anchor TMs, simulation scenarios and visualizations are written in the R language for statistical computing [Version 3.6.3, R Core Team, 2020]. To implement distributional anchor regression, note that the transformation model log-likelihood is concave w.r.t. $\vartheta$, unless all observations are right censored. The penalty resulting from the quadratic form $\text{\boldmath$r$}^{\top}\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}\text{\boldmath$r$}$ is convex if $r$, from Definition 2, is affine or convex [Boyd and Vandenberghe, 2004]. In case of exact observations in Lm and c-probit, the resulting penalty is convex and thus solvable by a constrained convex optimizer. Kook and Hothorn [2020] implement regularized transformation models under elastic net penalties in tramnet, using the domain-specific language optimizer CVXR [Fu et al., 2020]. From tramnet, we use the TM likelihood implementation. For the interval censored or right censored models (o-logit, c-logit) we fit the models using (stochastic) gradient descent (SGD) from package TensorFlow [Abadi et al., 2015]. The implementation of the interval censored log-likelihood for ordinal TMs was taken from Kook et al. [2020] and we used SGD with the Adam optimizer [Kingma and Ba, 2015] with learning rate $10^{-3}$, batch size $250$ and $200$ epochs. Parameters were initialized with the maximum likelihood estimate for $\xi=0$ obtained via tram::Polr() [Hothorn, 2020]. ## Appendix D Invalid Instruments and Shrinkage of Estimates In the linear IV setting an instrument is called weak if it has little impact on the explanatory variables $X$. Consequently, a two-stage least squares procedure will yield homogeneous predictions for $X$ in the first stage, because $\displaystyle\text{\boldmath$X$}\perp\text{\boldmath$A$}\implies\boldsymbol{\Pi}_{\text{$\mathbf{A}$}}\text{$\mathbf{X}$}\equiv\operatorname{const.}$ This leads to regressing the response on a matrix with constant columns, making it impossible to separate $\beta$ from an intercept. Thus, in case of weak instruments, the resulting estimates $\hat{\text{\boldmath$\beta$}}$ will shrink towards $0$ as $\xi\to\infty$ when using a causal regularizer. This explains the effect seen for the BostonHousing2 data in Section 4.1 and makes the diluted causal parameter impossible to study, because it is equal to constant 0. However, intermediary amounts of causal regularization yield a benefit in terms of worst-case LOEO CV (cf. Figure 4).
# Word Alignment by Fine-tuning Embeddings on Parallel Corpora Zi-Yi Dou, Graham Neubig Language Technologies Institute, Carnegie Mellon University <EMAIL_ADDRESS> ###### Abstract Word alignment over parallel corpora has a wide variety of applications, including learning translation lexicons, cross-lingual transfer of language processing tools, and automatic evaluation or analysis of translation outputs. The great majority of past work on word alignment has worked by performing unsupervised learning on parallel text. Recently, however, other work has demonstrated that pre-trained contextualized word embeddings derived from multilingually trained language models (LMs) prove an attractive alternative, achieving competitive results on the word alignment task even in the absence of explicit training on parallel data. In this paper, we examine methods to marry the two approaches: leveraging pre-trained LMs but fine-tuning them on parallel text with objectives designed to improve alignment quality, and proposing methods to effectively extract alignments from these fine-tuned models. We perform experiments on five language pairs and demonstrate that our model can consistently outperform previous state-of-the-art models of all varieties. In addition, we demonstrate that we are able to train multilingual word aligners that can obtain robust performance on different language pairs. Our aligner, AWESoME (Aligning Word Embedding Spaces of Multilingual Encoders), with pre-trained models is available at https://github.com/neulab/awesome-align. ## 1 Introduction Word alignment is a useful tool to tackle a variety of natural language processing (NLP) tasks, including learning translation lexicons (Ammar et al., 2016; Cao et al., 2019), cross-lingual transfer of language processing tools (Yarowsky et al., 2001; Padó and Lapata, 2009; Tiedemann, 2014; Agić et al., 2016; Mayhew et al., 2017; Nicolai and Yarowsky, 2019), semantic parsing (Herzig and Berant, 2018) and speech recognition (Xu et al., 2019). In particular, word alignment plays a crucial role in many machine translation (MT) related methods, including guiding learned attention (Liu et al., 2016), incorporating lexicons during decoding (Arthur et al., 2016), domain adaptation (Hu et al., 2019), unsupervised MT (Ren et al., 2020) and automatic evaluation or analysis of translation models (Bau et al., 2018; Stanovsky et al., 2019; Neubig et al., 2019; Wang et al., 2020). However, with neural networks advancing the state of the arts in almost every field of NLP, tools developed based on the 30-year-old IBM word-based translation models (Brown et al., 1993), such as GIZA++ (Och and Ney, 2003) or fast-align (Dyer et al., 2013), remain popular choices for word alignment tasks. Figure 1: Cosine similarities between subword representations in a parallel sentence pair before and after fine-tuning. Red boxes indicate the gold alignments. One alternative to using statistical word-based translation models to learn alignments would be to instead train state-of-the-art neural machine translation (NMT) models on parallel corpora, and extract alignments therefrom, as examined by Luong et al. (2015); Garg et al. (2019); Zenkel et al. (2020). However, these methods have two disadvantages (also shared with more traditional alignment methods): (1) they are directional and the source and target side are treated differently and (2) they cannot easily take advantage of large-scale contextualized word embeddings derived from language models (LMs) multilingually trained on monolingual corpora (Devlin et al., 2019; Lample and Conneau, 2019; Conneau et al., 2020), which have proven useful in other cross-lingual transfer settings (Libovickỳ et al., 2019; Hu et al., 2020b). In the field of word alignment, Sabet et al. (2020) have recently proposed methods to align words using multilingual contextualized embeddings and achieve good performance even in the absence of explicit training on parallel data, suggesting that these are an attractive alternative for neural word alignment. In this paper, we investigate if we can combine the best of the two lines of approaches. Concretely, we leverage pre-trained LMs and fine-tune them on parallel text with not only LM-based objectives, but also unsupervised objectives over the parallel corpus designed to improve alignment quality. Specifically, we propose a self-training objective, which encourages aligned words to have further closer contextualized representations, and a parallel sentence identification objective, which enables the model to bring parallel sentences' representations closer to each other. In addition, we propose to effectively extract alignments from these fine-tuned models using probability thresholding or optimal transport. We perform experiments on five different language pairs and demonstrate that our model can achieve state-of-the-art performance on all of them. In analysis, we find that these approaches also generate more aligned contextualized representations after fine-tuning (see Figure 1 as an example) and we can incorporate supervised signals within our paradigm. Importantly, we show that it is possible to train multilingual word aligners that can obtain robust performance even in zero-shot settings, making them a valuable tool that can be used out-of-the-box with good performance over a wide variety of language pairs. ## 2 Methods Formally, the task of word alignment can be defined as: given a sentence $\mathbf{x}=\langle x_{1},\cdots,x_{n}\rangle$ in the source language and its corresponding parallel sentence $\mathbf{y}=\langle y_{1},\cdots,y_{m}\rangle$ in the target language, a word aligner needs to find a set of pairs of source and target words: $A=\\{\langle x_{i},y_{j}\rangle:x_{i}\in\mathbf{x},y_{j}\in\mathbf{y}\\},$ where for each word pair $\langle x_{i},y_{j}\rangle$, $x_{i}$ and $y_{j}$ are semantically similar to each other within the context of the sentence. In the following paragraphs, we will first illustrate how we extract alignments from contextualized word embeddings, then describe our objectives designed to improve alignment quality. ### 2.1 Extracting Alignments from Embeddings Contextualized word embedding models such as BERT (Devlin et al., 2019) and RoBERTa (Liu et al., 2019) represent words using continuous vectors calculated in context, and have achieved impressive performance on a diverse array of NLP tasks. Multilingually trained word embedding models such as multilingual BERT can generate contextualized embeddings across different languages. These models can be used to extract contextualized word embeddings $h_{\mathbf{x}}=\langle h_{x_{1}},\cdots,h_{x_{n}}\rangle$ and $h_{\mathbf{y}}=\langle h_{y_{1}},\cdots,h_{y_{m}}\rangle$ for each pair of parallel sentences $\mathbf{x}$ and $\mathbf{y}$. Specifically, this is done by extracting the hidden states of the $i$-th layer of the model, where $i$ is an empirically-chosen hyper-parameter. Given these contextualized word embeddings, we propose two methods to calculate unidirectional alignment scores based on probability simplexes and optimal transport. We then turn these alignment scores into alignment matrices and reconcile alignments in the forward and backward directions. Figure 2: Extracting word alignments from multilingual BERT using probability thresholding (softmax). Red boxes denote the gold alignments. #### Probability Thresholding. In this method, for each word in the source/target sentence, we calculate a value on the probability simplex for each word in the aligned target/source sentence, and then select all values that exceed a particular threshold as ``aligned'' words. Concretely, taking inspiration from attention mechanisms (Bahdanau et al., 2015; Vaswani et al., 2017), we take the contextualized embeddings $h_{\mathbf{x}}$ and $h_{\mathbf{y}}$ and compute the dot products between them and get the similarity matrix: $S=h_{\mathbf{x}}h_{\mathbf{y}}^{T}.$ Then, we apply a normalization function $\mathcal{N}$ to convert the similarity matrix into values on the probability simplex $S_{\mathbf{xy}}=\mathcal{N}(S)$, and treat $S_{\mathbf{xy}}$ as the source- to-target alignment matrix. In this paper, we propose to use softmax and a sparse variant $\alpha$-entmax (Peters et al., 2019) to do the normalization. Compared with the softmax function, $\alpha$-entmax can produce sparse alignments for any $\alpha>1$ and assign non-zero probability to a short list of plausible word pairs, where a higher $\alpha$ will lead to a more sparse alignment. #### Optimal Transport. The goal of optimal transport (Monge, 1781; Cuturi, 2013) is to find a mapping that moves probability from one distribution to another, which can be used to find an optimal matching of similar words between two sequences (Kusner et al., 2015). Formally, in a discrete optimal transport problem, we are given two point sets $\\{{x_{i}}\\}_{i=1}^{n}$ and $\\{{y_{j}}\\}_{j=1}^{m}$ associated with their probability distributions $p_{\mathbf{x}}$ and $p_{\mathbf{y}}$ where $\sum_{i}p_{x_{i}}=1$ and $\sum_{j}p_{y_{j}}=1$. Also, a function $C({x_{i}},{y_{j}})$ defines the cost of moving point ${x_{i}}$ to ${y_{j}}$. The goal of optimal transport is to find a mapping that moves probability mass from $\\{{x_{i}}\\}_{i=1}^{n}$ to $\\{{y_{j}}\\}_{j=1}^{m}$ and the total cost of moving the mass between points is minimized. In other words, it finds the transition matrix $S_{\mathbf{xy}}$ that minimizes: $\sum_{i,j}C({x_{i}},{y_{j}}){S_{\mathbf{xy}}}_{ij},$ (1) where $S_{\mathbf{xy}}\mathbf{1}_{m}=p_{\mathbf{x}}$ and $S_{\mathbf{xy}}^{T}\mathbf{1}_{n}=p_{\mathbf{y}}$. The resulting transition matrix is self-normalized and sparse (Swanson et al., 2020), making it appealing alternative towards extracting alignments from word embeddings. In this paper, we propose to adapt optimal transport techniques to the task of word alignment. Concretely, we treat the parallel sentences $\mathbf{x}$ and $\mathbf{y}$ as two point sets and assume each word is uniformly distributed. The cost function is obtained by computing the pairwise distance (e.g. cosine distance) between $h_{\mathbf{x}}$ and $h_{\mathbf{y}}$, and all the distance values are scaled to [0, 1] with min-max normalization. The optimal transition matrix ${S_{\mathbf{xy}}}$ to Equation 1 can be calculated using the Sinkhorn- Knopp matrix scaling algorithm (Sinkhorn and Knopp, 1967). If the value of ${S_{\mathbf{xy}}}_{ij}$ is high, ${x_{i}}$ and ${y_{j}}$ are likely to have similar semantics and values that exceed a particular threshold will be considered as ``aligned''. #### Extracting Bidirectional Alignments. After we obtain both the source-to-target and target-to-source alignment probability matrices $S_{\mathbf{xy}}$ and $S_{\mathbf{yx}}$ using the previous methods, we can deduce the final alignment matrix by taking the intersection of the two matrices: $A=(S_{\mathbf{xy}}>c)(S_{\mathbf{yx}}^{T}>c),$ where $c$ is a threshold and $A_{ij}=1$ means $x_{i}$ and $y_{j}$ are aligned. Note that growing heuristics such as grow-diag-final (Och and Ney, 2000; Koehn et al., 2005) that are popular in statistical word aligners can also be applied in our alignment extraction algorithms, and we will demonstrate the effect of these heuristics in the experiment section. #### Handling Subwords. Subword segmentation techniques (Sennrich et al., 2016; Kudo and Richardson, 2018) are widely used in training LMs, thus the above alignment extraction methods can only produce alignments on the subword level. To convert them to word alignments, we follow previous work (Sabet et al., 2020; Zenkel et al., 2020) and consider two words to be aligned if any of their subwords are aligned. Figure 2 shows a concrete example of how we extract word-level alignments from a pre-trained embedding model. ### 2.2 Fine-tuning Contextualized Embeddings for Word Alignment While language models can be used to produce reasonable word alignments even without any fine-tuning (Sabet et al., 2020), we propose objectives that further improve their alignment ability if we have access to parallel data. #### Masked Language Modeling (MLM). Gururangan et al. (2020) suggest that we can gain improvements in downstream tasks by further pre-training LMs on the task datasets. Therefore, we propose to fine-tune the LMs with a masked language modeling objective on both the source and target side of parallel corpora. Specifically, given a pair of parallel sentences $\mathbf{x}$ and $\mathbf{y}$, we choose 15% of the token positions randomly for both $\mathbf{x}$ and $\mathbf{y}$, and for each chosen token, we replace it with (1) the [MASK] token 80% of the time (2) a random token 10% of the time and (3) unchanged 10% of the time. The model is trained to reconstruct the original tokens given the masked sentences $\mathbf{x}^{mask}$ and $\mathbf{y}^{mask}$: $L_{MLM}=\log p(\mathbf{x}\|\mathbf{x}^{mask})+\log p(\mathbf{y}\|\mathbf{y}^{mask}).$ (2) #### Translation Language Modeling (TLM). The MLM objective only requires monolingual data and the model cannot make direct connections between parallel sentences. To solve the issue, similarly to Lample and Conneau (2019), we concatenate parallel sentences $\mathbf{x}$ and $\mathbf{y}$ and perform MLM on the concatenated data. Compared with MLM, the translation language modeling (TLM) objective enable the model to align the source and target representations. Different from Lample and Conneau (2019), we feed source and target sentences twice in different orders instead of resetting the positions of target sentences: $\displaystyle L_{TLM}$ $\displaystyle=\log p([\mathbf{x};\mathbf{y}]\|[\mathbf{x}^{mask};\mathbf{y}^{mask}])$ (3) $\displaystyle+\log p([\mathbf{y};\mathbf{x}]\|[\mathbf{y}^{mask};\mathbf{x}^{mask}]).$ #### Self-training Objective (SO). We also propose a self-training objective for fine-tuning LMs which is similar to the EM algorithm used in the IBM models and the agreement constraints in Tamura et al. (2014). Specifically, at each training step, we first use our alignment extraction methods (described in Section 2.1) to extract the alignment $A$ for $\mathbf{x}$ and $\mathbf{y}$, then maximize the following objective: $L_{SO}=\sum_{i,j}A_{ij}\frac{1}{2}(\frac{S_{{\mathbf{xy}}_{ij}}}{n}+\frac{S_{{\mathbf{yx}}_{ij}}}{m}).$ (4) Intuitively, this objective encourages words aligned in the first pass of alignment to have further closer contextualized representations. In addition, because of the intersection operation during extraction, the self-training objective can ideally reduce spurious alignments and encourage the source-to- target and target-to-source alignments to be symmetrical to each other by exploiting their agreement (Liang et al., 2006). #### Parallel Sentence Identification (PSI). We also propose a contrastive parallel sentence identification loss that attempts to make parallel sentences more similar than mismatched sentence pairs (Liu and Sun, 2015; Legrand et al., 2016). This encourages the overall alignments of embeddings on both word and sentence level to be closer together. Concretely, we randomly select a pair of parallel or non-parallel sentences $\langle\mathbf{x}^{\prime},\mathbf{y}^{\prime}\rangle$ from the training data with equal probability. Then, the model is required to predict whether the two sampled sentences are parallel or not. The representation of the first [CLS] token is fed into a multi-layer perceptron to output a prediction score $s(\mathbf{x}^{\prime},\mathbf{y}^{\prime})$. Denoting the binary label as $l$, the objective function can be written as: $L_{PSI}=l\log s(\mathbf{x}^{\prime},\mathbf{y}^{\prime})+(1-l)\log(1-s(\mathbf{x}^{\prime},\mathbf{y}^{\prime})).$ (5) #### Consistency Optimization (CO). While the self-training objective can potentially improve the symmetricity between forward and backward alignments, following previous work on machine translation and multilingual representation learning (Cohn et al., 2016; Zhang et al., 2019; Hu et al., 2020a), we use an objective to explicitly encourage the consistency between the two alignment matrices. Specifically, we maximize the trace of $S_{\mathbf{xy}}^{\mathrm{T}}S_{\mathbf{yx}}$: $L_{CO}=-\frac{\text{trace}(S_{\mathbf{xy}}^{\mathrm{T}}S_{\mathbf{yx}})}{\min(m,n)}.$ (6) #### Our Final Objective. In summary, our training objective is a combination of the proposed objectives and we train the model with them jointly at each training step: $L=L_{MLM}+L_{TLM}+L_{SO}+L_{PSI}+\beta L_{CO},$ where $\beta$ is set to 0 or 1 in our experiments. \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|--- #Train Sents. \| 1.9M \| 1.1M \| 450K \| 444K \| 40K #Test Sents. \| 508 \| 447 \| 248 \| 582 \| 450 Table 1: Statistics of datasets. Model \| Setting \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|---\|--- Baseline SimAlign \| w/o fine-tuning \| 18.8 \| 7.6 \| 27.2 \| 46.6 \| 21.6 fast_align \| bilingual \| 27.0 \| 10.5 \| 32.1 \| 51.1 \| 38.1 eflomal \| bilingual \| 22.6 \| 8.2 \| 25.1 \| 47.5 \| 28.7 GIZA++ \| bilingual \| 20.6 \| 5.9 \| 26.4 \| 48.0 \| 35.1 Zenkel et al. (2020) \| bilingual \| 16.0 \| 5.0 \| 23.4 \| - \| - Chen et al. (2020) \| bilingual \| 15.4 \| 4.7 \| 21.2 \| - \| - Ours $\alpha$-entmax \| w/o fine-tuning \| 18.1 \| 5.6 \| 29.0 \| 46.3 \| 18.4 bilingual \| 16.1 \| 4.1 \| 23.4 \| 38.6 \| 15.4 multilingual ($\beta$ = 0) \| 15.4 \| 4.1 \| 22.9 \| 37.4 \| 13.9 multilingual ($\beta$ = 1) \| 15.0 \| 4.5 \| 20.8 \| 38.7 \| 14.5 \| zero-shot \| 16.0 \| 4.3 \| 28.4 \| 44.0 \| 13.9 softmax \| w/o fine-tuning \| 17.4 \| 5.6 \| 27.9 \| 45.6 \| 18.1 bilingual \| 15.6 \| 4.4 \| 23.0 \| 38.4 \| 15.3 multilingual ($\beta$ = 0) \| 15.3 \| 4.4 \| 22.6 \| 37.9 \| 13.6 multilingual ($\beta$ = 1) \| 15.1 \| 4.5 \| 20.7 \| 38.4 \| 14.5 \| zero-shot \| 15.7 \| 4.6 \| 27.2 \| 43.7 \| 14.0 Table 2: Performance (AER) of our models in bilingual, multilingual and zero- shot settings. The best scores for each alignment extraction method are in bold and the overall best scores are in italicized bold. ## 3 Experiments In this section, we first present our main results, then conduct several ablation studies and analyses of our models. ### 3.1 Setup #### Datasets. We perform experiments on five different language pairs, namely German-English (De-En), French-English (Fr-En), Romanian-English (Ro-En), Japanese-English (Ja-En) and Chinese-English (Zh-En). For the De-En, Fr-En, Ro-En datasets, we follow the experimental setting of previous work (Zenkel et al., 2019; Garg et al., 2019; Zenkel et al., 2020). The training and test data for Ro-En and Fr- En are provided by Mihalcea and Pedersen (2003). The Ro-En training data are also augmented by the Europarl v8 corpus (Koehn, 2005). For the De-En data, the Europarl v7 corpus is used as training data and the gold alignments are provided by Vilar et al. (2006). The Ja-En dataset is obtained from the Kyoto Free Translation Task (KFTT) word alignment data (Neubig, 2011), and the Japanese sentences are tokenized with the KyTea tokenizer (Neubig et al., 2011). The Zh-En dataset is obtained from the TsinghuaAligner website111http://nlp.csai.tsinghua.edu.cn/~ly/systems/TsinghuaAligner/TsinghuaAligner.html. We treat their evaluation set as the training data and use the test set in Liu and Sun (2015). The De-En, En-Fr, Zh-En datasets contain the distinction between sure and possible alignment links. The statistics of these datasets are shown in Table 1. We use the Ja-En development set to tune the hyper- parameters. #### Baselines. We compare our models with: • fast_align (Dyer et al., 2013): a popular statistical word aligner which is a simple, fast reparameterization of IBM Model 2. * • eflomal (Östling and Tiedemann, 2016): an efficient statistical word aligner using a Bayesian model with Markov Chain Monte Carlo (MCMC) inference. * • GIZA++ (Och and Ney, 2003; Gao and Vogel, 2008): an implementation of IBM models. Following previous work (Zenkel et al., 2020), we use five iterations each for Model 1, the HMM model, Model 3 and Model 4. * • SimAlign (Sabet et al., 2020): a BERT-based word aligner that is not fine- tuned on any parallel data. The authors propose three alignment extraction methods and we implement their IterMax model with default parameters. * • Zenkel et al. (2020) and Chen et al. (2020): two state-of-the-art neural word aligners based on MT models. #### Implementation Details. Our main results are obtained by using the probability thresholding method on the contextualized embeddings in the 8-th layer of multilingual BERT-Base (mBERT; Devlin et al. (2019)) and we will discuss this choice in our ablation studies. We use the AdamW optimizer (Loshchilov and Hutter, 2019) with a learning rate of 2e-5 and the batch size is set to 8. Following Peters et al. (2019), we set $\alpha$ to 1.5 for $\alpha$-entmax. The threshold $c$ is set to 0 for $\alpha$-entmax and 0.001 for softmax and optimal transport. Unless otherwise stated, $\beta$ is set to 0. We mainly evaluate the model performance using Alignment Error Rate (AER). ### 3.2 Main Results We first train our model on each individual language pair, then investigate if it is possible to train multilingual word aligners. #### Bilingual Model Performance. From Table 2, we can see that our softmax model can achieve consistent improvements over the baseline models, demonstrating the effectiveness of our proposed method. Surprisingly, directly extracting alignments from mBERT (the w/o fine-tuning setting) can already achieve better performance than the popular statistical word aligner GIZA++ on 4 out of 5 settings, especially in the Zh-En setting where the size of parallel data is small. #### Multilingual Model Performance. We also randomly sample 200k parallel sentence pairs from each language pair (except for Zh-En where we take all of its 40k parallel sentences) and concatenate them together to train multilingual word aligners. As shown in Table 2, the multilingually trained word aligners can achieve further improvements and they consistently outperform our bilingual word aligners and all the baselines even though the size of training data for each individual language pair is smaller. The results demonstrate that we can indeed obtain a neural word aligner that has state-of-the-art and robust performance across different language pairs. We also test the performance of our consistency optimization objective in this setting. We can see that incorporating this objective ($\beta$=1) can significantly improve the model performance on Ro- En, while it also deteriorates the Ja-En and Zh-En performance by a non- negligible margin. We find that this is because the CO objective can significantly improve the alignment recall while sacrificing the precisions, and our Ro-En dataset tends to favor models with high recall and the Ja-En and Zh-En datasets have an opposite tendency. #### Zero-Shot Performance. In this paragraph, we want to find out how our models perform on language pairs that it has never seen during training. To this end, for each language pair, we train our model with data of all the other language pairs and test its performance on the target language pair. Results in Table 2 demonstrate that training our models with parallel data on _other_ language pairs can still improve the model performance on the target language pair. This is a very important result, as it indicates that our model can be used as a off- the-shelf tool for multilingual word alignment for any language supported by the underlying embeddings, _regardless of whether parallel data has been used for training or not_. ### 3.3 Ablation Studies \| Component \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En \| Speed ---\|---\|---\|---\|---\|---\|---\|--- Prob. \| softmax \| 17.4 \| 5.6 \| 27.9 \| 45.6 \| 18.1 \| 33.22 $\alpha$-entmax \| 18.1 \| 5.6 \| 29.0 \| 46.3 \| 18.4 \| 32.36 OT \| Cosine \| 24.4 \| 15.7 \| 33.7 \| 54.0 \| 31.1 \| 3.36 Dot Product \| 25.4 \| 17.1 \| 34.1 \| 54.2 \| 30.9 \| 3.82 Euclidean \| 20.7 \| 15.1 \| 33.3 \| 53.2 \| 29.8 \| 3.05 Table 3: Comparisons of probability thresholding (Prob.) and optimal transport (OT) for alignment extraction. We try both softmax and $\alpha$-entmax for probability thresholding and different cost functions for optimal transport. We measure both the extraction speed (#sentences/seconds) and the alignment quality (AER) on five language pairs, namely German-English (De-En), French- English (Fr-En), Romanian-English (Ro-En), Japanese-English (Ja-En), and Chinese-English (Zh-En). The best scores are in bold. In this part, we compare the performance of different alignment extraction methods, pre-trained embedding models and training objectives. #### Alignment Extraction Methods. We first compare the performance of our two proposed alignment extraction methods, namely the probability thresholding and optimal transport techniques. We use the representations of the 8-th layer of mBERT following Sabet et al. (2020). As shown in Table 3, probability thresholding methods can consistently outperform optimal transport by a large margin on the five language pairs. In addition, probability thresholding methods are much faster than optimal transport. softmax is marginally better than $\alpha$-entmax, yet one advantage of $\alpha$-entmax is that we do not need to manually set the threshold. Therefore, we use both softmax and $\alpha$-entmax to obtain the main results. #### Pre-trained Embedding Models. In this paragraph, we investigate the performance of three different types of pre-trained embedding models, including mBERT, XLM (Lample and Conneau, 2019) and XLM-R (Conneau et al., 2020). For XLM, we have tried its three released models: 1) XLM-15 (MLM) pre-trained with MLM and supports 15 languages; 2) XLM-15 (MLM+TLM) pre-trained with both the MLM and TLM objectives and supports 15 languages; 3) XLM-100 (MLM) pre-trained with MLM and supports 100 languages. We use softmax to extract the alignments. Because XLM-15 does not support Japanese or Romanian, we only report the performance on the three other language pairs in Table 4. We take representations from different layers and report the performance of the best three layers. We can see that while XLM-15 (MLM+TLM) can achieve the best performance on De-En and Fr-En, the best layer is not consistent across language pairs. On the other hand, the optimal configurations for mBERT are consistent across language pairs. In addition, considering mBERT supports many more languages than XLM-15 (MLM+TLM), we will use mBERT in the following sections. Model \| Layer \| De-En \| Fr-En \| Zh-En ---\|---\|---\|---\|--- mBERT \| 7 \| 18.7 \| 6.1 \| 19.1 8 \| 17.4 \| 5.6 \| 18.1 9 \| 18.8 \| 6.1 \| 20.1 XLM-15 (MLM) \| 4 \| 21.1 \| 6.8 \| 25.3 5 \| 20.4 \| 6.1 \| 26.1 6 \| 23.2 \| 7.7 \| 33.3 XLM-15 (MLM+TLM) \| 4 \| 16.4 \| 4.9 \| 18.6 5 \| 16.2 \| 4.7 \| 23.7 6 \| 18.8 \| 5.7 \| 26.2 XLM-100 (MLM) \| 7 \| 20.5 \| 8.5 \| 30.8 8 \| 19.8 \| 8.2 \| 28.6 9 \| 19.9 \| 8.8 \| 29.3 XLM-R \| 5 \| 24.4 \| 10.3 \| 33.2 6 \| 23.1 \| 9.2 \| 30.7 7 \| 24.7 \| 11.5 \| 28.1 Table 4: Comparisons of different LMs in terms of AER. We extract alignments using softmax and take representations from different layers of LMs. The best scores for each individual model are in bold and the overall best scores are in italicized bold. Model \| Objective \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|---\|--- softmax \| All \| 15.3 \| 4.4 \| 22.6 \| 37.9 \| 13.6 2-7 \| All w/o MLM \| 15.3 \| 4.4 \| 22.8 \| 38.6 \| 13.7 \| All w/o TLM \| 15.5 \| 4.7 \| 22.9 \| 39.7 \| 14.0 \| All w/o SO \| 16.9 \| 4.8 \| 23.0 \| 39.1 \| 15.4 \| All w/o PSI \| 15.4 \| 4.4 \| 22.7 \| 37.9 \| 13.8 Table 5: Ablation studies on our training objectives in multilingual settings. #### Training Objectives. We also conduct ablation studies on each of our training objectives. We can see from Table 5 that the self-training objective can best improve the model performance. Also, the translation language modeling and parallel sentence identification objectives can marginally benefit the model. The masked language modeling objective, on the other hand, cannot always improve the model and can sometimes even deteriorate the model performance, possibly because the TLM objective already provides the model with sufficient supervision signals. ### 3.4 Analysis We conduct several analyses to better understand our models. Unless otherwise stated, we perform experiments on the softmax model using mBERT. #### Incorporating Supervised Signals. We investigate if our models can benefit from supervised signals. If we have access to word-level gold labels for word alignment, we can simply utilize them in our self-training objectives. Specifically, we can set $A_{ij}$ in Equation 4 to 1 if and only if they are aligned. In our experimental settings, we have gold labels for all the Zh-En sentences and 653 sentences from the Ja- En development set. Table 6 demonstrates that training our models with as few as 653 labeled sentences can dramatically improve the alignment quality, and combining labeled and unlabeled parallel data can further improve the model performance. This analysis demonstrate the generality of our models as they can also be applied in semi-supervised settings. Lang. \| Unsup. \| Sup. \| Semi-Sup. ---\|---\|---\|--- Zh-En \| 15.3 \| 12.5 \| - Ja-En \| 38.4 \| 31.6 \| 30.0 Table 6: Incorporating supervised word alignment signals into our model can further improve the model performance in terms of AER. #### Growing Heuristics. As stated in Section 2.1, because our alignment extraction methods essentially take the intersection of forward and backward alignments, growing heuristics can also be applied in our settings. The main motivation of growing heuristics is to improve the recall of the resulting alignments. While effective in statistical word aligners, as shown in Table 7, the growing heuristics only improve our alignment extraction method on the vanilla mBERT model in the Ro- En setting while degrading the model performance on all the other language pairs. After fine-tuning, the growing heuristics can only hurt the model performance, possibly because the self-training objective encourages the forward and backward alignments to be symmetrical. Based on these results, we do not adopt the growing heuristics in our models. Model \| Ext. \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|---\|--- mBERT \| X-En \| 24.7 \| 14.4 \| 31.9 \| 54.7 \| 27.4 En-X \| 22.6 \| 12.2 \| 32.0 \| 52.7 \| 29.9 softmax \| 17.4 \| 5.6 \| 27.9 \| 45.6 \| 18.1 gd \| 18.7 \| 9.2 \| 27.0 \| 48.5 \| 23.4 gd-final \| 18.6 \| 9.3 \| 26.9 \| 48.7 \| 23.2 Ours-Multi. \| X-En \| 20.2 \| 12.9 \| 25.4 \| 42.1 \| 19.3 En-X \| 18.1 \| 9.3 \| 25.9 \| 41.7 \| 23.5 softmax \| 15.3 \| 4.4 \| 22.6 \| 37.9 \| 13.6 gd \| 16.3 \| 8.1 \| 23.1 \| 38.2 \| 18.3 gd-final \| 16.5 \| 8.3 \| 23.2 \| 38.7 \| 18.5 Table 7: The grow-diag-final heuristic can only improve our alignment extraction method in the Romanian-English setting without fine-tuning. ``gd'' refers to grow-diag. Model \| Prec. % \| Rec. % \| F1 % ---\|---\|---\|--- BERT-En (zero-shot) \| 53.1 \| 54.3 \| 52.7 fast_align \| 51.5 \| 59.8 \| 55.2 GIZA++ \| 56.5 \| 64.1 \| 60.0 SimAlign \| 59.9 \| 67.6 \| 63.5 Ours \| 60.6 \| 68.5 \| 64.3 Table 8: Our model is also effective in an annotation projection setting where we train a BERT-based NER model on English data and test it on Spanish data. The best scores are in bold. Model \| En \| Fr \| Es \| De \| El \| Bg \| Ru \| Tr \| Ar \| Vi \| Th \| Zh \| Hi \| Sw \| Ur \| Ave. ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- mBERT \| 81.3 \| 73.4 \| 74.3 \| 70.5 \| 66.9 \| 68.2 \| 68.5 \| 59.5 \| 64.3 \| 70.6 \| 50.7 \| 68.8 \| 59.3 \| 49.4 \| 57.5 \| 65.5 Ours \| 81.5 \| 74.1* \| 74.9* \| 71.2* \| 67.1 \| 68.7* \| 68.6 \| 61.0* \| 66.2* \| 70.5 \| 53.8* \| 69.1 \| 59.8* \| 50.6* \| 58.6* \| 66.4* Table 9: Results of mBERT and our fine-tuned model on XNLI (Conneau et al., 2018). Our objectives can improve the model cross-lingual transfer ability. ``'' denotes significant differences using paired bootstrapping (p$<$0.05) . Figure 3: An example of extracting alignments from our fine-tuned model using softmax. Red boxes indicate the gold alignments. The fine-tuned model can generate more accurate alignments then vanilla mBERT (Figure 2). #### Annotation Projection. Word alignment has been a useful tool in cross-lingual annotation projection (Yarowsky et al., 2001; Nicolai and Yarowsky, 2019). Therefore, it would be interesting to see if our model can be beneficial in these settings. To this end, we evaluate our model and baselines on cross-lingual named entity recognition (NER). We train a BERT-based NER model on the CoNLL 2003 English data (Tjong Kim Sang and De Meulder, 2003) and test it on the CoNLL 2002 Spanish data (Tjong Kim Sang, 2002). We use Google Translate to translate Spanish test set into English, predict the labels using the NER model, then project the labels from English to Spanish using word aligners. From Table 8, we can see that our model is also better than baselines in this setting, demonstrating its usefulness in cross-lingual annotation projection. #### Sentence-Level Representation Transfer. We also test if the aligned representations are beneficial for sentence-level cross-lingual transfer. In doing so, we perform experiments on XNLI (Conneau et al., 2018), which evaluates cross-lingual sentence representations in 15 languages on the task of natural language inference (NLI). We train our models with the provided 10k parallel data on the 15 languages, fine-tune our model on the English NLI data, then test its performance on other languages. As shown in Table 9, our model can outperform the baseline, indicating the aligned word representations can also be helpful for sentence-level cross- lingual transfer. #### Alignment Examples. We also conduct qualitative analyses as shown in Figure 1, 2 and 3. After fine-tuning, the learned contextualized representations are more aligned, as the cosine distances between semantically similar words become closer, and the extracted alignments are more accurate. More examples are shown in Appendix 13. ## 4 Related Work Based on the IBM translation models (Brown et al., 1993), many statistical word aligners have been proposed (Vogel et al., 1996; Östling and Tiedemann, 2016), including the current most popular tools GIZA++ (Och and Ney, 2000, 2003; Gao and Vogel, 2008) and fast_align (Dyer et al., 2013). Recently, there is a resurgence of interest in neural word alignment (Tamura et al., 2014; Alkhouli et al., 2018). Based on NMT models trained on parallel corpora, researchers have proposed several methods to extract alignments from them (Luong et al., 2015; Zenkel et al., 2019; Garg et al., 2019; Li et al., 2019) and successfully build an end-to-end neural model that can outperform statistical tools (Zenkel et al., 2020). However, there is an inherent discrepancy between translation and word alignment: translation models are directional and the source and target side are treated differently, while word alignment is a non-directional task. Therefore, certain adaptations are required for translation models to perform word alignment. Another disadvantage of MT-based word aligners is that they cannot easily utilize contextualized embeddings. Using learned representations to improve word alignment have been investigated (Sabet et al., 2016; Pourdamghani et al., 2018). Recently, pre-trained LMs (Peters et al., 2018; Devlin et al., 2019; Brown et al., 2020) have proven to be useful in cross-lingual transfer (Libovickỳ et al., 2019; Hu et al., 2020b). In word alignment, Sabet et al. (2020) propose effective methods to extract alignments from multilingual LMs without explicit training on parallel data. In this work, we propose better alignment extraction methods and combine the best of the two worlds by fine- tuning contextualized embeddings on parallel data. There are also work on supervised neural word alignment (Stengel-Eskin et al., 2019; Nagata et al., 2020). However, supervised data are not always accessible, making their methods inapplicable in many scenarios. In this paper, we demonstrate that our model can incorporate supervised signals if available and perform semi-supervised learning, which is a more realistic and general setting. Some work on bilingual lexicon induction also share similar general ideas with ours. For example, Zhang et al. (2017) minimize the earth mover’s distance to match the embedding distributions from different languages. Similarly, Grave et al. (2019) present an algorithm to align point clouds with Procrustes (Schönemann, 1966) in Wasserstein distance for unsupervised embedding alignment. ## 5 Discussion and Conclusion We present a neural word aligner that achieves state-of-the-art performance on five diverse language pairs and obtains robust performance in zero-shot settings. We propose to fine-tune multilingual embeddings with objectives suitable for word alignment and develop two alignment extraction methods. We also demonstrate its applications in semi-supervised settings. We hope our word aligner can be a tool that can be used out-of-the-box with good performance over various language pairs. Future directions include designing better training objectives and experimenting on more language pairs. 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We evaluate the model performance using Alignment Error Rate (AER). ## Appendix B Analysis In this section, we conduct more analyses of our models. #### Monolingual Alignment. We also investigate how our models perform in monolingual alignment settings. Previous methods MacCartney et al. (2008); Yao et al. (2013a, b); Sultan et al. (2014) typically exploit external resources such as WordNet to tackle the problem. As shown in Table 10, mBERT can outperform previous methods in terms of recall and F1 without any fine-tuning. Our multilingually fine-tuned model can achieve better recall and slightly better F1 score than the vanilla mBERT model, and fine-tuning our model with supervised signals can achieve further improvements. Model \| Prec. % \| Rec.% \| F1 % ---\|---\|---\|--- Baseline Yao et al. (2013a) \| 91.3 \| 82.0 \| 86.4 Yao et al. (2013b) \| 90.4 \| 81.9 \| 85.9 Sultan et al. (2014) \| 93.5 \| 82.6 \| 87.6 Ours mBERT \| 87.0 \| 89.0 \| 88.0 Ours-Multilingual \| 87.0 \| 89.3 \| 88.1 Ours-Supervised \| 87.2 \| 89.8 \| 88.5 Table 10: Our model is also effective in monolingual alignment settings. #### Sensitivity Analysis. We also conduct a sensitivity analysis on the threshold $c$ for our softmax alignment extraction method. As shown in Table 11, our method is relatively robust to this threshold. In particular, after fine-tuning, the AERs change within 0.5% when varying the threshold. Model \| c. \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|---\|--- mBERT \| 1e-6 \| 17.3 \| 6.0 \| 27.2 \| 45.2 \| 18.9 1e-5 \| 17.3 \| 5.9 \| 27.4 \| 45.1 \| 18.6 1e-4 \| 17.3 \| 5.7 \| 27.6 \| 45.3 \| 18.3 1e-3 \| 17.4 \| 5.6 \| 27.9 \| 45.6 \| 18.1 1e-2 \| 17.7 \| 5.6 \| 28.4 \| 45.8 \| 18.2 1e-1 \| 18.1 \| 5.6 \| 28.9 \| 46.3 \| 18.3 5e-1 \| 18.4 \| 5.6 \| 29.5 \| 47.0 \| 18.7 Ours-Multilingual \| 1e-6 \| 15.4 \| 4.6 \| 22.7 \| 38.2 \| 14.1 1e-5 \| 15.4 \| 4.5 \| 22.7 \| 38.1 \| 14.0 1e-4 \| 15.3 \| 4.5 \| 22.6 \| 37.9 \| 13.9 1e-3 \| 15.3 \| 4.4 \| 22.6 \| 37.9 \| 13.8 1e-2 \| 15.3 \| 4.3 \| 22.7 \| 37.9 \| 13.8 1e-1 \| 15.4 \| 4.3 \| 22.8 \| 38.0 \| 13.8 5e-1 \| 15.4 \| 4.2 \| 23.0 \| 38.2 \| 13.9 Table 11: Our softmax alignment extraction method is relatively robust to the threshold $c$. (a) mBERT Itermax (b) mBERT softmax (c) Fine-tuned IterMax (d) Fine-tuned softmax Figure 4: Extracting alignments from our model using IterMaxSabet et al. (2020) and our softmax method from the vanilla and fine-tuned mBERT models. #### Comparisons with IterMax. IterMax is the best alignment extraction method in SimAlign Sabet et al. (2020). The results in the main paper have demonstrated that our alignment extraction methods are able to outperform IterMax. In Figure 4, we can see that the IterMax algorithm tends to sacrifice precision for a small improvements in recall, while our model can generate more accurate alignments. Model \| Objective \| De-En \| Fr-En \| Ro-En \| Ja-En \| Zh-En ---\|---\|---\|---\|---\|---\|--- Ours-Bilingual $\alpha$-entmax \| All \| 16.1 \| 4.1 \| 23.4 \| 38.6 \| 15.4 2-7 \| All w/o MLM \| 15.6 \| 4.2 \| 23.3 \| 38.8 \| 15.1 \| All w/o TLM \| 16.4 \| 4.3 \| 23.7 \| 40.1 \| 15.3 \| All w/o SO \| 17.8 \| 4.7 \| 23.9 \| 39.4 \| 16.3 \| All w/o PSI \| 16.5 \| 4.2 \| 23.1 \| 38.5 \| 15.4 softmax \| All \| 15.6 \| 4.4 \| 23.0 \| 38.4 \| 15.3 2-7 \| All w/o MLM \| 15.5 \| 4.2 \| 23.2 \| 38.9 \| 14.9 \| All w/o TLM \| 15.9 \| 4.5 \| 23.7 \| 40.1 \| 15.1 \| All w/o SO \| 17.4 \| 4.7 \| 23.2 \| 38.6 \| 16.3 \| All w/o PSI \| 15.6 \| 4.3 \| 23.1 \| 38.8 \| 15.4 Ours-Multilingual $\alpha$-entmax \| All \| 15.4 \| 4.1 \| 22.9 \| 37.4 \| 13.9 2-7 \| All w/o MLM \| 15.1 \| 4.2 \| 22.8 \| 37.8 \| 13.7 \| All w/o TLM \| 16.4 \| 4.4 \| 23.3 \| 39.7 \| 14.4 \| All w/o SO \| 17.5 \| 4.6 \| 23.6 \| 40.0 \| 15.6 \| All w/o PSI \| 15.5 \| 3.9 \| 23.0 \| 38.2 \| 14.1 softmax \| All \| 15.3 \| 4.4 \| 22.6 \| 37.9 \| 13.6 2-7 \| All w/o MLM \| 15.3 \| 4.4 \| 22.8 \| 38.6 \| 13.7 \| All w/o TLM \| 15.5 \| 4.7 \| 22.9 \| 39.7 \| 14.0 \| All w/o SO \| 16.9 \| 4.8 \| 23.0 \| 39.1 \| 15.4 \| All w/o PSI \| 15.4 \| 4.4 \| 22.7 \| 37.9 \| 13.8 Table 12: Ablation studies on training objectives. #### Ablation Studies on Training Objectives. Table 12 presents more ablation studies on our training objectives. We can see that the self training objective is the most effective one, with the translation language modeling objective being the second and the parallel sentence identification objective being the third. The masked language modeling objective can sometimes hurt the model performance, possibly because of the translation language modeling objective. #### Experiments on More Language Pairs. We also test our alignment extraction methods on other language pairs following the setting of Sabet et al. (2020) without fine-tuning as shown in Table 13.222Their English-Persian dataset is unavailable at the time of writing the paper. Model \| En-Cs \| En-Hi ---\|---\|--- GIZA++ \| 18.2 \| 51.8 SimAlign \| 13.4 \| 40.2 Ours (softmax, $c$=1e-3) \| 12.3 \| 41.2 Ours (softmax, $c$=1e-5) \| 12.7 \| 39.5 Ours (softmax, $c$=1e-7) \| 13.3 \| 39.2 Table 13: Performance on more language pairs. #### More Qualitative Examples. In addition to the examples provided in the main text, we also present some randomly sampled samples in Figure 5. We can clearly see that our model learns more aligned representations than the baseline model. Figure 5: Cosine similarities between subword representations in a parallel sentence pair before and after fine-tuning. Red boxes indicate the gold alignments.
# Towards Understanding How Readers Integrate Charts and Captions: A Case Study with Line Charts Dae Hyun Kim<EMAIL_ADDRESS>Stanford University/Tableau ResearchStanford/Palo AltoCalifornia94305, 94306 , Vidya Setlur <EMAIL_ADDRESS>Tableau ResearchPalo AltoCalifornia94306 and Maneesh Agrawala<EMAIL_ADDRESS>Stanford UniversityStanfordCalifornia94305 (2021) ###### Abstract. Charts often contain visually prominent features that draw attention to aspects of the data and include text captions that emphasize aspects of the data. Through a crowdsourced study, we explore how readers gather takeaways when considering charts and captions together. We first ask participants to mark visually prominent regions in a set of line charts. We then generate text captions based on the prominent features and ask participants to report their takeaways after observing chart-caption pairs. We find that when both the chart and caption describe a high-prominence feature, readers treat the doubly emphasized high-prominence feature as the takeaway; when the caption describes a low-prominence chart feature, readers rely on the chart and report a higher- prominence feature as the takeaway. We also find that external information that provides context, helps further convey the caption’s message to the reader. We use these findings to provide guidelines for authoring effective chart-caption pairs. Captions; line charts; visually prominent features; takeaways. ††copyright: rightsretained††journalyear: 2021††conference: CHI Conference on Human Factors in Computing Systems; May 8–13, 2021; Yokohama, Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI ’21), May 8–13, 2021, Yokohama, Japan††doi: 10.1145/3411764.3445443††isbn: 978-1-4503-8096-6/21/05††ccs: Human-centered computing Empirical studies in visualization ## 1\. Introduction Charts provide graphical representations of data that can draw a reader’s attention to various visual features such as outliers and trends. Readers are initially drawn towards the most visually salient components in the chart such as the chart title and the labels (Matzen et al., 2017). However, they eventually apply their cognitive processes to extract meaning from the most prominent chart features (Card et al., 1999; Tufte, 1990). Consider the line chart at the beginning of this article. What do you think are the main visual features of the chart and what are its key takeaways? Such charts are often accompanied by text captions that emphasize specific aspects of the data as chosen by the chart author. In some cases, the data emphasized in the caption corresponds to the most visually prominent features of the chart and in other cases it does not. Prior studies have shown that charts with captions can improve both recall and comprehension of some aspects of the underlying information, compared to seeing the chart or the caption text alone (Bransford, 1979; Nugent, 1983; Large et al., 1995; Hegarty and Just, 1993). But far less is known about how readers integrate information between charts and captions, especially when the data emphasized by the visually prominent features of the chart differs from the data that is emphasized in the caption. [An example line graph]An example line graph showing 30 year fixed mortgage rates. Consider the visually prominent features in our initial line chart and then consider each of the following caption possibilities one at a time. How do your takeaways change with each one? (1) The chart shows the 30-year fixed mortgage rate between 1970 and 2018. (2) The 30-year fixed mortgage rate increased slightly from 1997 to 1999. (3) The 30-year fixed mortgage rate reached its peak of 18.45% in 1981. (4) The 30-year fixed mortgage rate reached its peak of 18.45% in 1981 due to runaway inflation. The first caption simply describes the dimensions graphed in the chart and only provides redundant information that could be read from the axis labels. Automated caption generation tools often create such basic descriptive captions (Tableau Software, 2020; Microsoft Q & A, 2020). The next three captions each emphasizes aspects of the data corresponding to a visual feature of the chart (i.e., upward trend, peak) by explicitly mentioning the corresponding data point or trend. However, the second caption emphasizes a feature of low visual prominence – a relatively local and small rise in the chart between 1997 and 1999. The third caption describes the most visually prominent feature of the chart – the tallest peak that occurs in 1981. The fourth caption also describes this most visually prominent feature, but adds external information that is not present in the chart and provides context for the data. In this paper, we examine two main hypotheses - (1) When a caption emphasizes more visually prominent features of the chart, people are more likely to treat those features as the takeaway; even when a caption emphasizes a less visually prominent feature, people are still more likely to treat a more visually prominent feature in the chart as the takeaway. (2) When a caption contains external information for context, the information serves to further emphasize the feature described in the caption and readers are therefore more likely to treat that feature as the takeaway. We considered univariate line charts for our work because they are among the most common basic charts and are easily parameterizable, making them useful for the initial exploration of our hypotheses. We synthesized $27$ single charts with carefully chosen parameters and collected $16$ real-world single line charts to confirm the generalizability of our findings. We ran a data collection activity on the $43$ single-line charts, where we asked $219$ participants to mark visually prominent regions on the line charts. We generated text captions for the ranked set of prominent features using templates to control variations in natural language. Finally, we conducted a crowdsourced study with a new set of $2168$ participants to report their takeaways after seeing the chart-caption pairs. Our findings from the study support both of our hypotheses. Referring back to our initial line chart, when the caption mentions the most prominent feature as in the third caption (i.e., the peak in 1981), readers will probably take away information from that feature. When the caption mentions a less prominent feature as in the second caption (i.e., the increase from 1997 to 1999), there is a mismatch in the message between the chart and the caption. Readers will have a strong tendency to go with the message conveyed in the chart and take away information about the peak value. Finally, the external information about the peak value present in the fourth caption will reinforce the message in the caption and the readers will more likely take away information about the peak. These findings help better understand the relationship between charts and their captions when conveying information about certain aspects of the data to the reader. Based on these studies, we provide guidelines for authoring charts and captions together in order to emphasize the author’s intended takeaways. Visualization authors can more effectively convey their message to readers by ensuring that both charts and captions emphasize the same set of features. Specifically, authors could make visual features that are related to their key message, more prominent through visual cues (e.g., highlighting or zooming into a focus area, adding annotations) (Egeth and Yantis, 1997; Liang and Huang, 2010) or include external information in the caption to further emphasize the feature described in the caption. Often, an alternative chart representation may be more conducive to making certain visual features more prominent. ## 2\. Related Work Our work is related to two lines of research: (1) Cognitive Understanding of Charts and (2) Caption Generation Tools. ### 2.1. Cognitive Understanding of Charts The prevalence of text with visuals has led researchers to explore how readers specifically understand information in figures with accompanying text in several domains. Li et al. (Li et al., 2018) conducted studies to demonstrate that figures with text can convey essential information and better aid understanding than just text alone for scientific publications in a biomedical domain. Odell et al. (Ottley et al., 2016) demonstrated that having text that accurately describes important findings in medical diagnostic images can increase physicians’ speed and accuracy on Bayesian reasoning tasks while making life-critical judgments for patients. Xiong et al. (Xiong et al., 2019) showed that background knowledge can affect viewers’ visual perception of data as they tend to see the pattern in the data corresponding to the background knowledge as more visually salient. Kong et al. (Kong et al., 2018) explored the impact of titles on visualization interpretation with different degrees of misalignment between a visualization and its title. A title contains a miscued slant when the visualization emphasizes one side of the story through visual cues but the title’s message addresses the other (less emphasized) side of the story. Titles have a contradictory slant where the information conveyed in the title is not presented at all in the visualization. They observe that even though the title of a visualization may not be recalled, the title can still measurably impact the remembered contents of a chart. Specifically, titles with a contradictory slant trigger more people to identify bias compared to titles with a miscued slant, while visualizations are perceived as impartial by the majority of viewers (Kong et al., 2019). Elzer et al. (Elzer et al., 2005) conducted a study to better understand the extent to which captions contribute to recognizing the intended message of an information graphic for sight-impaired users. They find that the caption strengthens the intended message of the graphic. Carberry et al. (Carberry et al., 2006) showed that the communicative goals of infographics in digital libraries are often not repeated in the text of the articles. Their work looked into how information in the graphics could be better utilized for summarizing a document by employing a Bayesian network. However, this previous research has not explored the relationship between charts and their captions with respect to how they work together to emphasize certain aspects of the data to the reader. ### 2.2. Caption Generation Tools A number of visual analysis tools help users design charts and captions from an input data table (Cui et al., 2018; Demiralp et al., 2017; Hu et al., 2018; Vartak et al., 2015; Wills and Wilkinson, 2010). These captions generally only describe the data attributes and visual encodings that are in play in the charts and do not highlight key takeaways. Nevertheless, authors often include text with a chart to help emphasize an intended message to their audience. PostGraphe (Fasciano and Lapalme, 1996) generated reports integrating graphics and text from a list of user-defined intentions about the underlying data such as the comparison of variables. SAGE (Mittal et al., 1995) used natural language generation techniques to produce explanatory captions for information graphics. The system generates captions based on the structural and spatial relations of the graphical objects and their properties along with explanations describing the perceptual complexity of the data attributes in the graphics. SumTime (Yu et al., 2003) used pattern recognition techniques to generate textual summaries of time-series data. The iGRAPH-Lite system (Ferres et al., 2007) made information in a graphic accessible to blind users by using templates to provide textual summaries of what the graphic looks like. The summaries however, do not focus on the higher-level takeaway conveyed by the graphic. Chen et al. (Chen et al., 2019b) produced natural language descriptions for figures by identifying relations among labels present in the figures. Other work has explored natural language generation techniques for assembling multiple caption units together to form captions (Qian et al., 2020). Deep learning techniques based on neural networks automate caption generation tasks for news images (Chen et al., 2019a). Elzer et al. (Elzer et al., 2011) identified communicative signals that represent the intent of messages portrayed in basic bar charts by applying a Bayesian network methodology for reasoning about these signals and generating captions. Liu et al. (Liu et al., 2009; Wei et al., 2010; Liu et al., 2012) explored the integration of text analytics algorithms with interactive visualization tools to help users understand and interpret the summarization results. Contexifier (Hullman et al., 2013) automatically annotated visualizations of stock behavior with news article headlines taking into account visual salience contextual relevance, and key events from the articles. Kim et al. (Kim et al., 2020) introduced an automatic chart question answering pipeline that generates visual explanations that refer to visual features of charts using a template-based natural language generation approach. Voder (Srinivasan et al., 2019) generated data facts for visualizations with embellishments to help users interpret visualizations and communicate their findings. While an evaluation of that system suggested that interactive data facts aided users in interpreting visualizations, the paper did not specifically explore the interplay between data facts and the visualizations and their effects on the readers’ takeaways. These systems focus on helping authors with auto-generated text that can be associated with graphics; however, the work does not evaluate what information readers gather from the generated captions with their corresponding graphics. Our paper specifically explores how similarities and differences between what is visually emphasized in a line chart and textually emphasized in its caption, can affect what readers take away from the information when presented together. Future directions from our work could extend the functionality of chart authoring tools by providing automatic suggestions for captions as well as for chart presentation to help the reader take away information that is consistently emphasized by both the chart and caption. ## 3\. Study Figure 1. Our study pipeline. The inputs to the study are $27$ synthetic and $16$ real-world charts. Yellow boxes represent steps where we employed crowdsourcing. The green box indicates that the step did not involve crowdsourcing. [Our study pipeline.]Our study pipeline. The inputs to the study are 27 synthetic and 16 real-world charts. Yellow boxes represent steps where we employed crowdsourcing. The green box indicates that the step did not involve crowdsourcing. We conducted a crowdsourced study to understand how captions describing features of varying prominence levels and the effect of including or not including external information for context, interacts with the chart in forming the readers’ takeaways. Through an initial data collection activity, we asked participants to identify features in the line charts that they thought were visually prominent. We generated captions corresponding to those marked features of various levels of prominence. We then ran a study asking a new set of participants to type their takeaways after viewing a chart and caption pair. Figure 1 shows the study pipeline. ### 3.1. Datasets Figure 2. The $27$ data shapes generated for the study and their top three prominent features. Columns represent the nine possible global shapes and rows represent the three possible local outlier types. Here, ‘flat’, ‘inc’, and ‘dec’ denote flat, increasing, and decreasing trends respectively. ‘none’, ‘neg’, and ‘pos’ denote none, negative, and positive outlier types respectively. Red, green, and blue regions indicate the top three prominent features in order. [The 27 data shapes with top three prominent features.]The 27 data shapes generated for the study and their top three prominent features. Columns represent the nine possible global shapes and rows represent the three possible local outlier types. Here, ‘flat’, ‘inc’, and ‘dec’ denote flat, increasing, and decreasing trends respectively. ‘none’, ‘neg’, and ‘pos’ denote none, negative, and positive outlier types respectively. Red, green, and blue regions indicate the top three prominent features in order. Figure 3. The $16$ real-world charts. Red, green, and blue regions indicate the top three prominent features in order. [The 16 real-world charts with top three prominent features.]The 16 real-world charts. Red, green, and blue regions indicate the top three prominent features in order. We ran the study on two different datasets - (1) synthetically generated line charts that we designed to ensure good coverage of a variety of visual features that occur in line charts and (2) line charts gathered from real- world sources to serve as a more ecologically valid setting for our study. Synthetic Charts. We generated a set of synthetic line charts with common visual features (i.e., trends, extrema, and inflection points) while maintaining realistic global shapes. To keep the overall design space tractable, we limited global shapes to include at most two trends (i.e., up, down, and flat) and added at most one perturbation to induce features (e.g. inflection points) in either the positive or negative direction, resulting in a total of $27$ data shapes (Figure 2). To provide context to the charts, we labeled the x-axis with time unit values implying that the chart represents a time series. Specifically, we selected the start and end of the x-axis from the set of years {1900, 1910, 1920,…, 2020}. To label the y-axis, we chose a domain for the y-axis and its value range from the MassVis dataset (Borkin et al., 2013). Real-world Charts. To build a more ecologically representative dataset of line charts with various shapes, styles, and domains, we collected $16$ charts (Figure 3) from sources such as The Washington Post (Washington Post, 2020), Pew Research (Pew Research, 2020), Wikipedia (Wikipedia, 2020), and Tableau Public (Tableau Public, 2020). Because our study focuses on prominence arising from intrinsic features in line charts, we removed all graphical elements that could potentially affect the prominence of the features in the charts (e.g., text annotations, highlighting, and background shading). In addition, we removed all text except for the axis labels (e.g. chart titles) so that the captions serve as the primary source of text provided with the chart. We added axis labels to those charts without labels to ensure readability. ### 3.2. Identify Visually Prominent Features Figure 4. The line on the bottom left shows the prominence curve for the line chart above. From this curve, we obtain the most prominent (red), the second most prominent (green), and the third most prominent (blue) features in the chart. The $10$ caption variants (one of them being a no-caption variant) generated based on these prominent features, are shown on the right. The text colors indicate the types of fill-in values based on the caption templates; purple for dimensions, fuchsia for the feature description, blue for data values, and brown for the time period. [Generation of 10 caption variants based on the prominence curve.]The line on the bottom left shows the prominence curve for the line chart above. From this curve, we obtain the most prominent (red), the second most prominent (green), and the third most prominent (blue) features in the chart. The 10 caption variants (one of them being a no-caption variant) generated based on these prominent features, are shown on the right. The text colors indicate the types of fill-in values based on the caption templates; purple for dimensions, fuchsia for the feature description, blue for data values, and brown for the time period. To identify the most visually prominent features in our dataset, we recruited at least five workers from Amazon Mechanical Turk (Amazon Mechanical Turk, 2020) for each line chart and asked them to draw rectangular bounding boxes around the top three most prominent features in the chart. We also asked them to briefly describe each marked feature in their own words so that we could differentiate between trend and slope features versus peak, inflection, and other point features. In each trial of the data collection, we presented one of the $43$ line charts. Because we were seeking subjective responses, each participant completed only one trial to avoid biases that might arise from repeated exposure to the task. Participation was limited to English speakers in the U.S. with at least a 98% acceptance rate and $5000$ approved tasks. We payed a rate equivalent to $2 / 10 mins. We asked a total of $219$ participants (average of $5.09$ per chart) to label the top three features for a total of $657$ prominence boxes. We then aggregated all of the feature bounding boxes provided by first projecting each box onto the x-axis, to form a 1D interval (Figure 4 upper left). We weighted each interval inversely proportional to the ranking provided by the participant. Specifically, the top ranked feature bounding box for each participant was assigned a weight of $3$, while the 3rd ranked feature was assigned a weight of $1$. We noticed that bounding boxes corresponding to the same features were pretty consistent in the central regions although the exact boundary drawn by the participants varied. In order to boost the signal in the central regions while suppressing the noise in the boundary regions, we multiplied the weight assigned to each interval by a Gaussian factor centered at the interval and with standard deviation set to half the width of the interval. Summing all of the Gaussian weighted intervals, we obtained a prominence curve (Figure 4 bottom left). However, a region defined by a local maximum of the curve may not have an obvious one-to-one mapping with a feature in the chart because it roughly indicates a high prominence region instead of pinpointing a specific visual feature. We considered all the bounding boxes containing the region along with the participants’ text descriptions of the features to associate the local maximum to a certain feature. We iterated this process for the region around the top three local maximum to identify three prominent features. Results of the algorithm for the charts in our dataset are shown in Figures 2 and 3. ### 3.3. Caption Generation To carefully control the language used in the captions and keep the number of conditions manageable, we generated captions using templates that only vary the feature mentioned and whether external information is introduced. Using the templates, we produced the following caption variants: (1) two captions (one with and one without external information) for each of the top three visually prominent features identified earlier, (2) two captions (one with and one without external information) describing a minimally prominent feature that is neither an extremum nor an inflection point, and (3) a basic caption that simply describes the domain represented in the chart without describing a particular feature. We generated $10$ caption variants (including the no caption variant in which we presented a chart without caption) for each of the $43$ charts, providing a total of $430$ chart-caption pairs. We manually generated all the captions rather than using the original captions for the real-world charts to control for word use and grammatical structure. For real-world charts, we searched for information from the document that they originally appeared in, to extract information not present in the charts. In particular, we looked for information about potential reasons for trends or change (e.g. the external information included in the caption about the most prominent feature in Figure 4) or comparisons with a similar entity (e.g. comparison between Macron’s approval rating with Trump’s approval rating in the second most prominent feature in Figure 4). For synthetically generated charts and real-world charts that were not accompanied with additional information about their features, we referenced Wikipedia (Wikipedia, 2020) articles to create a plausible context. We employed simple language templates for caption generation to minimize the effects of linguistic variation (Table 1). The captions generated with the templates were allowed to vary in the features they described in the charts. To make the descriptions of the features appear natural, we included words the participants used to describe the features during the prominent feature collection phase. Because the participants usually described each of the features using a noun occasionally with an adjective modifier (e.g. “sharp increase”), we manually lemmatized the words and modified the forms to correctly fit into our template (e.g. “sharply increased” in the caption about the third most prominent figure in Figure 4). Feature \| Template ---\|--- Extremum \| [dimension] reached its [extrema-word] of [value] in [time-period]. Trend \| [dimension] [slope-word] in — between [time-period]. Inflection \| [dimension] started [slope-word] in [time-period]. Point \| [dimension] was [value] in [time-period]. Table 1. Examples of templates we employed for generating captions about specific features. The text colors indicate the types of fill-in values based on the caption templates; purple for dimensions, fuchsia for feature descriptions, blue for data values, and brown for time periods. Examples of filled in captions are in Figure 4 (right). ### 3.4. Collect Takeaways for Charts & Captions Figure 5. The procedure for collecting takeaways for chart-caption pairs. The images show simplified versions of the screen that the participants saw during each step. [The procedure for collecting takeaways for chart-caption pairs.]The procedure for collecting takeaways for chart-caption pairs. The images show simplified versions of the screen that the participants saw during each step. #### 3.4.1. Design We ran a between-subjects design study for collecting takeaways for charts and their captions. For each of the $43$ charts, we presented one of the ten variants (including the no caption variant) (examples in Figure 4): (1) [1st w/o ext] Caption for most prominent feature, no external info. (2) [1st w/ ext] Caption for most prominent feature, has external info. (3) [2nd w/o ext] Caption for 2nd most prominent feature, no external info. (4) [2nd w/ ext] Caption for 2nd most prominent feature, has external info. (5) [3rd w/o ext] Caption for 3rd most prominent feature, no external info. (6) [3rd w/ ext] Caption for 3rd most prominent feature, has external info. (7) [non-pro w/o ext] Caption for non-prominent feature, no external info. (8) [non-pro w/ ext] Caption for non-prominent feature, has external info. (9) [basic] Caption about domain represented in the chart and $x$-range (10) [no cap] No caption #### 3.4.2. Procedure The study began with a screening test to ensure that the participant had a basic understanding of line charts and could read values and encodings, extract extrema and trends, and compare values (Figure 5 first step). Only participants who passed this test were allowed to continue with the study. After they read the instructions, the participants were presented with a chart and a caption underneath the chart, similar to most charts in the real world (unless it is the no-caption variant) (Figure 5 second step). We did not impose a time constraint on the amount of time spent looking at the chart and the caption to allow participants sufficient time to read and digest the information at their own pace, like document reading in the real world. On the next screen for collecting takeaways, the chart and the caption were removed to constrain readers to provide the takeaways based on memory instead of simply re-reading from the chart and the caption. The participants were asked to list as many text takeaways as they could in the order of importance (Figure 5 third step). Finally, using a 5-point Likert scale, we asked how much they relied on the chart and caption individually when determining their takeaways. We asked each participant to provide takeaways for exactly one chart-caption pair to prevent potential biases from already having read a different caption about a chart. From $2168$ participants (average of $5.04$ per chart-caption pair), we collected a total of $4953$ takeaways (average of $2.28$ per participant). #### 3.4.3. Labeling Takeaways In order to analyze the takeaways, we manually labeled each takeaway with the corresponding chart feature described. Since participants often described multiple chart features in a single takeaway, we first split each takeaway into separate takeaways for each visual feature mentioned. At the end of this process, we identified on average $1.31$ features per takeaway. If the referenced feature was one of three most prominent features or the non- prominent feature we identified during caption generation, we labeled the takeaway with the corresponding feature, otherwise we labeled the takeaway as referring to an other feature. If the takeaway did not refer to any specific feature in the chart, we labeled the takeaway as a non-feature. Examples of non-feature takeaways include an extrapolation such as “The value will continue to rise after 2020” or a judgment such as “I should buy gold” when looking at a chart showing the price of gold over time. One of the authors labeled the features and discussed any confusing cases with the other authors to converge on the final label. ## 4\. Results Figure 6. Study results. Each column shows bar charts for each prominence level mentioned in the caption (i.e., the leftmost bar chart is for captions mentioning the 1st ranked visual feature, the next bar chart is for captions mentioning the 2nd ranked visual feature, while the rightmost bar chart is for the no-caption condition). Within a bar chart, each bar represents the percentage of takeaways mentioning the visual feature at that prominence level. For example, the leftmost bar in each bar chart represents the percentage of total takeaways that mention the top ranked takeaway. Each bar chart also reports the percentage of Other features and Non-features that were mentioned in the takeaways. These charts aggregate data for captions with and without external information. The percentages do not sum to $100\%$ as some takeaways mention multiple takeaways. [Study results.]Study results. Each column shows bar charts for each prominence level mentioned in the caption (i.e., the leftmost bar chart is for captions mentioning the 1st ranked visual feature, the next bar chart is for captions mentioning the 2nd ranked visual feature, while the rightmost bar chart is for the no-caption condition). Within a bar chart, each bar represents the percentage of takeaways mentioning the visual feature at that prominence level. For example, the leftmost bar in each bar chart represents the percentage of total takeaways that mention the top ranked takeaway. Each bar chart also reports the percentage of Other features and Non-features that were mentioned in the takeaways. These charts aggregate data for captions with and without external information. The percentages do not sum to $100\%$ as some takeaways mention multiple takeaways. The primary goal of our study is to understand what readers take away when charts and captions are presented together and how the emphasis on different prominent features and presence of external information affects the takeaways. We analyze our results with respect to two hypotheses: [H1] When captions emphasize more visually prominent features of the chart, people are more likely to treat the features as the takeaway; when a caption emphasizes a less visually prominent feature, people are less likely to treat that feature as the takeaway and more likely to treat a more visually prominent feature in the chart as the takeaway. [H2] When captions contain external information for context, the external information serves to further emphasize the feature presented in the caption and people are therefore more likely to treat that feature as the takeaway, compared to when the caption does not contain external information. Assessing H1. To evaluate H1, we examine how varying the prominence of a visual feature mentioned in a caption (independent variable), affects the visual feature mentioned in the takeaways (dependent variable). Figure 6 summarizes the study results for the synthetic charts (top row) and the real- world charts (bottom row). In general, these results suggest that when a caption mentions visual features of differing prominence levels, the takeaways also differ. Omnibus Pearson’s chi-squared tests confirm a significant difference between the bar charts for the 5 different caption conditions in both the synthetic ($\chi^{2}(20)=202.211$, $p<0.001$) and real world ($\chi^{2}(20)=207.573$, $p<0.001$) datasets. These results also suggest that when the caption mentions a specific feature, the takeaways also tend to mention that feature, when compared to the baseline ‘no-caption’ condition. \| Caption-Takeaway 1 \| Caption-Takeaway 2 \| \| ---\|---\|---\|---\|--- Source \| Caption \| Takeaway \| Caption \| Takeaway \| $Z$ \| $p$ Block 1. Takeaways mentioning feature in caption vs. without caption Synthetic \| 1st \| 1st \| no cap \| 1st \| $2.846$ \| $0.002^{}$ 2nd \| 2nd \| no cap \| 2nd \| $4.641$ \| $<0.001^{}$ 3rd \| 3rd \| no cap \| 3rd \| $3.643$ \| $0.001^{}$ non-pro \| non-pro \| no cap \| non-pro \| $6.195$ \| $<0.001^{}$ Real-world \| 1st \| 1st \| no cap \| 1st \| $1.660$ \| $0.049$ 2nd \| 2nd \| no cap \| 2nd \| $4.225$ \| $<0.001^{}$ 3rd \| 3rd \| no cap \| 3rd \| $3.347$ \| $<0.001^{}$ non-pro \| non-pro \| no cap \| non-pro \| $4.732$ \| $<0.001^{}$ Block 2. Between takeaways mentioning feature in caption Synthetic \| 1st \| 1st \| 2nd \| 2nd \| $1.782$ \| $0.037$ 2nd \| 2nd \| 3rd \| 3rd \| $0.705$ \| $0.044$ 3rd \| 3rd \| non-pro \| non-pro \| $8.989$ \| $<0.001^{}$ Real-world \| 1st \| 1st \| 2nd \| 2nd \| $3.708$ \| $<0.001^{}$ 2nd \| 2nd \| 3rd \| 3rd \| $0.363$ \| $0.358$ 3rd \| 3rd \| non-pro \| non-pro \| $5.940$ \| $<0.001^{}$ Block 3. When caption $=$ 1st: takeaway $=$ 1st vs. takeaway $\neq$ 1st Synthetic \| 1st \| 1st \| 1st \| 2nd \| $8.168$ \| $<0.001^{}$ 1st \| 1st \| 1st \| 3rd \| $8.275$ \| $<0.001^{}$ 1st \| 1st \| 1st \| non-pro \| $19.463$ \| $<0.001^{}$ Real-world \| 1st \| 1st \| 1st \| 2nd \| $9.981$ \| $<0.001^{}$ 1st \| 1st \| 1st \| 3rd \| $11.301$ \| $<0.001^{}$ 1st \| 1st \| 1st \| non-pro \| $11.536$ \| $<0.001^{}$ Block 4. When caption $\neq$ 1st: takeaway $=$ 1st vs. takeaway $=$ caption Synthetic \| 2nd \| 2nd \| 2nd \| 1st \| $3.829$ \| $<0.001^{}$ 3rd \| 3rd \| 3rd \| 1st \| $0.258$ \| $0.398$ non-pro \| 1st \| non-pro \| non-pro \| $8.342$ \| $<0.001^{}$ Real-world \| 2nd \| 2nd \| 2nd \| 1st \| $2.010$ \| $0.022$ 3rd \| 3rd \| 3rd \| 1st \| $2.521$ \| $0.006^{}$ non-pro \| 1st \| non-pro \| non-pro \| $5.454$ \| $<0.001^{}$ Table 2. Pairwise Z-test results of comparisons between various ratios of takeaways that mention a certain feature (third, fifth columns) when provided a caption describing a certain feature (second, fourth columns). The tests were one-sided with the alternative hypothesis that the ratio of takeaways for ‘Caption-Takeaway 1’ is greater than the ratio of takeaways for ‘Caption- Takeaway 2’. Asterisks indicate significance with Bonferroni correction. Figure 7. (Top row) Comparison of percentages of takeaways that mention the same feature as the caption for the synthetic (a) and real-world (b) datasets (i.e., darker bars on the left correspond to the red bar from Figure 6a, the green bar from 6b, the blue bar from 6c, and the grey bar from 6d), and percentages of takeaways that mention the feature in the no caption condition (i.e., the right lighter-hued bars in the chart correspond to the bars from Figure 6e). (Middle row) Percentage of takeaways mentioning the visual features at each prominence level when presented with the basic caption. (Bottom row) Dividing the left bars in charts (top row)a and (top row)b based on whether the caption contains external information (purple bars) or does not (olive bars). The leftmost Any bars show aggregates over all prominence levels. Asterisks indicate significant difference. [(Top row) Comparisons of results in Figure 6. (Middle row) Results with basic caption. (Bottom row) Percentage of takeaways mentioning the visual features at each prominence level when presented with the basic caption.](top row) Comparison of percentages of takeaways that mention the same feature as the caption for the synthetic (a) and real-world (b) datasets (i.e., darker bars on the left correspond to the red bar from Figure 6a, the green bar from 6b, the blue bar from 6c, and the grey bar from 6d), and percentages of takeaways that mention the feature in the no caption condition (i.e., the right lighter- hued bars in the chart correspond to the bars from Figure 6e). (middle row) Percentage of takeaways mentioning the visual features at each prominence level when presented with the basic caption. (bottom row) Dividing the left bars in charts (top row)a and (top row)b based on whether the caption contains external information (purple bars) or does not (olive bars). The leftmost Any bars show aggregates over all prominence levels. Asterisks indicate significant difference. Figures 7a and 7b collect the percentage of takeaways that mention the same feature as in the caption for the synthetic and the real-world datasets respectively (left darker bars) and compare them with the percentages corresponding to the no-caption case (lighter-hued bars on the right). We see that captions do play a role in forming takeaways and the takeaway is thus more likely to mention that feature (i.e., each darker bar in Figures 7a and 7b is usually longer than the corresponding lighter-hued bar to its right). Planned pairwise Z-tests with Bonferroni correction are shown in Table 2. Block 1 shows that the differences between the corresponding color bars are significant for the second most prominent, third most prominent, and non- prominent features. For the most prominent feature, we find that while a higher proportion of people mentioned the most prominent feature in their takeaways when the caption mentions it, the difference is only significant for the synthetic charts. We believe that this is possibly because people already include the most prominent features in their takeaways in the no-caption condition and the difference hence is not significant. While we confirmed that both the chart and caption play a role as to what the reader takes away from them, the key question is how the chart and the caption interact with each other – Do they have a synergistic effect when they emphasize the same feature? Which one wins over when they emphasize different features? Referring to Figure 6, we see the synergistic effect of the double- emphasis from the chart and caption when they emphasize the same feature (Figures 6a and 6f). In particular, the participants took away from the most prominent feature significantly more often than from any other feature in the chart (Table 2 Block 3). When the caption diverged from the chart and described a feature that was not prominent, the participants relied more on the chart and took away from the most prominent feature significantly more than the feature described in the caption (Table 2 Block 4, rows 3 and 6; Figures 6d and 6i). When the caption did not diverge as much and described the second or the third most prominent feature, the takeaways mentioned the feature described in the caption more than the most prominent feature (Table 2 Block 4, rows 1, 2, 4, and 5; Figures 6b, 6c, 6g, and 6h). However, the difference was smaller than the difference between the ratio of people who took away from the most prominent feature and the ratio of people who took away from any of the other features. We believe this result may be due to the fact that the charts still had more influence on the readers than the captions as the second and the third most prominent feature are still among the top prominent features and are among the features emphasized by the chart. We observe from Figure 7 that the chart also plays an important role in what people take away – when a caption mentions a higher-prominent feature, the takeaways more consistently mentions that feature. Specifically, we see that the bars for the higher-prominence features are taller than the bars for the lower-prominence features, indicating an increase in the effectiveness of chart in reinforcing the message in the caption. Planned pairwise Z-tests with Bonferroni correction between each subsequent pair of bars (red bar vs. green bar, green bar vs. blue bar, blue bar vs. gray bar) (Table 2 Block 2) find that the red bar vs. green bar is significant for real-world charts and the blue bar vs. gray bar is significant both synthetic and real-world charts, whereas the green bar vs. blue bar difference is not significant. We believe that the visual prominence levels for some of the top-ranked features are similar in several charts (i.e., the difference in prominence between the 1st and 2nd ranked features is small) in our dataset and this results in a smaller difference between them, although the trend is in the right direction. \| \| Reported Reliance ---\|---\|--- Source \| Caption Type \| Chart \| Caption Block 1. Overall Synthetic \| all \| $4.675\pm 0.670$ \| $2.624\pm 1.609$ Real-world \| all \| $4.536\pm 0.784$ \| $2.779\pm 1.679$ Block 2. Prominence Synthetic \| 1st \| $4.590\pm 0.711$ \| $3.249\pm 1.327$ 2nd \| $4.567\pm 0.814$ \| $3.082\pm 1.433$ 3rd \| $4.567\pm 0.726$ \| $3.059\pm 1.408$ non-pro \| $4.775\pm 0.549$ \| $2.447\pm 1.429$ \| basic \| $4.850\pm 0.377$ \| $2.593\pm 1.320$ Real-world \| 1st \| $4.494\pm 0.838$ \| $3.405\pm 1.481$ 2nd \| $4.462\pm 0.890$ \| $3.165\pm 1.359$ 3rd \| $4.503\pm 0.805$ \| $3.236\pm 1.354$ non-pro \| $4.595\pm 0.718$ \| $2.680\pm 1.545$ \| basic \| $4.628\pm 0.601$ \| $2.718\pm 1.568$ Block 3. External Information Synthetic \| w/o ext \| $4.679\pm 0.688$ \| $2.798\pm 1.402$ w/ ext \| $4.573\pm 0.728$ \| $3.110\pm 1.448$ Real-world \| w/o ext \| $4.606\pm 0.741$ \| $3.061\pm 1.481$ w/ ext \| $4.424\pm 0.875$ \| $3.194\pm 1.439$ Table 3. The reported reliance on the chart and the caption respectively on 5-point Likert scales. Block 1 shows the reported reliance across all the captions. Block 2 shows the reported reliance depending on the prominence of the feature described in the chart and Block 3 shows the reported reliance depending on the inclusion of external information. The values are reported in the form of $\mu\pm\sigma$. Table 3 shows average and standard deviation of how much the participants reported to have relied on the chart and the caption respectively on a 5-point Likert scale. The results in Table 3 Block 1 suggest that the participants drew information from both the chart and the caption when determining their takeaways, although they consistently relied on the chart more than the caption. These results potentially shed light on why participants took away more often from the chart than the caption when they began to diverge – they relied more on the chart than the caption. The results further suggest that the participants’ tendency to rely on the charts grew while their tendency to rely on the captions declined as the prominence of the feature described in the caption decreased (Table 3 Block 2). We found a significant drop in the self-reported reliance on the caption when the caption described a non- prominent feature compared to when it described the third-most prominent feature (synthetic: Mann-Whitney $U=28941$, $p<0.001$; real-world: Mann- Whitney $U=9666$, $p<0.001$) whereas the increase in the reported reliance on the chart when the caption described a non-prominent feature compared to when it described the third-most prominent feature was only significant with the synthetic charts (Mann-Whitney $U=32844.5$, $p<0.001$). Although the general trend is in the right direction, we did not find significant differences in the reliance scores when the caption mentioned one of the top three prominent features. This may be because the difference in prominence is not as great among these features as it is with the non-prominent feature. These results are in line with our findings from the takeaways; we find that when the chart contains a high-prominence visual feature, but the caption emphasizes a low- prominence feature, participants relied more on the chart and less on the caption. Considering all these results together suggests that we can accept our hypothesis H1 – readers take away from the highly prominent features when the chart and caption both emphasize the same feature and that their inclination to rely more on the most prominent feature instead of the feature described in the caption becomes greater when the caption describes a less prominent feature. H1 Additional Results. We also collected takeaways for charts with basic captions that describe the axes of the chart. (Figure 7 \- middle row). We find that the percentage of takeaways for each of the features is similar to that of the no-caption condition. In fact, Pearson’s chi-square test finds no significant difference between the takeaway histograms of the basic caption and the no-caption conditions (synthetic: $\chi^{2}(4)=1.564$, $p=0.815$; real-world: $\chi^{2}(4)=7.168$, $p=0.127$). While automated captioning tools (Tableau Software, 2020; Microsoft Q & A, 2020) generate captions corresponding to our basic captions, we were unable to find evidence that these captions affect what people take away. Such captions may help readers with accessibility needs; however, we believe further exploration will help future systems determine appropriate uses for such captions. Assessing H2. To evaluate H2, we examine whether including external content information in the caption makes it more likely for readers to take away the feature mentioned in the caption. We find that people are significantly more likely to mention the feature described in the caption when it includes external information than when it does not (Figures 7e and Figures 7f Any bars). A pairwise Z-test finds significant difference between these ratios (synthetic: $Z=2.273,p=0.011$; real-world: $Z=2.032,p=0.021$). In addition, the reported reliance on the chart and the captions shifted towards the captions with external information, which is in-line with our findings (Figure 3 Block 3). Specifically, the reported reliance on the chart was significantly lower with external information (synthetic: Mann-Whitney $U=137318$, $p<0.001$; real-world: Mann-Whitney $U=45292$, $p=0.001$); the reported reliance on the caption was higher with external information, but the difference was only significant for the synthetic charts (synthetic: Mann- Whitney $U=131594$, $p<0.001$; real-world: Mann-Whitney $U=48599.5$, $p=0.132$). The results together suggest that we can accept H2 that states that including external information in the caption helps reinforce the message in the caption and users are more likely to take away from the feature described in the caption. H2 Additional Results. Figure 7 (bottom row) breaks down the ratio of the takeaways that mention the feature described in the caption by level of prominence of the feature. The figure shows that there is usually an increase in the ratio of the takeaways that mentioned the feature described in the caption when the caption included external information for each level of prominence. Among the differences, we only found significant difference when the caption mentioned a non-prominent feature for synthetic charts ($Z=3.027$, $p=0.001$). Further study could shed light on the correlation between the prominence of the feature described in the caption and how external information affects the readers’ takeaways. ## 5\. Design Guidelines (a) “The cheap Yen and PM Abe’s tourism policy caused the number of tourists in Japan to steeply rise between 2011 and 2018.” (b) “Due to the 2008 Financial Crisis, the number of tourists in Japan decreased in 2009.” Figure 8. Examples of chart-caption pairs authored to emphasize the same feature in the data. (a) Both the caption and chart emphasize the sharp positive trend. (b) The original chart is modified to zoom into a portion of the time range and the feature is made more visually prominent with an annotation showing the dip in the number of tourists. The caption describes that dip with additional context. [Examples of chart-caption pairs authored to emphasize the same feature in the data together.]Examples of chart-caption pairs authored to emphasize the same feature in the data. (a) Both the caption and chart emphasize the sharp positive trend. (b) The original chart is modified to zoom into a portion of the time range and the feature is made more visually prominent with an annotation showing the dip in the number of tourists. The caption describes that dip with additional context. Our findings indicate that the readers will take away from the feature doubly emphasized by both the chart and caption if they provide a coherent message. However, when the chart and caption diverge in terms of the feature that they are emphasizing, readers are less likely to use information from the caption in their takeaways. To improve the efficacy of the chart-caption pair, authors could (1) design the chart to make the feature described in the caption more prominent and (2) include external information in the caption to give more context to the information in the caption. There are several ways for authors to emphasize aspects of the data in a chart so that readers’ attention is drawn to these visual features. One technique is to ensure that aspects of the data such as trends and outliers are presented at the right level of detail or interval range; too-broad of a measurement interval may hide a signal. For example, assume that we were given the chart in Figure 8(b)a with the caption in Figure 8(b)b. The decrease in 2009 is not very prominent because the large increase starting in 2011 overshadows the decrease. Zooming closer to the intended feature and cropping out irrelevant features (Figure 8(b)b), helps make the feature more visually prominent. However, when zooming into the data in this manner, authors must take precaution to avoid removing important information or rendering the chart misleading (O’Brien and Lauer, 2018; Pandey et al., 2015). A simple way to further facilitate effective chart reading is to enhance the visualization with highlighting and overlays such as annotations to guide the audience’s attention to the image area they are describing (Kong and Agrawala, 2012) (Figure 8(b)b). Sometimes, a different chart altogether may be more effective to emphasize a particular aspect of the data. For example, converting continuous data in line charts into discrete values could help emphasize individual values that the author would like to focus on. The consistency between the redesigned chart-caption pairs helps readers take away from the doubly emphasized feature (Figure 8(b)). ## 6\. Future Work Chart and caption authoring tools. We would like to explore how this work can provide interesting implications for _both_ chart and caption design to help the author effectively convey a specific point of view. Enhancements to visualization authoring tools could suggest chart design alternatives given a feature that the author would like to emphasize. Specifically, the system could go further by emphasizing features in the chart according to the main message the author wants to convey by automatically adding annotations to the chart, adding highlights, and adjusting levels of detail so that the chart and the caption deliver a concerted message. This will require formulating a high- level language specification that the authors can use to communicate to the system about their intents or a natural language processing module that can infer the authors’ intents based on the captions they write. Coordinating interaction between the chart and the caption such that hovering over the text in the caption would highlight the corresponding visual feature in the chart and vice-versa, is another interesting direction to pursue to help the reader. The resulting system would be a significant extension of the interactive document reader presented by Kong et al. and Kim et al. (Kong et al., 2014; Kim et al., 2018). On the captioning side, a system could classify basic captions, captions about high-prominence features, and captions about low- prominence features. Based on the classification, the system could suggest external information to further emphasize the information presented. Further exploration of caption variations. In this work, we use a template- based approach for generating captions to minimize the effect of the variation of natural language and to keep the experiment size reasonable. Simultaneously, we carefully vary the visual feature described in the caption and the presence of external information to best understand how people read captions and charts together to form their takeaways. Future work could study captions with various natural language expressions and different ways of emphasis. It would be useful to understand whether the relationship between multiple features in a caption (e.g., a simple list - “There were major dips in employment in 2008 and 2020.” or a comparison - “The dip in 2020 was greater than the dip in 2008.”) has an effect on what readers take away. Studying how our findings generalize to other types of external information (e.g., extrapolation, breakdown into subcategories) would be an interesting direction to pursue. Generalization to other chart types. Our work explores how readers take away information when presented with univariate line charts and captions. Basic chart types still have prominent features (e.g., extrema in bar charts, outliers in scatterplots) and less prominent features (e.g., a point in a cluster in scatterplots). We expect similar findings would hold for those other chart types. We leave it to future work to confirm this intuition. ## 7\. Conclusion In this paper, we examine what readers take away from both a chart and its caption. Our results suggest that when the caption mentions visual features of differing prominence levels, the takeaways differ. When the caption mentions a specific feature, the takeaways also tend to mention that feature. We also observed that when a caption mentions a visually prominent feature, the takeaways more consistently mention that feature. On the other hand, when the caption mentions a less prominent feature, the readers’ takeaways are more likely to mention the most prominent prominence features than the feature described in the caption. We also find that including external information in the caption makes the readers more likely to form their takeaways based on the feature described in the caption. From the results of our study, we propose guidelines to better design charts and captions together; using visual cues and alternative chart representations, visual features can be made more prominent and be further emphasized by their descriptions in the caption. Design implications from this work provide opportunities for the authoring of chart and caption pairs in visual analysis tools to effectively convey a specific point of view to the reader. ###### Acknowledgements. The authors thank the Stanford University HCI Group and Tableau Research for their feedback during the development of the studies. The authors also thank the reviewers for their feedback during the review cycle. This work is supported by NSF award III-1714647. ## References * (1) * Amazon Mechanical Turk (2020) Amazon Mechanical Turk 2020. Amazon Mechanical Turk. https://www.mturk.com. * Borkin et al. 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# Probabilistic Solar Power Forecasting: Long Short-Term Memory Network vs Simpler Approaches Vinayak Sharma<EMAIL_ADDRESS>Jorge Ángel González Ordiano Ralf Mikut Umit Cali University of North Carolina at Charlotte Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology Colorado State University Norwegian University of Science and Technology ###### Abstract The high penetration of volatile renewable energy sources such as solar, make methods for coping with the uncertainty associated with them of paramount importance. Probabilistic forecasts are an example of these methods, as they assist energy planners in their decision-making process by providing them with information about the uncertainty of future power generation. Currently, there is a trend towards the use of deep learning probabilistic forecasting methods. However, the point at which the more complex deep learning methods should be preferred over more simple approaches is not yet clear. Therefore, the current article presents a simple comparison between a long short-term memory neural network and other more simple approaches. The comparison consists of training and comparing models able to provide one-day-ahead probabilistic forecasts for a solar power system. Moreover, the current paper makes use of an open source dataset provided during the Global Energy Forecasting Competition of 2014 (GEFCom14). ###### keywords: GEFCom14, Neural Networks, Quantile Regressions, LSTM, Probabilistic Forecasting ## 1 Introduction Over the past couple of years solar power has become one of the most popular renewable energy sources (RES). Unfortunately, the generation of solar power depends completely on the Sun [5]. This dependency on weather adds uncertainty and variability to the generation of solar power. To deal with this uncertainty, solar forecasts are made in-order to predict the future power generation. Solar power forecasts can be categorized into deterministic and probabilistic forecasts [3]. Some examples of deterministic forecasting methods present in literature can be found in [1, 7, 13, 23, 24]. While deterministic forecasts predict only the expected future generation, probabilistic forecasts offer a description of the forecast uncertainty. This additional information helps in managing resources, as well as, in calculating risks associated with future decisions [4, 11]. Furthermore, economic benefits can also be gained from using probabilistic forecasts, since they improve the decision making capabilities within electricity markets [22]. Various methodologies to generate probabilistic solar power forecasts have been discussed in literature. For example, nearest neighbor approaches [25], vector auto-regressive (VAR) models [6], methods for estimating volatility [9], and ensemble models [2]. Additionally, examples of solar power probabilistic forecasting using deep learning techniques can also be found in literature, e.g., in [10]. However, even though deep learning methodologies have gained in popularity in the past couple of years, they have often under- performed in terms of accuracy when compared to other statistical forecasting techniques [19]. For this reason, the current article presents a small experiment with the goal of defining a starting point for understanding the limitations of deep learning probabilistic forecasting methodologies. To be more specific, the experiment consists in training, evaluating, and comparing solar power probabilistic forecasts based on quantile regressions [8] obtained using a long-short term memory (LSTM) neural network (i.e. a deep learning approach) and more simple techniques (i.e. polynomials and a fully connected artificial neural network). Furthermore, the open source dataset of the Global Energy Forecasting Competition of 2014 is used for the experiment. The remainder of the current paper is divided as follows. Section 2 presents a brief description of the various methods tested. Thereafter, Section 3 describes more in detail the conducted experiment. Afterwards, Section 4 presents the obtained results and finally, Section 5 offers the conclusion and outlook of this work. ## 2 Methods Quantile regressions are useful at estimating the uncertainty of a time series’ future. A finite time series $\\{P[k];k=1,\dots,K\\}$ is defined as a sequence of observations $P[k]$ measured at different points in time; with the timestep $k$ defining the order of the observation in the sequence and $K\in\mathbb{N}_{>0}$ representing the time series’ length. In turn, a quantile regression can be viewed as a model able to estimate a quantile with a probability $\tau\in(0,1)$ of a future value $P[k+H]$ at a forecast horizon $H\in\mathbb{N}_{>0}$. For instance, a quantile regression that takes auto-regressive and exogenous values as input can be defined as: $\hat{P}_{\tau}[k+H]=f(P[k],\dots,P[k-H_{\mathrm{l}}],\mathbf{u}^{T}[k],\dots,\mathbf{u}^{T}[k-H_{\mathrm{l}}];\hat{\boldsymbol{\theta}}_{\tau})\text{ ;}$ (1) where $\hat{P}_{\tau}[k+H]$ is the quantile estimate, $H_{\mathrm{l}}$ is the number of used lags, and $\mathbf{u}[k]$ represents a vector containing observations of exogenous time series at timestep $k$. Moreover, $\hat{\boldsymbol{\theta}}_{\tau}$ is a vector containing the estimated regression parameters, which are traditionally obtained through the minimization of the sum of pinball-losses [8]. Furthermore, one of the most important properties of quantile regressions is the fact that pairs of them can be combined to form intervals with a certain probability of containing a future time series’ value (i.e. probabilistic forecasts). Finally, more detailed information of the models used in the present article can be found in the following sections. ### 2.1 Simple Models #### 2.1.1 Polynomials Quantile regressions trained using a polynomial model are multiple linear quantile regressions, whose features can be raised to a maximal allowed degree. Some examples of this type of model can be found in [14]. #### 2.1.2 Fully Connected Artificial Neural Network The fully connected artificial neural network (FCANN) used in the present article is a simple multilayer perceptron [15] with only one hidden layer. The advantage of this model over the polynomials is the fact that it can more easily describe non-linear relations between its inputs and its desired output (i.e. the solar power time series’ future values). It needs to be mentioned, that the FCANN quantile regressions are trained using a method described in [12]. ### 2.2 Long Short-Term Memory Neural Network A Long Short-Term Memory (LSTM) neural network model [18] is part of the Recurrent Neural Network (RNN) family. An RNN is a neural network able to learn temporal dependencies in data. In other words, RNNs can establish a correlation between the previous data points and the current data point in the training sequence [17]. This property makes them ideal for solar power forecasting. However, in cases where long-term relationships need to be learned, traditional RNNs face the problem of gradient vanishing. LSTMs solve this issue by using an additional unit called a memory cell [18] that helps them in learning and explaining long-term relationships [10]. LSTM quantile regressions can be obtained using the pinball-loss as error function during training. ## 3 Experiment ### 3.1 Data The dataset used comes from the solar track of the Global Energy Forecasting Competition of 2014 (i.e. GEFCom14) [16]. It contains three different sets of time series with hourly power measurements of three solar power systems in Australia (normalized to values between 0 and 1), as well as, a number of corresponding weather forecast time series for the period of April $1^{st}$, 2012 to July $1^{st}$, 2014. In the present work, only the forecast weather time series containing forecasts of the solar surface radiation, solar thermal radiation, and top net solar radiation from the 1st zoneID are used. Additionally, the data of only one of the solar power systems is utilized; with 70% of the data used for training and 30% for testing. ### 3.2 Experiment Description The experimental setup, to compare the performance of the LSTM to that of the other models, consists in forecasting daily 99 quantiles (i.e. $\tau=0.01,0.02,$ $\dots,0.99$) of the next 24 hours of solar power generation (i.e. $H=24$, due to the time series’ hourly resolution). Furthermore, the same input data is used for all quantile regressions; i.e. the solar power measured over the past 24 hours and the forecast radiation values for the next day. The polynomial models used have maximal allowed degrees of one up to three, hence they are referred to as Poly1, Poly2, and Poly3. In turn, the simple FCANN models are multilayer perceptrons with one hidden layer and 10 hidden neurons with a tanh activation function. Additionally, a forward feature selection is applied to select the four most relevant features with which both the polynomial and FCANN models are later trained. Moreover, to improve the forecast accuracy, the night values are removed during training and automatically set to zero during testing. Note that all polynomial and FCANN models are trained using the MATLAB open-source toolbox SciXMiner [20]. Finally, an LSTM model with one input layer, one hidden layer, and one output layer is developed using the Keras API 111keras.io. The hidden layer consists of 100 hidden neurons with a sigmoid activation function. Additionally, dropout is applied to avoid overfitting the model. Notice that the neural network architecture (i.e. its hyper-parameters) was selected after conducting a series of preliminary experiments. The value used to evaluate the results on a test set of size $N$ is the pinball-loss averaged over all the estimated quantiles (as in [16]), i.e.: $\begin{split}Q_{\mathrm{PL}}&=\operatorname{mean}\\{Q_{\mathrm{PL},0.01},\dots,Q_{\mathrm{PL},\tau},\dots,Q_{\mathrm{PL},0.99}\\},\text{with}\\\ Q_{\mathrm{PL},\tau}&=\dfrac{1}{N}\sum^{K-H}_{k=1}\begin{dcases}(\tau-1)~{}(\hat{P}_{\tau}[k+H]-P[k+H])&\text{, if }P[k+H]>P_{\tau}[k+H]\\\ \tau~{}(\hat{P}_{\tau}[k+H]-P[k+H])&\text{, else}\end{dcases}\end{split}\text{ .}$ (2) In the previous equation, $Q_{\mathrm{PL},\tau}$ is the pinball-loss obtained by a quantile regression with probability $\tau$, while $Q_{\mathrm{PL}}$ is the average of the pinball-losses obtained by all estimated regressions. Please notice that a comparison based on computation time is excluded from the present article, as some models are created with MATLAB and others with Python. Nevertheless, due to its relevance, such a comparison is to be done in future related works. ## 4 Results The results on the test set from the above described experiments are presented in Table 1. Model \| Avg. Pinball-loss [%] ---\|--- Poly1 \| 1.70 Poly2 \| 1.59 Poly3 \| 1.66 FCANN \| 1.43 LSTM \| 1.43 Table 1: Average pinball-loss test set results from all the models As the contents of Table 1 show, the LSTM outperforms all of the polynomial models. Nonetheless, the difference in pinball-loss between the LSTM model and the best performing polynomial model is not significantly large, as it just amounts to $0.15\%$. Additionally, the FCANN model has in average the same performance as the more complex LSTM model. The underwhelming performance of the LSTM regressions may be caused by different reasons. For instance, their extensive need for a large training dataset, as it is known that deep learning methodologies need large amounts of data to accurately learn the relationship between the dependent and the independent variables [21]. Furthermore, the manually selected hyper-parameters may also be behind the LSTM’s underwhelming performance, as this manual selection does not assure that the optimal set of parameters is found. Another explanation could be that the existing real-world non-linearities can be covered by FCANN as good as by LSTM For the sake of illustration, Figure 1 depicts the interval forecasts obtained by the FCANN and LSTM models. Figure 1: Interval forecasts obtained with the FCANN and LSTM quantile regressions As can be seen in Figure 1, the LSTM intervals seem to be larger than the ones obtained by the FCANN regressions. Therefore it can be argued, that the LSTM may be overestimating in some degree the uncertainty. This aspect needs to be considered in future related works, if the accuracy of the herein LSTM-based probabilistic forecasts is to be improved. ## 5 Conclusion and Outlook The main contribution of the current article is to present a comparison between a long short-term memory (LSTM) model and other more simple approaches; specifically some polynomial models and a simple fully connected artificial neural network (FCANN). The comparison consists in obtaining and evaluating 24 hour ahead probabilistic solar forecasts. The experiment shows that the LSTM model performs slightly better than the polynomials and obtains the same results as the FCANN. Therefore, it can be argued that the complex LSTM may not always provide the best solution, at least not for the dataset evaluated in this paper. Henceforth, the current article recommends the use of simpler/classical forecasting methodologies as a preliminary benchmarking step before exploring more complex deep learning methods. Also, since the underwhelming performance of the LSTM may be caused by a sub- optimal selection of hyper-parameters, hyper-parameter selection via automated machine learning (AutoML) techniques has to be studied in future related works. Moreover, aspects like multiple runs of the neural networks and computation time need also to be taken into consideration in future experiments. At the same time, comparisons as the one presented herein for the case of probabilistic wind and/or load forecasts also need to be studied in the future. ## Acknowledgement The present contribution is supported by the Helmholtz Association under the Joint Initiative “Energy System 2050 — A Contribution of the Research Field Energy” ## References ## References * [1] M. Abuella and B. Chowdhury. Solar power forecasting using support vector regression. arXiv preprint arXiv:1703.09851, 2017. * [2] S. Alessandrini, L. D. Monache, S. Sperati, and G. Cervone. 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# Renormalization group improved pressure for hot and dense quark matter Jean-Loïc Kneur<EMAIL_ADDRESS>Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université Montpellier, 34095 Montpellier, France Marcus Benghi Pinto<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil Tulio E. Restrepo<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil CFisUC - Center for Physics of the University of Coimbra, Department of Physics, Faculty of Sciences and Technology, University of Coimbra, 3004-516 Coimbra, Portugal ###### Abstract We apply the renormalization group optimized perturbation theory (RGOPT) to evaluate the quark contribution to the QCD pressure at finite temperatures and baryonic densities, at next-to-leading order (NLO). Our results are compared to NLO and state-of-the-art higher orders of standard perturbative QCD (pQCD) and hard thermal loop perturbation theory (HTLpt). The RGOPT resummation provides a nonperturbative approximation, exhibiting a drastically better remnant renormalization scale dependence than pQCD, thanks to built-in renormalization group invariance consistency. At NLO, upon simply adding to the RGOPT-resummed quark contributions the purely perturbative NLO glue contribution, our results show a remarkable agreement with ab initio lattice simulation data for temperatures $0.25\lesssim T\lesssim 1\,{\rm GeV}$, with a remnant scale dependence drastically reduced as compared to HTLpt. ## I Introduction The complete prediction of the phase diagram describing strongly interacting matter transitions represents one of the major theoretical challenges in contemporary particle physics, despite the enormous progress achieved by lattice QCD (LQCD) numerical simulations. The main reason is that the well documented sign problem [1], which arises when finite chemical potential ($\mu$) values are considered, prevents LQCD to be reliably applied at intermediate to high finite baryonic densities, while at low densities the problem may be circumvented, e.g., by performing a Taylor expansion around vanishing chemical potential results. In particular, within the latter regime LQCD has been very successful in predicting [2] that a crossover occurs at a pseudocritical temperature close to $T_{pc}\approx 155\,{\rm MeV}$ when $\mu=0$. One alternative to describe the low temperature-high density domain is to employ effective quark theories [3], such as the Nambu–Jona-Lasinio model [4], evaluating physical quantities within some analytical nonperturbative framework (e.g., mean field theory, MFT). This approach predicts that the (chiral) phase transition at low-$T$ and finite $\mu$ is of the first kind [5] so that, as a byproduct, one should observe a critical end point (CP) signalled by a second order phase transition taking place at intermediate values of $T$ and $\mu$ where the first order transition boundary terminates. This intriguing possibility is about to be tested in heavy-ion collisions experiments by decreasing the beam energy, $\sqrt{s_{NN}}$, so that the baryonic density increases. In view of these experiments it is an unfortunate situation that theoretical predictions using the full QCD machinery cannot be consistently carried out with the currently available nonperturbative techniques. As already emphasized LQCD is plagued by the sign problem while analytical tools such as the large-$N$ approximation (which is related to MFT) may produce misleading results at criticality. On the other extreme standard thermal perturbation theory (PT) is unreliable at the relevant temperature and chemical potential ranges. Indeed, despite the asymptotic freedom (AF) property, its convergence can only be achieved at temperatures many orders of magnitude larger than the critical one. Even at intermediate temperatures, it is well-known that thermal PT is plagued by severe infrared divergences, and has to be resummed to be more compatible with strong coupling regimes (for pedagogical reviews and lecture notes see, e.g., Refs. [6, 7] and the very recent Ref. [8]). Yet, even the state-of-the-art, highest available order thermal PT [9, 10], that incorporates a suitable resummation of infrared divergences, becomes more poorly accurate at moderate to low temperatures. A very successful alternative resummation method is to systematically expand from the start around a quasiparticle mass [12, 11, 13], that also more directly avoids infrared divergences apart from improving convergence issues. This is actually close to analytical resummation approaches also used at zero- temperature, reminiscent of the traditional Hartree approximation and its variational generalizations, suitable to tackle the infrared divergence issues of massless theories. Basically one essentially deforms the original Lagrangian by a Gaussian mass term to be treated as an interaction, defining a modified perturbative expansion leading to a sequence of (variationally improved) approximations at successive orders. The latter approaches appear under various names in the literature, such as optimized perturbation theory (OPT) [14, 15, 16] (as we dub it here), linear $\delta$ expansion (LDE) [17], variational perturbation theory (VPT) [18], or screened perturbation theory (SPT) [12, 19] in the thermal context. Remark that adding a Gaussian term does not change the polynomial structure of the theory so that the process is compatible with the usual renormalization procedure. Already at NLO one usually goes beyond the simple Hartree approximation since the variational mass is “nonperturbatively” dressed by incorporating different topologies (exchange terms, vertex corrections, etc) order by order. Moreover, at leading order the OPT has the welcome property of exactly reproducing large-$N$ results [20]. As discussed, e.g., in Ref. [21] this technique has been used to describe successfully a variety of physical situations, involving phase transitions in different models. On the other hand, for thermal theories, the SPT method has been generalized over the past two decades in order to be compatible with Yang-Mills theories. This generalization was made possible thanks to the hard thermal loop (HTL) gauge- invariant effective Lagrangian originally built by Braaten and Pisarski [11]. The high temperature expansion based on the HTL Lagrangian, known as hard thermal loop perturbation theory (HTLpt) [13], has been employed in a series of applications up to NNLO (three-loops), to describe the QCD thermodynamics, considering both the glue [22] and quark [23, 24, 25] sectors at finite temperatures and baryonic densities. Given the intrinsic technical difficulties associated with the HTLpt evaluations, the NNLO state-of-the-art calculations performed typically in Refs. [24, 25] represents a remarkable achievement. Unfortunately it is worth noting a serious remnant issue, also plaguing standard PT but not sensibly reduced in HTLpt: namely the sensitivity to the arbitrary renormalization scale is observed to substantially increase when higher orders are considered. More precisely, as compared to PT the NNLO HTLpt predictions in Refs.[24, 25] are very close to the lattice results for temperatures down to $T\gtrsim 2\,T_{pc}$ for the commonly chosen “central” renormalization scale choice, $M=2\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$. However, even a moderate scale variation of a factor 2 dramatically affects the pressure and related thermodynamical quantities by relative variations of order 1 or more. It has been argued [25] that resumming logarithms may help to improve the situation but, as explained in Refs.[27, 26], it appears that the lack of renormalization group (RG) invariance is more basically rooted within the HTLpt approach. More recently an alternative combining the OPT framework with RG properties has been proposed: the renormalization group optimized perturbation theory (RGOPT)[28, 29, 27, 26]. The main novelty is that it restores RG invariance at all stages of the calculation, in particular when fixing the arbitrary variational mass parameter. At vanishing temperatures it has been used in QCD up to high (three and four-loop) orders to estimate the basic coupling $\alpha_{s}$ [29], predicting values very compatible with the world averages [30]. Also accurate values of the (vacuum) quark condensate were obtained at four-loop[31] and five-loop[32] orders. Concerning thermal theories the RGOPT has been applied to the simpler scalar $\phi^{4}$ model [27, 26] at NLO, as well as to the nonlinear sigma model (NLSM) [33]. In these thermal applications the RGOPT and PT/SPT predictions for the pressure have been compared, showing how the former approximation ameliorates the generic residual scale dependence of thermal perturbation theories at increasing perturbative orders. More recently we have evaluated the quark contribution to the QCD pressure at two-loop (NLO) at finite densities and vanishing temperatures, showing how the method improves over perturbative QCD (pQCD) [34]. In the present work we extend our approach to include the effects of a thermal bath. Note that applying the RGOPT readily to the glue contributions is beyond the present scope, due to some specific technical difficulties as briefly explained below (work in this direction is in progress [35]). Therefore in the present application the RGOPT resummation will be applied strictly only to the quark sector, while the gluons will be treated as in standard (NLO) PT. In the end both contributions will be combined in order to produce our complete final prediction for the NLO QCD pressure. The paper is organized as follows. In the next section we briefly review our starting point, the perturbative expressions considered for the (massive) quark pressure at NLO, for which the basic RGOPT construction is recalled. In Sec. III the RGOPT is precisely defined for the quark pressure up to NLO (two- loop). Details of our two possible prescriptions at NLO are described in Sec. IV (that may be skipped by the reader only interested in the main results). Then Sec.V illustrates our main results for the pressure, both for the pure quark sector and for the full QCD one. We compare our results with the NLO and state-of-the-art higher orders of both PT and HTLpt, and also to lattice data for the complete QCD pressure. Sec. VI contains our conclusions and perspectives. Finally, three appendices specify some formulas and additional details used in our analysis. ## II Quark Contribution to the QCD Pressure ### II.1 RG invariant perturbative pressure Figure 1: Feynman diagrams contributing to the perturbative quark pressure up to NLO ${\cal O}(g)$. At two-loop order-$g$ 111In all what follows we normalize for convenience the (running) coupling in the $\overline{\rm MS}$-scheme as $g(M)\equiv 4\pi\alpha_{S}(M)$., the contribution of (massive) quarks to the QCD perturbative pressure (the Feynman diagrams displayed in Fig. 1) can be obtained by combining the vacuum ($T=\mu=0$) results of Ref.[31] and the $T,\mu\neq 0$ results of Refs.[36, 37]. Considering the case of degenerate masses $m_{u}=m_{d}=m_{s}\equiv m$, the renormalized pressure in the $\overline{\rm MS}$ renormalization scheme, normalized per flavor, is $\displaystyle\frac{P^{PT}_{1}}{N_{f}\,N_{c}}$ $\displaystyle=$ $\displaystyle-\frac{m^{4}}{8\pi^{2}}\left(\frac{3}{4}-L_{m}\right)+2T^{4}J_{1}\left(\frac{m}{T},\frac{\mu}{T}\right)-3g\frac{m^{4}}{2\left(2\pi\right)^{4}}C_{F}\left(L_{m}^{2}-\frac{4}{3}L_{m}+\frac{3}{4}\right)$ (1) $\displaystyle-$ $\displaystyle gC_{F}\left\\{\left[\frac{m^{2}}{4\pi^{2}}\left(2-3L_{m}\right)+\frac{T^{2}}{6}\right]T^{2}J_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)+\frac{T^{4}}{2}J^{2}_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)+m^{2}T^{2}J_{3}\left(\frac{m}{T},\frac{\mu}{T}\right)\right\\}\;,$ where $g\equiv g(M)$, $L_{m}\equiv\ln(m/M)$, $M$ is the arbitrary renormalization scale, $C_{F}=(N_{c}^{2}-1)/(2N_{c})$, $N_{c}=3$, and $N_{f}=3$. The in-medium and thermal effects are included in the (dimensionless) single integrals: $J_{1}\left(\frac{m}{T},\frac{\mu}{T}\right)=\int\frac{d^{3}{\bf\hat{p}}}{\left(2\pi\right)^{3}}\left\\{\ln\left[1+e^{-\left(E_{p}+\frac{\mu}{T}\right)}\right]+\ln\left[1+e^{-\left(E_{p}-\frac{\mu}{T}\right)}\right]\right\\}\;,$ (2) with ${\bf\hat{p}}\equiv{\bf p}/T$, $E_{p}=\sqrt{{\bf\hat{p}}^{2}+\frac{m^{2}}{T^{2}}}$, $J_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)=\int\frac{d^{3}{\bf\hat{p}}}{\left(2\pi\right)^{3}}\frac{1}{E_{p}}\left[f^{+}(E_{p})+f^{-}(E_{p})\right]\;,$ (3) as well as in the double integral (after performing exactly the angular integration over $p\cdot q/(\|p\|\|q\|)$) $J_{3}\left(\frac{m}{T},\frac{\mu}{T}\right)=\frac{1}{(2\pi)^{4}}\int^{\infty}_{0}\int^{\infty}_{0}\frac{d\hat{p}\,\hat{p}\,d\hat{q}\,\hat{q}}{E_{p}E_{q}}\left\\{\Sigma_{+}\ln\left[\frac{E_{p}E_{q}-\frac{m^{2}}{T^{2}}-\hat{p}\hat{q}}{E_{p}E_{q}-\frac{m^{2}}{T^{2}}+\hat{p}\hat{q}}\right]+\Sigma_{-}\ln\left[\frac{E_{p}E_{q}+\frac{m^{2}}{T^{2}}+\hat{p}\hat{q}}{E_{p}E_{q}+\frac{m^{2}}{T^{2}}-\hat{p}\hat{q}}\right]\right\\}\;,$ (4) where $\displaystyle\Sigma_{\pm}$ $\displaystyle=$ $\displaystyle f^{+}\left(E_{p}\right)f^{\pm}\left(E_{q}\right)+f^{-}\left(E_{p}\right)f^{\mp}\left(E_{q}\right),$ (5) in terms of the Fermi-Dirac distributions for anti-quarks ($+$ sign) and quarks ($-$ sign), $f^{\pm}(E_{p})=\frac{1}{1+e^{(E_{p}\pm\frac{\mu}{T})}}\;,$ (7) where $\mu$ represents the quark chemical potential, which relates to the baryonic chemical potential via $\mu_{B}=3\mu$. In the present work we consider the case of symmetric quark matter and so do not distinguish the chemical potentials associated with the different flavors ($\mu_{s}=\mu_{u}=\mu_{d}\equiv\mu$). The generalization to the case of chemical equilibrium needed to impose, e.g., $\beta$ equilibrium should be straightforward. Also relevant for our purpose and comparisons is the well- known resulting two-loop pressure expression for strictly massless quarks (that simplifies considerably since the $J_{i}$ integrals reduce to simple analytical expressions in this case, given for completeness in Appendix A): $\frac{P^{PT}_{1}(m\to 0)}{P_{SB}(T,\mu)}=1-\frac{25g(M)}{42\pi^{2}}\,\left(\frac{1+\frac{72}{5}{\hat{\mu}}^{2}+\frac{144}{5}{\hat{\mu}}^{4}}{1+\frac{120}{7}{\hat{\mu}}^{2}+\frac{240}{7}{\hat{\mu}}^{4}}\right)$ (8) with ${\hat{\mu}}=\mu/(2\pi T)$. The Stefan-Boltzmann (SB) ideal gas limit reads $P_{SB}(T,\mu)=T^{4}N_{f}N_{c}\left(\frac{7\pi^{2}}{180}\right)\left(1+\frac{120}{7}{\hat{\mu}}^{2}+\frac{240}{7}{\hat{\mu}}^{4}\right)\;.$ (9) Coming back to the massive quark case, we next define the standard homogenous RG operator, $M\frac{d}{dM}=M\,\partial_{M}+\beta(g)\,\partial_{g}-m\,\gamma_{m}(g)\,\partial_{m}\;,$ (10) where our normalization convention for the QCD $\beta$-function and anomalous mass dimension $\gamma_{m}$ is $\beta\left(g\right)=-2b_{0}g^{2}-2b_{1}g^{3}+\mathcal{O}\left(g^{4}\right)\;,$ (11) $\gamma_{m}\left(g\right)=\gamma_{0}g+\gamma_{1}g^{2}+\mathcal{O}\left(g^{3}\right)\;,$ (12) where to two-loop order, $\displaystyle\left(4\pi\right)^{2}b_{0}$ $\displaystyle=$ $\displaystyle 11-\frac{2}{3}N_{f},$ (13) $\displaystyle\left(4\pi\right)^{4}b_{1}$ $\displaystyle=$ $\displaystyle 102-\frac{38}{3}N_{f},$ (14) $\displaystyle\gamma_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi^{2}}\;,\;\;\left(4\pi\right)^{4}\gamma_{1}=\frac{404}{3}-\frac{40}{9}N_{f}\;.$ (15) As is well known, the massive pressure (equivalently the massive vacuum energy) is lacking perturbative RG-invariance: it can be easily checked that applying Eq.(10) to Eq.(1) leaves a remnant term of leading order. Now to turn Eq.(1) into a (perturbatively) RG invariant quantity, we proceed as in Refs.[31, 27, 26] (or closer to the present case, as in Ref.[34]), by subtracting a finite zero-point contribution, $\frac{P^{RGPT}}{N_{c}N_{f}}=\frac{P^{PT}}{N_{c}N_{f}}-\frac{m^{4}}{g}\sum_{k}s_{k}g^{k}\;,$ (16) where the $s_{i}$ are determined at successive perturbative orders so that $M\frac{d}{dM}\left(\frac{P^{RGPT}}{N_{c}N_{f}}\right)=0\,,$ (17) up to neglected higher order terms. Since our evaluations are being carried up to NLO, to restore perturbative RG invariance at this order it is sufficient to add the first two $s_{0}$, $s_{1}$ coefficients that involve the coefficients of $\beta(g)$, $\gamma_{m}(g)$ through Eq.(10). One finds explicitly [31, 34] $s_{0}=-\left[(4\pi)^{2}(b_{0}-2\gamma_{0})\right]^{-1},$ (18) $s_{1}=-\frac{1}{4}\left[\frac{b_{1}-2\gamma_{1}}{4(b_{0}-2\gamma_{0})}-\frac{1}{12\pi^{2}}\right].$ (19) ### II.2 Implementing the RGOPT for the quark pressure The RGOPT requires to variationally deform the Lagrangian, by rescaling the coupling (consistently for every standard interaction terms), while also adding a modified Gaussian interpolating (mass) term. Explicitly, following the prescription [29, 34] ${\cal L}_{QCD}^{RGOPT}={\cal L}_{QCD}\|_{g\to\delta g}-m(1-\delta)^{a}{\overline{\psi}}_{q}\psi_{q},$ (20) where ${\cal L}_{QCD}$ is the standard massless QCD Lagrangian and $m$ is an arbitrary quark mass at this stage. This is equivalent to perform the substitution $m\to m(1-\delta)^{a}$ and $g\to\delta g$ in $P^{RGPT}$, Eq. (16), reexpanded in powers of $\delta$, and finally setting $\delta\to 1$ to recover the massless limit. At any finite order this leaves a residual $m$-dependence, that is appropriately fixed by a stationarity criterion[14], the mass optimization prescription (MOP): $\frac{\partial{P}^{RGOPT}}{\partial m}\Bigr{\|}_{\overline{m}}\equiv 0\;.$ (21) The next step is to fix the arbitrary exponent $a$ that we introduced in Eq.(20), by expanding to leading order-$\delta^{0}$ and requiring the resulting pressure to satisfy, since Eq.(21) is used, the reduced (massless) RG equation: $\left[M\partial_{M}+\beta(g)\partial_{g}\right]P=0\;.$ (22) This procedure yields $a=\frac{\gamma_{0}}{2b_{0}}\,,$ (23) that only depend on the universal (scheme-independent) LO RG coefficients, in agreement with previous RGOPT applications to which we refer for demonstration [29, 26, 34]. Accordingly at lowest nontrivial order the resulting RGOPT pressure is given, keeping all terms of formally one-loop order, by $\frac{P^{RGOPT}_{0}}{N_{f}N_{c}}=-\frac{2m^{4}}{\left(4\pi\right)^{2}}\left(\frac{3}{4}-L_{m}\right)-m^{4}\,s_{1}+2T^{4}J_{1}\left(\frac{m}{T},\frac{\mu}{T}\right)+\frac{m^{4}}{\left(4\pi\right)^{2}g\,b_{0}}\,.$ (24) Remark that the LO $s_{0}$ coefficient, Eq.(18), has produced the last term $\propto 1/b_{0}$ in Eq.(24) after algebraic simplifications. There is a subtlety here: as Eq.(19) shows $s_{1}$ involves two-loop RG coefficients and thus it is not mandatory to restore (perturbative) RG invariance at LO, that requires only $s_{0}\neq 0$ as explained. Yet, since $s_{1}$ enters the pressure formally at ${\cal O}(1)$, it appears sensible to include it also within our one-loop RGOPT result Eq.(24), incorporating in this way a priori more complete RG properties. (Actually, the difference between the LO prescriptions with $s_{1}\neq 0$ or taking more simply $s_{1}=0$ is not drastic). At the one-loop level the coupling runs according to the well-known expression $g\left(M\right)=\frac{1}{2b_{0}\,\ln(M/\Lambda_{\overline{\rm MS}})}.$ (25) Proceeding similarly at the next RGOPT order, the NLO pressure reads (after setting $\delta=1$) $\displaystyle\frac{P^{RGOPT}_{1}}{N_{f}N_{c}}$ $\displaystyle=$ $\displaystyle-\frac{m^{4}}{8\pi^{2}}\left(\frac{3}{4}-L_{m}\right)+2T^{4}\,J_{1}\left(\frac{m}{T},\frac{\mu}{T}\right)+\frac{m^{4}}{\left(2\pi\right)^{2}}\left(\frac{\gamma_{0}}{b_{0}}\right)\left(\frac{1}{2}-L_{m}\right)$ (26) $\displaystyle+$ $\displaystyle m^{2}\left(\frac{\gamma_{0}}{b_{0}}\right)T^{2}\,J_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)+\frac{m^{4}}{\left(4\pi\right)^{2}b_{0}}\left\\{\frac{1}{g}\left(1-\frac{\gamma_{0}}{b_{0}}\right)+\left[\left(b_{1}-2\gamma_{1}\right)\pi^{2}-\frac{\left(b_{0}-2\gamma_{0}\right)}{3}\right]\right\\}$ $\displaystyle-$ $\displaystyle 3gC_{F}\frac{m^{4}}{2\left(2\pi\right)^{4}}\left(L_{m}^{2}-\frac{4}{3}L_{m}+\frac{3}{4}\right)$ $\displaystyle-$ $\displaystyle gC_{F}\left\\{\left[\frac{m^{2}}{4\pi^{2}}\left(2-3L_{m}\right)+\frac{T^{2}}{6}\right]T^{2}J_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)+\frac{T^{4}}{2}J^{2}_{2}\left(\frac{m}{T},\frac{\mu}{T}\right)+m^{2}T^{2}\,J_{3}\left(\frac{m}{T},\frac{\mu}{T}\right)\right\\}.$ The exact two-loop (2L) running coupling, analogue of the one-loop Eq.(25), is obtained by solving for $g(M)$ the implicit relation (see, e.g., Ref. [38]) $\ln\frac{M}{\Lambda_{\overline{\text{MS}}}}=\frac{1}{2b_{0}\,g}+\frac{b_{1}}{2b_{0}^{2}}\ln\left(\frac{b_{0}g}{1+\frac{b_{1}}{b_{0}}g}\right),$ (27) for a given $\Lambda_{\overline{\text{MS}}}$ value (this also defines the $\Lambda_{\overline{\text{MS}}}$ basic scale in our normalization conventions). In the numerical illustrations below, we will use a value very close to the latest world average value[30], $\Lambda_{\overline{\rm MS}}=335\,{\rm MeV}$ for $N_{f}=3$, that equivalently corresponds to $\alpha_{s}(N_{f}=3,1.5\,{\rm GeV})\simeq 0.326$. (NB the latter $\alpha_{s}$ value precisely compares with the one taken in the literature for the NLO PT and HTLpt pressures[23]). ## III RG-optimized resummation ### III.1 One-loop RGOPT Before proceeding to our most relevant NLO results, derived basically from Eq.(26), it is useful to examine the probably more transparent RGOPT features at the lowest nontrivial ($\delta^{0}$) LO. We recall that at this order the pressure already satisfies the massless RG Eq.(22) exactly, via the RG-driven exponent Eq.(23) of the variationally modified Lagrangian, Eq.(20). Consequently the arbitrary mass $m$ may be fixed only by using the MOP Eq.(21). The latter acting on the LO pressure Eq.(24) can easily be cast into the form $\frac{1}{b_{0}\,g}+\ln\frac{m^{2}}{M^{2}}-1-16\pi^{2}s_{1}-8\pi^{2}\frac{T^{2}}{m^{2}}J_{2}\left({\frac{m}{T}},{\frac{\mu}{T}}\right)=0,$ (28) whose nontrivial solution gives a RG invariant dressed mass $\overline{m}(g,T,\mu)$, since the combination $1/(b_{0}\,g(M))+\ln m^{2}/M^{2}$ is trivially $M$-independent according to Eq.(25). (NB for more generality we keep $s_{1}$ unspecified at this stage, while for numerics below we will take $s_{1}\neq 0$ as given by Eq.(19)). Once inserting $\overline{m}$ in Eq.(24) it produces a (one-loop) exactly RG invariant pressure, that takes the compact form: $\frac{P^{RGOPT}_{0}}{N_{f}N_{c}}=2T^{4}\,J_{1}\left(\frac{\overline{m}}{T},\frac{\mu}{T}\right)+\frac{T^{2}}{2}{\overline{m}}^{2}J_{2}\left(\frac{\overline{m}}{T},\frac{\mu}{T}\right)-\frac{\overline{m}^{4}}{32\pi^{2}}\;,$ (29) where it is understood that $\overline{m}$ is the nontrivial solution of Eq.(28) 222Notice that the explicit dependence upon $s_{1}$ cancelled in $P^{RGOPT}_{0}$ Eq.(29) upon using Eq.(28), but the solution $\overline{m}$ of Eq.(28) does depend on $s_{1}$ as specified.. Some properties of the dressed mass $\overline{m}(g,T,\mu)$ may be more transparent from considering the above expressions in the high temperature approximation (and $\mu=0$ to simplify), upon using well-known $T\gg m$ limits of the thermal integrals [7] $J_{1},J_{2}$, given in Appendix A. This gives from Eq.(28) ${\overline{m}}^{2}(g,T,\mu=0)=T^{2}\frac{\pi^{2}}{3}\left[\frac{1}{2b_{0}g}-\ln\left(\frac{Me^{\gamma_{E}}}{\pi T}\right)-8\pi^{2}s_{1}\right]^{-1}\;\simeq\frac{3}{8}g\,T^{2}+{\cal O}(g^{2}),$ (30) or, equivalently using Eq.(25) ${\overline{m}}^{2}(T,\mu=0)=T^{2}\frac{\pi^{2}}{3}\left[\ln\left(\frac{\pi T}{e^{\gamma_{E}-\frac{53}{84}}\,\Lambda_{\overline{\rm MS}}}\right)\right]^{-1}\;,$ (31) where we used $8\pi^{2}s_{1}=-53/84$ for $N_{f}=3$. As seen in Eq.(30), for small coupling $\overline{m}$ admits a perturbative expansion having the expected form of a thermal screening mass. We stress however that $\overline{m}$ is unrelated to the perturbative Debye mass [36], which at one- loop order has the well-known expression (for $\mu=0$): $m_{PT}^{2}=\frac{g}{6}\,T^{2}+{\cal O}(g^{2}).$ (32) In contrast $\overline{m}$ in Eq.(30) represents an intermediate variational quantity, whose meaning is merely once being inserted in $P({\overline{m}},g,T,\mu)$ to define the (optimized) physical pressure at a given order. Remark that, upon embedding RG invariance properties via the subtraction terms in Eq.(16), leading to $\overline{m}$ in Eq.(28), the LO RGOPT pressure (29) involves nontrivial interaction terms. Indeed upon perturbatively reexpanding Eq.(29) using Eq.(30), it can be seen to resum arbitrary higher order contributions, although only those contributions induced by the specific leading order RG dependence333In the simpler $O(N)$ $\phi^{4}$ model, the analogous LO RGOPT[26] resums all large-$N$ contributions, reproducing the exactly known large-$N$ pressure[39], including nonanalytic terms $\sim\lambda^{3p/2}$, $p\geq 1$, typical of a boson gas pressure. . Accordingly at LO and in the high-$T$ approximation, using Eq.(25), Eq.(29) takes the simpler form $\frac{P^{RGOPT}_{0}}{P_{SB}}\simeq 1-\frac{5}{14}\left[\ln\left(\frac{\pi T}{e^{\gamma_{E}-\frac{53}{84}}\,\Lambda_{\overline{\rm MS}}}\right)\right]^{-1}\;,$ (33) normalized to the SB ideal quark gas $P_{SB}$ Eq.(9) (here for $\mu=0$). The fact that the higher order contributions may be absorbed essentially into a one-loop running coupling (for $\mu=0$ and high-$T$ limits) is a peculiar LO feature of our construction: as we will see below at NLO the more involved RG- induced higher order corrections are not so simply incorporated. Another RGOPT feature is manifest in Eq.(33): at high-$T$ the explicit $M$-dependence in Eq.(30) has been automatically traded for a dependence in $g(\sim\pi T/\Lambda_{\overline{\rm MS}})$, consequently from scale invariance, rather than being an extra convenient scale choice $M\sim\pi T$ to absorb $\ln(M/\pi T)$ terms like in more standard (non-resummed) thermal perturbative expansions. The LO pressure Eq.(29) is however not expected to give a very realistic approximation of the complete higher order pressure, as it only relies on LO RG-invariance properties embedded within an essentially free gas pressure. The LO dressed mass $\overline{m}$ of Eq.(28) with exact $T$-dependence is illustrated as function of $T$ in Fig. 3 (where it is also compared to NLO RGOPT dressed masses to be specified below). The corresponding pressure Eq.(29) is illustrated e.g. in Figs.5 and 6 for $\mu=0$ or in Figs. 9 and 10 for $\mu\neq 0$, where it is also compared with NLO RGOPT and other NLO results. We will next proceed to the more realistic NLO RGOPT pressure: most of the above features will be maintained, except that the scale invariance can only be achieved approximately beyond LO, as we will examine. ### III.2 Two-loop RGOPT At NLO the RG Eq.(22) is no longer automatically satisfied by Eq.(26) from Eq.(23), and can be thus considered as an independent constraint. Following [27, 33, 34] we can in principle use either the MOP Eq.(21) or the RG Eq.(22), defining two possible alternative dressed mass $\overline{m}(g,T,\mu)$: we will consider in the following both prescriptions, for completeness and comparison purposes. Accordingly the coupling $g(M)$ is simply determined from standard PT, i.e. with its running at (exact) two-loop order given by Eq.(27) and scale $M$ chosen as a combination of $\pi T$ and $\mu$ when both $T,\mu$ are non-zero. A drawback however is that NLO partly spoils the sought RG invariance properties: while at LO, as shown previously the running coupling Eq.(25) perfectly matches the RGOPT pressure Eq.(24) such as to produce exact (one-loop) scale invariance, at NLO the more involved running coupling Eq.(27) has no reason to exactly match the scale dependence of Eq.(26). Accordingly a remnant scale dependence inevitably arises at NLO, as we will exhibit by varying the scale by a factor 2 around central $M\sim 2\pi T$ (for $\mu=0$). Nevertheless, this RGOPT scale dependence is generically milder[27, 26, 33] than the one produced by standard PT, as will be further illustrated in the present analysis. This happens essentially since perturbative RG invariance is maintained by construction for an arbitrary variational mass, such that the RGOPT residual scale dependence is expected to remain moderate (and, to further decrease at NNLO) even for relatively low temperature where the resulting dressed thermal mass is not necessarily perturbatively screened. Using the standard PT running coupling also more directly compares with the same common prescription in other related thermal resummations approaches, like HTLpt typically. But one should keep in mind that identifying the arbitrary renormalization scale $M$ to be ${\cal O}(\pi T)$ is strictly valid only at sufficiently high temperatures. A known unwelcome feature of any related OPT/SPT approaches is the generally increasing number of possible solutions of e.g. Eq.(21) at increasing orders. In contrast in our construction Eq.(23) further guarantees that the only acceptable solutions are those matching[29] the perturbative asymptotic freedom (AF) behavior for $g\to 0$ at $T=0$: a simple but compelling criteria that often selects a unique solution, even at five-loop order so far explored [29, 31]. As it happens however regarding the NLO quark pressure Eq.(26), imposing either Eq.(21) or Eq.(22) both fail to readily give a real dressed mass $\overline{m}(g,T,\mu)$ for a substantial part of the physically relevant $T,\mu$ range. This is admittedly a technical burden of such methods, but the occurrence of complex variational solutions has no deeper physical meaning. Rather, it may be viewed to some extent as an accident of the specific $\overline{\rm MS}$ scheme in which the original perturbative coefficients were calculated, given that nonreal solutions are often expected upon exactly solving nonlinear equations, like in the present case solving for $m$ the NLO Eqs.(21) or (22). At the same time we wish to maintain these relations as exact as possible in order to capture RG resummation properties beyond PT. A crude escape could be simply to take the real part of the solutions, but that potentially loses some of the sought RG properties. The nonreal solution issue also occurred in the simpler $T=\mu=0$ case [29] as well as within the $T=0,\mu\neq 0$ cold quark matter application [34], where it was cured by performing a renormalization scheme change (RSC)[29]. The latter allows for the recovery of real solutions by modifying perturbative coefficients while keeping RG consistency by definition. Of course for such a solution to work the RSC should not be arbitrary, but fixed by a sensible prescription, and importantly such that it remains a moderate (i.e. perturbative) deviation from the original scheme. More specifically in [34] a relevant NLO RSC parameter $B_{2}$ was uniquely fixed by requiring collinearity of the RG and MOP curves in the $\\{m,g\\}$ plane (that precisely expresses the recovering of real solutions). Technically this implies to nullify the determinant of partial derivatives of the RG and MOP equations, and to solve the latter together with, e.g., Eq.(21) for $\\{B_{2},\overline{m}(B_{2},g)\\}$. While solving such a coupled system was easily manageable for the (entirely analytical) $T=0,\mu\neq 0$ NLO expressions in [34], it becomes numerically quite challenging for the rather involved $T,\mu\neq 0$ NLO dependence from Eq.(26). Therefore in the present study, seeking as much as possible for simplicity, we will exploit the RSC arbitrariness quite similarly to recover real solutions, but via simpler alternative prescriptions precisely defined in next Sec. IV. The reader mainly interested in concrete results for the thermodynamical quantities may skip this section proceeding directly to Sec.V. ## IV NLO prescriptions ### IV.1 Simple RSC parametrization Let us first specify for our later purposes the RSC to be used. Since one basically introduces a variational (quark) mass, the most natural and simplest RSC can be defined by modifying only the mass parameter: $m\to m^{\prime}(1+B_{2}g^{2})\,,$ (34) where a single $B_{2}$ coefficient parametrizes a perturbative NLO scheme change from the original ${\overline{\text{MS}}}$-scheme 444Eq.(34) has also the welcome property that it does not affect the definition of the reference QCD scale $\Lambda_{\overline{\text{MS}}}$, in contrast with a similar perturbative modification acting on the coupling (see Ref.[29] for details).. As is well-known, for a perturbative series truncated at order $g^{k}$ (like in the present case the original order-$g$ pressure Eq.(1)), different schemes differ formally by remnant terms of order ${\cal O}(g^{k+1})$, such that the difference between two schemes is expected to decrease at higher orders for sufficiently small coupling value. Note that we perform the perturbative RSC Eq.(34) consistently on the original PT expression (1) prior to its (nonperturbative) modification induced from Eq.(20) with the subsequent $\delta$-expansion. The net RSC modification to the pressure is to add an extra term, $-4g\,(m^{\prime})^{4}s_{0}B_{2}$, entering thus the resulting exact NLO Eq.(21) or Eq.(22). Thus Eq.(34) modifies the latter equations purposefully, now considering those equations as constraints for the arbitrary mass $m^{\prime}$, after the (nonperturbative) modifications from Eq.(20)555To avoid excessive notation proliferation, in what follows once having performed the replacement implied by Eq. (34) we simply rename $m^{\prime}\to m$ the variational mass to be determined from Eq.(21) or Eq.(22).. Accordingly $B_{2}$ may be considered as an extra variational parameter, quite similarly to $m$, thus to be fixed by a definite prescription as will be specified below. ### IV.2 AF-compatible NLO dressed mass solutions To identify some relevant properties of the sought dressed $\overline{m}(g,T,\mu)$ solutions we consider first the MOP Eq.(21) more explicitly at NLO, thus applied to Eq.(26). It is convenient to formally solve it in a first stage for $\ln[m^{2}/M^{2}]$, as that would give simply an exact quadratic equation at $T=\mu=0$. Accordingly the two equations (that are implicit in $m$ for $T,\mu\neq 0$) can be conveniently written, after straightforward algebra, as $-\ln\frac{m^{2}}{M^{2}}+B_{mop}\mp\frac{2\pi}{3g}\sqrt{D_{mop}}=0\;,$ (35) where for $T,\mu\neq 0$, $B_{mop}$ and $D_{mop}$ take a relatively compact form: $B_{mop}=-\frac{7\pi^{2}}{9g}+\frac{5}{6}+4\pi^{2}\left(J_{2}^{\prime}+\frac{T^{2}}{m^{2}}J_{2}\right),$ (36) $\displaystyle D_{mop}$ $\displaystyle=$ $\displaystyle 9\frac{\pi^{2}}{4}-\frac{47}{6}g-g^{2}\left(\frac{35}{16\pi^{2}}+288\frac{\pi^{2}}{7}B_{2}\right)$ (37) $\displaystyle+36\pi^{2}g^{2}\left(J_{2}^{\prime\,2}+\frac{T^{4}}{m^{4}}J_{2}^{2}\right)+9g(g-2\pi^{2})\left(\frac{T^{2}}{m^{2}}J_{2}-J_{2}^{\prime}\right)$ $\displaystyle+8\pi^{2}g^{2}\frac{T^{2}}{m^{2}}\left(3J_{2}-1\right)J_{2}^{\prime}-48\pi^{2}g^{2}\left(J_{3}^{\prime}+\frac{T^{2}}{m^{2}}J_{3}\right)\;,$ where $J_{i}\equiv J_{i}(m^{2}/T^{2},\mu/T)$, $J_{i}^{\prime}\equiv\partial_{x}J_{i}(x)$, (note that here $x\equiv m^{2}/T^{2}$). In Eq.(37) we explicitly separated the $T,\mu$-independent part within $D_{mop}$ in the very first line to make its $T,\mu\to 0$ limit clear (remark also that $D_{mop}(T=0)$ does not depend on $m$). One first property of Eq.(35) is exhibited from expanding it perturbatively to the first few terms. That gives $\ln\frac{\overline{m}^{2}}{M^{2}}(-)\simeq-\frac{16\pi^{2}}{9g}+\frac{139}{54}+8\pi^{2}\frac{T^{2}}{\overline{m}^{2}}J_{2}+{\cal O}(g),$ (38) and $\ln\frac{\overline{m}^{2}}{M^{2}}(+)\simeq\frac{2\pi^{2}}{9g}-\frac{49}{54}+8\pi^{2}J_{2}^{\prime}+{\cal O}(g).$ (39) One easily recognizes that, for $T\to 0$ the leading term for $g\to 0$ in $\overline{m}^{2}(-)$ has the correct AF behavior: $\ln\frac{\overline{m}^{2}}{M^{2}}(-)\sim-1/(b_{0}g)$, noting that $b_{0}=9/(16\pi^{2})$ (for $N_{f}=3$), which as recalled above is a compelling requirement of the RGOPT. In contrast the other $(+)$ solution has a wrong sign and coefficient, thus drastically in contradiction with AF for $g\to 0$. Therefore clearly only the above equation (35) with $(-)$ is to be selected. It is further instructive to investigate the behavior of those two solutions for $T\neq 0$, taking for simplicity the high-$T$ approximation (and $\mu=0$, see Eq.(52)). After straightforward algebra one obtains, for the first few perturbative expansion terms: $\frac{\overline{m}^{2}_{(-)}}{T^{2}}=\frac{3}{8}g\left[1-\frac{3}{8\pi^{2}}g\left(3L_{T}+\frac{85}{36}\right)\right]^{-1}+g^{2}\,\left(\frac{67}{288\pi^{2}}+6J_{3}(0,0)\right)+{\cal O}(g^{3}),$ (40) where we defined for short $L_{T}\equiv\ln\left(\frac{Me^{\gamma_{E}}}{\pi T}\right).$ (41) As seen the AF-compatible solution $\overline{m}(-)$ has a typical perturbative thermal screening mass behavior $m\sim\sqrt{g}\,T$, with a coefficient here mainly determined by RG properties (notice that the first order term is consistent with our LO above result, Eq.(30)). In contrast the non-AF-compatible Eq.(35) with $(+)$ has $\overline{m}(+)$ solutions for $T,\mu\neq 0$ having a coupling dependence that cannot be cast into the form of a perturbative expansion for small enough $g$. Moreover the corresponding real solutions generally give $m/T\gg 1$, unless $g$ is very large (see Appendix B). The latter features give further compelling reasons to rejecting this non-AF solution also for the $T,\mu\neq 0$ case. Thus as anticipated the AF-compatibility criteria leads to a unique MOP solution. The purely perturbative expansion Eq.(40) is however not expected to give a very good approximation for relatively low 666In particular the exact NLO $\overline{m}$ as obtained below can be such that $\overline{m}/T>1$ at sufficiently low $T$ (see Fig. 3), somewhat invalidating the high-$T$ approximation. $T$, and obviously not useful anyway for $\mu\neq 0$. Yet, before to proceed below with the more elaborate RSC prescription to solve exactly Eq.(35), it may be instructive to illustrate the results of using the simple perturbative solution Eq.(40), inserted in our NLO pressure Eq.(26), and with the resulting expression being truncated simply at first order in $g$ (i.e. this is accordingly the NLO generalization of Eq.(33)). This is shown in Fig. 2, compared to the true standard NLO massless quark PT pressure Eq.(8). This result represents a good consistency check of our procedure: the two pressures are not strictly identical but very close, since after expressing the optimized mass $\overline{m}(g)$, the RGOPT is expected to approximate a massless theory. (Note that replacing $m$ in Eq.(26) instead by, e.g., the standard thermal Debye mass Eq.(32), would give results more drastically departing from the massless PT pressure). Now more interestingly, the main purpose of the RGOPT is rather to provide higher order deviations from standard PT, induced by higher order RG-induced terms, as we will exhibit next. Figure 2: Perturbatively re-expanded NLO RGOPT pressure $P(T,\mu=0)$ (red band) compared with standard perturbative NLO pressure Eq.(8) (blue band), with scale dependence $\pi T\leq M\leq 4\pi T$. ### IV.3 NLO mass optimization prescription Going back to the exact MOP Eq.(35), Eq.(37) involves the RSC parameter $B_{2}$ as induced from Eq.(34). In the original $\overline{\rm MS}$ scheme, i.e. $B_{2}\equiv 0$, $D_{mop}$ from Eq.(37) can take negative values for not particularly large couplings777 For example for $T=\mu=0$ where only the first three terms of Eq.(37) are nonvanishing, $D_{mop}\leq 0$ for rather moderate $g\geq 2.64$, i.e $\alpha_{S}\geq 0.21$.. As anticipated above it therefore renders the (exact) $\overline{m}(g,T,\mu)$ solution not always real, except in a rather limited range of physically interesting $T$ and/or $\mu$ values. Remark however that since the (perturbatively leading) first term in Eq.(37) is positive, this lost of real $\overline{m}$ solutions arises solely when considering the exact Eq.(35): now since all our results were obtained from modifying perturbative NLO expressions, one may simply (re)expand Eq.(35) in perturbation, that gives a real expression at arbitrary orders (as partially illustrated by the first few orders of such an expansion in Eq.(40)). But it is soon realized that this is a poor approximation of the actual exact expression, even for $g$ slightly below the value at which $D_{mop}$ becomes negative. Accordingly it would partly lose the sought good RG properties, due to RG-induced contributions being perturbatively truncated. Now with $D_{mop}$ not too far from being positive, a more efficient way to recover real solutions is from an appropriately chosen $B_{2}$ value such that $D_{mop}>0$. Let us thus define precisely our prescription for the MOP Eq.(35): in a first stage we fix the arbitrary RSC parameter $B_{2}$ in Eq.(37) such that $D_{mop}>0$. Next, the resulting modified AF-matching Eq.(35) with $(-)$ is solved exactly (numerically) for $\overline{m}(g,T,\mu)$, recovering real solutions for practically most relevant $g$ values. Note that simply requiring $D_{mop}\geq 0$ does not give a unique prescription, but it happens to be rather constrained: first $D_{mop}=0$ is excluded, as it would spoil the crucial AF-compatibility of Eq.(38), that at least requires the LO (first) term of Eq.(37). On the other hand if $D_{mop}>0$ would be too large, the AF- matching $(-)$ Eq.(35) would take too negative values no longer giving a real solution (i.e., it cannot cross the $x$-axis). Since the problem comes from some negative terms within Eq.(37), a prescription that appears minimal is to fix $B_{2}$ such as to cancel solely the largest (in magnitude) $T,\mu$-independent negative term within Eq.(37), $-(47/6)g$. Explicitly that gives: $B_{2}=-\frac{329}{1728\pi^{2}\,g}\;.$ (42) The latter $B_{2}$ prescription is very simple, and the resulting $\overline{m}_{MOP}$ solution remains real for practically all physically relevant $g(T,\mu)$ values, while still including nontrivial higher order corrections induced from all remnant terms of Eq. (35). Other slightly different $B_{2}$-fixing prescriptions are possible for $T,\mu\neq 0$, but a notable property is that for different $B_{2}$ choices, that imply different exact $\overline{m}(B_{2})$ solutions, the resulting physical pressure $P(\overline{m}(B_{2}),B_{2},\cdots)$ Eq.(26) happens to be largely insensitive to those unphysical $B_{2}$ parameter choices provided that $\overline{m}(B_{2})$ remains real. This welcome feature is to be traced to the underlying RSC properties, together with the further perturbative screening from Eq.(40): ${\overline{m}^{2}}\sim(3/8)gT^{2}+{\cal O}(g^{2})$: as easily checked, $B_{2}$ from Eq.(34) only appears at higher order ${\cal O}(g^{3})$ both in the perturbatively expanded $\overline{m}$ Eq.(40) and corresponding re-expanded pressure from Eq.(26). In other words once $B_{2}$ is adjusted to recover a real $\overline{m}$ solution of Eq.(35), the discrepancies between possibly different $B_{2}$ prescriptions are somewhat hidden within perturbatively higher order terms. To close this subsection, we illustrate in Fig. 3 the resulting dressed thermal masses as function of the temperature, both at LO from Eq.(28), and NLO from MOP Eqs.(35), (42). As already mentioned their behavior is essentially that of screening thermal masses, except that those are determined from RG properties. We also compare in Fig. 3 with the similar dressed thermal mass as obtained from the alternative RG prescription, to be specified in next subsection. Figure 3: Exact LO RGOPT thermal mass (dot-dashed) compared with exact MOP and RG NLO thermal mass for $\pi\leq M\leq 4\pi T$ at $\mu_{B}=0$. Correspondingly Fig. 4 illustrates the relevant RSC deviation $B_{2}g^{2}$ in Eq.(34) resulting from Eq.(42) as function of $T$. As an important crosscheck, it shows that the departure from the original $\overline{\rm MS}$-scheme remains quite moderate. Figure 4: RSC parameter $B_{2}g^{2}(M)$ for the MOP and RG prescriptions for $\pi T\leq M\leq 4\pi T$ at $\mu_{B}=0$. ### IV.4 Alternative NLO RG prescription Alternatively, the other very relevant prescription, as anticipated in Sec.III.2, is to consider the RG Eq.(22) instead of the MOP Eq.(35) to determine the dressed mass $\overline{m}(g,T,\mu)$. Once expressed for $\ln(m^{2}/M^{2})$ it takes a similar quadratic form as Eq.(35), conveniently normalized as $-\ln\frac{m^{2}}{M^{2}}+B_{rg}\mp\frac{8\pi^{2}}{g}\sqrt{\frac{2}{3}D_{rg}}=0\;,$ (43) where explicitly $B_{rg}=-\frac{1}{b_{0}\,g}+\frac{172}{81}-\frac{64}{81}\left(\frac{4g}{9\pi^{2}}\right)\,\frac{1}{1+\frac{4g}{9\pi^{2}}}+8\pi^{2}\frac{T^{2}}{m^{2}}J_{2},$ (44) and $D_{rg}=-\left(\frac{3}{7}B_{2}+\frac{11}{384\pi^{4}}\right)g^{2}-\frac{g}{27}\frac{(4g+81\pi^{2})}{(4g+9\pi^{2})^{2}}+g^{2}\,\frac{T^{4}}{m^{4}}J_{2}\left(J_{2}-\frac{1}{6}\right)-g^{2}\,\frac{T^{2}}{m^{2}}J_{3}.$ (45) Now, similarly to the previous MOP Eq.(35), for $B_{2}=0$ one obtains generally nonreal solutions since in $D_{rg}$ some contributions happen to be negative. In contrast with Eq.(35) however, the crucial AF-matching for the RG solution is already guaranteed solely from the first term in (44), up to higher order terms. These features strongly suggest the prescription fixing the arbitrary RSC parameter $B_{2}$ as simply to fully cancel $D_{rg}$: $D_{rg}(B_{2})\equiv 0.$ (46) Eq.(46) determines $B_{2}$ trivially using Eq.(45), leading to a single real AF-compatible solution $\overline{m}_{RG}$ determined from the first two terms of Eq.(43), the latter being still an implicit equation in $m$ for $T,\mu\neq 0$ via $J_{2}$ entering Eq.(44). Eq.(46) may appear a rather peculiar choice, but there happen to be very few other choices to recover a real RG solution. We stress that for any (MOP or RG) prescriptions the resulting $\overline{m}(B_{2})$ is an intermediate variational parameter without much physical meaning outside its use in the pressure. Here the resulting $\overline{m}_{RG}(B_{2})$ still involves arbitrary higher order contributions, as well as nontrivial $T,\mu$ dependence via $B_{rg}$ in Eq.(44). Similarly as for the MOP above prescription, we have checked that for other $B_{2}$ choices, as long as being moderately different from Eq.(46), our numerical RG results for $T,\mu\neq 0$ are not strongly dependent upon those choices. The dressed exact thermal mass $\overline{m}_{RG}$ resulting from Eqs.(43),(46) is illustrated as function of the temperature in Fig. 3, and compared with the previously discussed LO mass from Eq.(28) and $\overline{m}_{MOP}$ from Eqs.(35), (42). As seen the dressed masses are numerically quite different, but such differences in the two alternative NLO variational masses are drastically reduced within the physical pressure as will be illustrated below. The corresponding RSC deviation $B_{2}g^{2}$ obtained from Eq.(46) is illustrated in Fig. 4 as function of $T$, and compared to the similar MOP $B_{2}g^{2}$ from Eq.(42). Note that despite the visible discrepancies between the two expressions, they are numerically not drastically different and both behave smoothly, except at very low $T\lesssim 0.5$ GeV: as already mentioned above the important feature is that the induced departure from the original $\overline{\rm MS}$-scheme remains moderate. ## V RGOPT pressure results at NLO To obtain the full benefit from the RGOPT, in particular the optimally reduced scale dependence, a price to pay as a result of the variational approach is to first solve exactly numerically for the dressed mass (either from Eq.(21) or alternatively Eq.(22)), prior to its use in the RGOPT pressure at NLO, Eq.(26). Such a procedure is moreover complicated by the onus of complex solutions, cured by the appropriate RSC as specified above in Sec. IV. But the relevant NLO expressions (35) or alternatively (43) are reasonably simple and the numerical procedure is straightforward. Before illustrating the resulting exact NLO RGOPT pressure, we start this section with another intermediate (more perturbative) prescription, to show the gradual improvement typically concerning the remnant scale dependence. ### V.1 A simple perturbative approximation The simplest we can do to recover real solutions without going through RSC considerations as elaborated previously in Sec.IV, while capturing at the same time more accurate $T,\mu$ dependence, is to expand $\overline{m}$ from the MOP Eq.(35) perturbatively to NLO ${\cal O}(g^{2})$, but keeping the exact thermal integrals in the resulting expression. This gives after simple algebra 888As an algebraic subtlety, one should first expand perturbatively the (AF- matching) Eq.(35) with $(-)$ before to solve it formally for $m^{2}/T^{2}$, otherwise one loses the latter AF-matching properties. $\frac{m^{2}_{MOP}}{T^{2}}=\frac{9}{2}gJ_{2}+g^{2}\left[\frac{17}{9}J^{\prime}_{2}(1-12J_{2})+\frac{34}{3}J_{3}+\left(\frac{20371}{1728\pi^{2}}-\frac{81}{32\pi^{2}}\ln\frac{m^{2}}{M^{2}}\right)J_{2}\right],$ (47) therefore still to be solved numerically as an implicit function since $J_{i}\equiv J_{i}(\frac{m}{T},\frac{\mu}{T})$. Figure 5: Comparison of NLO RGOPT quark pressure Eq.(26) with $\overline{m}(g^{2})$ (green, thin lines), LO RGOPT (dotdashed), NLO PT (blue, dashed), NLO HTLpt quark pressure (red, dotted) with scale dependence $\pi T\leq M\leq 4\pi T$ (bands) and central scale $M=2\pi T$ (lines) at $\mu_{B}=0$. The above expression readily gives a real solution, and allows to consider $\mu\neq 0$ within the thermal integrals (and within the running coupling as well) while still keeping a relatively simple “perturbative-like” expression. Inserting the solution of Eq.(47) into the RGOPT NLO quark pressure Eq.(26) (keeping also exact thermal integrals consistently in the latter), gives the results illustrated for $\mu=0$ in Fig. 5, compared with the standard NLO PT pressure Eq.(8), and also with the NLO HTLpt (quark) pressure. [NB for a consistent comparison with the latter at this stage, we have extracted only the quark contributions within the complete QCD NLO HTLpt pressure, which is not a trivial separation as in the case of NLO pQCD. Details are explained in Appendix C]. Alternatively, proceeding similarly with the RG Eq.(43) and (46) gives $\frac{m^{2}_{RG}}{T^{2}}=\frac{9}{2}gJ_{2}+\frac{g^{2}}{32\pi^{2}}\,\left(172-81\ln\frac{m^{2}}{M^{2}}\right)J_{2},$ (48) observing that the LO term and the $\ln M$ dependence are identical to those in Eq.(47). This illustrates that although the MOP and RG prescriptions are quite different if considering their exact determinations, perturbatively they differ only by ${\cal O}(g^{2})$ terms, thus formally higher order than the original NLO perturbative pressure from which they were both constructed. Moreover, inserting Eq.(48) within Eq.(26) gives almost identical results as in Fig. 5. Note also that in both Eqs.(47) and (48) the running $g(M)$ exactly cancels the $M$-dependence at ${\cal O}(g^{2})$, as easily checked using Eqs.(25), (15), and (52). As seen in Fig. 5 the RGOPT pressure with the (MOP or RG) $\overline{m}(g^{2})$ approximation has a more pronounced decrease, i.e. a departure from the ideal gas limit, than the standard NLO PT (pQCD) quark pressure and than LO RGOPT for moderate and low $T$ values, that is mainly traced to the higher order $g^{2}$ contributions in Eq.(47) or Eq.(48). Actually, it is rather closer to the higher orders standard pQCD pressure, as will be illustrated below, partly due to Eq.(26) and the thermal functions $J_{i}$ being kept exact. (If perturbatively reexpanded, the resulting pressure gets back closer to the NLO pQCD result). This is in contrast with the NLO HTLpt pressure, that remains very close to the ideal gas limit except at very low $T$ as seen in Fig. 5 999We mention that the NLO HTLpt pressure in Fig. 5 (and similarly below in Figs. 6-7, Figs. 9-10) is somewhat different than the results in Ref.[23], specially at very low $T$. This is due to considering here only its pure quark contributions, and partly also from using the exact Eq.(27) instead of a more approximate two-loop running expression used in [23].. In Fig. 5 the RGOPT pressure also exhibits a better renormalization scale dependence as compared with NLO pQCD (at least for $T>1$ GeV), although this is only a moderate improvement. Very similar results are obtained for $\mu\neq 0$, that we omit to illustrate. We will see below that the more elaborate untruncated RGOPT pressure, accounting for higher orders in $\overline{m}(g)$, has a more drastically improved scale dependence, which is a main expected RGOPT feature. ### V.2 Hot quark matter: $T\neq 0$, $\mu=0$ #### V.2.1 MOP prescription The resummation properties of the NLO RGOPT become more evident when one compares it with the standard perturbative one (pQCD) at the same NLO. We illustrate (first for $\mu=0$) the exact NLO RGOPT pressure $P(\overline{m},g,T,\mu)$ obtained from our first $\overline{m}_{MOP}$ prescription, defined by solving Eqs.(35),(42) (as explained in details Subsec.IV.3). In Fig. 6 the pressure is displayed as function of the temperature, compared with the LO RGOPT and the standard NLO pQCD Eq.(8), for the scale dependence $\pi T\leq M\leq 4\pi T$. The reduction of scale dependence stemming from the now exact (untruncated) NLO RGOPT appears substantial (about a factor $\sim 2$ improvement for e.g. $T\sim 1$ GeV). The HTLpt NLO (quark) pressure[23] is also shown in the same figure for comparison. We observe that the (NLO) quark HTLpt pressure has a small residual scale dependence for most $T$ values (which is partly a consequence of limiting it to the quark only contribution), but does not depart very much from the ideal gas limit, in contrast with the RGOPT pressure. This latter feature is similar concerning the complete QCD NLO HTLpt[23]), while a more drastic departure from the ideal gas is only obtained at NNLO for HTLpt[25]. Figure 6: RGOPT quark pressures as function of temperature at LO and NLO (MOP prescription) compared with standard NLO PT (pQCD) and NLO HTLpt pressures, with scale dependence $\pi T\leq M\leq 4\pi T$ at $\mu_{B}=0$. Figure 7: Same captions as for Fig.6 but with the RGOPT pressure obtained from alternative $\overline{m}_{RG}$ prescription Eqs.(43), (46). #### V.2.2 Alternative RG prescription Similarly to Fig. 6, we illustrate in Fig. 7 the exact NLO RGOPT pressure as obtained from the alternative $\overline{m}_{RG}$ prescription defined from solving Eqs.(43) and (46) (explained in details in Subsec.IV.4). As is seen the RGOPT reduction of remnant scale dependence is even more substantial than for the previous $\overline{m}_{MOP}$ prescription. The efficient reduction of remnant scale dependence with respect to standard NLO pQCD is also shown more quantitatively in Fig. 8, illustrating the maximal scale variations, $\Delta P/P\equiv(P(M=4\pi T)/P(M=\pi T)-1$, for the different approximations as indicated. Figure 8: $\Delta P/P\equiv P(M=4\pi T)/P(M=\pi T)-1$ as function of temperature (for $\mu_{B}=0$) for the different NLO RGOPT prescriptions compared to standard NLO pQCD, with scale dependence $\pi T\leq M\leq 4\pi T$. Despite the numerically quite different MOP and RG dressed mass (see Fig 3), the resulting physical pressures are much closer for the two prescriptions, except at very low $T$ values (i.e., very large coupling). This is a reasonable crosscheck of the moderate dependence upon the details of the optimization prescriptions, already observed here at NLO. For both the MOP and RG prescriptions lower pressure values are obtained at moderate temperatures as compared to LO RGOPT, NLO HTLpt and NLO pQCD in Figs 6, 7. ### V.3 Hot and dense quark matter Figure 9: RGOPT pressure as function of the temperature at LO and NLO (MOP prescription), compared with NLO pQCD and NLO HTLpt pressures, with scale variation $\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}\leq M\leq 4\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$ at $\mu_{B}=1.2$ GeV. Figure 10: Same captions as in Fig. 9 with alternative NLO RG prescription. We now consider a nonzero chemical potential values. Since the MOP (35), (42) and RG (43), (46) prescriptions are defined quite generically they can be readily applied to the more general $T,\mu\neq 0$ case. As a representative physical value we illustrate our results for $\mu_{B}=1.2$ GeV. For the renormalization scale variation range we take as is common $\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}\leq M\leq 4\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$ within the exact NLO running coupling Eq.(27). This gives the results for the pressure as a function of temperature as shown in Fig. 9 and Fig. 10 for the MOP and RG prescriptions respectively. As is seen, for this rather sizable $\mu_{B}$ value the qualitative picture is very similar to the $\mu_{B}=0$ case above: namely the remnant scale dependence reduction from RGOPT is drastic as compared to pQCD, and sensible departures with respect to both pQCD and HTLpt are obtained from resummation effects at relatively low temperatures. These results appear to support the robustness of the RGOPT for a more reliable exploration of hot and dense matter. ### V.4 Including glue contribution: confrontation to lattice results In principle a rather similar RGOPT treatment of the pure glue sector should be possible, generalizing within our framework the hard thermal loop (HTL) originally proposed in [11], that essentially introduces a gauge-invariant (non-local) effective Lagrangian properly describing a gluon (thermal) “mass” term in the HTLpt approximation. However, this requires technically the evaluation of presently unknown and quite involved thermal integrals, that we leave for future work [35]. Therefore as above anticipated in the present work we treat the pure glue contribution most conservatively in a standard perturbative manner. At the same NLO, the standard perturbative pure glue contribution has the well-known expression [40] $\frac{P^{PT}_{g}}{P_{g,SB}}=1-\frac{15}{4}\left(\frac{g}{4\pi^{2}}\right)+{\cal O}(g^{2}),$ (49) where the ideal gluon gas pressure is $P_{g,SB}=(8\pi^{2}/45)\,T^{4}$. Thus we simply add the perturbative NLO contribution Eq.(49) (properly normalized) to our NLO RGOPT quark contributions Eq.(26), and for the numerical illustrations below we normalize our results to the full ideal pressure of quarks plus gluons: $P_{SB}\to P_{q,SB}+P_{g,SB}$ 101010As a slight abuse of notation, note that in Figs. 5-10 where only quark contributions are included, $P_{SB}$ designates the sole quark ideal pressure Eq.(9), while in Figs.11-13 below $P_{SB}\equiv P_{q,SB}+P_{g,SB}$. Following the progressive elaboration levels as in the previously shown quark pressure approximations, we first illustrate in Fig.11 the results of using the simple perturbatively re-expanded approximation for $\overline{m}$, Eq.(47), for the quark contribution, but supplemented now by the NLO glue contribution, Eq.(49). The resulting RGOPT pressure is compared with both the (massless quark) state-of-the-art ${\rm N}^{3}$LO pQCD, which expression is taken from Ref. [9], and to available LQCD results from Ref. [41, 42, 43]. As is seen, adding the NLO PT glue contribution puts our results in the right ballpark of LQCD data, with clearly visible improvement as compared to pQCD, both for the central scale choice and resulting remnant scale uncertainty. (We also note that using instead the similar RG perturbative approximation Eq.(48) gives almost undistinguishable results from Fig. 11, illustrating the low order perturbative consistency of the two different MOP and RG prescriptions). Figure 11: RGOPT $P(\overline{m}(g^{2}))$ plus NLO $P^{PT}_{g}$ pressure as function of $T$ (green band) compared to (${\rm N}^{3}$LO, $g^{3}\ln g$) pQCD (light blue band), with scale dependence $\pi T\leq M\leq 4\pi T$, and to lattice data [41, 42, 43] at $\mu_{B}=0$. Figure 12: Full NLO RGOPT (MOP prescription) plus NLO $P^{PT}_{g}$ pressure as function of $T$ (grey band) compared to (${\rm N}^{3}LO\,g^{3}\ln g$) pQCD (light blue band), with scale dependence $\pi T\leq M\leq 4\pi T$, and to lattice data [41, 42, 43] at $\mu_{B}=0$. Figure 13: Full NLO RGOPT (RG prescription) plus NLO $P^{PT}_{g}$ pressure (brown band) compared to ${\rm N}^{3}{\rm LO}\,g^{3}\ln g$ pQCD (light blue band), NLO HTLpt (light green band) and NNLO HTLpt (light red band), with scale dependence $\pi T\leq M\leq 4\pi T$, and to lattice data [41, 42, 43] at $\mu_{B}=0$. Next in Figs.12 and 13, we illustrate similarly the results obtained upon adding the NLO PT glue contributions Eq.(49) to the NLO RGOPT quark pressure respectively for the (exact) MOP and RG prescriptions. These are compared with the state-of-the-art ${\rm N}^{3}$LO pQCD [9], and to LQCD results [41, 42, 43]. As seen the RGOPT results get closer to LQCD data, with a further reduced scale dependence, as compared to pQCD. In Fig.13 we compare in addition with both NLO [23] and the state-of-the-art NNLO HTLpt [25]. The pressure from the RG prescription gives the smallest residual scale uncertainties, and is in remarkable agreement with LQCD data in [41] for the central scale $M=2\pi T$, for temperatures as low as $T\sim 0.25$ GeV up to $T=1$ GeV, the highest value considered in [41]. (More precisely let us mention that for the five available LQCD points in [41] with $T>0.3\,{\rm GeV}$ the central scale agreement is at the few permille level, and even slightly better when considering their estimated continuum data). It is also in good agreement with more recent LQCD data [42] at intermediate $T$. The RGOPT pressure is somewhat closer to LQCD results from [41] than the NNLO HTLpt pressure for $0.5\,{\rm GeV}\lesssim T\lesssim 1\,{\rm GeV}$, while at higher $T$ values HTLpt is nearer to the results of [43], and RGOPT shows more sizeable differences of order $5-7\%$. A concomitant feature however is the visible tension between low [41] and higher $T$ [43] LQCD data in their common temperature range 111111We show LQCD data as given in publicly available files[41, 42, 43], that do not include systematic uncertainties.. Let us briefly mention that we have tried some variants of our prescriptions in order to check the stability of our results. First, the other RSC prescription to recover real solutions, mentioned above in Subsec. III.2 and used in Ref.[34], is to require the collinearity of the vectors tangent to the MOP and RG curves considered as functions of $(m,g)$ (see Eq.(4.7) of Ref.[34]). In the present $T\neq 0$ case it is however numerically much more involved than our simpler prescriptions above (in particular to identify the AF-compatible solutions at moderate and low $T$ values). Yet we could check that the resulting pressure is roughly similar to the one given by the MOP prescription in Figs. 6, 12. Next, we have also considered a variant of the RG prescription, by including the NNLO $\sim g\,m^{4}\,s_{2}$ subtraction term of Eq.(16), that is formally of NLO ${\cal O}(g)$ 121212 This variant is the next order analogue of including the NLO coefficient $s_{1}\neq 0$ within LO RGOPT, see e.g. Eq.(30).. The $s_{2}$ expression [31, 32] incorporates three-loop order RG coefficient dependence, thus for consistency we took a three-loop perturbative running coupling generalizing Eq.(27). We remark that the resulting pressure for this variant hardly shows visible differences with Figs. 13, reflecting a good stability, so that we omit to illustrate it. Another physical quantity of interest is the trace anomaly (or equivalently interaction measure). The latter has the well-known expression $\Delta\equiv\varepsilon-3P=T\frac{\partial P}{\partial T}-4P=T^{5}\,\partial(P/T^{4})/\partial T,$ (50) (where the second and third equalities are of course valid only for $\mu=0$). As previously we add the pure glue NLO PT expression to our RGOPT quark contribution. The result is illustrated, for our best RG prescription, in Fig.14 where it is compared to LQCD data[41, 42, 43] only. A very good agreement with LQCD results of [41, 42] is obtained for $0.3\,{\rm GeV}\lesssim T\lesssim 1\,{\rm GeV}$, while there are more visible differences with the higher $T$ results from [43]. Just for indication is also delineated the part of the remnant scale uncertainties originating solely from the RGOPT quark contributions (dashed lines) within the total uncertainties that also include the ones coming from the (standard) NLO PT glue contribution. Figure 14: NLO RGOPT (RG prescription) trace anomaly $\Delta\equiv\varepsilon-3P$ (including $\Delta^{PT}_{g}$) (brown band) compared to lattice data [41, 42, 43]. The additional dashed lines illustrate the scale uncertainty originating solely from RGOPT quark contributions within the full scale uncertainty added by $\Delta_{g}^{PT}$ (brown) band. To conclude this section it may be worth to recap the origin of the drastic differences between RGOPT and HTLpt, the latter being also basically a variational modification of the original QCD Lagrangian with mass terms, although based on the more elaborate HTL effective Lagrangian[11] (in particular with a thermal gluon mass parameter, $m_{D}$). There are essentially three important differences: * • First, the perturbative RG-restoring subtraction terms, like in Eq.(16) typically, are missing in HTLpt. Accordingly the latter lacks perturbative RG- invariance formally by a leading order term of the massive theory pressure, ${\cal O}(m^{4})\ln(M/m)$. Now since for any (gluon or quark) thermal masses, $m^{2}\sim\\#gT^{2}$, and HTLpt is also based on high temperature expansions, the latter uncancelled term is effectively only a three-loop order effect, thus largely screened and harmless at LO, and moderate even at NLO. In contrast this mismatch plainly resurfaces at NNLO HTLpt, presumably mainly explaining the large remnant scale dependence observed in Refs.[22, 24, 25]. * • Second, the interpolating Lagrangian used in HTLpt is linear, namely with an exponent $a=1$ in the HTL equivalent of Eq.(20), instead of our RG-determined Eq.(23). As we have shown[26] this generally spoils RG invariance even when the latter is fulfilled perturbatively by the original pressure. * • Finally, remark that upon choosing a variational mass prescription Eq.(21) in HTLpt (as was done e.g. in [23, 24]), nonreal $\overline{m}$ may occur, similarly to what happens for RGOPT (although it happens rather at NNLO in HTLpt). In NNLO HTLpt applications this issue is avoided simply by replacing the gluon $\overline{m}_{D}$ arbitrary mass by a perturbative thermal mass [22, 25], and taking the quark mass $\overline{m}_{q}=0$. However, enforcing perturbative masses is partly lacking the a priori more nonperturbative behaviour rooted in variational prescriptions. ## VI Conclusions and perspectives We have applied our RGOPT resummation approach at NLO at finite temperature and density for the QCD quark matter. As explained it generates nonperturbative approximations with consistent RG properties already at LO (one-loop). Our NLO results have been compared to NLO and state-of-the-art ${\rm N}^{3}\mbox{LO}$ pQCD predictions as well as to the state-of-the-art (NNLO) HTLpt results. Scale variations in the range $\pi T\leq M\leq 4\pi T$ show that at NLO the method reduces scale uncertainties drastically as compared to pQCD. Since RG properties are consistently embedded within the RGOPT, we stress that generically the scale uncertainty bands observed at NLO should further shrink by considering the NNLO, ${\cal O}(g^{2})$. Our two possible ‘MOP’ or ‘RG’ prescriptions reflect the often non-uniqueness of variational approaches, although here their respective solution is unique from the compelling AF-matching requirement. Moreover the visible prescription difference for the resulting dressed mass (see Fig.3) is perturbatively consistent at low orders (Eqs.(47), (48)), and is substantially reduced within the resulting physical pressures. Using the RG Eq.(22) prescription, that more directly embeds consistent RG properties, not surprisingly gives the best remnant scale dependence at NLO (at it also happened in other considered models [26]). Note that once a specific RSC is adjusted to recover real solutions, the discrepancies between possibly different RSC prescriptions are formally perturbatively higher order terms. Nevertheless since we consider all expressions exactly rather than perturbatively truncated, numerically the RSC has a moderate net effect on the final pressure results. As we have illustrated, any perturbative reexpansion of the exact solutions somehow degrades the scale dependence. Concerning the full QCD pressure, due to some present technical limitations in applying the RGOPT plainly to the glue sector, in this work we have adopted a simple-minded approach, adding the formally same order purely perturbative NLO glue contributions to the pure quark sector, resummed by RGOPT. We have confronted the resulting predictions for the QCD pressure with available LQCD results. For our best RG prescription the central scale $M=2\pi T$ results are in remarkable agreement with the LQCD results [41, 42] for temperatures as low as $T\sim 0.25$ GeV, which lies within the nonperturbative regime, up to $T=1$ GeV. The striking matching with LQCD results from Ref. [41] as seen in Fig.13 may be partly numerically accidental, but variants of our prescription, specifically the MOP pressure in Fig.12, still appears in very good agreement given our essentially NLO construction. Moreover the RG properties native to the RGOPT are not accidental in drastically reducing the scale dependence problem, particularly when comparing our NLO results to NNLO HTLpt. There are however some visible differences between our results and higher $1\,{\rm GeV}\lesssim T\lesssim 2\,{\rm GeV}$ LQCD data[43]. We remark that the LQCD pressure results in [41] and in Ref. [43] appear to be in tension in their common temperature range, while the trace anomaly shows more continuity 131313 Note that the LQCD simulations in Refs. [41, 42, 43] primarily calculate the trace anomaly $\Delta$, the pressure being derived by the integral method, i.e. essentially from numerically integrating the last equality in Eq.(50)., a feature that may call for more investigations independently of our results. When comparing with $2+1$ flavor LQCD as here illustrated, one may also keep in mind our presently not fully realistic approximation of $N_{f}=3$ degenerate flavors. As illustrated the RGOPT properties extend without degradation to sizable chemical potential values, $\mu_{B}=1.2$ GeV, that indicates the potential of our approach towards a more systematic exploration of hot and dense matter. Future applications may consider the inclusion of physical quark masses to generate a more realistic equation of state. ###### Acknowledgements. We thank Peter Petreczky for bringing the results of Ref. [43] to our attention. We thank Eduardo Fraga and Rudnei Ramos for related discussions. M.B.P. is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil), Process No. 303846/2017-8, and by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-(CAPES- Brazil)-Finance Code 001. This author also thanks the Charles Coulomb Laboratory, in Montpellier, for the hospitality. T.E.R. thanks the support and hospitality of CFisUC where part of this work was developed and acknowledges Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-Brazil) for PhD grants at different periods of time. This work was financed in part by INCT-FNA (Process No. 464898/2014-5). ## Appendix A high-$T$ limit We give here for completeness the well-known $T\gg m,\mu=0$ approximations (see e.g.[7, 36]) of the basic thermal integrals defined in Eqs (2)-(4): $2J_{1}(T\gg m,\mu=0)\approx\frac{7\pi^{2}}{180}-\frac{m^{2}}{12\,T^{2}}+\frac{m^{4}}{T^{4}}\frac{2}{(4\pi)^{2}}\left[\frac{3}{4}-\ln\left(\frac{me^{\gamma_{E}}}{\pi T}\right)\right]+{\cal O}\left(\frac{m^{6}}{T^{6}}\right)\;,$ (51) $J_{2}(T\gg m,\mu=0)\approx\frac{1}{12}+\frac{1}{4\pi^{2}}\frac{m^{2}}{T^{2}}\left[\ln\left(\frac{me^{\gamma_{E}}}{\pi T}\right)-\frac{1}{2}\right]+{\cal O}\left(\frac{m^{4}}{T^{4}}\right)\;,$ (52) and the more complicated genuine two-loop integral $J_{3}$ of Eq.(4) has a finite $m\to 0$ limit (however not analytically integrable to our knowledge, we give below its numerically integrated approximate value): $J_{3}\left(\frac{m}{T}\to 0,\frac{\mu}{T}=0\right)=\frac{4}{(2\pi)^{4}}\int_{0}^{\infty}d\hat{p}\int_{0}^{\infty}d\hat{q}\,n_{F}(\hat{p})n_{F}(\hat{q})\,\ln\left(\frac{\|\hat{p}-\hat{q}\|}{\hat{p}+\hat{q}}\right)+{\cal O}\left(\frac{m^{2}}{T^{2}}\right)\simeq-0.00129532+{\cal O}\left(\frac{m^{2}}{T^{2}}\right)$ (53) where $\hat{p},\hat{q}\equiv p/T,q/T$ and $n_{F}(p)=(e^{p}+1)^{-1}$ is the Fermi-Dirac distribution. ## Appendix B Numerical $\overline{m}$ solutions at NLO We discuss here in some details the behavior of the exact NLO numerical solutions for the two MOP or RG prescriptions as defined in Secs.IV.3, and IV.4. Note that using directly the MOP Eq.(35) or the RG Eq.(43) makes the AF solution identification obvious. Concerning the MOP Eq.(35), once $B_{2}$ is consistently determined by Eq.(42) such as to recover $D_{mop}>0$ in Eq.(37), one sees from the structure of (35) that $(-)$ (AF) solutions only exist if $-\ln(m^{2}/M^{2})+B_{mop}>0$, and conversely $(+)$ (non-AF) solutions only exist if $-\ln(m^{2}/M^{2})+B_{mop}<0$. Figure 15: AF and non-AF roots of MOP Eqs.(35), (42) for $M=2\pi T$ and for two representative $T$ values, $T=0.5$ GeV (dashed), $T=1$ GeV (thick). Once $M,g(M)$ are taken to be $T,\mu$-dependent via the perturbative running coupling Eq.(27), Eq.(35) becomes a function of $m/T$ and $g(T/\Lambda_{\overline{MS}})$. Despite the nonlinear dependence in $m/T$, at the level of Eq.(35) both the AF and non-AF solutions happen to be unique in their respective existence range. This is illustrated in Fig. 15 (for $\mu=0$) for two representative low to moderate temperatures, respectively $T=0.5$ and $T=1$ GeV, and for the central scale choice $M=2\pi T$. It is also clear that for any $T$ the smallest solution is the AF one: Indeed for $g(\pi T\leq M\leq 4\pi T)$, $-\ln(m^{2}/M^{2})+B_{mop}$ is a monotonically decreasing function of $m$ for fixed $T$, and is $>0$ (respectively $<0$) below (respectively above) a given $m_{0}$, such that necessarily $\overline{m}(\mbox{AF})<m_{0}<\overline{m}(\mbox{non-AF})$. The value of $m_{0}$ depends quite strongly on $T$ (and $M$): typically for the input corresponding to Fig. 15 with $M=2\pi T$, one finds $m_{0}\simeq 1.28(1.91)$ for $T=0.5(1.0)$ GeV respectively. (Note also that in Fig. 15 the non-AF solution is unrealistically large with respect to $T$, that also makes it easy to unambiguously select the correct AF-matching solutions). At $\mu=0$, following the AF-matching $\overline{m}$ of Eq.(35) continuously from $T=0$ to arbitrary $T$ is in principle possible, although only for a fixed scale $M$ (thus a fixed $g(M)$) unrelated to $T$, otherwise obviously at some small $M\sim\pi T$ one hits on $M\sim\Lambda_{\overline{MS}}$ where the perturbative coupling diverges. For sizable $\mu\neq 0$ the latter problem if avoided if defining as conventional $M\sim\pi\sqrt{T^{2}+\mu^{2}/\pi^{2}}$ (provided that one is not in the case of both $T\ll\mu$ and small $\mu$). Finally concerning the RG Eq.(43), both NLO solutions are already AF-matching, giving thus a unique solution upon using the prescription Eq.(46). Numerically the exact $\overline{m}_{RG}$ solution of Eq.(43) is somewhat larger than $\overline{m}_{MOP}$ for a given $T$, as illustrated in Fig. 3. ## Appendix C NLO and NNLO HTLpt expressions For completeness we specify here how the NLO[23] and NNLO[25] HTLpt pressure expressions were precisely used when compared with other results. In particular for consistent comparison purposes in Figs. 5,6, and Figs. 9,10 we aim to pin down the HTLpt equivalent of the sole quark contributions, as shown up to NLO in Fig. 1, but with the quark and gluon propagators and quark-gluon vertex replaced with HTL-dressed ones consistently. More precisely from first comparing Eq.(51) of [23] to the pure glue NLO HTLpt pressure (given e.g. in Eq.(4.8) of second Ref. in [22]), it is not difficult to single out all terms originating solely from the pure quark vacuum energy. 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[a]Ciaran Hughes # Theory Overview of Heavy Exotic Spectroscopy ###### Abstract This proceeding broadly overviews the current landscape of heavy exotic spectroscopy. Such work includes the composition of certain $X$, $Y$, and $Z$ states, and proceeds to discuss tetraquarks made exclusively of four quarks. The slides for this talk can be found at the following link. ## 1 What Defines a State to be Exotic? In order to review exotic states, it is first necessary to know what defines a state as exotic. By definition, being exotic is being non-conventional. So, what do we mean by a conventional state111A state is rigorously defined to be a pole singularity of the S-matrix.? A conventional state is defined to be any state which we understand “well enough”222This definition also applies to nuclear physics, where there also exists exotic nuclei.. Currently, mesons and baryons are the only states deemed to be conventional, since they are phenomenologically understood well enough in a potential model [1]. Everything else, such as four-quark states, are classified as exotic. This definition of conventional is ever changing, and if we understood a new class of hadrons “well enough”, we would include this new class under the conventional umbrella. For illustrative purposes, the current (as of 2020) status of exotic states in the charmonium sector is shown in Fig. 1 and 2 of Richard Lebed’s talk [2]. Taking all exotic states from all sectors, there have been 44 experimentally observed exotic candidates, and 15 experimentally established333See the non-$q\bar{q}$ mesons 2020 review from the PDG. Also, PDG defines an established signal as being seen by two independent experiments with greater than $5\sigma$ precision. exotic candidates. Colloquially, these exotic candidates are called $XYZ$ states, but the PDG has defined a naming convention444See Naming Scheme for Hadrons 2020 review from the PDG.. ### 1.1 What Can The Exotic States Be? Gell-Mann knew in 1964 that color confinement allowed a plethora of states like $\bar{q}gq$, $\bar{q}\bar{q}qq$, $\bar{q}qqqq$, etc, where $q$ represents a quark, and $g$ an excited gluon degree of freedom. For current exotic states, there are four exotic configurations largely being considered: 1. 1. Hybrids. These $\bar{q}gq$ configurations exist when there is an excited gluon in an octet representation which combines with a $\bar{q}q$ in an octet representation. Phenomenological models include the constituent gluon model and the flux tube model. 2. 2. Molecules. A state composed of two or more constituent hadrons, typically bound by a Yakawa like force. 3. 3. Hadro-Quarkonium. A constituent hadron core with a quark and an anti-quark cloud orbiting the core. 4. 4. Compact Tetraquarks. A four quark state composed of a tightly arranged $qq$ diquark and a separate $\bar{q}\bar{q}$ anti-diquark. The goal of exotic spectroscopy is to correctly match which exotic configurations map onto a specific experimental exotic signal. Certain models start from the proposition that exotic signals arise from a single particular exotic configuration, e.g, are solely a compact tetraquark. However, a single configuration frequently explains only certain aspects of the data, but not all. This has given rise to debate about which single configuration is most apt, for example to describe the $X(3872)$. Such debate disappears if we start from the view point that the most general solution consists of an admixture of multiple configurations. It is then not so surprising that exotic candidates exhibit complicated phenomena which can be explained by the mixture of configurations. ## 2 The $\chi_{c1}(3872)$ aka $X(3872)$ The $X(3872)$ was the first exotic candidate discovered, in 2003 by BELLE, and consequently is the most well known and studied. Its quantum numbers have been established to be $J^{PC}=1^{++}$, $Q=0$, and $I^{G}=0^{+}$. Its decay modes include $\mathcal{B}(\pi^{+}\pi^{-}J/\psi)>3.2\%$. Since charmonium states like the $J/\psi$ have suppressed annihilation effects, the $X(3872)$ must have at least $\bar{c}c$ valence components. Other modes include $\mathcal{B}(D^{0}\bar{D^{}})>30\%$, and $\mathcal{B}(\rho^{0}J/\psi)\sim\mathcal{B}(\omega J/\psi)$. The isospin violations in this latter decay enter during the decay process, and are caused [5] by the 8 MeV mass difference between the $D^{0}\bar{D^{}}$ and $D^{+}\bar{D^{-}}$ components of the $X(3872)$, i.e., isospin violations do not arise in the state itself, but during the decay process. One amazing piece of information about the $X(3872)$ is that its binding energy lies exactly on the $D^{0}\bar{D^{}}$ threshold, with a binding energy of $0\pm 180$ keV. It also has a very narrow width of $\Gamma<1.2$ MeV. Being so close to the two-particle S-wave threshold indicates cusp effects can occur, however a pure cusp explanation of the data has been ruled out [6]. Both a virtual state and bound state are compatible with experimental data [6]. In a potential model which only includes the $\bar{c}c$ component, the nearest state, the $\chi_{c1}(2P)$, is over 100 MeV too high [7]. However, when the $\bar{c}c$ is coupled to the two meson $D\bar{D}^{}$ component, then the energy of the $\chi_{c1}(2P)$ drops and is largely in line with the experimental $X(3872)$ [7]. This is the reason that the $X(3872)$ is considered to be an exotic candidate. Lattice QCD studies find a bound state pole with a binding energy of $\mathcal{O}(10)$ MeV at unphysical pion masses, and indicate that only $\bar{c}c$ and $D\bar{D}$ are important contributions to the wavefunction. They disfavor a diquark interpretation. Still, lattice QCD calculations with physical pion mass could find a different binding energy and pole structure. The most likely model of the $X(3872)$ is an appreciable mixture of a $\bar{c}c$ and a $D\bar{D}$. There has been an interesting proposal that highlights that we can experimentally measure the binding energy an order of magnitude more precisely by using a triangle singularity with low energy $D^{0}\bar{D}^{0}$s [8]. ## 3 The $\psi(4230)$ aka $Y(4230)$ aka $Y(4260)$ State In 2020, the PDG replaced the previously known $Y(4260)$ (found by BELLE in 2013) with the $Y(4230)$ (found by BESIII in 2017), due to the latter being more precise. The quantum numbers of $Y$ states are defined to be $J^{PC}=1^{--}$, $Q=0$, and $I^{G}=0^{-}$. The $Y(4230)$ decays to $\pi^{+}\pi^{-}J/\psi$. Because charmonium states are annihilation suppressed, the $Y(4230)$ must have a minimal $\bar{c}c$ valence component. However, the most notable statement about the $Y(4230)$ is its lack of open-charm decays into $D\bar{D}$ states - unlike excited charmonium states555Charmonium states above threshold decay into open-charm thresholds $2-3$ orders of magnitude more frequently than to hidden-charm thresholds.. The lack of open-charm decays implies that the $Y(4230)$ has to contain more than just $\bar{c}c$, making it an exotic candidate. Any model for the $Y(4230)$ needs to explain the lack of open-charm decays. One model suggests that the $Y(4230)$ is a $D\bar{D}_{1}(2420)$ molecular state. The needed $65$ MeV binding energy is within reach of a potential model [9], making it a viable option. As molecular states decay through their constituents, this explains the $Y(4230)=D\bar{D}_{1}(2420)\to D\pi^{+}D^{-}$ decay, as well as the $Y(4230)\to\gamma X(3872)$ decay (caused by a triangle singularity). However, by studying the $\pi\pi$ and $K\bar{K}$ distributions in the final states of $Y(4230)$ decays, [10] finds that the $Y(4230)$ needs to have a sizable but not dominant $SU(3)$-flavor octet component. This means that the $Y(4230)$ needs a $SU(3)$-flavor singlet component. The $Y(4230)$ is also consistent with a hybrid scenario. Here, lattice QCD energies of hybrids are around 180 MeV too high, but if systematic errors were included then the masses are roughly consistent [1]. Additionally, with potentials extracted from the lattice, pNRQCD finds a hybrid energy consistent with the $Y(4230)$ [1]. Due to hybrid dynamics, decays of hybrids into S-wave open charm are forbidden, making the S-P wave $D\bar{D}_{1}$ channel the dominant decay process. Moreover, heavy quark spin symmetry is broken more in hybrids than in quarkonia, occurring at $\mathcal{O}(\Lambda_{\text{QCD}}/m_{Q})$. This could help explain the observation of the $Y(4230)$ decaying into both the heavy quark spin triplet $\pi^{+}\pi^{-}J/\psi$ and the spin singlet $\pi^{+}\pi^{-}h_{c}$ at comparable rates. Given the current experimental data, the most likely model for the $Y(4230)$ is a mixture of a $D\bar{D}_{1}$ molecule and a hybrid, causing a bound state pole. ## 4 The $Z_{c}(3900)$ and $Z_{b}(10610)$ States The $Z_{b}(10610)$ has decay modes $\mathcal{B}((B\bar{B}^{})^{+})=86\%$, $\mathcal{B}(\Upsilon(nS)\pi^{+})\sim 3\%$. For the $Z_{c}(3900)$ we have $\mathcal{B}((D\bar{D}^{})^{\pm})/\mathcal{B}(J/\psi(nS)\pi^{\pm})=6.2$. Again, due to quarkonium annihilation effects being suppressed, and the $Z$ states being charged, the $Z$ states must have a minimal valence contribution consisting of four quarks $\bar{Q}Qq_{1}q_{2}$. $Z$ states are defined by the quantum numbers $Q=\pm 1,0$, $I^{G}=1^{+}$, and $J^{PC}=1^{+-}$. For the $Z_{b}(10610)$, its large branching fraction into the S-wave $B\bar{B}^{}$ threshold is explained by its small binding energy of $3(3)$ MeV relative to that threshold. In fact, recent lattice QCD work [11] extracted the potential between the $B$ and $\bar{B}^{}$ and found sizable attraction in the potential for small $r$. Certain parameterisations of the potential allow for a virtual state - which would be identified as the $Z_{b}(10610)$. It should not be surprising then that the $Z_{b}(10610)$ has been identified as a virtual state when examining the experimental data by [12]. As a shallow virtual state, it would be molecular [4], and potentially be a mixture of multiple molecular components including $B\bar{B}^{}$, $\pi\Upsilon$, etc. Similarly, the $Z_{c}(3900)$ is 13 MeV lower than the S-wave $D\bar{D}^{}$ threshold. A lattice QCD [1] calculation includes diquark and two meson type operators but does not find evidence of a bound state or narrow resonance. However, [13] indicates that this lattice work is consistent with a virtual state or broad resonance, which is also consistent with experimental data. To distinguish between these cases, smaller bin sizes and better energy resolution are needed in experimental data. For the spatial distribution, the $Z_{c}(3900)$ is most likely a multi- molecular system, where the mixing between the $\pi J/\psi(\rho\eta_{c})-D\bar{D}^{}$ is as important as the diagonal parts of the potential [14]. ## 5 Four Quark States Containing Two or More Heavy Quarks As we have seen, the most understood exotic states are likely a mixture of very different components, e.g., the $X(3872)$ is likely a mixture of $\bar{c}c$ and $D\bar{D}^{}$. This mixing can, and does, cause complicated experimental phenomena, and this in turn makes it difficult to exclude regions in the theoretical space of models. As such, we should first attempt to find an exotic state that is solely composed of a single component and understand this state fully. Afterwards, we can use this understanding when describing the multi-component exotic candidates. To make full use of our intuition, we should focus on bound states rather than virtual or resonance states. A bound state is likely when at least two of the quarks are heavy, as then there is a possibility of the formation of diquark or anti-diquark pairs due to the non-relativistic behavior. The $QQ$ diquark can be in a $\bar{3}/6$ representation with an attractive/repulsive potential. Phenomenologically, the attractive $QQ$ diquark may be deep enough in the heavy-quark potential to cause binding. For a heavy-light diquark $Qq$, there is no known mechanism derived from QCD that can cause this structure to exist, and so I will not discuss it further. ### 5.1 Bound $\bar{b}\bar{b}bb$ States The above reasoning prompted my coauthors and I to search for a bound $\bar{b}\bar{b}bb$ state using lattice QCD. We searched in three channels ($J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$) and used a full basis of two-meson and diquark anti-diquark interpolating operators. We did not find evidence for any bound state below the lowest S-wave thresholds, as shown in Fig. 9 of [15]. When a signal is not found, a bound needs to be set on the likelihood that you missed that signal. This is frequently not done in lattice QCD calculations, but should be. In Fig. 11 [15] we show the probability that we would have missed a bound state at a specific energy within our statistical precision. As such, we have ruled out the possibility of a $\bar{b}\bar{b}bb$ compact tetraquark bound state to $5\sigma$. ### 5.2 Resonance $\bar{c}\bar{c}cc$ States In 2020, LHCb found evidence [16] of a state which decays into 2$J/\psi$. Its mass was around 700 MeV above the 2$J/\psi$ threshold, and was called the $X(6900)$. Consequently, it would be composed of $\bar{c}\bar{c}cc$. Being above threshold, this state is a resonance. A compact tetraquark with (anti-) diquark constituents is the only likely model for the state due to the non- relativistic behavior of charm quarks in this energy region. The $X(6900)$ is very exciting as it is the first clear evidence for a state that can be explained by a diquark model which can be connected to QCD. There are a few noticeable features in Fig. 2 of the LHCb data. First, there are three bumps, and second, there is a dip around 6.8 GeV. All these features need to be explained by a model. As we have discussed, two body S-wave thresholds are important for exotic states. Some important S-wave thresholds are the $2\chi_{c0}$, $\chi_{c1}\chi{c0}$, and the $\Xi_{cc}\bar{\Xi}_{cc}$ open flavour threshold. As these compact tetraquark states should be narrow, the broad width of the experimental bumps hint that multiple states are contributing to the signal. LHCb have not yet performed an amplitude analysis which could distinguish nearby $J^{PC}$ states. What most models seem to agree on is that the $X(6900)$ is some combination of multiple 2S $0^{++}$ states. Then the 1S $0^{++}$ state could either be the first bump around $6.5$ GeV [17], or alternatively could be below the $2J/\psi$ threshold (where LHCb does not have data). If the 1S $0^{++}$ is below the $2J/\psi$ threshold, then the first bump would be some combination of 1P $0^{-+}$ states [18]. The dip is explained by destructive interference from the $2\chi_{c0}$ threshold becoming accessible. Further, such states are likely in the bottom sector. ### 5.3 Bound $bb\bar{u}\bar{d}$ State A $J^{P}=1^{+}$, $I=0$ exotic state composed of $bb\bar{u}\bar{d}$ quarks has been predicted to be bound within a multitude of different theoretical frameworks. Fig. 8 of [19] illustrates this consensus between lattice NRQCD calculations, potentials extracted from lattice QCD, and phenomenological model calculations. For a review talk of this state, and similar $Q_{1}Q_{2}\bar{q_{3}}\bar{q_{4}}$ exotic states, see Anthony Francis’ talk. In fact, there is a straightforward intuitive understanding why this state is bound. First, start with the two $b$-quarks. Assume they are in the infinitely heavy quark mass limit. Then they arrange themselves into an attractive $\bar{3}$ diquark. The $\bar{u}$ and $\bar{d}$ then form a light quark cloud that screens the $bb$ interaction. This is sufficient to form deep binding around $100$ MeV. Adding in finite $b$-quark mass corrections does not change this intuitive picture. Given the robustness of this prediction, a confirmation is needed from experiment, potentially via the $bb\bar{u}\bar{d}\to\Xi^{0}_{bc}\bar{p}$ or $\to B^{-}D^{+}\pi^{-}$ processes. If found, this would be the first bound state that is composed exclusively of four quarks, and would be a very useful leverage point in our understanding of exotics. ### 5.4 The Bound $D_{s0}(2317)$ State The $D_{s0}(2317)$ has many similar features to the $X(3872)$. First, prior to its discovery, it was expected to be the $j=\frac{1}{2}^{+}$ state composed of $c\bar{s}$ quarks, above threshold, and very broad through its decay to $DK$ [7]. However, experimentally, the $D_{s0}(2317)$ state is below the $D^{0}K^{+}$ threshold and very narrow, in contrast to quark model expectations. This is what makes it an exotic candidate. Secondly, when quark potential model calculations include the two-meson $DK$ couplings to the $c\bar{s}$ component, the eigenstate mass dramatically shifts downwards and in line with experimental results. Multiple lattice QCD calculations [20] have been performed, taking into account the various systematical errors, and find a bound state pole nearby the experimental mass. These lattice QCD results find that both the $c\bar{s}$ and two-meson $DK$ components are important, but the diquark operators are not. For these reasons, the $D_{s0}(2317)$ is a prototype to understanding the $X(3872)$. Yet, the $D_{s0}(2317)$ is easier to study theoretically. We can use our understanding of the $D_{s0}(2317)$ as input into understanding the $X(3872)$. It is useful to ask [21] what perturbations can be performed to the $D_{s0}(2317)$ system in order to understand exotics more quantitatively, and the $X(3872)$ specifically? With lattice QCD, we can explore how exotics change (both their mass, pole type, and residue) as we vary the quark mass - a task that experiment cannot perform. In this way, we can supplement experimental data with lattice QCD. Specifically, [21] advocates examining the $D_{s0}\to D~{}K$ process as a function of strange quark mass $m_{s}$, and using leading order heavy-quark effective field theory to describe heavy-light systems, and leading order chiral perturbation theory for light-light pseudoscalar mesons. If we take $m_{s}\to m_{s}-\epsilon$, then $M_{D}$ reduces by $-\epsilon$, while $M_{K}^{2}$ changes by $-B\epsilon$, with $B$ the leading order $\chi$-PT constant. Applying this logic to $D_{s0}\to D~{}K$, we see that we can cause $D_{s0}$ to sit right at threshold, and even to lie above threshold. Studying how the $D_{s0}$ changes would quantitatively illuminate the role of the S-wave threshold effects on exotic states. It would also be interesting to see the bound state pole move in the complex plane and become a virtual/resonance state as the threshold effects change as a function of $m_{s}$. We could even move the binding energy to be exactly that of the $X(3872)$, and understand this arch-typical state more quantitatively. Such a calculation is computationally feasible also. All the expensive pieces can be reused, and only the cheap unphysical strange quark propagators would need to be recomputed. Additionally, the systematic errors would be minimal. Consequently, such a program is imminently needed and has few minimal downsides. ## 6 Conclusion Taken together, we have explored the latest exotic state developments. This has included the experimentally established $X(3872)$, $Y(4230)$, $Z_{c}(3900)$, $Z_{b}(10610)$, the recently discovered all charm tetraquark $X(6900)$, and the $D_{s0}(2317)$. We also discussed the non-existence of a bound all-bottom tetraquark, how lattice QCD can be used to study slight perturbation of the $D_{s0}(2317)$ in order to quantitatively understand S-wave threshold effects, and how imminent results are needed from experiments to discover the bound $bb\bar{u}\bar{d}$ state. We did not get to discuss all of the interesting exotic states currently in the literature. However, given the physics we have learned, we can generalise our knowledge to a multitude of other states. For example, the $Z_{b}(10650)$ is likely the $B^{}\bar{B}^{}$ spin-partner of the $Z_{b}(10610)$. Similarly, the $Z_{c}(4020)$ is likely the $D^{}\bar{D}^{}$ spin-partner of the $Z_{c}(3900)$. The heavy flavour equivalent of the $X(3872)$, the $X_{b}$, has likely not been seen as the $\chi_{b1}(2P)$ is below the open-bottom threshold. This is in contrast to the charm case, which highlights the importance of the $\chi_{c1}(2P)$ for the $X(3872)$. The pentaquark $P_{c}$ is likely a molecule of $\bar{c}c$ and a nucleon. LHCb have released preliminary results about a exotic flavour $cs\bar{u}\bar{d}$ state, where the lattice QCD HadSpec collaboration finds suggestions of this state in the $I=0$, $J^{P}=0^{+}$ channel. There are other established states that all models fit accurately. More experimental data is needed to nullify some of the models describing these states. The exotic candidates in this class are the $Y(4360)$ (which could be the $D_{1}D^{}$ partner of the $Y(4230)$), the $Y(4660)$ (which could be the $D_{s}D_{s1}$strange partner of the $Y(4230)$), and the $Z(4430)$. The future of exotic spectroscopy is bright. With the BELLE and BES upgrades, the new PANDA experiment, and the continuous measurements from the LHC, further data continues to be collected and analysed. 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# Sparse expanders have negative curvature Justin Salez111CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016 Paris, France ###### Abstract We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible “at infinity”. We then transfer this result to finite graphs via local weak convergence. The same approach also applies to the Bacry-Emery curvature condition CD$(0,\infty)$, thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019). ###### Contents 1. 1 Introduction 1. 1.1 Non-negative curvature 2. 1.2 Main result 3. 1.3 Proof outline 2. 2 Random rooted graphs 1. 2.1 Local weak convergence 2. 2.2 Tightness, unimodularity and stationarity 3. 2.3 Spectral radius, entropy and the Liouville property 3. 3 Proof of the main result 1. 3.1 Setting the stage 2. 3.2 Non-negative curvature implies zero entropy 3. 3.3 Zero entropy implies poor spectral expansion ## 1 Introduction ### 1.1 Non-negative curvature The _Ricci curvature_ of a manifold is a fundamental concept in Riemannian geometry, see e.g. [28]. In two celebrated works [42, 43], Ollivier proposed a notion of curvature based on optimal transport which applies to arbitrary metric spaces, hence in particular to the discrete setting of graphs. Specifically, let $G=(V_{G},E_{G})$ be a locally finite connected graph. As usual, write $\deg_{G}(x)$ for the degree of a vertex $x$, and ${\rm d}_{G}(x,y)$ for the length of a minimal path from $x$ to $y$ in $G$. Let also $P_{G}\colon V_{G}\times V_{G}\to[0,1]$ denote the transition matrix of the lazy simple random walk on $G$, i.e. $\displaystyle P_{G}(x,y)$ $\displaystyle:=$ $\displaystyle\left\\{\begin{array}[]{ll}\frac{1}{2\deg_{G}(x)}&\textrm{if }\\{x,y\\}\in E_{G};\\\ \frac{1}{2}&\textrm{if }x=y;\\\ 0&\textrm{else}.\end{array}\right.$ The _Ollivier-Ricci curvature_ at an edge $\\{x,y\\}\in E_{G}$ is defined as $\displaystyle\kappa_{G}(x,y)$ $\displaystyle:=$ $\displaystyle 1-\mathcal{W}_{1}\left(P_{G}(x,\cdot),P_{G}(y,\cdot)\right),$ where $\mathcal{W}_{1}$ denotes the $L^{1}-$Wasserstein distance on $\mathcal{P}_{1}(V_{G},{\rm d}_{G})$, see (21) below. Note that the computation of $\kappa_{G}(x,y)$ amounts to solving a finite-dimensional linear optimization problem, and is therefore amenable to standard algorithmic techniques (see [22] for a beautiful interactive curvature calculator). The Ollivier-Ricci curvature of the whole graph is then defined as $\displaystyle\kappa(G)$ $\displaystyle:=$ $\displaystyle\inf_{\\{x,y\\}\in E_{G}}\kappa_{G}(x,y).$ This fundamental geometric quantity measures how distances are contracted, on average, under the action of $P_{G}$. When $\kappa(G)\geq 0$, the graph $G$ is called _non-negatively curved_. This is the case, for example, when $G$ is the Cayley graph of an abelian group, as witnessed by the obvious coupling that uses the same random generators for both trajectories. Non-negative curvature is equivalent to the requirement that $P_{G}$ is a contraction under the Wasserstein metric $\mathcal{W}_{1}$, and constitutes the essence of the powerful _path coupling method_ for bounding mixing times [18]. Consequences in terms of geometry, mixing, and concentration of measure have been massively investigated, and quantified by a variety of functional inequalities. The literature is too vast for an exhaustive account, and we refer the reader to the seminal papers [42, 43, 34, 30], the survey [44], and the more recent works [24, 41, 21, 32, 40] for details, variations, references, and open problems. In particular, the present work was motivated by the following long- standing question, due to Naor and Milman, and publicized by Ollivier [44, Problem T]. Recall that a _family of expanders_ is a sequence of finite graphs with uniformly bounded degrees, diverging sizes, and spectral gap bounded away from $0$. ###### Question 1 (Problem T in [44]). Is there a family of non-negatively curved expanders ? An instructive special class of graphs for which non-negative curvature is completely understood is that of cubic graphs. Specifically, it was shown in [22] that prism graphs and Möbius ladders are the only cubic graphs with non- negative Ollivier-Ricci curvature. Since these are not expanders, the answer to Question 1 is negative for cubic graphs. To the best of our knowledge, this is the only result in the direction of Question 1, despite the rich body of works on non-negative curvature. ### 1.2 Main result In the present paper, we answer Question 1 negatively in full generality, as well as its CD$(0,\infty)$ analogue raised by Cushing, Liu and Peyerimhoff [23, Conjecture 9.11], see Remark 1 below. Moreover, we show that the answer to Question 1 remains negative even if we significantly relax the required properties. Specifically, denote by $\Delta(G)$ the maximum degree of a finite graph $G$, and by $1\ =\ \lambda_{1}(G)\ \geq\ \lambda_{2}(G)\ \geq\ldots\ \geq\ \lambda_{N}(G)\ \geq\ 0,$ the $N=\|V_{G}\|$ ordered eigenvalues of its transition matrix $P_{G}$. With these notations, Question 1 simply asks whether there exist constants $\Delta\geq 1$, $\rho<1$ and arbitrary large graphs satisfying 1. (A) sparsity: $\Delta(G)\leq\Delta;$ 2. (B) spectral expansion: $\lambda_{2}(G)\leq\rho;$ 3. (C) non-negative curvature: $\kappa(G)\geq 0.$ Our main result says that no large graph can even come close to satisfying these three requirements. ###### Theorem 2 (Main result). Fix $\Delta\geq 1$ and $\rho\in(0,1)$. Then, there exists a constant $\varepsilon=\varepsilon_{\Delta,\rho}>0$ such that _every_ finite graph $G$ must satisfy one of the following conditions: * • either $G$ is far from satisfying the sparsity requirement (A), in the following sense: $\displaystyle\sum_{x\in V_{G}}\deg_{G}(x)\log\deg_{G}(x)$ $\displaystyle>$ $\displaystyle(\Delta\log\Delta)\|V_{G}\|;$ * • or $G$ is far from satisfying the expansion requirement (B), in the following sense: $\displaystyle\mathrm{card}\\{i\colon\lambda_{i}(G)>\rho\\}$ $\displaystyle\geq$ $\displaystyle\varepsilon\|V_{G}\|;$ * • or $G$ is far from satisfying the curvature requirement (C), in the following sense: $\displaystyle\mathrm{card}\\{e\in E_{G}\colon\kappa_{G}(e)<-\varepsilon\\}$ $\displaystyle\geq$ $\displaystyle\varepsilon\|E_{G}\|.$ Note that the conclusion is only meaningful for large graphs, since the second condition is trivially satisfied when $\|V_{G}\|\leq\frac{1}{\varepsilon}$. Here is an equivalent – but perhaps more intuitive – formulation. ###### Theorem 3 (Rephrasing). Let $G_{n}=(V_{n},E_{n}),n\geq 1$ be finite graphs with the sparsity property $\displaystyle\sup_{n\geq 1}\left\\{\frac{1}{\|V_{n}\|}\sum_{x\in V_{n}}\deg_{G_{n}}(x)\log\deg_{G_{n}}(x)\right\\}$ $\displaystyle<$ $\displaystyle\infty.$ (2) Suppose in addition that the Ollivier-Ricci curvature is almost non-negative on most edges, i.e. $\displaystyle\forall\varepsilon>0,\quad\frac{1}{\|E_{n}\|}\,\mathrm{card}\\{e\in E_{n}\colon\kappa_{G_{n}}(e)<-\varepsilon\\}$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle 0.$ (3) Then, a macroscopic proportion of eigenvalues of the transition matrix must accumulate near $1$: $\displaystyle\forall\rho<1,\quad\liminf_{n\to\infty}\left\\{\frac{1}{\|V_{n}\|}\,\mathrm{card}\\{i\colon\lambda_{i}(G_{n})\geq\rho\\}\right\\}$ $\displaystyle>$ $\displaystyle 0.$ (4) Here again, the theorem is only meaningful in the large-size limit $\|V_{n}\|\to\infty$, since the conclusion (4) trivially holds otherwise. The high-level message is that on large sparse graphs, non-negative curvature (in an even weak sense) induces extremely poor spectral expansion. This stands in stark contrast with the traditional idea – quantified by a broad variety of functional inequalities over the past decade – that non-negative curvature is associated with _good_ mixing behavior. ###### Remark 1 (Bacry-Emery curvature). Bacry and Emery [7, 8, 9] developed a different notion of non-negative curvature based on $\Gamma-$calculus and known as the _CD $(0,\infty)$_ condition, see also [33, 26]. Since this notion is local, our proof also applies, with the role of Theorem 11 being played by a recent result of Hua [27, Theorem 2]. Consequently, there is no family of expanders satisfying _CD $(0,\infty)$_, as conjectured by Cushing, Liu and Peyerimhoff [23, Conjecture 9.11]. We note that the weaker statement obtained by replacing _CD $(0,\infty)$_ with _CD $(0,n)$_ was recently established by Münch [39]. We warmly thank David Cushing, Shiping Liu and Florentin Münch for pointing this out. ###### Remark 2 (Laziness). The literature actually contains a whole family of variants $(\kappa_{\alpha})_{\alpha\in[0,1)}$ of the Ollivier-Ricci curvature $\kappa$, obtained by replacing the matrix $P_{G}$ with its $\alpha-$idle version: $\displaystyle P_{G}^{(\alpha)}$ $\displaystyle:=$ $\displaystyle(2-2\alpha)P_{G}+(2\alpha-1)\,{\rm Id}.$ There is even a continuous-time version $\kappa_{\star}:=\lim_{\alpha\to 1}\frac{\kappa_{\alpha}}{1-\alpha}$, proposed in [34] and largely adopted since then. In fact, it was later shown (see [19, Remark 5.4]) that $\frac{\kappa_{\alpha}}{1-\alpha}\leq\kappa_{\star}\ =\ 2\kappa,$ where $\kappa=\kappa_{1/2}$ is the version considered in the present paper. Consequently, our result is stated in the strongest possible form, and applies to all versions of the Ollivier-Ricci curvature. ###### Remark 3 (Eigenvectors). Our proof will actually reveal more than (4): not only are there many eigenvalues near $1$, but the corresponding eigenvectors furthermore charge most vertices significantly. In other words, the poor spectral expansion of non-negatively curved graphs is not restricted to any specific region: it applies everywhere. See Remark 6 for a precise statement. ### 1.3 Proof outline #### Proof outline. The most natural route towards Question 1 would consist in looking for a quantitative upper-bound on the spectral gap of a finite non-negatively curved graph, in terms of its size and maximum degree. Interestingly, we do _not_ pursue this approach here. Neither do we try to obtain asymptotic estimates along a sequence of sparse graphs $(G_{n})_{n\geq 1}$ with non-negative curvature. Instead, we work directly at the elegant level of _local weak limits_ of finite graphs, and exploit their built-in _stationarity_ to prove that non-negative curvature and spectral expansion are incompatible “at infinity”. This relies on the central concept of _asymptotic entropy_ , and its classical relations with the Liouville property and the spectral radius. We then transfer this incompatibility result to finite graphs via a relative- compactness argument. As far as we know, the idea of using local weak limits as a tool to deduce generic bounds on the mixing parameters of sparse Markov chains have not received much attention. We firmly believe that this viewpoint will have many applications. #### Further questions. The surprising “$\deg\log\deg$” requirement (2) is used to define the asymptotic entropy on which our whole argument relies. We do not know whether it is necessary for the conclusion (4) to hold, or whether it can be further relaxed. Note that some degree restriction is necessary, since the complete graph satisfies $\lambda_{2}(G)=\kappa(G)=1/2$, regardless of its size. Also, a drawback of our approach – as of any limit argument – is its non- quantitative nature. It would be interesting to find an explicit upper-bound (vanishing as $n\to\infty$) on the spectral gap of a non-negatively curved graph with $n$ vertices and maximum degree $\Delta$, i.e. to estimate $\displaystyle\gamma_{\Delta}(n)$ $\displaystyle:=$ $\displaystyle\max\\{1-\lambda_{2}(G)\colon\|V_{G}\|=n,\Delta(G)\leq\Delta,\kappa(G)\geq 0\\}.$ #### Organization of the paper. The remainder of the paper is organized as follows: Section 2 offers a brief, self-contained introduction to the framework of random rooted graphs. In particular, we recall the definition of local weak convergence (Section 2.1), introduce the key notions of _unimodularity_ , _stationarity_ and _tightness_ (Section 2.2), and gather important results on the _asymptotic entropy_ of random walks on stationary graphs (Section 2.3). Section 3 is devoted to the proof of the main result, which is reduced (in Section 3.1) to the following two main steps: 1. 1. Proving that non-negative curvature implies zero-entropy (Section 3.2). 2. 2. Proving that zero-entropy causes poor spectral expansion (Section 3.3). #### Acknowledgment. The author warmly thanks Itai Benjamini, David Cushing, Nicolas Curien, Shiping Liu, Russell Lyons, Florentin Münch and Pierre Pansu for many wonderful comments, connections and references. This work was partially supported by Institut Universitaire de France. ## 2 Random rooted graphs In this section, we provide a self-contained introduction to the framework of _local weak convergence_. This limit theory for sparse graphs was introduced by Benjamini and Schramm [14] and developed further by Aldous and Steele [2] and Aldous and Lyons [1]. The limit points are _random rooted graphs_ enjoying a powerful form of _stationarity_. They describe the “internal” geometry of large graphs, as seen from a uniformly chosen vertex. Local weak limits are often much more convenient to work with than the finite-graph sequences that they approximate, and have been shown to capture the asymptotic behavior of a number of natural graph parameters, see, e.g. [35, 17, 16, 3]. The present paper can be viewed as another illustration of the strength of this modern viewpoint. ### 2.1 Local weak convergence #### The space of rooted graphs. All graphs considered in this paper will be simple, undirected, countable, and locally finite. A _rooted graph_ is a pair $(G,o)$, where $G$ is a graph and $o$ is a distinguished vertex, called the _root_. Two rooted graphs $(G,o)$ and $(G^{\prime},o^{\prime})$ are _isomorphic_ , written $G\simeq G^{\prime}$, if there is a bijection $\phi\colon V_{G}\to V_{G^{\prime}}$ which preserves the root ($\phi(o)=o^{\prime}$) and the edges: $\displaystyle\forall x,y\in V_{G},\quad\\{x,y\\}\in E_{G}$ $\displaystyle\Longleftrightarrow$ $\displaystyle\left\\{\phi(x),\phi(y)\right\\}\in E_{G^{\prime}}.$ We let ${\mathscr{G}_{\bullet}}$ denote the set of connected rooted graphs, considered up to the isomorphism relation $\simeq$. To lighten the exposition, we will use the same notation $(G,o)$ for the rooted graph and its equivalence class. We write $\mathcal{B}_{t}(G,o)$ for the _ball of radius $t$ around the root_ in $G$, i.e. the (finite) rooted subgraph of $G$ induced by the set $\\{x\in V_{G}\colon{\rm d}_{G}(o,x)\leq t\\}$. We equip ${\mathscr{G}_{\bullet}}$ with the _local metric_ ${{\rm d}}_{\textsc{loc}}\colon{\mathscr{G}_{\bullet}}\times{\mathscr{G}_{\bullet}}\to[0,1]$, defined by $\displaystyle{{\rm d}}_{\textsc{loc}}((G,o),(G^{\prime},o^{\prime}))$ $\displaystyle:=$ $\displaystyle\frac{1}{1+r},\quad\textrm{ with }\quad r\ =\ \sup\\{t\geq 0\colon\mathcal{B}_{t}(G,o)\simeq\mathcal{B}_{t}(G^{\prime},o^{\prime})\\}.$ In words, two elements of ${\mathscr{G}_{\bullet}}$ are “close” to each other if one has to look “far away” from the root to distinguish them apart. It can be shown that $({\mathscr{G}_{\bullet}},{{\rm d}}_{\textsc{loc}})$ is a complete separable metric space. We equip it with its Borel $\sigma-$algebra, and call ${\mathscr{G}_{\bullet}}-$valued random variables _random rooted graphs_. #### Local weak convergence. Write $\mathcal{P}({\mathscr{G}_{\bullet}})$ for the space of Borel probability measures on ${\mathscr{G}_{\bullet}}$, equipped with the usual topology of weak convergence. If $G$ is an arbitrary finite graph, define its _local profile_ $\mathcal{L}_{G}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ to be the empirical distribution of all possible _rootings_ of $G$, i.e. $\displaystyle\mathcal{L}_{G}$ $\displaystyle:=$ $\displaystyle\frac{1}{\|V_{G}\|}\sum_{x\in V_{G}}\delta_{(G,x)},$ (5) where $(G,x)$ is here implicitly restricted to the connected component of $x$ if $G$ is not connected. Finally, if $G_{n}=(V_{n},E_{n}),{n\geq 1}$ are finite graphs whose local profiles $(\mathcal{L}_{G_{n}})_{n\geq 1}$ admit a limit $\mathcal{L}$ in $\mathcal{P}({\mathscr{G}_{\bullet}})$, we call $\mathcal{L}$ the _local weak limit_ of the sequence $(G_{n})_{n\geq 1}$, and write simply $\displaystyle G_{n}$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle\mathcal{L}.$ In words, $\mathcal{L}$ is the law of a random rooted graph which describes how the deterministic graph $G_{n}$ asymptotically looks when seen from a uniformly chosen root. More formally, $\displaystyle\frac{1}{\|V_{n}\|}\sum_{x\in V_{n}}f(G_{n},x)$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle\mathcal{L}\left[f(G,o)\right]\ \triangleq\ \int_{\mathscr{G}_{\bullet}}f\,{\rm d}\mathcal{L},$ (6) for each continuous, bounded observable $f\colon{\mathscr{G}_{\bullet}}\to\mathbb{R}$. The left-hand side can be thought of as a spatial average of “local contributions” from the various vertices of $G_{n}$. In short, local weak convergence allows one to conveniently replace the asymptotic analysis of such averages with the direct computation of an expectation at the root of a certain random graph. #### Local observables. The class of continuous functions on ${\mathscr{G}_{\bullet}}$ clearly contains (but is not restricted to) all $t-$_local_ observables $(t\geq 0)$, where $f\colon{\mathscr{G}_{\bullet}}\to\mathbb{R}$ is called $t-$_local_ if the value $f(G,o)$ is determined by the (isomorphic class of the) finite ball $\mathcal{B}_{t}(G,o)$. Here is a short list of examples, which will be used throughout the paper without notice: * • The root degree $(G,o)\mapsto\deg_{G}(o)$ is $1-$local. * • The minimum curvature at $o$, $(G,o)\mapsto\min_{x\sim o}\kappa_{G}(o,x)$ is $2-$local. * • For each $t\geq 0$, the return probability $(G,o)\mapsto P_{G}^{t}(o,o)$ is $t-$local (in fact, $(\lfloor t/2\rfloor+1)-$local). * • For each $t\geq 0$, the $t-$step entropy $(G,o)\mapsto-\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log P_{G}^{t}(o,x)$ is $t-$local. ### 2.2 Tightness, unimodularity and stationarity #### Tightness. One of the many reasons for the success of the local weak convergence framework (compared to other limit theories for sparse graphs) is the fact that every “reasonable” sequence of sparse graphs admits a local weak limit. The following tightness criterion, due to Benjamini, Lyons and Schramm, gives an honest mathematical content to this vague claim. Note, of course, that passing to sub-sequences is unavoidable. ###### Theorem 4 (Tightness, see Theorem 3.1 in [12]). Let $G_{n}=(V_{n},E_{n}),{n\geq 1}$ be finite graphs so that $\displaystyle\sup_{n\geq 1}\left\\{\frac{1}{\|V_{n}\|}\sum_{x\in V_{n}}\phi\left(\deg_{G_{n}}(x)\right)\right\\}$ $\displaystyle<$ $\displaystyle\infty,$ for some function $\phi\colon\mathbb{Z}_{+}\to\mathbb{R}_{+}$ satisfying $\phi(d)\gg d$ as $d\to\infty$. Then, $(G_{n})_{n\geq 1}$ has a subsequence which admits a local weak limit. In particular, this criterion applies to the sequence $(G_{n})_{n\geq 1}$ in Theorem 3, with $\phi(d)=d\log d$. This will ensure that we can “pass to the limit” and study the question of existence of non-negatively curved expanders directly at the level of local weak limits. #### Unimodularity. Local weak limits of finite graphs happen to enjoy a powerful distributional invariance, which is directly inherited from the fact that the root is equally likely to be any vertex under the local profile (5). More precisely, a measure $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ is called _unimodular_ if it satisfies $\displaystyle\mathcal{L}\left[\sum_{x\in V_{G}}f(G,o,x)\right]$ $\displaystyle=$ $\displaystyle\mathcal{L}\left[\sum_{x\in V_{G}}f(G,x,o)\right],$ (7) for every Borel function $f\colon{\mathscr{G}_{\bullet\bullet}}\to[0,\infty]$, where ${\mathscr{G}_{\bullet\bullet}}$ denotes the analogue of the space ${\mathscr{G}_{\bullet}}$ with two distinguished roots instead of one. Thinking of $f(G,o,x)$ as an amount of mass sent from $o$ to $x$, the identity (7) expresses the fact that the expected masses received and sent by the root coincide. This _Mass Transport Principle_ is clearly satisfied when $\mathcal{L}$ is the local profile of a finite graph, and is preserved under weak convergence. Thus, we obtain the following fundamental result. ###### Theorem 5 (Inherited unimodularity). All local weak limits of finite graphs are unimodular. Whether the converse holds is a notoriously hard open problem with deep implications, see [1, 25, 12]. Let us here record a first simple consequence of unimodularity, which will be useful. ###### Lemma 6 (Everything shows at the root, see Lemma 2.3 in [1]). Suppose that $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ is unimodular, and let $B\subseteq{\mathscr{G}_{\bullet}}$ be a Borel set such that $\mathcal{L}(B)=1$. Then we also have, $\displaystyle\mathcal{L}\left(\\{\forall x\in V_{G},\ (G,x)\in B\\}\right)$ $\displaystyle=$ $\displaystyle 1.$ ###### Proof. Just apply the Mass Transport Principle with $f(G,o,x)={\bf 1}_{(G,o)\notin B}$. ∎ #### Stationarity. Under a mild integrability condition and a trivial change of measure, unimodularity can be rephrased as _reversibility_ under a natural Markov chain on ${\mathscr{G}_{\bullet}}$. We will here only need the weaker notion of _stationarity_. Specifically, we say that a law $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ is _stationary_ if it is invariant for the Markov chain on ${\mathscr{G}_{\bullet}}$ which, at each step, keeps the underlying graph as it is and moves the root according to the transition matrix $P_{G}$. In other words, $\mathcal{L}$ is stationary if $\displaystyle\mathcal{L}\left[\sum_{x\in V_{G}}P^{t}_{G}(o,x)h(G,x)\right]$ $\displaystyle=$ $\displaystyle\mathcal{L}\left[h(G,o)\right],$ (8) for every Borel function $h\colon{\mathscr{G}_{\bullet}}\to[0,\infty]$ and every $t\geq 0$ (equivalently, for $t=1$). The relation with unimodularity is summed up in the following classical lemma (see, e.g. [10]). ###### Lemma 7 (Degree-biasing). Let $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ be a unimodular law with ${\deg}(\mathcal{L}):=\mathcal{L}[\deg_{G}(o)]<\infty.$ Then, the law $\widehat{\mathcal{L}}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ defined by the following change of measure is stationary: $\displaystyle{\rm d}{\widehat{\mathcal{L}}}(G,o)$ $\displaystyle:=$ $\displaystyle\frac{\deg_{G}(o)}{\deg(\mathcal{L})}\,{\rm d}\mathcal{L}(G,o).$ (9) ###### Proof. Apply the Mass Transport Principle to $\mathcal{L}$ with $f(G,o,x)=h(G,o){\bf 1}_{\\{x,o\\}\in E_{G}}$. ∎ ###### Remark 4 (Mutual absolute continuity). It follows from (9) that the original law $\mathcal{L}$ and its degree-biased version ${\widehat{\mathcal{L}}}$ are mutually absolutely continuous. In other words, we have $\displaystyle\mathcal{L}(B)=1$ $\displaystyle\Longleftrightarrow$ $\displaystyle{\widehat{\mathcal{L}}}(B)=1,$ for any Borel set $B\subseteq{\mathscr{G}_{\bullet}}$, allowing us to transfer results from one law to the other. ### 2.3 Spectral radius, entropy and the Liouville property Stationarity is a powerful property, because it enables the development of an _ergodic theory_ of random rooted graphs. See the inspiring works [37] on Galton-Watson trees, [10] on random rooted graphs, and [11] on general random environments. In particular, a classical application of Kingman’s sub-additive ergodic theorem allows one to define the (quenched) _asymptotic entropy_ of random walks on stationary random graphs, as recalled in the following lemma. ###### Lemma 8 (Entropy). Let $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ be stationary with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. Then the limit $\displaystyle\mathscr{H}(G,o)$ $\displaystyle:=$ $\displaystyle\lim_{t\to\infty}\frac{1}{t}\sum_{x\in V_{G}}P^{t}_{G}(o,x)\log\frac{1}{P_{G}^{t}(o,x)},$ exists $\mathcal{L}-$almost-surely and in $L^{1}({\mathscr{G}_{\bullet}},\mathcal{L})$, and does not depend on the choice of the root $o$. We will henceforth simply write $\mathscr{H}(G)$ instead of $\mathscr{H}(G,o)$, and call this the _entropy_ of $G$. ###### Proof. Let $(G,o)$ have law $\mathcal{L}$, and conditionally on $(G,o)$, let $X=(X_{t})_{t\geq 0}$ be a lazy simple random walk on $G$ starting from $X_{0}=o$. For $0\leq s\leq t$, define a non-negative random variable $Z_{s,t}$ by $\displaystyle Z_{s,t}$ $\displaystyle:=$ $\displaystyle\log\frac{1}{P_{G}^{t-s}(X_{s},X_{t})}.$ Note that $Z_{t,s}\stackrel{{\scriptstyle d}}{{=}}Z_{0,t-s}$. Indeed, for any Borel function $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$, we have by definition $\displaystyle{\mathbb{E}}\left[f(Z_{s,t})\right]$ $\displaystyle=$ $\displaystyle{\mathbb{E}}\left[\sum_{x,y\in V_{G}}P_{G}^{s}(o,x)P_{G}^{t-s}(x,y)f\left(\log\frac{1}{P_{G}^{t-s}(x,y)}\right)\right]$ $\displaystyle=$ $\displaystyle{\mathbb{E}}\left[\sum_{y\in V_{G}}P_{G}^{t-s}(o,y)f\left(\log\frac{1}{P_{G}^{t-s}(o,y)}\right)\right]$ $\displaystyle=$ $\displaystyle{\mathbb{E}}\left[f(Z_{0,t-s})\right],$ where the second line uses the stationarity (8) with $h(G,o)=\sum_{y}P_{G}^{t-s}(o,y)f\left(\log\frac{1}{P_{G}^{t-s}(o,y)}\right)$. Moreover, the trivial inequality $P_{G}^{t}(o,y)\geq P_{G}^{s}(o,x)P_{G}^{t-s}(x,y)$ readily implies the sub-additive property $\displaystyle Z_{0,t}$ $\displaystyle\leq$ $\displaystyle Z_{0,s}+Z_{s,t}.$ (10) Finally, the assumption $\mathcal{L}[\log\deg_{G}(o)]<\infty$ ensures that ${\mathbb{E}}[Z_{0,1}]<\infty$. Consequently, Kingman’s sub-additive ergodic theorem (see, e.g. [38, Theorem 14.44]) guarantees the existence of a non- negative, integrable random variable $Z_{\infty}$ such that almost-surely and in $L^{1}$, $\displaystyle\frac{Z_{0,t}}{t}$ $\displaystyle\xrightarrow[t\to\infty]{}$ $\displaystyle Z_{\infty}.$ Averaging this convergence over the random walk $X$ (i.e., taking conditional expectation given the random rooted graph) yields the existence of the limit $\mathscr{H}(G,o)$. By Lemma 6, the same is true if $o$ is replaced by any $x\in V_{G}$. Moreover, the sub-additive property (10) with $s=1$ shows that $\displaystyle\mathscr{H}(G,o)$ $\displaystyle\leq$ $\displaystyle\sum_{x\in V_{G}}P_{G}(o,x)\mathscr{H}(G,x),$ $\mathcal{L}-$almost-surely. Since $\theta\mapsto(\theta-a)_{+}$ is monotone and convex for $a\geq 0$, this inequality implies $\displaystyle\forall a\geq 0,\quad\left(\mathscr{H}(G,o)-a\right)_{+}$ $\displaystyle\leq$ $\displaystyle\sum_{x\in V_{G}}P_{G}(o,x)\left(\mathscr{H}(G,x)-a\right)_{+}.$ But the two sides have the same law by stationarity, so they must coincide $\mathcal{L}-$almost-surely. The fact that this is true for all $a\geq 0$ deterministically forces the equality $\mathscr{H}(G,x)=\mathscr{H}(G,o)$ for all neighbours $x$ of $o$, and hence for all $x\in V_{G}$ by Lemma 6. ∎ #### The Liouville property. One of the interests of asymptotic entropy lies in its relation with the Liouville property. A function $f\colon V_{G}\to\mathbb{R}$ is called _harmonic_ on $G$ if $P_{G}f=f$, where $\displaystyle\forall x\in V_{G},\quad(P_{G}f)(x)$ $\displaystyle:=$ $\displaystyle\sum_{y\in V_{G}}P_{G}(x,y)f(y).$ (11) This is trivially the case, in particular, when $f$ is constant. The graph $G$ has the _Liouville property_ if it admits no non-constant bounded harmonic function. For stationary random graphs, this functional-analytic property turns out to admit the following simple entropic characterization. ###### Theorem 9 (Entropic characterization of the Liouville property). The equivalence $\displaystyle\mathscr{H}(G)=0$ $\displaystyle\Longleftrightarrow$ $\displaystyle G\textrm{ has the Liouville property},$ holds almost-surely under any stationary law $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. This remarkable result has a long history: it originates with the pioneering works of Avez [5, 6, 4], and was then made famous in a celebrated paper of Kaimanovich and Vershik [31]. In the present setting of stationary random graphs, the implication $\Longrightarrow$ was established by Benjamini and Curien [10], and refined by Benjamini, Duminil-Copin, Kozma and Yadin [11]. The converse $\Longleftarrow$ was proved by Carrasco Piaggio and Lessa [20] (see also [13]), but under an additional growth assumption. Since this is the implication that we are going to use, we need to give more details. ###### Proof of Theorem 9. Fix a connected graph $G$, and let $X=(X_{t})_{t\geq 0}$ denote a lazy simple random walk on $G$ starting at some fixed vertex $o\in V_{G}$. Write ${\bf P}^{G}$ for its law, which is a probability measure on the product space $V^{\mathbb{Z}_{+}}_{G}$. On this space, let $\mathcal{I}$ denote the $\sigma-$field of all events which are invariant under the natural shift $(x_{t})_{t\geq 0}\mapsto(x_{t+1})_{t\geq 0}$. Then [38, Proposition 14.12] states that $\displaystyle G\textrm{ has the Liouville property}$ $\displaystyle\Longleftrightarrow$ $\displaystyle\mathcal{I}\textrm{ is ${\bf P}^{G}-$trivial}.$ On the other hand, writing $\mathcal{T}=\bigcap_{t=0}^{\infty}\sigma(x_{t},x_{t+1},\ldots)$ for the tail $\sigma-$field on $V^{\mathbb{Z}_{+}}_{G}$, we have $\displaystyle\mathcal{I}\textrm{ is ${\bf P}^{G}-$trivial}$ $\displaystyle\Longleftrightarrow$ $\displaystyle\mathcal{T}\textrm{ is ${\bf P}^{G}-$trivial},$ by Theorem [38, Theorem 14.18] and because $X$ is lazy. Finally, the equivalence $\displaystyle\mathcal{L}\left(\mathcal{T}\textrm{ is ${\bf P}^{G}-$trivial}\right)=1$ $\displaystyle\Longleftrightarrow$ $\displaystyle\mathcal{L}(\mathscr{H}(G)=0)=1,$ was proved in [10, Theorem 3.2] for any stationary law $\mathcal{L}$ with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. Thus, $\displaystyle\mathcal{L}(G\textrm{ has the Liouville property})=1$ $\displaystyle\Longleftrightarrow$ $\displaystyle\mathcal{L}\left(\mathscr{H}(G)=0\right)=1,$ (12) and this annealed statement will actually suffice for the present paper. However, deducing the quenched claim is easy, as we now explain. Define the events $A:=\\{G\textrm{ has the Liouville property}\\}$ and $B:=\\{\mathscr{H}(G)=0\\}$, and let $A\Delta B$ denote their symmetric difference. We want to show that $\displaystyle\mathcal{L}(A\Delta B)$ $\displaystyle=$ $\displaystyle 0,$ (13) for any stationary law $\mathcal{L}$ with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. We already know this if $A,B$ are $\mathcal{L}-$trivial, thanks to (12). Moreover, the events $A,B$ are clearly _root-invariant_ , in the sense that $\displaystyle(G,o)\in A$ $\displaystyle\Longrightarrow$ $\displaystyle\\{\forall x\in V_{G},(G,x)\in A\\}.$ Consequently, (13) holds under the extra assumption that _root-invariant events are $\mathcal{L}-$trivial_. But this is known as _ergodicity_ , and any stationary law can be decomposed as a mixture of ergodic laws, by [1, Theorem 4.7]. Thus, (13) extends to all stationary laws $\mathcal{L}$ with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. ∎ #### Spectral radius. The entropy $\mathscr{H}(G)$ is related to several other fundamental graph- theoretical quantities, such as the _speed_ , _growth_ , or _spectral radius_ , see [38]. Let us recall the last notion. Fix a rooted graph $(G,o)\in{\mathscr{G}_{\bullet}}$. For any $t,s\geq 0$, we trivially have $P_{G}^{t+s}(o,o)\geq P_{G}^{t}(o,o)P_{G}^{s}(o,o).$ By Fekete’s lemma, we deduce that the limit $\displaystyle\varrho(G,o)$ $\displaystyle:=$ $\displaystyle\lim_{t\to\infty}\left(P_{G}^{t}(o,o)\right)^{\frac{1}{t}},$ (14) exists in $(0,1]$. Moreover, the connectivity of $G$ together with the trivial inequality $\displaystyle P^{t+2s}_{G}(o,o)$ $\displaystyle\geq$ $\displaystyle P^{s}_{G}(o,x)P^{t}_{G}(x,x)P^{s}_{G}(x,o),$ shows that $\varrho(G,o)$ does not depend on the choice of the root $o$. Thus, we will henceforth simply write $\varrho(G)$, and call this quantity the _spectral radius_ of $G$. ###### Lemma 10 (Spectral radius vs entropy). The inequality $\displaystyle\mathscr{H}(G)$ $\displaystyle\geq$ $\displaystyle 2\log\frac{1}{\varrho(G)},$ holds almost-surely under any stationary law $\mathcal{L}$ with $\mathcal{L}[\log\deg_{G}(o)]<\infty$. ###### Proof. For any rooted graph $(G,o)$ and any $t\geq 0$, we have by concavity $\displaystyle\log\left(P_{G}^{2t}(o,o)\right)$ $\displaystyle=$ $\displaystyle\log\left(\sum_{x\in V_{G}}P_{G}^{t}(o,x)P^{t}_{G}(x,o)\right)$ $\displaystyle\geq$ $\displaystyle\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log P_{G}^{t}(x,o)$ $\displaystyle=$ $\displaystyle\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log P_{G}^{t}(o,x)+\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log\left(\frac{\deg_{G}(o)}{\deg_{G}(x)}\right),$ where the last line uses the reversibility $\deg_{G}(o)P_{G}^{t}(o,x)=\deg_{G}(x)P_{G}^{t}(x,o)$. Dividing by $-2t$ and taking the limit as $t\to\infty$ in $L^{1}({\mathscr{G}_{\bullet}},\mathcal{L})$ yields the claim, provided we can show that $\displaystyle\frac{1}{t}\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log\left(\frac{\deg_{G}(o)}{\deg_{G}(x)}\right)$ $\displaystyle\xrightarrow[t\to\infty]{L^{1}({\mathscr{G}_{\bullet}},\mathcal{L})}$ $\displaystyle 0.$ But this follows from the crude bound $\displaystyle\mathcal{L}\left[\left\|\sum_{x\in V_{G}}P_{G}^{t}(o,x)\log\left(\frac{\deg_{G}(o)}{\deg_{G}(x)}\right)\right\|\right]$ $\displaystyle\leq$ $\displaystyle\mathcal{L}\left[\sum_{x\in V_{G}}P_{G}^{t}(o,x)\left(\log\deg_{G}(o)+\log\deg_{G}(x)\right)\right]$ $\displaystyle=$ $\displaystyle 2\mathcal{L}\left[\log\deg_{G}(o)\right],$ where the second line simply uses the stationarity property (8) with $h(G,o)=\log\deg_{G}(o)$. ∎ ###### Remark 5 (Unimodular analogues). By Lemma 7 and Remark 4, all results in this section also apply to any unimodular law $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ with $\mathcal{L}[\deg_{G}(o)\log\deg_{G}(o)]<\infty$. ## 3 Proof of the main result We are now ready to prove our main result. We work with the formulation given in Theorem 3. Section 3.1 below reduces it to two key results, which are then proved in Sections 3.2 and 3.3. ### 3.1 Setting the stage Let $G_{n}=(V_{n},E_{n})$, $n\geq 1$ be finite graphs satisfying the assumptions of Theorem 3, i.e. $\displaystyle\sup_{n\geq 1}\left\\{\frac{1}{\|V_{n}\|}\sum_{x\in V_{n}}\deg_{G_{n}}(x)\log\deg_{G_{n}}(x)\right\\}$ $\displaystyle<$ $\displaystyle\infty;$ (15) $\displaystyle\forall\varepsilon>0,\quad\frac{1}{\|E_{n}\|}\,\mathrm{card}\\{e\in E_{n}\colon\kappa_{G_{n}}(e)<-\varepsilon\\}$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle 0.$ (16) Recall that our goal is to establish $\displaystyle\forall\rho\in(0,1),\quad\liminf_{n\to\infty}\left\\{\frac{1}{\|V_{n}\|}\,\mathrm{card}\\{i\colon\lambda_{i}(G_{n})>\rho\\}\right\\}$ $\displaystyle>$ $\displaystyle 0.$ (17) By (15) and Theorem 4, we may assume, upon extracting a subsequence if necessary, that $\displaystyle G_{n}$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle\mathcal{L},$ (18) for some $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$. Note that $\mathcal{L}$ is automatically unimodular by Theorem 5, and such that $\displaystyle\mathcal{L}\left[\deg_{G}(o)\log\deg_{G}(o)\right]$ $\displaystyle<$ $\displaystyle\infty.$ (19) Just like the degree, the curvature is a local notion, hence it also “passes to the limit”, i.e $\displaystyle\mathcal{L}\left(\kappa(G)\geq 0\right)$ $\displaystyle=$ $\displaystyle 1.$ (20) ###### Proof. As already mentioned, the observable $f\colon(G,o)\mapsto\min_{x\sim o}\kappa_{G}(o,x)$ is $2-$local, hence continuous on ${\mathscr{G}_{\bullet}}$. By the Portmanteau Theorem, we deduce that for any $\varepsilon>0$, $\displaystyle\mathcal{L}\left(f<-\varepsilon\right)$ $\displaystyle\leq$ $\displaystyle\liminf_{n\to\infty}\mathcal{L}_{G_{n}}\left(f<-\varepsilon\right)$ $\displaystyle=$ $\displaystyle\liminf_{n\to\infty}\left\\{\frac{1}{\|V_{n}\|}\mathrm{card}\\{o\in V_{n}\colon f(G_{n},o)<-\varepsilon\right\\}$ $\displaystyle\leq$ $\displaystyle\liminf_{n\to\infty}\left\\{\frac{2}{\|V_{n}\|}\mathrm{card}\\{e\in E_{n}\colon\kappa_{G_{n}}(e)<-\varepsilon\right\\}$ $\displaystyle=$ $\displaystyle\mathcal{L}\left[\deg_{G}(o)\right]\liminf_{n\to\infty}\left\\{\frac{1}{\|E_{n}\|}\mathrm{card}\\{e\in E_{n}\colon\kappa_{G_{n}}(e)<-\varepsilon\right\\},$ where the last inequality follows from the observation that $\frac{2\|E_{n}\|}{\|V_{n}\|}\to\mathcal{L}[\deg_{G}(o)]$, by the continuity and uniform integrability of $(G,o)\mapsto\deg_{G}(o)$. Sending $\varepsilon\to 0$ yields $\mathcal{L}(f<0)=0$, by (16). To conclude, we simply apply Lemma 6 to the event $B=\\{f\geq 0\\}$. ∎ The first crucial step in our proof consists in deducing from (20) that the entropy is zero under $\mathcal{L}$. This is the content of the following theorem, which will be proved in Section 3.2. ###### Theorem 11 (Non-negative curvature implies zero-entropy). The implication $\displaystyle\kappa(G)\geq 0$ $\displaystyle\Longrightarrow$ $\displaystyle\mathscr{H}(G)=0$ holds almost-surely under any stationary law $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ satisfying $\mathcal{L}\left[\log\deg_{G}(o)\right]<\infty$. In view of Remark 4, this result also applies to any unimodular law $\mathcal{L}\in\mathcal{P}({\mathscr{G}_{\bullet}})$ satisfying $\mathcal{L}\left[\deg_{G}(o)\log\deg_{G}(o)\right]<\infty$, hence in particular to the limit $\mathcal{L}$ in (18). Combining this with Lemma 10, we immediately deduce that our local weak limit satisfies $\displaystyle\mathcal{L}(\rho(G)=1)$ $\displaystyle=$ $\displaystyle 1.$ It turns out that this simple condition suffices to guarantee (17). This is the content of the following second result, established in Section 3.3 below, and which completes the proof of our main result. ###### Theorem 12 (Zero-entropy implies poor spectral expansion). Let $G_{n}=(V_{n},E_{n}),{n\geq 1}$ be finite graphs having local weak limit $\mathcal{L}$, and suppose that $\mathcal{L}\left(\rho(G)=1\right)=1$. Then, for any $\rho<1$, $\displaystyle\liminf_{n\to\infty}\left\\{\frac{1}{\|V_{n}\|}\,\mathrm{card}\left\\{i\colon\lambda_{i}(G_{n})>\rho\right\\}\right\\}$ $\displaystyle>$ $\displaystyle 0.$ In fact, a stronger statement about eigenvectors will be derived, as claimed in Remark 3. ### 3.2 Non-negative curvature implies zero entropy Consider a connected graph $G$ and two vertices $x,y\in V_{G}$. The proof of Theorem 11 relies on the following intuitive idea: if $G$ has non-negative curvature and bounded degrees, then it takes time $O({\rm d}_{G}^{2}(x,y))$ for two random walks starting at $x$ and $y$ to meet. This classical observation constitutes the very essence of the path coupling method of Bordewich and Dyer [18]. It was later re-discovered and further developed by Münch [40]. We will here prove a refinement that does not require bounded degrees, see Corollary 16 below. Write $\mathcal{B}_{x},\mathcal{B}_{y}$ for the balls of radius $1$ around $x$ and $y$, and recall that the Wassertein distance $\mathcal{W}_{1}\left(P_{G}(x,\cdot),P_{G}(y,\cdot)\right)$ is defined as $\displaystyle\mathcal{W}_{1}\left(P_{G}(x,\cdot),P_{G}(y,\cdot)\right)$ $\displaystyle=$ $\displaystyle\inf_{\pi}\left\\{\sum_{u\in\mathcal{B}_{x}}\sum_{v\in\mathcal{B}_{y}}\pi(u,v)\,{\rm d}_{G}(u,v)\right\\},$ (21) where the infimum runs over all probability distributions $\pi\in\mathcal{P}(\mathcal{B}_{x}\times\mathcal{B}_{y})$ with marginals $P_{G}(x,\cdot)$ and $P_{G}(y,\cdot)$. By compactness, the above infimum is actually achieved, and the minimizers will be called _optimal couplings_. As in [18, 40], our first task consists in showing that an optimal coupling can always be chosen so as to assign a “decent” probability to the “good” set $\displaystyle\Gamma$ $\displaystyle:=$ $\displaystyle\left\\{(u,v)\in\mathcal{B}_{x}\times\mathcal{B}_{y}\colon{\rm d}_{G}(u,v)<{\rm d}_{G}(x,y)\right\\}.$ The argument crucially uses the laziness of $P_{G}$ but is otherwise rather general. ###### Lemma 13 (Good optimal couplings). If $x\neq y$, then there is an optimal coupling $\pi$ such that $\displaystyle\pi\left(\Gamma\right)$ $\displaystyle\geq$ $\displaystyle\frac{1}{2}\max\left\\{\frac{1}{\deg_{G}(x)},\frac{1}{\deg_{G}(y)}\right\\}.$ ###### Proof. By compactness, we can find an optimal coupling $\pi$ which, among all optimal couplings, maximizes $\pi(\Gamma)$. Suppose for a contradiction that this “doubly optimal” coupling satisfies $\displaystyle\pi\left(\Gamma\right)$ $\displaystyle<$ $\displaystyle\frac{1}{2\deg_{G}(x)}.$ (22) The set ${A}:=\\{u\in\mathcal{B}_{x}\colon(u,y)\in\Gamma\\}$ is not empty, since it contains the first vertex on a geodesic from $x$ to $y$. Thus, $\pi(A\times\mathcal{B}_{y})\geq 1/(2\deg_{G}(x)).$ In view of (22), this forces $\pi((A\times\mathcal{B}_{y})\setminus\Gamma)>0$, i.e. $\displaystyle\exists(x_{0},y_{0})\in(A\times\mathcal{B}_{y})\setminus\Gamma,\quad\pi(x_{0},y_{0})$ $\displaystyle\geq$ $\displaystyle\varepsilon,$ (23) for some $\varepsilon>0$. On the other hand, we have $\pi(A\times\\{y\\})+\pi(A^{c}\times\\{y\\})=P_{G}(y,y)\ =\ \frac{1}{2}.$ This forces $\pi(A^{c}\times\\{y\\})>0$, because $\pi(A\times\\{y\\})\leq\pi(\Gamma)<\frac{1}{2}$. In other words, $\displaystyle\exists x_{1}\in A^{c},\quad\pi(x_{1},y)$ $\displaystyle\geq$ $\displaystyle\varepsilon,$ (24) provided $\varepsilon>0$ is chosen small enough. We now use the vertices $x_{0},y_{0},x_{1}$ found at (23)-(24) to construct a new coupling $\widehat{\pi}$ which contradicts the optimality of $\pi$. For all $(u,v)\in\mathcal{B}_{x}\times\mathcal{B}_{y}$, we set $\displaystyle\widehat{\pi}(u,v)$ $\displaystyle:=$ $\displaystyle\left\\{\begin{array}[]{ll}\pi(u,v)&\textrm{ if }u\notin\\{x_{0},x_{1}\\}\textrm{ and }b\notin\\{y_{0},y\\};\\\ \pi(u,v)-\varepsilon&\textrm{ if }(u,v)=(x_{0},y_{0})\textrm{ or }(u,v)=(x_{1},y);\\\ \pi(u,v)+\varepsilon&\textrm{ if }(u,v)=(x_{0},y)\textrm{ or }(u,v)=(x_{1},y_{0}).\end{array}\right.$ By construction, $\widehat{\pi}$ is non-negative on $\mathcal{B}_{x}\times\mathcal{B}_{y}$ and has the same marginals as $\pi$. Thus, it is a coupling of $P_{G}(x,\cdot),P_{G}(y,\cdot)$. This coupling is moreover optimal, since $\displaystyle\sum_{u\in\mathcal{B}_{x}}\sum_{v\in\mathcal{B}_{y}}{\rm d}_{G}(u,v)\left(\widehat{\pi}(u,v)-\pi(u,v)\right)$ $\displaystyle=$ $\displaystyle\varepsilon\left({\rm d}_{G}(x_{0},y)+{\rm d}_{G}(x_{1},y_{0})-{\rm d}_{G}(x_{0},y_{0})-{\rm d}_{G}(x_{1},y)\right)$ $\displaystyle\leq$ $\displaystyle\varepsilon\left({\rm d}_{G}(x,y)-1+{\rm d}_{G}(x_{1},y_{0})-{\rm d}_{G}(x,y)-{\rm d}_{G}(x_{1},y)\right)$ $\displaystyle\leq$ $\displaystyle 0,$ where the first inequality uses $x_{0}\in A$ and $(x_{0},y_{0})\notin\Gamma$, while the second uses the triangle inequality ${\rm d}_{G}(x_{1},y_{0})\leq{\rm d}_{G}(x_{1},y)+{\rm d}_{G}(y,y_{0})$. Finally, since $\Gamma$ contains $(x_{1},y)$ but not $(x_{0},y_{0}),(x_{1},y)$, we have $\displaystyle\widehat{\pi}(\Gamma)$ $\displaystyle\geq$ $\displaystyle\pi(\Gamma)+\varepsilon,$ contradicting the definition of $\pi$. Thus, (22) can not be true, and the claim follows by symmetry. ∎ We will also need the following technical lemma, which is of independent interest and quantifies the intuition that non-negative super-martingales that “move a lot” must “quickly” hit zero. ###### Lemma 14 (Non-negative super-martingales quickly hit zero). Let $\tau:=\inf\\{t\geq 0\colon Z_{t}=0\\}$ be the hitting time of zero by a non-negative super-martingale $Z=(Z_{t})_{t\geq 0}$. Suppose that $Z_{0}=z$, and that all increments $(Z_{t+1}-Z_{t})_{t\geq 0}$ are upper-bounded by a constant $K$. Then, $\displaystyle{\mathbb{P}}\left(\tau\geq t\right)$ $\displaystyle\leq$ $\displaystyle z\left(\frac{2a+K-z}{a^{2}}\right)+{\mathbb{P}}\left(\tau\geq t,\sum_{s=0}^{t-1}W_{s}<a^{2}\right),$ for all $t\in\mathbb{Z}_{+},a>0$, where $W_{s}={\mathbb{E}}\left[(Z_{s+1}-Z_{s})^{2}\|\mathscr{F}_{s}\right]$ and $(\mathscr{F}_{s})_{s\geq 0}$ is the underlying filtration. ###### Proof. First note that the process $Z$ is trivially square-integrable, because $Z_{t}\in[0,z+Kt]$ for each $t\geq 0$. Now fix $t\geq 0$ and $a>0$, and consider the bounded stopping time $\displaystyle\sigma$ $\displaystyle:=$ $\displaystyle\inf\left\\{s\geq 0\colon Z_{s}\geq a\right\\}\wedge t.$ Using the Optional Stopping Theorem, the non-negativity of $Z$ and the definition of $\sigma$, we have $\displaystyle z$ $\displaystyle\geq$ $\displaystyle{\mathbb{E}}\left[Z_{\sigma\wedge\tau}\right]$ $\displaystyle\geq$ $\displaystyle{\mathbb{E}}\left[Z_{\sigma\wedge\tau}{\bf 1}_{(\sigma<\tau\wedge t)}\right]$ $\displaystyle\geq$ $\displaystyle a{\mathbb{P}}\left(\sigma<\tau\wedge t\right).$ On the other hand, observe that for all $s\geq 0$, we may rewrite $W_{s}$ as $\displaystyle W_{s}$ $\displaystyle=$ $\displaystyle{\mathbb{E}}\left[Z_{s+1}^{2}-Z_{s}^{2}\|\mathscr{F}_{s}\right]+2Z_{s}{\mathbb{E}}[Z_{s}-Z_{s+1}\|\mathscr{F}_{s}].$ Note that the second conditional expectation is non-negative by assumption. Moreover, we have $Z_{s}\leq a$ on the event $\\{\sigma>s\\}$, which is in $\mathscr{F}_{s}$. Thus, $\displaystyle W_{s}{\bf 1}_{\sigma>s}$ $\displaystyle\leq$ $\displaystyle{\mathbb{E}}\left[\left(Z_{s+1}^{2}-Z_{s}^{2}\right){\bf 1}_{\sigma>s}\|\mathscr{F}_{s}\right]+2a{\mathbb{E}}\left[\left(Z_{s}-Z_{s+1}\right){\bf 1}_{\sigma>s}\|\mathscr{F}_{s}\right].$ Taking expectations and summing over all $s\geq 0$, we obtain $\displaystyle{\mathbb{E}}\left[\sum_{s=0}^{\sigma-1}W_{s}\right]$ $\displaystyle\leq$ $\displaystyle{\mathbb{E}}\left[Z_{\sigma}^{2}\right]-2a{\mathbb{E}}[Z_{\sigma}]-z^{2}+2az$ $\displaystyle\leq$ $\displaystyle(K+a-z)z,$ where the second inequality follows from the observations that $Z_{\sigma}\leq K+a$ and ${\mathbb{E}}[Z_{\sigma}]\leq z$. Let us now use these two estimates to conclude. By union bound, we have $\displaystyle{\mathbb{P}}\left(\tau\geq t\right)$ $\displaystyle\leq$ $\displaystyle{\mathbb{P}}\left(\sigma<\tau\wedge t\right)+{\mathbb{P}}\left(\sigma\wedge\tau\geq t\right)$ $\displaystyle\leq$ $\displaystyle{\mathbb{P}}\left(\sigma<\tau\wedge t\right)+{\mathbb{P}}\left(\tau\geq t,\sum_{s=0}^{\sigma-1}W_{s}\geq\sum_{s=0}^{t-1}W_{s}\right)$ $\displaystyle\leq$ $\displaystyle{\mathbb{P}}\left(\sigma<\tau\wedge t\right)+{\mathbb{P}}\left(\sum_{s=0}^{\sigma-1}W_{s}\geq a^{2}\right)+{\mathbb{P}}\left(\tau\geq t,\sum_{s=0}^{t-1}W_{s}<a^{2}\right)$ $\displaystyle\leq$ $\displaystyle\frac{z}{a}+\frac{(K+a-z)z}{a^{2}}+{\mathbb{P}}\left(\tau\geq t,\sum_{s=0}^{t-1}W_{s}<a^{2}\right).$ This is exactly the claimed bound. ∎ Combining these two lemmas, we may now deduce the following estimate, which exploits non-negative curvature to control the action of $P_{G}$ on the variations of bounded observables. ###### Proposition 15 (Variational estimate via non-negative curvature). Let $G$ be a connected graph with $\kappa(G)\geq 0$. Then, for any $f\colon V_{G}\to[-1,1]$, any vertices $x,y\in V_{G}$, and any $a>0,t\in\mathbb{Z}_{+}$, $\displaystyle{\|P^{t}_{G}f(x)-P^{t}_{G}f(y)\|}$ $\displaystyle\leq$ $\displaystyle\frac{8{\rm d}_{G}(x,y)}{a}+2{\mathbb{P}}\left(\sum_{s=0}^{t-1}\frac{1}{\deg_{G}(X_{s})}<2a^{2}\right),$ where $X$ denotes a lazy random walk on $G$ starting from $x$. ###### Proof. Let $(X,Y)$ be the Markov chain on $V_{G}\times V_{G}$ which, from any state $(x,y)\in V_{G}\times V_{G}$, draws the next state according to the “good” optimal coupling of $P_{G}(x,\cdot),P_{G}(y,\cdot)$ described in Lemma 13. We use the standard notations ${\mathbb{P}}_{(x,y)}(\cdot),{\mathbb{E}}_{(x,y)}[\cdot]$ to specify the choice of the initial state. Since the two coordinates $X,Y$ are marginally distributed as lazy random walks on $G$, we have $\displaystyle\left\|P^{t}_{G}f(x)-P^{t}_{G}f(y)\right\|$ $\displaystyle=$ $\displaystyle\left\|{\mathbb{E}}_{x,y}\left[f(X_{t})\right]-{\mathbb{E}}_{x,y}\left[f(Y_{t})\right]\right\|$ $\displaystyle\leq$ $\displaystyle{\mathbb{E}}_{x,y}\left[\left\|f(X_{t})-f(Y_{t})\right\|\right]$ $\displaystyle\leq$ $\displaystyle 2{\mathbb{P}}_{x,y}\left(X_{t}\neq Y_{t}\right)$ $\displaystyle\leq$ $\displaystyle 2{\mathbb{P}}_{x,y}\left(\tau>t\right),$ where $\tau=\inf\\{t\geq 0\colon X_{t}=Y_{t}\\}$ denotes the meeting time of the two walkers. Note that $\tau$ is also the hitting time of zero by the non- negative process $Z=(Z_{t})_{t\geq 0}$ defined as follows: $\displaystyle\forall t\geq 0,\quad Z_{t}$ $\displaystyle:=$ $\displaystyle{\rm d}_{G}(X_{t},Y_{t}).$ We claim that $Z$ is a super-martingale w.r.t. the natural filtration $(\mathscr{F}_{t})_{t\geq 0}$ associated with $(X,Y)$. Indeed, by the Markov property and the optimality of the chosen couplings, this claim reduces to $\displaystyle\mathcal{W}_{1}\left(P_{G}(x,\cdot),P_{G}(y,\cdot)\right)$ $\displaystyle\leq$ $\displaystyle{\rm d}_{G}(x,y),$ for all $x,y\in V_{G}$. But this inequality readily follows from the assumption $\kappa_{G}(x,y)\geq 0$ in the case $\\{x,y\\}\in E_{G}$, and it then automatically extends to all $x,y\in V_{G}$ by the triangle inequality of $\mathcal{W}_{1}(\cdot,\cdot)$ (see, e.g., [45]). On the other hand, Lemma 13 ensures that on the event $\\{\tau>t\\}$, $\displaystyle{\mathbb{E}}_{x,y}\left[(Z_{t+1}-Z_{t})^{2}\|\mathscr{F}_{t}\right]$ $\displaystyle\geq$ $\displaystyle\frac{1}{2\deg_{G}(X_{t})}.$ Finally, note that the distance between the two walkers can not increase by more than $2$ at each step. Thus, we may invoke Lemma 14 to conclude that $\displaystyle{\mathbb{P}}_{x,y}\left(\tau\geq t\right)$ $\displaystyle\leq$ $\displaystyle 2{\rm d}_{G}(x,y)\left(\frac{a+1}{a^{2}}\right)+{\mathbb{P}}_{x,y}\left(\sum_{s=0}^{t-1}\frac{1}{\deg_{G}(X_{s})}<2a^{2}\right)$ $\displaystyle\leq$ $\displaystyle\frac{4{\rm d}_{G}(x,y)}{a}+{\mathbb{P}}_{x,y}\left(\sum_{s=0}^{t-1}\frac{1}{\deg_{G}(X_{s})}<2a^{2}\right),$ where the second line follows from the first if $a\geq 1$, and is trivial otherwise. ∎ In particular, this applies to any bounded harmonic function $f$, after a trivial normalization. Since $P^{t}_{G}f=f$ for all $t\geq 0$, we may send $t\to\infty$ and then $a\to\infty$ in the resulting estimate to obtain the following key result, which ensures that non-negatively curved graphs satisfy the Liouville property, provided they have a “decent proportion” of vertices with “reasonable” degree. ###### Corollary 16 (Liouville property and non-negative curvature). Let $G$ be a connected graph with $\kappa(G)\geq 0$. Fix $o\in V_{G}$ and suppose that the simple random walk $X$ on $G$ starting from $o$ satisfies $\displaystyle{\mathbb{P}}\left(\sum_{t=0}^{\infty}\frac{1}{\deg_{G}(X_{t})}=\infty\right)$ $\displaystyle=$ $\displaystyle 1.$ (26) Then, $G$ has the Liouville property. A simple situation where the above condition trivially holds is that where $G$ has bounded degrees. In that case, the Liouville property was recently established by Jost, Münch, and Rose [29]. Our relaxation allows for arbitrary large degrees, as long as the random walk can avoid them from times to times. This is the case under any stationary law by Birkhoff’s Ergodic Theorem, allowing us to prove Theorem 11. ###### Proof of Theorem 11. Let $(G,o)$ have law $\mathcal{L}$ and, conditionally on $(G,o)$, let $X$ be a lazy random walk starting from the root. Then the process $Z=(Z_{t})_{t\geq 0}$ defined by $\displaystyle\forall t\geq 0,\quad Z_{t}$ $\displaystyle:=$ $\displaystyle\frac{1}{\deg_{G}(X_{t})}$ is stationary, in the usual sense that its law is invariant under the shift $(z_{t})_{t\geq 0}\mapsto(z_{t+1})_{t\geq 0}$ on $[0,1]^{\mathbb{Z}_{+}}$. Thus, Birkhoff’s Ergodic Theorem (see, e.g. [38, Theorem 14.43]) ensures that $\displaystyle\frac{1}{t}\sum_{s=0}^{t-1}Z_{s}$ $\displaystyle\xrightarrow[t\to\infty]{}$ $\displaystyle{\mathbb{E}}[Z_{1}\|\mathscr{I}],$ almost-surely, where $\mathscr{I}$ is the invariant $\sigma-$algebra. Since $Z_{1}$ is almost-surely positive, we deduce $\displaystyle\sum_{s=0}^{\infty}Z_{s}$ $\displaystyle=$ $\displaystyle\infty,$ almost-surely. In other words, the random graph $(G,o)$ satisfies (26) almost- surely. By the above corollary, this implies that $G$ has the Liouville property almost-surely on the event $\\{\kappa(G)\geq 0\\}$. By Theorem 9, we conclude that $\mathscr{H}(G)=0$ almost-surely on the same event. ∎ ### 3.3 Zero entropy implies poor spectral expansion This final section is devoted to proving Theorem 12, which relates the eigenvalues of finite graphs to the spectral radius of their local weak limits. If $G$ is a finite graph, the $N=\|V_{G}\|$ eigenvalues $\lambda_{1}(G)\geq\ldots\geq\lambda_{N}(G)$ of its transition matrix $P_{G}$ can be conveniently encoded into a probability measure $\mu_{G}\in\mathcal{P}([0,1])$, called the _empirical eigenvalue distribution_ of the matrix $P_{G}$: $\displaystyle\mu_{G}$ $\displaystyle:=$ $\displaystyle\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}(G)}.$ It turns out that the large-size asymptotics of this fundamental object can be understood directly at the level of local weak limits. When $P_{G}$ is replaced with the more standard adjacency matrix, this classical observation is the starting point of a rich and well-established theory, see the comprehensive introductory survey [15] by Bordenave, and the references therein. #### Local spectral measures. The transition kernel $P_{G}$ of a graph $G$ can be viewed as a linear operator acting via (11) on the Hilbert space $\displaystyle\ell^{2}(G)$ $\displaystyle:=$ $\displaystyle\left\\{f\in\mathbb{C}^{V_{G}}\colon\sum_{o\in V_{G}}\deg_{G}(o)\|f(o)\|^{2}<\infty\right\\},$ with inner product $\langle f,g\rangle=\sum_{o\in V_{G}}\deg_{G}(o)\overline{f(o)}g(o)$. The stochasticity, laziness and reversibility $\displaystyle\sum_{y\in V_{G}}P_{G}(x,y)=1,\qquad P_{G}(x,x)\geq 1/2,\qquad\deg_{G}(x)P_{G}(x,y)=\deg_{G}(y)P_{G}(y,x),$ easily (and classically) imply that $P_{G}$ is a positive contraction on $\ell^{2}(G)$, i.e. $\displaystyle\forall f\in\ell^{2}(G),\qquad 0\ \leq\ \langle f,P_{G}f\rangle\ \leq\langle\ f,f\rangle.$ In particular, for each $o\in V_{G}$, the spectral theorem for self-adjoint operators ensures the existence of a _local spectral measure_ $\mu_{(G,o)}\in\mathcal{P}([0,1])$, characterized by the moment identity $\displaystyle\forall t\geq 0,\quad\int_{0}^{1}\lambda^{t}\mu_{(G,o)}({\rm d}\lambda)$ $\displaystyle=$ $\displaystyle P^{t}_{G}(o,o).$ (27) As we will now see, $\mu_{(G,o)}$ can be interpreted as the local contribution of $o$ to the spectrum of $P_{G}$. Local spectral measures are a powerful tool to investigate the mixing properties of graphs, see [36]. #### The finite case. When $G$ is finite with $N$ vertices, there is an orthonormal basis $(\phi_{1},\ldots,\phi_{N})$ of $\ell^{2}(G)$ consisting of eigenvectors of $P_{G}$ with eigenvalues $\lambda_{1}(G),\ldots,\lambda_{N}(G)$, and we easily find $\displaystyle\mu_{(G,o)}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}\deg_{G}(o)\|\phi_{i}(o)\|^{2}\delta_{\lambda_{i}(G)}.$ (28) Thus, the local spectral measure $\mu_{(G,o)}$ is a mixture of Dirac masses located at the various eigenvalues of $P_{G}$, and weighted by the squared amplitudes of the corresponding eigenvectors at $o$. Moreover, thanks to the orthonormality of $(\phi_{1},\ldots,\phi_{N})$, the identity (28) readily implies $\displaystyle\mu_{G}$ $\displaystyle=$ $\displaystyle\frac{1}{\|V_{G}\|}\sum_{o\in V_{G}}\mu_{(G,o)}.$ (29) In other words, the empirical eigenvalue distribution of a finite graph $G$ coincides with the spatial average of its local spectral measures. #### Spectral continuity. In light of (6), it is tempting to pass to the limit in the formula (29) along a convergent sequence of finite graphs $(G_{n})_{n\geq 1}$. This is made rigorous by the following continuity principle. As usual, $\mathcal{P}([0,1])$ is here equipped with the topology of weak convergence. ###### Lemma 17 (Spectral continuity). The map $(G,o)\mapsto\mu_{(G,o)}$ is continuous on ${\mathscr{G}_{\bullet}}$. In particular, if a sequence of graphs $(G_{n})_{n\geq 1}$ admits a local weak limit $\mathcal{L}$, then $\displaystyle\mu_{G_{n}}({\rm d}\lambda)$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle\mu_{\mathcal{L}}({\rm d}\lambda)\ :=\ \mathcal{L}\left[\mu_{(G,o)}({\rm d}\lambda)\right].$ ###### Proof. For each fixed $t\geq 0$, the observable $(G,o)\mapsto P^{t}_{G}(o,o)$ is clearly $t-$local, hence continuous. In particular, via the identity (27), the convergence $(G_{n},o_{n})\to(G,o)$ in ${\mathscr{G}_{\bullet}}$ implies $\displaystyle\forall t\geq 0,\quad\int_{0}^{1}\lambda^{t}\,\mu_{(G_{n},o_{n})}({\rm d}\lambda)$ $\displaystyle\xrightarrow[n\to\infty]{}$ $\displaystyle\int_{0}^{1}\lambda^{t}\,\mu_{(G,o)}({\rm d}\lambda).$ (30) Since convergence in $\mathcal{P}([0,1])$ is equivalent to the convergence of moments, we conclude that $\mu_{(G_{n},o_{n})}\xrightarrow[n\to\infty]{}\mu_{(G,o)}$, and the continuity is proved. Similarly, the second claim is obtained by applying (6) to the $t-$local observable $f\colon(G,o)\mapsto P^{t}_{G}(o,o)$, for each $t\geq 1$. ∎ ###### Corollary 18 (Unit spectral radius implies poor spectral expansion). Let $G_{n}=(V_{n},E_{n}),{n\geq 1}$ be finite graphs having a local weak limit $\mathcal{L}$ such that $\mathcal{L}(\rho(G)=1)=1$. Then, for any $0\leq\rho<1$, $\displaystyle\liminf_{n\to\infty}\,\mu_{G_{n}}\left([\rho,1]\right)$ $\displaystyle>$ $\displaystyle 0.$ (31) Moreover, we have the refinement $\displaystyle\sup_{n\geq 1}\,\frac{\left\|\left\\{x\in V_{n}\colon\mu_{(G_{n},x)}([\rho,1])\leq\varepsilon\right\\}\right\|}{\|V_{n}\|}$ $\displaystyle\xrightarrow[\varepsilon\to 0]{}$ $\displaystyle 0.$ (32) ###### Proof. Fix $0\leq\rho<1$. By the second part of Lemma 17 and the Portmanteau Theorem, we have $\displaystyle\liminf_{n\to\infty}\mu_{G_{n}}([\rho,1])$ $\displaystyle\geq$ $\displaystyle\mathcal{L}\left[\mu_{(G,o)}((\rho,1])\right].$ (33) On the other hand, comparing (27) with the definition of the spectral radius, we see that $\rho(G)$ is exactly the supremum of the support of $\mu_{(G,o)}$, for any $(G,o)\in{\mathscr{G}_{\bullet}}$. In other words, $\displaystyle\mu_{(G,o)}((\rho,1])>0$ $\displaystyle\Longleftrightarrow$ $\displaystyle\rho(G)>\rho.$ In particular, since $\mathcal{L}(\rho(G)=1)=1$, the right-hand side of (33) is positive, as desired. To prove the second claim, note that the continuity of $(G,o)\mapsto\mu_{(G,o)}$ implies that the event $F_{\varepsilon}=\left\\{\mu_{(G,o)}([\rho,1])\leq\varepsilon\right\\}$ is closed in ${\mathscr{G}_{\bullet}}$. Consequently, the convergence $G_{n}\to\mathcal{L}$ implies $\displaystyle\limsup_{n\to\infty}\mathcal{L}_{G_{n}}(F_{\varepsilon})$ $\displaystyle\leq$ $\displaystyle\mathcal{L}(F_{\varepsilon}),$ and the right-hand side tends to $\mathcal{L}(F_{0})\leq\mathcal{L}(\rho(G)\leq\rho)=0$ as $\varepsilon\to 0$. The limsup can then be replaced with a sup, since for each $n\geq 1$, $\mathcal{L}_{G_{n}}(F_{\varepsilon})$ decreases monotonically to $0$ with $\varepsilon$. ∎ ###### Remark 6 (Corollary 18 vs Theorem 12). The statement (31) asserts that a macroscopic proportion of eigenvalues of $G_{n}$ accumulate in $[\rho,1]$, which is exactly the conclusion of Theorem 12. The refinement (32), on the other hand, constitutes a rigorous formalization of the “delocalization” announced in Remark 3. To see this, recall that for any graph $G$ with $N$ vertices, we have by (28), $\displaystyle\mu_{(G,x)}([\rho,1])$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}\deg_{G}(x)\|\phi_{i}(x)\|^{2}{\bf 1}_{\lambda_{i}(G)\geq\rho}.$ In words, the number $\mu_{(G,x)}([\rho,1])\in[0,1]$ measures the cumulative squared amplitude at $x$ of all the basis eigenvectors corresponding to “bad” eigenvalues (those in $[\rho,1]$). In particular, the set $\\{x\in V_{G}\colon\mu_{(G,x)}([\rho,1])\leq\varepsilon\\}$ represents the region where these “bad” eigenvectors have a small cumulative squared amplitude. 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# Cyclotomic expansions for $\mathfrak{gl}_{N}$ link invariants via interpolation Macdonald polynomials Anna Beliakova Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland<EMAIL_ADDRESS>and Eugene Gorsky Department of Matematics University of California, Davis One Shields Avenue, Davis CA 95616, USA<EMAIL_ADDRESS> ###### Abstract. In this paper we construct a new basis for the cyclotomic completion of the center of the quantum $\mathfrak{gl}_{N}$ in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the $\mathfrak{gl}_{N}$ Reshetikhin–Turaev link invariants and the universal $\mathfrak{gl}_{N}$ knot invariant; 2) an explicit construction of the unified $\mathfrak{gl}_{N}$ invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for $\mathfrak{sl}_{2}$. In addition, we give a simple proof of the fact that the universal $\mathfrak{gl}_{N}$ invariant of any evenly framed link and the universal $\mathfrak{sl}_{N}$ invariant of any $0$-framed algebraically split link are $\Gamma$-invariant, where $\Gamma=Y/2Y$ with the root lattice $Y$. ## 1\. Introduction In a series of papers [1, 2, 15] Habiro, the first author et al. defined unified invariants of homology 3-spheres that belong to the Habiro ring and dominate Witten–Reshetikhin–Turaev (WRT) invariants. Unified invariants provide an important tool to study structural properties of the WRT invariants. In [3, 5] they were used to prove integrality of the $\mathfrak{sl}_{2}$ WRT invariants for all 3-manifolds at all roots of unity. The theory of unified invariants for $\mathfrak{sl}_{2}$ is based on cyclotomic expansions for the colored Jones polynomial and for the universal knot invariant constructed as follows. Given a framed oriented link $L$ in the 3-sphere, we open its components to obtain a bottom tangle $T$, presented by a diagram $D$ (see Figure 1). For a ribbon Hopf algebra $U_{q}\mathfrak{g}$, the universal link invariant $J_{L}(\mathfrak{g};q)$ is obtained by spliting $D$ intro elementary pieces: crossings, caps and cups and then by associating to them $R^{\pm 1}$-matrices, and pivotal elements, respectively. Figure 1. An example of the clasp bottom tangle For a knot $K$, $J_{K}(\mathfrak{g};q)$ belongs to (some completion of) the center $\mathcal{Z}(U_{q}\mathfrak{g})$. In the easiest case $\mathfrak{g}=\mathfrak{sl}_{2}$, the center is generated by the Casimir $C$. For a $0$-framed knot $K$, Habiro showed that there are coefficients $a_{m}(K)\in\mathbb{Z}[q^{\pm 1}]$ such that (1) $\displaystyle J_{K}(\mathfrak{sl}_{2};q)=\sum^{\infty}_{m=0}a_{m}(K)\,\sigma_{m}\;\quad\text{with}\;\quad\sigma_{m}=\prod_{i=1}^{m}\left(C^{2}-(q^{i}+q^{-i}+2)\right).$ Replacing $C^{2}$ in (1) by its value $q^{n}+q^{-n}+2$ on the $n$-dimensional irreducible representation $V_{n-1}$, we get the $n$-colored Jones polynomial of $K$ (normalized to 1 for the unknot) (2) $J_{K}(V_{n-1},q)=\sum_{m=0}^{\infty}(-1)^{m}q^{-\frac{m(m+1)}{2}}a_{m}(K)\,(q^{1+n};q)_{m}(q^{1-n};q)_{m}$ where $(a;q)_{m}=(1-a)(1-aq)\dots(1-aq^{m-1})$. Equation (2) is known as a cyclotomic expansion of the colored Jones polynomial. Thus, Habiro’s series (1) dominates all colored Jones polynomials of $K$. To prove the fact that $J_{K}(\mathfrak{sl}_{2};q)$ belongs to the even part of $\mathcal{Z}(U_{q}\mathfrak{sl}_{2})$, generated by $C^{2}$, Habiro used the whole power of the theory of bottom tangles developed in [16]. In this paper we give a simple proof for the “evenness” of the universal invariant of algebraically split links for all quantum groups of type $A$. Recall that $U_{q}\mathfrak{g}$ has a natural action of a finite group $\Gamma=Y/2Y$ where $Y$ is the root lattice of $\mathfrak{g}$. For $\mathfrak{g}=\mathfrak{gl}_{N}$, $\Gamma=\mathbb{Z}_{2}^{N}$ and for $\mathfrak{g}=\mathfrak{sl}_{N}$, $\Gamma=\mathbb{Z}_{2}^{N-1}$. ###### Theorem 1.1. The universal $\mathfrak{gl}_{N}$ invariant of any evenly framed link is $\Gamma$-invariant. The universal $\mathfrak{sl}_{N}$ invariant of any 0-framed algebraically split link is $\Gamma$-invariant. The quantum group $U_{q}\mathfrak{gl}_{N}$ admits a finite dimensional irreducible representation $V(\lambda)$ with highest weight $v^{\lambda}$ for any partition $\lambda=(\lambda_{1}\geq\dots\geq\lambda_{N})$ with $N$ parts and $v^{2}=q$. To prove Theorem 1.1 we extend the Reshetikhin-Turaev invariants to tangles colored with representations $L(\zeta)\otimes V(\lambda)$ where $L(\zeta)$ is a one-dimensional representation of $U_{q}\mathfrak{gl}_{N}$ for $\zeta\in\Gamma$. Then the claim follows from the comparison of the $\mathfrak{gl}_{N}$ Reshetikhin-Turaev link invariants colored with $L(\zeta)\otimes V(\lambda)$ and $V(\lambda)$. The next main result of the paper establishes an explicit basis in the $\Gamma$-invariant part of the center $\mathcal{Z}$ of $U_{q}\mathfrak{gl}_{N}$. It generalizes Habiro’s basis $\\{\sigma_{m}\,\|\,m\in\mathbb{N}\\}$ for the even part of $\mathcal{Z}(U_{q}\mathfrak{sl}_{2})$. ###### Theorem 1.2. There exists a family of central elements $\sigma_{\lambda}\in\mathcal{Z}$ labeled by partitions $\lambda$ with at most $N$ parts with the following properties: * (a) $\sigma_{\lambda}$ is $\Gamma$-invariant and annihilates $L(\zeta)\otimes V(\mu)$ for all $\zeta\in\Gamma$ and partitions $\mu$ with at most $N$ parts not containing $\lambda$; * (b) $\sigma_{\lambda}$ does not annihilate $V(\lambda)$ and acts on it by an explicit scalar (see Theorem 8.2). The proof uses the theory of interpolation Macdonald polynomials developed in [23, 24, 29, 30, 31, 32, 36]. This theory allows one to reconstruct a symmetric function $f(x_{1},\ldots,x_{N})$ from its values at special points $x_{i}=q^{-\mu_{i}-N+i}$ where $\mu$ is an arbitrary partition with at most $N$ parts. The connection between the center of $U_{q}\mathfrak{gl}_{N}$ and symmetric functions goes through the quantum Harish-Chandra isomorphism, and we interpret $f(q^{-\mu_{1}-N+1},\dots,q^{-\mu_{N}})$ as the scalar by which the element of the center $f$ acts on the irreducible representation $V({\mu})$. Interpolation Macdonald polynomials then correspond to a natural basis in the center of $U_{q}\mathfrak{gl}_{N}$. The polynomials $\sigma_{\lambda}$ yield a basis in the $\Gamma$-invariant parts of both the center $\mathcal{Z}$ and its completion (a function in the completion is determined by its values on all finite-dimensional representations). We use a formula of Okounkov [29] to give explicit expansion of a given central element $z$ in the basis $\sigma_{\lambda}$ in terms of the scalars by which $z$ acts on all finite-dimensional representations $V({\lambda})$. This leads to an expansion of the universal knot invariant in the basis $\sigma_{\lambda}$, where the coefficients are related to Reshetikhin-Turaev invariants of the same knot colored by $V({\mu})$ via an explicit triangular matrix $(d_{\lambda,\mu})$ which does not depend on the knot. ###### Theorem 1.3. For any evenly framed knot $K$, there exist Laurent polynomials $a_{\lambda}(K)\in\mathbb{Z}[q,q^{-1}]$ such that the universal invariant of $K$ has the following expansion: (3) $J_{K}(\mathfrak{gl}_{N};q)=\sum_{\lambda}a_{\lambda}(K)\,\sigma_{\lambda}\ .$ Moreover, the coefficients $a_{\lambda}(K)$ can be computed in terms of the Reshetikhin-Turaev invariants as follows: $a_{\lambda}(K)=\sum_{\mu\subset\lambda}{d_{\lambda,\mu}(q^{-1})}\;J_{K}(V(\mu),q)$ where the coefficients $d_{\lambda,\mu}(q)$ are defined in Theorem 10.17. We prove Theorem 1.3 as Proposition 8.7. We would like to emphasize that the fact that $a_{\lambda}(K)$ are Laurent polynomials in $q$ is highly nontrivial. Indeed, we have computed the tables of coefficients $d_{\lambda,\mu}(q)$ for $\mathfrak{gl}_{2}$ in Section 11.4 and these are complicated rational functions, so a priori $a_{\lambda}(K)$ are rational functions as well. Theorem 1.3 thus encodes certain divisibility properties for the linear combinations of colored invariants of $K$. We refer to Section 11.5 for the explicit computation of the coefficients $a_{\lambda}(K)$ for the figure eight knot. We call (3) a cyclotomic expansion of the universal $\mathfrak{gl}_{N}$ knot invariant. The name cyclotomic is justified by the fact that (3) has well- defined evaluations at any root of unity by Lemma 10.29 below. Note that for $N=2$ and a $0$-framed knot, our expansion does not coincide with that of Habiro, simply because if an element $z\in U_{q}\mathfrak{gl}_{2}$ is central and $\Gamma$-invariant, it does not imply $z$ has a decomposition in even powers of the Casimir. Therefore, our cyclotomic expansion is rather a generalization of $F_{\infty}$ in [37] or [4, eq.(3.14)], both having interesting application in the theory of non semisimple invariants of links and 3-manifolds. For our next application, assume $M$ is an integral homology 3-sphere obtained by $\mathbf{\varepsilon}$-surgery on an $\ell$-component algebraically split $0$-framed link $L$ with $\mathbf{\varepsilon}\in\\{\pm 1\\}^{\ell}$. Following Habiro–Le, we define an $\mathfrak{gl}_{N}$ unified invariant $I(M)$ as $I(M)=\langle\,r^{\otimes\mathbf{\varepsilon}},J_{L}(\mathfrak{gl}_{N};q)\,\rangle\ $ where $r$ is the $\mathfrak{gl}_{N}$ ribbon element and $\langle\cdot,\cdot\rangle$ is the Hopf pairing. In the case of $\mathfrak{sl}_{N}$ Habiro–Le proved [19] that the unified invariant belongs to a cyclotomic completition of the polynomial ring $\widehat{\mathbb{Z}[q]}:={\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\frac{\mathbb{Z}[q]}{((q;q)_{n})}\ $ known as Habiro ring. Using interpolation, we are able to express $I(M)$ in terms of special linear combinations of Reshetikhin–Turaev invariants of $L$, called Kirby colors. For this we diagonalize the Hopf pairing, i.e. find a basis $P_{\mu}$ that is orthonormal to $\sigma_{\lambda}$ and orthogonal to $V(\lambda)$ with respect to the Hopf pairing. This allows us to give explicit formulas for the universal Kirby colors $\omega_{\pm}$ (see (25)) in the basis $P_{\mu}$ and to prove the following result. ###### Theorem 1.4. The unified invariant $I(M)=J_{L}(\omega_{\epsilon_{1}},\dots,\omega_{\epsilon_{\ell}})\;\in\;\widehat{\mathbb{Z}[q]}$ belongs to the Habiro ring and dominates $\mathfrak{gl}_{N}$ WRT invariants of $M_{\pm}$ at all roots of unity. Moreover, $I(M)$ is equal to the $\mathfrak{sl}_{N}$ Habiro–Le invariant of $M_{\pm}$. To prove that $I(M)$ is equal to the $\mathfrak{sl}_{N}$ Habiro–Le invariant we show the equality of the universal $\mathfrak{gl}_{N}$ and $\mathfrak{sl}_{N}$ invariants for $0$-framed algebraically split links, and the fact that the $\mathfrak{gl}_{N}$ and $\mathfrak{sl}_{N}$ twist forms $x\mapsto\langle r^{\pm 1},x\rangle$ on them coincide. It follows that $I(M)$ belongs to the Habiro ring. Then we establish invariance of Kirby colors $\omega_{\pm}$ under Hoste moves (a version of Fenn–Rourke moves between algebraically split links) in Lemma 9.1, and finally deduce the equality $I(M)=J_{L}(\omega_{\epsilon_{1}},\dots,\omega_{\epsilon_{\ell}})$. The main advantage of Theorem 1.4 compared to Habiro–Le approach is the interpretation of $I(M)$ as the Reshetikhin–Turaev invariant of $L$ colored by $\omega_{\mathbf{\varepsilon}}$. This leads to various striking divisibility results and allows us to extend our cyclotomic expansion to links. ###### Corollary 1.5. Given an $\ell$ component algebraically split $0$-framed link $L$, then for all but finitely many partitions $\lambda_{i}$ with $1\leq i\leq\ell$, there exist positive integers $n=n(\lambda_{i},N)$, such that $J_{L}(P^{\prime}_{\lambda_{1}},\dots,P^{\prime}_{\lambda_{\ell}})\in(q;q)_{n}\,\mathbb{Z}[q,q^{-1}]\ $ where $P^{\prime}_{\lambda}=v^{\|\lambda\|}\,\dim_{q}V(\lambda)\,P_{\lambda}$ is a scalar multiple of $P_{\lambda}$. This is a generalization of the famous integrability theorem in [15, Thm. 8.2]. The authors do not know any direct proof of Corollary 1.5 without using the theory of unified invariants. Based on Corollary 9.4 we obtain a cyclotomic expansion for the Reshetikhin-Turaev invariants of $L$: (4) $J_{L}(\lambda_{1},\dots,\lambda_{\ell})=v^{\sum_{i}\|\lambda_{i}\|}\sum_{\mu_{i}\subset\lambda_{i}}\prod^{l}_{j=1}c_{\lambda_{j},\mu_{j}}(q^{-1})\,J_{L}(P^{\prime}_{\mu_{1}},\dots,P^{\prime}_{\mu_{\ell}})$ where the matrix $\left[c_{\lambda,\mu}(q)\right]_{\lambda,\mu}:=\left[F_{\lambda}(q^{-\mu_{i}-N+i})\right]_{\lambda,\mu}$ is the inverse of $\left[d_{\lambda,\mu}(q)\right]_{\lambda,\mu}$. This generalizes equation $(8.2)$ in [15]. In addition, in the case of knot surgeries we give a direct proof of the fact that $I(M_{\pm})=J_{L}(\omega_{\pm})\;\in\;\widehat{\mathbb{Z}[v]}$ by using our cyclotomic expansion and the interpolation theory. Finally, we would like to comment on potential ideas for categorification of these results. The ring of symmetric polynomials in $N$ variables is naturally categorified by the category of annular $\mathfrak{gl}_{N}$-webs, with morphisms given by annular foams [6, 33, 34, 13, 11]. By the work of the second author and Wedrich [13], one can interpret it as a symmetric monoidal Karoubian category generated by one object $E$ corresponding to a single essential circle. The symmetric polynomials are then categorified by the Schur functors of $E$. We expect the categorified interpolation polynomials to correspond to interpolation Macdonald polynomials where $q$ plays the role of quantum grading and $t$ of the homological grading (after some change of variables). We recall the general definitions and properties of these polynomials from [29] in Appendix. The key obstacle for categorification of interpolation polynomials is that they are not homogeneous. Therefore one needs to enrich the category and allow additional morphisms between $E$ and identity. On the other hand, the conjectures of the second author, Negut and Rasmussen ([12], see [10, 11] for further discussions) relate a version of the annular category to the derived category of the Hilbert scheme of points on the plane. The interpolation Macdonald polynomials appear in that context as well [7]. The paper is organized as follows. After recalling the definitions, we compare the Reshetikhin–Turaev invariants of tangles colored by $V(\lambda)$ and $L(\zeta)\otimes V(\lambda)$ in Section 4. In the next two sections we summarize known results about the center of $U_{q}\mathfrak{gl}_{N}$, define its completion and prove Theorem 1.1 in Section 6.2. The remaining results are proven in Sections 8, 9 assuming some facts about interpolation. In the last sections we develop the theory of the interpolation Macdonald polynomials, starting from the one variable case. We define multi-variable interpolation polynomials, state and prove their properties in Section 10.2. Next, we solve the interpolation problem in two ways, one using the approach of Okounkov (Theorem 10.17), and another using Hopf pairing (see (38)). We study divisibility of $F_{\lambda}(q^{a_{1}},\ldots,q^{a_{n}})$ by quantum factorials in Section 10.5 (see Lemma 10.29). Section 11 is focused on various stability properties of the interpolation polynomials such as adding a column to a partition $\lambda$ (Proposition 11.8) and changing $N$ for a fixed Young diagram $\lambda$. In particular, in Proposition 11.5 we describe a HOMFLY-PT analogue of the interpolation polynomials depending on an additional parameter $A=q^{N}$. We provide lots of examples and tables of interpolation polynomials, especially for $\mathfrak{gl}_{2}$. In Appendix A we collect some additional known facts about the interpolation Macdonald polynomials and the Habiro ring. ## Acknowledgments The authors would like to thank Pavel Etingof, Kazuo Habiro, Thang Le, Matt Hogancamp, Andrei Okounkov and Paul Wedrich for the useful discussions. We thank Satoshi Nawata for his comments on the first version of the paper and for bringing the reference [22] to our attention. Our work was partially supported by the NSF grants DMS-1700814 (E.G.), DMS-1760329 (E.G.) and the NCCR SwissMAP (A.B.). ## 2\. Notations and conventions ### 2.1. $q$-binomial formulas Throughout the paper we will use the following notations for the $q$-series. The $q$-Pochhammer symbols are defined as $(a;q)_{m}=\prod_{i=0}^{m-1}(1-aq^{i}),\ (a;q)_{\infty}=\prod_{i=0}^{\infty}(1-aq^{i}),\ m\geq 0.$ It is easy to see that $(a;q)_{m+k}=(a;q)_{m}(aq^{m};q)_{k},\ (a;q)_{m}=\frac{(a;q)_{\infty}}{(aq^{m};q)_{\infty}}.$ We will use two normalizations for $q$-binomial coefficients defined as follows: $\\{a\\}_{q}=1-q^{a},\ [a]_{q}=\frac{\\{a\\}_{q}}{\\{1\\}_{q}},\ [a]_{q}!=[1]_{q}\cdots[a]_{q},\ \binom{a}{b}_{q}=\frac{[a]_{q}!}{[b]_{q}![a-b]_{q}!}\ .$ Note that $[a]_{q}=\frac{(q;q)_{a}}{(1-q)^{a}},\ \binom{a}{b}_{q}=\frac{(q;q)_{a}}{(q;q)_{b}(q;q)_{a-b}}.$ Finally, the $q$-binomial formula gives $(a;q)_{m}=\sum_{j=0}^{m}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{m}{j}_{q}a^{j}.$ Let us also define symmetric $q$-numbers. For this we chose $v$ such that $v^{2}=q$ and set $\\{a\\}=v^{a}-v^{-a},\quad[a]:=\frac{\\{a\\}}{\\{1\\}},\quad\left[\begin{array}[]{cc}\\!a\\!\\\ \\!b\\!\end{array}\right]=\frac{\\{a\\}!}{\\{b\\}!\\{a-b\\}!}\ .$ We will use all these formulas throughout the paper without a reference. ### 2.2. Partitions We will work with partitions $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\ldots\lambda_{N})$ which we will identify with the corresponding Young diagrams in French notation, where the rows have length $\lambda_{i}$. Transpose diagram to $\lambda$ is denoted by $\lambda^{\prime}$, and $\|\lambda\|=\sum\lambda_{i}$. Given a box in a Young diagram, we define its arm, co-arm, leg and co-leg as in Figure 2. $l^{\prime}$$l$$a^{\prime}$$a$ Figure 2. Arm, co-arm, leg and co-leg We define the hook length as $h(\square)=a(\square)+l(\square)+1$, and the content $c(\square)=a^{\prime}-l^{\prime}$. Let $n(\lambda)=\sum(i-1)\lambda_{i}=\sum_{\square}l^{\prime}(\square)=\sum_{\square}l(\square),$ then $n(\lambda^{\prime})=\sum\frac{\lambda_{i}(\lambda_{i}-1)}{2}=\sum_{\square}a^{\prime}(\square)=\sum_{\square}a(\square).$ The content of $\lambda$ is defined as $c(\lambda)=\sum_{\square}c(\square)=n(\lambda^{\prime})-n(\lambda).$ Let $\bar{\lambda}_{i}=\lambda_{i}+N-i$ for $1\leq i\leq N$, then we have the following identity (5) $\prod_{\Box\in\lambda}(1-t^{h(\Box)})=\frac{\prod_{i\geq 1}\prod^{\bar{\lambda}_{i}}_{j=1}\left(1-t^{j}\right)}{\prod_{i<j}\left(1-t^{\bar{\lambda}_{i}-\bar{\lambda}_{j}}\right)}$ and we define (6) $\displaystyle D_{N}(\lambda)$ $\displaystyle=\sum_{i=1}^{N}\frac{(\bar{\lambda}_{i})(\bar{\lambda}_{i}-1)}{2}=\sum_{i}\frac{\lambda_{i}(\lambda_{i}-1)}{2}+\sum_{i}(N-i)\lambda_{i}+\sum_{i=1}^{N}\binom{N-i}{2}$ (7) $\displaystyle=n(\lambda^{\prime})+(N-1)\|\lambda\|-n(\lambda)+\binom{N}{3}=c(\lambda)+(N-1)\|\lambda\|+\binom{N}{3}.$ ## 3\. Quantum groups ### 3.1. Quantum $\mathfrak{gl}_{N}$ The quantum group $\mathcal{U}=U_{q}\mathfrak{gl}_{N}$ is a $\mathbb{C}(v)$-algebra generated by $E_{1},\ldots,E_{N-1}$, $F_{1},\ldots,F_{N-1}$, $K^{\pm 1}_{1},\ldots,K_{N}^{\pm 1}$ satisfying the following relations: (8) $K_{i}E_{i}=vE_{i}K_{i},\ K_{i}F_{i}=v^{-1}F_{i}K_{i},\ K_{i+1}E_{i}=v^{-1}E_{i}K_{i+1},\ K_{i+1}F_{i}=vF_{i}K_{i+1}$ (9) $[E_{i},F_{j}]=\delta_{ij}\frac{K_{i}K_{i+1}^{-1}-K_{i+1}K_{i}^{-1}}{v-v^{-1}},\ [K_{i},K_{j}]=0,$ (10) $E_{i}^{2}E_{j}-[2]E_{i}E_{j}E_{i}+E_{j}E_{i}^{2}=0\ \text{if}\ \|i-j\|=1\ \text{and}\ [E_{i},E_{j}]=0\ \text{otherwise}$ and analogously for $F_{i}$, where $v^{2}=q$. To simplify the notation we set $\mathcal{K}_{i}:=K_{i}K^{-1}_{i+1}$. Then the Hopf algebra structure on $\mathcal{U}$ (i.e. coproduct, antipode and counit) can be defined as follows: $\Delta(E_{i})=E_{i}\otimes 1+\mathcal{K}_{i}\otimes E_{i},\ \Delta(F_{i})=1\otimes F_{i}+F_{i}\otimes\mathcal{K}_{i}^{-1},\ \Delta(K_{i}^{\pm 1})=K_{i}^{\pm 1}\otimes K_{i}^{\pm 1},$ $S(K_{i}^{\pm 1})=K_{i}^{\mp 1},\ S(E_{i})=-E_{i}\mathcal{K}_{i}^{-1},\ S(F_{i})=-\mathcal{K}_{i}F_{i}$ $\varepsilon(K_{i}^{\pm 1})=1,\varepsilon(E_{i})=\varepsilon(F_{i})=0.$ Usually $\mathcal{U}$ is considered as a subalgebra of $\mathcal{U}_{h}$ that is an $h$-adically complete $\mathbb{C}[[h]]$-algebra topologically generated by $E_{i}$, $F_{i}$ and $H_{j}$ for $1\leq i\leq N-1$ and $1\leq j\leq N$ with $v=\exp{h/2},\quad K_{i}=v^{H_{i}}=\exp{hH_{i}/2}$ satisfying (9), (10) and $H_{i}E_{i}-E_{i}H_{i}=E_{i},\;\;H_{i}F_{i}-F_{i}H_{i}=-F_{i},\;\;H_{i+1}E_{i}-E_{i}H_{i+1}=-E_{i},\;\;H_{i+1}F_{i}-F_{i}H_{i+1}=F_{i}$ replacing (8). Rewriting the defining relations in terms of the generators $e_{i}={E_{i}}(v-v^{-1}),\quad F_{i}^{(n)}=\frac{F_{i}^{n}}{[n]!}\quad\text{and}\quad K_{j}\quad\text{for}\quad 1\leq i\leq N-1,\quad 1\leq j\leq N$ we obtain an integral version $\mathcal{U}_{\mathbb{Z}}$ as a Hopf algebra over $\mathbb{Z}[v,v^{-1}]\subset\mathbb{C}(v)\subset\mathbb{C}[[h]]$. The quantum group $\mathfrak{gl}_{N}$ has a fundamental representation $\mathbb{C}^{N}$ with basis $v_{1},\ldots,v_{N}$ such that $K_{i}v_{j}=v^{\delta_{ij}}v_{j},\ E_{i}v_{j}=\begin{cases}v_{i}&\text{if}\ j=i+1\\\ 0&\text{otherwise}\\\ \end{cases},F_{i}v_{j}=\begin{cases}v_{i+1}&\text{if}\ j=i\\\ 0&\text{otherwise}.\\\ \end{cases}$ It generates a braided monoidal category with simple objects $V(\lambda)$, where $\lambda$ is a partition with at most $N$ parts. These are highest weight modules where $K_{i}$ act on the highest weight vector by $v^{\lambda_{i}}$. The fundamental representation corresponds to $\lambda=(1)$. The representations $V(\lambda)$ have integral basis where $\mathcal{U}_{\mathbb{Z}}$ acts by $\mathbb{Z}[v,v^{-1}]$-valued matrices. ### 3.2. Ribbon structure The Hopf algebra $\mathcal{U}_{h}$ admits a ribbon Hopf algebra structure (see e.g. [8, Cor. 8.3.16]). The universal $R$-matrix has the form $\mathcal{R}=D\Theta$ where the diagonal part $D$ and the quasi-$R$-matrix are defined as follows $D=v^{\sum^{N}_{i=1}H_{i}\otimes H_{i}}\quad\text{and}\quad\Theta=\sum_{\mathbf{n}\in\mathbb{N}^{N-1}}F_{\mathbf{n}}\otimes e_{\mathbf{n}}$ where for any sequence of non-negative integers $\mathbf{n}=(n_{1},\dots,n_{N-1})$, the elements $e_{\mathbf{n}}$ and $F_{\mathbf{n}}$ are defined by equations (66) and (67) in [19] and form topological bases of the positive and negative parts in the triangular decomposition of $\mathcal{U}_{\mathbb{Z}}$. The inverse matrix $\mathcal{R}^{-1}=\iota(\Theta)D^{-1}$ is obtained by applying the involution $\iota:v\to v^{-1}$. The ribbon element and its inverse have the form (11) $r=\sum_{\mathbf{n}}F_{\mathbf{n}}\;\mathcal{K}_{\mathbf{n}}\;r_{0}\;e_{\mathbf{n}}\quad\text{and}\quad r^{-1}=\sum_{\mathbf{n}}\iota(F_{\mathbf{n}})\;\mathcal{K}_{-\mathbf{n}}\;r^{-1}_{0}\;\iota(e_{\mathbf{n}})$ where $r_{0}=K_{-2\rho}v^{-\sum_{i}H^{2}_{i}}$ and $K_{-2\rho}=\prod^{N}_{i=1}K_{i}^{2i-N-1}$ is the pivotal element. Here for any sequence of integers $\mathbf{n}\in\mathbb{Z}^{N-1}$ we set $\mathcal{K}_{\mathbf{n}}=\prod_{i}\mathcal{K}^{n_{i}}_{i}$, and denote by $\rho=\left(\frac{N-1}{2},\frac{N-3}{2},\ldots,\frac{1-N}{2}\right)=\frac{1-N}{2}(1,\ldots,1)+(N-1,N-2,\ldots,0)$ the half sum of all positive roots. Using the central element $K=\prod^{N}_{i=1}K_{i}$, we can write the previous definitions as follows: $r^{-1}_{0}=K^{N}\;\prod^{N}_{i=1}K^{-2i}_{i}\;v^{\sum_{i}H_{i}(H_{i}+1)},\quad K_{-2\rho}=K^{-N-1}\prod^{N}_{i=1}K^{2i}_{i}\ .$ The central element $r^{-1}$ acts on $V(\lambda)$ by the multiplication with $\theta_{V(\lambda)}=v^{(\lambda,\lambda+2\rho)}=v^{N\|\lambda\|}q^{c(\lambda)}$ where $(\lambda,\mu)=\sum^{N}_{i=1}\lambda_{i}\mu_{i}$, $c(\lambda)$ is the content of $\lambda$ and $v^{2}=q$. ### 3.3. Even part of $\mathcal{U}$ The algebra $\mathcal{U}$ has a natural grading by $\Gamma=\mathbb{Z}_{2}^{N}=\\{\pm 1\\}^{N}$ where $\zeta=(\zeta_{1},\ldots,\zeta_{N})\in\Gamma$ acts on $K_{i}$ by $\zeta_{i}$, on $E_{i}$ by 1 and on $F_{i}$ by $\zeta_{i}\zeta_{i+1}$. It is easy to see that the defining relations are preserved under this action. Following [19], we call an element of $\mathcal{U}_{N}$ even or $\Gamma$-invariant if it is preserved under the action of $\Gamma$. Let us denote by $\mathcal{U}^{\text{ev}}_{\mathbb{Z}}$ a $\mathbb{Z}[q,q^{-1}]$-subalgebra of $\mathcal{U}_{\mathbb{Z}}$ generated by $e_{i}$, $F^{(n)}_{i}\mathcal{K}_{i}$ and $K^{2}_{j}$ for $1\leq i\leq N-1$ and $1\leq j\leq N$. It is easy to check that $\mathcal{U}^{\text{ev}}_{\mathbb{Z}}$ is $\Gamma$-invariant. The action of $\Gamma$ descends on the category $\mathit{Rep}(\mathcal{U})$ of all finite-dimensional representations. Given $\zeta=(\zeta_{1},\ldots,\zeta_{N})\in\Gamma$, we can define a one-dimensional representation $L(\zeta)$ where $E_{i}$ and $F_{i}$ act by zero, and $K_{i}$ act by $\zeta_{i}$. We can also define representation $V(\lambda)\otimes L(\zeta)$ where $K_{i}$ act on the highest weight vector by $\zeta_{i}v^{\lambda_{i}}$. ###### Lemma 3.1. The action of $\mathcal{U}$ on $V(\lambda)\otimes L(\zeta)$ agrees with the $\Gamma$-twisted action of $\mathcal{U}$ on $V(\lambda)$. ###### Proof. Indeed, $\Delta(F_{i})=1\otimes F_{i}+F_{i}\otimes\mathcal{K}_{i}^{-1}$, so $F_{i}$ acts on $V(\lambda)\otimes L(\zeta)$ via $F_{i}\otimes\mathcal{K}_{i}^{-1}=F_{i}\zeta_{i}\zeta_{i+1}$. Similarly, $E_{i}$ acts on $V(\lambda)\otimes L(\zeta)$ via $E_{i}\otimes 1=E_{i}$ and $K_{i}$ acts via $K_{i}\otimes K_{i}=K_{i}\zeta_{i}.$ ∎ ### 3.4. The subalgebra $U_{q}\mathfrak{sl}_{N}$ We define $U_{q}\mathfrak{sl}_{N}$ as a subalgebra of $\mathcal{U}$ generated by $E_{i},F_{i}$ and $\mathcal{K}^{\pm 1}_{i}:=K^{\pm 1}_{i}K^{\mp 1}_{i+1}$ for $1\leq i\leq N-1$. The Hopf algebra $U_{q}\mathfrak{sl}_{N}$ also admits an integral version $\mathcal{U}_{\mathbb{Z}}\mathfrak{sl}_{N}$ generated by $e_{i},\;\;F^{(n)}_{i}\quad\text{and}\quad\mathcal{K}^{\pm 1}_{i}$ over $\mathbb{Z}[q,q^{-1}]$. The braiding $\mathcal{R}=D^{\prime}\Theta$ with $\Theta$ as for $\mathfrak{gl}_{N}$, but different diagonal part $D^{\prime}=v^{\sum^{N-1}_{i=1}\frac{{\mathcal{H}}_{i}\otimes{\mathcal{H}}_{i}}{2}}\quad\text{where}\quad{\mathcal{H}}_{i}=H_{i}-H_{i+1}\ .$ The ribbon element is defined by (11) with $r_{0}=K_{-2\rho}\prod^{N-1}_{i=1}v^{{-\mathcal{H}}_{i}^{2}/2}$. The pivotal element $K_{-2\rho}$ does not change. Note that the $\Gamma$-invariant part of $U_{q}\mathfrak{sl}_{N}$ generated by $e_{i}$, $F^{(n)}_{i}\mathcal{K}_{i}$ and $\mathcal{K}^{2}_{j}$ for $1\leq i,j\leq N-1$ has a smaller Cartan part than its $\mathfrak{gl}_{N}$ analogue. ###### Example 3.2. For $N=2$ the product $K_{1}K_{2}$ is central. By denoting $\mathcal{K}=K_{1}K_{2}^{-1},E=E_{1},F=F_{1}$ we get the standard presentation for $U_{q}(\mathfrak{sl}_{2})$: $\mathcal{K}E=v^{2}E\mathcal{K},\;\mathcal{K}F=v^{-2}F\mathcal{K},\;[E,F]=\frac{\mathcal{K}-\mathcal{K}^{-1}}{v-v^{-1}}.$ ### 3.5. Universal invariant Lawrence, Reshetikhin, Ohtsuki and Kauffman constructed quantum group valued universal link invariants. As it was already mentioned in the introduction, the universal invariant of a link is defined by splitting a diagram of its bottom tangle into elementary pieces and by associating $R$-matrices and pivotal elements to them. For more details and references we recommend to consult [16, Sec. 7.3]. However, we admit here the convention from [19, Sec. 2.7] and write the contributions from left to right along the orientation of each component. ## 4\. Ribbon structure on $\mathit{Rep}(\mathcal{U})$ The aim of this section is to compare the Reshetikhin-Turaev invariants of a bottom tangle whose components are colored with $V(\lambda)$ and $V(\lambda)\otimes L(\zeta)$. This will be later used to prove Theorem 1.1. Let us denote by $\mathcal{R}_{\mathbb{Q}}$ the representation ring of $\mathit{Rep}(\mathcal{U})$ over $\mathbb{Q}(v)$. Given an $l$ component link $L$, Reshetikhin–Turaev functor associated with Lie algebra $\mathfrak{g}$ provides a $\mathbb{Q}(v)$-multilinear map $\displaystyle J_{L}:\mathcal{R}_{\mathbb{Q}}\times\dots\times\mathcal{R}_{\mathbb{Q}}$ $\displaystyle\to\mathbb{Q}(v)$ $\displaystyle(\mu_{1},\dots,\mu_{l})$ $\displaystyle\mapsto\bigotimes_{i}\mathrm{Tr}^{V(\mu_{i})}_{q}\left(J_{L}(\mathfrak{g};q)\right)=:J_{L}(\mathfrak{g};\mu_{1},...,\mu_{l})$ normalized to $\prod_{i}\dim_{q}(V(\mu_{i}))$ for the $0$-framed $(\mu_{1},\dots,\mu_{l})$-colored unlink. In cases when $\mathfrak{g}$ is fixed in the context, we will remove it from the notation for simplicity. Note that in the case of a knot, we have $J_{K}(\lambda)=\dim_{q}(V(\lambda))J_{K}(V(\lambda),q)$ where the last invariant is the colored Jones polynomial used in Introduction and normalized to be 1 for the unknot. The universal $R$-matrix defines a braiding between the representations $V(\lambda)$. We can extend this braiding to $\mathit{Rep}(\mathcal{U})$ as follows. Clearly, $L(\zeta)\otimes L(\zeta^{\prime})\simeq L(\zeta\zeta^{\prime})$ and we define the braiding between $L(\zeta)$ and $L(\zeta^{\prime})$ to be trivial. Let $V$ be a finite-dimensional representation of $\mathcal{U}$ where the eigenvalues of $K_{i}$ are integral powers of $v$. Given $\zeta\in\Gamma$ we consider a $\mathbb{C}$-linear map $T_{V}(\zeta):V\to V$ which acts by $\prod\zeta_{i}^{a_{i}}$ on the weight subspace of $V$ where $K_{i}$ acts as $v^{a_{i}}$. ###### Lemma 4.1. The maps $c_{\zeta,V}:=\mathrm{swap}\circ(\mathrm{Id}\otimes T_{V}(\zeta)):L(\zeta)\otimes V\to V\otimes L(\zeta)$ with inverses $c_{V,\zeta}:=\mathrm{swap}\circ(T_{V}(\zeta)\otimes\mathrm{Id}):V\otimes L(\zeta)\to L(\zeta)\otimes V$ define a braiding on $\mathit{Rep}(\mathcal{U})$. ###### Proof. First, let us check that $\text{swap}\circ(\mathrm{Id}\otimes T_{V}(\zeta))$ intertwines the actions of $\mathcal{U}$ on both sides. Indeed, let $v\in V$ be a vector with weight $(v^{a_{1}},\ldots,v^{a_{N}})$, then $E_{i}v$ has weight $(v^{a_{1}},\ldots,v^{a_{i}+1},v^{a_{i+1}-1},\ldots,v^{a_{N}})$ while $F_{i}v$ has weight $(v^{a_{1}},\ldots,v^{a_{i}-1},v^{a_{i+1}+1},\ldots,v^{a_{N}})$. Let $\bullet$ denote the basis vector in $L(\zeta)$, then $c_{\zeta,V}E_{i}(\bullet\otimes v)=c_{\zeta,V}\left(\zeta_{i}\zeta_{i+1}\bullet\otimes E_{i}(v)\right)=\zeta_{1}^{a_{1}}\cdots\zeta_{i}^{a_{i}}\zeta_{i+1}^{a_{i+1}}\cdots\zeta^{a_{N}}_{N}E_{i}(v)\otimes\bullet,\\\ c_{\zeta,V}F_{i}(\bullet\otimes v)=c_{\zeta,V}\left(\bullet\otimes F_{i}(v)\right)=\zeta_{1}^{a_{1}}\cdots\zeta_{i}^{a_{i}-1}\zeta_{i+1}^{a_{i+1}+1}\cdots\zeta^{a_{N}}_{N}F_{i}(v)\otimes\bullet,\\\ c_{\zeta,V}K_{i}(\bullet\otimes v)=c_{\zeta,V}\left(\zeta_{i}\bullet\otimes K_{i}(v)\right)=\zeta_{1}^{a_{1}}\cdots\zeta_{i}^{a_{i}+1}\cdots\zeta^{a_{N}}_{N}K_{i}(v)\otimes\bullet,\\\ $ while $E_{i}c_{\zeta,V}(\bullet\otimes v)=E_{i}(\zeta_{1}^{a_{1}}\cdots\zeta_{N}^{a_{N}}v\otimes\bullet)=\zeta_{1}^{a_{1}}\cdots\zeta^{a_{N}}_{N}E_{i}(v)\otimes\bullet,\\\ F_{i}c_{\zeta,V}(\bullet\otimes v)=F_{i}(\zeta_{1}^{a_{1}}\cdots\zeta_{N}^{a_{N}}v\otimes\bullet)=\zeta_{1}^{a_{1}}\cdots\zeta_{i}^{a_{i}-1}\zeta_{i+1}^{a_{i+1}+1}\cdots\zeta^{a_{N}}_{N}F_{i}(v)\otimes\bullet,\\\ K_{i}c_{\zeta,V}(\bullet\otimes v)=K_{i}(\zeta_{1}^{a_{1}}\cdots\zeta_{N}^{a_{N}}v\otimes\bullet)=\zeta_{1}^{a_{1}}\cdots\zeta_{i}^{a_{i}+1}\cdots\zeta^{a_{N}}_{N}K_{i}(v)\otimes\bullet.\\\ $ Next, we observe that $T_{V}(\zeta)T_{V}(\zeta^{\prime})=T_{V}(\zeta\zeta^{\prime})$ and $T_{U\otimes V}(\zeta)=T_{U}(\zeta)\otimes T_{V}(\zeta)$, so $c_{\zeta,V}$ indeed defines a braiding. Even more concretely, we get the braiding as the composition (12) $c_{L(\zeta)\otimes V,L(\zeta^{\prime})\otimes U}:L(\zeta)\otimes V\otimes L(\zeta^{\prime})\otimes U\xrightarrow{c_{V,\zeta^{\prime}}}L(\zeta)\otimes L(\zeta^{\prime})\otimes V\otimes U=L(\zeta^{\prime})\otimes L(\zeta)\otimes V\otimes U\xrightarrow{c_{V,U}}\\\ L(\zeta^{\prime})\otimes L(\zeta)\otimes U\otimes V\xrightarrow{c_{\zeta,U}}L(\zeta^{\prime})\otimes U\otimes L(\zeta)\otimes V.$ ∎ The representations $L(\zeta)$ are self-dual, and it is easy to see that the braiding $c_{\zeta,V}$ is compatible with changing $V$ to $V^{}$. Therefore, $\mathit{Rep}(\mathcal{U})$ with objects $L(\zeta)\otimes V$ form a pivotal braided monoidal category. The quantum dimension of $L(\zeta)$ equals to the trace of the action of the pivotal element, which is $(\prod_{i}\zeta_{i})^{N+1}$. The twist coefficient $\theta_{L(\zeta)}$ is defined as the action of the ribbon element on $L(\zeta)$, and is given by $(\prod_{i}\zeta_{i})^{N}$. ###### Lemma 4.2. $\mathit{Rep}(\mathcal{U})$ is a ribbon category with twist $\theta_{L(\zeta)\otimes V}=\theta_{L(\zeta)}\theta_{V}$. ###### Proof. By definition $\theta_{L(\zeta)\otimes V}=c_{\zeta,V}\theta_{L(\zeta)}\theta_{V}c_{V,\zeta}=\theta_{L(\zeta)}\theta_{V}$. ∎ ### 4.1. Braiding in $\mathit{Rep}(U_{q}\mathfrak{sl}_{N})$ In this section, we study the action of $\Gamma$ and the corresponding braiding for $U_{q}\mathfrak{sl}_{N}$, starting from $N=2$. Similarly to the previous section, $U_{q}\mathfrak{sl}_{2}$ has a one dimensional representation $L(-1)$ where $E$ and $F$ act by 0 and $\mathcal{K}$ acts by $-1$. The action of $U_{q}\mathfrak{sl}_{2}$ on $L(-1)\otimes V$ is equivalent to $\mathbb{Z}_{2}$-twisted action on $V$ where $\mathbb{Z}_{2}$ scales $E$ by 1 and $F,\mathcal{K}$ by $-1$. One can attempt to define a braiding for $U_{q}\mathfrak{sl}_{2}$. Since $E$ and $F$ shift the weights by $2$, it is easy to see that the analogue of $T_{V}$ should act by $(\sqrt{-1})^{a}$ on a subspace with weight $v^{a}$, and it does not square to identity. Nevertheless, it squares to $\pm\text{id}$ on each irreducible representation. This means that braiding relations on $\mathit{Rep}(U_{q}\mathfrak{sl}_{2})$ hold up to sign. To pin down this sign, we define the sign automorphism $\Sigma_{V}$ which acts by $(-1)^{a}$ on a subspace with weight $v^{a}$. Since $E,F$ shift the weight by $\pm v^{2}$, $\Sigma_{V}$ commutes with the action of $U_{q}\mathfrak{sl}_{2}$ on $V$. The operator $\Sigma_{V}$ acts on the irreducible representation $V(n)$ by a scalar $(-1)^{n}$. Also, it is easy to see that $\Sigma_{V\oplus W}=\Sigma_{V}\oplus\Sigma_{W}$ and $\Sigma_{V\otimes W}=\Sigma_{V}\otimes\Sigma_{W}$. ###### Lemma 4.3. The operators $T_{V}$ and $\Sigma_{V}$ satisfy the following properties: (a) We have $T_{V}^{2}=\Sigma_{V},\ c_{L(-1),V}=c^{-1}_{L(-1),V}(1\otimes\Sigma_{V})=(\Sigma_{V}\otimes 1)c^{-1}_{L(-1),V}$ * (b) Let $c_{V,W}:V\otimes W\to W\otimes V$ be the braiding, then $c_{V,W}(\Sigma_{V}\otimes 1)=(1\otimes\Sigma_{V})c_{V,W},\ c_{V,W}(1\otimes\Sigma_{W})=(\Sigma_{W}\otimes 1)c_{V,W}$ * (c) We have $c_{L(-1),V\otimes W}=c_{L(-1),V}\circ c_{L(-1),W}.$ * (d) The braiding with $L(-1)$ satisfies Yang-Baxter equation, that is, the following diagram commutes: ${L(-1)\otimes V\otimes W}$${V\otimes L(-1)\otimes W}$${V\otimes W\otimes L(-1)}$${L(-1)\otimes W\otimes V}$${W\otimes L(-1)\otimes V}$${W\otimes V\otimes L(-1)}$$\scriptstyle{c_{L(-1),V}}$$\scriptstyle{c_{V,W}}$$\scriptstyle{c_{L(-1),W}}$$\scriptstyle{c_{V,W}}$$\scriptstyle{c_{L(-1),W}}$$\scriptstyle{c_{L(-1),V}}$ ###### Proof. Part (a) is clear. To prove (b), observe that the action of $U_{q}\mathfrak{sl}_{2}\otimes U_{q}\mathfrak{sl}_{2}$ on $V\otimes W$ commutes with both $\Sigma_{V}\otimes 1$ and $1\otimes\Sigma_{V}$, and the $R$-matrix is an element of the completion of $U_{q}\mathfrak{sl}_{2}\otimes U_{q}\mathfrak{sl}_{2}$. Given a pair of vectors $u\in V,w\in W$ such that $Ku=v^{i}u$ and $Kw=v^{j}w$, we get $K(u\otimes w)=v^{i+j}u\otimes w$, so $T_{V\otimes W}=T_{V}\otimes T_{W}$. Since $c_{L(-1),V}=\mathrm{swap}\circ(\mathrm{Id}\otimes T_{V})$, we get the desired relation. Finally, (d) follows from (c). ∎ We can generalize the above results to representations of $U_{q}\mathfrak{sl}_{N}$ as follows. For $\zeta\in\mathbb{Z}_{2}^{N-1}$ there is a one-dimensional representation $L(\zeta)$ of $U_{q}(\mathfrak{sl}_{N})$ where $E_{i},F_{i}$ act by 0 and $\mathcal{K}_{i}=K_{i}K_{i+1}^{-1}$ act by $\zeta_{i}$ ($1\leq i\leq N-1$). Given a representation $V$ where all weights of $\mathcal{K}_{i}$ are integral powers of $v$, we can define an operator $T_{\zeta,V}:V\to V$ which acts by $\zeta^{A^{-1}\mathbf{a}}$ on a subspace where $\mathcal{K}_{i}$ acts by $v^{a_{i}}$. Here $A$ is the Cartan matrix for $\mathfrak{sl}_{N}$ given by (13) $A=\left(\begin{matrix}2&-1&0&\ldots&0\\\ -1&2&-1&\ldots&0\\\ 0&-1&2&\ldots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\ldots&2\\\ \end{matrix}\right)$ and $\mathbf{a}=(a_{1},\ldots,a_{N-1})$. Note that $\det(A)=N$, so $A^{-1}$ has rational entries with denominator $N$ and one needs to choose an $N$-th root of $(-1)$ to define $\zeta^{A^{-1}\mathbf{a}}$. Define $\Sigma_{\zeta,V}=T^{2}_{\zeta,V}$. ###### Lemma 4.4. The operators $T_{\zeta,V}$ and $\Sigma_{\zeta,V}$ satisfy the following properties: * (a) $T_{\zeta,V}E_{i}=\zeta_{i}E_{i}T_{\zeta,V},T_{\zeta,V}F_{i}=\zeta_{i}F_{i}T_{\zeta,V}$ * (b) $\Sigma_{\zeta,V}$ commutes with the action of $U_{q}\mathfrak{sl}_{N}$ on $V$ * (c) The map $c_{L(\zeta),V}=\mathrm{swap}\circ(\mathrm{Id}\otimes T_{\zeta,V}):L(\zeta)\otimes V\to V\otimes L(\zeta)$ is a morphism of $U_{q}\mathfrak{sl}_{N}$-representations * (d) The maps $T_{\zeta,V}$ and $\Sigma_{\zeta,V}$ satisfy all equations in Lemma 4.3 with $L(-1)$ changed to $L(\zeta)$. ###### Proof. (a) The operator $F_{i}$ changes the weight $\mathbf{a}=(a_{1},\ldots,a_{N-1})$ by $Ae_{i}$, so if $\mathcal{K}_{i}v=v^{a_{i}}v$ then $T_{\zeta,V}F_{i}(v)=\zeta^{A^{-1}(\mathbf{a}+Ae_{i})}F_{i}v=\zeta^{A^{-1}\mathbf{a}+e_{i}}F_{i}v=\zeta_{i}F_{i}T_{\zeta,V}(v).$ The proof for $E_{i}$ is similar. Part (b) immediately follows from (a). For (c), we observe that the action of $E_{i}$ on $L(\zeta)\otimes V$ is the same as the action on $V$, while the actions of $F_{i},\mathcal{K}_{i}$ are twisted by $\zeta_{i}$. On the other hand, the action of $F_{i}$ on $V\otimes L(\zeta)$ is the same as the action on $V$, while the actions of $E_{i},\mathcal{K}_{i}$ are twisted by $\zeta_{i}$. Therefore by (a) the operator $c_{L(\zeta),V}$ intertwines the actions of $U_{q}\mathfrak{sl}_{N}$ on $L(\zeta)\otimes V$ and $V\otimes L(\zeta)$. Finally, the proof of the rest of Lemma 4.3 extends to $U_{q}\mathfrak{sl}_{N}$ verbatim. ∎ ###### Remark 4.5. The above construction of $T_{\zeta,V}$ and $\Sigma_{\zeta,V}$ can be extended to an arbitrary semisimple Lie algebra with Cartan matrix $A$. The action of $\Sigma_{\zeta,V}$ can be interpreted in terms of projection of the weight lattice to its quotient by the root lattice. We draw a tangle colored by a representation $V=V(\lambda)$ using solid lines, and a tangle colored by $L(\zeta)$ by dotted lines. If a component is colored by $L(\zeta)\otimes V$, we draw a dotted line on the left of a solid line and parallel to it. The crossings between solid and dotted lines correspond to $c^{\pm}_{L(\zeta),V}$ depicted in Figure 3. Note that unlike $\mathfrak{gl}_{N}$ case, $c_{L(\zeta),V}$ does not square to identity and we have to distinguish under- and over-crossings between solid and dotted lines. This allows us to define Reshetikhin-Turaev invariants for framed tangles colored by representations of $U_{q}\mathfrak{sl}_{N}$ of the form $L(\zeta)\otimes V(\lambda)$. Using the notations as in Figure 3, we can visualize the statement of Lemma 4.4 in Figure 4. $\Sigma$ Figure 3. The operators $c_{L(\zeta),V}$, $c^{-1}_{L(\zeta),V}$ and $\Sigma_{\zeta}$. (a)$=$$\Sigma$$=$$\Sigma$$=$$\Sigma$ (b)$\Sigma$$=$$\Sigma$ (d) Figure 4. Diagrammatics for Lemma 4.4 ###### Theorem 4.6. (a) Let $L$ be an algebraically split $0$-framed link with $\ell$ components. Then for arbitrary partitions $\lambda_{1},\ldots,\lambda_{\ell}$ and $\zeta_{1},\ldots,\zeta_{\ell}\in\Gamma$ the following identity of Reshetikhin–Turaev invariants holds: (14) $J_{L}\left({\mathfrak{sl}_{N}};V(\lambda_{1})\otimes L(\zeta_{1}),\ldots,V(\lambda_{\ell})\otimes L(\zeta_{\ell})\right)=\\\ J_{L}\left({\mathfrak{sl}_{N}};V(\lambda_{1}),\ldots,V(\lambda_{\ell})\right)\cdot\dim_{q}L(\zeta_{1})\cdots\dim_{q}L(\zeta_{\ell}),$ where $\dim_{q}L(\zeta_{i})=\mathrm{Tr}^{L(\zeta_{i})}_{q}(1)=\pm 1$. (b) Let $L$ be an arbitrary link with evenly framed components, if $N$ is odd. Then (14) holds for $\mathfrak{gl}_{N}$ Reshetikhin–Turaev invariants. ###### Proof. (a) We use the results of Lemmas 4.3 and 4.4 and the above diagrammatic notation. By Lemma 4.3(a), we can change crossings between dotted and solid lines at a cost of placing $\Sigma_{\zeta}$ on solid lines. By doing this iteratively, we can make all dotted lines to be above solid lines. At this stage, each solid component of $L$ acquires several copies of $\Sigma_{\zeta}$ and $\Sigma_{\zeta}^{-1}$ at various places of the link diagram. The number of these copies (with signs) equals the linking number between this component and the dotted part which is even by our assumption. By Lemma 4.3(b) we can combine all these copies of $\Sigma_{\zeta}$ together and cancel out. Finally, using Lemma 4.3(d), we can separate the dotted and solid links. By changing the crossings in the dotted link, we transform it to the $0$-framed unlink. Therefore the invariant of the solid link equals $J_{L}\left({\mathfrak{sl}_{N}};V(\lambda_{1}),\ldots,V(\lambda_{\ell})\right)$ while the invariant of the dotted link equals $\dim_{q}L(\zeta_{1})\cdots\dim_{q}L(\zeta_{\ell})$. The proof of (b) is similar, except that $\Sigma_{V}$ is trivial for all $V$. As before we can unknot dotted components. Now the ribbon element acts on $L(\zeta)$ by $\theta_{L(\zeta)}=(\prod_{i}\zeta_{i})^{N}$, and hence, any (even if $N$ is odd) number of them acts by $1$. The result follows. ∎ ## 5\. Center of $\mathcal{U}$ Let $\mathcal{Z}$ be the center of $\mathcal{U}_{\mathbb{Z}}$. In this section we recall the main facts known about $\mathcal{Z}$. ### 5.1. Harish-Chandra isomorphism Let $(\mathcal{U}^{0}_{\mathbb{Z}})^{S_{N}}:=\mathbb{Z}[v,v^{-1}][{K^{\pm 1}_{1}},\ldots,K_{N}^{\pm 1}]^{S_{N}}$ be the Cartan part of $\mathcal{U}_{\mathbb{Z}}$ invariant under the Weyl group action. After a multiplication by an appropriate power of the central element $K:=\prod^{N}_{i=1}K_{i}$, each element of $(\mathcal{U}^{0})^{S_{N}}$ can be viewed as a symmetric function in $N$ variables. This allows to identify $(\mathcal{U}^{0}_{\mathbb{Z}})^{S_{N}}$ with the ring of symmetric functions divided by powers of the elementary symmetric polynomial $e_{N}=K$. In the classical case, this ring can be identified with the center using the Harish- Chandra isomorphism. After quantization, the image of the Harish-Chandra homomorphism belongs to $\mathit{Sym}=\mathbb{Z}[v^{\pm 1},e_{N}^{-1}][x_{1},\dots,x_{N}]^{S_{N}}$ where $x_{i}=K_{i}^{2}$ (compare e.g. [20, Ch. 6]). In this section we will furthermore identify $\mathit{Sym}$ with the Grothendieck ring $\mathcal{R}$ of $\mathit{Rep}(\mathcal{U})$ with coefficients $\mathbb{Z}[v^{\pm 1}]$. First, the character map $\mathit{ch}:\mathcal{R}\to\mathit{Sym}$ sends a representation $U$ to its character $\mathit{ch}(U)$. Clearly, $\mathit{ch}(U\oplus V)=\mathit{ch}(U)+\mathit{ch}(V)$ and $\mathit{ch}(U\otimes V)=\mathit{ch}(U)\mathit{ch}(V)$, so $\mathit{ch}$ is a ring homomorphism. The character of $V(\lambda)$ equals the Schur function $s_{\lambda}(x_{1},\dots,x_{N})$, while the character of $L(\zeta)$ equals $\zeta_{1}\cdots\zeta_{N}$. The Harish-Chandra map $\mathit{hc}:\mathcal{Z}\to\mathit{Sym}$ is defined as follows. Let $\phi$ be a central element in $\mathcal{U}$, it acts in the Verma module $\Delta(\lambda)$ by some scalar $\phi\|_{\Delta(\lambda)}$. We define $\mathit{hc}(\phi)$ to be the polynomial in $\mathit{Sym}$ defined by the condition $\mathit{hc}(\phi)(q^{\rho+\lambda})=\phi\|_{\Delta(\lambda)}\quad\text{for all}\quad\lambda$ where $\rho=\left(\frac{N-1}{2},\frac{N-3}{2},\ldots,\frac{1-N}{2}\right)$. Note that the product $\phi\phi^{\prime}$ acts on $\Delta(\lambda)$ by the product of the corresponding scalars, so $\mathit{hc}$ is also a ring homomorphism. It is known to be an isomorphism (see e.g. [20, Ch. 6]). Finally, the map $\xi:\mathcal{R}\to\mathcal{Z}$ is defined by $\xi=\mathit{hc}^{-1}\circ\mathit{ch}$. It is a composition of two ring homomorphisms and hence a ring homomorphism too. Hence, we get the commutative diagram: ${\mathcal{R}}$${\mathcal{Z}}$${\mathit{Sym}}$$\scriptstyle{\mathit{ch}}$$\scriptstyle{\xi}$$\scriptstyle{\mathit{hc}}$ In Lemma 5.3 we will show that $\xi$ actually coincides with the Drinfeld map. ###### Example 5.1. The central element $K=K_{1}\cdots K_{N}$ acts on $V(\lambda)$ by a scalar $v^{\sum\lambda_{i}}$. Since $\sum\rho_{i}=0$, we get $\mathit{hc}(K_{1}\cdots K_{N})=y_{1}\cdots y_{N}$. ###### Example 5.2. The center of $U_{q}\mathfrak{sl}_{2}$ is generated by the Casimir element: $C=(v-v^{-1})^{2}FE+v\mathcal{K}+v^{-1}\mathcal{K}^{-1}$ It acts on a representation $V_{m}$ by $v^{m+1}+v^{-m-1}$, so $\mathit{hc}(C)=y+y^{-1}$ (note that $v^{\rho}=v$ in this case). On the other hand, $\mathit{ch}(V_{1})=y+y^{-1}$, so $\xi(V_{1})=C$, where $V_{1}$ the 2-dimensional representation. Similarly, we can consider the corresponding central element in $U_{q}\mathfrak{gl}_{2}$ defined by $C_{\mathfrak{gl}_{2}}=(v-v^{-1})^{2}FE+vK_{1}K_{2}^{-1}+v^{-1}K_{1}^{-1}K_{2}.$ It acts on a representation $V(\lambda)$ by a scalar $v^{1+\lambda_{1}-\lambda_{2}}+v^{-1-\lambda_{1}+\lambda_{2}}=\frac{y_{1}}{y_{2}}+\frac{y_{2}}{y_{1}},\ \quad y_{1}=v^{1/2+\lambda_{1}},y_{2}=v^{-1/2+\lambda_{2}},$ so $\mathit{hc}(C_{\mathfrak{gl}_{2}})=\frac{y_{1}}{y_{2}}+\frac{y_{2}}{y_{1}}=\frac{y^{2}_{1}+y^{2}_{2}}{y_{1}y_{2}}=e^{-1}_{2}(y_{1},y_{2})(x_{1}+x_{2})$. ### 5.2. Hopf pairing The Hopf pairing $\langle U,V\rangle$ of two representations $U,V\in\mathcal{R}$ is defined as the Reshetikhin–Turaev invariant of the Hopf link with components labeled by $U$ and $V$. This is a symmetric bilinear pairing on $\mathcal{R}$. The map $\xi$ is related to the Hopf pairing as follows: ###### Lemma 5.3. The Hopf pairing on $\mathcal{R}$ can be computed as $\langle U,V\rangle=\mathrm{Tr}_{q}^{U}(\xi(V)).$ ###### Proof. Consider the Drinfeld map $D$ [9] which sends a representation $V$ to a central element corresponding to the universal invariant of the following tangle: $D(V):=$$V$ By e.g. [14, eq. (20)] (see also [19, Proposition 8.19] and references therein) the eigenvalue of $D(V)$ on the irreducible representation $V(\lambda)$ equals $\mathit{ch}(q^{\lambda+\rho})$ where $\mathit{ch}$ is the character of $V$. By the definition of the Harish-Chandra map, this means that $\mathit{hc}(D(V))=\mathit{ch}(V)$, and $D(V)=\mathit{hc}^{-1}(ch(V))=\xi(V),$ so $\xi$ agrees with the Drinfeld map. Now $\langle U,V\rangle=\mathrm{Tr}_{q}^{U}(D(V))=\mathrm{Tr}_{q}^{U}(\xi(V))$ or more precisely, $\langle V(\lambda),V(\mu)\rangle=s_{\lambda}(q^{\mu+\rho})s_{\mu}(q^{\rho})\quad\text{where}\quad\dim_{q}V(\mu)=s_{\mu}(q^{\rho}).$ ∎ Using the Drinfeld isomorphism $\xi$ we can extend the Hopf pairing to the center by setting $\langle z_{1},z_{2}\rangle:=\langle\xi^{-1}(z_{1}),\xi^{-1}(z_{2})\rangle\quad\text{for any}\quad z_{1},z_{2}\in\mathcal{Z}\ .$ ## 6\. Cyclotomic completion and the universal invariant The universal invariant of a link belongs a priori to a (completed) tensor product of copies of $\mathcal{U}_{h}$, rather than $\mathcal{U}$ or $\mathcal{U}_{\mathbb{Z}}$, due to the diagonal part of the $R$-matrix. The aim of this section is to define a certain completion of $\mathcal{U}_{\mathbb{Z}}$ and its tensor powers, such that the universal $\mathfrak{gl}_{N}$ invariant of evenly framed links belongs to it. Since the action of $\Gamma$ extends to the completion, this will allow us to speak about $\Gamma$-invariance of $J_{L}(\mathfrak{gl}_{N};q)$. ### 6.1. Cyclotomic completion of $\mathcal{U}_{\mathbb{Z}}$ Given $n\in\mathbb{N}$, we define a family of two-sided ideals $\mathcal{U}^{(n)}_{\mathbb{Z}}$ as the minimal filtration such that $\mathcal{U}^{(n)}_{\mathbb{Z}}\mathcal{U}^{(m)}_{\mathbb{Z}}\subset\mathcal{U}^{(m+n)}_{\mathbb{Z}}$ and $(q;q)_{n},\;\;e^{n}_{i},\;\;f_{n}(K^{2}_{j})\in\mathcal{U}^{(n)}_{\mathbb{Z}}$ for any $1\leq i\leq N-1$ and $1\leq j\leq N$ where $f_{n}(x)=(x;q)_{n}$. In other words, $\mathcal{U}^{(n)}_{\mathbb{Z}}$ is the two-sided ideal generated by the products (15) $(q;q)_{a}\;e_{\mathbf{m}}\;f_{c_{1}}(K^{2}_{1})\cdots f_{c_{N}}(K^{2}_{N}),\quad\text{with}\quad a+\sum_{i}m_{i}+\sum_{i}c_{i}=n\ .$ ###### Lemma 6.1. We have $\Delta\left(f_{n}\left(K^{2}_{i}\right)\right)=\sum_{a=0}^{n}\binom{n}{a}_{q}f_{a}(K^{2}_{i})\otimes K_{i}^{2(n-a)}f_{n-a}(K^{2}_{i}).$ ###### Proof. We prove Lemma by induction in $n$. For $n=0$ it is clear. The induction step follows from the identities $f_{n+1}(K^{2}_{i})=f_{n}(K_{i})(1-q^{n}K^{2}_{i})$ and $\Delta(1-q^{n}K^{2}_{i})=1\otimes 1-q^{n}K^{2}_{i}\otimes K^{2}_{i}=(1-q^{a}K^{2}_{i})\otimes q^{n-a}K^{2}_{i}+1\otimes(1-q^{n-a}K^{2}_{i}).$ ∎ ###### Proposition 6.2. a) $\mathcal{U}^{(n)}_{\mathbb{Z}}$ is the left ideal generated by (15). b) $\mathcal{U}^{(n)}_{\mathbb{Z}}$ form a Hopf algebra filtration, that is $\Delta\,\mathcal{U}^{(n)}_{\mathbb{Z}}\subset\,\sum_{i+j=n}\,\mathcal{U}^{(i)}_{\mathbb{Z}}\otimes\mathcal{U}^{(j)}_{\mathbb{Z}}$. c) Assume that $\lambda_{i}\leq k$ for all $i$. Given arbitrary $m$, there exists $n=n(k,m)$ such that the elements of $\mathcal{U}^{(n)}_{\mathbb{Z}}$ act on the integral basis of $V(\lambda)$ by matrices divisible by $(q;q)_{m}$. ###### Proof. a) Observe that by Lemma 10.5 we get $f_{n}(q^{s}K^{2}_{i})\in\mathcal{U}^{(n)}_{\mathbb{Z}}$ for all integer $s$. Now the statement follows from the identities $f_{n}(K^{2}_{i})F_{i}^{(s)}=F_{i}^{(s)}f_{n}(q^{-s}K^{2}_{i}),\ f_{n}(K^{2}_{i+1})F_{i}^{(s)}=F_{i}^{(s)}f_{n}(q^{s}K^{2}_{i+1})$ and $f_{n}(K^{2}_{i})e^{s}_{i}=e^{s}_{i}f_{n}(q^{s}K^{2}_{i}),\ f_{n}(K^{2}_{i+1})e^{s}_{i}=e^{s}_{i}f_{n}(q^{-s}K^{2}_{i+1}).$ b) Follows from the identity $\Delta(e_{j}^{m})=\sum_{i=0}^{m}\binom{m}{i}_{q}e_{j}^{m-i}\mathcal{K}^{i}\otimes e_{j}^{i}$ and Lemma 6.1. c) By (a), it is sufficient to check the statement for $e_{i}^{n}$ and $f_{n}(K^{2}_{i})$. If $\lambda_{i}\leq k$ then for $n>k$ $e_{i}^{n}$ annihilates $V(\lambda)$, while $f_{n}(K^{2}_{i})$ acts on a vector with weight $(v^{\lambda_{1}},\ldots,v^{\lambda_{N}})$ by $f_{n}(q^{\lambda_{i}})=(q^{\lambda_{i}};q)_{n}$ which is divisible by $(q;q)_{n}$. ∎ By Proposition 6.2(b), the filtration $\mathcal{U}_{\mathbb{Z}}=\mathcal{U}^{(0)}_{\mathbb{Z}}\supset\mathcal{U}^{(1)}_{\mathbb{Z}}\supset\dots\mathcal{U}^{(n)}_{\mathbb{Z}}\supset\dots$ is a Hopf algebra filtration of $\mathcal{U}_{\mathbb{Z}}$ with respect to a descending filtration of ideals $I_{n}=((q;q)_{n})$ in $\mathbb{Z}[v,v^{-1}]$ in the sense of [18, Sec. 4]. Hence, the completion $\widehat{\mathcal{U}}:={\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\;\frac{\mathcal{U}_{\mathbb{Z}}}{\mathcal{U}^{(n)}_{\mathbb{Z}}}$ is a complete Hopf algebra over the ring $\widehat{\mathbb{Z}[v]}:={\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\;\frac{\mathbb{Z}[v]}{((q;q)_{n})}.$ We refer to [18, Section 4] for details. Analogously, we define the $\Gamma$-invariant subalgebra $\widehat{\mathcal{U}^{\text{ev}}}:={\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\;\frac{\mathcal{U}^{\text{ev}}_{\mathbb{Z}}}{\mathcal{U}^{(n)}_{\mathbb{Z}}}$ as a complete Hopf algebra over the Habiro ring $\widehat{\mathbb{Z}[q]}$. Let us now extend the completion to the tensor powers of $\mathcal{U}_{\mathbb{Z}}$. For this we define the filtration for $\mathcal{U}_{\mathbb{Z}}^{\otimes l}$ for $l\geq 1$ as follows ${\mathcal{F}}_{n}(\mathcal{U}_{\mathbb{Z}}^{\otimes l})=\sum^{l}_{i=1}\mathcal{U}_{\mathbb{Z}}^{\otimes i-1}\otimes\mathcal{U}^{(n)}_{\mathbb{Z}}\otimes\mathcal{U}_{\mathbb{Z}}^{\otimes l-i}$ and the completed tensor product $\mathcal{U}_{\mathbb{Z}}^{\hat{\otimes}l}$ with respect to this filtration will be the image of the homomorphism ${\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\;\frac{\mathcal{U}_{\mathbb{Z}}^{\otimes l}}{{\mathcal{F}}_{n}(\mathcal{U}_{\mathbb{Z}}^{\otimes l})}\;\;\to\;\;\mathcal{U}_{h}^{{\otimes}k}$ where on the right hand side we use the $h$-adically completed tensor product. ### 6.2. Hopf pairing and universal invariants Let us denote by $c\in\mathcal{U}_{h}{\otimes}\,\mathcal{U}_{h}$ the double braiding or the universal invariant of the clasp tangle in Figure 1, given by $c=(S\,\otimes\,\text{id})\,\mathcal{R}_{21}\mathcal{R}\ .$ The main point about this element is that it is dual to the Hopf pairing or the quantum Killing form (compare [19, Sec. 4]). Hence, after writing $c=\sum_{i}c(i)\otimes c^{\prime}(i)$ the Hopf pairing is defined by setting (16) $\langle c(i),c^{\prime}(j)\rangle:=\delta_{ij}$ Restricting to the Cartan part this gives us (compare [19, Lemma 3.12]) (17) $\displaystyle D^{-2}=\prod^{N}_{i=1}q^{-H_{i}\otimes H_{i}}=\prod^{N}_{i=1}\sum_{n_{i}}(-1)^{n_{i}}\frac{h^{n_{i}}}{n_{i}!}H_{i}^{n_{i}}\,\otimes\,H_{i}^{n_{i}}$ and hence, $\langle H_{i}^{n},H_{j}^{m}\rangle=\delta_{ij}\delta_{nm}(-1)^{n}\frac{n!}{h^{n}}$. We deduce that $\langle K_{i}^{2},K_{j}^{2}\rangle=q^{-1}$ or, more generally, $\langle K_{i}^{2a},K_{j}^{2b}\rangle=\delta_{ij}q^{-ab}$ defines the Hopf pairing on the $\Gamma$-invariant part of the Cartan. In Section 10 we construct another basis for the Cartan given by $\prod^{N}_{i=1}f_{n_{i}}(K^{2}_{i})$ such that $\langle f_{n},f_{m}\rangle=\delta_{nm}(-1)^{n}q^{-n}(q;q)_{n}$. In this new basis, we can rewrite the Cartan part of the clasp element as follows: (18) $D^{-2}=\sum_{\mathbf{n}\in\mathbb{N}^{N}}\prod^{N}_{i=1}\frac{(-1)^{n_{i}}q^{n_{i}}}{(q;q)_{n_{i}}}f_{n_{i}}(K^{2}_{i})\otimes f_{n_{i}}(K^{2}_{i})$ For $\mathfrak{sl}_{N}$ similar computations will give $(D^{\prime})^{-2}=\sum_{\mathbf{n}\in\mathbb{N}^{N-1}}\prod^{N-1}_{i=1}\frac{(-1)^{n_{i}}q^{n_{i}}}{(q;q)_{n_{i}}}f_{n_{i}}(\mathcal{K}_{i})\otimes f_{n_{i}}(\mathcal{K}^{2}_{i})\ $ (compare Section B.1 in [19]). Let us denote by $\text{Inv}\,(\mathcal{U})=\\{u\in\mathcal{U}\,\|\,x\vartriangleright u=\epsilon(x)u\quad\forall x\in\mathcal{U}\\}$ the invariant part of $\mathcal{U}$ under the adjoint action $x\vartriangleright u:=x_{(1)}uS(x_{(2)})$ in Sweedler notation. The main advantage of the usage of bottom tangles in the definition of $J_{L}(\mathfrak{gl}_{N};q)$ is that in this case $J_{L}(\mathfrak{gl}_{N};q)\in\text{Inv}\,(\mathcal{U})$ (compare [15, Sec.4.3]). As a corollary, we get the following: ###### Proposition 6.3. Given an $l$-component evenly framed link $L$, the universal invariant $J_{L}(\mathfrak{gl}_{N};q)$ is a well defined element of $\text{\rm Inv}\,\left(\widehat{\mathcal{U}}^{\hat{\otimes}l}\right)$. ###### Proof. By definition, $J_{K}$ is obtained by multiplying together elementary pieces, such as $F_{\mathbf{n}}$, $e_{\mathbf{n}}$, $K^{\pm 1}_{2\rho}$, $D^{\pm 1}$, and by then taking a sum over all indices. The linking between different components and framing will make appear powers of $D^{\pm 2}$ that we can decompose using the basis elements $f_{n}(K^{2}_{i})$ of the completion by (18). Note that we can collect all diagonal contributions of each component by using formulas like $D(E_{i}\otimes 1)D^{-1}=E_{i}\otimes\mathcal{K}_{i}\quad\text{and}\quad D(1\otimes F_{j})D^{-1}=\mathcal{K}^{-1}_{j}\otimes F_{j}\ .$ Since framing is assumed to be even, we will have an even number of $D$-parts. Hence using (18) and the explicit form of the quasi $R$-matrix $\Theta$, we get the claim. ∎ ###### Remark 6.4. For $\mathfrak{sl}_{N}$ we can build the same completion after replacing $K_{i}$ with $\mathcal{K}_{i}$. Then the arguments in the proof of Proposition 6.3 will show us that for any algebraically split link the universal invariants belongs to this completion. Proof of Theorem 1.1. Using Proposition 6.3 and remark above, we can define the action of $\Gamma$ on each component of $J_{L}(\mathfrak{g};q)$ separately. We will denote by $J_{L}^{\zeta_{1},\ldots,\zeta_{\ell}}(\mathfrak{g};q)$ the result of this action. Then we have $J_{L}\left(V(\lambda_{1})\otimes L(\zeta_{1}),\ldots,V(\lambda_{\ell})\otimes L(\zeta_{\ell})\right)=$ $\bigotimes^{l}_{i=1}\mathrm{Tr}^{V(\lambda_{i})\otimes L(\zeta_{i})}_{q}\left(J_{L}(\mathfrak{g};q)\right)=\bigotimes^{l}_{i=1}\mathrm{Tr}^{V(\lambda_{i})}_{q}\left(J_{T}^{\zeta_{1},\ldots,\zeta_{\ell}}(\mathfrak{g};q)\right)\cdot\dim_{q}L(\zeta_{1})\cdots\dim_{q}L(\zeta_{\ell}).$ The second equation follows from Lemma 3.1. By Theorem 4.6 we conclude that $J_{L}^{\zeta_{1},\ldots,\zeta_{\ell}}\left(\lambda_{1},\ldots,\lambda_{\ell}\right)=J_{L}\left(\lambda_{1},\ldots,\lambda_{\ell}\right)$ for all $\lambda_{1},\ldots,\lambda_{\ell}$ under the assumptions of Theorem 1.1, therefore $J_{L}(\mathfrak{g};q)=J_{L}^{\zeta_{1},\ldots,\zeta_{\ell}}(\mathfrak{g};q)$ and hence, $J_{L}(\mathfrak{g};q)$ is $\Gamma$-invariant under the same assumptions. $\hfill\Box$ ###### Corollary 6.5. For any $\ell$-component evenly framed link $L$, $J_{L}(\mathfrak{gl}_{N};q)$ belongs to the $\Gamma$-invariant part of $\text{\rm Inv}\left(\widehat{\mathcal{U}}^{\hat{\otimes}\ell}\right)$. Moreover, for every $0$-framed algebraically split link $L$, $J_{L}(\mathfrak{gl}_{N};q)=J_{L}(\mathfrak{sl}_{N};q)\ .$ ###### Proof. The first statement is the direct consequence of Theorem 1.1. The second one follows from the fact that the only difference in the definitions of both invariants is in the diagonal part of the $R$-matrix, that does not contribute since the linking matrix vanishes and the rules for moving of $D$ and $D^{\prime}$ along a component of the link coincide. ∎ ### 6.3. Twist forms Let us denote by $\widehat{\mathcal{Z}}$ the center of $\widehat{\mathcal{U}}$. In what follows, we will be particularly interested in the following twist forms ${\mathcal{T}}_{\pm}:\widehat{\mathcal{Z}}\to\widehat{\mathcal{Z}}\quad\text{ given by}\quad{\mathcal{T}}_{\pm}(z):=\langle r^{\pm 1},z\rangle$ the Hopf pairing with the ribbon element. On the $\Gamma$-invariant Cartan part they are easy to compute, given the Hopf pairing between the generators $H_{i}$ in Section 6.2 . We have (19) ${\mathcal{T}}_{\pm}(K_{2\mathbf{a}})=\langle r^{\pm 1}_{0},K_{2\mathbf{a}}\rangle=v^{\pm(\mathbf{a},2\rho-\mathbf{a})}\in\mathbb{Z}[v,v^{-1}]\ $ for any $\mathbf{a}\in\mathbb{Z}^{N}$. Now equation (16) allows to extend the twists form to $\widehat{\mathcal{U}}^{\text{ev}}$ as follows: ${\mathcal{T}}_{\pm}(F_{\mathbf{m}}\mathcal{K}_{\mathbf{m}}K_{2\mathbf{a}}e_{\mathbf{n}})=\delta_{\mathbf{m},\mathbf{n}}q^{(\rho,\sum_{i}n_{i}\alpha_{i})}v^{\pm(\mathbf{a},2\rho-\mathbf{a})}\in\mathbb{Z}[v,v^{-1}]\ $ where $\alpha_{i}=e_{i}-e_{i+1}$ are the simple roots. Observe that after restriction to $\mathcal{U}_{q}^{\text{ev}}\mathfrak{sl}_{N}$, i.e. replacing $K_{2\mathbf{a}}$ with $\mathcal{K}_{2\mathbf{b}}$ in the above formula, the result belong to $\mathbb{Z}[q,q^{-1}]$ and coincide with [19, eq. (102)] for any $\mathbf{b}\in\mathbb{Z}^{N-1}$. ## 7\. Habiro’s basis for $\mathcal{Z}(U_{q}\mathfrak{sl}_{2})$ In this section we summarize Habiro’s results for $\mathfrak{sl}_{2}$ in the way suitable for our generalization. Habiro [15] defined a remarkable family of central elements in $\mathcal{Z}(U_{q}\mathfrak{sl}_{2})$: (20) $\sigma_{m}:=\prod^{m}_{i=1}\left(C^{2}-(v^{i}+v^{-i})^{2}\right)=\prod_{i=1}^{m}(C-v^{i}-v^{-i})(C+v^{i}+v^{-i}).$ Since $C$ acts on the $(j+1)$-dimensional representation $V_{j}$ by a scalar $v^{j+1}+v^{-j-1}$, the polynomial $\sigma_{m}$ is completely characterized by the following properties: * (a) (Parity) $\sigma_{m}$ is $\Gamma=\mathbb{Z}_{2}$-invariant. * (b) (Vanishing) $\sigma_{m}$ annihilates the representations $V_{j}$ for $j<m$. * (c) (Normalization) $\sigma_{m}$ acts on the representation $V_{m}$ by a scalar (21) $\prod^{m}_{i=1}\left((v^{m+1}+v^{-m-1})^{2}-(v^{i}+v^{-i})^{2}\right)\ .$ Note that parity implies that $\sigma_{m}$ also annihilates the representations $L(-1)\otimes V_{j}$ for $j<m$. By using the Harish-Chandra isomorphism, we can alternatively consider the polynomials $T_{m}(y):=\mathit{hc}(\sigma_{m}):=\prod^{m}_{i=1}(yv^{i}-y^{-1}v^{-i})(yv^{-i}-y^{-1}v^{i})=(-1)^{m}\prod^{m}_{i=1}q^{-i}(1-y^{2}q^{i})(1-y^{-2}q^{i})$ which are characterized by the following properties: * (a) (Parity) $T_{m}$ is $\mathbb{Z}_{2}$-invariant, that is, $T_{m}(-y)=T_{m}(y)$ * (b) (Vanishing) $T_{m}(\pm v^{j+1})=0$ for $j<m$ * (c) (Normalization) $T_{m}(v^{m+1})$ is given in (21). Habiro proved that $\\{\sigma_{m}\\}_{m\geq 0}$ form a basis in (a certain completion of) the $\Gamma$-invariant part of the center. Hence, the elements $S_{m}=\xi^{-1}(\sigma_{m})$, given by $S_{m}:=\prod_{i=1}^{m}(V_{1}-v^{i}-v^{-i})(V_{1}+v^{i}+v^{-i})$ form a basis of $\mathcal{R}$. We will show that $P_{n}=\prod_{i=0}^{n-1}(V_{1}-v^{2i+1}-v^{-2i-1})\in\mathcal{R}$ is a dual basis to $\\{S_{m}\\}_{m\geq 0}$ with respect to the Hopf pairing. The following is a slight reformulation of [15, Prop. 6.3]. ###### Lemma 7.1. We have $\langle P_{n},S_{m}\rangle=\frac{\\{2n+1\\}!}{\\{1\\}}\delta_{n,m}\ .$ ###### Proof. Clearly, one has $\xi(P_{n})=\prod_{i=0}^{n-1}(C-v^{2i+1}-v^{-2i-1})$ which annihilates $V_{2i}$ for $i<n$. We have the following cases: 1) For $n<m$ we have $\langle P_{n},S_{m}\rangle=\mathrm{Tr}_{q}^{P_{n}}(\sigma_{m})$. Since $P_{n}$ is in span of $V_{i}$ for $i\leq n$ and $\sigma_{m}$ annihilates all these, we get $\langle P_{n},S_{m}\rangle=0$. 2) For $m<n$ we have $\langle P_{n},S_{m}\rangle=\mathrm{Tr}_{q}^{S_{m}}(\xi(P_{n}))$. Since $S_{m}$ is in span of $V_{2i}$ for $i\leq n$ and $\langle P_{n},V_{2i}\rangle=\\{i+n\\}\dots\\{i-n+1\\}[2i+1]\ .$ Hence $P_{n}$ annihilates all these, we get $\langle P_{n},S_{m}\rangle=0$. 3) Finally, for $n=m$ we observe that $P_{n}$ has a unique copy of $V_{n}$ and $\langle P_{n},S_{n}\rangle=\langle V_{n},S_{n}\rangle=\mathrm{Tr}_{q}^{V_{n}}(\sigma_{n})$ which is easy to compute. ∎ We can use the above results to compute the coefficients in the decomposition of any central element into $\\{\sigma_{m}\\}_{m\geq 0}$. ###### Lemma 7.2. Let $\phi$ be a $\mathbb{Z}_{2}$-invariant element in $\mathcal{Z}(U_{q}\mathfrak{sl}_{2})$ which acts on $V_{j}$ by a scalar $\phi_{j}$. Then $\phi=\sum a_{n}\sigma_{n},\;\text{where}\;\;a_{n}=\sum_{i=0}^{n}(-1)^{n-i}\frac{\\{2i+2\\}\\{i+1\\}}{\\{n+i+2\\}!\\{n-i\\}!}\;\phi_{i}\ .$ ###### Proof. We have ([15, Lemma 6.1]) $P_{n}=\sum_{i=0}^{n}(-1)^{n-i}\frac{[2i+2]}{[n+i+2]}\left[\begin{array}[]{cc}\\!2n+1\\!\\\ \\!n+1+i\\!\end{array}\right]V_{i}.$ If $\phi=\sum a_{m}\sigma_{m}$ then $a_{n}=\frac{\\{1\\}}{\\{2n+1\\}!}\mathrm{Tr}_{q}^{P_{n}}(\phi)=\frac{\\{1\\}}{\\{2n+1\\}!}\sum_{i=0}^{n}(-1)^{n-i}\frac{[2i+2]}{[n+i+2]}\left[\begin{array}[]{cc}\\!2n+1\\!\\\ \\!n+1+i\\!\end{array}\right]\mathrm{Tr}_{q}^{V_{i}}(\phi)=$ $\sum_{i=0}^{n}(-1)^{n-i}\frac{\\{2i+2\\}\\{1\\}}{\\{n+i+2\\}!\\{n-i\\}!}\dim_{q}(V_{i})\phi_{i}.$ Using $\dim_{q}(V_{i})=[i+1]$ we obtain the result. ∎ Habiro proved that for any $0$-framed knot $K$, there exist $a_{n}(K)\in\mathbb{Z}[q,q^{-1}]$ such that $J_{K}(\mathfrak{sl}_{2};q)=\sum_{n\geq 0}a_{n}(K)\,\sigma_{n}$ known as a cyclotomic expansion of the colored Jones polynomial of the knot $K$. ## 8\. New basis for the center of $\widehat{\mathcal{U}}$ Recall that $\widehat{\mathcal{Z}}$ is the center of the completion $\widehat{\mathcal{U}}$. In this section we construct the basis $\\{\sigma_{\lambda}\\}_{\lambda}$ of the $\Gamma$-invariant part of $\widehat{\mathcal{Z}}$. Furthermore, we explicitly define its dual $\\{P_{\lambda}\\}_{\lambda}$ with respect to the Hopf pairing. This allows us to construct the cyclotomic expansion of $J_{K}(\mathfrak{gl}_{N};q)$ for any $0$-framed knot $K$. The proof uses the existence and properties of interpolation Macdonald polynomials [29] which are summarized in the following theorem. ###### Theorem 8.1. There is a family of symmetric polynomials $F_{\lambda}(x_{1},\ldots,x_{N};q)$ such that: * (a) $F_{\lambda}$ is in the span of Schur functions $s_{\mu}$ for $\mu\leq\lambda$ with the leading term $F_{\lambda}=(-1)^{\|\lambda\|+\binom{N}{2}}q^{D_{N}(\lambda)}s_{\lambda}+\ldots\ .$ * (b) $F_{\lambda}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}})=0$ unless $\mu$ contains $\lambda$. * (c) $F_{\lambda}(q^{-\lambda_{1}-N+1},\ldots,q^{-\lambda_{N}})=(-1)^{\binom{N}{2}}q^{n(\lambda)+\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-h(\square)})$. * (d) Any function $F$ in the completion can be written as (22) $F(x_{1},\ldots,x_{N})=\sum_{\lambda,\;\mu\subset\lambda}d_{\mu,\lambda}(q)F(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}})F_{\lambda}(x_{1},\ldots,x_{N};q)$ where $d_{\lambda,\mu}$ are explicit coefficients prescribed by Theorem 10.17. We discuss the definition and give more details on interpolation Macdonald polynomials in Section 10. ###### Theorem 8.2. There exists a family of central elements $\sigma_{\lambda}\in\mathcal{Z}$ with the following properties: * (a) $\sigma_{\lambda}$ is $\Gamma$-invariant and annihilates $L(\zeta)\otimes V(\mu)$ for all $\mu$ not containing $\lambda$ and $\zeta\in\Gamma$. * (b) $\mathit{hc}(\sigma_{\lambda})$ is in the span of $s_{\mu}(x_{1},\ldots,x_{N})$ for $\mu\leq\lambda$, with the leading term $\mathit{hc}(\sigma_{\lambda})=(-1)^{\|\lambda\|+\binom{N}{2}}v^{(N-1)\|\lambda\|}q^{D_{N}(\lambda)}s_{\lambda}+\ldots\ .$ * (c) $\sigma_{\lambda}$ acts on $V(\lambda)$ by a scalar $\sigma_{\lambda}\|_{V(\lambda)}=(-1)^{\binom{N}{2}}q^{-n(\lambda)-\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{h(\square)})\ .$ ###### Proof. Define $\sigma_{\lambda}=\mathit{hc}^{-1}(g_{\lambda})$, where $g_{\lambda}(x_{1},\dots,x_{N})=F_{\lambda}(v^{N-1}x_{1},\ldots,v^{N-1}x_{N};q^{-1})$. Then $\sigma_{\lambda}$ is clearly $\Gamma$-invariant and $\sigma_{\lambda}\|_{L(\zeta)\otimes V({\mu})}=g_{\lambda}(\zeta_{i}\cdot v^{\mu_{i}+\rho_{i}})=F_{\lambda}(q^{(\mu_{1}+N-1)},\ldots,q^{\mu_{N}};q^{-1}).$ Indeed, if $y_{i}=\zeta_{i}\cdot v^{\mu_{i}+\rho_{i}}=\zeta_{i}v^{(\mu_{i}-\frac{N-1}{2}+N-i)}$ then $v^{N-1}y_{i}^{2}=q^{(\mu_{i}+N-i)}$. Now $F_{\lambda}(q^{(\mu_{1}+N-1)},\ldots,q^{\mu_{N}};q^{-1})$ vanishes unless $\mu$ contains $\lambda$, and has the nonzero value prescribed by the previous theorem for $\mu=\lambda$. ∎ Let us define $\mathcal{R}_{\mathbb{Q}}:=\mathcal{R}\otimes\mathbb{Q}(v)$ by extending the coefficient ring $\mathbb{Z}[v^{\pm 1}]$ of $\mathcal{R}$ to the rational functions in $v$. ###### Theorem 8.3. Define the following formal elements of $\mathcal{R}_{\mathbb{Q}}$ $P_{\lambda}=\sum_{\mu\subset\lambda}\frac{d_{\lambda,\mu}(q^{-1})}{\dim_{q}V(\mu)}\;V(\mu)\in\mathcal{R}_{\mathbb{Q}},$ then one has (23) $\langle P_{\lambda},\sigma_{\nu}\rangle:=\mathrm{Tr}^{P_{\lambda}}_{q}(\sigma_{\nu})=\delta_{\lambda,\nu}\ .$ ###### Proof. First, let us write the interpolation formula (22) for $F=F_{\nu}$: $F_{\nu}(x_{1},\ldots,x_{N};q)=\sum_{\mu\subset\lambda}d_{\lambda,\mu}(q)F_{\nu}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}};q)F_{\lambda}(x_{1},\ldots,x_{N};q),$ so $\sum_{\mu\subset\lambda}d_{\lambda,\mu}(q)F_{\nu}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}};q)=\delta_{\lambda,\nu}.$ By changing $q$ to $q^{-1}$ we get (24) $\sum_{\mu\subset\lambda}d_{\lambda,\mu}(q^{-1})F_{\nu}(q^{\mu_{1}+N-1},\ldots,q^{\mu_{N}};q^{-1})=\delta_{\lambda,\nu}.$ Now $\mathrm{Tr}^{V(\mu)}_{q}(\sigma_{\nu})=\dim_{q}(V(\mu))\;g_{\nu}(q^{\mu_{1}+N-1},\ldots,q^{\mu_{N}})$, hence $\mathrm{Tr}^{P_{\lambda}}_{q}(\sigma_{\nu})=\sum_{\mu\subset\lambda}\frac{d_{\lambda,\mu}(q^{-1})}{\dim_{q}V({\mu})}\mathrm{Tr}^{V({\mu})}_{q}(\sigma_{\nu})=\;\delta_{\lambda,\nu}\ .$ ∎ Next, we would like to study the integrality properties of the universal knot invariant. ###### Lemma 8.4. (a) Let $\sigma\in\mathcal{U}^{\text{ev}}_{\mathbb{Z}}$. Then $\sigma=(K_{1}\cdots K_{N})^{-2s}\sum a_{\lambda}\sigma_{\lambda}$ with $a_{\lambda}\in\mathbb{Z}[q,q^{-1}]$. (b) Given $k$ and $m$, there exists $n=n(k,m)$ such that for all $\Gamma$-invariant central elements $\sigma$ in the ideal $\mathcal{U}^{(n)}_{\mathbb{Z}}$ the coefficients $a_{\lambda}$ are divisible by $(q;q)_{m}$ for $\|\lambda\|\leq k$. ###### Proof. (a) Recall that Harish-Chandra transform $\mathit{hc}$ identifies the $\Gamma$-invariant part of the center of $\mathcal{U}_{\mathbb{Z}}$ with the space of symmetric functions in $x_{1},\ldots,x_{N}$ with coefficients in $\mathbb{Z}[q,q^{-1}]$. Since $F_{\lambda}$ is a polynomial with top degree part equal to the Schur polynomial (up to a monomial in $q$), we can write $(x_{1}\cdots x_{N})^{s}f(x_{1},\ldots,x_{N})=\sum_{\lambda}a_{\lambda}F_{\lambda}(x_{1},\ldots,x_{N};q^{-1})$ and the result follows. (b) If $\sigma$ is in the ideal $\mathcal{U}^{(n)}_{\mathbb{Z}}$ for sufficiently large $n$, then by Proposition 6.2 its matrix elements in the integral basis of $V(\lambda)$ are divisible by $(q;q)_{m}$. By definition of Harish-Chandra transform, this implies that the values $f(q^{-\lambda_{1}-N+1},\ldots,q^{-\lambda_{N}})$ are divisible by $(q;q)_{m}$ and hence by the interpolation formula (22) the coefficients $a_{\lambda}$ are divisible by $(q;q)_{m}$ as well. ∎ ###### Corollary 8.5. The center of the completion $\widehat{\mathcal{U}}$ is isomorphic to the completion of the space of symmetric polynomials with coefficients in $\widehat{\mathbb{Z}[v]}$ with respect to the basis $F_{\lambda}$. ###### Proof. By Lemma 8.4 any element of the center of $\widehat{\mathcal{U}}$ can be written as an infinite series $\sum a_{\lambda}F_{\lambda}$ with coefficients in $\widehat{\mathbb{Z}[v]}$, up to a factor $(x_{1}\cdots x_{N})^{-s}$. By Corollary 11.10 the multiplication by $(x_{1}\cdots x_{N})^{-s}$ preserves the space of such series. ∎ ###### Corollary 8.6. Any $\sigma\in\widehat{\mathcal{U}^{\text{ev}}}$ can be written as an infinite sum $\sigma=\sum a_{\lambda}\sigma_{\lambda}$ with coefficients $a_{\lambda}=\mathrm{Tr}^{P_{\mu}}_{q}(\sigma)\in\widehat{\mathbb{Z}[q]}$. ###### Proposition 8.7. The universal knot invariant admits an expansion $J_{K}(\mathfrak{gl}_{N};q)=\sum_{\lambda}a_{\lambda}(K)\sigma_{\lambda}\quad\text{with}\quad a_{\lambda}(K)=\sum_{\mu\subset\lambda}{d_{\lambda,\mu}(q^{-1})}\,J_{K}(V(\mu),q)\in\mathbb{Z}[q,q^{-1}]\ $ called a cyclotomic expansion of the universal $\mathfrak{gl}_{N}$ knot invariant. Proposition 8.7 implies Theorem 1.3 in Introduction. Note that the knot invariant $J_{K}(V(\mu),q)$ is normalized to be 1 for the unknot. ###### Proof. By Corollary 6.5, $J_{K}(\mathfrak{gl}_{N};q)$ is a central element in $\widehat{\mathcal{U}^{\text{ev}}}$, so it can be written as $\sigma=\sum a_{\lambda}\sigma_{\lambda}$ with coefficients $a_{\lambda}\in\widehat{\mathbb{Z}[q]}$. On the other hand, the value of $J_{K}$ on any representation $V_{\lambda}$ is in $\mathbb{Z}[q,q^{-1}]$, so by the interpolation formula (22) the coefficients $a_{\lambda}$ can be written as rational functions with numerators in $\mathbb{Z}[q,q^{-1}]$ and cyclotomic denominators. By Proposition 12.1 this implies that $a_{\lambda}\in\mathbb{Z}[q,q^{-1}]$. The explicit formula for $a_{\lambda}$ is obtained by taking Hopf pairing with $P_{\mu}$ and observing that $\mathrm{Tr}^{V({\mu})}_{q}\left(J_{K}(\mathfrak{gl}_{N};q)\right)=\dim_{q}(V(\mu))J_{K}(V(\mu),q)$ according to our normalization. ∎ The last result shows that $a_{\lambda}(K)=\mathrm{Tr}_{q}^{P_{\lambda}}(J_{K}(\mathfrak{gl}_{N};q))\in\mathbb{Z}[q^{\pm 1}]$, even through the coefficients $d_{\lambda,\mu}(q)$ are rational functions in $q$ (compare Example 10.23). ## 9\. Unified invariants of integral homology 3-spheres This section is devoted to our main application of the previous results — a construction of the unified invariants for integral homology 3-spheres. We start with few auxiliary results. Let us denote by $P^{\prime}_{\lambda}=v^{-\|\lambda\|}\dim_{q}V(\lambda)\sum_{\mu\subset\lambda}\frac{d_{\lambda,\mu}(q^{-1})}{\dim_{q}V(\mu)}\;V(\mu)\in\mathcal{R}_{\mathbb{Q}}$ and define (25) $\omega_{\pm}=\sum_{\lambda}(-1)^{\|\lambda\|+\binom{N}{2}}q^{\mp c(\lambda)}q^{w_{\pm}(\lambda)}P^{\prime}_{\lambda}\in\widehat{\mathcal{R}}_{\mathbb{Q}}\quad\text{with}\quad\begin{array}[]{ll}w_{+}(\lambda)&=D_{N}(\lambda)\\\ w_{-}(\lambda)&=D_{N}(\lambda)+N\|\lambda\|\end{array}$ where $c(\lambda)$ is the content of $\lambda$. The next Lemma implies that $\omega_{\pm}$ is the universal Kirby color for $(\pm 1)$-surgery. ###### Lemma 9.1. For any $x\in\widehat{\mathcal{R}}_{\mathbb{Q}}$, we have (26) $\langle\omega_{\pm},x\rangle=J_{U_{\mp}}(x)=\langle r^{\pm 1},\xi(x)\rangle$ where $J_{U_{\pm}}(x)$ is the Reshetikhin–Turaev invariant of the $(\pm 1)$-framed unknot colored by $x$. ###### Proof. It is enough to check (26) for the basis elements $x=V(\nu)$. We compute $\displaystyle\langle P^{\prime}_{\lambda},V(\nu)\rangle$ $\displaystyle=v^{-\|\lambda\|}\dim_{q}V(\lambda)\sum_{\mu\subset\lambda}\frac{{d_{\lambda,\mu}(q^{-1})}}{\dim_{q}V(\mu)}\;\langle V(\mu),V(\nu)\rangle=v^{-\|\lambda\|}\dim_{q}V(\lambda)\sum_{\mu\subset\lambda}{d_{\lambda,\mu}(q^{-1})}s_{\nu}(q^{\mu_{i}+N-i})$ $\displaystyle=\dim_{q}V(\lambda)\sum_{\mu\subset\lambda}{d_{\lambda,\mu}(q^{-1})}C_{\nu}F_{\nu}(q^{\mu_{i}+N-i})={C_{\lambda}}\delta_{\lambda,\nu}\dim_{q}V(\lambda)$ where we used Lemma 5.3, equation (24) and the expansion $s_{\nu}=(-1)^{\|\lambda\|+\binom{N}{2}}q^{-D_{N}(\lambda)}v^{(1-N)\|\lambda\|}F_{\nu}+\text{lower terms}$ and hence, $C_{\lambda}=(-1)^{\|\lambda\|+\binom{N}{2}}q^{-D_{N}(\lambda)}v^{-N\|\lambda\|}\ .$ Using this computation it is easy to check that (27) $\langle\omega_{\pm},V(\nu)\rangle=v^{\mp N\|\nu\|}q^{\mp c(\nu)}\dim_{q}V(\nu)=v^{\mp(\nu,\nu+2\rho)}\dim_{q}V(\nu)=\mathrm{Tr}^{V(\nu)}_{q}(r^{\pm 1})$ is equal to $J_{U_{\mp}}(V(\nu))$. ∎ From the following computation for $V^{\prime}(\nu)=\frac{V(\nu)}{\dim_{q}(V(\nu))}$ $\langle\omega_{+}\omega_{-},V^{\prime}(\nu)\rangle=\langle\omega_{+},V^{\prime}(\nu)\rangle\langle\omega_{-},V^{\prime}(\nu)\rangle=v^{(\nu,\nu+2\rho)}v^{-(\nu,\nu+2\rho)}=\langle 1,V^{\prime}(\nu)\rangle$ we see that $\omega_{+}$ and $\omega_{-}$ are inverse to each other in the algebra $\widehat{\mathcal{R}}_{\mathbb{Q}}$ isomorphic to $\widehat{\mathcal{Z}}_{\mathbb{Q}}$. A direct consequence of the above Lemma and the fusion rules is the following result. ###### Theorem 9.2. Let $L\cup K=L_{1}\cup L_{2}\dots\cup L_{l}\cup K$ be an $(l+1)$ component algebraically split $0$-framed link such that $K$ is the unknot. We denote by $L_{(K,\pm 1)}$ the framed link in $S^{3}$ obtained from $L$ by $\pm 1$-surgery along $K$, then for any $p_{1},\dots,p_{l}\in\mathcal{R}$ $J_{L\cup K}(p_{1},\dots,p_{l},\omega^{\pm 1})=J_{L_{(K,\pm 1)}}(p_{1},\dots,p_{l}).$ ###### Proof. The proof is given in [15, Thm. 9.4]. ∎ ### 9.1. Construction of the unified invariants Without loss of generality, we can assume that an integral homology 3-sphere $M$ is obtained by $\mathbf{\varepsilon}$-surgery on an $\ell$ component algebraically split link $L$, where $\mathbf{\varepsilon}\in\\{\pm 1\\}^{\ell}$. For $\mathfrak{sl}_{N}$ Habiro and Le defined a unified invariant of $M$ as follows $I^{\text{HL}}(M):=\mathcal{T}^{\prime}_{\mathbf{\varepsilon}}(J_{L}(\mathfrak{sl}_{N};q))\;\in\;\widehat{\mathbb{Z}[q]}\quad\text{ where}\quad\mathcal{T}^{\prime}_{\varepsilon}=\bigotimes^{\ell}_{i=1}\mathcal{T}^{\prime}_{\varepsilon_{i}}\ $ is the $\mathfrak{sl}_{N}$ twist form. They also proved that $I^{\text{HL}}(M)$ belongs to the Habiro ring [19]. For $\mathfrak{gl}_{N}$ we define the unified invariant of $M$ similarly $I(M):=\mathcal{T}_{\mathbf{\varepsilon}}(J_{L}(\mathfrak{gl}_{N};q))\;\in\;\widehat{\mathbb{Z}[q]}\quad\text{ where}\quad\mathcal{T}_{\varepsilon}=\bigotimes^{\ell}_{i=1}\mathcal{T}_{\varepsilon_{i}}\ $ by using the $\mathfrak{gl}_{N}$ twist forms. ###### Theorem 9.3. For any integral homology $3$-sphere $M$, $I(M)=J_{L}(\omega_{\varepsilon_{1}},\dots,\omega_{\varepsilon_{l}})\in\;\widehat{\mathbb{Z}[q]}$ Moreover, its evaluation at any root of unity coincides with the $\mathfrak{sl}_{N}$ Witten–Reshetikhin–Turaev invariant of $M$. This implies Theorem 1.4 from Introduction. ###### Proof. By Corollary 6.5, for any algebraically split $0$-framed link $L$ we have $J_{L}(\mathfrak{gl}_{N};q)=J_{L}(\mathfrak{sl}_{N};q).$ Hence, as explained in Section 6.3, the $\mathfrak{gl}_{N}$ and $\mathfrak{sl}_{N}$ twist forms on $J_{L}$ do coincide. This implies $I(M)=I^{\text{HL}}(M)$. Since the Habiro–Le invariant is known to belong to the Habiro ring and to evaluate at a root of unity to the Witten-Reshetikhin- Turaev (WRT) one, it remain to show $I(M)=J_{L}(\omega_{\epsilon_{1}},\dots,\omega_{\epsilon_{\ell}})$. We prove this claim in two steps. Step 1: Assume $\ell=1$, then $J_{L}(\omega_{\pm})=I(M)$ by Lemma 9.1. Step 2: For any $x=x_{1}\otimes\dots\otimes x_{\ell}\in\text{Inv}(\widehat{\mathcal{U}}_{\mathbb{Z}}^{\hat{\otimes}\ell})$ we define $a_{k}$ for $k=0,1,\dots,\ell$ and $b_{k}=1,\dots,\ell$ as follows: $a_{k}=\prod^{k}_{i=1}\langle r^{\epsilon_{i}},x_{i}\rangle\prod^{\ell}_{j=k+1}\langle\omega_{\epsilon_{j}},x_{j}\rangle,\quad b_{k}=x_{k}\prod^{k-1}_{i=1}\langle r^{\epsilon_{i}},x_{i}\rangle\prod^{\ell}_{j=k+1}\langle\omega_{\epsilon_{j}},x_{j}\rangle.$ Then $a_{k-1}=\langle\omega_{\epsilon_{k}},b_{k}\rangle\quad\text{and}\quad a_{k}=\langle r^{\epsilon_{k}},b_{k}\rangle.$ where we identify $\omega_{\pm}$ with their image under $\xi$ for simplicity. Since $b_{k}\in\mathcal{Z}_{\mathbb{Q}}$, we have $a_{k}=a_{k-1}$ by Step 1 for $k=1,2,\dots,\ell$. Hence, we have $a_{0}=a_{\ell}$ which is our claim. ∎ Theorem 9.3 has striking consequences. Indeed, for any $\lambda\in\mathcal{R}$ let us denote $\mathcal{P}_{\lambda}=\text{\rm Span}_{\mathbb{Z}[q^{\pm 1}]}\\{P^{\prime}_{\mu}\,\|\,\lambda\subset\mu\\},\quad\mathcal{P}=\mathcal{P}_{\emptyset}\quad\text{and}\quad\widehat{\mathcal{P}}:={\lim\limits_{\overleftarrow{\hskip 5.69054pt\lambda\hskip 5.69054pt}}}\;\;\frac{\mathcal{P}}{\mathcal{P}_{\lambda}}.$ For any framed link $L$, the Reshetikhin–Turaev functor provides a $\mathbb{Q}(v)$-multilinear map $J_{L}:\mathcal{R}_{\mathbb{Q}}\times\dots\times\mathcal{R}_{\mathbb{Q}}\to\mathbb{Q}(v)$. For any algebraically split $0$-framed link $L$, Theorem 9.3 implies that its restriction to $\mathcal{P}$ provides a $\mathbb{Z}[q^{\pm 1}]$-multilinear map $J_{L}:\mathcal{P}\times\dots\times\mathcal{P}\to\mathbb{Z}[q,q^{-1}]$ inducing $J_{L}:\widehat{\mathcal{P}}\times\dots\times\widehat{\mathcal{P}}\to\widehat{\mathbb{Z}[q]}\ .$ This leads to a generalization of the famous integrability theorem in [15, Thm. 8.2]. ###### Corollary 9.4. Given an $\ell$ component algebraically split $0$-framed link $L$, then for all but finitely many partitions $\lambda_{i}$ with $1\leq i\leq\ell$, there exist positive integers $n=n(\lambda_{i},N)$, such that $J_{L}(P^{\prime}_{\lambda_{1}},\dots,P^{\prime}_{\lambda_{\ell}})\in(q;q)_{n}\mathbb{Z}[q,q^{-1}]\ .$ It would be interesting to have a direct proof of Corollary 9.4 without using the theory of unified invariants. Based on Corollary 9.4 we can give a cyclotomic expansion of the Reshetikhin–Turaev invariant of $L$ as follows: (28) $J_{L}(\lambda_{1},\dots,\lambda_{\ell})=v^{\sum_{i}\|\lambda_{i}\|}\sum_{\mu_{i}\subset\lambda_{i}}\prod^{\ell}_{j=1}c_{\lambda_{j},\mu_{j}}(q^{-1})\,J_{L}(P^{\prime}_{\mu_{1}},\dots,P^{\prime}_{\mu_{\ell}})$ where the matrix $\left[c_{\lambda,\mu}(q)\right]_{\lambda,\mu}:=\left[F_{\lambda}(q^{-\mu_{i}-N+i})\right]_{\lambda,\mu}$ is the inverse of $\left[d_{\lambda,\mu}(q)\right]_{\lambda,\mu}$ by Theorem 10.17 below. This generalizes equation $(8.2)$ in [15]. ### 9.2. Few direct arguments Our proof of the fact that $I(M)$ belongs to the Habiro ring is based on the result that $I^{\text{HL}}(M)\in\widehat{\mathbb{Z}[q]}$ proven in [19] on more than 100 pages. Given the complexity of their argument, we decided to collect here different facts that can be shown without reference to [19]. ###### Theorem 9.5. Assume $M_{\pm}$ is obtained by $(\pm 1)$-surgery on the knot $K$, then $I(M_{\pm})=J_{K}(\omega_{\pm})\;\in\;\widehat{\mathbb{Z}[v]}$ belongs to the Habiro ring. ###### Proof. By Theorem 1.3 we know $J_{K}(\mathfrak{gl}_{N};q)=\sum_{\mu}a_{\mu}(K)\sigma_{\mu}\quad\text{ with}\quad a_{\mu}(K)\in\mathbb{Z}[q,q^{-1}]$ The fact that $I(M_{\pm})$ belongs to the Habiro ring easily follows from the claim that $\mathcal{T}_{\pm}(\sigma_{\mu})$ is divisible by $(v;v)_{m}$ for some $m$ depending on $\mu$ and $N$. Let us prove this claim. By (19) the Hopf pairing with $r^{\pm 1}$ replaces an element $x^{k_{i}}_{i}$ with $v^{Q_{\pm}(k_{i})}$ where $Q_{\pm}$ is a quadratic form. By Lemma 10.27 we can rewrite $\sigma_{\mu}$ as a linear combination of $\prod^{d}_{i=1}f_{n_{i}}(q^{s_{i}}x_{i})$ such that $\sum_{i}n_{i}=\|\mu\|$, $d=N(N+1)/2$ and $s_{i}\in\mathbb{Z}$. Moreover, each $f_{n}(q^{a}x_{i})$ is divisible by $f_{n}(v^{a}y_{i})$ where $y_{i}^{2}=x_{i}$ and hence belongs to the ideal $I_{n}$ of $\mathbb{Z}[v^{\pm 1},y^{\pm 1}_{i}]$ characterized in Proposition 2.1 of [3]. The result follows now from [3, Theorem 2.2]. The number $m$ we are looking for is $\left\lfloor\frac{\|\mu\|}{N(N+1)}\right\rfloor$. ∎ Combining previous results we obtain an explicit expression for the unified invariant for knot surgeries: (29) $I_{M_{\pm}}=J_{K}(\omega_{\pm})=\mathrm{Tr}_{q}^{\omega_{\pm}}J_{K}(\mathfrak{gl}_{N};q)=\sum_{\lambda}(-1)^{\|\lambda\|+\binom{N}{2}}q^{\mp c(\lambda)}q^{w_{\pm}(\lambda)}\,J_{K}(P^{\prime}_{\lambda})$ Assuming that $I(M)=J_{L}(\omega_{\varepsilon_{1}},\dots,\omega_{\varepsilon_{l}})$ is well defined, its topological invariance can be shown directly as follows. Since $I(M)$ depends only on the isotopy class of $L$, it remains to check its invariance under Hoste moves (a version of Fenn-Rourke moves between algebraically split links). Without loss of generality, we can assume that the last component is an unknot, then the statement follows from Theorem 9.2. Assuming $I(M)$ belongs to the Habiro ring, and as such has well defined evaluations at roots of unity [17, Thm. 6.3] we can use the same trick as above to show that for any root of unity $\zeta$ $\text{ev}_{\zeta}I(M)=\text{WRT}(M,\zeta).$ Let us recall that the WRT invariant is obtained from $J_{L}(\mathfrak{sl}_{N};q)$ by taking trace along each component with the Kirby color $\Omega_{\pm}=\frac{\sum_{\lambda}v^{\pm(\lambda,\lambda+2\rho)}\dim_{q}V(\lambda)}{\sum_{\mu}v^{\pm(\mu,2\rho+\mu)}\dim^{2}_{q}V(\mu)}\,V(\lambda)$ where the sums are taken over all $\lambda,\mu\in\mathcal{R}^{\text{fin}}=\\{\lambda\|\dim_{\zeta}V(\lambda)\neq 0\\}$ and $v^{2}$ is evaluated to $\zeta$. Hence, we need to show that for any $x$ in the ad-invariant part of the completed $\ell$th tensor power of $\widehat{\mathcal{U}}_{\mathbb{Z}}$, we have $\mathrm{Tr}_{q}^{\Omega_{\varepsilon}}(x)\stackrel{{\scriptstyle\zeta}}{{=}}\mathrm{Tr}_{q}^{\omega_{\varepsilon}}(x)\quad\quad\forall x\in\text{Inv}(\widehat{\mathcal{U}}_{\mathbb{Z}}^{\hat{\otimes}\ell})$ where $\stackrel{{\scriptstyle\zeta}}{{=}}$ means the equality after evaluation $v^{2}=\zeta$. We will prove this fact in two steps. Step 1: Assume $\ell=1$, in this case $\text{Inv}\,\widehat{\mathcal{U}}_{\mathbb{Z}}=\widehat{\mathcal{Z}}$ with basis given by $z_{\lambda}=\xi(V(\lambda))$. Since $\Omega_{\pm}$ is invariant under Hoste moves, we have $\mathrm{Tr}_{q}^{\Omega_{\pm}}\left(z_{\nu}\right)=\langle\Omega_{\pm},V(\nu)\rangle=\text{ev}_{\zeta}\left(v^{\mp(\nu,\nu+2\rho)}\dim_{q}V(\nu)\right)$ where we interpret the left hand side as a Hopf link with components colored by $\Omega_{\pm}$ and $V(\nu)$, and the right hand side is the result of the sliding. Comparing this computation with (27), we deduce that at roots of unity the actions of $\Omega_{\pm}$ and $\omega_{\pm}$ do coincide on $\xi(\mathcal{R}^{\text{fin}})$, and they vanish on $x\in\widehat{\mathcal{Z}}\setminus\xi(\mathcal{R}^{\text{fin}})$ after evaluation. Step 2: Define $a_{k}$ for $k=0,1,\dots,\ell$ and $b_{k}=1,\dots,\ell$ as follows: $a_{k}=\bigotimes^{k}_{j=1}\mathrm{Tr}_{q}^{\Omega_{\varepsilon_{j}}}\otimes\bigotimes^{\ell}_{j=k+1}\mathrm{Tr}_{q}^{\omega_{\varepsilon_{j}}}(x),\quad b_{k}=\bigotimes^{k-1}_{j=1}\mathrm{Tr}_{q}^{\Omega_{\varepsilon_{j}}}\otimes 1\otimes\bigotimes^{\ell}_{j=k+1}\mathrm{Tr}_{q}^{\omega_{\varepsilon_{j}}}(x).$ Then $a_{k-1}=\mathrm{Tr}_{q}^{\omega_{\varepsilon_{k}}}(b_{k})\quad\text{and}\quad a_{k}=\mathrm{Tr}_{q}^{\Omega_{\varepsilon_{k}}}(b_{k}).$ Since $b_{k}\in\mathcal{Z}_{\mathbb{Q}}$, we have $a_{k}\stackrel{{\scriptstyle\zeta}}{{=}}a_{k-1}$ by Step 1 and Lemma 9.1 for $k=1,2,\dots,\ell$. Hence, we have $a_{0}\stackrel{{\scriptstyle\zeta}}{{=}}a_{\ell}$ which is our claim. ## 10\. Interpolation polynomials In this section we summarize the theory of interpolation Macdonald polynomials. ### 10.1. One variable case Consider the space of polynomials in one variable $x$ over $\mathbb{C}(q)$ with the following bilinear form $(x^{k},x^{m})=q^{-km}.$ Let us define polynomials $f_{m}(x),m=0,1,\ldots$ by the equation $f_{0}(x)=1$ and (30) $f_{m}(x)=(x;q)_{m}=(1-x)\cdots(1-xq^{m-1})\quad\text{for}\quad m\geq 1.$ Clearly, $f_{m}(x)$ is a degree $m$ polynomial with leading term $(-1)^{m}q^{\frac{m(m-1)}{2}}x^{m}$, so $\\{f_{m}\\}_{m\geq 0}$ form a basis in $\mathbb{Z}_{q,q^{-1}}[x]$. Our next aim is to show that this basis is orthogonal. Observe that $f_{m}(q^{-k})=0$ for $k<m$. ###### Lemma 10.1. We have $(f_{m}(x),f_{k}(x))=\delta_{km}q^{-m}(q;q)_{m}$ ###### Proof. First, observe that $(g(x),x^{k})=g(q^{-k})$ for any polynomial $g(x)$. Therefore for $m>k$ we have $(f_{m}(x),x^{k})=f_{m}(q^{-k})=0$, so $(f_{m}(x),g(x))=0$ for any polynomial $g(x)$ of degree strictly less than $m$. In particular, $(f_{m}(x),f_{k}(x))=0$ for $m>k$ and $(f_{m}(x),f_{m}(x))=(-1)^{m}q^{\frac{m(m-1)}{2}}(f_{m}(x),x^{m})=(-1)^{m}q^{\frac{m(m-1)}{2}}f_{m}(q^{-m})=$ $(-1)^{m}q^{\frac{m(m-1)}{2}}(1-q^{-m})\cdots(1-q^{-1})=q^{-m}(1-q^{m})\cdots(1-q).$ ∎ ###### Lemma 10.2. The transition matrix between the monomial basis $x^{a}$ and the basis $f_{b}(x)$ has the following form: (31) $x^{a}=\sum_{b\leq a}k_{a,b}f_{b}(x),\quad k_{a,b}=(-1)^{b}q^{-ab+\frac{b(b+1)}{2}}\binom{a}{b}_{q}.$ ###### Proof. To find the coefficients we compute the pairing $(f_{b}(x),x^{a})$, then using orthogonality we obtain $k_{a,b}=\frac{(f_{b}(x),x^{a})}{(f_{b}(x),f_{b}(x))}=\frac{f_{b}(q^{-a})}{(f_{b}(x),f_{b}(x))}.$ For $a\geq b$ from Lemma 10.1 we get $(f_{b}(x),f_{b}(x))=q^{-b}(q;q)_{b},$ while $\displaystyle f_{b}(q^{-a})$ $\displaystyle=(1-q^{-a})\cdots(1-q^{-a+b-1})=(-1)^{b}q^{-ab+\frac{b(b-1)}{2}}(1-q^{a})\cdots(1-q^{a-b+1})$ $\displaystyle=(-1)^{b}q^{-ab+\frac{b(b-1)}{2}}\frac{(q;q)_{a}}{(q;q)_{a-b}}.$ and the equation follows. ∎ Our next goal is to expand arbitrary polynomial $f(x)$ in the basis $f_{m}(x)$. This can be done in two different ways. First, we can expand $f(x)$ in the monomial basis and apply (31). Alternatively, we can apply Newton interpolation method: if $f(x)=\sum a_{m}f_{m}(x)$ then $f(q^{-j})=\sum_{m\geq j}a_{m}f_{m}(q^{-j}),$ which is a triangular system of equations for the unknown coefficients $a_{m}$. Thus knowing $f(q^{-j})$ one can at least theoretically reconstruct the coefficients $a_{m}$. This can be made explicit by the following: ###### Lemma 10.3. We have (32) $f(x)=\sum_{m=0}^{\infty}a_{m}f_{m}(x),\ a_{m}=\frac{1}{(f_{m},f_{m})}\sum_{j=0}^{m}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{m}{j}_{q}f(q^{-j}).$ ###### Proof. By $q$-binomial theorem we have (33) $f_{m}(x)=\sum_{j=0}^{m}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{m}{j}_{q}x^{j}.$ Now $a_{m}=\frac{(f,f_{m})}{(f_{m},f_{m})}=\frac{1}{(f_{m},f_{m})}\sum_{j=0}^{m}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{m}{j}_{q}(f,x^{j}).$ Finally, $(f,x^{j})=f(q^{-j})$. ∎ ###### Remark 10.4. Equation (33) can be interpreted as an explicit inverse of the matrix in (31). One can consider completion $\widehat{\mathbb{Z}_{q}[x]}$ of the space of polynomials with respect to the basis $f_{m}(x)$. In this completion, infinite sums $\sum_{m=0}^{\infty}a_{m}f_{m}(x)$ are allowed. Newton interpolation method and (32) identify this completion with the space of distributions on the interpolation nodes $1,q^{-1},\ldots$. We will need the following lemma. ###### Lemma 10.5. We have $(x-q^{s})(x-q^{s+1})\cdots(x-q^{s+m-1})=\\\ \sum_{j=0}^{m}(-1)^{j}q^{-jm+\binom{j+1}{2}}\binom{m}{j}_{q}(1-q^{s+j})\cdots(1-q^{s+m-1})f_{j}(x).$ ###### Proof. We prove it by induction in $m$. For $m=1$ we get $x-q^{s}=-(1-x)+(1-q^{s})=-f_{1}+(1-q^{s})f_{0}.$ For the step of induction we observe (34) $\displaystyle(x-q^{s+m})f_{j}(x)$ $\displaystyle=-q^{-j}(1-q^{j}x)f_{j}(x)+(q^{-j}-q^{s+m})f_{j}(x)$ $\displaystyle=-q^{-j}f_{j+1}(x)+q^{-j}(1-q^{s+m+j})f_{j}(x).$ Using (34), it is easy to identify the coefficient at $f_{j}(x)$ in $(x-q^{s+m})\sum(-1)^{j}q^{-jm+\binom{j+1}{2}}\binom{m}{j}_{q}(1-q^{s+j})\cdots(1-q^{s+m-1})f_{j}(x)$ as $-q^{-j+1}(-1)^{j-1}q^{-(j-1)m+\binom{j}{2}}\binom{m}{j-1}_{q}(1-q^{s+j-1})\cdots(1-q^{s+m-1})\\\ +q^{-j}q^{-jm+\binom{j+1}{2}}\binom{m}{j}_{q}(1-q^{s+j})\cdots(1-q^{s+m-1})(1-q^{s+m+j})\\\ =-q^{-j(m+1)+\binom{j+1}{2}}(1-q^{s+j})\cdots(1-q^{s+m-1})\\\ \times\left[q^{m-j+1}\binom{m}{j-1}_{q}(1-q^{s+j-1})+\binom{m}{j}_{q}(1-q^{s+m+j})\right].$ It remains to notice that $q^{m-j+1}\binom{m}{j-1}_{q}(1-q^{s+j-1})+\binom{m}{j}_{q}(1-q^{s+m+j})$ $=\left[q^{m-j+1}\binom{m}{j-1}_{q}+\binom{m}{j}_{q}\right]-q^{s+m}\left[\binom{m}{j-1}_{q}+q^{j}\binom{m}{j}_{q}\right]$ $=\binom{m+1}{j}_{q}-q^{s+m}\binom{m+1}{j}_{q}=(1-q^{s+m})\binom{m+1}{j}_{q}.$ ∎ ###### Remark 10.6. If we set a formal variable $y=q^{s}$ in Lemma 10.5, then we get the identity $(x-y)(x-qy)\cdots(x-yq^{m-1})=\sum_{j=0}^{m}(-1)^{j}q^{-jm+\binom{j+1}{2}}\binom{m}{j}_{q}f_{m-j}(yq^{j})f_{j}(x).$ This is a $q$-analogue of the binomial identity $(x-y)^{m}=\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}(1-y)^{m-j}(1-x)^{j}.$ ### 10.2. Multi-variable case: polynomials Let us generalize the above results to the case of $N$ variables. The pairing has the form $(x_{1}^{a_{1}}\cdots x_{N}^{a_{N}},x_{1}^{b_{1}}\cdots x_{N}^{b_{N}})=q^{-\sum a_{i}b_{i}}=(x_{1}^{a_{1}},x_{1}^{b_{1}})\cdots(x_{N}^{a_{N}},x_{N}^{b_{N}}).$ Note that for $\mathbf{x}=(x_{1},\dots,x_{N})$ $(g(\mathbf{x}),x_{1}^{b_{1}}\cdots x_{N}^{b_{N}})=g(q^{-b_{1}},\ldots,q^{-b_{N}}).$ Consider the products $f_{k_{1},\ldots,k_{N}}(\mathbf{x})=f_{k_{1}}(x_{1})\cdots f_{k_{N}}(x_{N}).$ Since $f_{k}(x)$ give a basis in $\mathbb{C}(q)[x]$, the polynomials $f_{k_{1},\ldots,k_{N}}$ give a basis in $\mathbb{C}(q)[x_{1},\ldots,x_{N}]$. Clearly, $(f_{k_{1},\ldots,k_{N}},x_{1}^{b_{1}}\cdots x_{N}^{b_{N}})=0\ \textrm{unless}\ b_{i}\geq k_{i}\ \textrm{for all}\ i.$ ###### Lemma 10.7. We have $(f_{k_{1},\ldots,k_{N}},f_{m_{1},\ldots,m_{N}})=0$ unless $k_{i}=m_{i}$ for all $i$. ###### Proof. Suppose that $k_{i}>m_{i}$ for some $i$. Since $f_{m_{1},\ldots,m_{N}}$ contains only monomials of the form $x_{1}^{b_{1}}\cdots x_{N}^{b_{N}}$ with $b_{i}\leq m_{i}$, we have $(f_{k_{1},\ldots,k_{N}},x_{1}^{b_{1}}\cdots x_{N}^{b_{N}})=0$ for all such monomials and hence $(f_{k_{1},\ldots,k_{N}},f_{m_{1},\ldots,m_{N}})=0$. ∎ Next, we would like to describe the basis in symmetric polynomials. It will be labeled by partitions $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N})$ with at most $N$ parts. We define (35) $F_{\lambda}(\mathbf{x})=\frac{\det(f_{\lambda_{i}+N-i}(x_{j}))}{\prod_{i<j}(x_{i}-x_{j})}.$ Clearly, the numerator in (35) is antisymmetric in $x_{i}$, so it is divisible by $\prod_{i<j}(x_{i}-x_{j})$ and the ratio is a symmetric function. It is easy to see that $F_{\lambda}(\mathbf{x})$ is a non-homogeneous polynomial of degree $\|\lambda\|$, and the top degree component equals $(-1)^{\|\lambda\|+\binom{N}{2}}q^{D_{N}(\lambda)}s_{\lambda}$ where $s_{\lambda}$ is the Schur function and $D_{N}(\lambda)$ is defined by (6). The function $F_{\lambda}(\mathbf{x})$ is known as a special case of a factorial Schur function [26, 27, 28], it is also a specialization of nonsymmetric Macdonald polynomials described below. ###### Lemma 10.8. Suppose that $b_{1}>\ldots>b_{N}$. Then $F_{\lambda}(q^{-b_{1}},\ldots,q^{-b_{N}})=0$ unless $b_{i}\geq\lambda_{i}+N-i$ for all $i$. ###### Proof. Suppose that $b_{j}<\lambda_{j}+N-j$ for some $j$, then for all $i\leq j$ and $\ell>j$ one has $\lambda_{i}+N-i\geq\lambda_{j}+N-j>b_{j}\geq b_{\ell}$, so $f_{\lambda_{i}+N-i}(q^{-b_{\ell}})=0$. This implies $\det[f_{\lambda_{i}+N-i}(q^{-b_{\ell}})]_{i,\ell=1}^{N}=0$. On the other hand, since $b_{i}\neq b_{j}$ the denominator $\prod_{i<j}(q^{-b_{i}}-q^{-b_{j}})$ does not vanish. ∎ ###### Corollary 10.9. If $\mu$ is another partition then we can define $b_{i}=\mu_{i}+N-i$, and conclude that $F_{\lambda}(q^{-\mu_{i}-N+i})=0$ unless $\mu_{i}\geq\lambda_{i}$ for all $i$, that is, partition $\mu$ contains $\lambda$. ###### Example 10.10. Suppose that $\lambda=(1)$, then $F_{(1)}$ is a symmetric function of degree 1 with leading term $(-1)^{1+\binom{N}{2}}q^{D_{N}(1)}s_{(1)}=q^{D_{N}(1)}\sum x_{i}$. We have $D_{N}(1)=N-1+\binom{N}{3}$, so $F_{(1)}(x_{1},\ldots,x_{N})=(-1)^{1+\binom{N}{2}}q^{N-1+\binom{N}{3}}\sum x_{i}+c.$ To find the constant $c$, we observe that by Corollary 10.9 we get $F_{(1)}(q^{-N+1},q^{-N+2},\ldots,1)=0$, so $c=(-1)^{\binom{N}{2}}q^{N-1+\binom{N}{3}}(q^{-N+1}+q^{-N+2}+\ldots+1)=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}[N]_{q}.$ ###### Lemma 10.11. We have $F_{\lambda}(q^{-\lambda_{i}-N+i})=(-1)^{\binom{N}{2}}q^{n(\lambda)+\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-h(\square)}),$ where $h(\square)$ is the hook length of a box $\square$ in the Young diagram corresponding to $\lambda$. ###### Proof. Since the sequence $\lambda_{i}+N-i$ is strictly decreasing, we have $f_{\lambda_{j}+N-j}(q^{-\lambda_{i}-N+i})=0$ for $j>i$ and $f_{\lambda_{i}+N-i}(q^{-\lambda_{i}-N+i})=\\{\lambda_{i}+N-i\\}_{q^{-1}}!$ and $\det(f_{\lambda_{j}+N-j}(q^{-\lambda_{i}-N+i}))=\prod_{i}\\{\lambda_{i}+N-i\\}_{q^{-1}}!\ .$ On the other hand, $\prod_{i<j}(q^{-\lambda_{i}-N+i}-q^{-\lambda_{j}-N+j})=(-1)^{\binom{N}{2}}q^{-\sum(\lambda_{j}+N-j)(j-1)}\prod_{i<j}(1-q^{-\lambda_{i}+i+\lambda_{j}-j}).$ and the statement follows now from formula (5) and the identity $\sum(\lambda_{j}+N-j)(j-1)=n(\lambda)+\binom{N}{3}.$ ∎ ###### Example 10.12. For arbitrary $N$ and $\lambda=(1)$ we computed in Example 10.10 that $F_{(1)}=(-1)^{1+\binom{N}{2}}q^{\binom{N}{3}}(q^{N-1}(x_{1}+\ldots+x_{N})-[N]_{q}).$ Hence, $F_{(1)}(q^{-N},q^{-N+2},\ldots,1)=q^{\binom{N}{3}}(q^{N-1}(q^{-N}+q^{-N+2}+\ldots+1)-[N]_{q})=$ $(-1)^{1+\binom{N}{2}}q^{\binom{N}{3}}(q^{-1}-1)=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}(1-q^{-1}).$ We summarize the above results in the following proposition: ###### Proposition 10.13. [29] There exists a unique collection of nonhomogeneous symmetric polynomials $F_{\lambda}(x_{1},\ldots,x_{N})$ with the following properties: * • $F_{\lambda}(x_{1},\ldots,x_{N})$ has degree $\|\lambda\|$. * • $F_{\lambda}(q^{-\mu_{i}-N+i})=0$ for all partitions $\mu$ not containing $\lambda$. * • $F_{\lambda}(q^{-\lambda_{i}-N+i})=(-1)^{\binom{N}{2}}q^{n(\lambda)+\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-h(\square)}).$ We will denote the value $F_{\lambda}(q^{-\lambda_{i}-N+i})=(-1)^{\binom{N}{2}}q^{n(\lambda)+\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-h(\square)})$ by $c_{\lambda,\lambda}$. ###### Lemma 10.14. Suppose that $q$ is a root of unity. Then $c_{\lambda,\lambda}$ vanishes for all but finitely many partitions $\lambda$. ###### Proof. Observe that $\prod_{\square\in\lambda}(1-q^{-h(\square)})$ is divisible by $\prod_{i}[\lambda_{i}-\lambda_{i+1}]_{q}!$ and $\sum_{i=1}^{N}i(\lambda_{i}-\lambda_{i+1})=\|\lambda\|.$ This means that for some $i$ we must have $i(\lambda_{i}-\lambda_{i+1})\geq\frac{\|\lambda\|}{N},\ \lambda_{i}-\lambda_{i+1}\geq\frac{\|\lambda\|}{iN}\geq\frac{\|\lambda\|}{N^{2}},$ and $c_{\lambda,\lambda}$ is divisible by $(1-q)\cdots(1-q^{\lfloor\frac{\|\lambda\|}{N^{2}}\rfloor})$. If $q^{s}=1$ then it vanishes for $\|\lambda\|\geq sN^{2}$. ∎ ###### Remark 10.15. A partition is called an $s$-core if none of its hook lengths is divisible by $s$. The $s$-core partitions play an important role in representation theory of symmetric groups in finite characteristic, and of Hecke algebras at roots of unity [21]. If $q^{s}=1$ then clearly $c_{\lambda,\lambda}(q)\neq 0$ if and only if $\lambda$ is an $s$-core. Although there are infinitely many $s$-cores, Lemma 10.14 shows that there are finitely many $s$-cores with at most $N$ rows. For example, for $s=2$ the $2$-cores are “staircase partitions” $\lambda=(k,k-1,\ldots,1)$, and the maximal $2$-core with at most $N$ rows has size $N+(N-1)+\ldots+1=\binom{N+1}{2}$. ### 10.3. Multi-variable case: interpolation One can use the polynomials $F_{\lambda}$ to solve the following interpolation problem. ###### Problem 10.16. Find a symmetric function $f=\sum a_{\lambda}F_{\lambda}$ given its values $f(q^{-\mu_{i}-N+i})$ for all $\mu$. We have $f(q^{-\mu_{i}-N+i})=\sum a_{\lambda}F_{\lambda}(q^{-\mu_{i}-N+i})$ This is a linear system on $a_{\lambda}$ with the triangular matrix (36) ${\sf C}=\left[c_{\lambda,\mu}\right]_{\lambda,\mu},\ c_{\lambda,\mu}(q):=F_{\lambda}(q^{-\mu_{i}-N+i})$ It is clear from Proposition 10.13 that to find $a_{\lambda}$ for a given $\lambda$ it is sufficient to know all coefficients $c_{\mu,\nu}$ for $\mu\subset\nu\subset\lambda$. In [29] Okounkov computed the inverse matrix $D={\sf C}^{-1}$ which allows one to explicitly compute the coefficients $a_{\lambda}$. ###### Theorem 10.17. [29] Define $c^{}_{\lambda,\mu}(q)=c_{\lambda,\mu}(q^{-1})$ and $\operatorname{cont}(\lambda)=n(\lambda)-n(\lambda^{\prime})$. Then $D=\left[d_{\lambda,\mu}\right]_{\lambda,\mu},\ d_{\lambda,\mu}=(-1)^{\|\mu\|-\|\lambda\|}q^{\operatorname{cont}(\lambda)-\operatorname{cont}(\mu)}\frac{c^{}_{\lambda,\mu}}{c_{\mu,\mu}c^{}_{\lambda,\lambda}}$ and $a_{\mu}=\sum_{\lambda\subset\mu}d_{\lambda,\mu}f(q^{-\lambda_{i}-N+i})=\frac{1}{c_{\mu,\mu}}\sum_{\lambda\subset\mu}(-1)^{\|\mu\|-\|\lambda\|}q^{\operatorname{cont}(\lambda)-\operatorname{cont}(\mu)}\frac{c^{}_{\lambda,\mu}}{c^{}_{\lambda,\lambda}}f(q^{-\lambda_{i}-N+i}).$ ###### Example 10.18. If $\lambda=\mu$ then clearly $d_{\lambda,\mu}=\frac{1}{c_{\lambda,\lambda}}$. ###### Example 10.19. We have $F_{(\emptyset)}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}$, so $c_{(\emptyset),(\emptyset)}=c_{(\emptyset),(1)}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}},\quad c^{}_{(\emptyset),(\emptyset)}=c^{}_{(\emptyset),(1)}=(-1)^{\binom{N}{2}}q^{-\binom{N}{3}}.$ Since $c_{(1),(1)}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}(1-q^{-1})=(-1)^{\binom{N}{2}+1}q^{\binom{N}{3}-1}(1-q),$ we get $d_{(\emptyset),(1)}=\frac{(-1)^{\binom{N}{2}}q^{-\binom{N}{3}+1}}{(1-q)}.$ So the first two terms of interpolation series have the following form: $f(x_{1},\ldots,x_{N})=(-1)^{\binom{N}{2}}q^{-\binom{N}{3}}f(q^{1-N},q^{2-N},\ldots,1)F_{(\emptyset)}(\mathbf{x})+\\\ \frac{(-1)^{\binom{N}{2}+1}q^{-\binom{N}{3}+1}}{1-q}\left[-f(q^{1-N},q^{2-N},\ldots,1)+f(q^{-N},q^{2-N},\ldots,1)\right]F_{(1)}(\mathbf{x})+\ldots$ ###### Example 10.20. For $N=1$ and $a\geq b$ we have $c_{(b),(a)}=f_{b}(q^{-a})=(1-q^{-a})\cdots(1-q^{-a+b-1})$ hence $c^{}_{(b),(a)}=(1-q^{a})\cdots(1-q^{a-b+1})$ Now $\frac{c^{}_{(b),(a)}}{c^{}_{(b),(b)}}=\frac{(1-q^{a})\cdots(1-q^{a-b+1})}{(1-q^{b})\cdots(1-q)}=\binom{a}{b}_{q},$ and $d_{(b),(a)}=(-1)^{a-b}q^{\frac{b(b-1)}{2}-\frac{a(a-1)}{2}}\frac{c^{}_{(b),(a)}}{c_{(a),(a)}c^{}_{(b),(b)}}=\frac{(-1)^{a-b}}{c_{(a),(a)}}q^{\frac{b(b-1)}{2}-\frac{a(a-1)}{2}}\binom{a}{b}_{q},$ which matches (32). ###### Example 10.21. Let $N=2$, $\lambda=(1)$ and $\mu=(3,2)$. We have $F_{\lambda}=q(x_{1}+x_{2})-(1+q)$, so $c_{\lambda,\mu}=F_{\lambda}(q^{-4},q^{-2})=(-q-1+q^{-1}+q^{-3}),\ c^{}_{\lambda,\mu}=q^{3}+q-1-q^{-1},\\\ $ and using Lemma 10.11 $c_{\lambda,\lambda}=-(1-q^{-1}),\ c^{}_{\lambda,\lambda}=-(1-q),$ $c_{\mu,\mu}=-q^{2}(1-q^{-1})^{2}(1-q^{-2})(1-q^{-3})(1-q^{-4})=q^{-9}(1-q)^{2}(1-q^{2})(1-q^{3})(1-q^{4}).$ Now $d_{\lambda,\mu}=q^{-2}\frac{c^{}_{\lambda,\mu}}{c_{\mu,\mu}c^{}_{\lambda,\lambda}}=-q^{6}\frac{q^{4}+q^{2}-q-1}{(1-q)^{3}(1-q^{2})(1-q^{3})(1-q^{4})}.$ ### 10.4. Hopf pairing We have a symmetric bilinear form $(\cdot,\cdot)$ on $\mathbb{Z}[x_{1},\ldots,x_{N}]^{S_{N}}$ defined by its values on Schur polynomials $(s_{\lambda},s_{\mu})=s_{\lambda}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}})s_{\mu}(q^{-N+1},\ldots,1).$ It is closely related to the Hopf pairing $\langle\cdot,\cdot\rangle$ for $\mathcal{R}=\mathit{Rep}(\mathcal{U})$ defined in Section 5.2. Note that $(f,s_{\mu})=f(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}})s_{\mu}(q^{-N+1},\ldots,1).$ for any symmetric function $f$. ###### Proposition 10.22. We have (37) $(F_{\lambda},F_{\nu})=\delta_{\lambda,\nu}q^{-\|\lambda\|+2\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{N+c(\square)}),$ so the Hopf pairing is diagonal in the basis $\\{F_{\lambda}\\}_{\lambda}$. ###### Proof. We have $(F_{\lambda},s_{\mu})=F_{\lambda}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N}})s_{\mu}(q^{-N+1},\ldots,1)=0$ unless $\lambda\subset\mu$. On the other hand, $F_{\nu}$ can be expanded in $s_{\mu}$ for $\mu\preceq\nu$, so $(F_{\lambda},F_{\nu})$ vanishes unless there exists $\mu\preceq\nu$ such that $\lambda\subset\mu$, in particular, $\lambda\preceq\nu$. Since the Hopf pairing is symmetric, $(F_{\lambda},F_{\nu})$ vanishes unless $\lambda\preceq\nu$ and $\nu\preceq\lambda$, so $\lambda=\nu$. Finally, $(F_{\lambda},F_{\lambda})=(-1)^{\|\lambda\|+\binom{N}{2}}q^{D_{N}(\lambda)}(F_{\lambda},s_{\lambda})=(-1)^{\|\lambda\|+\binom{N}{2}}q^{D_{N}(\lambda)}F_{\lambda}(q^{-\lambda_{1}-N+1},\ldots,q^{-\lambda_{N}})s_{\lambda}(q^{-N+1},\ldots,1).$ Now $F_{\lambda}(q^{-\lambda_{1}-N+1},\ldots,q^{-\lambda_{N}})=(-1)^{\binom{N}{2}}q^{n(\lambda)+\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-h(\square)})$ while $s_{\lambda}(q^{-N+1},\ldots,1)=q^{-n(\lambda)}\prod_{\square\in\lambda}\frac{(1-q^{-N-c(\square)})}{(1-q^{-h(\square)})}.$ hence $F_{\lambda}(q^{-\lambda_{1}-N+1},\ldots,q^{-\lambda_{N}})s_{\lambda}(q^{-N+1},\ldots,1)=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}\prod_{\square\in\lambda}(1-q^{-N-c(\square)})=$ $(-1)^{\|\lambda\|+\binom{N}{2}}q^{-N\|\lambda\|-c(\lambda)+\binom{N}{3}}(1-q^{N+c(\square)}).$ On the other hand, $D_{N}(\lambda)=c(\lambda)+(N-1)\|\lambda\|+\binom{N}{3}$. ∎ This provides us with a different perspective for the interpolation problem. Suppose that we have a Schur expansion for $F_{\lambda}$: $F_{\lambda}=\sum_{\mu\preceq\lambda}b_{\lambda,\mu}s_{\mu}.$ Then for an arbitrary symmetric function $f(x_{1},\ldots,x_{N})$ we can write $f=\sum_{\lambda}\frac{(f,F_{\lambda})}{(F_{\lambda},F_{\lambda})}F_{\lambda}=\sum_{\lambda}\sum_{\mu\preceq\lambda}b_{\lambda,\mu}\frac{(f,s_{\mu})}{(F_{\lambda},F_{\lambda})}F_{\lambda}=\sum_{\lambda}\sum_{\mu\preceq\lambda}\frac{b_{\lambda,\mu}s_{\mu}(q^{-N-i})}{(F_{\lambda},F_{\lambda})}f(q^{-\mu_{i}-N+i})F_{\lambda},$ and the interpolation coefficient is equal to (38) $d_{\lambda,\mu}=\frac{b_{\lambda,\mu}s_{\mu}(q^{-N+i})}{(F_{\lambda},F_{\lambda})}.$ ###### Example 10.23. For $N=2$ and $\lambda=(3,2)$ we have $F_{(3,2)}=q^{2}(1-x_{1})(1-qx_{1})(1-x_{2})(1-qx_{2})(q^{3}(x_{1}+x_{2})-(1+q))=\\\ q^{7}s_{3,2}-q^{6}(1+q)s_{3,1}-q^{4}(1+q+q^{2}+q^{3})s_{2,2}+q^{6}s_{3,0}+q^{3}(1+q+q^{2}+q^{3})(1+q)s_{2,1}-\\\ q^{3}(1+q+q^{2}+q^{3})s_{2,0}-q^{2}(1+q+q^{2}+q^{3})(1+q)s_{1,1}+(q^{5}+q^{4}+2q^{3}+q^{2})s_{1,0}-(q^{3}+q^{2}).$ Also $(F_{3,2},F_{3,2})=-q^{-5}(1-q^{4})(1-q^{3})(1-q^{2})^{2}(1-q)$ Therefore the interpolation coefficient for $\lambda=(3,2)$ and $\mu=(1,0)$ equals $d_{(3,2),(1,0)}=(q^{5}+q^{4}+2q^{3}+q^{2})\frac{s_{1,0}(q^{-1},1)}{(F_{3,2},F_{3,2})}=$ $-\frac{(q^{5}+q^{4}+2q^{3}+q^{2})(1+q^{-1})}{q^{-5}(1-q^{4})(1-q^{3})(1-q^{2})^{2}(1-q)}=-\frac{q^{6}(q^{4}+q^{2}-q-1)}{(1-q^{4})(1-q^{3})(1-q^{2})(1-q)^{3}}.$ This agrees with Example 10.21. ### 10.5. Divisibility Given a polynomial $f(x)$, define $\partial_{xy}(f):=\frac{f(x)-f(y)}{x-y}.$ Observe that $\partial_{xy}(fg)=\frac{f(x)-f(y)}{x-y}g(x)+f(y)\frac{g(x)-g(y)}{x-y}=\partial_{xy}f\cdot g(x)+f(y)\cdot\partial_{xy}(g).$ More generally, we have (39) $\displaystyle\partial_{xy}(f_{1}\cdots f_{k})$ $\displaystyle=\partial_{xy}(f_{1})f_{2}(x)\cdots f_{k}(x)+f_{1}(y)\partial_{xy}(f_{2})f_{3}(x)\cdots f_{k}(x)+\ldots$ $\displaystyle+f_{1}(y)f_{2}(y)\cdots\partial_{xy}(f_{k}).$ ###### Example 10.24. For $f_{n}(x)=(1-x)\cdots(1-q^{n-1}x)$, note that $\partial_{xy}(1-q^{i}x)=-q^{i}$, so we get $\displaystyle F_{n,0}(x,y)$ $\displaystyle=\partial_{x,y}f_{n+1}(x)=\sum_{i=0}^{n}(1-y)\cdots(1-q^{i-1}y)[\partial_{x,y}(1-q^{i}x)](1-q^{i+1}x)\cdots(1-q^{n}x)$ $\displaystyle=\sum_{i=0}^{n}f_{i}(y)\cdot(-q^{i})f_{n-i}(q^{i+1}x).$ ###### Example 10.25. For example, $F_{1,0}(x,y)=q(x+y)-(1+q)=q(y-1)+(qx-1)=-[qf_{1}(y)+f_{1}(qx)].$ Similarly, $\displaystyle F_{2,0}(x,y)$ $\displaystyle=-q^{3}(x^{2}+xy+y^{2})+(q+q^{2}+q^{3})(x+y)-(1+q+q^{2})$ $\displaystyle=-[(1-qx)(1-q^{2}x)+q(1-q^{2}x)(1-y)+q^{2}(1-y)(1-qy)]$ $\displaystyle=-[f_{2}(qx)+qf_{1}(x)f_{2}(y)+q^{2}f_{2}(y)].$ ###### Corollary 10.26. For all integers $a$ and $b$ the value $F_{n,0}(q^{a},q^{b})$ is divisible by $\left(\left\lfloor\frac{n}{2}\right\rfloor\right)_{q}!$ ###### Proof. Let $k=\left\lfloor\frac{n}{2}\right\rfloor$. In the above equation either $i\geq k$ or $n-i\geq k$, so each term in the sum is either divisible by $f_{k}(q^{i+1+a})$ or by $f_{k}(q^{b})$, so by $q$-binomial theorem it is divisible by $(k)_{q}!$ ∎ More generally, let $\partial_{i}=\partial_{x_{i},x_{i+1}}$ then it is well known that $\partial_{i}$ satisfy braid relations, so one can define $\partial_{w}$ for any permutation $w$. Furthermore, $F_{\lambda}(x_{1},\ldots,x_{N})=\partial_{w_{0}}[f_{\lambda_{1}+N-1}(x_{1})\cdots f_{\lambda_{N}}(x_{N})],$ where $w_{0}=(N\ N-1\ \ldots 1)$ is the longest element in $S_{N}$. ###### Lemma 10.27. For all $\lambda$ one can write $F_{\lambda}(x_{1},\ldots,x_{N})$ as the sum where each term has the form (40) $f_{j_{1}}(q^{s_{1}}x_{m_{1}})\cdots f_{j_{d}}(q^{s_{d}}x_{m_{d}}),\ \text{where}\ j_{1}+\ldots+j_{d}=\|\lambda\|\ \text{and}\ d=\binom{N+1}{2}.$ Here the indices $m_{i}$ might repeat arbitrarily. ###### Proof. From (39) and Example 10.24 it is clear that $\partial_{i}$ applied to a product (40) with $\ell$ factors produces a sum of similar products with $\ell+1$ factors. We start from a product of $N$ factors, and $\partial_{w}$ is a composition of $\binom{N}{2}$ operators $\partial_{i}$, so the terms in the resulting sum have $N+\binom{N}{2}=\binom{N+1}{2}$ factors. Also, each $\partial_{i}$ decreases the degree by 1, so $j_{1}+\ldots+j_{d}=\sum(\lambda_{i}+N-i)-\binom{N}{2}=\|\lambda\|.$ ∎ ###### Remark 10.28. A more careful analysis of this proof leads to a combinatorial formula for $F_{\lambda}$ where the terms are labeled by semistandard tableaux, but we do not need it here. This is a $q$-analogue of the expansion of a Schur function in the monomial basis. ###### Lemma 10.29. For any sequence of integers $a_{1},\ldots,a_{N}$ the value $F_{\lambda}(q^{a_{1}},\ldots,q^{a_{N}})$ is divisible by $(k)_{q}!$ where $k=\left\lfloor\frac{\|\lambda\|}{\binom{N+1}{2}}\right\rfloor$. ###### Proof. In each term (40) there are $d=\binom{N+1}{2}$ indices $j_{1},\cdots,j_{d}$ which add up to $\|\lambda\|$, so at least one of these indices is greater than $\|\lambda\|/d$. It remains to notice that $f_{j}(q^{a})$ is divisible by $(q)_{j}!$ for all integers $a$. ∎ The following lemma gives a rough description of the expansion (41) $F_{\lambda}(x_{1},\ldots,x_{N})=\sum_{m_{1},\ldots,m_{N}}b_{m_{1},\ldots,m_{k}}f_{m_{1}}(x_{1})\cdots f_{m_{N}}(x_{N}).$ of the symmetric interpolation polynomial $F_{\lambda}$ in terms of nonsymmetric ones. ###### Lemma 10.30. Given $k$, for sufficiently large $\|\lambda\|$ for all terms of the expansion (41) either the coefficient $b_{m_{1},\ldots,m_{k}}$ is divisible by $(k)_{q}!$ or there exists $m_{i}\geq k$ for some $1\leq i\leq N$. ###### Proof. We follow the same logic as in Lemma 10.29. For $\|\lambda\|>2k\binom{N+1}{2}$ every term (40) is divisible by $f_{2k}(q^{s}x_{i})$ for some $s$ and $i$. By Lemma 10.5 this can be further decomposed into terms which are divisible by $(j)_{q}!f_{2k-j}(x_{i})$, and either $j$ or $2k-j$ is greater than or equal to $k$. Overall, we presented $F_{\lambda}(x_{1},\ldots,x_{N})=A(k)_{q}!+\sum B_{i}f_{k}(x_{i})$ for some polynomials $A$ and $B_{i}$. It remains to notice that the polynomial $B_{i}f_{k}(x_{i})$ can be presented as the sum of $f_{m_{1}}(x_{1})\cdots f_{m_{N}}(x_{N})$ where $m_{i}\geq k$. ∎ ## 11\. Stability of interpolation and the case $N=2$ ### 11.1. Stability of interpolation matrices In this section study the dependence of the interpolation polynomials on $N$. As above, if partition $\lambda$ has less than $N$ parts we can complete it with zeroes. We denote by $F_{\lambda;N}(x_{1},\ldots,x_{N})$ the corresponding polynomial in $N$ variables. ###### Lemma 11.1. Let $\lambda$ be a partition with at most $N$ parts. Then $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)=\begin{cases}(-1)^{N-1}q^{\binom{N-1}{2}}F_{\lambda;N-1}(qx_{1},\ldots,qx_{N-1})&\text{if}\ \lambda_{N}=0\\\ 0&\text{otherwise}.\end{cases}$ ###### Proof. Let $\mu$ be a partition with at most $N-1$ parts. Then by Proposition 10.13 $F_{\lambda;N}(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N-1}-1},1)=0$ unless $\mu$ contains $\lambda$. If $\lambda_{N}>0$ then this never happens and $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)=0$. If $\lambda_{N}=0$ we write $L(x_{1},\ldots,x_{N-1})=F_{\lambda;N-1}(qx_{1},\ldots,qx_{N-1})$. We have $L(q^{-\mu_{1}-N+1},\ldots,q^{-\mu_{N-1}-1})=F_{\lambda;N-1}(q^{-\mu_{1}-(N-1)+1},\ldots,q^{-\mu_{N-1}})$ which vanishes unless $\mu$ contains $\lambda$, so by Proposition 10.13 $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)$ is proportional to $L(x_{1},\ldots,x_{N-1})$. Finally, at $\mu=\lambda$ we can use Lemma 10.11 to determine the coefficient. ∎ ###### Remark 11.2. We can also prove the lemma using the explicit determinantal formula. Indeed, $f_{\lambda_{i}+N-i}(1)=0$ unless $f_{\lambda_{i}+N-i}=0$ which is equivalent to $i=N$ and $\lambda_{N}=0$. Therefore for $\lambda_{N}\neq 0$ the last row in the matrix $f_{\lambda_{i}+N-i}(x_{j})$ vanishes (where $x_{N}=1$), and $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)=0$. For $\lambda_{N}=0$ we have $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)=\frac{\det\left[f_{\lambda_{i}+N-i}(x_{j})\right]_{i,j=1}^{N-1}}{\prod_{i<j\leq N-1}(x_{i}-x_{j})\prod_{i\leq N-1}(x_{i}-1)}.$ Note that $f_{k+1}(x)=(1-x)f_{k}(qx)$, so $f_{\lambda_{i}+N-i}(x_{j})=(1-x_{j})f_{\lambda_{i}+(N-1)-i}(qx_{j})$ Therefore $F_{\lambda;N}(x_{1},\ldots,x_{N-1},1)=\frac{\prod_{i=1}^{n}(1-x_{i})\det\left[f_{\lambda_{i}+(N-1)-i}(qx_{j})\right]_{i,j=1}^{N-1}}{\prod_{i<j\leq N-1}(x_{i}-x_{j})\prod_{i\leq N-1}(x_{i}-1)}=$ $(-1)^{N-1}q^{\binom{N-1}{2}}F_{\lambda;N-1}(qx_{1},\ldots,qx_{N-1}).$ ###### Corollary 11.3. Let $c_{\lambda,\mu}^{(N)}$ be the coefficient defined in previous section for symmetric functions in $N$ variables. Then the expressions $(-1)^{\binom{N}{2}}q^{-\binom{N}{3}}c^{(N)}_{\lambda,\mu},(-1)^{\binom{N}{2}}q^{\binom{N}{3}}c^{(N)}_{\lambda,\mu},(-1)^{\binom{N}{2}}q^{\binom{N}{3}}d^{(N)}_{\lambda,\mu}$ are independent of $N$ (provided that $\lambda$ and $\mu$ have at most $N$ parts). ###### Example 11.4. For one-row partitions $\lambda=(b)$ and $\mu=(a)$ the interpolation coefficients are given by the formulas in Example 10.20 up to a monomial factor. The above results allow us to describe Schur expansion of interpolation polynomials: ###### Proposition 11.5. We have (42) $F_{\lambda}^{(N)}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}\sum_{\mu\subset\lambda}\overline{b_{\lambda,\mu}}A^{\|\mu\|}\prod_{\square\in\lambda\setminus\mu}(1-Aq^{c(\square)})s_{\lambda}^{(N)}$ where $A=q^{N}$ and the coefficients $\overline{b_{\lambda,\mu}}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}d^{(N)}_{\lambda,\mu}q^{-\|\lambda\|-\|\mu\|-n(\mu)}\prod_{\square\in\mu}(1-q^{h(\square)})$ do not depend on $N$. ###### Proof. It follows from (38) that $F_{\lambda}=\sum b_{\lambda,\mu}s_{\mu},\ b_{\lambda,\mu}=\frac{d_{\lambda,\mu}(F_{\lambda},F_{\lambda})}{s_{\mu}(q^{-N+i})}.$ Since $d_{\lambda,\mu}$ vanishes unless $\mu\subset\lambda$, the same is true for $b_{\lambda,\mu}$. By Corollary 11.3 the product $\overline{d_{\lambda,\mu}}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}d_{\lambda,\mu}$ does not depend on $N$, and we can use the formulas $(F_{\lambda},F_{\lambda})=q^{-\|\lambda\|+2\binom{N}{3}}\prod_{\square\in\lambda}(1-Aq^{c(\square)}),$ $s_{\mu}(q^{-N+i})=q^{n(\mu)-(N-1)\|\mu\|}\prod_{\square\in\mu}\frac{(1-Aq^{c(\square)})}{(1-q^{h(\square)})}.$ to write $b_{\lambda,\mu}=(-1)^{\binom{N}{2}}q^{-\binom{N}{3}}\overline{d_{\lambda,\mu}}\cdot q^{-\|\lambda\|+2\binom{N}{3}-n(\mu)+N\|\mu\|-\|\mu\|}\prod_{\square\in\lambda}(1-Aq^{c(\square)})\prod_{\square\in\mu}\frac{(1-q^{h(\square)})}{(1-Aq^{c(\square)})}.$ The result follows. ∎ ###### Corollary 11.6. The one-row interpolation polynomials have the following Schur expansion: $F_{(m)}^{(N)}=(-1)^{\binom{N}{2}}q^{\binom{N}{3}}\sum_{\mu\subset\lambda}(-1)^{j}q^{\frac{j(j-3)}{2}}A^{j}\frac{(1-Aq^{j})\cdots(1-Aq^{m-1})}{(1-q)\cdots(1-q^{m-j})}h_{j}^{(N)}=$ $(-1)^{\binom{N}{2}}q^{\binom{N}{3}}\sum_{\mu\subset\lambda}(-1)^{j}q^{\frac{j(j-3+2N)}{2}}\binom{N+m-1}{m-j}_{q}h_{j}^{(N)}.$ Here $h_{j}^{(N)}=s_{(j)}^{(N)}$ are complete symmetric functions in $N$ variables. ###### Proof. For $\lambda=(m)$ and $N=1$ we have $f_{m}(x)=\sum_{j=0}^{m}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{m}{j}_{q}x^{j}.$ By writing $A=q$ and $\mu=(j)$ we get $A^{\|\mu\|}\prod_{\square\in\lambda\setminus\mu}(1-Aq^{c(\square)})=q^{j}(1-q^{j+1})\cdots(1-q^{m}),$ so $\overline{b_{(m),(j)}}=(-1)^{j}\frac{q^{\frac{j(j-3)}{2}}}{(1-q)\cdots(1-q^{m-j})}.$ ∎ ###### Remark 11.7. The HOMFLY-PT limit of interpolation polynomials in Proposition 11.5 appears to be related to the results and conjectures in [22], it would be interesting to find a precise connection. ### 11.2. Adding a column It is well known that in symmetric functions in $N$ variables one has the identity $s_{\lambda+1^{N}}=x_{1}\cdots x_{N}\cdot s_{\lambda}.$ Here $\lambda+1^{N}=(\lambda_{1}+1,\ldots,\lambda_{N}+1)$ and the corresponding Young diagram is obtained from the Young diagram for $\lambda$ by adding a vertical column. For interpolation polynomials we have two different generalizations of this identity: the first relates $F_{\lambda+1^{N}}$ to $F_{\lambda}$ and the second describe the action of the multiplication by $x_{1}\cdots x_{N}$. ###### Proposition 11.8. We have $F_{\lambda+1^{N}}(x_{1},\ldots,x_{N})=q^{\binom{N}{2}}\prod_{i=1}^{N}(1-x_{i})F_{\lambda}(qx_{1},\ldots,qx_{N})$. More generally, (43) $F_{\lambda+k^{N}}(x_{1},\ldots,x_{N})=q^{k\binom{N}{2}}\prod_{i=1}^{N}f_{k}(x_{i})F_{\lambda}(q^{k}x_{1},\ldots,q^{k}x_{N}).$ ###### Proof. We have $f_{m+1}(x)=(1-x)f_{m}(qx)$, therefore $\det\left[f_{\lambda_{i}+1+N-i}(x_{j})\right]=\det\left[(1-x_{j})f_{\lambda_{i}+N-i}(qx_{j})\right]=\prod_{j=1}^{N}(1-x_{j})\det\left[f_{\lambda_{i}+N-i}(x_{j})\right].$ Since each factor $(x_{i}-x_{j})$ in the denominator gets multiplied by $q$ after changing $x_{i}\to qx_{i}$, this implies the first equation. Now (43) can be obtained by applying it $k$ times. ∎ Let $e_{i}$ denote the $i$-th basic vector in $\mathbb{Z}^{N}$ with $1$ at $i$-th position and $0$ at other positions. Given $I\subset\\{1,\ldots,n\\}$, we define $e_{I}=\sum_{i\in I}e_{i}$. ###### Proposition 11.9. We have $x_{1}\cdots x_{N}F_{\lambda}(x_{1},\ldots,x_{N})=q^{-\|\lambda\|-\binom{N}{2}}\sum_{I\subset\\{1,\ldots,n\\}}(-1)^{\|I\|}F_{\lambda+e_{I}}(x_{1},\ldots,x_{N}).$ Here we use the convention that $F_{\lambda+e_{I}}=0$ unless the entries of $\lambda+e_{I}$ are non-increasing (that is, $\lambda+e_{I}$ is a partition). ###### Proof. We have $f_{m+1}(x)=f_{m}(x)(1-q^{m}x)$, so $xf_{m}(x)=q^{-m}(f_{m}(x)-f_{m+1}(x)).$ Therefore $x_{1}\cdots x_{N}\det\left[f_{\lambda_{i}+N-i}(x_{j})\right]=\det\left[x_{j}f_{\lambda_{i}+N-i}(x_{j})\right]=$ $\det\left[q^{-\lambda_{i}-N+i}(f_{\lambda_{i}+N-i}(x_{j})-f_{\lambda_{i}+1+N-i}(x_{j}))\right].$ ∎ ###### Corollary 11.10. Consider the completion of the space of symmetric functions with coefficients in $\mathbb{Z}[q,q^{-1}]$ with respect to the basis $F_{\lambda}$. Then the operator of multiplication by $x_{1}\cdots x_{N}$ is invertible in this completion and its inverse is given by the equation $(x_{1}\cdots x_{N})^{-1}F_{\lambda}(x_{1},\ldots,x_{N})=q^{\binom{N}{2}}\sum_{v\in\mathbb{Z}_{\geq 0}^{N}}q^{\|\lambda\|+v}F_{\lambda+v}(x_{1},\ldots,x_{N}).$ ###### Proof. Define the operators $A_{i}$ by $A_{i}(F_{\lambda})=F_{\lambda+e_{i}}$, and $p_{i}(F_{\lambda})=q^{\lambda_{i}}F_{\lambda}$ for $i=1,\ldots,N$. Clearly, $[A_{i},A_{j}]=[p_{i},p_{j}]=[A_{i},p_{j}]$ for $i\neq j$ and by Proposition 11.9 we have $x_{1}\cdots x_{N}=q^{-\binom{N}{2}}\prod_{i}(1-A_{i})p^{-1}_{i},$ hence $(x_{1}\cdots x_{N})^{-1}=q^{\binom{N}{2}}\prod_{i}p_{i}(1+A_{i}+A_{i}^{2}+\ldots).$ ∎ ###### Example 11.11. For $N=1$ and $\lambda=(0)$ we get a curious identity $x^{-1}=\sum_{m=0}^{\infty}f_{m}(x)q^{m}$ We can check this identity directly, by computing the values of both sides at $q^{-j}$ for all $j$. Denote $u_{j}=\sum_{m=0}^{\infty}f_{m}(q^{-j})q^{m}=\sum_{m=0}^{j}f_{m}(q^{-j})q^{m}.$ Then $u_{j+1}=1+q(1-q^{-j-1})u_{j}$ and $u_{0}=1$, so it is easy to see that $u_{j}=q^{j}$. ### 11.3. Interpolation polynomials for $\mathfrak{gl}_{2}$ In this subsection we describe the interpolation polynomials for $\mathfrak{gl}_{2}$ explicitly. By definition, we have polynomials $F_{\lambda}(x_{1},x_{2})$ where $\lambda_{1}\geq\lambda_{2}$: $F_{\lambda_{1},\lambda_{2}}(x_{1},x_{2})=\frac{1}{x_{1}-x_{2}}\left\|\begin{matrix}f_{\lambda_{1}+1}(x_{1})&f_{\lambda_{1}+1}(x_{2})\\\ f_{\lambda_{2}}(x_{1})&f_{\lambda_{2}}(x_{2})\end{matrix}\right\|$ Let us consider the case $\lambda_{2}=0$ first, and write $\lambda_{1}=k$. Then $F_{k,0}(x_{1},x_{2})=\frac{1}{x_{1}-x_{2}}\det\left\|\begin{matrix}f_{k+1}(x_{1})&f_{k+1}(x_{2})\\\ 1&1\end{matrix}\right\|=\frac{f_{k+1}(x_{1})-f_{k+1}(x_{2})}{x_{1}-x_{2}}.$ Let $h_{i}(x_{1},x_{2})=\frac{x_{1}^{i+1}-x_{2}^{i+1}}{x_{1}-x_{2}}.$ Recall that $f_{k+1}(x)=\sum_{j=0}^{k+1}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{k+1}{j}_{q}x^{j}$, so $F_{k,0}(x_{1},x_{2})=\sum_{j=1}^{k+1}(-1)^{j}q^{\frac{j(j-1)}{2}}\binom{k+1}{j}_{q}h_{j-1}(x_{1},x_{2}),$ compare with Corollary 11.6. We just replace each $x^{j}$ in the expression for $f_{k+1}(x)$ by $h_{j-1}(x_{1},x_{2})$. ###### Example 11.12. We have $f_{1}(x)=1-x,\ f_{2}(x)=(1-x)(1-qx)=1-(1+q)x+qx^{2},$ $f_{3}(x)=(1-x)(1-qx)(1-q^{2}x)=1-(1+q+q^{2})x+(q+q^{2}+q^{3})x^{2}-q^{3}x^{3}$ so $F_{0,0}(x_{1},x_{2})=-1,\ F_{1,0}(x_{1},x_{2})=q(x_{1}+x_{2})-(1+q),\ $ $F_{2,0}=-q^{3}(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2})+(q+q^{2}+q^{3})(x_{1}+x_{2})-(1+q+q^{2}).$ By Proposition 11.8 we have $F_{\lambda_{1},\lambda_{2}}(x_{1},x_{2})=q^{\lambda_{2}}f_{\lambda_{2}}(x_{1})f_{\lambda_{2}}(x_{2})F_{\lambda_{1}-\lambda_{2},0}(q^{\lambda_{2}}x_{1},q^{\lambda_{2}}x_{2}).$ In particular, for $(\lambda_{1},\lambda_{2})=(k,k)$ we have $F_{k,k}(x_{1},x_{2})=q^{k}f_{k}(x_{1})f_{k}(x_{2}).$ Also, by Lemma 11.1 we get (44) $F_{\lambda_{1},\lambda_{2}}(x_{1},1)=\begin{cases}-f_{\lambda_{1}}(qx_{1})&\text{if}\ \lambda_{2}=0\\\ 0&\text{otherwise}.\end{cases}$ ### 11.4. Interpolation tables for $\mathfrak{gl}_{2}$ For the reader’s convenience, we have computed the polynomials $F_{\lambda}(x_{1},x_{2})$ and the corresponding interpolation matrices using Sage [35]. First, we present $F_{\lambda}$ in Schur basis: $F_{0}=-1,\quad F_{1}=qs_{1}-(q+1),\quad F_{2}=-q^{3}s_{2}+(q^{3}+q^{2}+q)s_{1}-(q^{2}+q+1),\\\ F_{1,1}=-qs_{1,1}+qs_{1}-q=-q(1-x_{1})(1-x_{2})\\\ F_{3}=q^{6}s_{3}-(q^{6}+q^{5}+q^{4}+q^{3})s_{2}+(q^{5}+q^{4}+2q^{3}+q^{2}+q)s_{1}-(q^{3}+q^{2}+q+1)\\\ F_{2,1}=q^{3}s_{2,1}-q^{3}s_{2}-(q^{3}+q^{2}+q)s_{1,1}+(q^{3}+q^{2}+q)s_{1}-(q^{2}+q)\\\ F_{3,1}=-q^{6}s_{3,1}+q^{6}s_{3}+(q^{6}+q^{5}+q^{4}+q^{3})s_{2,1}-(q^{6}+q^{5}+q^{4}+q^{3})s_{2}-\\\ (q^{5}+q^{4}+2q^{3}+q^{2}+q)s_{1,1}+(q^{5}+q^{4}+2q^{3}+q^{2}+q)s_{1}-(q^{3}+q^{2}+q)\\\ F_{2,2}=-q^{4}s_{2,2}+(q^{4}+q^{3})s_{2,1}-q^{3}s_{2}-(q^{4}+q^{3}+q^{2})s_{1,1}+(q^{3}+q^{2})s_{1}-q^{2}\\\ F_{3,2}=q^{7}s_{3,2}-(q^{7}+q^{6})s_{3,1}-(q^{7}+q^{6}+q^{5}+q^{4})s_{2,2}+q^{6}s_{3}+(q^{7}+2q^{6}+2q^{5}+2q^{4}+q^{3})s_{2,1}-\\\ (q^{6}+q^{5}+q^{4}+q^{3})s_{2}-(q^{6}+2q^{5}+2q^{4}+2q^{3}+q^{2})s_{1,1}+(q^{5}+q^{4}+2q^{3}+q^{2})s_{1}-(q^{3}+q^{2})\\\ F_{3,3}=-q^{9}s_{3,3}+(q^{9}+q^{8}+q^{7})s_{3,2}-(q^{8}+q^{7}+q^{6})s_{3,1}-(q^{9}+q^{8}+2q^{7}+q^{6}+q^{5})s_{2,2}+q^{6}s_{3}+\\\ (q^{8}+2q^{7}+2q^{6}+2q^{5}+q^{4})s_{2,1}-(q^{6}+q^{5}+q^{4})s_{2}-(q^{7}+q^{6}+2q^{5}+q^{4}+q^{3})s_{1,1}+(q^{5}+q^{4}+q^{3})s_{1}-q^{3}$ Next, we list the values of the evaluations $c_{\lambda,\mu}=F_{\lambda}(q^{-\mu_{1}-1},q^{-\mu_{2}})$ for various $\lambda$ and $\mu$ in Tables 1, 2, 3 below. The resulting matrix ${\sf C}=(c_{\lambda,\mu})$ is upper-triangular, with diagonal entries prescribed by Lemma 10.11. Zero entries correspond to pairs $(\lambda,\mu)$ where $\mu$ does not contain $\lambda$. The entry corresponding to $(\lambda,\mu)=((1),(3,2))$ is marked in bold, it is divisible by $1-q$ but does not factor any further. Using either Theorem 10.17 or equation (38), one can easily reconstruct the inverse matrix $D={\sf C}^{-1}$, and we list part of it in Table 4 (see Examples 10.21 and 10.23 for more computations). Note that by Corollary 11.3 this determines the coefficients $c_{\lambda,\mu}$ and $d_{\lambda,\mu}$ for $\lambda\subset\mu\subset(3,3)$ and arbitrary $N$. ### 11.5. Link invariants for $\mathfrak{gl}_{2}$ We can use the interpolation tables to expand the invariants of simple knots in the basis $F_{\lambda}$. Indeed, the colored $\mathfrak{gl}_{2}$ invariants are determined by the colored $\mathfrak{sl}_{2}$ invariants (that is, colored Jones polynomial) by the formula $J_{K}(V(\lambda_{1},\lambda_{2}),q)=J_{K}(V_{\lambda_{1}-\lambda_{2}},q).$ The coefficients $a_{\lambda}(K)$ are then determined by Theorem 1.3 $a_{\lambda}(K)=\sum_{\mu\subset\lambda}d_{\lambda,\mu}(q^{-1})J_{K}(V(\mu),q).$ For example, for the figure eight knot we have the following values of the colored Jones polynomial: $J_{K}(V_{0},q)=1=J_{K}(V({1,1}),q),\ J_{K}(V_{1},q)=J_{K}(V({2,1}),q)=1+q^{2}+q^{-2}-q-q^{-1},$ $J_{K}(V_{2},q)=1+q^{3}+q^{-3}-q-q^{-1}+(q^{3}+q^{-3}-q-q^{-1})(q^{3}+q^{-3}-q^{2}-q^{-2}).$ Using the values of $d_{\lambda,\mu}$ from Table 4 (and changing $q$ to $q^{-1}$) we obtain $a_{0}(K)=-J_{K}(V_{0},q)=-1,\ a_{1}(K)=-\frac{q^{-1}}{1-q^{-1}}J_{K}(V_{0},q)+\frac{q^{-1}}{1-q^{-1}}J_{K}(V_{1},q)=q^{-2}(q^{3}-1),$ $a_{2}(K)=-\frac{q^{-2}}{(1-q^{-1})(1-q^{-2})}J_{K}(V_{0},q)+\frac{q^{-2}}{(1-q^{-1})^{2}}J_{K}(V_{1},q)-\\\ \frac{q^{-3}}{(1-q^{-1})(1-q^{-2})}J_{K}(V_{2},q)=q^{-6}(-q^{9}+q^{5}+q^{4}-q^{3}-1),$ $a_{1,1}(K)=-\frac{q^{-3}}{(1-q^{-1})(1-q^{-2})}J_{K}(V_{0},q)+\frac{q^{-2}}{(1-q^{-1})^{2}}J_{K}(V_{1},q)-\\\ \frac{q^{-2}}{(1-q^{-1})(1-q^{-2})}J_{K}(V({1,1}),q)=q^{-2}(q^{2}+q+1),$ $a_{2,1}(K)=-\frac{q^{-4}}{(1-q^{-1})^{2}(1-q^{-3})}J_{K}(V_{0},q)+\frac{q^{-3}}{(1-q^{-1})^{3}}J_{K}(V_{1},q)-\\\ \frac{q^{-4}}{(1-q^{-1})^{2}(1-q^{-2})}J_{K}(V_{2},q)-\frac{q^{-3}}{(1-q^{-1})^{2}(1-q^{-2})}J_{K}(V({1,1}),q)+\\\ \frac{q^{-4}}{(1-q^{-1})^{2}(1-q^{-3})}J_{K}(V({2,1}),q)=q^{-6}(-q^{8}-q^{7}-q^{6}-q^{5}+q^{4}+2q^{3}+q^{2}+q+1).$ Using Tables 1, 2, 3, one can similarly compute the values of $a_{\lambda}(K)$ for all $\lambda\subset(3,3)$ and verify that these are indeed Laurent polynomials in $q$. $\lambda\backslash\mu$ \| (0) \| (1) \| (2) \| (1,1) ---\|---\|---\|---\|--- (0) \| $-1$ \| $-1$ \| $-1$ \| $-1$ (1) \| $0$ \| $q^{-1}(1-q)$ \| $q^{-2}(1-q^{2})$ \| $q^{-1}(1-q^{2})$ (2) \| $0$ \| $0$ \| $-q^{-3}(1-q)(1-q^{2})$ \| $0$ (1,1) \| $0$ \| $0$ \| $0$ \| $-q^{-2}(1-q)(1-q^{2})$ (3) \| $0$ \| $0$ \| $0$ \| $0$ (2,1) \| $0$ \| $0$ \| $0$ \| $0$ (3,1) \| $0$ \| $0$ \| $0$ \| $0$ (2,2) \| $0$ \| $0$ \| $0$ \| $0$ (3,2) \| $0$ \| $0$ \| $0$ \| $0$ (3,3) \| $0$ \| $0$ \| $0$ \| $0$ Table 1. Evaluations of interpolation polynomials: matrix $C=(c_{\lambda,\mu})$ $\lambda\backslash\mu$ \| (3) \| (2,1) \| (3,1) ---\|---\|---\|--- (0) \| $-1$ \| $-1$ \| $-1$ (1) \| $q^{-3}(1-q^{3})$ \| $q^{-2}(1-q^{3})$ \| $q^{-3}(1-q^{4})$ (2) \| $-q^{-5}(1-q^{2})(1-q^{3})$ \| $-q^{-3}(1-q)(1-q^{3})$ \| $-q^{-5}(1-q^{2})(1-q^{4})$ (1,1) \| $0$ \| $-q^{-3}(1-q)(1-q^{3})$ \| $-q^{-4}(1-q)(1-q^{4})$ (3) \| $q^{-6}(1-q)(1-q^{2})(1-q^{3})$ \| $0$ \| $q^{-6}(1-q)(1-q^{2})(1-q^{4})$ (2,1) \| $0$ \| $q^{-4}(1-q)^{2}(1-q^{3})$ \| $q^{-6}(1-q)(1-q^{2})(1-q^{4})$ (3,1) \| $0$ \| $0$ \| $-q^{-7}(1-q)^{2}(1-q^{2})(1-q^{4})$ (2,2) \| $0$ \| $0$ \| $0$ (3,2) \| $0$ \| $0$ \| $0$ (3,3) \| $0$ \| $0$ \| $0$ Table 2. Matrix $C=(c_{\lambda,\mu})$, continued $\lambda\backslash\mu$ \| (2,2) \| (3,2) \| (3,3) ---\|---\|---\|--- (0) \| $-1$ \| $-1$ \| $-1$ (1) \| $q^{-2}(1+q)(1-q^{2})$ \| $\mathbf{q^{-3}(-q^{4}-q^{3}+q^{2}+1)}$ \| $q^{-3}(1+q)(1-q^{3})$ (2) \| $-q^{3}(1-q^{2})(1-q^{3})$ \| $-q^{-5}(1-q^{3})(1-q^{4})$ \| $-q^{-5}(1+q)(1-q^{3})^{2}$ (1,1) \| $-q^{-4}(1-q^{2})(1-q^{3})$ \| $-q^{-5}(1-q^{2})(1-q^{4})$ \| $-q^{-6}(1-q^{3})(1-q^{4})$ (3) \| $0$ \| $q^{-6}(1-q)(1-q^{3})(1-q^{4})$ \| $q^{-6}(1-q^{2})(1-q^{3})(1-q^{4})$ (2,1) \| $q^{-5}(1-q^{2})^{2}(1-q^{3})$ \| $q^{-7}(1-q^{2})(1-q^{3})(1-q^{4})$ \| $q^{-8}(1+q)(1-q^{2})(1-q^{3})(1-q^{4})$ (3,1) \| $0$ \| $-q^{-8}(1-q)(1-q^{2})(1-q^{3})(1-q^{4})$ \| $-q^{-9}(1-q^{2})(1-q^{3})^{2}(1-q^{4})$ (2,2) \| $-q^{-6}(1-q)(1-q^{2})^{2}(1-q^{3})$ \| $-q^{-8}(1-q)(1-q^{2})(1-q^{3})(1-q^{4})$ \| $-q^{-10}(1-q^{2})(1-q^{3})^{2}(1-q^{4})$ (3,2) \| $0$ \| $q^{-9}(1-q)^{2}(1-q^{2})(1-q^{3})(1-q^{4})$ \| $q^{-11}(1-q^{2})^{2}(1-q^{3})^{2}(1-q^{4})$ (3,3) \| $0$ \| $0$ \| $-q^{-12}(1-q)(1-q^{2})^{2}(1-q^{3})^{2}(1-q^{4})$ Table 3. Matrix $C=(c_{\lambda,\mu})$, continued $\lambda\backslash\mu$ \| (0) \| (1) \| (2) \| (1,1) \| (2,1) ---\|---\|---\|---\|---\|--- (0) \| $-1$ \| $-\frac{q}{1-q}$ \| $-\frac{q^{2}}{(1-q)(1-q^{2})}$ \| $-\frac{q^{3}}{(1-q)(1-q^{2})}$ \| $-\frac{q^{4}}{(1-q)^{2}(1-q^{3})}$ (1) \| $0$ \| $\frac{q}{1-q}$ \| $\frac{q^{2}}{(1-q)^{2}}$ \| $\frac{q^{2}}{(1-q)^{2}}$ \| $\frac{q^{3}}{(1-q)^{3}}$ (2) \| $0$ \| $0$ \| $-\frac{q^{3}}{(1-q)(1-q^{2})}$ \| $0$ \| $-\frac{q^{4}}{(1-q)^{2}(1-q^{2})}$ (1,1) \| $0$ \| $0$ \| $0$ \| $-\frac{q^{2}}{(1-q)(1-q^{2})}$ \| $-\frac{q^{3}}{(1-q)^{2}(1-q^{2})}$ (2,1) \| $0$ \| $0$ \| $0$ \| $0$ \| $\frac{q^{4}}{(1-q)^{2}(1-q^{3})}$ Table 4. Interpolation matrix $D=(d_{\lambda,\mu})=C^{-1}$ ## 12\. Appendix Here we collect some useful definitions and facts about Habiro’s ring and interpolation Macdonald polynomials. ### 12.1. Habiro’s ring The Habiro ring [17] is defined as $\widehat{\mathbb{Z}[q]}:={\lim\limits_{\overleftarrow{\hskip 5.69054ptn\hskip 5.69054pt}}}\;\frac{\mathbb{Z}[q]}{((q;q)_{n})}$ Any element of $\widehat{\mathbb{Z}[q]}$ can be presented (not uniquely) as infinite series $f(q)=\sum_{n=0}^{\infty}f_{n}\,(q;q)_{n},\quad f_{n}\in\mathbb{Z}[q].$ Evaluations of such $f(q)$ at all roots of unity are well defined, since if $q^{s}=1$ one has $f(q)=\sum_{n=0}^{s-1}f_{n}(q)_{n}$. It is easy to expand every $f(q)\in\widehat{\mathbb{Z}[q]}$ into formal power series in $(q-1)$, denoted by $T(f)$ and called the Taylor series of $f(q)$ at $q=1$. One important property of the Habiro ring is that any $f\in\widehat{\mathbb{Z}[q]}$ is uniquely determined by its Taylor series. In other words, the map $T:\widehat{\mathbb{Z}[q]}\to\mathbb{Z}[[q-1]]$ is injective [17, Thm 5.4]. In particular, $\widehat{\mathbb{Z}[q]}$ is an integral domain. Moreover, every $f\in\widehat{\mathbb{Z}[q]}$ is determined by the values of $f$ at any infinite set of roots of unity of prime power order. Because of these properties, Habiro ring is also known as a ring of analytic functions at roots of unity. Since $\cap_{n\geq 0}I_{n}=0$ with $I_{n}=(q;q)_{n}\mathbb{Z}[q]$, the natural map $\mathbb{Z}[q]\to\widehat{\mathbb{Z}[q]}$ is injective. The image of $q$ under this map is invertible, and the inverse is given by $q^{-1}=\sum_{n=1}^{\infty}q^{n}(q;q)_{n},$ compare with Example 11.11. This implies that there is an injective map $\mathbb{Z}[q,q^{-1}]\to\widehat{\mathbb{Z}[q]}$. The following result is proved in [17, Proposition 7.5], but we give a slightly different proof here for the reader’s convenience. We will denote by $\Phi_{n}(q)$ the $n$th cyclotomic polynomial $\Phi_{n}(q)=\prod_{(a,n)=1}\left(q-\zeta^{a}_{n}\right)$ where $\zeta_{n}$ is any primitive $n$th root of unity. ###### Proposition 12.1. Suppose that $f(q)\in\widehat{\mathbb{Z}[q]}$ and $f(q)h(q)\in\mathbb{Z}[q,q^{-1}]$ for some product of cyclotomic polynomials $h(q)=\Phi_{n_{1}}(q)\cdots\Phi_{n_{r}}(q)$. Then $f(q)\in\mathbb{Z}[q,q^{-1}]$. ###### Proof. Let us denote $g(q)=f(q)h(q)\in\mathbb{Z}[q,q^{-1}]$, we prove the statement by induction in $r$. For $r=1$ we get $h(q)=\Phi_{n}(q)$ and $g(q)=f(q)\Phi_{n}(q)$, so for any primitive $n$-th root of unity $\zeta_{n}$ we have $g(\zeta_{n})=f(\zeta_{n})\Phi_{n}(\zeta_{n})=0$, so $g(q)=\alpha(q)\Phi_{n}(q)$ for some $\alpha\in\mathbb{Z}[q,q^{-1}]$. This implies $(f(q)-\alpha(q))\Phi_{n}(q)=0$, and since $\widehat{\mathbb{Z}[q]}$ is an integral domain we get $f(q)=\alpha(q)$. For $r>1$ we get $f(q)\Phi_{n_{1}}(q)\cdots\Phi_{n_{r}}(q)\in\mathbb{Z}[q,q^{-1}],$ so by the above $f(q)\Phi_{n_{1}}(q)\cdots\Phi_{n_{r-1}}(q)\in\mathbb{Z}[q,q^{-1}],$ and by the assumption of induction $f(q)\in\mathbb{Z}[q,q^{-1}]$. ∎ ### 12.2. Interpolation Macdonald polynomials We consider partitions with at most $N$ parts. ###### Theorem 12.2. [23, 24, 29, 30, 31, 32, 36] There exists unique up to scalar factors family of symmetric polynomials $I_{\lambda}(x_{1},\ldots,x_{N};q,t)$ with the following properties: (a) $I_{\lambda}(q^{-\mu_{i}}t^{N-i})=0$ unless $\mu$ contains $\lambda$ * (b) $I_{\lambda}(q^{-\lambda_{i}}t^{N-i})\neq 0$ * (c) $I_{\lambda}$ is a nonhomogeneous polynomial of degree $\|\lambda\|$, and its degree $\|\lambda\|$ part is proportional to the Macdonald polynomial $P_{\lambda}(x_{1},\ldots,x_{N};q,t)$. The polynomials $I_{\lambda}$ are called interpolation Macdonald polynomials. In fact, the properties (a) and (b) already uniquely determine $I_{\lambda}$ (up to a scalar), and their existence follows from the fact that $q^{-\lambda_{i}}t^{N-i}$ for a nondegenerate grid in the sense of [31]. Part (c) is then a deep property of these polynomials. 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# Moderate Deviations in Triangle Count Joe Neeman and Charles Radin and Lorenzo Sadun Joe Neeman Department of Mathematics The University of Texas at Austin Austin, TX 78712<EMAIL_ADDRESS>Charles Radin Department of Mathematics The University of Texas at Austin Austin, TX 78712<EMAIL_ADDRESS>Lorenzo Sadun Department of Mathematics The University of Texas at Austin Austin, TX 78712<EMAIL_ADDRESS> ###### Abstract. We prove moderate deviations bounds for the lower tail of the number of triangles in a $\mathcal{G}(n,m)$ random graph. We show that the probability of decreasing the triangle density by $t$, with $n^{-3/4}\ll t\ll 1$, is $\exp(-\Theta(n^{2}t^{2/3}))$; we find the leading coefficient in the exponent for $m\geq\frac{1}{2}\binom{n}{2}$, and estimate it otherwise. This complements results of Goldschmidt et al., who showed that for $n^{-3/2}\ll t\ll n^{-1}$, the probability is $\exp(-\Theta(n^{3}t^{2}))$. That is, moderate deviations behave much like small deviations for $t\ll n^{-1}$ and much like large deviations for $n^{-3/4}\ll t$. We conjecture a sharp change between the two regimes at $t=\Theta(n^{-3/4})$, which we associate with a single large negative eigenvalue of the adjacency matrix becoming responsible for almost all of the triangle deficit. Our results can be interpreted as finite size effects in phase transitions in constrained edge-triangle graphs. This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813, and by a fellowship from the Alfred P. Sloan Foundation. ## 1\. Introduction We prove moderate deviations bounds for the lower tail of the number of triangles in a $\mathcal{G}(n,m)$ random graph, for deviations larger than those of Goldschmidt et al. [1] but smaller than large deviations, which are of order the mean of the triangle density. For instance, with the notation that $\tau(G)$ is the triangle density of a $\mathcal{G}(n,m)$ graph $G$ where $n\to\infty$ and $m=p\binom{n}{2}+O(1)$, for some $1/2\leq p<1$ that is fixed as $n\to\infty$ and $n^{-3/4}\ll t\ll 1$, we prove (see Theorem 1) that (1) $\Pr\left(\tau(G)\leq p^{3}-t\right)=\exp\left(-\frac{\ln\frac{p}{1-p}}{2(2p-1)}t^{2/3}n^{2}+o(t^{2/3}n^{2})\right).$ The number of triangles in a random graph is a fundamental and surprisingly important random variable in the study of probabilistic combinatorics. The probabilistic behavior of these triangle counts is at least partially responsible for the development of many important methods related to concentration inequalities for dependent random variables, including Janson’s inequality [2], the entropy method [3], martingale difference techniques in random graphs, and others [4]. The traditional point of view, as exemplified by the seminal paper by Janson and Rucínski [5], holds that the lower tail of the triangle count is easy to characterize while the upper tail is hard. This view stems at least partly from the fact that most earlier works studied the $\mathcal{G}(n,p)$ model, in which edges appear independently, each with probability $p$. In the $\mathcal{G}(n,m)$ model, in which the number of edges is fixed at $m$, the situation is rather more subtle. For example, one can easily see that under $\mathcal{G}(n,p)$, the number of triangles, $T(G)$, satisfies $\operatorname{Var}(T(G))=\Theta(n^{4})$, while under $\mathcal{G}(n,m)$, $\operatorname{Var}(T(G))=\Theta(n^{3})$. The distinction between the two models – especially in the lower tail – becomes even more pronounced at larger deviations. This can be intuitively explained by the fact that in $\mathcal{G}(n,p)$ one can easily “depress” the triangle count simply by reducing the number of edges: a graph $G$ with edge number $\|E(G)\|\approx q\binom{n}{2}$ will typically have triangle density $\tau\approxeq q^{3}$, and the probability of seeing such a graph under $\mathcal{G}(n,p)$ is of the order $\exp(-\Theta(n^{2}(p-q)^{2}))$; it follows that under $\mathcal{G}(n,p)$ we have (2) $\Pr(\tau(G)\leq\mathbb{E}\tau(G)-t)\geq\exp(-\Omega(n^{2}t^{2})).$ Under $\mathcal{G}(n,m)$, large deficits in the triangle density are much rarer than they are in $\mathcal{G}(n,p)$. At the scale of constant-order deficits, this was noticed in [6, 7], where it is proved that for $t=\Theta(1)$ and $\mathcal{G}(n,m)$ with $m=\Theta(n^{2})$, (3) $\Pr(\tau(G)\leq\mathbb{E}\tau(G)-t)=\exp(-\Theta(n^{2}t^{2/3})).$ (They also found the exact leading-order term in the exponent when $m=\frac{1}{2}\binom{n}{2}+o(n^{2})$ and bounded the leading-order coefficient for all other values of $m$.) At the other end of the scale, a recent result of Goldschmidt et al. [1] showed that for $n^{-3/2}\ll t\ll n^{-1}$ the lower triangle tail has a different behavior: (4) $\Pr(\tau(G)\leq\mathbb{E}\tau(G)-t)=\exp(-\Theta(n^{3}t^{2})).$ (Again, they also found the exact leading-order term in the exponent.) Since $t\leq\Theta(n^{-3/2})$ is within the range of the Central Limit Theorem this leaves open the case of $n^{-1}\ll t\ll 1$. Noting that the two exponential rates (namely $n^{2}t^{2/3}$ and $n^{3}t^{2}$) cross over at $t=\Theta(n^{-3/4})$, it is natural to guess that (5) $\Pr(\tau(G)\leq\mathbb{E}\tau(G)-t)=\begin{cases}\exp(-\Theta(n^{3}t^{2}))&\text{if $t\ll n^{-3/4}$},\\\ \exp(-\Theta(n^{2}t^{2/3}))&\text{if $n^{-3/4}\ll t\ll 1$}.\end{cases}$ We prove the second of these two cases (see Theorem 2); the first remains a conjecture. We also prove some structural results on graphs with $\tau(G)\leq\mathbb{E}\tau(G)-\omega(n^{-3/4})$. These structural results provide a plausible explanation for the importance of $t=\Theta(n^{-3/4})$, namely that it is the threshold at which a single large negative eigenvalue of the adjacency matrix becomes responsible for almost all of the triangle deficit. ## 2\. Context and references We are concerned with random graphs, more specifically with $\mathcal{G}(n,m)$, the uniform distribution on graphs on $n$ nodes with $m$ edges, rather than the more common model $\mathcal{G}(n,p)$, in which the edges are independent with probability $p$. For either of these random graph models one can study the distribution of subgraph counts, for instance triangle counts or triangle density (density being the quotient of the count by the count in the complete graph) with which we are concerned. Results on the probability of deviations of triangle density from the mean fall into three classes by size: small deviations, on the order of the standard deviation, large deviations, on the order of the mean, and moderate deviations, of intermediate size. Our main results concern the moderate regime of deviations of triangle density in $\mathcal{G}(n,m)$, in which we prove, among other things, that deviations near but below the large class are qualitatively different from deviations near but above the small class. We know of no other results of this sort, for $\mathcal{G}(n,m)$ or $\mathcal{G}(n,p)$. For small deviations there is a long history under the name Central Limit Theorem. There are also many papers on moderate and large deviations of subgraph counts. As background, more specifically for results discussed here, we suggest the following: [8, 9, 10, 11, 12, 13, 14, 15, 16] and references within them for a broader view. As our results are strongly colored by large deviations we note in particular [17]. For convenience we note some common asymptotics notation. We use $f=o(g)$ or $f\ll g$ to mean $\lim\|f(n)\|/g(n)=0$, $f=O(g)$ to mean $\lim\sup f(n)/g(n)<\infty$, $f=\Omega(g)$ to mean $\lim\inf f(n)/g(n)>0$, $f=\omega(g)$ or $f\gg g$ to mean $\lim\|f\|/g=\infty$, and $f=\Theta(g)$ to mean both $f=O(g)$ and $f=\Omega(g)$. The phrase “with high probability” means “with probability converging to 1 as $n\to\infty$,” and we also make use of probabilistic asymptotic notation: “$f=O(g)$ with high probability” means that for every $\epsilon>0$ there exists $C>0$ with $\limsup\Pr(f\geq Cg)\leq\epsilon$; “$f=o(g)$ with high probability” means that for every $\epsilon>0$, $\|f\|/g\leq\epsilon$ with high probability; and analogously for $\Omega$ and $\omega$. More specifically we are studying the triangle density of $\mathcal{G}(n,m)$ graphs in the range $\tau(G)=p^{3}-t$ for $n^{-3/4}\ll t\ll 1$. The case $0\leq t\leq\Omega(n^{-3/2})$ is within the range of the Central Limit Theorem (and it is covered by Janson’s more general work on subgraph statistics [18]). The range $n^{-3/2}\ll t\ll n^{-1}$ is studied by [1]; they showed that in this regime (6) $\Pr(\tau(G)\leq p^{3}-t)=\exp\left(-\frac{t^{2}n^{3}}{2\sigma_{p}^{2}}(1+o(1))\right),$ where $\sigma_{p}^{2}=\operatorname{Var}(\tau(G))/n^{3}$, which is of constant order. They also show an upper bound for larger $t$: for $n^{-3/2}\ll t\ll 1$, (7) $\Pr(\tau(G)\leq p^{3}-t)=\exp\left(-\Omega(t^{2}n^{3})\right).$ We show that this is not tight for $n^{-3/4}\ll t\ll 1$. For example, we show that in this range, (8) $\Pr(\tau(G)\leq p^{3}-t)=\exp\left(-\Theta(t^{2/3}n^{2})\right).$ In the case $p\geq\frac{1}{2}$, we also derive more detailed results (see Theorem 1): we identify the leading constant in the exponent and prove some results on the spectrum of the adjacency matrix of $G$. ### 2.1. Related work on random graphs Besides the work of [1], there is related work on large deviation principles (LDPs) for more general statistics, and LDPs for sparser graphs, notably in [19]. Moderate deviations in triangle count in $\mathcal{G}(n,m)$ can be seen from a different vantage based on [20]. That paper follows a series of works [6, 7, 24, 22, 23, 25, 26] on the asymptotics of ‘constrained’ random graphs, in particular the asymptotics of $\mathcal{G}(n,m,t)$, the uniform distribution on graphs on $n$ nodes constrained to have $m$ edges and $t$ triangles. A large deviation principle, using optimization over graphons, a variant of the seminal work [27] by Chatterjee and Varadhan on large deviations in $\mathcal{G}(n,p)$, was used to prove various features of phase transitions between asymptotic ‘phases’, phases illustrated by the entropy-optimal graphons. (See also [28].) But in [20] numerical evidence showed that the transitions could be clearly seen in finite systems, using constrained graphs with as few as 30 vertices. From this perspective moderate deviations in triangle count can be understood as finite size effects in a phase transition. Asymptotically, entropy goes through a sharp ridge as the edge density/triangle density pair $(\varepsilon,\tau)$ passes through $(\varepsilon,\varepsilon^{3})$ (Thms. 1.1,1.2 in [7]), and moderate deviations quantify how the sharp ridge rounds off at finite node number, somewhat as an ice cube freezing in water has rounded edges. The focus thus shifts to the infinite system, where emergent phases are meaningful, away from $\mathcal{G}(n,m,t)$ or $\mathcal{G}(n,m)$. ### 2.2. Related work on random matrices Since we are studying the spectrum of the adjacency matrix, our methods mainly come from random matrix theory. Specifically, we are interested in large deviations of eigenvalues of the random adjacency matrices coming from our random graphs. The study of large deviations of eigenvalues is an active topic, but the results we aim for are somewhat atypical. Traditionally, “large deviations” refers to deviations on the order of the mean, so large deviations results for random matrices typically consider the event that the largest eigenvalue of a symmetric $n\times n$ matrix with i.i.d. mean-zero, variance-$\sigma^{2}$ entries is of order $\alpha\sqrt{n}$ for $\alpha>2\sigma$; this is because the typical value of the largest eigenvalue is of order $2\sigma\sqrt{n}$. However, because an eigenvalue of order $n^{\beta}$ contributes $n^{3\beta}$ to the triangle count, and because we are interested in triangle deviation of orders $n^{9/4}$ through $n^{3}$, we are necessarily interested in much larger eigenvalues. Another difference in our work is that we consider several large eigenvalues simultaneously. This is because we need to consider the possibility that the triangle count is affected by several atypically large eigenvalues instead of just one. In related work, * • Guionnet and Husson [29] showed an LDP for the largest eigenvalue for a family of random matrices that includes Rademacher matrices, which is essentially the case that we consider when $p=\frac{1}{2}$. * • Augeri [30] showed an LDP for the largest eigenvalue for random matrices whose entries have heavier-than-Gaussian tails. * • Battacharya and Ganguly [31] showed an LDP for the largest eigenvalue of an Erdős-Rényi graph. Their setting differs from the others in that the random matrices they consider are not centered (which makes a big difference when studying the largest eigenvalue). * • Augeri, Guionnet, and Husson [32] showed an LDP for the largest eigenvalue for most random matrices with subgaussian elements. These are essentially the same random matrices that we consider, with the main difference being that they are looking at eigenvalues of size $\Theta(\sqrt{n})$. ## 3\. Triangle counts Our general setting is: we let $A$ be the adjacency matrix of a $\mathcal{G}(n,m)$ graph, where $n\to\infty$ and $m=p\binom{n}{2}+O(1)$, for some $p\in\mathbb{R}$ that is fixed as $n\to\infty$. We denote by $\tau(A)$ the triangle density of $A$, and by $\lambda_{n}(A)$ the smallest eigenvalue of $A$. We prove two theorems governing asymptotic behavior as $n\to\infty$ and $n^{-3/4}\ll t\ll 1$. The first is a strong result for $\frac{1}{2}\leq p<1$. ###### Theorem 1. If $\frac{1}{2}\leq p<1$ and $n^{-3/4}\ll t\ll 1$ then (9) $\Pr\left(\tau(A)\leq p^{3}-t\right)=\exp\left(-\frac{\ln\frac{1-p}{p}}{2(1-2p)}t^{2/3}n^{2}+o(t^{2/3}n^{2})\right),$ with the convention that $\frac{\ln\frac{1-p}{p}}{1-2p}=2$ when $p=\frac{1}{2}$. Moreover, conditioned on $\tau(A)\leq p^{3}-t$, with high probability we have (10) $\lambda_{n}^{3}(A)=-tn^{3}(1-o(1))$ and $\lambda_{n-1}^{3}(A)\geq-o(tn^{3})$. The second result, for $0<p\leq\frac{1}{2}$, is weaker. ###### Theorem 2. If $0<p\leq\frac{1}{2}$ and $n^{-3/4}\ll t\ll 1$ then $\Pr\left(\tau(G)\leq p^{3}-t\right)$ is bounded above by (11) $\exp\left(-\frac{\ln\frac{p}{1-p}}{2(2p-1)}t^{2/3}n^{2}+o(t^{2/3}n^{2})\right)$ and bounded below by (12) $\exp\left(-\frac{1}{2p(1-p)}t^{2/3}n^{2}+o(t^{2/3}n^{2})\right).$ Moreover, conditioned on $\tau(A)\leq p^{3}-t$, with high probability we have (13) $\lambda_{n}^{3}(A)=-\Omega(tn^{3}).$ Together, these theorems show that $p(\tau(A)\leq p^{3}-t)=\exp(-\Theta(t^{2/3}n^{2}))$ for all $0<p<1$. ### 3.1. Centering the matrix The main point of this section is that when considering the lower tail for triangle counts in $\mathcal{G}(n,m)$ graphs, it suffices to look at eigenvalues of the centered adjacency matrix. This might sound obvious, but there are two subtleties: 1. (1) It is important that we are looking at the lower tail, because the upper tail probabilities are controlled by perturbations to the largest eigenvector; this is exactly the eigenvector that gets destroyed when we center the adjacency matrix, so the eigenvalues of the centered adjacency matrix don’t give much information about the upper tail probabilities. 2. (2) It is important that we are looking at $\mathcal{G}(n,m)$ and not $\mathcal{G}(n,p)$, because – as discussed in the introduction – in $\mathcal{G}(n,p)$ the entropically favorable way to reduce the triangle count is to reduce the number of edges; again, this primarily affects the largest eigenvector and so is not related to the centered adjacency matrix. ###### Lemma 3. Let $A$ be the adjacency matrix of a graph with $n$ vertices and $m$ edges. For any $p\in\mathbb{R}$, (14) $\operatorname{tr}[(A-p\mathbf{1}+pI)^{3}]=\operatorname{tr}[A^{3}]-p^{3}n^{3}+p^{3}n+6mp(np-2p+1)+3p^{3}n(n-1)-3p\sum_{i}d_{i}^{2},$ where $d_{i}$ is the degree of vertex $i$. Or, if $p=m/\binom{n}{2}$, then (15) $\operatorname{tr}[(A-p\mathbf{1}+pI)^{3}]\leq\operatorname{tr}[A^{3}]-p^{3}n^{3}+p^{3}n+6mp.$ Note that if $A$ is sampled from $\mathcal{G}(n,m)$ and $p=m/\binom{n}{2}$ then $\mathbb{E}A=p\mathbf{1}-pI$, and so the quantity of interest in Lemma 3 is in fact the centered adjacency matrix $A-\mathbb{E}A$. ###### Proof. We expand everything in gory detail: $\displaystyle\operatorname{tr}[(A-p\mathbf{1}+pI)^{3}]-\operatorname{tr}[A^{3}]+p^{3}\operatorname{tr}[\mathbf{1}^{3}]-p^{3}\operatorname{tr}[I^{3}]$ $\displaystyle=3\operatorname{tr}[-pA^{2}\mathbf{1}+p^{2}A\mathbf{1}^{2}+pA^{2}I+p^{2}AI^{2}+p^{3}\mathbf{1}^{2}I-p^{3}\mathbf{1}I^{2}-2p^{2}A\mathbf{1}I]$ $\displaystyle=3\operatorname{tr}[-pA^{2}\mathbf{1}+np^{2}A\mathbf{1}+pA^{2}+p^{2}A+np^{3}\mathbf{1}-p^{3}\mathbf{1}-2p^{2}A\mathbf{1}]$ $\displaystyle=3(-p\operatorname{tr}[A^{2}\mathbf{1}]+2(n-2)p^{2}m+2pm+0+n^{2}p^{3}-np^{3}),$ where we used the fact that $\operatorname{tr}[A\mathbf{1}]$ is the sum of the entries of $A$ and $\operatorname{tr}[A^{2}]$ is the sum of squares of entries of $A$; since $A$ is an adjacency matrix, both of these are $2m$. Finally, (16) $\operatorname{tr}[A^{2}\mathbf{1}]=\sum_{i,k}(A^{2})_{ik}=\sum_{i,j,k}A_{ij}A_{jk}=\sum_{j}d_{j}^{2}.$ This proves the equality. For the inequality, Cauchy-Schwarz implies that $\sum_{i}d_{i}^{2}\geq\frac{1}{n}(\sum_{i}d_{i})^{2}=\frac{4m^{2}}{n}=2mp(n-1)$. After applying this inequality and rewriting $3p^{3}n(n-1)$ as $6p^{2}m$, we obtain the inequality. ∎ Combining Lemma 3 with the observation that $\mathbb{E}\operatorname{tr}[A^{3}]=p^{3}n^{3}+O(n^{2})$ when $A$ is the adjacency matrix of a $\mathcal{G}(n,m)$ graph, we arrive at the following consequence: ###### Corollary 4. Let $A$ be the adjacency matrix of a $\mathcal{G}(n,m)$ graph and let $\tilde{A}=A-\mathbb{E}A$. For any $t\geq 0$, (17) $\Pr(\operatorname{tr}[A^{3}]\leq\mathbb{E}\operatorname{tr}[A^{3}]-t)\leq\Pr(\operatorname{tr}[\tilde{A}^{3}]\leq-t+O(n^{2}))$ For an inequality in the other direction, note that by the same argument as in Lemma 3, as long as $\sum_{i}d_{i}^{2}\leq n^{3}p^{2}+D$, we have (18) $\operatorname{tr}[(A-p\mathbf{1}+pI)^{3}]=\operatorname{tr}[A^{3}]-p^{3}n^{3}+O(D+n^{2}).$ ###### Corollary 5. With the notation of Corollary 4, if $D=\Omega(n^{2})$ then (19) $\Pr\left(\operatorname{tr}[A^{3}]\leq\mathbb{E}\operatorname{tr}[A^{3}]-t\right)\geq\Pr\left(\operatorname{tr}[\tilde{A}^{3}]\leq-t-\Omega(D)\text{ and }\sum_{i}d_{i}^{2}\leq n^{3}p^{2}+D\right).$ ## 4\. Large deviations for eigenvalues of random matrices In this section and beyond, we let $A$ denote a generic random matrix and we estimate the most positive eigenvalues of $A$. Since we are looking at lower tails, the most important such matrix to keep in mind is minus the centered adjacency matrix, previously denoted $\tilde{A}$ or $A-\mathbb{E}A$. This is the same as plus the centered adjacency matrix of a random graph with edge density $q=1-p$. The proof of Theorem 1 ($p\geq\frac{1}{2}$) thus relies on results for $q\leq\frac{1}{2}$, while the proof of Theorem 2 ($p\leq\frac{1}{2}$) relies on results for $q\geq\frac{1}{2}$. ###### Definition 6. For a random variable $\xi$, its cumulant-generating function is (20) $\Lambda_{\xi}(s)=\ln\mathbb{E}\exp(s\xi)$ whenever the expectation exists; when the expectation does not exist, we set $\Lambda_{\xi}(s)=+\infty$. ###### Definition 7. The random variable $\xi$ is _subgaussian_ if there exists a constant $C$ such that $\Lambda_{\xi}(t)\leq Ct^{2}$ for every $t\in\mathbb{R}$. Note that according to our definition, a subgaussian random variable has mean zero (since if $\Lambda_{\xi}(t)$ is finite on a neighborhood of 0 then $\Lambda_{\xi}(0)=0$ and $\Lambda_{\xi}^{\prime}(0)=\mathbb{E}\xi$, and so if $\mathbb{E}\xi$ is non-zero then one cannot have $\Lambda_{\xi}(t)\leq Ct^{2}$ on a neighborhood of 0). Note also that if $\mathbb{E}\xi=0$ and $\\|\xi\\|_{\infty}<\infty$ then $\xi$ is subgaussian. ###### Definition 8. For a function $f:\mathbb{R}\to\mathbb{R}$, its Legendre transform is the function $f^{}:\mathbb{R}\to\mathbb{R}\cup\\{+\infty\\}$ defined by (21) $f^{}(y)=\sup_{x\in\mathbb{R}}\\{xy-f(x)\\}$ Some basic properties of the Legendre transform include: * • If $f\leq g$ then $f^{}\geq g^{}$. * • If $f$ is convex then $f^{*}=f$. • If $f(x)=cx^{2}$ then $f^{}(x)=\frac{x^{2}}{4c}$. Our goal in this note is to establish large deviations principles for extreme eigenvalues and singular values of random matrices. We will consider a symmetric $n\times n$ random matrix $A_{n}$ (or sometimes just $A$) having i.i.d. upper-diagonal entries and zero diagonal entries. The letter $\xi$ will always denote a random variable that is distributed as an upper-diagonal element of $A$, and we will always assume that $\xi$ is subgaussian. We write $\lambda_{i}(A)$ for the eigenvalues of $A$ (in non-increasing order) and $\sigma_{i}(A)$ for the singular values of $A$ (in non-increasing order). For the definition of a large deviations principle (LDP), we refer to [36, Chapter 27]. ###### Theorem 9. Let $\xi$ be a subgaussian random variable. For any integer $k\geq 1$ and any sequence $m_{n}$ satisfying $\sqrt{n}\ll m_{n}\ll n$, the sequence (22) $\frac{1}{m_{n}}(\sigma_{1}(A_{n}),\dots,\sigma_{k}(A_{n}))$ satisfies an LDP with speed $m_{n}^{2}$ and good rate function $I:\mathbb{R}_{+}^{k}\to[0,\infty)$ given by (23) $I(x)=\frac{\|x\|^{2}}{2}\inf_{s\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}.$ If we assume in addition that the function $s\mapsto\frac{\Lambda^{}(s)}{s^{2}}$ achieves its infimum at some $s\geq 0$, then the sequence (24) $\frac{1}{m_{n}}(\lambda_{1}(A_{n}),\dots,\lambda_{k}(A_{n}))$ satisfies an LDP with speed $m_{n}^{2}$ and the same good rate function $I$ as above. If $A_{n}$ is the centered adjacency matrix of $\mathcal{G}(n,q)$ then it is covered by Theorem 9, where $\xi$ is the random variable taking the values $-q$ and $1-q$ with probabilities $1-q$ and $q$ respectively. In this case, we have (25) $\Lambda_{\xi}^{}(s)=D(q+s,q):=(q+s)\ln\frac{q+s}{q}+(1-q-s)\ln\frac{1-q-s}{1-q},$ with the understanding that $\Lambda_{\xi}^{}(s)=+\infty$ whenever $q+s\not\in(0,1)$. It is not hard to check – and we will do it in Section 5.5 – that $\frac{\Lambda_{\xi}^{}(s)}{s^{2}}$ achieves its infimum at some $s\geq 0$ if and only if $q\leq\frac{1}{2}$. In the case that $\frac{\Lambda^{}(s)}{s^{2}}$ saturates its infimum only at negative $s$ (corresponding to $q>\frac{1}{2}$ in the Bernoulli example), we are not able to show an LDP for the eigenvalues. Note, however, that $\sum_{i}\sigma_{i}^{2}(A)\geq\sum_{i}\lambda_{i}^{2}(A)$ and so our LDP for singular values provides an upper bound: it implies, for example, that (26) $\frac{1}{m_{n}^{2}}\ln\Pr\left(\sqrt{\sum_{i}\lambda_{i}^{2}(A_{n})}>m_{n}t\right)\leq-\frac{t^{2}}{2}\inf_{s\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}+o(1)$ On the other hand, we can also easily show the lower bound (27) $\frac{1}{m_{n}^{2}}\ln\Pr\left(\sqrt{\sum_{i}\lambda_{i}^{2}(A_{n})}>m_{n}t\right)\geq-\frac{t^{2}}{2}\inf_{s\geq 0}\frac{\Lambda^{}(s)}{s^{2}}-o(1),$ but the assumption that $\frac{\Lambda^{}(s)}{s^{2}}$ saturates its infimum only at negative $s$ implies that these bounds are non-matching. There are natural examples (including the Bernoulli example mentioned above) where $s^{-2}\Lambda^{}(s)$ is increasing for $s\geq 0$. In this case, (28) $\inf_{s\geq 0}s^{-2}\Lambda^{}(s)=\lim_{s\to 0}s^{-2}\Lambda^{}(s)=\frac{1}{2}(\Lambda^{})^{\prime\prime}(0)=\frac{1}{2\mathbb{E}\xi^{2}},$ and so our lower bound (for simplicity, focusing only on the case $k=1$) becomes (29) $\frac{1}{m_{n}^{2}}\ln\Pr\left(\lambda_{1}(A_{n})>m_{n}t\right)\geq-\frac{t^{2}}{4\mathbb{E}\xi^{2}}-o(1).$ When $\xi$ has a Gaussian distribution, this turns out to be sharp, but we show that it is not sharp in general. ###### Theorem 10. In the setting of Theorem 9, if $\mathbb{E}\xi^{3}<0$ and $\lim_{s\to\infty}s^{-2}\Lambda(s)=0$ then there exists some $\eta>0$ such that for any $t>0$, (30) $\lim_{n\to\infty}\frac{1}{m_{n}^{2}}\ln\Pr\left(\lambda_{1}(A_{n})>m_{n}t\right)>-(1-\eta)\frac{t^{2}}{4\mathbb{E}\xi^{2}}.$ In particular, the assumptions of Theorem 10 are satisfied for the (centered) Bernoulli random variable with $q>\frac{1}{2}$ mentioned above. For our applications to random graphs, we require a version of Theorem 9 for random bits chosen without replacement. Specifically, we consider the Erdős- Rényi random graphs $\mathcal{G}(n,m)$, where $m$ is an integer satisfying $\|m-q\binom{n}{2}\|=O(1)$ (and $q\in(0,1)$ is fixed). ###### Theorem 11. Fix $q\in(0,1)$ and let $A_{n}$ be the centered adjacency matrix of a $\mathcal{G}(n,m)$ random graph with $\|m-q\binom{n}{2}\|=O(1)$. For any integer $k\geq 1$ and any sequence $m_{n}$ satisfying $\sqrt{n}\ll m_{n}\ll n$, the sequence (31) $\frac{1}{m_{n}}(\sigma_{1}(A_{n}),\dots,\sigma_{k}(A_{n}))$ satisfies an LDP with speed $m_{n}^{2}$ and good rate function $I:\mathbb{R}_{+}^{k}\to[0,\infty)$ given by $I(x)=\frac{\|x\|^{2}}{2}\cdot\frac{\ln\frac{1-q}{q}}{1-2q}$ (or $I(x)=\|x\|^{2}$ when $q=\frac{1}{2}$). If, in addition, $q\leq\frac{1}{2}$ then the sequence (32) $\frac{1}{m_{n}}(\lambda_{1}(A_{n}),\dots,\lambda_{k}(A_{n}))$ also satisfies an LDP with the same speed and rate function. ## 5\. Upper bound Most of this work is concerned with handling the triangle-count contribution of very negative eigenvalues, but we also need to show that there is no significant contribution from the rest. For this, we will use a deviation inequality from [33]: ###### Theorem 12. Assume that $\\|\xi\\|_{\infty}<\infty$, and let $f:\mathbb{R}\to\mathbb{R}$ be a 1-Lipschitz, convex function. Define $X_{n}=\frac{1}{n}\sum_{i=1}^{n}f(n^{-1/2}\lambda_{i}(A_{n}))$. Then there is a universal constant $C<\infty$ such that for any $\delta\gg n^{-1}$, (33) $\Pr(\|X_{n}-\mathbb{E}X_{n}\|\geq\delta)\leq C\exp\left(-\frac{n^{2}\delta^{2}}{C\\|\xi\\|_{\infty}^{2}}\right).$ The main observation is that in the regime we are interested in (namely, eigenvalues or singular values of order $\omega(\sqrt{n})$), the probability of large eigenvalues can be controlled by a union bound over the potential eigenvectors. Let $\mathcal{M}_{k}$ be the set of $n\times n$ matrices with rank at most $k$ and Frobenius norm at most 1. Let $\mathcal{M}_{k}^{+}\subset\mathcal{M}_{k}$ consist of those matrices that are symmetric and positive semidefinite. ###### Lemma 13. For any symmetric matrix $A$, (34) $\left(\sum_{i=1}^{k}\max\\{0,\lambda_{i}(A)\\}^{2}\right)^{1/2}=\sup_{M\in\mathcal{M}_{k}^{+}}\langle A,M\rangle.$ For any matrix $A$, (35) $\left(\sum_{i=1}^{k}\sigma_{i}(A)^{2}\right)^{1/2}=\sup_{M\in\mathcal{M}_{k}}\langle A,M\rangle.$ ###### Proof. Let $UDU^{T}=A$ be an eigen-decomposition of $A$ (where $D$ is diagonal and $U$ is orthogonal), and let $\tilde{D}$ be $D$ but with all but the $k$th- largest diagonal entries set to zero. Define (36) $M=\frac{U\tilde{D}U^{T}}{\\|\tilde{D}\\|_{F}}=\frac{U\tilde{D}U^{T}}{\left(\sum_{i=1}^{k}\lambda_{i}(A)^{2}\right)^{1/2}}.$ Then $M\in\mathcal{M}_{k}^{+}$ and $\langle A,M\rangle=\\|\tilde{D}\\|_{F}=\left(\sum_{i=1}^{k}\lambda_{i}(A)^{2}\right)^{1/2}$. This proves one direction of the first claim. For the other direction, take any $M\in\mathcal{M}_{k}^{+}$, and decompose $A$ as $A_{+}-A_{-}$, where $A_{+}$ and $A_{-}$ are positive semi-definite and the non-zero eigenvalues of $A_{+}$ are the positive eigenvalues of $A$. Then (37) $\langle A,M\rangle\leq\langle A_{+},M\rangle\leq\\|A_{+}\\|_{F}\\|M\\|_{F}\leq\sqrt{\sum_{i=1}^{k}\lambda_{i}(A_{+})^{2}}=\sqrt{\sum_{i=1}^{k}\lambda_{i}(A)^{2}}.$ This proves the first claim. The proof of the second claim is identical, but uses a singular value decomposition instead of an eigen-decomposition. ∎ Hence, in order to prove the upper bounds in Theorem 9, it suffices to control (38) $\Pr\left(\sup_{M\in\mathcal{M}_{k}^{+}}\langle A,M\rangle>tn^{\alpha}\right).$ The first step is to replace the supremum with a finite maximum. ### 5.1. The net argument ###### Definition 14. For a subset $\mathcal{N}$ of a metric space $(X,d)$, we say that $\mathcal{N}$ is an $\epsilon$-net of $X$ if for every $x\in X$ there exists $y\in\mathcal{N}$ with $d(x,y)\leq\epsilon$. ###### Lemma 15. Let $\mathcal{N}\subset\mathcal{M}_{k}$ be an $\epsilon$-net (with respect to $\\|\cdot\\|_{F}$) for $\epsilon<\frac{1}{2}$. Then for any symmetric matrix $A$, (39) $\sup_{M\in\mathcal{M}_{k}}\langle A,M\rangle\leq\frac{1}{1-2\epsilon}\sup_{N\in\mathcal{N}}\langle A,N\rangle.$ ###### Proof. Fix $M\in\mathcal{M}_{k}$, and choose $N\in\mathcal{N}$ with $\\|N-M\\|_{F}\leq\epsilon$. Note that $N-M$ has rank at most $2k$, and it has at most $k$ positive eigenvalues and $k$ negative eigenvalues. Letting $\epsilon M_{0}$ and $\epsilon M_{1}$ be the positive and negative parts of $N-M$, we have $\\|M_{0}\\|_{F}\leq\\|N-M\\|_{F}/\epsilon\leq 1$ and (similarly $\\|M_{1}\\|_{F}\leq 1$). In other words, we can decompose (40) $M=N+\epsilon M_{0}+\epsilon M_{1}$ with $N\in\mathcal{N}$ and $M_{0},M_{1}\in\mathcal{M}_{k}$. We continue this construction recursively: for every finite binary string $v$ and matrix $M_{v}\in\mathcal{M}_{k}$, we can find $N_{v}\in\mathcal{N}$ and $M_{v0},M_{v1}\in\mathcal{M}_{k}$ such that (41) $M_{v}=N_{v}+\epsilon M_{v0}+\epsilon M_{v1}.$ Recursing this construction $m$ levels, it follows that (with $S_{m}$ being the set of binary strings of length $m$ and $\|v\|$ denoting the length of the string $v$) (42) $M=\sum_{\ell=0}^{m-1}\sum_{v\in S_{\ell}}\epsilon^{-\|v\|}N_{v}+\epsilon^{-m}\sum_{v\in S_{m}}M_{v}.$ Since $\|S_{m}\|=2^{m}$ and each $M_{v}$ has $\\|M_{v}\\|_{F}\leq 1$, the remainder term converges to zero and we can continue this construction to the limit: (43) $M=\sum_{\ell=0}^{\infty}\sum_{v\in S_{\ell}}\epsilon^{-\|v\|}N_{v},$ where the outer sum converges in Frobenius norm. Taking the inner product with $A$, note that Cauchy-Schwarz and the convergence of the sum imply that the inner product and summation can be exchanged: (44) $\langle M,A\rangle=\sum_{\ell=0}^{\infty}\sum_{v\in S_{\ell}}\epsilon^{-\|v\|}\langle N_{v},A\rangle\leq\sum_{\ell=0}^{\infty}\|S_{\ell}\|\epsilon^{-\ell}\sup_{N\in\mathcal{N}}\langle N,A\rangle=\frac{1}{1-2\epsilon}\sup_{N\in\mathcal{N}}\langle N,A\rangle.$ ∎ The construction in Lemma 15 approximates the supremum over $M\in\mathcal{M}_{k}$, which is enough for most of what we will do. In some cases, we will want the supremum over $M\in\mathcal{M}_{k}^{+}$ instead, but that can be handled also: ###### Lemma 16. Let $\mathcal{N}\subset\mathcal{M}_{k}$ and $\mathcal{N}^{+}\subset\mathcal{M}_{k}^{+}$ be $\epsilon$-nets (with respect to $\\|\cdot\\|_{F}$) for $\epsilon<\frac{1}{2}$ Then for any symmetric matrix $A$, (45) $\sup_{M\in\mathcal{M}_{k}^{+}}\langle A,M\rangle\leq\sup_{N^{+}\in\mathcal{N}^{+}}\langle A,N^{+}\rangle+\frac{2\epsilon}{1-2\epsilon}\sup_{N\in\mathcal{N}}\langle A,N\rangle.$ ###### Proof. Fix $M\in\mathcal{M}_{k}^{+}$ and choose $M_{0}\in\mathcal{N}^{+}$ such that $\\|M_{0}-M\\|_{F}\leq\epsilon$. Then $\frac{M_{0}-M}{\epsilon}$ has rank at most $2k$ and Frobenius norm at most $1$. Hence, we can write $M_{0}-M=\epsilon N_{0}+\epsilon N_{1}$, where $N_{0},N_{1}\in\mathcal{M}_{k}$. It follows that (46) $\langle A,M\rangle=\langle A,M_{0}\rangle+\epsilon\langle A,N_{0}\rangle+\epsilon\langle A,N_{1}\rangle,$ and we conclude by applying Lemma 15 to $\langle A,N_{0}\rangle$ and $\langle A,N_{1}\rangle$. ∎ We have shown that to approximate the supremum it suffices to take a good enough net. In order to put this together with a union bound, we need a bound on the size of a good net. Our starting point is the following basic bound in Euclidean space [35, Corollary 4.2.13] ###### Lemma 17. The unit Euclidean ball in $\mathbb{R}^{d}$ admits an $\epsilon$-net (with respect to the Euclidean metric) $\mathcal{N}$ satisfying $\|\mathcal{N}\|\leq(3/\epsilon)^{d}$. ###### Corollary 18. There is a constant $C$ such that for any $0<\epsilon<1$, there is an $\epsilon$-net (with respect to Frobenius norm) for $\mathcal{M}_{k}$ of size at most $(C/\epsilon)^{2nk}$ and an $\epsilon$-net for $\mathcal{M}_{k}^{+}$ of size at most $(C/\epsilon)^{k}$. ###### Proof. Let $\tilde{\mathcal{N}}$ be an $(\epsilon/2)$-net for the set of $n\times k$ matrices with Frobenius norm at most one. Since this space is isometric to $\mathbb{R}^{nk}$ with the Euclidean norm, Lemma 17 implies that we can choose such a $\tilde{\mathcal{N}}$ with $\|\tilde{\mathcal{N}}\|\leq(C/\epsilon)^{nk}$. Now let $\mathcal{N}=\\{XY^{T}:X,Y\in\tilde{\mathcal{N}}\\}$. Then $\|\mathcal{N}\|\leq\|\tilde{\mathcal{N}}\|^{2}\leq(C/\epsilon)^{2nk}$. It remains to show that $\mathcal{N}$ is an $\epsilon$-net. Since $\\|XY^{T}\\|_{F}\leq\\|X\\|_{F}\\|Y\\|_{F}$, it follows that every $N\in\mathcal{N}$ has $\\|N\\|_{F}\leq 1$; also, each $N\in\mathcal{N}$ clearly has rank at most $k$. Now choose an arbitrary $M\in\mathcal{M}_{k}$ and write $M=AB^{T}$ for $n\times k$ matrices $A$ and $B$ of Frobenius norm at most 1 (for example, this can be done using a singular value decomposition). Choose $X,Y\in\tilde{\mathcal{N}}$ with $\\|X-A\\|_{F}\leq\frac{\epsilon}{2}$ and $\\|Y-B\\|_{F}\leq\frac{\epsilon}{2}$. Then $\displaystyle\\|XY^{T}-M\\|_{F}$ $\displaystyle\leq\\|XY^{T}-AY^{T}\\|_{F}+\\|AY^{T}-AB^{T}\\|_{F}$ $\displaystyle\leq\\|X-A\\|_{F}+\\|Y^{T}-B^{T}\\|_{F}$ $\displaystyle\leq\epsilon.$ To construct an $\epsilon$-net of $\mathcal{M}_{k}^{+}$, take $\tilde{\mathcal{N}}$ be as above and let $\mathcal{N}=\\{XX^{T}:X\in\tilde{\mathcal{N}}\\}$. Then $\|\mathcal{N}\|\leq\|\tilde{\mathcal{N}}\|$, and the proof that $\mathcal{N}$ is an $\epsilon$-net of $\mathcal{M}_{k}^{+}$ is essentially the same as the proof above, the only change being that every $M\in\mathcal{M}_{k}^{+}$ can be written as $M=AA^{T}$ for an $n\times k$ matrix $A$ of Frobenius norm at most 1. ∎ Applying a union bound over these nets gives the main result of this section: singular values and eigenvalues of $A$ can be controlled in terms of the deviations of linear functions of $A$. The main point here is that (as we will show in the next section) if $t\gg\sqrt{n}$ then the $O(nk\ln\frac{1}{\epsilon})$ terms are negligible compared to the other terms. ###### Proposition 19. Let $A$ be a symmetric $n\times n$ random matrix with i.i.d. entries. For any integer $k\geq 1$, any $0<\epsilon<\frac{1}{2}$, and any $t>0$, (47) $\ln\Pr\left(\sum_{i=1}^{k}\sigma_{i}^{2}(A)>t\right)\leq\sup_{M\in\mathcal{M}_{k}}\ln\Pr\left(\langle A,M\rangle\geq(1-2\epsilon)t\right)+O(nk\ln\frac{1}{\epsilon}).$ ###### Proof. For the first inequality, let $\mathcal{N}$ be an $\epsilon$-net for $\mathcal{M}_{k}$ according to Corollary 18. By Lemma 13 and Lemma 15 $\displaystyle\Pr\left(\sum_{i=1}^{k}\sigma_{i}^{2}(A)>t\right)$ $\displaystyle=\Pr\left(\sup_{M\in\mathcal{M}_{k}}\langle A,M\rangle>t\right)$ $\displaystyle\leq\Pr\left(\max_{N\in\mathcal{N}}\langle A,N\rangle>(1-2\epsilon)t\right).$ By a union bound, $\displaystyle\Pr\left(\max_{N\in\mathcal{N}}\langle A,N\rangle>(1-2\epsilon)t\right)$ $\displaystyle\leq\sum_{N\in\mathcal{N}}\Pr\left(\langle A,N\rangle>(1-2\epsilon)t\right)$ $\displaystyle\leq\|\mathcal{N}\|\sup_{M\in\mathcal{M}_{k}}\Pr\left(\langle A,M\rangle>(1-2\epsilon)t\right),$ which, by our bound on $\|\mathcal{N}\|$, completes the proof of the first claim. ∎ We remark that it is possible to prove a version of Proposition 19 for eigenvalues also, giving an upper bound on $\Pr(\sum\lambda_{i}^{2}(A)>t)$ in terms of (48) $\sup_{M^{+}\in\mathcal{M}_{k}^{+}}\Pr\left(\langle A,M^{+}\rangle\geq t\right).$ This can in principle give a better bound on the eigenvalues than for the singular values. The issue is that we do not know how to exploit the additional information that we are testing $A$ against a positive semidefinite matrix. ### 5.2. Hoeffding-type argument Using a Hoeffding-type argument, we can get a sharp upper bound on (49) $\sup_{M\in\mathcal{M}_{k}}\ln\Pr\left(\langle A,M\rangle\geq t\right)$ for any $k$ and any $t$ (in fact, the sharp upper bound turns out not to depend on $k$). ###### Lemma 20. If $\xi$ is subgaussian then (50) $4\sup_{s\in\mathbb{R}}\frac{\Lambda_{\xi}(s)}{s^{2}}=\left(\inf_{s\in\mathbb{R}}\frac{\Lambda_{\xi}^{}(u)}{u^{2}}\right)^{-1}<\infty.$ ###### Proof. The fact that $\sup_{s\in\mathbb{R}}\frac{\Lambda(s)}{s^{2}}<\infty$ is the definition of subgaussianity. To show the claimed identity, let $L=\sup_{t\in\mathbb{R}}\frac{\Lambda_{\xi}(t)}{t^{2}}$ and define $M_{L}(s)=Ls^{2}$. Clearly, $\Lambda(s)\leq M_{L}(s)$ for all $s\in\mathbb{R}$. It follows that $\Lambda^{}(u)\geq M_{L}^{}(u)=\frac{u^{2}}{4L}$; in other words, (51) $\frac{\Lambda^{}(u)}{u^{2}}\geq\frac{M_{L}^{}(u)}{u^{2}}=\frac{1}{4L}$ for all $u$. This shows that (52) $4\sup_{s\in\mathbb{R}}\frac{\Lambda(s)}{s^{2}}\geq\left(\inf_{u\in\mathbb{R}}\frac{\Lambda^{}(u)}{u^{2}}\right)^{-1}.$ For the other direction, suppose that for some $L^{\prime}$ we have $\Lambda^{}(u)\geq\frac{u^{2}}{4L^{\prime}}=M^{1/(4L)^{\prime}}(u)$ for every $u$. Then (since $\Lambda$ is convex) $\Lambda(t)=\Lambda^{}(t)\leq M_{1/(4L^{\prime})}^{}(t)=L^{\prime}t^{2}$ for every $t$. The definition of $L$ ensures that $L^{\prime}\geq L$, and this shows the other direction of the claim. ∎ ###### Proposition 21. Let $\xi$ be a random variable with globally-finite moment-generating function, and define (53) $\Lambda(s)=\ln\mathbb{E}\exp(s\xi)$ to be the cumulant-generating function of $\xi$. Let $A$ be a symmetric random matrix with zero diagonal, and with upper-diagonal elements distributed independently according to $\xi$. Define $\ell^{}=\sup_{s>0}\frac{\Lambda(s)}{s^{2}}$. Then (54) $\sup_{\\|M\\|_{F}\leq 1}\Pr(\langle A,M\rangle>t)\leq\exp\left(-\frac{t^{2}}{8\sup_{s>0}\frac{\Lambda(s)}{s^{2}}}\right)=\exp\left(-\frac{t^{2}}{2}\inf_{s>0}\frac{\Lambda^{}(s)}{s^{2}}\right).$ ###### Proof. Since $\langle A,M\rangle=\langle A,(M+M^{T})/2\rangle$ and since $\\|(M+M^{T})/2\\|_{F}\leq\\|M\\|_{F}$, it suffices to consider only symmetric matrices $M$. Let $m=\frac{n}{n-1}{2}$ and let $\xi_{1},\dots,\xi_{m}$ be the upper-diagonal elements of $A$, in any order. Let $\\|M\\|\leq 1$ be symmetric, with upper-diagonal entries $a_{1},\dots,a_{m}$. Then $\langle A,M\rangle=2\sum_{i=1}^{m}a_{i}\xi_{i}$, and so (for any $s>0$) $\displaystyle\Pr(\langle A,M\rangle>t)$ $\displaystyle=\Pr\left(\sum a_{i}\xi_{i}>t/2\right)$ $\displaystyle=\Pr\left(e^{s\sum a_{i}\xi_{i}}>e^{st/2}\right)$ $\displaystyle\leq e^{-st/2}\mathbb{E}e^{s\sum a_{i}\xi_{i}}$ $\displaystyle=\exp\left(\sum_{i}\Lambda(sa_{i})-st/2\right),$ where the inequality follows from Markov’s inequality. Now, $\sum_{i=1}^{m}a_{i}^{2}\leq\frac{1}{2}\\|M\\|_{F}^{2}\leq\frac{1}{2}$, and so if we set $\ell^{}=\sup_{r>0}\frac{\Lambda(r)}{r^{2}}$ then (55) $\sum_{i}\Lambda(sa_{i})=\sum_{i}\frac{\Lambda(sa_{i})}{(sa_{i})^{2}}(sa_{i})^{2}\leq s^{2}\sum_{i}\ell^{}a_{i}^{2}\leq\frac{s^{2}\ell^{}}{2}.$ Hence, (56) $\Pr(\langle A,M\rangle>t)\leq\exp\left(\frac{s^{2}\ell^{}}{2}-\frac{st}{2}\right),$ and the first claim follows by optimizing over $s$. The second claim follows immediately from Lemma 20. ∎ Putting Proposition 21 together with Proposition 19, we arrive at the following upper bound for singular values: ###### Corollary 22. Let $A$ be a symmetric $n\times n$ random matrix with i.i.d. upper diagonal entries. Assuming that the entries are subgaussian and have cumulant- generating function $\Lambda$, let $L=\inf_{s\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}$. Then for any integer $k$ and any $t>0$, if $t^{2}L>2nk$ then (57) $\ln\Pr\left(\sqrt{\sum_{i=1}^{k}\sigma_{i}^{2}(A)}>t\right)\leq-\frac{t^{2}L}{2}+O\left(nk\ln\frac{t^{2}L}{nk}\right).$ ###### Proof. We combine Proposition 21 and Proposition 19, setting $\epsilon=\frac{nk}{t^{2}L}$ (which is less than $\frac{1}{2}$ by assumption). This yields an upper bound of (58) $-\frac{t^{2}L}{2}+O\left(nk+nk\ln\frac{t^{2}L}{nk}\right),$ and the $nk$ term can be absorbed in the final term. ∎ ###### Remark 23. Note that the argument leading to Corollary 22 applies even when the entries $\xi_{ij}$ are not identically distributed as long as $L\leq\inf_{s}\frac{\Lambda_{ij}^{}(s)}{s^{2}}$ for every $i$, where $\Lambda_{ij}$ is the cumulant-generating function of $\xi_{ij}$. ### 5.3. Lower bound In this section, we give a lower bound that matches the upper bound of Corollary 22 whenever $\sqrt{n}\ll t\ll n$. The starting point is the lower bound of Cramér’s theorem [36, Theorem 27.3] ###### Theorem 24. Let $\xi$ be a mean-zero random variable with everywhere-finite cumulant- generating function $\Lambda_{\xi}$. Let $\xi_{1},\dots,\xi_{m}$ be independent copies of $\xi$. Then (59) $\frac{1}{m}\ln\Pr\left(\sum_{i=1}^{m}\xi_{i}>mt\right)\to-\Lambda^{}(t)$ as $m\to\infty$. ###### Proposition 25. In the setting of Corollary 22, suppose in addition that the function $s\mapsto s^{-2}\Lambda^{}(s)$ achieves its minimum at some finite $s\in\mathbb{R}$. Then for any $1\ll t\ll n^{2}$ and for any $w_{1},\dots,w_{k}>0$, we have (60) $\ln\Pr\left(\sum_{i=1}^{k}w_{i}\sigma_{i}(A_{n})>\|w\|\sqrt{t}\right)\geq-\frac{tL}{2}-o(t).$ If $s\mapsto s^{-2}\Lambda^{}(s)$ achieves its minimum at some $s\geq 0$, then for any $1\ll t\ll n^{2}$ and for any $w_{1},\dots,w_{k}>0$, we have (61) $\ln\Pr\left(\sum_{i=1}^{k}w_{i}\lambda_{i}(A_{n})>\|w\|\sqrt{t}\right)\geq-\frac{tL}{2}-o(t).$ Choosing an arbitrary $w_{1},\dots,w_{k}$ and applying the Cauchy-Schwarz inequality, Proposition 25 implies the same lower bounds on $\ln\Pr(\sum_{i}\sigma_{i}^{2}(A_{n})>t)$ and $\ln\Pr(\sum_{i}\lambda_{i}^{2}(A_{n})>t)$. In particular, it really is a lower bound that matches the upper bound of Corollary 22. ###### Proof. Fix $t$ and assume that $\frac{\Lambda^{}(s)}{s^{2}}$ achieves its minimum at $s_{}\in\mathbb{R}$. Actually, we will assume $s_{}\neq 0$; the case $s_{}=0$ is easily handled by replacing $s_{}$ with $\epsilon>0$ everywhere, and then sending $\epsilon\to 0$. Fix $w_{1},\dots,w_{k}$ and assume $\sum_{i}w_{i}^{2}=t$; because the statement of the proposition is homogeneous in $w$, this is without loss of generality. Now choose the smallest integers $\ell_{1},\dots,\ell_{k}$ so that $\ell_{i}-1\geq\frac{w_{i}}{s_{}}$. We write $\|\ell\|^{2}$ for $\sum_{i}\ell_{i}^{2}$, and note that $\|\ell\|^{2}\geq\frac{1}{s_{}^{2}}\sum_{i}w_{i}^{2}=\frac{t}{s_{}^{2}}$, meaning that $1\ll\|\ell\|^{2}\ll n^{2}$. Let $M$ be a block-diagonal matrix, whose non-zero entries are all equal to $s_{}$, appearing in blocks of size $\ell_{i}\times\ell_{i}$ for $i=1,\dots,k$. (The fact that $\sum_{i}\ell_{i}\leq\sqrt{k}\|\ell\|\ll n$ implies that these blocks do indeed fit into an $n\times n$ matrix.) Then $M$ has rank $k$, and the singular values of $M$ are $s_{}\ell_{i}$ for $i=1,\dots,k$; note that our choices of $\ell_{i}$ ensure that $w_{i}\leq\sigma_{i}(M)\leq w_{i}+2s_{}$. Moreover, if we set $m=\sum_{i}\frac{\ell_{i}(\ell_{i}-1)}{2}$ (which is also an integer, and counts the number of non-zero upper-diagonal elements of $M$) then $\langle A,M\rangle$ is equal in distribution to $2s_{}\sum_{i=1}^{m}\xi_{i}$. Hence, (62) $\Pr\left(\langle A,M\rangle>t\right)=\Pr\left(\operatorname{sgn}(s_{})\sum_{i=1}^{m}\xi_{i}>\frac{t}{2\|s_{}\|}\right).$ Now, $m=\frac{1}{2}\|\ell\|^{2}-\frac{1}{2}\sum_{i}\ell_{i}$, while on the other hand (63) $\frac{t}{s_{}^{2}}=\frac{\sum_{i}w_{i}^{2}}{s_{}^{2}}\leq\sum_{i}(\ell_{i}-1)^{2}=\|\ell\|^{2}-2\sum_{i}\ell_{i}+2k.$ Since $\sum_{i}\ell_{i}\geq\|\ell\|\gg 1$, we have $\frac{t}{2s_{}^{2}}\leq m$ for sufficiently large $n$. Going back to our probability estimates, we have $\displaystyle\ln\Pr\left(\langle A,M\rangle>t\right)$ $\displaystyle=\ln\Pr\left(\operatorname{sgn}(s_{})\sum_{i=1}^{m}\xi_{i}>\frac{t}{2\|s_{}\|}\right)$ $\displaystyle\geq\ln\Pr\left(\operatorname{sgn}(s_{})\sum_{i=1}^{m}\xi_{i}>m\|s_{}\|\right)$ $\displaystyle=-m\Lambda^{}(s_{})+o(m)$ $\displaystyle=-\frac{t\Lambda^{}(s_{})}{2s_{}^{2}}-o(t),$ where the second-last equality follows by Cramér’s theorem (applied to the random variables $-\xi_{i}$ in case $s_{}<0$). By the Cauchy-Schwarz inequality we have (64) $\langle A,M\rangle\leq\sum_{i=1}^{k}\sigma_{i}(A)\sigma_{i}(M)\leq\sum_{i=1}^{k}\sigma_{i}(A)(w_{i}+2s_{})\\\ \leq\sum_{i=1}^{k}\sigma_{i}(A)w_{i}+2s_{}\sqrt{k}\sqrt{\sum_{i=1}^{k}\sigma_{i}^{2}(A)},$ and hence (65) $\Pr\left(\langle A,M\rangle>t\right)\leq\Pr\left(\sum_{i=1}^{k}\sigma_{i}(A)w_{i}>t-t^{2/3}\right)+\Pr\left(\sum_{i=1}^{k}\sigma_{i}^{2}(A)>\frac{t^{4/3}}{4s_{}^{2}k}\right).$ By Corollary 22, the second probability is of order $\exp(-\Omega(t^{4/3}))$, and hence (66) $\ln\Pr\left(\sum_{i=1}^{k}\sigma_{i}(A)w_{i}>t-t^{2/3}\right)\geq(1-o(1))\ln\Pr\left(\langle A,M\rangle>t\right)\geq-\frac{t\Lambda^{}(s_{})}{2s_{}^{2}}-o(t).$ Substituting in $t=\|w\|\sqrt{t}$ in place of $t-t^{2/3}$, the extra error term can be absorbed in the $o(t)$ term. For the second claim, simply note that if $s_{}>0$ then the matrix $M$ is positive semi-definite. Denoting $\lambda_{i}^{+}(A)=\max\\{0,\lambda_{i}(A)\\}$, we replace (64) by (67) $\langle A,M\rangle\leq\sum_{i=1}^{k}\lambda_{i}^{+}(A)\lambda_{i}(M)\leq\sum_{i=1}^{k}\lambda_{i}^{+}(A)(w_{i}+2s_{})\leq\sum_{i=1}^{k}\lambda_{i}^{+}(A)w_{i}+2s_{}\sqrt{k}\sqrt{\sum_{i=1}^{k}\sigma_{i}^{2}(A)},$ and the rest of the proof proceeds as before. ∎ There are a few extra useful facts that we can extract from the proof of Proposition 25, namely that we have explicit candidates for extremal eigenvectors and singular vectors. We will state these just for the largest eigenvector, but of course they also hold in other situations. ###### Corollary 26. Assume that $s\mapsto s^{-2}\Lambda^{}(s)$ achieves its minimum at some $s_{}\geq 0$. For $1\ll t\ll n^{2}$, let $\ell=\lceil 1+\sqrt{t}/s_{}\rceil$ and define $v\in\mathbb{R}^{n}$ by $v_{1},\dots,v_{\ell}=s_{}^{1/2}t^{-1/4}$ and $v_{\ell+1},\cdots,v_{n}=0$. Then $\|v\|\leq 1+o(1)$ and (68) $\ln\Pr(v^{T}A_{n}v\geq t)\geq-\frac{t^{2}L}{2}-o(t^{2}).$ Corollary 26 is immediate from the proof of Proposition 25, because in the case $k=1$ and $w_{1}=\sqrt{t}$, the $M$ that we constructed in that proof is exactly $\sqrt{t}vv^{T}$. ### 5.4. The LDP Putting together Corollary 22 and Proposition 25, we complete the proof of the LDP (Theorem 9). Take a sequence $m_{n}$ satisfying $\sqrt{n}\ll m_{n}\ll n$, and set $X=\frac{1}{m_{n}}(\sigma_{1}(A_{n}),\dots\sigma_{k}(A_{n}))$. Let $E\subset\mathbb{R}^{k}$ be any closed set, and let $t=\inf_{x\in E}\|x\|$. Then $\frac{1}{m_{n}}(\sigma_{1}(A_{n}),\dots,\sigma_{k}(A_{n}))\in E$ implies that $\sum\sigma_{i}^{2}(A_{n})>m_{n}^{2}t^{2}$. By Corollary 22, $\ln\Pr\left(X\in E\right)\leq\ln\Pr\left(\sum_{i=1}^{k}\sigma_{i}^{2}(A_{n})>m_{n}^{2}t^{2}\right)\\\ \leq-\frac{m_{n}^{2}t^{2}L}{2}+O\left(n\ln\frac{m_{n}^{2}}{n}\right)=-\frac{m_{n}^{2}t^{2}L}{2}+o(m_{n}^{2}).$ On the other hand, if $E\subset\mathbb{R}^{k}$ is open, then choose any $w\in E$. Since $E$ is open, there is some $\epsilon>0$ so that if $\langle x,w\rangle\geq\|w\|^{2}$ and $\|x\|^{2}\leq\|w\|^{2}+\epsilon$ then $x\in E$. Now, Proposition 25 implies that (69) $\ln\Pr\left(\langle X,w\rangle\geq\|w\|^{2}\right)=\ln\Pr\left(\sum_{i}\sigma_{i}(A_{n})w_{i}\geq m_{n}\|w\|^{2}\right)\geq-\frac{m_{n}^{2}\|w\|^{2}L}{2}-o(m_{n}^{2})$ On the other hand, Corollary 22 implies that $\ln\Pr\left(\|X\|^{2}>\|w\|^{2}+\epsilon\right)=\ln\Pr\left(\sum_{i}\sigma_{i}^{2}(A_{n})\geq m_{n}^{2}(\|w\|^{2}+\epsilon)\right)\\\ \leq-\frac{m_{n}^{2}(\|w\|^{2}+\epsilon)L}{2}-o(m_{n}^{2}).$ In particular, $\Pr(\|X\|^{2}>\|w\|^{2}+\epsilon)$ is dominated by $\Pr(\langle X,w\rangle\geq\|w\|^{2})$, implying that (70) $\ln\Pr(X\in E)\geq\ln\Pr\left(\langle X,w\rangle\geq\|w\|^{2}\text{ and }\|X\|^{2}\leq\|w\|^{2}+\epsilon\right)\geq-\frac{m_{n}^{2}\|w\|^{2}L}{2}-o(m_{n}^{2}).$ Since this holds for arbitrary $w\in E$, it implies the lower bound in the LDP. The second part of Theorem 9 follows the exact same argument, only it uses the second part of Proposition 25. ### 5.5. The case of $\mathcal{G}(n,m)$ We next consider the case of Theorem 11. The first observation is that $q\leq\frac{1}{2}$ if and only if $\Lambda^{}(s)/s^{2}$ achieves its minimum at some non-negative $s$. ###### Lemma 27. If $\xi=-q$ with probability $1-q$ and $\xi=1-q$ with probability $q$ then $\Lambda^{}$ (the convex conjugate of $\xi$’s cumulant generating function) satisfies (71) $\inf_{s\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}=\frac{\ln\frac{1-q}{q}}{1-2q},$ and the minimum is uniquely attained at $s=1-2q$. ###### Proof. We recall that $\Lambda^{}(s)=D(q+s,q)$ where (72) $D(r,q)=r\ln\frac{r}{q}+(1-r)\ln\frac{1-r}{1-q}$ (with the convention that $D(r,q)=+\infty$ for $r\not\in(0,1)$). Note that $D(r,q)$ is non-negative, convex, and has a double-root at $r=q$. Fix $q$ and define (73) $L(r)=\frac{D(r,q)}{(r-q)^{2}}=\frac{\Lambda^{}(r-q)}{(r-q)^{2}}$ (defined by continuity at $r=q$); our task is then to minimize $L$. We compute (74) $L^{\prime}(r)=-\frac{(q+r)\ln\frac{r}{q}+(2-q-r)\ln\frac{1-r}{1-q}}{(r-q)^{3}}=:-\frac{F(r)}{(r-q)^{3}}.$ Then $\displaystyle F^{\prime}(r)$ $\displaystyle=\ln\frac{r}{q}-\ln\frac{1-r}{1-q}+\frac{q}{r}-\frac{1-q}{1-r}$ $\displaystyle F^{\prime\prime}(r)$ $\displaystyle=(r-q)\left(\frac{1}{r^{2}}-\frac{1}{(1-r)^{2}}\right).$ In particular, $F^{\prime\prime}$ has exactly two roots on $(0,1)$: at $r=\frac{1}{2}$ and at $r=q$ (counting with multiplicity in case $q=\frac{1}{2}$). It follows that $F$ has at most 4 roots on $(0,1)$. On the other hand, we can easily see that $F(q)=F^{\prime}(q)=F^{\prime\prime}(q)=F(1-q)=0$. Hence, $F(r)$ has a triple- root at $r=q$ and a single root at $r=1-q$, and no other roots. Since $r=q$ is only a triple-root, $L^{\prime}(q)\neq 0$, and it follows that $r=1-q$ is the only root of $L^{\prime}(r)$. It follows that $L(r)$ is minimized at either $r=0$, $r=1$, or $r=1-q$. The possible minimum values are therefore (75) $x:=q^{-2}\ln\frac{1}{1-q},\qquad y:=(1-q)^{-2}\ln\frac{1}{q},\qquad\text{or }z:=\frac{\ln\frac{1-q}{q}}{1-2q}.$ We will show that $z$ is the smallest one. By symmetry in $q$ and $1-q$, it suffices to show that $z\leq x$ for all $q$. Now, (76) $q^{2}(1-2q)(z-x)=q^{2}\ln\frac{1-q}{q}+(1-2q)\ln(1-q)=(1-q)^{2}\ln(1-q)-q^{2}\ln q.$ Let $f(q)=(1-q)^{2}\ln(1-q)-q^{2}\ln q$, and we need to show that $f(q)<0$ for $0<q<\frac{1}{2}$ and $f(q)>0$ for $\frac{1}{2}<q<1$. In fact, since $f(q)=-f(1-q)$, it suffices to show only one of these. Finally, note that $f(0)=f(\frac{1}{2})=0$, and $f^{\prime\prime}(q)>0$ for $0<q<\frac{1}{2}$, and it follows that $f(q)<0$ for $0<q<\frac{1}{2}$. ∎ To complete the proof of Theorem 11, it is enough to show that the upper bound of Corollary 22 and the lower bound of Proposition 25 still hold in this setting; then the proof of the LDP proceeds exactly as in the proof of Theorem 9. Checking Corollary 22 is trivial: recalling that $A_{n}$ is the centered adjacency matrix of $\mathcal{G}(n,m)$ for $\|m-q\binom{n}{2}\|=O(1)$, we let $\tilde{A}_{n}$ be the centered adjacency matrix of $\mathcal{G}(n,q)$. Note that the distribution of $A_{n}$ is equal to the distribution of $\tilde{A}_{n}$, conditioned on the event that $\tilde{A}_{n}$ has exactly $m$ positive entries on the upper diagonal; call this event $E$. By Stirling’s approximation, $\Pr(E)=\Omega(n^{-1})$, and it follows that for any event $F$, (77) $\Pr(A_{n}\in F)=\Pr(\tilde{A}_{n}\in F\mid E)\leq\frac{\Pr(A_{n}\in F)}{\Pr(E)}\leq O(n\Pr(\tilde{A}_{n}\in F)).$ In other words, $\ln\Pr(A_{n}\in F)\leq\ln\Pr(\tilde{A}_{n}\in F)+O(\ln n)$, and so Corollary 22 immediately implies the same upper bound for $\mathcal{G}(n,m)$. For the lower bound, we need to look into the proof of Proposition 25. Recall that in the proof of Proposition 25, we constructed a matrix $M$ with $O(t)=o(n^{2})$ non-zero entries, all of which had the same value. For the $\mathcal{G}(n,q)$ adjacency matrix $\tilde{A}_{n}$, $\langle\tilde{A}_{n},M\rangle$ has a (scaled and translated) binomial distribution; for the $\mathcal{G}(n,m)$ adjacency matrix $A_{n}$, $\langle A_{n},M\rangle$ has a (scaled and translated) hypergeometric distribution. Now, if $H_{k,n,r}$ denotes a hypergeometric random variable with population size $n$, $k$ successes, and $r$ trials; and if $B_{q,r}$ denotes a binomial random variable with success probability $q$ and $r$ trials; then one easily shows using Stirling’s approximation that (78) $\|\ln\Pr(H_{k,n,r}=s)-\ln\Pr(B_{k/n,r}=s)\|=O(r^{2}/n).$ In the setting of Proposition 25, the number of trials $r$ is the number of non-zero elements in $M$, and since $r^{2}/n=O(t^{2}/n)=o(t)$, we have (79) $\ln\Pr(\langle A_{n},M\rangle>t)\geq\ln\Pr(\langle\tilde{A}_{n},M\rangle>t)-o(t).$ With this lower bound, we can follow the rest of the proof of Proposition 25 to complete the proof of Theorem 11. ## 6\. Proof of Theorem 10 Next, we consider the case that $\frac{\Lambda^{}(s)}{s^{2}}$ does not achieve its infimum at any $s>0$, and we construct an example showing that taking $s\to 0$ does not yield the sharp bound. The basic idea is to use the first part of Lemma 13, by producing a positive semi-definite matrix $M$ and giving a lower bound on the tails of $\langle A,M\rangle$. The main challenge is to find a good matrix satisfying the positive definiteness constraint: in Proposition 25 we chose a matrix taking only one non-zero value, specifically, $s_{}\in\operatorname{argmin}\frac{\Lambda^{}(s)}{s^{2}}$. The issue, of course, is that if $s_{}$ is negative then such matrix cannot be positive semi-definite. Instead, we will construct a rank-1 matrix taking four different non-zero values. Consider a sequence $a_{1},\dots,a_{n}$ whose non-zero elements take $m$ different values, $\alpha b_{1},\dots,\alpha b_{m}$, with $\alpha b_{i}$ repeated $\tilde{m}_{i}=\beta m_{i}(1+o(1))$ times respectively (the addition of the error term just allows us to deal with the fact that matrices have integer numbers of rows and columns). We will think of $m_{i}$ and $b_{i}$ as being fixed, while $\alpha$ and $\beta$ depend on the tail bound that we want to show, with $\alpha$ being small and $\beta$ being large. Then for any $t=\sum_{i=1}^{m}t_{i}$, (80) $\Pr\left(\sum_{i}a_{i}\xi_{i}>t\right)\geq\prod_{i=1}^{m}\Pr\left(\sum_{j=1}^{\lceil\tilde{m}_{i}\rceil}\xi_{j}>t/(\alpha b_{i})\right)$ and so Theorem 24 implies that if $\frac{t_{i}}{\alpha\beta m_{i}b_{i}}=\Theta(1)$ then (81) $\ln\Pr\left(\sum_{i}a_{i}\xi_{i}>t\right)\geq-\beta\sum_{i}m_{i}\Lambda^{}\left(\frac{t_{i}}{\alpha\beta m_{i}b_{i}}\right)-o\left(\beta\sum_{i}m_{i}\right)$ Our goal will be to choose the parameters $m_{i},b_{i},\alpha,\beta$, and $t_{i}$ to make the right hand side large. First, we will treat $m_{i}$ and $b_{i}$ as given, and optimize over $t_{i}$, $\alpha$, and $\beta$. We will enforce the constraints $\sum_{i}t_{i}=t$ and $\sum_{i}a_{i}^{2}=\alpha^{2}\beta\sum_{i}m_{i}b_{i}^{2}=2$. Define $\displaystyle\beta$ $\displaystyle=t^{2}\frac{\sum_{i}m_{i}b_{i}^{2}}{2\left(\sum_{i}m_{i}b_{i}\Lambda^{\prime}(b_{i})\right)^{2}}$ $\displaystyle\alpha$ $\displaystyle=2\left(\beta\sum_{i}m_{i}b_{i}^{2}\right)^{-1/2}=\frac{\sum_{i}m_{i}b_{i}\Lambda^{\prime}(b_{i})}{t\sum_{i}m_{i}b_{i}^{2}}$ $\displaystyle t_{i}$ $\displaystyle=\alpha\beta m_{i}b_{i}\Lambda^{\prime}(b_{i}).$ With these choices, we have (82) $\alpha^{2}\beta=\frac{2}{\sum_{i}m_{i}b_{i}^{2}},$ meaning that (83) $\sum_{i}a_{i}^{2}=\alpha^{2}\beta\sum_{i}m_{i}b_{i}^{2}=2$ and (84) $\sum_{i}t_{i}=\alpha\beta\sum_{i}m_{i}b_{i}\Lambda^{\prime}(b_{i})=t.$ (These turn out to be the optimal choices of $\alpha,\beta$, and $t$, although we do not need to show this, since any choice will give us a bound.) Plugging these parameters into (81), we obtain (85) $\ln\Pr\left(\sum_{i}a_{i}\xi_{i}>t\right)\geq-\frac{t^{2}}{2}\cdot\frac{\sum_{i}m_{i}b_{i}^{2}\cdot\sum_{i}m_{i}\Lambda^{}(\Lambda^{\prime}(b_{i}))}{\left(\sum_{i}m_{i}b_{i}\Lambda^{\prime}(b_{i})\right)^{2}}-o(t^{2}),$ where the $o(t^{2})$ term depends on the parameters $m_{i}$ and $b_{i}$. Next, we will define the parameters $m_{i}$ and $b_{i}$. Take $\epsilon,\delta>0$, and define $\displaystyle m_{1}$ $\displaystyle=\frac{1}{\epsilon^{2}}$ $\displaystyle b_{1}$ $\displaystyle=\epsilon$ $\displaystyle m_{2}$ $\displaystyle=2\frac{\epsilon}{\delta^{3}}$ $\displaystyle b_{2}$ $\displaystyle=-\delta$ $\displaystyle m_{3}$ $\displaystyle=\frac{\epsilon^{4}}{\delta^{6}}$ $\displaystyle b_{3}$ $\displaystyle=\frac{\delta^{2}}{\epsilon},$ and note that it is possible to define a positive semi-definite integral kernel taking the value $b_{i}/2$ on a set of measure $2m_{i}$, simply by starting with a function taking the values $\sqrt{\epsilon}$ and $-\delta/\sqrt{\epsilon}$ on sets of size $1/\epsilon$ and $\epsilon/\delta^{3}$ respectively, and then taking the outer product of that function with itself. It follows that if $\epsilon$ and $\delta$ are fixed and $\beta$ is large (and $\alpha$ is arbitrary), then we can define a rank-1 p.s.d. matrix ($M$, say) with $(1+o(1))2\beta m_{i}$ entries taking the value $\alpha b_{i}/2$; note that $\\|M\\|_{F}^{2}=\frac{1+o(1)}{2}\alpha\beta^{2}\sum_{i}m_{i}=1+o(1)$. Since $A$ is a symmetric matrix with $\xi$ on the upper diagonal, this will yield (86) $\langle A,M\rangle=\sum_{i}a_{i}\xi_{i}$ where $(a_{i})$ is a sequence containing $(1+o(1))\beta m_{i}$ copies of $\alpha b_{i}$. We will first choose a small $\delta$ and then choose a smaller $\epsilon$. The error terms in the following analysis are taking this into account, so for example we may write $\epsilon^{2}\delta^{-k}=o(\epsilon)$ no matter how large $k$ is. Our next task is to compute the various expressions in (85), in terms of $\epsilon$ and $\delta$. Before doing so, we observe some basic properties of the Legendre transform. ###### Lemma 28. Assume that $f$ is convex and differentiable and $\lim_{x\to\infty}\frac{f(x)}{x^{2}}=0$. Then $\lim_{x\to\infty}\frac{f^{}(f^{\prime}(x))}{x^{2}}=0$. ###### Proof. Fix $x$ and let $y=f^{\prime}(x)$. By the definition of $f^{}$, we can write (87) $f^{}(y)=\sup_{z}\\{zy-f(z)\\},$ and note that the supremum is attained at $x=z$ (because the derivative is zero, and the expression being supremized is concave). Hence, (88) $f^{}(f^{\prime}(x))=xf^{\prime}(x)-f(x).$ Convexity of $f$ implies that $f^{\prime}$ is non-decreasing, and so $f(x)=o(x^{2})$ implies that $f^{\prime}(x)=o(x)$ as $x\to\infty$. Hence, $f^{}(f^{\prime}(x))=xf^{\prime}(x)-f(x)=o(x^{2})$. ∎ ###### Lemma 29. If $f$ is convex with $f(0)=f^{\prime}(0)=0$ and $f^{\prime\prime}(0)>0$, and if both $f$ and $f^{}$ are $\mathcal{C}^{4}$ in a neighborhood of $0$, then (89) $f^{}(f^{\prime}(\epsilon))=f^{\prime\prime}(0)\frac{\epsilon^{2}}{2}+((f^{})^{\prime\prime\prime}(0)(f^{\prime\prime})^{3}(0)+3f^{\prime\prime\prime}(0))\frac{\epsilon^{3}}{6}+O(\epsilon^{4})$ as $\epsilon\to 0$ ###### Proof. This is nothing but Taylor’s theorem and a computation. Setting $g=f^{}$, we compute (90) $\frac{d}{d\epsilon}g(f^{\prime}(\epsilon))=g^{\prime}(f^{\prime}(\epsilon))f^{\prime\prime}(\epsilon),$ and then (91) $\frac{d^{2}}{d\epsilon^{2}}g(f^{\prime}(\epsilon))=g^{\prime\prime}(f^{\prime}(\epsilon))(f^{\prime\prime}(\epsilon))^{2}+g^{\prime}(f^{\prime}(\epsilon))f^{\prime\prime\prime}(\epsilon),$ and finally (92) $\frac{d^{3}}{d\epsilon^{3}}g(f^{\prime}(\epsilon))=g^{\prime\prime\prime}(f^{\prime}(\epsilon))(f^{\prime\prime}(\epsilon))^{3}+3g^{\prime\prime}(f^{\prime}(\epsilon))f^{\prime\prime}(\epsilon)f^{\prime\prime\prime}(\epsilon)+g^{\prime}(f^{\prime}(\epsilon))f^{\prime\prime\prime\prime}(\epsilon).$ Our assumptions on $f$ ensure that $g^{\prime}(0)=0$, and hence the first- order term vanishes, the second-order term at $\epsilon=0$ becomes (93) $g^{\prime\prime}(0)(f^{\prime\prime}(0))^{2},$ and the third-order term at $\epsilon=0$ becomes (94) $g^{\prime\prime\prime}(0)(f^{\prime\prime}(0))^{3}+3g^{\prime\prime}(0)f^{\prime\prime}(0)f^{\prime\prime\prime}(0).$ Finally, note that $g^{\prime\prime}(0)f^{\prime\prime}(0)=1$. ∎ Note that $\Lambda$ satisfies the assumptions on $f$ in Lemmas 28 (because we assumed that $\Lambda(s)=o(s^{2})$) and 29 (because every cumulant-generating function defined on a neighborhood of zero is $\mathcal{C}^{\infty}$ in a neighborhood of zero). Note that $\Lambda$ and $\Lambda^{}$ both have a second-order root at zero. Define (95) $L=\Lambda^{\prime\prime}(0)>0.$ Expanding out the parameters in (85), we have (96) $\sum_{i}m_{i}b_{i}^{2}=1+2\frac{\epsilon}{\delta}+\frac{\epsilon^{2}}{\delta^{2}}$ for the first term in the numerator. The second term in the numerator is $\displaystyle\sum_{i}m_{i}\Lambda^{}(\Lambda^{\prime}(b_{i}))$ $\displaystyle=\frac{1}{\epsilon^{2}}(\Lambda^{}\circ\Lambda^{\prime})(\epsilon)+2\frac{\epsilon}{\delta^{3}}(\Lambda^{}\circ\Lambda^{\prime})(-\delta)+\frac{\epsilon^{4}}{\delta^{6}}(\Lambda^{}\circ\Lambda^{\prime})(\delta^{2}/\epsilon).$ According to Lemma 28 and our assumptions on $\Lambda$, the last term is $o(\epsilon^{2})$. Applying Lemma 29 to the other terms, we have $\displaystyle\sum_{i}m_{i}\Lambda^{}(\Lambda^{\prime}(b_{i}))$ $\displaystyle=\frac{L}{2}+M\frac{\epsilon}{6}+L\frac{\epsilon}{\delta}-M\frac{\epsilon}{3}+O(\epsilon^{2}+\epsilon\delta)$ $\displaystyle=\frac{L}{2}\left(1+\frac{2\epsilon}{\delta}\right)-M\frac{\epsilon}{6}+O(\epsilon^{2}+\epsilon\delta),$ where (97) $M=(\Lambda^{})^{\prime\prime\prime}(0)L^{3}+3\Lambda^{\prime\prime\prime}(0).$ For the denominator in (85), we ignore the $i=3$ contribution, giving a lower bound of $\displaystyle\sum_{i}m_{i}\Lambda^{\prime}(b_{i})$ $\displaystyle\geq\frac{\Lambda^{\prime}(\epsilon)}{\epsilon}-2\frac{\epsilon\Lambda^{\prime}(-\delta)}{\delta^{2}}$ $\displaystyle=\Lambda^{\prime\prime}(0)+\frac{\epsilon}{2}\Lambda^{\prime\prime\prime}(0)+O(\epsilon^{2})+2\frac{\epsilon}{\delta}\Lambda^{\prime\prime}(0)-\epsilon\Lambda^{\prime\prime\prime}(0)+O(\epsilon\delta)$ $\displaystyle=L\left(1+2\frac{\epsilon}{\delta}\right)-\frac{\epsilon}{2}\Lambda^{\prime\prime\prime}(0)+O(\epsilon^{2}+\epsilon\delta).$ Putting everything together, $\displaystyle\frac{\sum_{i}m_{i}\Lambda^{\prime}(b_{i})\cdot\sum_{i}m_{i}\Lambda^{}(\Lambda^{\prime}(b_{i}))}{\left(\sum_{i}m_{i}\Lambda^{\prime}(m_{i})\right)^{2}}$ $\displaystyle=\frac{\left(1+\frac{2\epsilon}{\delta}+O(\epsilon^{2})\right)\left(\frac{L}{2}\left(1+\frac{2\epsilon}{\delta}\right)-\frac{\epsilon M}{6}+O(\epsilon^{2}+\epsilon\delta)\right)}{\left(L\left(1+\frac{2\epsilon}{\delta}\right)-\frac{\epsilon\Lambda^{\prime\prime\prime}(0)}{2}+O(\epsilon^{2}+\epsilon\delta)\right)^{2}}$ $\displaystyle=\frac{\frac{L}{2}-\frac{\epsilon M}{6}+O(\epsilon^{2}+\epsilon\delta)}{L^{2}-\epsilon L\Lambda^{\prime\prime\prime}(0)+O(\epsilon^{2}+\epsilon\delta)}$ $\displaystyle=\frac{1}{2L}-\frac{\epsilon M}{6L^{2}}+\frac{\epsilon\Lambda^{\prime\prime\prime}(0)}{2L^{2}}+O(\epsilon^{2}+\epsilon\delta)$ $\displaystyle=\frac{1}{2L}-\frac{\epsilon(\Lambda^{})^{\prime\prime\prime}(0)L}{6}+O(\epsilon^{2}+\epsilon\delta),$ and in particular it is possible to choose $\delta$ and $\epsilon$ so that this quantity is at most $(1-\eta)\frac{1}{2L}$ for some $\eta>0$. Going back to (85) and recalling that the sequence $a_{i}$ can be realized as the elements of a rank-1 p.s.d. matrix, $M$ say, with $\\|M\\|_{F}=1+o(1)$, we have shown that (98) $\ln\Pr(\lambda_{1}(A_{n})>t)\geq\ln\Pr(\langle A,M\rangle>t\\|M\\|_{F})\geq-(1-\eta)\frac{t^{2}}{4L}-o(t^{2}).$ Replacing $t$ by $m_{n}t$ and recalling that $L=\Lambda^{\prime\prime}(0)=\mathbb{E}\xi^{2}$ completes the proof of Theorem 10. ## 7\. Back to triangle counts We now turn to the proofs of Theorems 1 and 2. The proofs are very similar, so we devote most of this section to proving Theorem 1 and then indicate what changes must be made to obtain Theorem 2. Our eigenvalue LDP (Theorem 9) allows us to control the triangle-count contribution from a constant number of very extreme eigenvalues, but in order to fully characterize the behavior of the triangle-count, we also need to handle the other eigenvalues. We will do this in two steps: we use Theorem 12 to control the contribution of the bulk eigenvalues, and then Corollary 22 to show that the triangle count cannot be determined by $\omega(1)$ largish eigenvalues. Bear in mind that we will be applying our eigenvalue LDP to $\mathbb{E}A-A$, where $A$ is the adjacency matrix, because Theorem 11 is for the positive eigenvalues of centered matrices, and we are interested in the negative eigenvalues here. ### 7.1. The contribution of the bulk We consider two functions $f_{1}$ and $f_{2}$, where (99) $f_{1}(x)=\begin{cases}0&\text{if $x<0$}\\\ x^{3}&\text{if $0\leq x<\sqrt{K}$}\\\ 3Kx-2K^{3/2}&\text{if $x\geq\sqrt{K}$}\end{cases}$ and $f_{2}(x)=-f_{1}(-x)$. Then both $f_{1}$ and $f_{2}$ are $3K$-Lipschitz functions; also, $f_{1}$ is convex and $f_{2}$ is concave. The following lemma is the main technical result of this section. Essentially, it says that changing the triangle-count using non-extreme eigenvalues carries a substantial entropy cost. ###### Lemma 30. Let $A_{n}$ be the centered adjacency matrix of a $\mathcal{G}(n,m)$ graph. There is a universal constant $C$ such that if $K\geq C$ then (100) $\Pr\left(\sum_{i:\lambda_{i}(A_{n})\geq-\sqrt{Kn}}\lambda_{i}^{3}(A_{n})<-\delta-O(n^{2})\right)\leq\exp\left(-\Omega\left(\frac{\delta^{2}}{n^{3}K^{2}}\right)\right).$ ###### Proof. We will prove the claim when $A_{n}$ is the centered adjacency matrix of a $\mathcal{G}(n,p)$ graph, with $p=m/\binom{n}{2}$. The result for $\mathcal{G}(n,m)$ follows from the fact that a $\mathcal{G}(n,m)$ graph can be obtained by starting from $\mathcal{G}(n,p)$ and conditioning on the (probability $\Omega(1/n)$) event that there are exactly $m$ edges. Note that (101) $f_{1}(x)+f_{2}(x)\leq\begin{cases}0&\text{if $x<-\sqrt{K}$}\\\ x^{3}&\text{if $x\geq-\sqrt{K}$}.\end{cases}$ Hence, (102) $\sum_{i}(f_{1}+f_{2})(n^{-1/2}\lambda_{i}(A_{n}))\leq n^{-3/2}\sum_{i:\lambda_{i}(A_{n})\geq-\sqrt{Kn}}\lambda_{i}^{3}(A_{n}).$ Since $-f_{2}$ is convex, Theorem 12 applies to both $f_{1}$ and $f_{2}$, giving (103) $\Pr\left(\frac{1}{n}\operatorname{tr}[(f_{1}+f_{2})(n^{-1/2}A_{n})]\leq\frac{1}{n}\mathbb{E}\operatorname{tr}[(f_{1}+f_{2})(n^{-1/2}A_{n})]-s\right)\leq 2\exp(-\Omega(n^{2}s^{2}/K^{2}))$ whenever $s=\omega(K/n)$. From the inequality above, we also have (104) $\Pr\left(\sum_{i:\lambda_{i}(A_{n})\geq-\sqrt{Kn}}\lambda_{i}^{3}(A_{n})\leq n^{3/2}\mathbb{E}\operatorname{tr}(f_{1}+f_{2})(n^{-1/2}A_{n})-s\right)\leq 2\exp\left(-\Omega\left(\frac{s^{2}}{K^{2}n^{3}}\right)\right).$ It remains to control $\mathbb{E}\operatorname{tr}[(f_{1}+f_{2})(n^{-1/2}A_{n})]$; specifically, we want to show that $\mathbb{E}\operatorname{tr}(f_{1}+f_{2})(n^{-1/2}A_{n})$ is close to $n^{-3/2}\mathbb{E}\operatorname{tr}(A_{n}^{3})$ (which is $O(\sqrt{n})$). But note that $\|\operatorname{tr}[(f_{1}+f_{2})(n^{-1/2}A_{n})-n^{-3/2}A_{n}^{3}]\|\\\ \leq n^{-3/2}\sum_{i:\|\lambda_{i}\|>\sqrt{Kn}}\|\lambda_{i}(A_{n})\|^{3}\leq n^{-1/2}\|s_{1}(A_{n})\|^{3}1_{\\{\|s_{1}(A_{n})\|>\sqrt{Kn}\\}},$ where $s_{1}(A_{n})$ is the largest singular value of $A_{n}$. But Proposition 21 implies that if $K$ is sufficiently large then $\mathbb{E}[\|s_{1}(A_{n})\|^{3}1_{\\{\|s_{1}(A_{n})\|>\sqrt{Kn}\\}}]\leq\exp(-\Omega(\sqrt{n}))$. Hence, (105) $\Pr\left(\sum_{i:\lambda_{i}(A_{n})\geq-\sqrt{Kn}}\lambda_{i}^{3}(A_{n})\leq n^{3/2}\mathbb{E}\operatorname{tr}(A_{n}^{3})-s-\exp(-\Omega(\sqrt{n}))\right)\leq 2\exp\left(-\Omega\left(\frac{s^{2}}{K^{2}n^{3}}\right)\right).$ Finally, note that $\mathbb{E}\operatorname{tr}(A_{n}^{3})=O(n^{2})$. ∎ We will be interested in applying Lemma 30 when $\delta\gg n^{9/4}$. In this case, the $O(n^{2})$ term becomes negligible and the probability bound is at most $\exp(-\omega(n^{3/2}))$. ### 7.2. Many large negative eigenvalues There is one situation that we still need to handle: the possibility that there are $\omega(1)$ eigenvalues smaller than $-\Omega(\sqrt{n})$, and $\omega(1)$ of these eigenvalues contribute to the triangle count. The first observation is that although Corollary 22 is written for a fixed _number_ of singular values, it can be easily transferred to an inequality for singular values above a certain threshold. ###### Corollary 31. With the notation of Corollary 22, if $\sigma_{i}=\sigma_{i}(A)$ are the singular values of $A$ then (106) $\ln\Pr\left(\sqrt{\sum_{\sigma_{i}>\sqrt{Kn}}\sigma_{i}^{2}}\geq t\right)\leq-\frac{t^{2}L}{2}+O\left(\frac{t^{2}}{K}\ln K\right)$ The same bound holds if $A$ is the centered adjacency matrix of a $\mathcal{G}(n,m)$ graph. ###### Proof. Set $k=\lceil t^{2}/(Kn)\rceil$ and observe that if $s_{1},\dots,s_{k}\geq\sqrt{Kn}$ then $\sum_{i=1}^{k}\sigma_{i}^{2}\geq t$. Hence, we either have (107) $\sum_{\sigma_{i}>\sqrt{Kn}}\sigma_{i}^{2}\leq\sum_{i=1}^{k}\sigma_{i}^{2},$ or else $\sum_{i=1}^{k}\sigma_{i}^{2}\geq t$. It follows that (108) $\ln\Pr\left(\sqrt{\sum_{\sigma_{i}>\sqrt{Kn}}\sigma_{i}^{2}}\geq t\right)\leq\ln\Pr\left(\sqrt{\sum_{i=1}^{k}\sigma_{i}^{2}}\geq t\right),$ and we conclude by applying Corollary 22 with our choice of $k$. Finally, if $A$ is the centered adjacency matrix of a $\mathcal{G}(n,m)$ graph then we use the same argument that was used to extend Corollary 22 to the $\mathcal{G}(n,m)$ case, namely that a $\mathcal{G}(n,m)$ graph can be obtained by conditioning a $\mathcal{G}(n,p)$ graph on an event of $\Omega(n^{-1})$ probability. ∎ ### 7.3. The upper bound in Theorem 1 Let $A$ be the adjacency matrix of a $\mathcal{G}(n,m)$ graph and recall that $\tau(A)=\frac{\operatorname{tr}[A^{3}]}{n(n-1)(n-2)}=\frac{\operatorname{tr}[A^{3}]}{n^{3}}+O(1/n)$. Let $\tilde{A}=A-\mathbb{E}A$; by Corollary 4, (109) $\Pr(\tau(A)\leq p^{3}-t)=\Pr(\operatorname{tr}[A^{3}]\leq n^{3}p^{3}-n^{3}t+O(n^{2}))\leq\Pr(\operatorname{tr}[\tilde{A}^{3}]\leq-n^{3}t+O(n^{2})).$ Writing out $\operatorname{tr}[\tilde{A}^{3}]=\sum_{i}\lambda_{i}^{3}(\tilde{A})$, choose $K=\omega(1)$ and $\epsilon=o(1)$ such that $K/\epsilon=o(n^{1/2}t^{2/3})$. Applying Lemma 30 to $-\tilde{A}$ gives (110) $\Pr\left(\sum_{i:\lambda_{i}\geq-\sqrt{Kn}}\lambda_{i}^{3}(\tilde{A})<-\epsilon tn^{3}\right)\leq\exp\left(-\Omega\left(\frac{\epsilon^{2}t^{2}n^{3}}{K^{2}}\right)\right)=\exp(-\omega(n^{2}t^{2/3}))$ On the other hand, Jensen’s inequality implies that (111) $\left\|\sum_{i:\lambda_{i}<-\sqrt{Kn}}\lambda_{i}^{3}\right\|\leq\left(\sum_{i:\lambda_{i}<-\sqrt{Kn}}\lambda_{i}^{2}\right)^{3/2}\leq\left(\sum_{i:\sigma_{i}>\sqrt{Kn}}\sigma_{i}^{2}\right)^{3/2},$ where $\lambda_{i}=\lambda_{i}(\tilde{A})$ and $\sigma_{i}=\sigma_{i}(\tilde{A})$. Recall here that $L=\inf_{s\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}$, where $\Lambda$ is the cumulant-generating function of an entry of $\tilde{A}$. Lemma 27 (with $q=1-p$) implies that $L=\frac{\ln\frac{p}{1-p}}{2p-1}$. By Corollary 31 (and taking into account the fact that $\epsilon=o(1)$ and $K=\omega(1)$), $\displaystyle\Pr\left(\sum_{i:\lambda_{i}<-\sqrt{Kn}}\lambda_{i}^{3}(\tilde{A})<-(1-\epsilon)tn^{3}\right)$ $\displaystyle\leq\Pr\left(\sqrt{\sum_{i:\sigma_{i}>\sqrt{Kn}}\sigma_{i}^{2}}>(1-\epsilon)^{1/3}t^{1/3}n\right)$ $\displaystyle\leq\exp\left(-\frac{L}{2}t^{2/3}n^{2}+o(t^{2/3}n^{2})\right).$ Combined with (110), this yields (112) $\ln\Pr\left(\operatorname{tr}[\tilde{A}^{3}]\leq- tn^{3}\right)\leq-\frac{Lt^{2/3}n^{2}}{2}(1+o(1)).$ Now we apply (109), noting that $n^{3}t=\omega(n^{2})$, and so $n^{3}t+O(n^{2})=n^{3}t(1+o(1))$, to get (113) $\ln\Pr(\tau(A)\leq p^{3}-t)\leq-\frac{Lt^{2/3}n^{2}}{2}(1+o(1)).$ This completes the proof of the upper bound in Theorem 1 but let us also note two other facts that we can easily extract from the proof. From (110) we see that only the extremely negative eigenvalues contribute to the triangle deviation: ###### Corollary 32. Conditioned on $\tau(A)\leq p^{3}-t$, $\sum_{i:\lambda_{i}\leq-\Omega(\sqrt{n})}\lambda_{i}^{3}(\tilde{A})\leq- tn^{3}(1-o(1))$ with high probability. The other piece of information we can extract from our proof is that the vertex degrees of a triangle-deficient graph are close to constant. ###### Corollary 33. Conditioned on $\tau(A)\leq p^{3}-t$, if $d_{1},\dots,d_{n}$ are the vertex degrees of the graph then with high probability (114) $\sum_{i}(d_{i}-pn)^{2}=o(tn^{3}).$ ###### Proof. Since $\sum_{i}d_{i}=2m=n^{2}p+O(n)$, we have $\sum_{i}(d_{i}-pn)^{2}=\sum_{i}d_{i}^{2}-p^{2}n^{3}+O(n^{2})$. Going back to the proof of Lemma 3, we have (115) $\operatorname{tr}[\tilde{A}^{3}]\leq\operatorname{tr}[A^{3}]-p^{3}n^{3}-3p\sum_{i}(d_{i}-pn)^{2}+O(n^{2}).$ It follows that (116) $\left\\{\tau(A)\leq p^{3}-t\right\\}\subseteq\left\\{\operatorname{tr}[\tilde{A}^{3}]\leq- tn^{3}-3p\sum_{i}(d_{i}-pn)^{2}+O(n^{2})\right\\}.$ Hence, if $tn^{3}\gg n^{2}$ then $\ln\Pr\left(\tau(A)\leq p^{3}-t\text{ and }\sum_{i}(d_{i}-pn)^{2}\geq\epsilon tn^{3}\right)\\\ \leq\ln\Pr\left(\operatorname{tr}[\tilde{A}^{3}]\leq- tn^{3}(1+\Omega(\epsilon))\right)\leq-\frac{Lt^{2/3}n^{2}}{2}(1+\Omega(\epsilon)).$ In particular, (117) $\Pr\left(\tau(A)\leq p^{3}-t\text{ and }\sum_{i}(d_{i}-pn)^{2}\geq\epsilon tn^{3}\right)=o\left(\Pr\left(\tau(A)\leq p^{3}-t\right)\right),$ and claim follows. ∎ ### 7.4. The lower bound in Theorem 1 In showing the lower bound of Theorem 1, we need to apply Corollary 5 (instead of Corollary 4 as in the upper bound), and therefore we need to control $\operatorname{tr}[\tilde{A}^{3}]$ and $\sum_{i}d_{i}^{2}$ simultaneously. To do this, take $v$ and $\ell$ as in Corollary 26 (applied with $t^{2/3}n^{2}$ in place of $t$). Now, let $\xi_{1},\dots,\xi_{\binom{n}{2}}$ be some ordering of the upper diagonal of $\tilde{A}$, ordered so that the first $\xi_{1},\dots,\xi_{\binom{\ell}{2}}$ correspond to the upper diagonal of the upper-left $\ell\times\ell$ principal submatrix. Then $\langle\tilde{A},vv^{T}\rangle=2\sum_{i=1}^{\binom{\ell}{2}}v_{1}^{2}\xi_{i}$, and so conditioning on $\langle\tilde{A},vv^{T}\rangle<-t^{1/3}n$ is equivalent to conditioning on $\sum_{i=1}^{r}\xi_{i}<-\frac{t^{1/3}n}{2v_{1}^{2}}$ (where we have set $r=\binom{\ell}{2}$). Let $\Omega$ be the event that $\langle\tilde{A},vv^{T}\rangle\leq-t^{1/3}n$. To prove the lower bound of Theorem 1, we show three properties: 1. (1) $\ln\Pr(\Omega)\geq-\frac{t^{2/3}n^{2}L}{2}(1+o(1))$. 2. (2) Conditioned on $\Omega$, $\sum_{i}d_{i}^{2}=n^{3}p^{2}+O(n^{2})$ with probability at least $\exp(-o(t^{2/3}n^{2}))$. 3. (3) Conditioned on $\Omega$, $\operatorname{tr}[\tilde{A}^{3}]\leq- tn^{3}(1-\epsilon)$ with probability at least $1-\exp(-\Omega(\epsilon^{2/3}t^{2/3}n^{2}))$. To complete the proof of the lower bound in Theorem 1 from these three facts, we choose $\epsilon\to 0$ slowly enough that the probability of failure in item (3) is at most half the probability of success in item (2). It then follows that with probability at least $\exp(-t^{2/3}n^{2}L/2(1+o(1))$, $\Omega$ holds together with properties (2) and (3). In particular, we have shown that (118) $\ln\Pr\left(\operatorname{tr}[\tilde{A}^{3}]\leq- tn^{3}-\Omega(n^{2})\text{ and }\sum_{i}d_{i}^{2}\leq n^{3}p^{2}+O(n^{2})\right)\geq-\frac{t^{2/3}n^{2}L}{2}(1+o(1)),$ and so the lower bound of Theorem 1 follows from Corollary 5. Note that the first claimed property follows immediately from Corollary 26. Next we prove the second property, noting that $\ell=\Theta(t^{1/3}n)$ and so $\ell^{3}/n=O(tn^{2})=o(t^{2/3}n^{2})$. ###### Lemma 34. Conditioned on $\Omega$, we have $\sum_{i}d_{i}^{2}=n^{3}p^{2}+O(n^{2})$ with probability at least $\exp(-O(\ell^{3}/n))$. ###### Proof. We decompose $A$ block-wise as $A=\begin{bmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{bmatrix}$ where $A_{11}$ has size $\ell\times\ell$. Note that the event $\Omega$ amounts to conditioning on the number of ones in $A_{11}$, and that the number of ones in $A_{11}$ is between 0 and $\ell^{2}$. For some $0\leq s\leq\ell^{2}$, let $\Omega_{s}$ be the event that $A_{11}$ has $s$ ones; we will prove the claim conditioned on each $\Omega_{s}$ and then it follows for $\Omega$. Let $\tilde{\Omega}_{s}$ be the event that $A_{12}$ has $\ell pn-s+\eta$ ones for $\|\eta\|\leq 1$ (where the $\eta$ term is merely to account for integrality). Conditioned on $\Omega_{s}\cap\tilde{\Omega}_{s}$, $d_{1},\dots,d_{\ell}$ have expectation $pn\pm O(1)$, and it follows that $d_{\ell+1},\dots,d_{n}$ also have expectation $pn\pm O(1)$. Note that – conditioned on $\Omega_{s}\cap\tilde{\Omega}_{s}$, each $d_{i}$ is distributed as either a hypergeometric random variable or a sum of two independent hypergeometric random variables; it follows from standard concentration results that conditioned on $\Omega_{s}\cap\tilde{\Omega}_{s}$, $\sum_{i}d_{i}^{2}=n^{3}p^{2}+O(n^{2})$ with high probability. It therefore suffices to lower bound $\Pr(\tilde{\Omega}_{s}\mid\Omega_{s})$, which is exactly the probability that a hypergeometric random variable – with $\binom{n}{2}-\binom{\ell}{2}$ population, $p\binom{n}{2}-\Theta(\ell^{2})$ successes, and $n\ell$ trials – deviates from its mean by order $\Theta(\ell^{2})$. Since a deviation of order $\Theta(\ell^{2})$ is a deviation of $\Theta(\ell^{3/2}n^{-1/2})$ standard deviations, Stirling’s approximation implies that $\Pr(\tilde{\Omega}_{s}\mid\Omega_{s})=\exp(-O(\ell^{3}/n)).$ ∎ Finally, we prove the third property claimed above. ###### Lemma 35. For any $\epsilon>0$, $\Pr(\Omega\text{ and }\operatorname{tr}[\tilde{A}^{3}]\geq-\epsilon tn^{3})\leq\exp(-\Omega(\epsilon^{2/3}t^{2/3}n^{2})).$ ###### Proof. On the event $\Omega$, we have $\lambda_{n}(\tilde{A})\leq-t^{1/3}n$, and so in order to have $\Omega$ and $\operatorname{tr}[\tilde{A}^{3}]\geq tn^{3}(1-a_{n})$ we must have $\sum_{i=1}^{n-1}\lambda_{i}(\tilde{A})^{3}\geq a_{n}tn^{3}.$ Now, Lemma 30 implies that only extremely large eigenvalues can contribute: for a sufficiently large $K$ we have $\Pr\left(\sum_{i:\lambda_{i}(\tilde{A})\geq\sqrt{Kn}}\lambda_{i}^{3}(\tilde{A})\geq a_{n}tn^{3}\right)\leq\exp(-\Omega(a_{n}^{2}t^{2}n^{3})),$ which is less than $\exp(-\omega(n^{2}t^{2/3}))$ if $a_{n}\to 0$ sufficiently slowly. In order to control the contribution of the large eigenvalues, consider the symmetric random matrix $B$ having independent upper-diagonal entries, with each entry $B_{ij}$ having the same distribution as $\tilde{A}_{ij}$ given $\Omega$. Because the distribution of $\tilde{A}$ given $\Omega$ can be obtained by conditioning $B$ on an event of probability $\Omega(1/n)$, it suffices to show that $\Pr\left(\sum_{i:\lambda_{i}(B)\geq\sqrt{Kn}}\lambda_{i}(B)^{2}\geq\epsilon^{2/3}t^{2/3}n^{2}\right)\leq\exp(-\Omega(\epsilon^{2/3}t^{2/3}n^{2})),$ But this follows because $\mathbb{E}B=\mathbb{E}[\tilde{A}\mid\Omega]$ is negative semi-definite and we may apply Remark 23 to $B-\mathbb{E}B$. ∎ ### 7.5. The two extreme eigenvalues In proving the upper bound on $\Pr(\tau(A)\leq p^{3}-t)$, we applied the inequality $\sum_{i}\|a_{i}\|^{3}\leq(\sum_{i}a_{i}^{2})^{3/2}$ to the collection of most-negative eigenvalues. In order to understand how these most negative eigenvalues are actually distributed, observe that in order for the inequality above to be an equality, all but one of the terms in the sum must be zero. Made quantitative, this observation implies that in order for our probability upper bound to be tight, the smallest eigenvalue must dominate the others. In what follows, we write $\\|a\\|_{p}^{p}$ for $\sum_{i}\|a_{i}\|^{p}$. ###### Lemma 36. Let $a_{1},\dots$ be a sequence of non-negative numbers, in non-increasing order. For $\epsilon>0$, if (119) $\sum_{i\geq 2}a_{i}^{3}\geq\epsilon a_{1}^{3}$ then (120) $\\|a\\|_{2}^{2}\geq(1+\epsilon)^{1/3}\\|a\\|_{3}^{2}.$ ###### Proof. If $\sum_{i\geq 2}a_{i}^{3}\geq\epsilon a_{1}^{3}$ then $\\|a\\|_{\infty}^{3}=a_{1}^{3}\leq\frac{\\|a\\|_{3}^{3}}{1+\epsilon}$. Then $\\|a\\|_{3}^{3}\leq\\|a\\|_{\infty}\\|a\\|_{2}^{2}\leq(1+\epsilon)^{-1/3}\\|a\\|_{3}\\|a\\|_{2}^{2}$, and the claim follows. ∎ Applying Lemma 36 to the most negative eigenvalues of $\tilde{A}$ allows us to show that the eigenvalues of $\tilde{A}$ satisfy the claims that Theorem 1 makes for the eigenvalues of $A$. ###### Corollary 37. In the setting of Theorem 1, for any $\epsilon>0$, conditioned on $\tau(A)\leq p^{3}-t$ we have (121) $\lambda_{n}^{3}(\tilde{A})\leq-(1-\epsilon)tn^{3}\text{ and }\lambda_{n-1}^{3}(\tilde{A})\geq-\epsilon tn^{3}$ with high probability. ###### Proof. Let $S=\\{i:\lambda_{i}(\tilde{A})\leq-\Omega(\sqrt{n})\\}$. By Corollary 32, for any $\delta>0$, conditioned on $\tau(A)\leq p^{3}-t$ we have (122) $\sum_{i\in S}\lambda_{i}^{3}(\tilde{A})\leq-(1-\delta)tn^{3}$ with high probability. On this event, we either have $\lambda_{n}^{3}(\tilde{A})\leq-(1-\delta-\epsilon)tn^{3}$ or $\sum_{i\in S\setminus\\{n\\}}\lambda_{i}^{3}(\tilde{A})\leq-\epsilon tn^{3}$. We will show that for some $\delta=\Omega(\epsilon)$, (123) $\Pr\left(\sum_{i\in S}\lambda_{i}^{3}(\tilde{A})\leq-(1-\delta)tn^{3}\text{ and }\sum_{i\in S\setminus\\{n\\}}\lambda_{i}^{3}(\tilde{A})\leq-\epsilon tn^{3}\right)$ is much smaller than $\Pr(\tau(A)\leq p^{3}-t)$; this will imply the claim. Indeed, applying Lemma 36 to the sequence of $\|\lambda_{i}\|$ for $i\in S$, we see that if (124) $\sum_{i\in S}\lambda_{i}^{3}(\tilde{A})\leq-(1-\delta)tn^{3}\text{ and }\sum_{i\in S\setminus\\{n\\}}\lambda_{i}^{3}(\tilde{A})\leq-\epsilon tn^{3}$ then (125) $\sum_{i\in S}\lambda_{i}^{2}(\tilde{A})\geq(1+\epsilon)^{1/3}(1-\delta)t^{2/3}n^{2}\geq(1+\Omega(\epsilon))t^{2/3}n^{2},$ where the last inequality follows by choosing a sufficiently small $\delta=\Omega(\epsilon)$. But Corollary 31 implies that $\displaystyle\Pr\left(\sum_{i\in S}\lambda_{i}^{2}(\tilde{A})\geq(1+\Omega(\epsilon))t^{2/3}n^{2}\right)$ $\displaystyle\leq\exp\left(-(1+\Omega(\epsilon))(1-o(1))\frac{t^{2/3}n^{2}L}{2}\right)$ $\displaystyle=o(\Pr(\tau(A)\leq p^{3}-t)),$ where the final bound follows from the lower bound of Theorem 1. ∎ To complete the proof of Theorem 1, we need to pass from the eigenvalues of $\tilde{A}$ to the eigenvalues of $A$; recall that $A=\tilde{A}+p\mathbf{1}-pI$. Since $p\mathbf{1}\geq 0$, we have (126) $\lambda_{n-1}(A)\geq\lambda_{n-1}(\tilde{A})-p,$ and so $\lambda_{n-1}(\tilde{A})\geq-o(tn^{3})$ implies the same for $\lambda_{n-1}(A)$. For $\lambda_{n}$, we have $\lambda_{n}(A)\leq\lambda_{n}(\tilde{A}+p\mathbf{1})$ and so it remains to take care of the $p\mathbf{1}$ term. Let $v$ be an eigenvector with eigenvalue $\lambda_{n}(\tilde{A})$ satisfying $\|v\|^{2}=1$. Then $\|\langle v,\tilde{A}\mathbf{1}\rangle\|=\|\lambda_{n}(\tilde{A})\|\|\langle v,\mathbf{1}\rangle\|$; conditioned on $\tau(A)\leq p^{3}-t$, this is $(1+o(1))tn^{3}\|\langle v,\mathbf{1}\rangle\|$ with high probability. On the other hand $\tilde{A}\mathbf{1}=(A-p\mathbf{1}+pI)\mathbf{1}=d-p(n-1)\mathbf{1}$, where $d$ whose entries are the vertex degrees. By Corollary 33, conditioned on $\tau(A)\leq p^{3}-t$ we have $\|d-pn\mathbf{1}\|^{2}=o(tn^{3})$, and since $\|\mathbf{1}\|^{2}=n=o(tn^{3})$, we also have $\|\tilde{A}\mathbf{1}\|^{2}=\|d-p(n-1)\mathbf{1}\|^{2}=o(tn^{3})$. Hence, (127) $(1+o(1))tn^{3}\|\langle v,\mathbf{1}\rangle\|=\|\langle v,\tilde{A}\mathbf{1}\rangle\|\leq\|\tilde{A}\mathbf{1}\|=o(tn^{3}),$ and it follows that $\|\langle v,\mathbf{1}\rangle\|=o(1)$. Finally, note that (128) $v^{T}(\tilde{A}+p\mathbf{1})v=v^{T}\tilde{A}v+p\|\langle v,\mathbf{1}\rangle\|^{2}=\lambda_{n}(\tilde{A})+o(1),$ and so by Rayleigh’s criterion it follows that $\lambda_{n}(\tilde{A}+p\mathbf{1})\leq\lambda_{n}(\tilde{A})+o(1)$. This completes the proof of Theorem 1. ### 7.6. Theorem 2 Like Theorem 1, Theorem 2 has three elements: an upper bound on the probability of a moderate deviation, a lower bound, and a bound on the most negative eigenvalue of the adjacency matrix. The upper bound is proved exactly as in the proof of Theorem 1. The singular values of the eigenvalues are controlled by the rate function involving $\inf_{x\in\mathbb{R}}\frac{\Lambda^{}(s)}{s^{2}}$, which we have already established to be $\frac{\ln\frac{1-p}{p}}{2(1-2p)}$. Upper bounds on singular values then give upper bounds on eigenvalues. The entire argument is independent of whether $p\geq\frac{1}{2}$ or $p<\frac{1}{2}$. The proof of the lower bound in Theorem 2 is similar to that of the lower bound in Theorem 1, except that we use the vector $v=(\frac{1}{\sqrt{n}},\dots,\frac{1}{\sqrt{n}},-\frac{1}{\sqrt{n}},\dots,-\frac{1}{\sqrt{n}})$. For this $v$, Cramér’s theorem shows that $\ln\Pr(\langle\tilde{A},vv^{T}\rangle\leq-t^{1/3}n)\geq-\frac{t^{2/3}n^{2}}{2p(1-p)}(1+o(1))$, and the rest of the proof proceeds as before. (Note that we cannot use the vector $v$ constructed in the proof of Theorem 10, because Lemma 35 fails for that $v$: the matrix $\mathbb{E}[\tilde{A}\mid\Omega]$ is not negative semi- definite.) For the claim about the eigenvalue, we use Lemma 36: fix $\eta>0$ and $K>0$ and consider the event $\Omega$ on which $\sum_{\lambda_{i}(\tilde{A})\leq-\sqrt{Kn}}\lambda_{i}^{3}(\tilde{A})\leq- tn^{3}$ but $\lambda_{n}^{3}(\tilde{A})\geq-\frac{1}{1+\eta}tn^{3}$. According to Lemma 36, on this event we have $\sum_{\lambda_{i}(\tilde{A})\leq-\sqrt{Kn}}\lambda_{i}^{2}(\tilde{A})\geq(1+\eta)^{1/3}t^{2/3}n^{2}.$ By Corollary 31 (for a sufficiently slowly growing $K=\omega(1)$), $\ln\Pr(\Omega)\leq-\frac{t^{2/3}n^{2}(1+\eta)^{1/3}\ln\frac{p}{1-p}}{2(2p-1)}+o(t^{2/3}n^{2}),$ which, for sufficiently large $\eta$ (depending on $p$) implies that $\ln\Pr(\Omega)\leq-(1+\Omega(1))\frac{t^{2/3}n^{2}}{2p(1-p)}.$ It follows from the lower bound in Theorem 2 that $\Pr(\Omega\mid\tau\leq p^{3}-t)\to 0$. 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# RADAR: Run-time Adversarial Weight Attack Detection and Accuracy Recovery Jingtao Li1, Adnan Siraj Rakin1, Zhezhi He2, Deliang Fan1, Chaitali Chakrabarti1 1{jingtao1, asrakin, dfan<EMAIL_ADDRESS><EMAIL_ADDRESS>1School of Electrical Computer and Energy Engineering, Arizona State University, Tempe, AZ, 85287 2Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai ###### Abstract Adversarial attacks on Neural Network weights, such as the progressive bit- flip attack (PBFA), can cause a catastrophic degradation in accuracy by flipping a very small number of bits. Furthermore, PBFA can be conducted at run time on the weights stored in DRAM main memory. In this work, we propose RADAR, a Run-time adversarial weight Attack Detection and Accuracy Recovery scheme to protect DNN weights against PBFA. We organize weights that are interspersed in a layer into groups and employ a checksum-based algorithm on weights to derive a 2-bit signature for each group. At run time, the 2-bit signature is computed and compared with the securely stored golden signature to detect the bit-flip attacks in a group. After successful detection, we zero out all the weights in a group to mitigate the accuracy drop caused by malicious bit-flips. The proposed scheme is embedded in the inference computation stage. For the ResNet-18 ImageNet model, our method can detect 9.6 bit-flips out of 10 on average. For this model, the proposed accuracy recovery scheme can restore the accuracy from below 1% caused by 10 bit flips to above 69%. The proposed method has extremely low time and storage overhead. System- level simulation on gem5 shows that RADAR only adds $<$1% to the inference time, making this scheme highly suitable for run-time attack detection and mitigation. ###### Index Terms: Neural networks, weight attack, run-time detection, protection ## I Introduction Neural networks have been widely adopted in image recognition, natural language processing, medical diagnosis and autonomous driving tasks. The security and trustworthiness of neural networks directly affect the safety of these applications making this study even more important. Neural network models have been shown to be vulnerable to various types of attacks. Adversarial input attack, which manipulates the inputs fed to the neural network model, such as FGSM [1], can cause serious misclassification. Recently, adversarial weight attack with malicious weight bit-flips, aka. PBFA [2], on ResNet-18 model was able to degrade ImageNet classification accuracy to below 0.2% with only 13 bit-flips. Furthermore, [3] showed how weight attacks can be mounted at run-time to circumvent protection schemes that perform detection periodically. There is only a handful of techniques that provide some level of security against adversarial weight attacks. The passive defense method in [4] applies regularization to make the weights more resistant to weight attacks. However it is incapable of detecting whether an attack has occurred or not. Error correction code (ECC) based schemes proposed in [5, 6] provide protection against random soft-errors but not against adversarial attacks. Standard data integrity checking methods such as MD5, CRC can perform detection but with high overhead. These are generic techniques and do not exploit the characteristics of the neural network model or the specifics of the attacks that are launched against the networks. As a countermeasure to PBFA, we propose RADAR, a Run-time adversarial weight Attack Detection and Accuracy Recovery scheme. It operates on weights that are fetched from DRAM to on-chip cache for inference computation. RADAR leverages the PBFA characteristics to derive a simple checksum based technique that has excellent error detection and accuracy recovery performance. The weights in a layer are reorganized into groups for the checksum computation, where each group has weights that were originally $k$ locations apart, $k>1$. The checksum is computed on the weights in a group that have been masked using a secret key and is used to derive a 2-bit signature. At run-time, the 2-bit signature of a group is compared with the secure signature to detect possible errors. Once an error is flagged, all weights in that group are replaced with zeroes. The storage and the time overhead of this method is very small compared to the RADAR-free inference baseline. Our contributions can be summarized as follows: * • We present RADAR, a low latency and storage overhead run-time scheme that can provide effective detection and recovery on state-of-the-art adversarial weight attack on DNN, namely, Progressive Bit-Flip Attack (PBFA). * • RADAR computes addition checksum on a group of masked weights that were originally interspersed to derive a 2-bit signature. Use of interleaved weights and masking helps achieve a high detection ratio of 96.1% on a 10-bit PBFA attack on ResNet-18 model. * • RADAR employs a simple scheme where all weights in a group are set to zero if that group has been flagged with an error. For ResNet-18 on ImageNet, the accuracy drop due to PBFA can be recovered from 0.18% to greater than 60% when the partition size is 512. * • System-level Gem5 simulations show that the time cost of RADAR is $<$1% of the inference time for ResNet-18. The overhead to store the signature is only 5.6 KB for ResNet-18, making it feasible to be stored securely on-chip. ## II PRELIMINARIES ### II-A DNN Attack, Defense & Detection. Recent developments of memory fault injection techniques on hardware [7, 8] have made directly attacking model parameters, such as DNN weights, at run time feasible. Among them, row-hammer attack which causes bit-flips in Dynamic Random-Access Memory (DRAM) through repeatedly activating DRAM rows, is the most popular one [7, 9, 10]. Adversarial weight attack [11, 12, 2] corrupts the neural network weights directly to achieve certain attacking goals [12, 2]. A recently developed adversarial weight attack, known as Progressive Bit- Flip Attack (PBFA), identifies vulnerable bits based on gradient information and degrades DNN classification accuracy to random guess level [2]. To mitigate affect of adversarial weight attacks, attack defense mechanisms have also been investigated [4]. For instance, [4] uses binarization or a relaxed version of the binarization technique to handle PBFA attacks. This method increases the resistance to the PBFA attack and is a good passive defense method. DNN soft error detection schemes can be used to detect small perturbations in weights [13]. Error Correction Codes (ECC)-based techniques [5, 6, 14] have been shown to correct soft errors in neural network models. However, rowhammer attack can be used to compromise ECC codewords in DRAM, making these methods not as effective. ## III Threat Model Fig. 1 describes the threat model in this work. At the software end, the attacker uses PBFA in [2] to identify the vulnerable bits, and at the hardware end, the attacker performs fault injection via DRAM row-hammer attack by mounting the vulnerable bits at run-time, thus corrupting the stored weights. We consider DNNs with 8-bit quantized weights as in [2]. Figure 1: Software and hardware aspects of the threat model. ### III-A Hardware Assumptions Row-hammer attack has been demonstrated to be very effective in corrupting DRAM contents [7, 9, 10]. The neural network weight parameters are are very large in size (MB$\sim$GB) and hence stored in DRAM. Recent work in [10] has demonstrated how DRAM weights can be attacked using rowhammer in practice. We consider all weight attacks are physically implemented by DRAM row-hammer attack on PBFA identified vulnerable weight bits. Since every bit flip attack costs time and effort, we assume that the attacker stops the bit flip attacks after causing a significant accuracy drop. We also assume that the attacker is unable to attack data stored securely in on-chip SRAM. Additionally we assume that the attacker cannot corrupt the system kernels (otherwise the attacker would be able to break the system [15]). ### III-B Software Assumptions We only consider Progressive Bit-Flip Attack (PBFA) [2] since it is the strongest adversarial weight attack technique to date. It causes the DNN to malfunction with the fewest number of bit-flips. We argue performing random bit-flip is too weak to be considered as an attack. It has already been demonstrated in [2] that randomly flipping 100 bits merely degrades the accuracy by less than 1%. To perform PBFA, the attacker has access to the network architecture and parameters, e.g., weight, bias, etc. (white box assumption). Such information can be acquired by acting as a benign user or revealed through side-channels [16]. To perform BFA, we assume the attacker has a small dataset with roughly similar distribution as the training data to get accurate gradient information. Additionally, we assume that the attacker has some knowledge of the defense mechanism (aka checksum) but does not know of the secret key used for generating masked weights or the interleaving strategy. ### III-C Characteristics of PBFA. PBFA is a very powerful attack that can severely degrade the accuracy with only a few bit flips. Our experiments show that, on average, with 10 bit- flips, the accuracy of a trained 8-bit ResNet-20 model on CIFAR-10 dataset can drop from above 90% to 18.01% and the accuracy of an 8-bit ResNet-18 model on ImageNet can drop from around 70% to 0.18%. To derive an effective detection scheme for PBFA-based attacks, we first did an in-depth characterization of the attack. We generated multiple sets of PBFA bit profiles and did a statistical analysis. Specifically, we performed 100 rounds of PBFA with 10 bit-flips per round on ResNet-20 model and ResNet-18 model, saved the profiles of the vulnerable bits in each round, and computed the statistics. TABLE I: Number of PBFA Attacks in Different Bit Positions over 100 rounds \| MSB (0 $\rightarrow$ 1) \| MSB (1 $\rightarrow$ 0) \| others ---\|---\|---\|--- ResNet-20 \| 334 \| 666 \| 0 ResNet-18 \| 16 \| 897 \| 87 Observation 1. The PBFA attack exploits the non-uniformity in the importance of some bits over others in a quantized representation. PBFA always chooses to flip the Most Significant Bit (MSB) in a weight. Table I shows that the MSBs are targeted (334+666)/1000 times for ResNet-20 and (16+897)/1000 times for ResNet-18. Thus a low overhead detection scheme should target detecting bit- flips in the MSB position. Observation 2. The vulnerable bits identified by PBFA have a scattered spatial distribution. In this experiment, we partition the weights into groups with $G$ weights in a group, and count the number of bits picked by PBFA in each group. Fig. 2 shows that the proportion of multiple vulnerable bits inside one group is very low when $G$ is small (relative to the model size), and the proportion grow in a super-linear manner for larger group sizes. This indicates that vulnerable bits are scattered across groups rather than being clustered in a group. Figure 2: Proportion of occurrences of multiple vulnerable bits in the same group. Observation 3. The bit-flip locations in PBFA are more likely to occur on weights that have very small values. As shown in Table II, most of the bit- flips happen on weights that have values in the range (-32, 32). Thus, after the bit-flip, the weight value will be in either (96, 127) or (-128, -96) range. We believe that the large weight change is the main cause of severe accuracy drop in PBFA attacks. TABLE II: Frequency of targeted weights in different ranges Range \| (-128, -32) \| (-32, 0) \| (0, 32) \| (32, 127) ---\|---\|---\|---\|--- ResNet-20 \| 85 \| 595 \| 249 \| 71 ResNet-18 \| 16 \| 860 \| 76 \| 27 ## IV RADAR Scheme: Detection We assume that the weight values are loaded from DRAM main memory into on-chip caches and then processed. A well-designed computing scheme maximizes the weight reuse so that each weight is accessed only once. Since the main memory is vulnerable to rowhammer attack, the weights stored there could be compromised and so detection has to be performed on all weights that are loaded into cache prior to processing. In order to embed detection in the inference process, the proposed method has to have the following properties: * • Low timing overhead. The time to perform detection adds to the total inference computation time and thus has to be as small as possible. * • Low storage overhead. The golden signature that is required for detection has to be small enough to be stored in the secure on-chip memory. ### IV-A Checksum-based Signature Calculation Popular detection schemes based on CRC or SEC-DED have high storage overhead (Section VII.B) and are not applicable. We adopt an addition-based checksum scheme [17] for its simplicity, and add interleaving of weights and checksum on masked weights to improve the attack resistance. Specifically, we compute $M$, the sum of $G$ weights in a group and derive a two-bit signature $S_{i,j}=\\{S_{A},S_{B}\\}$ from $M$ for $i$-th layer, $j$-th group in the following way: $S_{A}=\lfloor M/256\rfloor\%2;\hskip 36.135ptS_{B}=\lfloor M/128\rfloor\%2$ (1) In equation 1, the $\lfloor\cdot\rfloor$ denotes the floor function and $\%$ denotes the remainder function. Note that the binarization step can be simply implemented as bit truncation in hardware. Similar to the parity code, $S_{B}$ can detect any odd number of bit-flips on MSBs of a group of $G$ weights. From Fig. 2 we see that most groups have single bit-flips which can be detected by the parity bit $S_{B}$ 100% of the time. Also, when the group size is large, multiple bits in a group could be flipped. Since $S_{B}$ is blind to any even number of bit-flips, we include a second bit, $S_{A}$, which can only detect double bit-flips if they occur in the same direction, i.e., the bit-flips are of the form (0$\rightarrow$1, 0$\rightarrow$1) or (1$\rightarrow$0, 1$\rightarrow$0). However, flips of the form (0$\rightarrow$1, 1$\rightarrow$0) cannot be detected since they do not change the value of $M$. Next we show how this weakness can be addressed by computing the checksum on weights that are masked and interleaved. We argue that it is less efficient to incorporate more bits into the signature, such as one more bit to protect the MSB-1 position by computing $S_{C}=\lfloor M/64\rfloor$. This is because attacking MSB-1 would require the attacker to flip more bits to achieve the same attacking performance as discussed in section VIII. ### IV-B Improving attack detection capability of checksum We adopt the following two features to improve the attack detection capability of the simple addition checksum approach. 1\. Masking Weights in Checksum Computation: We argue that simply performing addition checksum to derive the signature makes the system risky. We use a randomly generated secret key as a mask on a group of weights to determine whether or not to take its two’s complement or not during the summation (lines 4-6 of Algorithm 1). The secret key is $N_{k}$ bits long and changes from layer to layer. Increasing $N_{k}$ can reduce the probability of the sequence of operations being guessed correctly by the attacker but comes with a high implementation cost. We set $N_{k}=16$, and the $2^{16}$ different combinations provide for sufficient security. 2\. Interleaving Weights for Checksum Computation. Given the double bit error blindness of addition checksum, the attacker can intentionally target multiple bits in the same group to attack in order to bypass the detection. So we compute the checksum on a group of weights, where the weights in a group were originally $m$ locations apart, $m>1$. This process is referred to as interleaving, a well known technique that is used to handle detection of burst errors in communication systems. The basic interleaving process is shown in Fig. 3. For the case when there are $N$ groups where each group consists of weights that were originally $N_{W}$ locations apart, the $k$-th group consists of weights $k+N_{W}\times l$, where $0\leq l<N$ and $0\leq k<N_{W}$. So for $N=16$, $N_{W}=8$, group $0$ consists of weights in locations $0,8,16,\ldots,120$ as shown in the figure. We choose $N_{W}=G$ and add an additional offset of $3$ in all our experiments. The interleaving distance can be kept as secret and stored in the secure on- chip SRAM. It can be different from one layer to the next making it even harder for the attacker to know which two bits are in the same group. We will show that the interleaving strategy not only addresses the security concern, but also improves the detection of multiple bit-flips. Figure 3: Basic interleaving strategy in checksum calculation. The checksum is calculated on a group of interleaved weights. Algorithm 1 Pseudo-code of signature calculation 1:8-bit fixed point weight tensor $B_{i}$ in layer $i$. $B_{i,j}$ denotes $j$th group of weights in layer $i$, $K_{i}$ is the secret key for layer $i$, and $N_{W}$ is the parameter for interleaving. $N$ denotes the total number of groups of size $G$ in layer $i$. 2:Signature $S_{i,j}$ 3: 4:function SignCal($B_{i},~{}G$) 5: $B_{i}^{}$ $\leftarrow$ Interleave ($B_{i}$, $N_{W}$) $\triangleright$ Interleaving 6: for $j$ = 0 : $N$ do 7: $B_{i,j}^{}[t]$ $\leftarrow$ $B_{i}^{}[j\times G:(j+1)\times G]$ $\triangleright$ Grouping 8: for $t$ = 0 : $G$ do 9: sign $\leftarrow$ $K_{i}.next()$ $\triangleright$ Secret Key 10: if sign $==$ 0 then: 11: $B_{i,j}^{}[t]$ $\leftarrow$ $-B_{i,j}^{}[t]$ $\triangleright$ Two’s Complement 12: end if 13: end for 14: $M$ $\leftarrow$ $\sum_{t=0}^{G-1}B_{i,j}^{}[t]$$\triangleright$ Summation 15: $S_{i,j}$ = Binarize(M, 2) $\triangleright$ Signature 16: end for 17: return $S_{i}$ 18:end function ### IV-C Overall Algorithm The overall detection scheme is described in Algorithm 1. The weights in a layer, $B_{i}$, are reorganized into groups with weights that are originally interspersed. The secret key is applied on the interleaved weights to determine the sign of each weight (referred to as masking) in the checksum computation. The 2-bit signature for each group is the binarized summation of the scaled weights. The signatures $S_{i,j}$ of each group $B_{i,j}$ in $B_{i}$ are calculated and stored as golden signature in on-chip memory. During run-time, a fresh set of signatures is computed for every new chunk of data that is fetched from the cache. The detection is performed by comparing the computed signature with the golden signature. ## V RADAR Scheme: Recovery If an attack does occur, a successful detection can help halt the system to stop making decisions, wait for downloading a clean copy of weights or let the secondary system take over. This may result in significant increase in timing overhead so next we describe a very simple recovery scheme that can recover most of the model accuracy instantly. In the PBFA analysis described in Section III.C, we see that PBFA attacks the MSB of a small weight and converts it to a large value. It is this large change in weight value that causes a direct change in the ReLU activation and thus has a dramatic effect on the output. So we locate the groups where the bit-flips have occurred using the proposed fine-grain detection scheme and then set all the weights in that group to 0. A de-interleaving step is required when interleaving is applied prior to checksum calculation during the weight update so that the original weight organization is not affected. Since most of the weights in a group have small values and are clustered around 0, setting all the weights in the group to 0 works well especially if the partition size is small. In Section VI we demonstrate that this scheme helps regain most of the accuracy lost due to PBFA for ResNet-18 and ResNet-20 models. ## VI Experiments ### VI-A Settings We demonstrate the effectiveness of our detection and protection scheme for image classification using two popular datasets: CIFAR-10 [18] and ImageNet [19]. We use a 8-bit quantized ResNet-20 for CIFAR-10 dataset and 8-bit quantized ResNet-18 for ImageNet [20]. ResNet-20 model is trained from scratch for 200 epochs using Adam, with a learning rate of 0.01 and decay of 0.0001. We use a pre-trained model for ResNet-18 and fine-tune it for 20 epochs using SGD. ### VI-B Detection Performance We set the number of bit-flips to be equal to 10 per attack since this is sufficient to cause a significant performance degradation and calculate the number of bit-flips that were detected. We perform each attack 100 times. The detection performance with and without the interleaving strategy is shown in Fig. 4. For the ResNet-20 model (shown in blue), the detection performance without interleave approaches 10/10 when $G$ is small, and drops to around 7/10 when $G=64$. This is consistent with the observation in Fig. 2 where we show that when $G$ is large, the proportion of multiple bits in the same group sharply increases thereby harming the detection performance. With interleaving, the detection performance for large group size is better because of the reduction in the number of multiple bit-flip cases. For the ResNet-18 model (shown in red), we observe that interleaving results in a very high 9.5/10 detection ratio even when the group size is large. Figure 4: Average number of detected bit-flips (out of 10 bit flip attacks) using PBFA. The group size G is swept from 4 to 64 for ResNet-20 model and 64 to 1024 for ResNet-18. We also investigate the probability of failing to detect an attack on the MSBs. We consider a layer with 512 weights and perform $10^{6}$ rounds of bit- flips with 10 random bit-flips on MSB position per round. We find that for group size $G=32$, the detection miss rate is $10^{-5}$; for group size $G=16$, the detection miss rate further reduces to $10^{-6}$. Since the number of weights in a group is much larger than the toy example, we conclude that the proposed detection scheme exposes even smaller risk for full-fledged networks. ### VI-C Recovery Performance To evaluate recovery performance, we consider number of bit-flips (N_BF) of 5 and 10. For ResNet-20 and ResNet-18 we compare choices of different group sizes with and without interleaving. Recall that in the proposed recovery technique, the weights in a group are set to zero if a bit-flip is successfully detected. So in this experiment, we check the test accuracy of the revised model obtained by setting the weights of a group to 0 if that group detected bit-flips. TABLE III: Accuracy Recovery of the RADAR scheme Model \| $N_{BF}$ \| Test Accuracy (%) ---\|---\|--- Baseline \| w.o. interleave/ with interleave ResNet-20 \| 0 \| 90.15 \| G = 8 \| G = 16 \| G = 32 5 \| 40.72 \| 82.66/85.64 \| 76.39/83.72 \| 68.06/73.35 10 \| 18.01 \| 80.86/81.07 \| 70.53/77.96 \| 61.62/61.32 ResNet-18 \| 0 \| 69.79 \| G = 128 \| G = 256 \| G = 512 5 \| 5.66 \| 66.60/67.51 \| 65.12/66.15 \| 62.89/62.87 10 \| 0.18 \| 62.69/66.33 \| 59.95/64.96 \| 57.46/60.69 The results for ResNet-20 and ResNet-18 are shown in Table III. For ResNet-20, the accuracy after the attack drops to as low as 18.01% with N_BF=10. After recovery, the accuracy can climb up to 81% when $G=8$ and 62% when $G=32$. Similarly for ResNet-18, the accuracy drops to 0.18% with N_BF=10 and climbs to 66% when $G=128$ and 61% when $G=512$. We see that the accuracy recovery is consistently better when the interleaving strategy is used. Fig. 5 further illustrates the test accuracies for ResNet-18. While there is a mild performance drop when $G$ increases, the accuracy recovery is consistently quite high. Figure 5: Accuracy recovery performance on ResNet-18 model (ImageNet). ## VII Discussion ### VII-A Tradeoff between recovery & storage We use gem5 [21] to evaluate the timing overhead of RADAR. We use a 8-core processor build in gem5. Each core is instantiated as an Arm Cortex-M4F core and runs at 1GHz frequency. The system is equipped with a two-level cache: L1-32KB and L2-64KB. The layer information and weights are obtained from the pre-trained ResNet-20 and ResNet-18 models. Our detection and recovery procedure, RADAR, is embedded in the computations of every layer. For RADAR scheme with group size $G$, we use padding if the number of weights in a layer is not divisible by $G$. To choose a good group size, we study the tradeoffs between recovery performance and storage overhead. Fig. 6 plots recovery accuracy as a function of storage overhead for ResNet-18 and ResNet-20 models. For ResNet-20, the best accuracy-storage tradeoff occurs at $G=8$. The accuracy under 10 bit- flips is still over 80% and the storage overhead for the signature bits is 8.2 KB, which can be easily stored on-chip. For ResNet-18, $G=512$ seems to be the best choice. The accuracy can be kept at over 60% for 10 bit-flip attack and the storage overhead is just 5.6 KB. Figure 6: Test accuracy after recovery vs. Storage overhead of proposed RADAR scheme under PBFA with N_BF = 10 on ResNet-20 and ResNet-18 models. TABLE IV: Time Overhead of RADAR \| Original \| RADAR \| Overhead ---\|---\|---\|--- ResNet-20 \| 66.3ms \| 68.7ms (69.8ms) \| 3.56% (5.27%) ResNet-18 \| 3.268s \| 3.287s (3.328s) \| 0.58% (1.83%) The time overhead for RADAR on ResNet-20 with $G=8$ and ResNet-18 with $G=512$ for batch size of 1 is shown in Table IV. The time overhead with interleaving is shown in brackets. The overhead is quite small – 5.27% for ResNet-20 and 1.83% for ResNet-18 with interleaving. The time overhead can be further reduced in a multi-batch inference setting, where each chunk of weights is loaded once and used many times. ### VII-B Comparison with Related work RADAR has been designed to address the strongest adversarial weight attack to date, i.e., PBFA, via a fast and light weight checksum algorithm that has a very low storage overhead. Other options to perform single and double bit-flip detection for general data integrity checking include Cyclic Redundancy Check (CRC) [22] and Hamming Code [23] based Double-bit Error Detection. To provide for recovery, both codes require significantly higher storage overhead. For instance Hamming code requires 7 bits for 64 bits of data (corresponding to group size of 8) and 13 bits for 4096 bits (corresponding to group size of 512). Similarly, to achieve a HD=3, CRC needs 7 bits and 13 bits for group size of 8 and 512 respectively. We compare the performance of RADAR with the competitive CRC schemes. Table V shows the total inference time, the overhead time for detection ($\Delta$) and the storage overhead for ResNet-20 when G=8 and ResNet-18 when G=512. For ResNet-18, when G=512, CRC-13 has a time overhead if 0.317s compared to 0.060s. The storage overhead of CRC-13 is 36.4KB, compared to 5.6KB which is required by RADAR. If only the MSBs were to be protected, we would require CRC-10 which has a time overhead of 0.315s and storage overhead of 28.0KB, which is still significantly larger than RADAR. TABLE V: Overhead comparison with CRC techniques Scheme \| ResNet-20; G=8 \| ResNet-18; G=512 ---\|---\|--- Time/$\Delta$ \| Storage \| Time/$\Delta$ \| Storage CRC \| 84.2ms/17.9ms \| 28.7KB \| 3.585s/0.317s \| 36.4KB RADAR \| 69.8ms/3.5ms \| 8.2KB \| 3.328s/0.060s \| 5.6KB ## VIII Knowledgeable Attacker Next we assume that the attacker knows that a checksum-based scheme has been used to detect attacks on MSBs. However, the attacker does not know the secret key that is used for masking the weights and/or the interleaving strategy. Flip multiple bits in a group. In addition to attacking the 10 bits identified by PBFA, the attacker could add in another 10 bits to flip (20 bit-flips in total). These bit flips would be of the form (0 $\rightarrow$ 1 and 1 $\rightarrow$ 0) to evade detection. As shown in Fig. 7, the detection performance without interleaving drops greatly (lower blue line) causing the accuracy recovery to be low as well. By applying interleaving, the detection ratio can be keep at a level similar to the traditional PBFA case. Also, the accuracy is much higher when group size is small. Figure 7: Detection and accuracy recovery performance on ResNet-20 model against knowledgeable attackers. Avoid flipping MSB. The proposed checksum-based detection scheme with a 2-bit signature cannot detect bit-flips on less significant bits as effectively as it can on MSB. However, many more bits are required to launch a successful attack if only MSB-1 or lower significant bits are allowed to be flipped. For instance, the attacker needs around 30 bit-flips on MSB-1 bits (compared to 10 bit-flips on MSB) for comparable accuracy degradation on the ResNet-20 model. We address attacks on MSB-1 bits by utilizing a 3-bit signature computed by binarizing $M$ to 3 bits. While this method has a higher storage overhead (3-bit signature vs 2-bit signature), it can successfully detect errors due to attacks on MSB-1 bits. ## IX Conclusion In this work, we propose RADAR, a low overhead run-time adversarial weight attack detection and accuracy recovery scheme for PBFA. We show that the RADAR scheme has very low timing overhead and can be embedded in the inference process. A thorough analysis of the PBFA attack characteristics helped derive a simple error detection and accuracy recovery scheme. A 2-bit signature is computed using addition checksum of a group of interspersed weights and compared with the golden signature to determine whether a PBFA attack had been launched on that group or not. We show that RADAR has superior detection performance; it can consistently detect over 9.5 bit-flips out of 10 injected bit-flips. A simple but effective accuracy recovery scheme is built on top of this detection scheme. It can recover the accuracy from 0.18% back to above 60% for a ResNet-18 model on ImageNet. The gem5 simulation shows that for ResNet-18 model, the RADAR scheme only increases the inference time by $<$2%, making this scheme highly suitable for run time detection and protection. ## References * [1] Ian J Goodfellow et al. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014. * [2] Adnan Siraj Rakin et al. Bit-flip attack: Crushing neural network with progressive bit search. 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# Autocart - spatially-aware regression trees for ecological and spatial modeling Ethan Ancell, Brennan Bean (Utah State University) ## 1 Abstract Many ecological and spatial processes are complex in nature and are not accurately modeled by linear models. Regression trees promise to handle the high-order interactions that are present in ecological and spatial datasets, but fail to produce physically realistic characterizations of the underlying landscape. The “autocart” (autocorrelated regression trees) R package extends the functionality of previously proposed spatial regression tree methods through a spatially aware splitting function and novel adaptive inverse distance weighting method in each terminal node. The efficacy of these autocart models, including an autocart extension of random forest, is demonstrated on multiple datasets. This highlights the ability of autocart to model complex interactions between spatial variables while still providing physically realistic representations of the landscape. ## 2 Introduction Accurate characterizations of ecological variables have important implications for researchers and stakeholders in agriculture, watershed sciences, natural resources, and environmental sciences [1]. Periodic land surveys to get accurate measurements for ecological variables are expensive. For example, in 2018 the United States Department of Agriculture spent over $70 million dollars in soil surveys [2]. Many of these ecological variables are highly interactive. As an example, soil moisture is affected by precipitation, soil composition, temperature, elevation, and the surrounding ecosystem, yet the individual relationships between each variable and soil moisture cannot be considered in isolation. The complexity of these high ordered interactions make it extremely difficult to accurately predict with traditional spatial interpolation models [3]. This is particularly true in Utah, where sharp changes in elevation create drastic changes in climate over short distances. Traditional spatial methods such as kriging [4] create smooth maps of soil properties, but such approaches fail to adequately handle high-order interactions among explanatory variables. Such shortcomings emphasize the need to model interactions among the soil variables to mitigate this problem. The need for non-linear predictions of soil properties is well-documented [5]. Machine-learning techniques promise to remedy under-performing linear models, due to their flexibility in characterizing complex and high-order interactions [6]. There is a lot of existing research in the realm of applying machine-learning algorithms to spatial ecological data [7, 8, 9, 10, 11, 12]. Soil property mapping that utilizes machine-learning has also been extensively studied [13, 5, 14]. One particularly promising method are regression trees [15], which model high-order interactions in a way that is easy to interpret without requiring a massive amount of data like other machine-learning approaches. Unfortunately for spatial data, traditional tree-based algorithms such as regression trees have no way of accounting for the relationship between observations that is explained by their spatial distribution. Coordinate information such as longitude and latitude can be naively included as a predictor in a tree-based model, but this leads to physically unrealistic predictions, often with sharp and discrete jumps in predictions across the geographical region. An example of these visual artifacts can be seen in Figure 2, which shows an attempt to use the random forests algorithm to model soil moisture in Utah. This figure shows a sharp, discrete jump in the predicted soil moisture at approximately 41.5 degrees latitude. Figure 1: Prediction map of soil moisture from a default implementation of Random Forests in Utah. Explicitly including longitude/latitude as predictors in the tree-based method yields a clear jump in predictions moving from north to south that is not physically realistic. The visual artifacts in Figure 2 are a symptom of the overfitting that is present in machine-learning methods when coordinates are included as explicit predictors [16]. Entirely omitting coordinate information from the prediction process may remove the visual artifacts, but ignoring the spatial component can compromise accuracy. Coordinate information is especially useful when analyzing data exhibiting spatial autocorrelation, which occurs when observations are related to each other as a function of distance [17]. The discussion and definitions surrounding the concept of spatial autocorrelation can be argued to directly follow from Tobler’s first law of geography, which states ”everything is related to everything else, but near things are more related than distant things” [18]. Properly accounting for spatial autocorrelation in the modeling process is a powerful way to improve predictions in data that exhibit this property [19]. Another problem that exists when analyzing a spatial dataset that covers a large region is the inability to make the assumption that the distribution of the variable of interest remains consistent across the entire space [20, 21]. Regression trees as a means to decompose a global spatial process into smaller local regions have been studied, including the effort by [21] which discusses the use of hierarchical clustering trees for this purpose. This paper seeks to extend a machine-learning algorithm to handle coordinate information in an appropriate way, avoiding the problems of overfitting and visual artifacts while fully economizing on the predictive power that resides in coordinate information. Initial attempts to create models of this type include regression tree extensions by [22, 23], and a random forest extension by [24]. In the ensuing sections, we propose a tree-based algorithm called “autocart” (autocorrelative regression trees) intended to decompose a global spatial process into local regions in a hierarchical and automatic way, building upon the work proposed by [22] and [23]. The terminal nodes of the tree can be used for the simulation of a local spatial process using inverse- distance weighting interpolation. The result is a predictive process that harnesses the predictive power of both interpolation and regression trees. ## 3 Existing work ### 3.1 Classification and regression trees In the traditional regression tree algorithm, we create partition rules on a dataset to predict some response variable according to a set of splits on the predictor variables [15]. The regression tree algorithm falls under the paradigm of supervised learning, where we use labeled training data to form rules for the prediction of new unlabeled training data. Formally, we predict a class or response $Y$ from predictor variables $X_{1},X_{2},\dots,X_{p}$ by growing a binary tree. We form a prediction with the grown tree by applying a test on one of the predictor variables at each internal node of the tree, and depending on the outcome of the test, we move to either the right or left child of the internal node and proceed to apply the next test to one of our predictor variables. The final prediction for an observation with predictors $X_{1},X_{2},\dots,X_{p}$ is made upon arriving at a terminal node of the binary tree, using the average of the response variable of the training observations that were a part of the terminal node during training. Figure 2: An example of a regression tree. The response variable for this tree is the percentage of water by volume found in a particular soil sample. To make a prediction, a set of predictors $\\{X_{1},X_{2},\dots,X_{p}\\}$ are passed into the tree. As an example, if we had predictors $\text{slope}=0.54$ and $\text{silt}=32$, then we would make a prediction by starting at the top node of the tree, going left because $\text{slope}=0.54<0.62$, and then going right because $\text{silt}<42$. We would arrive at the terminal node, and then make the prediction $y=6.7$. The 13% in the terminal node indicates that 13% of the training data for the building of the tree resides in that node. While growing the tree, the algorithm searches for the decision rule $X_{p}<x$ out of all possible decision rules, such that predictive accuracy is maximized on the training dataset. This partitioning is done recursively, such that each two child nodes that are created as part of a split form the basis for the next splitting rule. To maximize predictive accuracy on the training data at each step of the algorithm, the decision rule splits the data into the two halves that minimize the residual sum of squares (RSS) in the newly formed children nodes. $RSS=\sum_{i}(y_{i}-\bar{y})^{2}$ where $y_{i}$ are the observations of the labeled response value at this node, and $\bar{y}=\frac{1}{n}\sum_{i}{y_{i}}$ is the average of the labeled response variables within the node. The minimization of $RSS$ described above can be most efficiently calculated by maximizing the variance between the nodes. This variance is calculated as $SS_{B}=n_{l}(\bar{y}_{l}-\bar{y})^{2}+n_{r}(\bar{y}_{r}-\bar{y})^{2}$ where $n_{l}$ and $n_{r}$ are the sample sizes in the “left” and “right” partitions respectively, $\bar{y}_{l}$ and $\bar{y}_{r}$ are the average values of the response in the left and right children, and $\bar{y}$ is the average of the response over all observations in the parent node where the split is being made. $SS_{B}$ is compared to the total variance of the node if no split had been made: $SS_{T}=\sum_{i}(y_{i}-\bar{y})^{2}$ The tree makes splits by maximizing the ratio between $SS_{B}$ and $SS_{T}$. The ratio of $SS_{B}$ and $SS_{T}$ is encapsulated in the so-called “objective function”, the measure of goodness or utility of each potential split. $g_{rss}=SS_{B}/SS_{T}$ (1) ### 3.2 Spatial extensions to regression trees Breiman’s CART algorithm discussed in section 3.1 is aspatial, meaning that it does not consider the geographic distribution of the measurements. As discussed in section 2, coordinate information can be included as predictor variables in the model, but often leads to physically unrealistic predictions as seen in Figure 2. Including coordinate information as explicit predictors in machine-learning models for spatial data also leads to an overfitting of the training dataset [16]. Simply excluding coordinate information from the predictive process may reduce the overfitting of the training dataset, but this may curtail the predictive power that lies in coordinate information, especially in spatial datasets with spatial autocorrelation. An appropriate handling of coordinate information is to prevent the tree from making splits based upon coordinate information, but allow the tree to track coordinate information for each sample to use as part of the predictive process. The following techniques assume that the coordinate vector $\mathbf{s}=(x,y)$ is available for all samples in the training dataset. This restructure frames the regression tree prediction process as being fueled by both the information encapsulated in the predictor variables $\\{X_{1},X_{2},\dots,X_{p}\\}$ as well as the geographic location expressed as a coordinate vector $(x,y)$. An extension to the regression tree algorithm was proposed by [22]. In this extension, the objective function in the regression tree algorithm is formed by a linear combination of the objective function $g_{rss}$ described in Equation 1 and another objective function $g_{ac}$ that optimizes for measures of spatial autocorrelation in the partitions. It is defined as $g=(1-\alpha)g_{rss}+\alpha g_{ac},\quad\quad\alpha\in[0,1]$ (2) where $\alpha$ is a user-set parameter that weights minimizing the RSS versus maximizing the autocorrelative statistic. Please note that all cited equations have been converted into a common notation for convenience. A popular statistic intended to measure the presence of spatial autocorrelation in a group of observations is Global Moran’s I (i.e, Moran’s I) [25]. Moran’s I requires a vector of the response variable $Y$ where we measure spatial autocorrelation. We also require a spatial weights matrix that reflects the intensity of the spatial relationship between all pairwise observations in the group. Moran’s I of a response variable $Y$ for a group of observations $G$ is defined to be $I_{Y}=\dfrac{n\sum_{i}\sum_{j}w_{ij}(y_{i}-\bar{y})(y_{j}-\bar{y})}{W\sum_{i}(y_{i}-\bar{y})^{2}}$ (3) where $n$ is the total number of observations that are indexed by $i$ and $j$ in the group $G$; $w_{ij}$ is the spatial weights matrix entry that represents the weight between observations $i$ and $j$ ($w_{ij}$ is 0 when $i=j$), and $W=\sum_{i}\sum_{j}w_{ij}$. $y_{i}$ is the response variable of interest at observation $i$, and $\bar{y}=\frac{1}{n}\sum_{i}{y_{i}}$ is the mean of the response variable in the group of observations $G$. Under the null hypothesis of no spatial autocorrelation, the expected value of Moran’s I for the response $Y$ in group $G$ is $E(I_{Y})=\dfrac{-1}{n-1}.$ The statistic $I$ typically lies in the range $[-1,1]$. Values of $I$ that are significantly above $E(I)$ indicate positive spatial autocorrelation, where values significantly below $E(I)$ indicate negative spatial autocorrelation [17]. A critical choice is the spatial weighting scheme to produce the spatial weights matrix entries $w_{ij}$ used in Moran’s I. The following Gaussian similarity measure connecting observations $i$ and $j$ is used by [22]. $w_{ij}=\begin{cases}e^{-\frac{\text{dist}(i,j)^{2}}{b^{2}}},&\text{if dist}(i,j)<b\\\ 0,&\text{if dist}(i,j)\geq b\end{cases}$ (4) where $\text{dist}(\cdot)$ is a distance metric between two observations $i$ and $j$, and $b$ is a spatial-bandwidth parameter that reflects the spatial distance at which no spatial influence is assumed between observations. Other methods for choosing $w_{ij}$ in the weights matrix for the calculation of Moran’s I are viable, and the best scheme for picking $w_{ij}$ may be dataset dependent. For a typical use case, this Gaussian weighting scheme is sufficient. In order to simultaneously maximize $I_{Y}$ in both partitions of the data, a fair weighting of $I_{Y}$ in each half according to the number of observations is required. This can be expressed as $\tilde{I}_{Y}=\dfrac{n_{L}\cdot I_{YL}+n_{R}\cdot I_{YR}}{n}$ (5) where $n_{L}$ and $n_{R}$ represent then number of observations in arbitrary “left” and “right” partitions. $I_{YL}$ and $I_{YR}$ are the value of the Moran’s I statistic from equation 3 for the individual left and right partitions, and $n=n_{L}+n_{R}$ is the total number of observations in the node being split. The nature of $\tilde{I}_{Y}\in[-1,1]$ requires a re-scaling to $[0,1]$ to ensure an even match against the residual sum of squares objective function $g_{rss}$. The autocorrelative objective function can be defined as $g_{ac}=\dfrac{\tilde{I}_{Y}+1}{2}$ which finds its way into final weighted objective function from equation 2: $g=(1-\alpha)g_{rss}+\alpha g_{ac},\quad\quad\alpha\in[0,1].$ By weighting $g_{ac}$ higher with $\alpha$, the tree is more likely to choose splits that create partitions where the observations in the partition exhibit the property of spatial autocorrelation. This is in contrast to traditional regression trees which simply seek to minimize the residual sum of squares. The motivation underlying this spatial adaptation is that the consideration of $g_{ac}$ at the expense of $g_{rss}$ is worth the investment: the creation of spatial partitions that reflect self-contained units exhibiting a spatial pattern. [22] showed that for some datasets, weighting $g_{ac}$ higher resulted in gains in predictive accuracy with cross-validation. On the other hand, weighting $g_{ac}$ too high has its own problems. If $g_{rss}$ is not weighted enough, then the prediction that is formed by the mean of the response value in the terminal node $\bar{y}_{T}$ is not representative of the region as a whole and leads to poor predictions. The best strategy is to pick the value of $\alpha$ such that an optimal balance is struck between creating partitions that exhibit spatial autocorrelation while still maintaining the power of $\bar{y}_{T}$ as a baseline prediction. [23] builds upon the work in [22] by proposing a novel data mining framework for geophysical data called interpolative clustering, in which interpolation of the response variable of the training data can be used to supplement the prediction $\bar{y}_{T}$ of a regression tree. As discussed in Section 2, the end goal of these modified regression trees is to decompose the global spatial landscape into focused sub-regions. [23] observed that the predictive power of the tree can be improved by simulating a local spatial process contained in a terminal node of the tree using an interpolative method such as inverse-distance weighting. Previously, we formed a prediction for a new observation by considering the arithmetic mean of the response variable at the terminal node of the tree $\bar{y}_{T}$. Now, we supplement that prediction with an interpolative simulation of the spatial pattern of the training observations in the terminal node. In more formal terms, to make a prediction we run the predictor variables $\\{X_{1},X_{2},\dots,X_{p}\\}$ through the splitting rules of the tree as normal, and note the resulting terminal node $T$ that the prediction falls into. Next, we make the prediction using an interpolation method, using only the training data in the terminal node as the reference points for interpolation. We denote the coordinates of the new prediction with $\mathbf{s}=(x,y)$, whose final prediction is made with $\hat{Y}(\mathbf{s})=\begin{cases}\dfrac{\sum_{i=1}^{n_{T}}w_{i}(\mathbf{s}_{Ti})y_{Ti}}{\sum_{i=1}^{n_{T}}w_{i}(\mathbf{s})},&\text{if }\text{dist}(\mathbf{s},\mathbf{s_{Ti}})\neq 0\text{ for all }i\\\ y_{i},&\text{if }\text{dist}(\mathbf{s},\mathbf{s_{i}})=0\text{ for some }i\end{cases}$ (6) where $n_{T}$ is the total number of training observations in the terminal node $T$, $y_{Ti}$ are the labeled response values of the training data in $T$, $\mathbf{s}_{Ti}=(x_{i},y_{i})$ are the coordinate locations of the training data in $T$, and $\text{dist}(\cdot)$ is some spatial distance metric. $w_{i}$ is the spatial weight that is assigned between the interpolated location $\mathbf{s}$ and the known response locations $\mathbf{s}_{Ti}$. $w_{i}$ is calculated with inverse distance weighting: $w_{i}(\mathbf{s})=\dfrac{1}{\text{dist}(\mathbf{s},\mathbf{s}_{Ti})^{p}}$ (7) where $\mathbf{s}$ is the interpolated point, $\mathbf{s}_{Ti}$ are the known points in $T$, $\text{dist}(\cdot)$ is a distance metric from the known point to the interpolated point (this may be Euclidean distance in the case of projected data or also great circle distance in the case of latitude/longitude coordinates), and $p\in\mathbb{R}^{+}$ is the power parameter set by the user. A higher power parameter will highly weight close observations in relation to further observations, compared to a lower power parameter where further observations have more influence in the final prediction $\hat{Y}(\mathbf{s})$. The optimal choice for $p$ is a function of the strength of the underlying spatial distribution in nature, but $p=2$ is commonly used. The interpolative step described in equations 6 and 7 is most effective when used in combination with the objective function $g_{ac}$. Creating partitions and terminal nodes that exhibit spatial autocorrelation is beneficial for inverse-distance weighting interpolation as its efficacy relies upon the assumption that the correlation between observations decreases as the distance between them increases. ## 4 Autocart: extensions to the spatial regression tree In this section we propose further extensions to the methodologies introduced in Section 3.2. We refer to the final tree algorithm as the “autocart” algorithm (autocorrelative regression trees), which is publicly available as an R statistical software package at the following URL: github.com/ethanancell/autocart ### 4.1 An adaptive approach to picking the power parameter $p$ In Section 3.2, inverse distance weighting at the terminal nodes of the regression tree is discussed as a way to supplement the prediction of the tree. Here, we propose a novel approach to picking the power parameter $p$ in equation 7 for the inverse-distance weights. If we consider these spatial regression trees to do the part of automatically decomposing a global landscape into local areas where the spatial pattern may differ, then it is unfair to state that the optimal choice of the power parameter $p$ is constant across all terminal nodes representing local regions. A local region may exhibit a stronger dependence between closely neighboring observations than other local regions, necessitating a varying $p$. In order to assess the strength of a spatial relationship in some terminal node $T$, we can reuse the Moran’s I statistic from equation 3. A terminal node exhibiting a stronger correlation between closely neighboring observations will have a comparatively higher value of $I_{Y}$. In a region where a stronger correlation is noted between closely neighboring observations, it is sensible to pick $p$ to be higher so that the weights defined in equation 7 give greater weight to neighboring observations as compared to far observations. In a region where a weak correlation is noted between closely neighboring observations, it is sensible to pick $p$ to be lower so that the resulting prediction catches on to the trend of the region as a whole, rather than relying upon inappropriate confidence in the predictive power of close observations, indicated by the low value of $I_{Y}$. Let $M$ represent the set of $I_{Yi}$ calculated for each terminal node in the regression tree. In the case that $I_{Yi}<E(I_{Yi})$ for the terminal node, there is no value in interpolation as the correlation between observations close in space is not significant. The ranged value of $p$ will be only based upon the values $I_{Yi}\in M$ where spatial autocorrelation is observed. We denote this new set with $\tilde{M}=\\{I_{Yi}\in M:I_{Yi}>E(I_{Yi})\\}.$ To make a prediction for the coordinate $\mathbf{s}=(x,y)$, we first run the accompanying predictor variables $\\{X_{1},X_{2},\dots,X_{p}\\}$ through the tree to find which terminal node $\mathbf{s}$ belongs to. Once we have found the correct terminal node $T$, we make the prediction in a similar way to equation 6: $\hat{Y}(\mathbf{s})=\begin{cases}y_{i},&\text{if dist}(\mathbf{s},\mathbf{s}_{i})=0\text{ for some }i\\\ \bar{y}_{T},&\text{if }I_{T}<E(I_{T})\text{ and dist}(\mathbf{s},\mathbf{s}_{i})\neq 0\text{ for all }i\\\ \dfrac{\sum_{i=1}^{n_{T}}w_{i}(\mathbf{s})y_{i}}{\sum_{i=1}^{n_{T}}w_{i}(\mathbf{s})},&\text{if }I_{T}>E(I_{T})\text{ and dist}(\mathbf{s},\mathbf{s}_{i})\neq 0\text{ for all }i\end{cases}$ (8) where $I_{YT}$ represent the observed value of the Moran’s I statistic for the training observations in the terminal node $T$ with respect to the response variable $Y$, and $E(I_{YT})$ is the expected value of $I_{YT}$. $\bar{y}_{T}$ denotes the observed mean of the response variable $Y$ of the training observations in $T$. $w_{i}$ remains much the same as in equation 7, the difference being we use the varying power parameter $p_{T}$: $w_{i}(\mathbf{s})=\dfrac{1}{\text{dist}(\mathbf{s},\mathbf{s}_{i})^{p_{T}}}.$ (9) The varying power parameter $p_{T}$ is a monotonic increasing function that maps from $[\min(\tilde{M}),\max(\tilde{M})]$ to $[p_{1},p_{2}]$, where $p_{1}$ and $p_{2}$ are user-set parameters that indicate the range of power parameters. The terminal node that exhibits the most significant value of $I_{YT}$ will use $p_{2}$ for its power parameter, and the terminal node with the least significant (yet above expected) value of $I_{YT}$ will use $p_{1}$. One choice of $p_{T}:[\min(\tilde{M}),\max(\tilde{M})]\mapsto[p_{1},p_{2}]$ is the following linear function: $p_{T}=\dfrac{(I_{T}-\min(\tilde{M}))(p_{2}-p_{1})}{\max(\tilde{M})-\min(\tilde{M})}+p_{1}$ (10) As $p_{1}$ and $p_{2}$ are set by the user, it is crucial that the user have a sense of an appropriate range of values for $p$ in the context of their particular dataset. ### 4.2 An objective function for the encouragement of spatially-compact terminal nodes Section 3.2 described an objective function that encourages high values of spatial autocorrelation within the internal nodes of the tree. Section 3.2 also describes the use of interpolation at the terminal nodes of the tree to supplement the prediction of $\bar{y}_{T}$. In this section, we propose another possible objective function for the tree and explore the possibility of weighting this new objective function alongside the objective function $g_{ac}$ described in section 3.2. When using interpolation as part of the predictive process, it is desired that terminal nodes of the tree create sub-regions of the data that are ideal for interpolation. Excessive overlap in the regions create by the terminal nodes is not ideal for interpolation, as much of the final prediction is weighted by distant observations while ignoring other observations that are geographically close but not in the same terminal node. In this section, another objective function $g_{sc}$ for the encouragement of spatially-compact internal and terminal nodes is introduced. At an arbitrary level of the splitting process, define the total sum of squared pairwise distances within the node $N$ to be $TSS_{D}=\sum_{\mathbf{s}_{i}\in N}\sum_{\mathbf{s}_{j}\in N}\text{dist}(\mathbf{s}_{i},\mathbf{s}_{j})^{2}.$ Consider an arbitrary partition of the data in the node $N$ that produces a “left” and “right” partition, sub-scripted by $l$ and $r$ respectively. Let $N_{l}$ be the set of all training observations in the left partition, and $N_{r}$ be the set of all training observations in the right partition. Define the between sum of squared differences for the partitions to be $BSS_{D}=\sum_{\mathbf{s}_{i}\in N_{l}}\sum_{\mathbf{s}_{j}\in N_{r}}\text{dist}(\mathbf{s}_{i},\mathbf{s}_{j})^{2}.$ As a sort of “spatial extension” to a 1-way anova, the total sum of squares of distances is composed of the sum of all between sums of squared differences and the sum of all within sums of squared differences. This is represented as $TSS_{D}=BSS_{D}+WSS_{D}$ (11) where $WSS_{D}=\sum_{\mathbf{s}_{i}\in N_{l}}\sum_{\mathbf{s}_{j}\in N_{l}}\text{dist}(\mathbf{s}_{i},\mathbf{s}_{j})^{2}+\sum_{\mathbf{s_{i}}\in N_{r}}\sum_{\mathbf{s}_{j}\in N_{r}}\text{dist}(\mathbf{s}_{i},\mathbf{s}_{j})^{2}.$ Minimizing $WSS_{D}$ encourages spatially compact regions resulting from the split, minimizing uncertainty in the interpolation step. Due to the identity in equation 11, this is possible by maximizing $BSS_{D}$. The previous objective functions $g_{rss}$ and $g_{ac}$ from sections 3.1 and 3.2 respectively indicate a more desirable split when $g_{rss}$ and $g_{ac}$ are higher in value and closer to 1. As $\dfrac{BSS_{D}}{TSS_{D}}\in[0,1]$, it is natural and intuitive to define the goodness of spatial compactness $g_{sc}$ as: $g_{sc}=\dfrac{BSS_{D}}{TSS_{D}}.$ (12) We can include $g_{sc}$ in the linear combination of previously discussed objective functions with the weighting parameter $\beta$: $g=(1-\alpha-\beta)g_{rss}+\alpha g_{ac}+\beta g_{sc},\quad\quad\text{where }\alpha,\beta\in[0,1]\text{ and }\alpha+\beta\leq 1.$ (13) Thus, the revised regression tree algorithm is to search through all predictor values $X_{1},X_{2},\dots,X_{p}$ looking for the splitting rule $x<X_{i}$ for some $i$ that maximizes the objective function $g$ of equation 13 at each recursive partitioning of the data. ## 5 Autoforest - a Random Forest extension to autocart trees The revised objective function in equation 13 and the interpolative process discussed in section 4 are promising ways to improve upon the predictions of regression trees when applied to continuous spatial data. Random Forests are a powerful extension of classification and regression trees. Random Forests increase the predictive accuracy of classification and regression trees by minimizing the variance of predictions by producing a “forest” of trees, each trained using a bootstrapped sample of training data and a random subset of the predictor variables to split on at each node. The averaging of predictions that occurs in Random Forests greatly improve upon the predictive power of a single regression tree [26]. The creation of a “forest” of autocart trees is proposed and discussed in this section. Let us denote a single autocart tree as a function $f_{A}$, where a prediction is made by running a set of predictors $\\{x_{1},x_{2},\dots,x_{p}\\}$ through the splitting rules in the tree (trained by maximizing the objective function $g$ at each recursive partition), and then assigning the final prediction either by the average of the response variable in the terminal node (previously referred to as $\bar{y}_{T}$), or by an interpolative rule $\hat{Y}(\mathbf{s})$ as in equation 6 or equation 8. We create a forest of $k$ autocart trees by creating the set of trees $F=\\{f_{A_{1}},f_{A_{2}},\dots,f_{A_{k}}\\}$. A prediction is made by running the set of predictors $\\{x_{1},x_{2},\dots,x_{p}\\}$ through each tree, and then using the arithmetic mean of the prediction of all trees in $F$: $\hat{Y}=\dfrac{1}{k}\sum_{i=1}^{k}f_{A_{i}}(\\{x_{1},x_{2},\dots,x_{p}\\}).$ Each regression tree $f_{A_{i}}$ is trained with $\dfrac{2n}{3}$ training observations randomly sampled from all $n$ observations without replacement. Note that this differs from standard Random Forests where $n$ observations are sampled with replacement. In this spatial adaption, repeat observations with identical coordinate information causes problems in the spatial weights matrix, as this results in an “infinite” weight. As all $n$ records have an equal chance of being chosen, the bias of $f_{A_{i}}$ is not affected, especially across all $k$ trees. Additionally, in each tree only $m$ predictors are selected from ${X_{1},X_{2},\dots,X_{p}}$ at each node. $m$ is a user-set parameter, but can be safely defaulted to $\lceil\frac{p}{3}\rceil$. The autoforest extension to the autocart tree is a way to imbue the Random Forest algorithm with spatial data while refraining from explicitly including coordinates as a predictor in the splitting. (Note: The software implementation of autoforest currently chooses $m$ predictor variables to split on at each tree rather than at each node. This was done for ease of implementation, but a future version of the R package will resolve this issue.) ## 6 Results and comparison of model architectures ### 6.1 Datasets Tested An optimal dataset for the autocart algorithm would include coordinate information for all training observations, and the prediction of a continuous response variable over the region represented by the dataset. Additionally, all predictor variables used to train the autocart tree must be available at all locations where predictions are desired, as techniques to infer the value of a missing predictor variable $X_{i}$ are not covered in this paper. If the autocart algorithm is to be used in a mapping setting, then gridded data across the entire mapped region is required. #### 6.1.1 September 2017 Utah soil moisture This dataset contains the average soil moisture level recorded in moisture per cubed centimeter for 195 remote sensing stations across the state of Utah. Gridded 30-year PRISM Climate Normals [27] are used, including the 30-year normals for maximum vapor pressure deficit, mean annual temperature, and mean annual precipitation. Additionally, a digital elevation map of Utah from the PRISM Climate Normals is used to obtain the elevation predictor and the derived slope and aspect predictors. The 30-year PRISM Climate Normals and derived data are selected for their gridded nature and possible environmental relation to soil moisture. Variables \| Description ---\|--- sm_8 \| The proportion of water per $\text{cm}^{3}$ of soil elev \| The elevation of the location in meters slope \| A unitless “rise over run” measurement of the surface angle of the location. This is calculated from the “elev” model. aspect \| The compass orientation of the slope at a point measured in degrees, where 0 and 360 degrees is north, 90 degrees is east, etc. min_vpd and max_vpd \| The 30-year estimate of minimum / maximum vapor pressure deficit measured in kilopascals (kPa) min_temp, max_temp, and mean_temp \| The 30-year estimate of minimum, maximum, and mean temperature measured in degrees Fahrenheit mean_dewpoint \| The 30-year estimate of the mean dew point temperature in degrees Fahrenheit precip \| The 30-year estimate of annual precipitation in inches #### 6.1.2 Utah 2017 Ground snowload dataset This dataset contains the 50 year ground snow load at a variety of measurement stations across the state of Utah [28]. Predictors are obtained from gridded PRISM 30-year Climate Normals [27]: Variables \| Description ---\|--- yr50 \| The response variable: measures the designed ground snow load at the site in kilopascals (kPa) ELEVATION \| The elevation of the measurement site in meters PPTWT \| The total amount of precipitation in a year in inches MCMT \| The mean temperature of the coldest month in the year in Celsius MWMT \| The mean temperature of the warmest month in the year in Celsius To fix the skewed distribution of the yr50 variable, a log transform was taken of the response. #### 6.1.3 Kenya Poverty Mapping This dataset contains variables related to mapping the presence of poverty in various states of Kenya [29, 30]. The variables in the dataset include the following: Variables \| Description ---\|--- POORDENS \| The number of poor people per $\text{km}^{2}$ AREA \| The area of the active community group in Kenya FLIVDEN \| The density of livestock expressed in total livestock units/$\text{km}^{2}$ ACCWAT \| The percentage of area within one hour walking distance of a permanent natural water source. PTOWNKM \| Distance from the shopping center in each sublocation to the nearest major towns by road, in kms. GROUPDENS \| The total number of active community groups, including non-governmental organizations and community based organizations. LATITUDE \| The latitude of the centroid of the community group (obtained from accompanying shapefile) LONGITUDE \| The longitude of the centroid of the community group (obtained from accompanying shapefile) ### 6.2 Results We use cross-validation to assess the predictive accuracy of each model. In cross-validation, we divide the data into $k$ disjoint partitions known as “folds”. The model is trained on $k-1$ folds, and then used to predict the response variable $Y$ of the withheld fold. The predictions $\hat{Y}$ of the model can be compared to the real response $Y$ for an assessment of the performance of the model. We repeat the training on $k-1$ folds of the data $k$ times, withholding a different fold each time, such that all data in the training data eventually has the chance to be withheld and compared to a prediction from the model where the fold was absent. This strategy of withholding information at each step provides an estimate of a model’s ability to predict new observations and discourages over-fitting the input data. Cross-validation is the gold standard for the assessment of a model when a separate testing dataset is not available. One choice in forming the $k$ folds is to randomly select observations from the training data to be a part of each fold. However, the autocorrelation present in spatial datasets can cause traditional cross-validation to under- estimate the true model error. [16] discusses this issue and presents a solution for the cross-validation of spatial data known as “spatial blocking”. In this setup, the folds in cross-validation are formed by creating chunks of neighboring observations, which limits the opportunity for geographically close neighbors to be a part of different folds in cross-validation. If the spatial blocks that form the cross-validation folds are too large, then the predictive power of the model may be underestimated, as in a realistic setting predictions may be often made at sites that are very close to the training observations. Additionally, if we consider the regression tree models discussed in this paper to be a tool for decomposing a global spatial process into separate local processes, then withholding all data in a large spatial block region may inadvertently eradicate the local spatial region’s representation. On the other hand, if the spatial blocks are too small, then the number of folds may be very large and dramatically increase the computation time required to perform the cross-validation procedure. In the absence of well-defined rules regarding the geographical size of the cross validation folds, groups were constructed through the process of trial and error through a consideration of the maximum geographical distance between within-group observations. In the case of both the Utah 2017 snow and soil datasets, a distance of 15km was chosen. For the Utah 2017 snow dataset, this yields around hundred sub-groups which were then consolidated into 10 larger groups for use in cross-validation. Once a vector of predictions from the model has been created for all 10 folds, the results of each algorithm on each dataset are evaluated with the root mean square error (RMSE). This is a common metric to assess the predictive accuracy of cross-validation for continuous regression problems. $RMSE=\sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(\hat{y}_{i}-y_{i})^{2}}$ where $n$ is the number of observations in the dataset used for cross- validation, $y_{i}$ is the $i$th element of the true response vector of the training data, and $\hat{y}_{i}$ is the $i$th element of the prediction vector made by the model from 10-fold cross-validation. #### 6.2.1 A tuning method for autocart As the autocart function contains several tune-able parameters (namely $\alpha$, $\beta$, and the spatial bandwidth ’$b$’), we need to be careful how we select the optimal choice of parameters during cross-validation. The “best” performing choices of $\alpha$, $\beta$, and $b$ over all the training data may be different than the best performing choices for a random subset of the data, such as a subset used in cross-validation. In a realistic scenario, we predict new data that was not a part of the training data, and thus we do not have the labeled response variable $y$ to tune the parameters with. Thus, instead of tuning the parameters $\alpha$, $\beta$, and $b$ over all the data and then performing cross-validation, we tune the parameters to the 9 withheld groups, and then predict onto the last “test” group. In this way, the optimal choices for $\alpha$, $\beta$, and $b$ will likely vary each time we withhold a group. In the next section, the cross-validated accuracy of autocart is obtained using this tuning method. #### 6.2.2 Dataset results To test the datasets, 6 different methods are used: * • “Simple” Prediction: This is a baseline prediction method that ensures a machine-learning method is providing an improvement over an overly-simplistic model. The simple prediction is formed by taking the average of the response variable in the nine withheld groups, and then using that average to predict for the withheld group in cross-validation. * • Regression trees: Simple regression trees that are introduced at the beginning of the paper. The trees are pruned to the point with the best cross-validated accuracy. * • Autocart with $p=2$: An autocart interpolation tree using the power parameter of $p=2$. * • Autocart with $p_{1}=0.5,p_{2}=3.0$: An autocart interpolation tree using the ranged power parameter $p_{1}$ and $p_{2}$, meant to provide a comparison to the unranged power parameter $p=2$. * • Random forest with ntree = 100: A random forest made of 100 regression trees. * • Autoforest with ntree = 100: An autoforest made up of 100 autocart trees. In each column representing a dataset, the RMSE of the “best performing” model is written in bold font. RMSE of spatial cross-validation --- \| Dataset Method \| September 2017 Soil (proportion of water composition per $\text{cm}^{3}$) \| Utah 2017 Snow (log of 50 year ground snow load avg in kPa) \| Kenya Poverty Mapping (log of number of poor residents per $\text{km}^{2}$) “Simple Prediction” \| 0.0882 \| 0.8890 \| 1.219 Regression trees \| 0.1082 \| 0.3445 \| 1.255 Autocart with $p=2$ \| 0.0962 \| 0.3097 \| 0.966 Autocart with $p_{1}=0.5,p_{2}=3.0$ \| 0.0935 \| 0.3089 \| 0.989 Random forest with $\text{ntree}=100$ \| 0.0871 \| 0.2845 \| 0.933 Autoforest with $\text{ntree}=100$ \| 0.0842 \| 0.3003 \| 0.993 ## 7 Discussion of Results ### 7.1 Inadequacies in the soil moisture datasets In the ”September 2017 Soil” dataset, we observe that the simple regression using the average of the response in the nine withheld groups was nearly the best performing method, outperforming all except Random Forests by a slim margin. This highlights an inadequacy in the data, as none of the tested machine- learning methods are capable of learning the patterns in the labeled response variable given the set of gridded predictor variables. The following are possible explanations for the poor performance of the models on the data: 1. 1. The variation in soil moisture may be much more “localized” than previously thought. As the soil moisture data is only available at 195 sites in the 2017 soil moisture dataset, there is a strong possibility that there does not exist enough data for the machine-learning models to appropriately characterize the patterns in the landscape. 2. 2. The given gridded predictor variables have no relationship with soil moisture. One requirement for candidate predictor variables is that they are available as high resolution gridded maps for the area of interest. The number of candidate variables satisfying this requirement is limited. Thus, there may be variables better suited for soil moisture prediction, but are unusable in their current forms. 3. 3. The data may be contaminated. Some of the sampling locations are yield unusually high soil moisture measurements. This may be the result of an improper inclusion of irrigation site data (yielding an artificially high measurement of soil moisture) or perhaps an anomalous rainy day. Such anomalies defy relationships that may otherwise exist between soil moisture and the response variables, but there is no current way to know which observations should be removed due to human intervention in the soil moisture content. Datasets covering other time periods were supplied by the Plants, Soils, and Climates department at USU. However, these datasets contained considerably less soil moisture remote sensing stations, around 95 as opposed to the 195 in the September 2017 datasets. The timeline required to obtain the additional soil moisture information fell outside the scope of this current project. All other datasets with the 95 remote sensing sites exhibited the same problems as the September 2017 soil moisture dataset: overall poor performance from tree-based machine-learning methods using the PRISM gridded climate normals. This was the same regardless of time period observed. Tested time periods included the average of all summer month soil moisture measurements since 2016, the average of soil moisture measurements in individual months, weeks, and days. Future research will require a re-evaluation of the candidate predictor variables for soil moisture as well as methods to detect and remove data anomalies associated with irrigated sites. ### 7.2 “Smoothness” of the maps: Characterizing a complex landscape Using the cross-validated RMSE score is not the only way we can evaluate the performance of these methods. In Section 2 and Figure 2, it is mentioned that the ultimate aim of these methods is to characterize a complex landscape in a physically realistic way. As an example, in the field of meteorology, advection schemes are a way to model the horizontal flow of heat or matter through a certain geographic region. Advection schemes are based upon the gradients (i.e. local rates of change) which must be relatively smooth to avoid numerical precision issues in modeling. Having an extreme jump in a modeled surface can lead to an extreme local gradient, thereby disrupting the small-scale meteorology. While smoothness is not a necessary condition of predictions on all spatial datasets, most spatial and ecological datasets benefit from predictions with realistic transitions in modeled values across space. In a spatial or ecological setting, it may be the case that we reject a method that has a slightly higher measure of predictive accuracy in favor of a method with similar accuracy and physically realistic output. In this sense, the method that appears to be providing the map with the most detail and physical realism across the geographical landscape may be judged to be the “superior” method, as there is strong evidence it is characterizing the patterns inherently present in the data. We are unaware of any methods that formally balance accuracy with smoothness. In practice, this balance will be domain specific. A critical parameter in the formation of a regression tree is the choice of pruning: the depth that the tree is grown to. A pruned regression tree may have few terminal nodes, thus yielding a “chunky” landscape with sharp and discrete breaks across the space. It would unfair to say that a pruned regression tree does a poor job of characterizing a complex landscape. To circumvent this unfair comparison, in the mapping process both the simple regression tree and the autocart tree are not pruned. The stopping criteria for the autocart and regression tree growth are the same, so we are ensured that there will be a similar number of terminal nodes in each tree. #### 7.2.1 2017 Utah Soil Moisture Although the RMSE of all machine-learning methods tested are less than impressive when compared against the “simple” regression discussed in Section 6.2.2, the maps generated by the methods are promising: (a) Regression trees (b) Autocart with ranged power parameter $p_{1}=0.5,p_{2}=3.0$ Figure 3: Prediction map of average soil moisture (proportion of water per $\text{cm}^{3}$) in September 2017 In Figure 3b, we observe the characterization of a more complex landscape, as compared to Figure 3a. One noticeable “halo” of the interpolative effect can be seen at approximately (112 W, 41 N). (a) Random forests (b) Autoforest with $\text{ntree}=100$ and $p_{1}=0.5,p_{2}=3.0$ Figure 4: Prediction map of average soil moisture (proportion of water per $\text{cm}^{3}$) in September 2017 with ensemble methods. In both the Random Forest and Autoforest maps, we see an improved characterization of the complex landscape with physically realistic predictions as compared to the traditional regression-tree based approaches. Although there is a discernible difference between the two maps, the difference is not as stark as with Figures 3a and 3b. It is important to mention that Figure 4a differs from Figure 2 from the beginning of the paper in that longitude/latitude are not included as predictors in the model. As the coordinate information is not explicitly included as a predictor, we do not observe the same visual artefacts that we did in Figure 2. #### 7.2.2 2017 Utah ground snow load (a) Regression trees (b) Autocart with ranged power parameter $p_{1}=0.5,p_{2}=3.0$ Figure 5: Prediction map of 50-year ground snow load average (log of kPa) ground snow load in Utah as of 2017 In a similar way to Figure 3, we observe that Figure 5b is slightly more smooth and complex in nature. The smoothness of the autocart tree can once again be primarily attributed to the interpolation step at the terminal node of the tree. The efficacy of both the regression tree and autocart tree is observed here, as the patterns in the map directly reflect the mountains of Utah. It is no surprise that high snow loads are observed in mountainous locations. (a) Random forests (b) Autoforest with $\text{ntree}=100$ and $p_{1}=0.5,p_{2}=3.0$ Figure 6: Prediction map of 50-year ground snow load average (log of kPa) ground snow load in Utah as of 2017 with ensemble methods In the prediction map of Random Forests and Autoforest, we see very few differences. Primarily, we observe a slightly more “patchy” area in southwestern Utah in Figure 6a when compared with the smooth southwestern Utah area in Figure 6b. ## 8 Developed Software All software and source code surrounding the ideas in this paper are available at https://github.com/ethanancell/autocart. All methods are implemented in the R statistical software environment [31]. The autocart package utilizes the Rcpp [32, 33, 34], RcppArmadillo [35], RcppParallel [36], fields [37], and mgcv [38, 39, 40, 41, 42] R packages. Regression tree and random forest testing were performed using the rpart [43] and randomForest [44] packages respectively. ## 9 Future Work * • Resolving possible issues in the soil moisture dataset As mentioned in Section 7, a future research direction would be in obtaining gridded predictor variables that have a stronger relationship with the measure of soil moisture. As it currently stands, the PRISM 30-year normals and other derived predictors don’t appear to strongly characterize soil moisture. In addition, it would be helpful to have a human examination of the data so that contaminated sites (such as measurements taken on irrigated land) can be removed. The inclusion of these sites can tamper with a possible underlying relationship between predictor variables and soil moisture. * • Another ensemble method of autocart trees In addition to a “forest” of autocart trees, another interesting research direction would be in creating a boosted autocart trees. Boosted trees are another tree-ensemble method [45]. Boosted trees are very popular and powerful tools for both regression and classification. It may be the case that a boosted ensemble of autocart trees does comparatively better than a random forest ensemble of autocart trees. * • An automatic selection of the power parameter Wherever the ranged or unranged power parameter exists, some speculation and knowledge on the part of the researcher is required to pick a good value of $p$ or $p_{1}$ and $p_{2}$. It may be possible to include an algorithm that can automatically pick up on the weight of spatial strength in a particular region. * • An objective function representing the interaction between $g_{ac}$ and $g_{sc}$ It may be the case that the objective function $g_{sc}$ described in Section 4.2 is only useful in regions with strong autocorrelation. There may be some predictive value in selecting splits that maximize an objective function such as $\tilde{g}=g_{ac}\cdot g_{sc}\cdot I_{\\{g_{ac}>\lambda\\}}$ where $I$ is the indicator function and $\lambda$ is some threshold of autocorrelation that needs to be met before the interaction objective function $g_{ac}g_{sc}$ is used in evaluating the split. * • Using $g_{ac}$ and $g_{sc}$ in a non-tree-based machine-learning method In many cases, other machine-learning methods are easily adaptable with a change in the objective function. The objective functions $g_{ac}$ and $g_{sc}$ described in this paper could be used as objective functions for other methods such as neural networks or support vector machines. * • A formal evaluation of the smoothness of a region In Section 7.2, the smoothness of the maps is compared with a visual assessment. The visual comparison of these maps could be supplemented with a numerical test of “smoothness” over a region. ## 10 Conclusion Complex spatial datasets pose a difficult challenge to existing machine- learning methods are they are not equipped to handle spatial information in a natural way. Autocart regression trees provide a way to imbue a traditional regression tree with coordinate information in a natural way through the use of spatially-aware objective functions. The autocart tree is also capable of traditional interpolation in the terminal nodes of the tree using an adaptive inverse distance weighting technique. Spatially-aware regression trees such as autocart also show a level of promise in providing results with a high measure of predictive accuracy on spatial datasets. In addition, the mapping result of autocart trees exhibit many desirable properties such as smoothness in the modeled response variable over the geographical region. In many cases, the mapped result of an autocart tree is very similar to that obtained by a random forest, yet retains a simpler model form. Although the autocart method was originally created to respond to the unique challenges of modeling soil moisture in a climatically-diverse state like Utah, as of now this method does not show a particularly strong increase in predictive accuracy over a very simple regression formed by averaging the available data. It is suspected that this may be the product of a lack of soil moisture data availability, contamination of irrigation data, or an unfortunate lack of predictive power in available gridded climate variable predictors. Due to the simple nature of the autocart tree’s structure, it can be easily included in an ensemble method such as a random forest of autocart trees. 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11institutetext: Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia 22institutetext: SIA, Sapienza Universita di Roma, Rome, Italy # Black hole shadow to probe modified gravity A. Stepanian 11 Sh. Khlghatyan 11 V.G. Gurzadyan 1122 (Received: date / Revised version: date) ###### Abstract We study the black hole’s shadow for Schwarzschild-de Sitter and Kerr-de Sitter metrics with the contribution of the cosmological constant $\Lambda$. Based on the reported parameters of the M87* black hole shadow we obtain constraints for the $\Lambda$ and show the agreement with the cosmological data. It is shown that, the coupling of the $\Lambda$-term with the spin parameter reveals peculiarities for the photon spheres and hence for the shadows. Within the parametrized post-Newtonian formalism the constraint for the corresponding $\Lambda$-determined parameter is obtained. ###### pacs: 98.80.-kCosmology ## 1 Introduction The release of the M87* massive black hole (BH) shadow image by the EHT Collaboration M87a ; M87a1 ; M87a2 ; M87b marked a new phase of study of a number of physical effects occurring in the very centers of galactic nuclei. The BH’s shadow image is used to constraint both the BH parameters including its spin within General Relativity (GR), as well as the properties of the accretion disk, see M87c ; M87d ; M87e and references therein. The shadow parameters appear to be efficient also in testing for strong-field conditions the parametrized post-Newtonian (PPN) formalism in first order approximation PPN . In this Letter we consider the possibility to use the M87* shadow available information to constraint the Schwarzschild-de Sitter (SdS) and Kerr-de Sitter (KdS) BHs, i.e. taking into account the cosmological constant $\Lambda$ in the metric. The $\Lambda$-term in the spherically symmetric metric is arising also in the modified weak-field General Relativity (GR) which enables to consider the common nature of the dark energy and dark matter G ; GS1 ; GS2 ; GS3 . That approach also suggests a test for the potential deviations from GR having in view of the value of the cosmological constant $\Lambda=1.11\times 10^{-52}m^{-2}$ Pl using the gravity lensing observations GSlens . We use key properties of the photon orbits in the presence of non-zero $\Lambda$ for SdS and KdS metrics and of the shadows, to obtain the constraints on the numerical value of $\Lambda$. We also analyze the role of the $\Lambda$ term for the shadow properties within the parametrized post- Newtonian formalism. ## 2 Constraining the value of $\Lambda$ from the shadow The Schwarzschild-de Sitter metric, i.e. the spherically symmetric metric with non-vanishing $\Lambda$-term has the following form R $ds^{2}=\left(1-\frac{2GM}{rc^{2}}-\frac{\Lambda r^{2}}{3}\right)c^{2}dt^{2}-\left(1-\frac{2GM}{rc^{2}}-\frac{\Lambda r^{2}}{3}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}.$ (1) Importantly, it has been shown that the $\Lambda$-term does not affect the null geodesics Is ; Con1 ; L . Thus, it can be checked that, comparing to $\Lambda=0$ case the photon sphere will remain unchanged i.e. $r_{ph}=\sqrt{g_{tt}}\left(\frac{d\sqrt{g_{tt}}}{dr}\|_{r_{ph}}\right)^{-1}=\frac{3GM}{c^{2}}.$ (2) However, due to the presence of the $\Lambda$ term in the $g_{tt}$ component, the same is not true for the radius of the BH’s shadow which is equal to $r_{sh}=\frac{r_{ph}}{\sqrt{g_{tt}(r_{ph})}}.$ (3) In this sense, comparing to Schwarzschild metric where the shadow is $r_{sh,sh}=3\sqrt{3}\frac{GM}{c^{2}},$ (4) we get $r_{sh,\Lambda}=\frac{\frac{3GM}{c^{2}}}{\sqrt{\frac{1}{3}-3(\frac{GM}{c^{2}})^{2}\Lambda}}.$ (5) Thus, by taking the recently reported values of M87* BH shadow we can find constraints on the numerical value of the cosmological constant. Namely, the upper limits for $\Lambda$ can be obtained based on two different reported values. First, we get the upper limit based on the numerical values for $r_{sh}=42\pm 3\mu as$. In this case the constraint is equal to $1+\frac{\mathbb{E}(r_{sh})}{r_{sh}}\geq\frac{r_{sh,\Lambda}}{r_{sh,sh}}=\frac{1}{\sqrt{1-9\left(\frac{GM}{c^{2}}\right)^{2}\Lambda}}.$ (6) Next, we repeat the same analysis this time by considering the mass of M87* i.e. $M=(6.5\pm 0.7)\times 10^{9}M_{\odot}$. It should be noticed that we cannot take into account both errors simultaneously, since the mass of the BH and the radius of the shadow are dependent on each other. Consequently, for these two cases we get $\Lambda\leq 1.542\times 10^{-28}m^{-2},\quad\Lambda\leq 2.214\times 10^{-28}m^{-2}.$ (7) The obtained limits, as we see, are close to each other and are in agreement with the cosmological value of $\Lambda$. ## 3 Schwarzschild-de Sitter vs Kerr-de Sitter BHs For Schwarzschild-de Sitter metric in Eq.(5) we get $r_{sh,\Lambda}\to\infty$ once $9\left(\frac{GM}{c^{2}}\right)^{2}\Lambda=1.$ (8) It can be checked that, this relation is also the condition of having the so- called extreme SdS BH solution. Indeed, it is known the event horizons of the SdS metric i.e. Eq.(1) are equal to $r_{1}=\frac{2}{\sqrt{\Lambda}}cos\left(\frac{1}{3}cos^{-1}\left(\frac{3GM\sqrt{\Lambda}}{c^{2}}\right)+\frac{\pi}{3}\right),\quad r_{2}=\frac{2}{\sqrt{\Lambda}}cos\left(\frac{1}{3}cos^{-1}\left(\frac{3GM\sqrt{\Lambda}}{c^{2}}\right)-\frac{\pi}{3}\right),\quad r_{3}=-(r_{1}+r_{2}).$ (9) However, once the condition in Eq.(8) is satisfied, instead of two (positive and real) event horizons we will have one horizon of radius equal to $r_{1}=r_{2}=\frac{1}{\sqrt{\Lambda}}.$ (10) Interestingly, one can check that at $r=\frac{1}{\sqrt{\Lambda}}$ the gravitational attraction of Newtonian term in Eq.(1) will be completely balanced by the repulsive force produced by $\Lambda$ term i.e. $\left(\frac{GM}{r^{2}}-\frac{\Lambda c^{2}r}{3}\right)\bigg{\|}_{r=\frac{1}{\sqrt{\Lambda}}}=0.$ (11) Moreover, it is commonly believed that in more realistic astrophysical cases a BH is described not by Schwarzschild metric, but by Kerr metric, where the spin parameter of BH i.e. $a=\frac{J}{Mc}$ is also taken into account. Indeed, the presence of $a$ as the indicator of BH’s intrinsic angular momentum leads to several interesting results which have been studied extensively in the literature (see e.g. K1 ; K2 ; K3 ). Accordingly, for Kerr BH instead of one photon orbit, one will have two of them with the following radii $r_{1}=\frac{2GM}{c^{2}}(1+cos(\frac{2}{3}cos^{-1}(-\frac{\|a\|c^{2}}{GM}))),\quad r_{2}=\frac{2GM}{c^{2}}(1+cos(\frac{2}{3}cos^{-1}(+\frac{\|a\|c^{2}}{GM}))).$ (12) Clearly, for $a=0$ the two solutions will coincide and the solution of Schwarzschild case in Eq (2) is recovered. Considering our analysis, here we are interested to include the $\Lambda$ in the metric of rotating BHs. Namely, in our case this will be the Kerr-de Sitter (KdS) metric which is defined as follows $ds^{2}=\frac{\Delta_{r}}{\rho^{2}L^{2}}\left(cdt-a\sin^{2}\theta d\phi\right)^{2}-\frac{\rho^{2}}{\Delta_{r}}dr^{2}-\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2}-\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}L^{2}}\left(acdt-(r^{2}+a^{2})d\phi\right)^{2},$ (13) where $\displaystyle\Delta_{r}=\left(1-\frac{\Lambda r^{2}}{3}\right)(r^{2}+a^{2})-\frac{2GMr}{c^{2}},$ (14) $\displaystyle\Delta_{\theta}=\left(1+\frac{a^{2}\Lambda\cos^{2}\theta}{3}\right),$ $\displaystyle L=\left(1+\frac{a^{2}\Lambda}{3}\right),$ $\displaystyle\rho^{2}=r^{2}+a^{2}\cos^{2}\theta.$ In this sense, KdS metric describes the geometry of spacetime when a single axially symmetric object is immersed in de Sitter background KdS1 ; KdS2 ; KdS3 . As a result, the photon sphere for KdS metric can be obtained by solving the following cubic equation $3\left(1+\frac{1}{3}\Lambda a^{2}\right)^{2}r^{3}+6(\frac{GM}{c^{2}})\left(\Lambda a^{2}-3\right)r^{2}+27(\frac{GM}{c^{2}})^{2}r-12\left(\frac{GM}{c^{2}}\right)a^{2}=0$ (15) The key point of the above equation is the coupling of $a$ and $\Lambda$. Namely, we have no free $\Lambda$-term which means that for $a=0$ the equation will be reduced to standard Schwarzschild case. Clearly, this fact illustrates that for spherically symmetric BHs, no matter $\Lambda$ is vanishing or not, the photon sphere will be equal to $3\frac{GM}{c^{2}}$. But in axially symmetric case we have the coupling of $\Lambda$ term with spin parameter and as a result of that in contrast to spherically symmetric case, the photon sphere of Kerr and KdS BHs will be different. However, the differences for Kerr and KdS BHs for astrophysical configurations such as M87* are too small to be detected via current observational methods. In fact, the difference between Eq.(15) and pure Kerr case is the presence of $\frac{1}{3}\Lambda a^{2}$ and $\Lambda a^{2}$ in the 3rd and 2nd degrees of the polynomial, respectively. Similarly, the difference between Kerr and Schwarzschild case is arisen due to $12\left(\frac{GM}{c^{2}}\right)a^{2}$. Nevertheless, the main point is that, while considering the current value of cosmological constant and the parameters of a BH such as M87, it turns out the contribution of $\Lambda a^{2}$ is too small to be observed at the typical astrophysical scales. In particular, for M87 this contribution is around $10^{-27}$ which is far smaller than $\frac{GM}{c^{2}}a^{2}$ in the last term of Eq.(15). Here, the essential point is that, the nature of photon orbits in both Kerr and KdS BHs is identical. Indeed, since the roots of Eq.(15) can be considered as the radii of photon orbits, finding the real and positive roots is of the main importance. Meantime, it should be noticed that the number of real and positive solutions depends on the sign of $\Lambda a^{2}-3$. Based on the current astrophysical data for a BH of M87* type this value will be definitely negative. Thus, one can state that there are three positive solutions. Furthermore, it can be shown that the value of the first positive root is smaller than the radius of the outer event horizon of BH. Namely, for the data of M87* the solutions of full Eq.(15) for radii in the presence of positive $\Lambda$, are (in meters) $r_{1}=5.124\times 10^{12},\quad r_{2}=1.5\times 10^{13},\quad r_{3}=3.767\times 10^{13}.$ (16) On the other hand, by taking the metric of KdS according to Eq.(13) it becomes clear that the event horizons of BH are obtained by solving the following equation $\Delta_{r}=\left(1-\frac{\Lambda r^{2}}{3}\right)(r^{2}+a^{2})-\frac{2GMr}{c^{2}}=0.$ (17) Consequently, for M87* the radii of event horizons (in meters) will be $EH_{1}=-1.643\times 10^{26},\quad EH_{2}=5.434\times 10^{12},\quad EH_{3}=1.383\times 10^{13},\quad EH_{4}=1.643\times 10^{26}.$ (18) Following the geometry of KdS BH it turns out that the $EH_{4}$ is the so- called “cosmological horizon” and is located far beyond the other two horizons i.e. $EH_{2}$ and $EH_{3}$ which are regarded as the inner and outer event horizons of KdS in the same analogy with Kerr BH. Meantime, the negative $EH_{1}$ is interpreted as the dual of $EH_{4}$ which is located on the other side of BH’s ring singularity and becomes important only during the KdS solution’s mathematical extension. Accordingly, by comparing the Eqs.(16,18) we find $r_{1}<EH_{3}<r_{2},r_{3}<EH_{4}.$ (19) Thus, similar to the Kerr case, here two photon orbits will be formed outside the BH. Indeed, the presence of these two solutions for Kerr BH can be regarded as a manifestation of frame dragging effect. Namely, the frame dragging for KdS metric in Eq.(13) is defined as $\Omega=-\frac{g_{t\phi}}{g_{\phi\phi}},$ (20) so while the photon at inner orbit i.e. $r_{2}$ moves in the same direction of BH’s spin, the motion at the outer orbit $r_{3}$ is in the opposite direction which is due to the presence of BH’s intrinsic spin. In this sense, we can conclude that the presence of $\Lambda$ term in the BH equations only corrects the radii of photon orbits while the nature of the frame dragging effect itself and particularly the formation of photon orbits remains intact. Finally, since the well-known Lense-Thirring (LT) precession can be obtained from Eq.(20) in the slow rotation limit, our above statement about the effect of $\Lambda$ on the frame dragging effect can be regarded as the continuation of investigation of LT precession in the presence of $\Lambda$ SKG . It was shown that then an additional term is appeared which contains $\Lambda$ $\Omega_{LT}=\frac{2GJ}{c^{2}r^{3}}+\frac{\Lambda J}{3M},$ (21) and which can be interpreted as a correction to the so-called gravito–gyromagnetic ratio. However, in the above relation the $\Lambda$ is coupled to the mass and angular momentum of the rotating object. In other words, in the presence of a positive $\Lambda$ the LT precession is corrected according to Eq.(21) and no new effect is reported. This is due to the fact that, without the rotation of the central object the $\Lambda$ itself cannot cause any precession at all. In PPN it has been proposed that based on the data of BH’s shadow one can get constraints on the variables of parametrized post-Newtonian formalism. Namely, by considering the expansion to the $r^{-3}$ order term one has $g_{tt}=1-\frac{2GM}{c^{2}r}+\frac{2(\beta-\gamma)(GM)^{2}}{c^{4}r^{2}}-\frac{2\zeta(GM)^{3}}{c^{6}r^{3}}.$ (22) Accordingly, since $\beta-\gamma\approx 0$, the $r_{ph}$ and $r_{sh}$ in the $M=c=G=1$ units will be written as $r_{ph}=3+\frac{5}{9}\hat{\zeta},\quad r_{sh}=3\sqrt{3}\left(1+\frac{1}{9}\hat{\zeta}\right).$ (23) By substituting the numerical values, we find $\hat{\zeta}_{\Lambda}=4.77\times 10^{-26},$ as the order of the potential discrepancy from the standard GR. The value of $\hat{\zeta}$ is a measure of deviations from GR and it is also a criterion to put constraints on the free parameters of different modified theories of gravity. ## 4 Conclusions We studied the effect of the cosmological constant $\Lambda$ on the shadow of BH for Schwarzschild-de Sitter and Kerr-de Sitter metrics. The recently reported data of M87* BH shadow was used to obtain the constraints on the numerical value of $\Lambda$. Namely, two upper limits have been obtained which are based on the error limits of the shadow and the BH mass. Comparing both limits with the current value of $\Lambda$ we have shown that there is no inconsistency among those. It should be noticed that, the importance of such kind of analysis is not to directly observe the effect of the modified term, but to check the validity of the modified theory of gravity according to reported data. Then, it is revealed that the condition for having the extreme case for BH shadow is equivalent to the formation of extreme SdS BH. Furthermore, we analyzed the structure of the Kerr BH in the presence of non-zero $\Lambda$. It is shown that while in case of spherically symmetric BHs the radius of photon sphere remains unchanged for both Schwarzschild and SdS BHs, for axially symmetric case due to the coupling of spin parameter $a$ with $\Lambda$ the radii are changed. However, similar to Lense-Thirring precession, the nature of the frame dragging effect does not change in the presence of $\Lambda$. Namely, the radii are modified and again as in the Kerr BH case, two photon orbits i.e. one prograde and other retrograde, are formed. We also checked the potential numerical deviation due to the $\Lambda$-term from standard GR based on the PPN formalism. We obtained the corresponding $\hat{\zeta}_{\Lambda}$ parameter, which although being too small to be observed, can be regarded as an indication of deviation from GR. ## References * (1) The Event Horizon Telescope Collaboration, ApJL, 875, L1 (2019) * (2) The Event Horizon Telescope Collaboration, ApJL, 875, L3 (2019) * (3) The Event Horizon Telescope Collaboration, ApJL, 875, L5 (2019) * (4) The Event Horizon Telescope Collaboration, ApJL, 875, L6 (2019) * (5) D. Garofalo, Annalen der Physik, 532, 1900480 (2020) * (6) F.H. Vincent et al, A&A, in press, arXiv:2002.09226 * (7) V.I. Dokuchaev, N. O. Nazarova, arXiv:2010.01885 * (8) D. Psaltis et al. (EHT Collaboration), Phys. Rev. Lett. 125, 141104 (2020) * (9) V.G. Gurzadyan, Eur. Phys. J. Plus, 134, 98 (2019) * (10) V.G. Gurzadyan, A. Stepanian, Eur. Phys. J. C, 78, 632 (2018) * (11) V.G. Gurzadyan, A. Stepanian, Eur. Phys. J. C, 79, 169 (2019) * (12) V.G. Gurzadyan, A. Stepanian, Eur. Phys. J. Plus, 134, 98 (2019) * (13) P.A.R. 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# THE ROLE OF STRONG GRAVITY AND THE NUCLEAR EQUATION OF STATE ON NEUTRON-STAR COMMON-ENVELOPE ACCRETION A. Miguel Holgado Department of Physics and McWilliams Center for Cosmology, Carnegie Mellon University, Pittsburgh, PA, 15213, USA Department of Astronomy and National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana IL, 61801, USA Hector O. Silva Max- Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, D-14476 Potsdam, Germany Department of Physics and Illinois Center for Advanced Studies of the Universe, University of Illinois at Urbana- Champaign, Urbana IL, 61801, USA Paul M. Ricker Department of Astronomy and National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, Urbana IL, 61801, USA Department of Physics and Illinois Center for Advanced Studies of the Universe, University of Illinois at Urbana- Champaign, Urbana IL, 61801, USA Nicolás Yunes Department of Physics and Illinois Center for Advanced Studies of the Universe, University of Illinois at Urbana-Champaign, Urbana IL, 61801, USA ###### Abstract Common-envelope evolution is important in the formation of neutron star binaries within the isolated binary formation channel. As a neutron star inspirals within the envelope of a primary massive star, it accretes and spins up. Because neutron stars are in the strong-gravity regime, they have a substantial relativistic mass deficit, i.e., their gravitational mass is less than their baryonic mass. This effect causes some fraction of the accreted baryonic mass to convert into neutron star binding energy. The relativistic mass deficit also depends on the nuclear equation of state, since more compact neutron stars will have larger binding energies. We model the mass growth and spin-up of neutron stars inspiraling within common-envelope environments and quantify how different initial binary conditions and hadronic equations of state affect the post-common-envelope neutron star’s mass and spin. From these models, we find that neutron star mass growth is suppressed by $\approx 15-30\%$. We also find that for a given amount of accreted baryonic mass, more compact neutron stars will spin-up faster while gaining less gravitational mass, and vice versa. This work demonstrates that a neutron star’s strong gravity and nuclear microphysics plays a role in neutron-star-common-envelope evolution, in addition to the macroscopic astrophysics of the envelope. Strong gravity and the nuclear equation of state may thus affect both the population properties of neutron star binaries and the cosmic double neutron star merger rate. neutron stars — common envelope evolution — accretion — nuclear physics — compact objects – compact binary stars – interacting binary stars ††software: matplotlib (Hunter, 2007), numpy (Walt et al., 2011), scipy (Virtanen et al., 2020), MESA: v12778 (Paxton et al., 2010, 2013, 2015, 2018, 2019) ## 1 Introduction Neutron stars (NSs) as well as double neutron star (DNS) systems are versatile laboratories for multiple disciplines, including (but not limited to) astrophysics, nuclear physics, and gravitational physics. Our knowledge of DNS population properties as well as the nuclear equation of state (EoS) has greatly improved as we are entering a data-rich era for NS observations. The LIGO-Virgo Collaboration (LVC) has observed GWs from NS mergers, providing constraints on the NS tidal deformability (LVC, 2017, 2019) and new insights on the DNS mass distribution (LVC, 2020). NICER X-ray timing observations of pulsars have provided the first constraints on the NS compactness (Miller et al., 2019b; Riley et al., 2019). Radio pulsar timing has revealed the most massive NS to date from the Green Bank Telescope (Cromartie et al., 2020) and has also revealed a DNS with the lowest asymmetric mass ratio of $0.78\pm 0.03$ observed to date from the Arecibo Observatory (Ferdman et al., 2020). A DNS that forms in isolation must survive two supernova explosions and one or more common-envelope (CE) phases (e.g., Andrews et al., 2015; Tauris et al., 2017; Andrews & Mandel, 2019). In the context of CE evolution, NSs have been treated as point masses that accrete some fraction of their pre-CE mass, similar to white dwarfs and black holes (e.g., Belczynski et al., 2002a, b; Voss & Tauris, 2003; Dewi et al., 2006; Osłowski et al., 2011; Dominik et al., 2012; Belczynski et al., 2018; Chruslinska et al., 2018; Giacobbo & Mapelli, 2018; Vigna-Gómez et al., 2020; Kruckow, 2020). The NS’s strong gravity and nuclear EoS, however, result in a relativistic mass deficit, where the gravitational mass is significantly less than the total baryonic mass. This binding-energy effect has been previously studied in the context of NS accretion in low-mass X-ray binaries (Alécian & Morsink, 2004; Lavagetto et al., 2005; Bagchi, 2011). Early theoretical studies of NS mass growth during CE evolution predicted that accretion would be substantial enough to cause NSs to collapse into black holes (e.g., Chevalier, 1993; Brown, 1995; Fryer et al., 1996; Armitage & Livio, 2000; Brown et al., 2000). Global 3D hydrodynamic CE simulations, however, have found typical accretion rates to be less than the Hoyle- Lyttleton (HL) rate (e.g., Ricker & Taam, 2012). Moreover, MacLeod & Ramirez- Ruiz (2015a) have found from local 3D wind-tunnel simulations that envelope density gradients may substantially suppress the accretion rate (MacLeod & Ramirez-Ruiz, 2015a). These results imply that NSs are much more likely to survive the CE phase (e.g., MacLeod & Ramirez-Ruiz, 2015b; Holgado et al., 2018) instead of collapsing into black holes. Further wind-tunnel studies have provided more insights into how the local density gradient and flow properties are correlated, where such correlations occur, and to what extent such correlations hold (MacLeod et al., 2017; De et al., 2020; Everson et al., 2020). General-relativistic 2D wind-tunnel simulations with a relativistic plasma have also been carried out to characterize accretion and drag on compact-object scales (Cruz-Osorio & Rezzolla, 2020). Building on these general-relativistic models towards 3D and further capturing the plasma conditions relevant to massive-star interiors is certainly well motivated. In addition to these studies of accretion and drag local to the compact object, the global numerical modeling of NS-CE evolution has been steadily progressing with 1D hydrodynamic (Fragos et al., 2019) and 3D hydrodynamic models (Law-Smith et al., 2020). As such numerical models improve in complexity, it may soon be of interest to consider how CE evolution may be sensitive to additional physics, which itself is an open question. Given the current observational constraints on the nuclear EoS, we here investigate how a NS’s macrosopic properties affects its mass-growth and spin-up during the CE inspiral, and before the primary explodes and forms another NS. In addition to focusing on the role of strong gravity and the nuclear EoS, we approximate the pertinent aspects of the accretion and local dynamical friction, which isolates the full complexities of the macroscopic CE physics. ## 2 Methods We consider a primary massive star with mass $M_{\star}$ and radius $R_{\star}$ orbiting a companion NS with initial mass $M_{\rm NS,0}$ that rotates rigidly with an initial angular frequency $\Omega_{0}$. For the system to be in the NS-CE phase, we also initialize the orbit at a separation $a_{0}$ that is equal to the radius of the primary massive star, $a_{0}=R_{\star}$. The primary’s radius $R_{\star}$ will depend on its evolutionary stage, where we consider here the base and tip of the red-giant branch (RGB). During the CE phase, the inspiral is driven by local dynamical friction, causing the NS to accrete matter and spin-up. If enough energy is injected into the CE, it will be ejected, thus leaving a less massive primary star and a spun-up NS at a closer separation; the DNS then would form after the primary goes supernova. A second CE, however, may occur before the primary helium star forms the second NS (e.g., Dewi et al., 2002; Ivanova et al., 2013; Romero- Shaw et al., 2020; Galaudage et al., 2021), though we leave such considerations for future work. ### 2.1 Neutron Star Equation of State, Stellar Structure, Accretion, and Spin-Up Even for the highest spinning pulsars observed to date, such NSs can be considered as slowly-rotating objects, meaning that rotation can thus be treated as a small perturbation $\epsilon$ to the Tolman-Oppenheimer-Volkoff (TOV) solution for non-rotating NSs (Tolman, 1939; Oppenheimer & Volkoff, 1939). Here, $\epsilon\equiv\Omega/\Omega_{\rm k}$ is a dimensionless spin parameter, where $\Omega$ is the angular spin frequency of the star, and $\Omega_{\rm k}$ is the Keplerian angular spin frequency $\Omega_{\rm k}=\sqrt{GM_{\rm TOV}/R_{\rm TOV}^{3}}$, with $M_{\rm TOV}$ and $R_{\rm TOV}$ the mass and radius of our NS if it were not rotating. We solve for the structure of slowly rotating NSs to second-order in $\epsilon\ll 1$ using the Hartle-Thorne approximation (Hartle, 1967; Hartle & Thorne, 1968) with the same set of 46 hadronic EoSs from Silva et al. (2020, Appendix A). This set of EoSs is simultaneously consistent with the LIGO-Virgo observations of GW170817 (LVC, 2017) and the NICER observation of PSR J0030+0451 (Miller et al., 2019a; Riley et al., 2019). For a given EoS, and a chosen value of the central density and spin frequency $\Omega$, the second-order in $\epsilon$ solution to the Einstein equations in the Hartle-Thorne approximation allows us to calculate macroscopic properties of the star (e.g., Berti et al., 2005). These properties include the spin- corrected mass $M_{\rm NS}=M_{\rm TOV}+\epsilon^{2}\delta M$, the spin- corrected equatorial radius $R_{\rm NS}=R_{\rm TOV}+\epsilon^{2}\delta R$, the leading-order-in-spin moment of inertia $I_{\rm NS}$ and the spin-corrected dimensionless gravitochemical potential $\Phi_{\rm NS}$ (Alécian & Morsink, 2004). In the context of accreting NSs, the gravitochemical potential can be interpreted as a susceptibility to changes in baryon number or the fraction of baryon mass that gets converted into gravitational mass. In the non-rotating limit, $\Phi_{\rm NS}$ simplifies to $\lim_{\Omega\to 0}\Phi_{\rm NS}=\Phi_{\rm TOV}=\sqrt{1-2{\cal C}_{\rm TOV}}=\sqrt{1-\frac{2GM_{\rm TOV}}{c^{2}R_{\rm TOV}}}\ ,$ (1) where ${\cal C}_{\rm TOV}$ is the compactness of a given non-rotating NS. We will use $\Phi_{\rm NS}$ for our calculations, where we elaborate in Appendix B how this is calculated with our EoS catalog. Equation 1 provides a fast approximation for population synthesis or as a sub-grid prescription for global hydrodynamic simulations. We later compare in §3 how well this approximation compares to using $\Phi_{\rm NS}$, with a more detailed quantification shown in Appendix D. We plot in Figure 1 the gravitochemical potential $\Phi_{\rm TOV}$ versus the gravitational mass $M_{\rm TOV}$ for non-rotating NSs, as predicted from our EoS catalog. Figure 1: Gravitochemical potential vs. gravitational mass for non-rotating NSs. Each curve corresponds to a different EoS in our catalog that is consistent with both the latest LIGO-Virgo and NICER constraints and is able to produce a NS with $M_{\rm max}/M_{\odot}\geq 1.96$. The color of each curve corresponds to the nondimensional NS binding energy $\|{\cal B}_{\rm TOV}\|/(M_{\rm TOV}c^{2})$. For $\Phi=1$, all of the accreted baryonic mass contributes to the gravitational mass growth. NSs with larger gravitational masses and with larger compactness will convert a larger fraction of the accreted baryonic mass into binding energy instead of gravitational mass. Each curve represents $\Phi_{\rm TOV}$ for a different EoS, with different colors corresponding to different dimensionless NS binding energy $\|{\cal B}_{\rm TOV}\|/(M_{\rm TOV}c^{2})$. Observe that the gravitochemical potential decreases as the gravitational mass increases, and also it decreases as the NS compactness increases. Since the gravitochemical potential is inversely related to the binding energy, this figure tells us that binding energy conversion is enhanced for higher mass or higher compactness NSs. As the NS mass and spin increase during the CE inspiral, we are then able to track the temporal evolution of all of NS macroscopic quantities. For example, as the NS accretes, its gravitational mass responds to the baryon mass accretion rate $\dot{M}_{\rm b}$ as well as the angular momentum that the accreted mass carries. The resulting NS gravitational-mass accretion rate is $\dot{M}_{\rm NS}=\frac{\partial M_{\rm NS}}{\partial M_{\rm b}}\dot{M}_{\rm b}+\frac{\partial M_{\rm NS}}{\partial J_{\rm NS}}\dot{J}_{\rm NS}=\Phi_{\rm NS}\dot{M}_{\rm b}+\frac{\Omega}{c^{2}}\dot{J}_{\rm NS}\ ,$ (2) where $c$ is the speed of light, and $J_{\rm NS}=I_{\rm NS}\Omega$ is the NS spin angular momentum. Similarly, as the NS accretes, its spin angular momentum will also change, as given by $\dot{J}_{\rm NS}=\dot{I}_{\rm NS}\Omega+I_{\rm NS}\dot{\Omega}=\frac{{\rm d}I_{\rm NS}}{{\rm d}M_{\rm NS}}\dot{M}_{\rm NS}\Omega+I_{\rm NS}\dot{\Omega}\ .$ (3) With this at hand, we can now solve for the temporal evolution of the angular frequency and the gravitational mass. We assume the NS accretes from a Keplerian accretion disk, where matter captured within the NS accretion radius carries angular momentum and spirals several orders of magnitude down to the scale of several NS radii. Approximating the total torque as the accretion torque, $\dot{J}_{\rm NS}\approx\dot{M}_{\rm NS}\sqrt{GM_{\rm NS}R_{\rm NS}}$ (e.g., Brown et al., 2000), in Eqs. 2 and 3, and for now ignoring other external torques on the NS, we then find $\dot{M}_{\rm NS}=\frac{\Phi_{\rm NS}\dot{M}_{\rm b}}{1-\sqrt{GM_{\rm NS}R_{\rm NS}}\,({\Omega}/{c^{2}})}\ ,$ (4) and $\dot{\Omega}=\frac{\Phi_{\rm NS}\dot{M}_{\rm b}}{I_{\rm NS}}\frac{\sqrt{GM_{\rm NS}R_{\rm NS}}-\Omega\,({{\rm d}I_{\rm NS}}/{{\rm d}M_{\rm NS}})}{1-\sqrt{GM_{\rm NS}R_{\rm NS}}\,({\Omega}/{c^{2}})}\ .$ (5) The evolution equations (4) and (5) are generic for any slowly rotating NS accreting from a Keplerian disk. In general, however, the accretion and NS’s angular-momentum evolution may be more complex. Such complications can arise if the NS’s magnetic field pressure is comparable to the pressure of the radiation and accreting plasma, or if there is feedback from the accretion itself (e.g., Soker et al., 2019; Grichener & Soker, 2019; López-Cámara et al., 2020). For a given EoS, we can then find the right-hand sides of the above equations as a function of $M_{\rm NS}$ and $\Omega$, which leads to a closed system of ordinary differential equations, once $\dot{M}_{\rm b}$ is prescribed. In the CE inspiral context, the baryon mass accretion rate depends on the primary star’s envelope structure, which we discuss in the following subsection. ### 2.2 Primary massive-star models, common envelope accretion, inspiral, and ejection We evolve single massive stars with the MESA (v12778) stellar-evolution code (Paxton et al., 2010, 2013, 2015, 2018, 2019) to obtain their interior structure. We consider a total of 6 primary red-giant stars with masses of $M_{\star}/M_{\odot}=(12,12,16,16,20,20)$ with respective radii $R_{\star}/R_{\odot}=(173,594,322,672,872,1247)$. Here, the smaller radii at a given mass corresponds to the RGB base, while the larger radii at a given mass corresponds to the RGB tip. For our CE inspiral calculations, we take the envelope structure to be constant in time. As a NS inspirals in the CE, the envelope plasma supersonically flows past the NS and may be captured within the NS’s accretion radius $R_{\rm a}=2GM_{\rm NS}/v^{2}$, where $v$ is the upstream flow velocity, which, in the NS’s rest frame is the orbital velocity. If the upstream flow is homogeneous, then from Hoyle-Lyttleton (HL) theory (Hoyle & Lyttleton, 1939), the accretion rate and local drag force obey $\displaystyle\dot{M}_{\rm HL}$ $\displaystyle=\pi R_{\rm a}^{2}\rho v=\frac{4\pi\rho G^{2}M_{\rm NS}^{2}}{v^{3}}\ ,$ (6a) $\displaystyle F_{\rm d,HL}$ $\displaystyle=\dot{M}_{\rm HL}v=\pi R_{\rm a}^{2}\rho v^{2}=\frac{4\pi\rho G^{2}M_{\rm NS}^{2}}{v^{2}}\ ,$ (6b) where $\rho$ is the upstream mass density. For NS accretion in stellar- envelope environments, the density and temperature may be high enough for neutrino cooling, such that the accretion rate exceeds the Eddington limit (Houck & Chevalier, 1991). The envelope’s local density scale height may be comparable in size to the NS accretion radius, which breaks the symmetry that HL theory assumes and thus requires a treatment of this effect. To model the accretion and drag, we use the fitting formulae from De et al. (2020, see their Appendix A). The accretion and drag coefficients, $C_{\rm a}$ and $C_{\rm d}$ are defined such that the baryonic mass accretion rate and the local drag force are $\displaystyle\dot{M}_{\rm b}$ $\displaystyle=C_{\rm a}\dot{M}_{\rm HL}\,,\quad C_{\rm a}=C_{\rm a}({\cal M},q,R_{\rm sink})\,,$ (7a) $\displaystyle F_{\rm d}$ $\displaystyle=C_{\rm d}F_{\rm d,HL}\,,\quad C_{\rm d}=C_{\rm d}({\cal M},q)\ .$ (7b) These coefficients are both functions of the upstream Mach number ${\cal M}$ and the mass ratio $q$ between the compact object and the enclosed mass within the orbit. The accretion coefficient also depends on the sink radius $R_{\rm sink}$, given that the wind-tunnel simulations only resolve the accretion flow up to a sphere with radius $0.05R_{\rm a}$ surrounding the point-mass accretor. Thus, some fraction of matter that flows into the region within $0.05R_{\rm a}$ ultimately ends up accreting onto the NS. For each EoS, we use $R_{\rm NS}$ as the sink radius. In Appendix C, we describe in more detail how we compute these accretion and drag coefficients. We plot in Figure 2 the stellar profiles of the density, upstream Mach number, polytropic exponent, and envelope binding energy (panels A, B, C, and D, respectively) for the primary masses of $M_{\star}/M_{\odot}\in(12,16,20)$ that we consider here. Figure 2: MESA stellar models. Panel A: density profiles for primary stellar masses of $M_{\star}/M_{\odot}=(12,12,16,16,20,20)$ and respective radii of $R_{\star}/R_{\odot}=(173,594,322,672,872,1247)$. Masses of $M_{\star}/M_{\odot}=(12,16,20)$ correspond to blue, orange, and green colors, respectively. Solid and dashed lines correspond to models at the base and tip of the RGB, respectively. Panel B: the upstream Mach number ${\cal M}=v/c_{\rm s}$ for each stellar model (formatted in the same manner) for a NS companion with $M_{\rm NS}=1.4M_{\odot}$. Panel C: the polytropic exponent $\gamma$ for each stellar model. Horizontal magenta lines are shown for $\gamma=4/3$ and $\gamma=5/3$ (dot-dashed and dotted, respectively). Panel D: envelope binding energy profiles (absolute values) for each stellar model. For a given evolutionary stage and for most of the primary’s radii, the $\delta_{\rho}$ parameter decreases as the primary’s mass increases, such that the accretion rate will be greater and result in higher accreted mass. Primary stars that are smaller in size will have higher envelope binding energy, which thus requires more energy dissipation during the CE phase in order for successful envelope ejection. In §3, we quantify how much more NSs accrete when in envelopes with higher binding energies compared to less bound envelopes. Given the primary’s envelope structure, we can now model the CE inspiral as follows. We approximate the orbital inspiral with Newtonian gravity (Blanchet, 2014), given that on the scales of CE evolution, gravity is weak and the orbital velocities are non-relativistic, i.e., $v_{\rm orb}/c\ll 1$. With this in mind, the orbital energy throughout the inspiral is $E=-GM_{\rm NS}m_{\star}/(2a)$, where $m_{\star}=m_{\star}(a)=\int_{0}^{a}4\pi\rho r^{2}\,{\rm d}r$ is the mass enclosed within the NS’s separation from the primary’s center. The orbital velocity at any given time obeys $v^{2}=G\left[M_{\rm NS}+m_{\star}(a)\right]/a$, since we consider the inspiral to be quasi-circular. The change in the binary orbital energy as the NS inspirals thus obeys ${\rm d}E=\frac{\partial E}{\partial m_{\star}}{\rm d}m_{\star}+\frac{\partial E}{\partial M_{\rm NS}}{\rm d}M_{\rm NS}+\frac{\partial E}{\partial a}{\rm d}a\ ,$ (8) such that we can then solve for ${\rm d}a/{\rm d}t$ $\frac{{\rm d}a}{{\rm d}t}=a\left(-\frac{\dot{E}}{E}+\frac{\dot{m}_{\star}}{m_{\star}}+\frac{\dot{M}_{\rm NS}}{M_{\rm NS}}\right)=\frac{a}{1-4\pi\rho a^{3}/m_{\star}}\left(-\frac{\dot{E}}{E}+\frac{\dot{M}_{\rm NS}}{M_{\rm NS}}\right)\ ,$ (9) where $\dot{m}_{\star}=4\pi\rho a^{2}\dot{a}$ since we assume a static envelope. We take the energy decay rate to be the drag luminosity $\dot{E}=-F_{\rm d}(\delta_{\rho})v$, which dominates over the gravitational- wave luminosity from the orbital motion, and which can be obtained using both Eqs. (6b) and (7b). We summarize our integration procedure as follows. We first precompute the NS properties shown in Eqs. (4) and (5) as well as Appendix B for each EoS in our catalog, which are then stored as tables to interpolate from at each timestep of an orbital integration. We then explicitly integrate Eqs. (4), (5), and (9) to obtain $M_{\rm NS}$, $\Omega$, and $a$ throughout the CE inspiral. The NS properties at each point in the NS’s evolution correspond to a Hartle-Thorne NS with gravitational mass $M_{\rm NS}$ and spin $\Omega$ that we obtain from our pre-computed tables. Our orbital integrations are carried out for each of our 6 primary stellar models and for each EoS in our catalog, varying the initial NS gravitational mass $M_{\rm NS,0}$ and initial NS spin $\Omega_{0}$. We terminate these orbital integrations when the dissipated orbital energy $\Delta E_{\rm orb}=E(a)-E(a_{0})$ is equal to the primary envelope’s binding energy $E_{\rm env,bind}$ given by $E_{\rm env,bind}=\int_{m_{\star}(r)}^{m_{\star}(R_{\star})}\left(u-\frac{GM_{\star}(r)}{r}\right)\,{\rm d}M\ ,$ (10) where $u$ is the stellar fluid’s internal energy and where the integration coordinate is the primary’s mass coordinate. This amounts to assuming a CE efficiency parameter (e.g., Webbink, 1984) of $\alpha_{\rm CE}=1$. ## 3 Results ### 3.1 NS mass gain and spin-up We plot the NS evolution for the often fiducial case of a pre-CE NS mass of $1.4M_{\odot}$ and a primary of $12M_{\odot}$ in the top panel of Figure 3. For this case, the primary is taken to be at the RGB base such that $a_{0}=173R_{\odot}$ and we take the initial NS spin to be $\Omega_{0}/(2\pi)=50$ Hz. Figure 3: NS mass gain and spin-up. The initial pre-CE system considered here is a primary star at the base of the RGB with a mass $M_{\star}=12M_{\odot}$ with a companion NS that has an initial mass $M_{\rm NS,0}=1.4M_{\odot}$ and initial spin $\Omega_{0}/(2\pi)=50$ Hz. Top panel: gain in gravitational mass vs. orbital separation. The orange and black curves correspond to $\Phi<1$ and $\Phi=1$ (with binding energy vs. without), respectively, where each curve corresponds to a different EoS. Bottom panel: the final spin-up $\Delta\Omega_{\rm f}/(2\pi)$ vs. the final gravitational mass gain $\Delta M_{\rm NS,f}$ for each EoS and the same initial pre-CE parameters, except with a varying initial NS spin. The circle and diamond points are for $\Phi=1$ and $\Phi<1$, respectively. The color of each data point corresponds to a different initial NS spin of $\Omega_{0}/(2\pi)=$ (10, 100, 200, 500) Hz with blue, orange, green, and red, respectively. The black curves corresponds to $\Phi=1$, i.e., not accounting for NS binding energy. Each black and orange curve corresponds to a different EoS in our catalog. In all cases, the NS accretes no more than a few percent of its pre-CE mass, due to the suppressed accretion rate from the envelope density gradient. The gravitational-mass gain as well as the spin-up further decreases, since some of the accreted baryon mass-energy is converted into binding energy. In the bottom panel of Figure 3, we plot the final spin-up $\Delta\Omega_{\rm f}/(2\pi)=(\Omega_{\rm f}-\Omega_{0})/(2\pi)$ and the final gravitational-mass gain for each EoS model and for varying initial NS spins of $\Omega_{0}/(2\pi)=(10,50,100,200,500)$ Hz as blue, orange, green, red, and purple points, respectively. Higher initial NS spins increase the NS binding energy, such that less gravitational mass is gained and the spin-up decreases. With different EoSs, there is an anti-correlation between the mass gain and spin-up, where an EoS that allows for higher gravitational-mass gain results in a lower spin-up when starting with the same initial NS spin. A larger increase in $\Delta M_{\rm NS}$ is a result of a larger $\Phi_{\rm NS}$, i.e., higher baryon mass converted to gravitational mass. The gravitochemical potential $\Phi_{\rm NS}$ is proportional to the inertia $I_{\rm NS}$, such that less compact NSs are harder to spin up because they have higher $\Phi_{\rm NS}$ and higher $I_{\rm NS}$. Conversely, more compact NSs will gain less gravitational mass and spin-up faster because they have lower $\Phi_{\rm NS}$ and lower $I_{\rm NS}$. ### 3.2 Parameter survey Given that the relativistic mass deficit is greater for more massive NSs (see Figure 1), we then vary the pre-CE NS mass. We run inspirals for the following set of pre-CE NS masses $M_{0}/M_{\odot}\in[1.2,1.8]$ with a step size of $0.1M_{\odot}$. We plot in the top panels of Figure 4 the mean accreted masses when varying the EoS as solid lines with the shaded region corresponding to the $\pm 2\sigma$ deviation. The dashed lines correspond to $\Phi=1$, i.e., taking the accreted gravitational mass to be equivalent to the accreted baryonic mass. Figure 4: Varying initial NS masses, primary masses, and envelope structures. Top row: the accreted gravitational mass at the end of our orbital integrations for the 6 primary stellar models, initial NS gravitational masses ranging in $M_{0}/M_{\odot}\in[1.2,1.8]$ with spacings of $\delta M=0.1M_{\odot}$, and an initial NS spin of 50 Hz. The left, middle, and right columns correspond to primary stellar masses of $M_{\star}/M_{\odot}=(12,16,18)$, respectively. The top and bottom rows correspond to the accreted mass and the spin-up, respectively. The blue and orange curves correspond to primary stellar models at the base and tip of the RGB, respectively. The dashed lines correspond to $\Phi=1$, i.e., taking the accreted gravitational mass to be equivalent to the accreted baryonic mass. The width of the dashed line encompasses the $\pm 2\sigma$ region. The solid lines with shaded bands correspond to the mean and the $\pm 2\sigma$ deviation, respectively, of our predicted accreted NS masses including binding energy from our catalog of 46 EoSs. The width of the dashed line encompasses the $\pm 2\sigma$ region. In the bottom panels of Figure 4, we plot the corresponding spin-up $\Delta\Omega_{\rm f}=(\Omega_{\rm f,0}-\Omega_{0})/(2\pi)$. An increasing pre-CE NS mass results in a systematically decreasing accreted NS mass across all of our models. This is because at constant $\alpha_{\rm CE}$, having a more massive NS results in a larger dissipated orbital energy, such that envelope ejection is achieved at wider separations and such that the accreted baryonic mass is reduced compared to lower-mass NSs. It remains to be seen whether or not this trend will hold in global 3D hydrodynamic CE simulations when the initial NS mass is varied. Models at the RGB base result in higher accreted mass and spin-up, which is due to the larger envelope binding energy from their smaller sizes as compared to the RGB tip (bottom panel of Figure 2). As previously shown in Figure 1, NSs with higher gravitational mass will convert a larger fraction of the accreted mass into binding energy. We plot in Figure 5 the distributions of the ratio of the accreted gravitational mass to the accreted baryonic mass from our RGB base models. Figure 5: Ratio distributions. Distributions of the ratio of the accreted gravitational mass to the accreted baryonic mass $\Delta M_{\rm NS}/\Delta M_{\rm b}$. These distributions are represented with a kernel density estimator. Without accounting for NS binding energy, $\Delta M_{\rm NS}/\Delta M_{\rm b}=1$. The left, middle, and right panels correspond to primary stellar masses of $M_{\star}/M_{\odot}=(12,16,18)$ at the RGB base, respectively. In each panel, each distribution from bottom ascending to top is for initial NS gravitational masses of $M_{\rm NS,0}/M_{\odot}=(1.2,1.4,1.6,1.8)$, respectively. The green and magenta curves correspond to initial NS spins of 50 Hz and 200 Hz, respectively. The black dashed curve corresponds to the $\Phi_{\rm TOV,0}$ distribution from our EoS catalog, i.e., evaluating Equation 1 with the initial TOV mass and radius. Distributions for the RGB tip case will be similar, though the separation between distributions at the same initial NS mass will be smaller since the accreted baryonic mass for the RGB tip cases is smaller than the RGB base cases (see Figure 4). This is quantified in Appendix D and Figure 6. In each panel, each distribution from bottom ascending to top is for initial NS gravitational masses of $M_{\rm NS,0}/M_{\odot}=(1.2,1.4,1.6.,1.8)$, respectively. The green and magenta curves correspond to initial NS spins of 50 Hz and 200 Hz, respectively. We also plot a black dashed curve that corresponds to using $\Phi_{\rm TOV,0}$ as a fast approximation, i.e., the gravitochemical potential of the non-rotating NS at its initial properties. Higher initial spins tend to decrease $\Delta M_{\rm NS}/\Delta M_{\rm b}$, though the model with $M_{\rm NS,0}=1.8M_{\odot}$ and $M_{\star}=20M_{\odot}$ at the RGB base exhibits opposite behavior, albeit slight. Since lower-mass NSs accrete more gravitational mass compared to the higher-mass NSs in our models, the differences between the ratio distributions at various spins is also higher as well. Distributions for the RGB tip case will be similar, though the separation between distributions at the same initial NS mass will be smaller since the RGB tip cases resulted in less mass gain (Figure 4). ## 4 Discussion and Conclusions We have investigated here how NS binding energy affects NS-CE accretion, which plays a role in forming DNSs that merge within a Hubble time. We find that the gravitational-mass gain and spin-up is systematically reduced and that this effect is enhanced for higher-mass NSs. We also find that more compact NSs will gain less gravitational mass and spin-up faster due to having a lower $\Phi_{\rm NS}$ and a lower $I_{\rm NS}$ compared to less compact NSs. The strongest assumption from our model is that the envelope remains static throughout the inspiral. Realistically, the envelope is expected to respond and readjust in structure as the NS inspirals deeper toward the primary’s core. The accretion, which we have focused on in this work, is still expected to be some small fraction of the pre-CE NS mass. There will still be density gradients within the envelope that break BHL symmetry and the accreting material still needs to overcome the angular momentum barrier over multiple length scales. The amount of NS mass gain and spin-up we obtain with this modeling approach may be testable with Galactic DNS observations (e.g., Osłowski et al., 2011). For millisecond pulsars, spin-period derivatives corresponding to a spindown timescale of order a Hubble time would be ideal. If a phase-transition to quark matter happens in NS interiors, a new branch of stable stars with the same masses, but smaller radii relative to their hadronic counterparts can appear (e.g., Gerlach, 1968; Kampfer, 1981; Glendenning & Kettner, 2000; Montana et al., 2019). These have been called “twin-stars” and due to their larger compactness, the effects we present here would be further enhanced in comparison to the purely hadronic NSs we studied. We leave a more systematic investigation of these aspects for future work. This work demonstrates that a NS’s strong gravity and nuclear microphysics play a role in NS-CE evolution in addition to the macroscopic astrophysics of the envelope. Strong gravity and the nuclear EoS thus may affect the population properties of NS binaries and the cosmic double NS merger rate. Our results may further inform binary population synthesis models, 1D hydrodynamic CE inspiral coupled to stellar evolution, and global 3D hydrodynamic CE simulations. We thank the anonymous referee for comments and suggestions that led to improvements of this work. We thank Cole Miller for insightful discussions and for detailed feedback on our manuscript. A.M.H. is supported by the McWilliams Postdoctoral Fellowship. 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(2020) for purely hadronic NSs, including ALF2, APR3, APR4, BCPM, BSP, BSR2, BSR2Y, BSk20, BSk21, BSk22, BSk23, BSk24, BSk25, BSk26, DD2, DD2Y, DDHd, DDME2, DDME2Y, ENG, FSUGarnet, G3, GNH3, IOPB, K255, KDE0v1, MPA1, Model1, Rs, SINPA, SK272, SKOp, SKa, SKb, SLY2, SLY230a, SLY4, SLY9, SLy, SkI2, SkI3, SkI4, SkI6, SkMP, WFF1, and WFF2 (Read et al., 2009; Kumar & Landry, 2019). ## Appendix B The Neutron Star Gravitochemical Potential For a NS with spin parameter $\epsilon=\Omega/\Omega_{*}$, the gravitochemical potential is defined as (Alécian & Morsink, 2004) $\Phi_{\rm NS}=\sqrt{e^{\nu}(1+2h)}\ ,$ (B1) where $\nu$ and $h$ are both metric functions related to the metric tensor via $g_{tt}=-e^{\nu}(1+2h)$ in the Hartle-Thorne approximation. The metric function $\nu$ is a quantity of ${\cal{O}}(\epsilon^{0})$, and thus, it is obtained by solving the TOV equations. The metric correction $h$ is a quantity of ${\cal{O}}(\epsilon^{2})$, so we then write it as $h=\epsilon^{2}\delta h$, such that the non-rotating limit is recovered as $\epsilon\to 0$ and $\delta h$ remains finite. In order to evaluate the gravitochemical potential $\Phi_{\rm NS}$, we need to solve for the function $h$, which therefore requires that we solve the Einstein equations at second-order in the small rotation expansion (Hartle, 1967). Performing a Legendre decomposition, we can write $\delta h(r)=\delta h_{0}(r)+\delta h_{2}(r)P_{2}(\cos\theta)\ ,$ (B2) where $\theta$ is the polar angle from the equator, and $P_{2}$ is the second- order Legendre polynomial. Matching the interior and the exterior solutions at the NS surface allows us to find an exact solution for $\delta h_{0}(r)$ at the NS surface, namely $\delta h_{0}(R_{\rm TOV})=-\frac{\delta M}{R_{\rm TOV}-2M_{\rm TOV}}+\frac{\delta J^{2}}{R_{\rm TOV}^{3}(R_{\rm TOV}-2M_{\rm TOV})}\ .$ (B3) Here, $\delta J$ is the NS angular momentum at the Keplerian angular spin frequency. The function $\delta h_{2}(r)$ generally obeys $\delta h_{2}\ll\delta h_{0}$, such that when this function is scaled by $\epsilon^{2}$, which, for this work obeys $\epsilon^{2}\ll 1$, the contribution from the $\epsilon^{2}\delta h_{2}$ component to Equation B1 is effectively negligible. We thus take $\delta h(R_{\rm NS})\approx\delta h_{0}(R_{\rm NS})$, such that $h(R_{\rm NS})\approx\epsilon^{2}\delta h_{0}(R_{\rm TOV})=-\frac{\epsilon^{2}\delta M}{R_{\rm TOV}-2M_{\rm TOV}}+\frac{\epsilon^{2}\delta J^{2}}{R_{\rm TOV}^{3}(R_{\rm TOV}-2M_{\rm TOV})}\ .$ (B4) ## Appendix C Accretion and Drag Coefficients De et al. (2020) present fitting formulae for the accretion rate and drag within a non-relativistic background plasma. They consider two polytropic exponents of $\gamma=4/3$ and $\gamma=5/3$, where the coefficients for each fitting formula are given in their Tables A1 and A2. Given that our stellar models for the massive primaries have polytropic exponents that predominantly obey $4/3\leqslant\gamma\leqslant 5/3$ (see Figure 2), we compute the accretion and drag coefficients by weighting both the $C_{\rm ad,4/3}$ and $C_{\rm ad,5/3}$ formulae as $\displaystyle C_{\rm a}$ $\displaystyle=\xi\left(w_{4/3}C_{\rm a,4/3}+w_{5/3}C_{\rm a,5/3}\right)\ ,$ (C1a) $\displaystyle C_{\rm d}$ $\displaystyle=w_{4/3}C_{\rm d,4/3}+w_{5/3}C_{\rm d,5/3}\ ,$ (C1b) where $\displaystyle w_{4/3}=1-3(\gamma-4/3)\ ,$ (C2a) $\displaystyle w_{5/3}=1-3(5/3-\gamma)\ ,$ (C2b) and where $\xi$ is defined as $\xi\equiv(R_{\rm NS}/0.05R_{\rm a})^{0.33}\ .$ (C3) For $\gamma<4/3$, we use $C_{\rm ad,4/3}$. The factor $\xi$ approximates the fraction of matter flowing into the sink radius that ultimately accretes onto the NS. Given that these wind-tunnel models do not resolve the flow past a sink radius $R_{\rm sink}=0.05R_{\rm a}$, the matter falling into this sink volume is likely to be an upper estimate of the NS’s accreted baryons. De et al. (2020) estimate how the accretion rate depends on the sink radius and fit a power-law dependence $\dot{M}\propto\left(R_{\rm sink}/R_{\rm a}\right)^{\alpha_{\dot{M}}}$, where $\alpha_{\dot{M}}\approx 0.33$ with a scatter of order tens of percent. ## Appendix D Kullback-Leibler Divergence For a given NS-CE system evolution with an initial primary star with mass $M_{\star}$ and radius $R_{\star}$ and an initial NS with mass $M_{\rm NS,0}$ and spin $\Omega_{0}$, we define $p$ as distribution of $\Delta M_{\rm NS}/\Delta M_{\rm b}$ predicted from our EoS catalog and semi-analytic models. We also define $q$ as the distribution of $\Phi_{\rm TOV,0}$ from our EoS catalog, i.e., evaluating Equation 1 at the initial NS parameters. Given these two distributions, we can compute the Kullback-Leibler divergence, given by ${\cal D}(p\|\|q)=\int p(x)\ln\left(\frac{p(x)}{q(x)}\right)\ {\rm d}x\ ,$ (D1) where $x\equiv\Delta M_{\rm NS}/\Delta M_{\rm b}$. The distributions $p$ and $q$ are approximated as a kernel-density estimate of the samples for each model. One can interpret the KL divergence between $p$ and $q$ as the information loss when using $q$ to approximate $p$. Conversely, it can be interpreted as the information gained by using $p$ in place of $q$. Directly using $\Phi_{\rm TOV,0}$ as a fast approximation in other models such as population synthesis or as a subgrid prescription for global 3D hydrodynamic simulations may be acceptable as long as $\Delta M_{\rm NS}/M_{\rm NS,0}\lesssim 1\%$ and if the initial NS spin is low enough. To quantify the information loss from this approximation, we compute the KL divergences (Kullback & Leibler, 1951, Appendix D) between our semi-analytic inspiral models and using Equation 1 at the initial NS properties over a range of initial NS spins: $\Omega_{0}/(2\pi)=(10,20,50,80,100,150,200,300,500)$ Hz. We plot the KL divergences for each of our models in Figure 6. Figure 6: KL divergence vs. initial NS spin. The KL divergence between $\Delta M_{\rm NS}/\Delta M_{\rm b}$ and $\Phi_{\rm TOV,0}$ evaluated at varying initial NS spins of $\Omega_{0}/(2\pi)=(10,20,50,80,100,150,200,300,500)$ Hz. The top and bottom rows are for stellar models at the RGB base and tip, respectively, with each column for primary stellar masses of $M_{\star}/M_{\odot}=(12,16,18)$, respectively. The blue, orange, green, and red curves correspond to initial NS gravitational masses of $M_{\rm NS,0}/M_{\odot}=(1.2,1.4,1.6,1.8)$. For KL divergences $\lesssim 0.1$, the information loss is considered to be small, while KL divergences $\gtrsim 1$ corresponds to a large information loss. For initial NS spins of $\lesssim 200$ Hz, the KL divergence is $\lesssim 0.1$, corresponding to a small information loss and thus $\Phi_{\rm TOV,0}$ being a reasonable approximation if used in other models.
11institutetext: European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago, Chile 11email<EMAIL_ADDRESS>22institutetext: Université Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 33institutetext: Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile 44institutetext: Escuela de Ingeniería Industrial, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile 55institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France 66institutetext: Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 77institutetext: Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA 88institutetext: Department of Physics and Astronomy, Bucknell University, Lewisburg, PA 17837 # A search for a 5th planet around HR 8799 using the star-hopping RDI technique at VLT/SPHERE Z. Wahhaj 1155 J. Milli 1122 C. Romero 1122 L. Cieza 3344 A. Zurlo 334455 A. Vigan 55 E. Peña 11 G. Valdes 11 F. Cantalloube 66 J. Girard 77 B. Pantoja 88 (Received June 29, 2020; accepted December 20, 2020) ###### Abstract Context. The direct imaging of extrasolar giant planets demands the highest possible contrasts ($\Delta$H $\gtrsim$10 magnitudes) at the smallest angular separations ($\sim 0.1^{\prime\prime}$) from the star. We present an adaptive optics observing method, called star-hopping, recently offered as standard queue observing (service mode) for the SPHERE instrument at the VLT. The method uses reference difference imaging (RDI) but unlike earlier works, obtains images of a reference star for PSF subtraction, within minutes of observing the target star. Aims. We aim to significantly gain in contrast over the conventional angular differencing imaging (ADI) method, to search for a fifth planet at separations less than 10 au, interior to the four giant planets of the HR 8799 system. The most likely semi-major axes allowed for this hypothetical planet, estimated by dynamical simulations in earlier work, were 7.5 and 9.7 au within a mass range of 1–8 $M_{Jup}$. Methods. We obtained 4.5 hours of simultaneous low resolution integral field spectroscopy (R$\sim$30, Y–H band with IFS) and dual-band imaging (K1 and K2-band with IRDIS) of the HR 8799 system, interspersed with observations of a reference star. The reference star was observed for about one-third of the total time, and generally needs to be of similar brightness ($\Delta$R$\lesssim$1 magnitude) and separated on sky by $\lesssim$1–2o. The hops between stars were made every 6–10 minutes, with only 1 minute gaps in on-sky integration per hop. Results. We did not detect the hypothetical fifth planet at the most plausible separations, 7.5 and 9.7 au, down to mass limits of 3.6 and 2.8 $M_{Jup}$ respectively, but attained an unprecedented contrast limit of 11.2 magnitudes at 0.1′′. We detected all four planets with high signal-to-noise ratios. The YJH spectra for planets $c$, $d$ were detected with redder H-band spectral slopes than found in earlier studies. As noted in previous works, the planet spectra are matched very closely by some red field dwarfs. Finally, comparing the current locations of the planets to orbital solutions, we found that planets $e$ and $c$ are most consistent with coplanar and resonant orbits. We also demonstrated that with star-hopping RDI, the contrast improvement at 0.1′′ separation can be up to 2 magnitudes. Conclusions. Since ADI, meridian transit and the concomitant sky rotation are not needed, the time of observation can be chosen from within a 2–-3 times larger window. In general, star-hopping can be used for stars fainter than R=4 magnitudes, since for these a reference star of suitable brightness and separation is usually available. The reduction software used in this paper has been made available online1. ###### Key Words.: exoplanets – adaptive optics ## 1 Introduction 11footnotetext: https://github.com/zwahhaj/starhopping. Radial velocity (RV) surveys have revealed to us the exoplanet population orbiting within $\sim$5 au of their parent stars (Mayor et al., 2011; Fernandes et al., 2019). Transit techniques have done the same for the population of closer-in planets ($\lesssim$1 au), providing us a glimpse of their atmospheres as inferred from their spectra (Howard et al., 2010; Dong & Zhu, 2013; Madhusudhan, 2019). Direct imaging on the other hand has found more than a dozen planets orbiting farther than 10 au from their stars (http://exoplanet.eu/). Direct imaging and interferometry are the only methods that allow us to obtain spectra of exoplanets separated by more than a few au from their host stars (Bonnefoy et al., 2014, 2016). Direct imaging is also the only technique that captures protoplanetary disks in the act of forming planets (Keppler et al., 2018; Müller et al., 2018; Haffert et al., 2019). Moreover, it has shown us fully formed planetary systems with their left-over dusty planetesimal disks (Lagrange et al., 2012), and captured these dust- producing rocky disks at various stages over their lifetime (e.g., Boccaletti et al., 2018, 2020; Wahhaj et al., 2016). Studies of systems like HR 8799 with its four planets can offer us a glimpse at possible early (age $<$ 30 Myrs) architectures (Marois et al., 2010), perhaps at a stage prior to major planet-migration and scattering (Chatterjee et al., 2008; Crida, 2009; Raymond et al., 2010). However, the extrasolar Jupiter and Saturn analogs are mostly still hidden from us, orbiting in the glare of their parent stars between 5 and 10 au (Fernandes et al., 2019). Fortunately, a giant planet at an age of 30 Myr can be a hundred times brighter than at 300 Myr (e.g., Allard et al., 2012a). With direct imaging, we are trying to detect the younger component of this hidden population, bridging the unexplored gap to connect to the RV and transit exoplanet populations closer in. In fact, some of the state-of-the-art direct imaging surveys have nearly completed and yielded a few more giant planets, fainter and orbiting closer to their stars than in earlier surveys, but mostly they report that the regions beyond 10 au rarely have planets more massive than 3–5 $M_{Jup}$. (Nielsen et al., 2019; Chauvin et al., 2017; Macintosh et al., 2015). Especially for gound-based instruments, the success of the direct imaging technique, imaging dozens of exoplanets and protoplanetary disks has been mainly due to angular and spectral difference imaging (ADI, SDI and ASDI; Liu, 2004; Marois et al., 2006; Sparks & Ford, 2002; Wahhaj et al., 2011). Without point spread function (PSF) differencing, within minutes we hit a wall in terms of sensitivity because of quasi static speckles in adaptive optics images. Speckles essentially mimic astronomical point sources, integrating more like signal than noise. The ADI, SDI and other related techniques decouple the speckles from the real signal, allowing them to be isolated and subtracted. However, these techniques are hampered by the self-subtraction problem (Marois et al., 2006). Since the decoupling of speckles and astronomical signal is never complete, there is inevitably some self- subtraction of signal. This can be manageable for planets moderately separated from the star, where we just lose sensitivity depending on the subtraction algorithm used (e.g., Wahhaj et al., 2013, 2015). However, for planet-star separations of 1–2 resolution elements and extended structures like circumstellar disks, the signal can be completely subtracted or the morphology significantly altered or completely masked (Milli et al., 2012). Reference difference imaging (RDI), a possible solution, has been routinely used in space telescope observations (eġ\̇@@bibref{missing}{missing}{missing}{missing}AuthorsPhrase1Year1999ApJ…525L..53W, 2016ApJ…817L…2C, ), as the PSF is quite stable over successive orbits of the telescope. However, RDI is not often used in ground-based observing where PSFs change significantly over hours. This is because, prior to extreme AO, the PSF of other stars could not closely match the target PSFs in speckle similarity, especially if the reference star images were not obtained the same night as the science target. Nevertheless, impressive ground-based results on quite a few targets have been achieved (Lagrange et al., 2009; Xuan et al., 2018; Ruane et al., 2019; Bohn et al., 2020). In the more recent efforts, reference PSFs were obtained 30 mins to hours apart and the telescope operator would have to manually setup the guiding for each target change, costing significant human effort and photon dead-time. Starting recently at VLT/SPHERE, we now offer fast automated RDI available in queue mode for the first time, requiring only a one minute gap for each target change, a technique monikered star- hopping RDI. To demonstrate the power of this new observing mode, and to look for new planets closer to the star, we targeted HR 8799, the home of the four giants. HR 8799 is a young main-sequence star (age 20–160 Myrs; Cowley et al., 1969; Moór et al., 2006; Marois et al., 2008; Hinz et al., 2010; Zuckerman et al., 2011; Baines et al., 2012) at a distance of 41.29$\pm$0.15 pc (Gaia Collaboration, 2018). The space motions of the star suggest membership in the Columbia moving group (age 30–40 Myr; Torres et al., 2008; Zuckerman et al., 2011; Bell et al., 2015; Geiler et al., 2019). It has four directly imaged giant planets at projected distances of 15, 27, 43, and 68 au (Marois et al., 2008, 2010). Upper-limit to the masses from orbital stability requirements and the derived luminosities assuming an age of $\sim$30 Myrs suggest that the planet masses are 5–7 $M_{Jup}$ (Marois et al., 2010; Currie et al., 2011; Sudol & Haghighipour, 2012). Interior and exterior to the planets, warm dust at 6–10 au and an exo-Kuiper Belt beyond 100 au have been detected (Sadakane & Nishida, 1986; Su et al., 2009; Hughes et al., 2011; Matthews et al., 2014; Booth et al., 2016). Thus, it is likely that the planets formed in a circumstellar disk, instead of directly from a protostellar cloud as in binary or multiple star formation. However, it is currently a theoretical challenge to form so many massive planets in a single system. The total system architecture and stability, considering the age, mass and debris disk formation history have been studied in some detail (see Goździewski & Migaszewski, 2009, 2014, 2018; Reidemeister et al., 2009; Su et al., 2009; Fabrycky & Murray-Clay, 2010; Moro-Martín et al., 2010; Galicher et al., 2011; Marleau & Cumming, 2014; Matthews et al., 2013; Booth et al., 2016; Konopacky et al., 2016; Wilner et al., 2018; Geiler et al., 2019). HR 8799 is a star of the $\lambda$ Bootis type (indicating an iron poor atmosphere), and also a $\gamma$ Dor variable, indicating small surface-pulsations perhaps also due to some accretion-associated chemical peculiarity (Saio et al., 2018; Saio, 2019; Takata et al., 2020). Spectra of the planets has been obtained in the NIR bands with Keck/OSIRIS (Barman et al., 2011, 2015; Konopacky et al., 2013), Project 1640 at Palomar (Oppenheimer et al., 2013), VLT/NACO (Janson et al., 2010), GPI (Ingraham et al., 2014) and SPHERE (Zurlo et al., 2016; Bonnefoy et al., 2016). The comparison of the spectra to brown dwarfs, cool field objects and current atmospheric models suggest patchy thin and thick clouds of uncertain height, non-equilibrium chemistry, and a dusty low-gravity atmosphere (Marois et al., 2008; Currie et al., 2011; Madhusudhan et al., 2011; Skemer et al., 2012; Marley et al., 2012; Morley et al., 2012; Apai et al., 2013; Buenzli et al., 2015). Given the theoretical challenge in explaining such a massive multi-planet and debris disk system with detailed and specific information, and the prospect of finding additional planets (Goździewski & Migaszewski, 2014, 2018) the system deserves a deeper look. We describe our SPHERE study of HR 8799 in the following sections. The reduction software used in this paper can be found online 111https://github.com/zwahhaj/starhopping. ## 2 Observations ### 2.1 Telescope and instrument control for Star-hopping The goal of star-hopping on VLT/SPHERE is to switch from recording adaptive optics corrected images of the science star to the reference star with only a $\sim$1 minute gap. Thus the usual help from the human operator to setup the guide star for the primary mirror’s active optics correction, typically a 5 minute interaction, should be restricted to once per star, thus two times in total. This would allow us to make hops between science and reference star every $\sim$10 minutes without much loss in photon collecting efficiency, and ensuring minimal change in the PSF shape in the elapsed time. We do not provide an exact calculation for the optimum hopping frequency as it depends strongly on how the seeing and coherence time vary over the observation. However, we found in our observations that PSF similarity drops $\sim$2% every 10 minutes (see Section 3.3). This is significant as the sensitivity reached depends non-linearly on the PSF subtraction quality. Thus, we recommend observing the science target for 10 minutes, then hopping to the reference star and observing it for 5 mins, repeating the cycle as needed. To preserve PSF similarity and for time-efficiency, the AO loops would not be re-optimized when changing stars, and thus the reference star would need to have an R-magnitude (mag) within 1 mag of the science star, to ensure similar AO performance. While, we do not have strong constraints on the color of the reference star, again similar brightness (within 1 mag) in the observing wavelength is important. This is because the adaptive optics performance need to be similar and the signal-to-noise of the reference images need to be comparable or better. Also, the reference star would need to be within 1 to 2 degrees of the science star, so that the main mirror’s shape-changes at the new pointing would not result in large changes in PSF properties. Fortunately, for the vast majority of stars fainter than $R\sim 4$ mags a suitable reference star can be found, making star-hopping practical for $R\sim$ 4–13 mag stars. The solution required new software to be designed for telescope control and new template software to be written for the observing sequence and instrument, i.e., SPHERE’s control. For SPHERE, we designed two new acquisition templates called starhop and hopback which are only responsible for moving the telescope between the two stars and store relevant setup information so that subsequent hops can be made automatically. Thus a typical observing sequence would be: 1) Normal acquisition of science star with desired instrumental mode and setup, 2) An observing template lasting a few minutes, 3) Acquisition of reference star a few degrees away, with the starhop template, 4) Another observing template, 5) Quick return to the science star using the hopback template lasting $\sim$1 minute, 6) Another observing template 7) Quick return to reference star using the hopback template again, 8) As many iterations of steps 4 to 7 as desired. All three types of acquisitions constitute a full preset of the telescope, i.e., the primary mirror’s shape and the secondary’s pointing are set by a look-up table, then a guide star is selected (automatic for hopback) for accurate pointing corrections, continuous active optics corrections for the main mirror shape are activated using the guide star. However, human operators only assist with the first (normal) acquisition and the starhop acquisition, especially in the selection of the guide star and related setup. The starhop template stores all parameters required for these setups for the first star, moves (presets) to the second star, lets the operator assist in the second acquisition, and then stores all the parameters for the second acquisition. Small telescope offsets for fine-centering made by the operator when positioning the star on the instrument detector, are also recorded. Thus, the hopback template already has the relevant parameters saved and can automatically hop back and forth between the two stars, taking only $\sim$1 minute each time. ### 2.2 HR 8799 observations We observed HR 8799 as part of a director’s discretionary time (DDT) proposal, to test the performance limits of star-hopping with RDI on SPHERE. The SPHERE instrument (Beuzit et al., 2019), installed at the Nasmyth Focus of unit telescope 3 (UT3) at the VLT, is a state-of-the-art high-contrast imager, polarimeter and spectrograph, designed to find and characterize exoplanets. It employs an extreme adaptive optics system, SAXO (Fusco et al., 2005, 2006; Petit et al., 2012; Sauvage et al., 2016), with 41$\times$41 actuators (1377 active in the pupil) for wavefront control, a low read noise EMCCD running at 1380 Hz, a fast (800 Hz bandwidth) tip-tilt mirror (ITTM) for pupil stabilization, extremely smooth toric mirrors (Hugot et al., 2012), and a differential tip-tilt loop for accurate centering in the NIR. This system can deliver H-band strehl ratios for bright stars (R$<$9) of up to 90% and continue to provide AO correction for stars as faint as R$=$14 mags. SPHERE also provides coronagraphs for diffraction suppression, including apodized Lyot coronagraphs (Soummer, 2005) and achromatic four-quadrants phase masks (Boccaletti et al., 2008). It is comprised of three subsystems: the infrared dual-band imager and spectrograph (IRDIS; Dohlen et al., 2008), an integral field spectrograph (IFS; Claudi et al., 2008) and the Zimpol imaging polarimeter (ZIMPOL; Schmid et al., 2018). We observed HR 8799 in the IRDIFS extended mode (Zurlo et al., 2014), where IRDIS K1 and K2-band images and IFS Y–H spectra are obtained simultaneously (Vigan et al., 2010). The IRDIFS data was obtained in three 1.5 hour observing blocks (OBs), one block on the night of October 31, 2019 and two contiguous blocks on the night of November 1, 2019. We used the N_ALC_YJ_S coronagraph with a central obscuration of radius 73 mas, which is not ideal for the maximum contrast in K-band but ensures that any object at 100 mas separation would not be partially obscured. With IRDIS we used 8s exposures, while with IFS we used 32s. We also obtained short-exposure unsaturated non-coronagraphic observations of the primary star for flux calibration, which we will call FLUX observations henceforth. The datasets can be found in the ESO archive under program ID 2103.C-5076(A) and container IDs: 2622640, 2623891 and 2623923. Each container represents a separation epoch, consisting of several OBs alternating between HR 8799 and the reference star. The reference star, HD 218381 (spectral type K0 vs F0V for HR 8799), is separated 0.55o from HR 8799 and is 0.52 mag fainter than it in R-band but 0.75 mag brighter in H-band. In total, we had 1440 IRDIS exposures for HR 8799 and 830 for the reference star. With IFS, we had 190 exposures for HR 8799 and 114 for the reference star. The observing conditions were average, with a coherence time of 4.7$\pm$1.3 ms, a seeing of 0.9$\pm$0.15′′, and a windspeed of 2.1–7.7 m/s without the low-wind effect (Milli et al., 2018). The total sky-rotations were 23.8o on the first night and 53.4o on the second night. ## 3 Data Reduction ### 3.1 IFS reduction and contrast limit estimates Since our main motivation is to achieve sensitivities to fainter planets than earlier observations, we begin by estimating the detection limits of our data set and post-processing method. The detection limits are estimated by comparison to simulated planets which undergo the same reduction processes as the real planets. The measurement and analysis of the real planets in the system are presented afterwards. For the basic reduction calibrations, we used SPHERE pipeline version 0.36.0 and scripts by (Vigan et al., 2015, http://astro.vigan.fr/tools.html) The IFS data sets from all 3 epochs were combined to form a cube of 7254 images, 186 images in each of the 39 wavelength channels. In each image, 16 simulated companions were inserted with offsets wrt. to the star, given by separations: 0.1$"$ to 1.6$"$ with steps of 0.1$"$ and position angles increments of 90o with each step. The simulated companions were made from the FLUX exposures of the primary appropriately scaled in intensity. Since these sources were given constant chromatic contrast, i.e. the same spectra as the host star, we did not apply any spectral differencing in the reduction described below. The contrasts of these sources were chosen to be roughly 2 mags brighter than a preliminary contrast limit estimate for the data set. The reference PSF data set consisted of 4446 images. All science and reference images were unsharp-masked, i.e., each image was convolved with a Gaussian of FWHM 0.1$"$ (roughly twice the image resolution) and subtracted from the original to produce an image where most large scale spatial features like diffuse stellar light has been removed (eġ\̇@@bibref{missing}{missing}{missing}{missing}AuthorsPhrase1Year1999PASP..111..587R,2013ApJ…779…80W, ). A diagonally oriented stripe pattern was found in all the IFS images, which we were unable to remove in the basic calibrated images. A zero-valued image passed through the basic calibration also yielded this pattern, found to be independent of the channel wavelength. Thus the pattern is likely an artefact of the pipeline. The output pattern image was bad-pixel cleaned and unsharp- masked to prepare it to be subtracted from the science images. Two annular regions were defined to optimize PSF subtraction, i.e., minimize the residual RMS in each region. These two annuli had inner and outer radii of 0.075$"$ and 0.67$"$, and 0.67$"$ and 1.33$"$ respectively. The science images were median- combined without de-rotation to reveal the background stripe pattern more clearly. Then we obtained the best intensity-scaled pattern images for the inner and outer annuli, which we in turn subtracted from each science image, to perform a preliminary removal of the pattern. Next, for each science image, we computed the best linear combination of reference images that reduced the RMS in the two annular regions separately, similar to the LOCI algorithm (Lafrenière et al., 2007), but a much simpler version since optimization is done only over the two large annuli. We then took the difference of the science image and this composite reference image, and further applied an azimuthal profile removal filter as described in Wahhaj et al. (2013). All the difference images were median-combined again to check for any residual striped pattern, and remove it again by the same procedure as before. Generally, we see a consistent but modest improvement in contrast ($\sim$ 0.2 mag) with the use of image filters (e.g. unsharp masking), and so we recommend their use. Also, we notice fewer artifacts, e.g. fewer PSF residuals in these reductions. However, as data sets may differ in PSF morphology, we also recommend studying reductions without applying such filters, even when trying to detect faint point sources. Figure 1: Left: An IFS Y–H band reduced image showing simulated planets which are recovered with high SNR. The source recovered closest to the star indicates a contrast limit of 11.2 mags at 0.1′′ projected separation. Right: An IRDIS K1+K2 band reduced image also showing simulated planets at the same separations, all recovered with high SNR. The same contrast at 0.1′′ was reached with IRDIS also. The planets were inserted into the basic calibrated data (flat-fielded, dark-subtracted and bad pixel corrected) All real planets have been masked out. The color scale is linear with intensity. Figure 2: Contrast limits achieved in the IFS and IRDIS data sets, estimated by flux comparison to simulated planets recovered post-reduction. Figure 3: IFS and IRDIS images from star-hopping RDI reductions shown with same scale and orientation (North is up, East is left). Left: SNR map of the IFS Y–H band reduced image, showing only the real planets. The azimuthal filtering creates the dark negative arcs around the planets. They are more pronounced in the IFS reduction as more images were combined here than for the IRDIS reduction. Right: SNR map of the IRDIS K1+K2 band reduced image, showing only the real planets. The star, at the center of the black circle, is masked by the coronagraph. No new planets are detected in the newly probed region around 0.1′′ separation above the contrast limit of 11.2 mags. Next, the images were derotated to align the sky with North up and East left orientation and median-combined. A signal-to-noise map is made for the final reduced image (Figure 1), where the pixels in annular rings of width 4 pixels are divided by the robust standard deviation in that region. The robust value is taken to mitigate the effect of the simulated planets on the RMS. The signal-noise-ratio of each recovered simulated planet was then compared to its input contrast to calculate the 5$\sigma$-contrast limit achieved at the separation, like so $Contrast=InputContrast\times SNR/5$. The 5$\sigma$-contrasts achieved in this RDI-only reduction at 0.1$"$, 0.2$"$, 0.4$"$ and 0.8$"$ separations were 11.2, 13.5, 14.4 and 15 mags, corresponding to mass limits of 6.5, 3.1, 2.3 and 1.8 $M_{Jup}$ respectively, as estimated from BT-Settl models assuming an age of 30 Myrs (Allard et al., 2012b). The contrast curve is shown in Figure 2. The reduction showing only the real planets (without simulated planet insertions) is shown in Figure 3. No new planets are detected. ### 3.2 IRDIS reduction and contrast limit estimates The IRDIS reductions with simulated planets were done in a similar way to the IFS reductions. Since there were less images to process, we opted to use a more sophisticated but also more computation intensive reduction method. The simulated planets were inserted in the basic calibrated data at the same offsets with respect to the star as before. The planets inserted were $\sim$1 mag brighter than the 5$\sigma$ detection limit. For this exercise, we did not correct the relative rotational offset between IFS and IRDIS, so the PAs of the real HR 8799 planets do not agree between the two reduced images in Figure 1. There were 1443 good science images in the three datasets combined and 828 reference images. The images were first unsharp-masked. Next, we calculated the residual rms between all pairs of science and reference images, after intensity scaling to minimize the rms between 70 mas and 270 mas. For each science image, the best 16 reference images (more would worsen signal loss) were linearly combined by LOCI for subtraction to minimize the residual rms separately in annular rings covering the whole image. Each target annulus, where the subtraction was actually done, had width 200mas. But the reference annuli, where LOCI tried to minimize the residual rms, started 25mas outside the target annuli and extended outwards to the cover the rest of the image. This was done to mitigate over-subtraction and signal loss. We chose these parameters mostly by trial and error. The azimuthal filtering, de-rotation and combination of all the difference images, and the contrast limit estimates were done in the same way as in the IFS reduction. The final reduced images (with and without simulated companions) and the contrast performance are shown in Figures 1, 3 and 2, respectively. The IRDIS contrast limit is 11.2 mags at 0.1′′ which is equal to the IFS limits, but IFS fares $\sim$0.5 mags better at larger separations. ### 3.3 Comparison of RDI and ADI IRDIS detection limits For a comparison of typical ADI and RDI IRDIS observations we use only the first of the three data sets, totalling 1.5 hours of execution time, since this is slightly longer that the typical observation length (1 hour) at the VLT. The data set constitutes 481 science images and for RDI, 276 reference images. The total sky rotation in the sciences images was 24o. We performed 3 different ADI-based reductions which we call ADI-LOCI-F1, ASDI-LOCI-F10 and ASDI-PCA-F10. The ADI-LOCI-F1 is the same as the RDI reduction in terms of reference image selection and reference sector size and the use of LOCI, except that the references were restricted to those with more relative rotation than one-half FWHM (found by trial). The ASDI-LOCI-F10 reduction (ASDI is Angular and Spectral Difference Imaging) was performed on a data set with simulated companions which were made 10 times fainter (thus labeled F10) in the K2 channel than in K1 channel, allowing aggressive spectral differencing and a potential contrast gain over ADI. Since reference images could have companions both spectrally and rotationally displaced, only the combined displacement need to be more than one-half FWHM. The ASDI-PCA-F10 reduction was performed on the same data set as that of ASDI-LOCI-F10. The reduction parameters were again optimized by trial and error. We used principal component analysis (PCA) to construct the subtraction PSFs with 5 components (See Soummer et al., 2012). However, for each science image, and for each annular sub-component of the image (same as the reductions above) only selected subsets of the science images were chosen as input for the PCA – residual rms were calculated after subtracting all science image pairs, the best 30 matches (with least rms) that had more relative rotation than one-half FWHM were chosen, if less than 30 appropriate matches were found then the relative rotation criteria was relaxed to down to one-fourth FWHM, but no further, to allow input images for the PSF construction. This more selective approach to PCA helps to reduce the signal self-subtraction expected in ADI, and our tests supported this assumption, yielding significantly better results than PCA alone. The RDI reduction (see top of section 3.2) was repeated for the same 1.5 hour data set used in the ADI reduction. In Figure 4 we compare the RDI and the ADI-LOCI-F1 reduction. The simulated planets inside 0.3′′ separation are much better recovered in the star-hopping RDI reduction. In the ADI reduction, the innermost planet at 0.1′′ is not recovered at all, while the one at 0.2′′ is barely recovered. Contrast curves were calculated from the signal to noise ratio of the recovered simulated planets as before. The contrast improvement of RDI over the three ADI reductions, more than 2 mags at 0.1′′ separation, is shown in Figure 5 as a difference between the two contrast curves. The improvement will of course vary with the total amount of sky rotation in the science images. Figure 4: Comparison of star-hopping RDI versus ADI reductions of IRDIS K1+K2 band data injected with flat spectrum simulated planets. The inner two simulated planets are not successfully recovered in the ADI reduction, while they are clearly detected in the RDI reductions. The third simulated planet is recovered significantly better in the star-hopping RDI reduction. All real planets have been masked out. The color scale is linear with intensity. Figure 5: RDI contrast improvement over ADI or ASDI, estimated from the SNR of recovered simulated companions from an IRDIS data set. The star-hopping RDI technique yields detections limits more than 2 mags fainter than ADI at 0′′.1 separation from the target star. The green line shows the case for a K1/K2 companion flux ratio of 10, and very similar algorithms for RDI and ASDI, except that the ASDI reduction is fine-tuned to minimize self-subtraction. The blue line similarly shows RDI$-$ADI difference for equal K1, K2 flux. The red line shows the RDI improvment against the best PCA-based ADI reduction for a K1/K2 flux ratio of 10. The LOCI and PCA reductions are described in section 3.3. Figure 7 illustrates why star-hopping RDI performs so much better than ADI. It shows the residual fractional rms (RFR) for each science image as a function of relative rotation, i.e., the remaining rms between 0.1′′–0.3′′ separations after subtraction of another science or reference star image, divided by the original rms in each science image. Specifically, $RFR_{i}=RMS(s_{i}-o_{j})/RMS(s_{i})$, where $s_{i}$ is a science image, $o_{j}$ is another science or reference star image and $RMS$ is computed between 0.1′′–0.3′′ separations. The RFRs post-RDI subtraction had a 2$\sigma$ range of 0.32–0.78. We see that although the science images provide better- matched PSFs in general, the images that can be used with minimal self- subtraction are much fewer and much poorer matches than the RDI reference set. Thus, the reference star images constitute a superior set for constructing subtraction PSFs. In Figure 6 we show that artificially increasing the field rotation for the RDI reduction (1.5 hour data set) before coadding the images does not improve the contrast significantly. Thus the speckle residuals are comparable to white noise as more rotation does not seem to result in additional smoothing. We estimate no improvement at 0.1′′, $\sim$0.2 mag improvement farther out when comparing rotations of 140o to 20o, and 0.5 mag improvement at 1′′, when comparing rotations of 140o to 2o. The reductions were done by mutliplying the actual position angles of the images by specific factors that would achieve total field rotaions of 2o to $\sim$140o (distributed logarithmically), before coadding the images. Figure 6: Artificially increasing the field rotation for an RDI reduction before coadding the images does not improve the contrast significantly (see section 3.3). The legend gives the total rotation of the reduction for each contrast curve. At small separations (0.1–0.2′′) we see no improvement, as contrasts are not correlated with rotation angles. At larger separations, we see a maximum of 0.5 mag improvement between the minimum and maximum rotations, 2o and 143o, but only 0.3 mag improvement between 18o and 143o. During star-hopping tests on the night of August 8, 2019, we obtained 8 images for each of a pair of stars, HD 196963 and HD 196081, which are separated by $\sim$1.75o. Since this pair has a much larger angular separation, we can use the RFR from this data set to gauge whether there is significant degradation in PSF similarity. Fortunately, the 2$\sigma$ range of the RFR was 0.33–0.53, indicating that star-hopping is still very effective for such large separations. It should be noted that the coherence time was only 1.9–2.1 msec for these observations, compared to 2.5–7.2 msec for the HR 8799 observations. Although we have low statistics for such a performance, these results shows that even in poor to average conditions, star-hopping RDI can be effective for a pair of stars separated by almost 2o. Figure 7: The comparison of PSF similarity between reference-science and science-science pairs. The residual fractional rms of difference images are plotted as a function of relative position angle/rotational offset. The black dots represent science-science subtractions, the blue dots represent science- reference subtractions, the red dots represent science-science differences with acceptable self-subtraction. For the science-reference points, the relevant quantity is the time difference, which in our case has an almost linear relationship to the PA difference. ### 3.4 $JH$-band spectra from IFS The spectra of planets $c,d$ and $e$ were extracted with an aperture size of 3 pixels for all IFS channels. The spectra for planets $d$ and $e$ were corrected for flux loss by comparing them to three flat contrast sources (uniform contrast across wavelength) per planet inserted at the planets’ separations, but at different PAs (offset from the planets by 30o to 270o). These simulated planets are just the IFS $FLUX$ exposures scaled appropriately in intensity. They were inserted at 10 mags of contrast, which is somewhat brighter than the real planets. Since planet $c$ was detected at the edge of the IFS detector where simulated planets could not be inserted, we used the same comparison sources for planets $c$ and $d$. The simulated planets undergo the same reduction process as the real planets, and their fluxes are extracted using the same aperture sizes, and thus their systematic fractional flux error are the same. We verified this by checking that the spectrum recovered from the simulated companions did indeed have a uniform contrast. Thus the planet spectra is calculated as $B_{PR}(\lambda)=\frac{F_{PR}(\lambda)}{F_{PS}(\lambda)}\times 10^{-4}B_{S}(\lambda)$ (1) where $F_{PR}$ and $F_{PS}$ are the real and simulated planet aperture fluxes respectively, and $B_{S}$ is the stellar spectra. Here, the fractional flux losses for the real planet are fully accounted for in the ratio, $F_{PR}(\lambda)/F_{PS}(\lambda)$. The flux corrected spectra for planets $d$ and $e$ are shown in Figure 8 along with that of the particularly red L6 object 2MASS J2148+4003 (Looper et al., 2008) for comparison. All three spectra are much redder towards the $H$-band, in comparison to typical late L-types. Although not as red, the dusty dwarves of the field population also have redder than average spectra(see Zurlo et al., 2016; Stephens et al., 2009; Gagné et al., 2014). It should be noted that the spectra do differ somewhat in shape from earlier publications, (e.g., Zurlo et al., 2016). This could be because the spectra we present here are the first not to be effected by signal self-subtraction due to ADI or SDI processing. The most notable differences from earlier spectra (see Figure 9) are less defined peaks at 1.1$\leavevmode\nobreak\ \mu$m, and for planet $d$ in 2019, a gentler slope towards 1.6 $\mu$m. The absence of the peak at 1.1 $\mu$m is quite common among observed late L-type (see Figure 3 of Bonnefoy et al., 2016, for example), and also seen in the spectra of 2MASS J2148+4003. However, we also note that the 2016 spectral slopes towards 1.6 $\mu$m are very similar to planet $e$ in 2019. Although, the higher fluxes at 1.6 $\mu$m are rarer among such L-types, it would explain the earlier discrepancy between IRDIS and IFS fluxes near the $H$-band (Zurlo et al., 2016). We could not estimate an accurate flux normalization for the spectra of planet $c$ as it was detected near the edge of the detector, so we show its spectra normalized to 1 at 1.25 $\mu$m in Figure 10. We do not pursue this further, as accurate $JH$-band photometry has already been provided in past publications. However, the shape of the planet’s spectra is reliably detected and show’s an even redder $J-H$ color than planets $d$ and $e$. Although such red spectra are not common, a very similar slope (flux doubling between 1.25 and 1.6$\mu$m) was seen in the L7 object, VHS J125601257 b (Gauza et al., 2015). This L7 object, a planetary candidate companion to a brown dwarf, is also thought to have a dusty atmosphere with thick clouds (see Bonnefoy et al., 2016, for a discussion). Figure 8: The spectra for planets $d$ and $e$ compared with that of the L6 object, 2MASS J2148+4003 from Looper et al. (2008). The planet spectra have been divided by the stellar flux at 1.25 $\mu$m to show the contrast at that wavelength. The L6 object spectra was scaled to match planet $e$ at 1.25 $\mu$m. The shaded regions indicate the 1$\sigma$ error ranges of the spectra. The wavelength range 1.37–1.45 $\mu$m which is dominated by telluric lines is not shown. Figure 9: The RDI-extracted spectra for planets $d$ and $e$ in 2019 compared with their ADI-extracted spectra from 2016 as reported in Zurlo et al. (2016). The 2016 planet spectra to the 2019 have been matched at 1.25 $\mu$m for easier comparison for their respective shapes. The shaded regions indicate the 1$\sigma$ error ranges of the spectra. The wavelength range 1.37–1.45 $\mu$m which is dominated by telluric lines is not shown. Figure 10: The RDI-extracted spectra for planets $c$ compared with that of the L6 object, 2MASS J2148+4003 from Looper et al. (2008) and L7 object VHS J125601257 b from Gauza et al. (2015). The wavelength range 1.37–1.45 $\mu$m which is dominated by telluric lines is not shown. ### 3.5 The HR 8799 debris disk Booth et al. (2016), using the ALMA millimeter array, detected a broad debris ring, extending from $\sim$145 au to $\sim$430 au with an inclination of 40$\pm$6o and a position angle of 51$\pm$8o. Prior to this, Su et al. (2009) inferred from the spectral energy distribution of the system that a planetesimal belt extending from 100 and 300 au separation was the source of blow-out grains extending out to $\sim$1500 au. Thus the inner radius of the disk could start as close as 2.5′′ and the outer radius could be as far as 11′′ from the star. It is expected that RDI reductions would be a major improvement over ADI for detections of disks with large angular extents, as self-subtraction in these cases is a severe problem for ADI. To detect the disk, we repeated the IRDIS RDI reduction without simulated companions or any prior image filtering (used to enhance speckle subtraction), as these remove all extended emission. We only used the K1-band images as the K2-band have much higher background. Detecting disks which are close to azimuthally symmetric in the plane of the sky, and extended over several arcseconds is a challenge very different from planet recovery, as the expected signal area is most of the image and the background area is perhaps non-existent. The image sectors used for PSF subtraction cannot be small, as this would remove extended signal. So, we used one large annulus extending from 0.4′′ to 2′′ separations to cover most of the PSF halo. The final reduction is shown in Figure 11, but no disk emission was detected down to a 5$\sigma$ contrast of 14.1 magnitudes beyond 2.5′′ separations. The non-detection is not surprising given the marginal detection of the much brighter 49 Cet debris disk with SPHERE (Choquet et al., 2017). The fractional disk luminosity of HR 8799 is 8$\times$10-5 (Su et al., 2009) versus 9$\times$10-4 for 49 Cet (Moór et al., 2015). The inner radius of the disks start at roughly 2′′ separation for both (Choquet et al., 2017; Booth et al., 2016), with expected physical separations of 100–150 au. The two stars have similar spectral types (F0–A1) with very similar $H$-band magnitudes (5.3–5.5 mag). Figure 11: A IRDIS reduction without any prior image filtering to search for an extended circumstellar disk beyond angular separations of 2.5′′ (to $>$6′′) from the star. ALMA observations by Booth et al. (2016) indicate that the disk should have a position angle of 51$\pm$8o and an inclination of 40$\pm$6o. We do not detect any disk down to a contrast limit of 14.1 magnitudes. Some faint thermal emission from the detector background is seen in the lower right, but not in the expected orientation of the known disk. North is up and East is to the left. ### 3.6 IRDIS K1, K2 band photometry The photometry of the four planets were extracted by comparison with simulated planets in a similar way to the IFS spectra. For each of the four planets, three simulated planets were inserted into the dataset with a contrast of 10 mags, at the same separation as the real planets, but with large PA offsets (30 to 270o). The relative aperture photometry was done similar to IFS, but with aperture radius 4 pixels, because of the larger FWHM in the K-band. The recovered photometry are all brighter than the Zurlo et al. (2016) measurements by about 0.1 mag (see Table 1). The standard deviation in the contrasts estimates for the three reference simulated planets is less than 0.03 mags. The dominant contrast uncertainty comes from the measurement of the AO-corrected stellar PSF core flux, which is measured only once every 1.5 hours. ### 3.7 Astrometric measurements and comparison to orbital models The IRDIS data set for science images were separately reduced by the SPHERE data center (Delorme et al., 2017) which treated it as an ordinary pupil- tracking sequence. The data center applied the optimal distortion correction methods consistent with Maire et al. (2016), to produce a basic-calibrated data set with high astrometric fidelity (3–4 mas). These images were then reduced using the high-contrast imaging algorithm, ANDROMEDA (Cantalloube et al., 2015), to produce astrometric measurements (see Table 1) for the four known HR 8799 planets. We also compared the recovered coordinates for the real planets between the RDI and ADI reductions, and found that the planet locations agreed to within 2.7 mas, smaller than the errors estimated in Table 1. An exhaustive orbital fitting effort is being currently undertaken by Zurlo et al. (in preparation) including all extant astrometry. Moreover, extensive work has been done to find orbital solutions to the prior astrometry for this system, so we just compare our latest measurements to the viable orbits computed by Wang et al. (2018). From millions of orbits generated by a monte carlo method, they generated 3 sets of solutions: 1) the orbits are forced to be coplanar and have 1:2:4:8 orbital commensurabilities, 2) no coplanarity but with low eccentricity and period commensurabilities as before, 3) with no additional constraints. In Figure 12, we overlay our astrometry on orbital solution sets 1 and 3. Although, the latest points are consistent with both sets of solutions, planets $e$ and $c$ fall close to the expected position in the dynamically stable set, but a bit far from the mean expected location in the unconstrained set of orbits. Thus, the coplanar orbits with period commensurabilities are favored in our comparisons. Survival of the four planets and even a hypothetical fifth planet is possible for the lifetime of the system ($>$ 30 Myrs), but requires the period commensurabilities mentioned above. In fact, this was needed even when only planets $b$, $c$ and $d$ were known (Goździewski & Migaszewski, 2009; Reidemeister et al., 2009; Fabrycky & Murray-Clay, 2010; Marshall et al., 2010). Such dynamical models envision that the four planets were formed at larger separations and migrated inwards. This would allow the very similar chemical compositions indicated by their spectra, as opposed to more variation expected if they had formed insitu (Marois et al., 2010). The most likely semi-major axes allowed for the hypothetical inner planet $f$, estimated by Goździewski & Migaszewski (2014, 2018) were 7.5 au and 9.7 au, with dynamical constraints on the masses of 2–8 $M_{Jup}$ and 1.5–5 $M_{Jup}$ respectively. The IFS contrasts we achieved at these separation were 13.05 and 13.86 mags, corresponding to estimated masses of 3.6 $M_{Jup}$ and 2.8 $M_{Jup}$ respectively (assuming an age of 30 Myr), from the BT-Settl models (Allard et al., 2012b). Thus, the planet may still exist with a mass of 2–3.6 $M_{Jup}$ at 7.5 au or 1.5–2.8 $M_{Jup}$ at 10 au. Table 1: Astrometry and photometry of the four HR 8799 planets. planet \| $\rho$ (mas) \| $\sigma_{\rho}$(mas) \| PA \| $\sigma_{PA}$ \| SNR \| $\Delta$K1 (mag) \| $\Delta$K2 (mag) \| Mass ($M_{J}$) ---\|---\|---\|---\|---\|---\|---\|---\|--- e \| 406 \| 4 \| 302.72o \| 0.04o \| 41 \| 10.8$\pm$0.02 \| 10.63$\pm$0.03 \| 8${}^{+7}_{-2}$ d \| 686 \| 4 \| 231.38o \| 0.006o \| 83 \| 10.7$\pm$0.02 \| 10.47$\pm$0.02 \| 8${}^{+7}_{-2}$ c \| 958 \| 3 \| 335.86o \| 0.05o \| 96 \| 10.8$\pm$0.02 \| 10.53$\pm$0.03 \| 8${}^{+7}_{-2}$ b \| 1721 \| 4 \| 69.05o \| 0.04o \| 47 \| 11.89$\pm$0.01 \| 10.75$\pm$0.01 \| 6${}^{+7}_{-1}$ The mass estimates are from the PHOENIX BT-Settl atmospheric models (Baraffe et al., 2015), assuming an age of 30${}^{+130}_{-10}$ Myrs. However, the most dynamically stable orbital solutions from Wang et al. (2018) set much tighter limits: a mass of $5.8\pm 0.5\leavevmode\nobreak\ M_{J}$ for planet $b$, and $7.2\pm 0.6\leavevmode\nobreak\ M_{J}$ for the other planets. Figure 12: Top: The November 1, 2019 epoch astrometry overlaid as gray diamonds on the most dynamically stable orbital solutions from Wang et al. (2018) (see their Figure 4), where coplanarity and 1:2:4:8 period commensurabilities were imposed. The black dots represent earlier measured astrometry for the four planets. Bottom: Same points overlaid on the orbital solutions without the additional constraints. The 2019 locations for planets $e$ and $c$ are more consistent with the dynamical stable family of orbits. ## 4 Conclusions In this paper, we successfully used the new star-hopping RDI technique to detect all four known planets of the HR 8799 system, and significantly improved on the contrast limits attained previously with ADI, at separations less than 0.4′′. This technique of moving quickly to a reference star to capture a similar AO PSF for differencing, with only a 1 minute gap in photon collection, can now be used in service mode at the VLT with all the observing modes available on the SPHERE instrument. Using star-hopping RDI, we demonstrated the contrast improvement at 0.1′′ separation can be up to 2 mags, while at larger separations the improvement can be $\approx$1–0.5 mags, results which are comparable to those of Ruane et al. (2019). With this technique there is no need for any local sidereal time constraints during observations, which is usually essential for ADI observations. This means that the observing window for typical targets can be expanded by a factor of 2–3. Moreover, star-hopping can usually be used for stars fainter than R$=$4 mag, as for these a reference star of comparable brightness can be found within 1–2 degrees (closer is better). Indeed we found comparable PSF similarity for a pair of stars 1.75o apart. The technique provides significant contrast improvement mainly because of two reasons: the usable PSF, those without significant self-subtraction or flux loss from PSF subtraction 1) occur closer in time and thus are more similar to the target image than in ADI and 2) are more numerous than in ADI as they are spread uniformly over the whole sequence, rather than only available after significant sky rotation. The benefit for extended object like disks will be the most impactful, as in ADI the self-subtraction artefacts can result in significant change in their apparent morphology. In our SPHERE observations of HR 8799, we did not detect planet $f$ at the most plausible locations, 7.5 and 9.7 au, down to mass limits of 3.6 and 2.8 $M_{Jup}$, respectively. Also, we did not detect any new candidate companions, even at the smallest observable separation, 0.1′′ or $\approx$ 4.1 au, where we attained a contrast limit of 11.2 mags or 6 $M_{Jup}$ in K1+K2-band (6.5 $M_{Jup}$ in JHK-band using BT-Settl models from Allard et al., 2012a). However, we detected all 4 planets in K1+K2-band with SNR of 41, 83, 96 and 47 for planets $e$, $d$, $c$ and $b$, respectively. The YJH spectra for planets $c$, $d$, $e$ were detected with very red colors. Our spectra of planet $c$ has higher SNR than earlier observations (P1640, Oppenheimer et al., 2013; Pueyo et al., 2015). Planets $c$ and $d$ spectra have some differences with respect to earlier observations. Particularly, the spectral slope is redder in the H-band, which is significant as that part of the spectra has the highest SNR. This could be due to real evolution of the atmosphere of the planets over the past few years. Previous work has already shown that the spectra are difficult to find close matches with current compositional models due to inadequate understanding of cloud properties and non-equilibrium chemistry (Bonnefoy et al., 2016). However, the spectra are matched very closely by some red field dwarfs and a planetary mass companion to a brown dwarf (VHS J125601257 b; Gauza et al., 2015). We disk not detect the debris disk seen by ALMA (Booth et al., 2016), but this is not surprising given that the much brighter debris disk of a comparable system, 49 Cet, was only marginally detected by SPHERE (Choquet et al., 2017). Finally, comparing the current locations of the planets to orbital solutions from Wang et al. (2018), we found that planets $e$ and $c$ are more consistent with coplanar and resonant orbits than without such restrictions. In summary, the star-hopping RDI technique significantly boosts SPHERE’s detection capabilities both for planets and circumstellar disks, and should contribute to high-impact exoplanet science, as the technique is brought to other telescope facilities. ###### Acknowledgements. This work has made use of the the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/LAM/CESAM (Marseille), OCA/Lagrange (Nice), Observatoire de Paris/LESIA (Paris), and Observatoire de Lyon. 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# Hubble Space Telescope Imaging of Isolated Local Volume Dwarfs GALFA-Dw3 and Dw4 P. Bennet Physics & Astronomy Department, Texas Tech University, Box 41051, Lubbock, TX 79409-1051, USA Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA D. J. Sand Department of Astronomy/Steward Observatory, The University of Arizona, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA D. Crnojević University of Tampa, Department of Chemistry, Biochemistry, and Physics, 401 West Kennedy Boulevard, Tampa, FL 33606, USA D. R. Weisz University of California, Berkeley, Department of Astronomy, 501 Campbell Hall # 3411, Berkeley, CA 94720-3411,USA N. Caldwell Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA P. Guhathakurta UCO/Lick Observatory, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA J. R. Hargis Space Telescope Science Institute, 3800 San Martin Drive, Baltimore, MD, 21208, USA A. Karunakaran Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6, Canada B. Mutlu-Pakdil Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA Department of Astronomy and Astrophysics, University of Chicago, Chicago IL 60637, USA E. Olszewski Department of Astronomy/Steward Observatory, The University of Arizona, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA J. J. Salzer Department of Astronomy, Indiana University, 727 East Third Street, Bloomington, IN 47405, USA A. C. Seth Department of Physics and Astronomy, University of Utah, 115 South 1400 East, Salt Lake City, Utah 84112, USA J. D. Simon Observatories of the Carnegie Institution for Science, Pasadena, California 91101, USA K. Spekkens Department of Physics and Space Science, Royal Military College of Canada P.O. Box 17000, Station Forces Kingston, ON K7K 7B4, Canada Department of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6, Canada D. P. Stark Department of Astronomy/Steward Observatory, The University of Arizona, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA J. Strader Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA E. J. Tollerud Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA E. Toloba Department of Physics, University of the Pacific, 3601 Pacific Avenue, Stockton, CA 95211, USA B. Willman LSST and Steward Observatory, 933 North Cherry Avenue, Tucson, AZ 85721, USA (Received January 20, 2020) ###### Abstract We present observations of the dwarf galaxies GALFA Dw3 and GALFA Dw4 with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST). These galaxies were initially discovered as optical counterparts to compact HI clouds in the GALFA survey. Both objects resolve into stellar populations which display an old red giant branch, younger helium burning, and massive main sequence stars. We use the tip of the red giant branch method to determine the distance to each galaxy, finding distances of 7.61${}_{-0.29}^{+0.28}$ Mpc and 3.10${}_{-0.17}^{+0.16}$ Mpc, respectively. With these distances we show that both galaxies are extremely isolated, with no other confirmed objects within $\sim$1.5 Mpc of either dwarf. GALFA Dw4 is also found to be unusually compact for a galaxy of its luminosity. GALFA Dw3 & Dw4 contain HII regions with young star clusters and an overall irregular morphology; they show evidence of ongoing star formation through both ultraviolet and H$\alpha$ observations and are therefore classified as dwarf irregulars (dIrrs). The star formation histories of these two dwarfs show distinct differences: Dw3 shows signs of a recently ceased episode of active star formation across the entire dwarf, while Dw4 shows some evidence for current star formation in spatially limited HII regions. Compact HI sources offer a promising method for identifying isolated field dwarfs in the Local Volume, including GALFA Dw3 & Dw4, with the potential to shed light on the driving mechanisms of dwarf galaxy formation and evolution. Dwarf galaxies (416), Dwarf irregular galaxies (417), Galaxy distances (590), HST photometry (756), Red giant tip (1371), Star formation (1569) ††journal: ApJ††journal: ApJ††facilities: HST (ACS), WIYN:0.9m, GALEX, SWIFT††software: Numpy, Astropy (The Astropy Collaboration et al., 2018), DOLPHOT (Dolphin, 2000) ## 1 Introduction The Lambda Cold Dark Matter model for structure formation has been very successful at reproducing observations of large scale structures; however challenges emerge at sub-galactic scales (for a recent review, see Bullock & Boylan-Kolchin, 2017, and the references therein). Some of these challenges can be examined by switching focus from dwarf galaxies in nearby groups (McConnachie et al., 2018; Crnojević et al., 2019; Bennet et al., 2019, 2020; Carlsten et al., 2020; Mao et al., 2020) to isolated field galaxies within the Local Volume (Sand et al., 2015; McQuinn et al., 2015b; Tollerud et al., 2016). Examining these isolated, gas rich, dwarf galaxies is critical to our understanding of dwarf galaxy formation and testing dark matter theories. They are the faintest/least massive galaxies we know of that have never interacted with a massive galaxy halo, and thus have never felt the effects of tidal/ram pressure stripping (Spekkens et al., 2014; Wetzel et al., 2015). They are a more controlled experiment for understanding other mechanisms which drive the star formation history (SFH) and metallicity of a dwarf galaxy, for instance supernova-driven winds, or infall of pristine gas from the local environment (McQuinn et al., 2013). By characterizing their resolved stellar populations, it becomes possible both to obtain the present-day structural parameters for these galaxies and to characterize their SFHs, providing constraints on their pasts (McQuinn et al., 2015a; Tollerud et al., 2016; McQuinn et al., 2020). Additionally, these gas rich galaxies potentially trace the full dwarf galaxy population at the outskirts of the Local Group and other similar low-density environments, a regime where the numbers and properties of these dwarfs are just starting to be compared directly with numerical simulations (Tikhonov & Klypin, 2009; Garrison-Kimmel et al., 2014, 2019; Tollerud & Peek, 2018). In this work, we will examine the isolated Local Volume dwarf galaxies GALFA Dw3 and Dw4. These objects were discovered as part of an archival search for optical counterparts to HI clouds (Giovanelli et al., 2010) discovered in the ALFALFA (Adams et al., 2013) and GALFA (Saul et al., 2012) surveys by Sand et al. (2015), and were both confirmed to have H$\alpha$ emission at a velocity consistent with the HI detection. The key properties of GALFA Dw3 and Dw4 are listed in Table 1. An outline of the paper follows. In Section 2, we describe the HST photometry and artificial star tests (ASTs), as well as supplemental observations of the dwarfs. In Section 3, we derive distances to GALFA Dw3 and Dw4 via the Tip of the Red Giant Branch (TRGB) method. In Section 4, we examine the observational properties of the dwarfs in the HST imaging and derive their physical properties. In Section 5, we discuss the star formation histories based on their HST color-magnitude diagrams (CMDs), as well as supplemental H$\alpha$ and ultraviolet (UV) images. In Section 6, we discuss the environment of the dwarfs and potential analogs within the Local Volume. Finally we summarize and conclude in Section 7. ## 2 Data Overview ### 2.1 Hubble Space Telescope Observations The HST observations of GALFA Dw3 & Dw4 were taken as part of program GO-14676 (Cycle 24, PI Sand). Both Dw3 & Dw4 were observed for a single orbit with the Advanced Camera for Surveys (ACS)/Wide Field Camera (WFC), using the F606W and F814W filters. We did not dither to fill in the WFC CCD chip gap, as each dwarf easily fit into one chip. The total exposure time was 1062 s for each filter on both Dw3 & Dw4. Color composites of these images are shown in Figure 1. We perform PSF-fitting photometry on the provided .flt images using the DOLPHOT v2.0 photometric package (with the ACS module), a modified version of HSTphot (Dolphin, 2000). For this work we use the suggested input parameters from the DOLPHOT/ACS User’s Guide111http://americano.dolphinsim.com/dolphot/dolphotACS.pdf, including corrections for charge transfer efficiency losses and default aperture corrections based around a 0.5” aperture. Quality cuts are then applied using the following criteria: the derived photometric errors must be $\leq$0.3 mag in both bands, the sum of the crowding parameter in both bands is $\leq$1 and the square of the sum of the sharpness parameter in both bands is $\leq$0.075. Detailed descriptions of these parameters can be found in Dolphin (2000). For this analysis, we correct these extracted magnitudes for foreground extinction and reddening using the Schlafly & Finkbeiner (2011) calibration of the Schlegel et al. (1998) dust maps (we note that GALFA Dw4 suffers from significant extinction due to its proximity to the plane of the Galaxy, E(B-V)=0.531 mag). We estimate photometric uncertainties using ASTs in a CMD region covering the full range of observed stars, from blue Main Sequence (MS) features to regions redward of the red giant branch (RGB). The fake stars have a similar color- magnitude distribution to that of the observed sources, except for a deeper extension at faint magnitudes (down to $\sim$2 mag fainter than the faintest real recovered stars), so as to take into account those faint objects that are upscattered in the observed CMD due to noise. The AST photometry is derived in exactly the same way as for the real data, and the same quality cuts and calibration are applied. The resulting CMDs can be seen in Figure 2. The completeness and uncertainties for Dw4 appear to be worse than that of Dw3, but this is solely because of the higher extinction associated with Dw4; they are identical in uncorrected apparent magnitude space. We assessed the crowding for each field. Visual inspection of GALFA Dw3 showed clearly separated point sources throughout the main body of the dwarf. GALFA Dw4 required more careful examination, with possible crowding in the blue knots in the southeast and northwest ends of the dwarf (see Figure 1). Examination of the potentially crowded regions showed similar completeness levels to those found in the rest of the dwarf when using standard photometry, and visual inspection showed no obviously missed point sources in the region in question. We also made standard changes to the photometry recommended for crowded regions, namely setting the parameter FitSky=3 (for more details please see the DOLPHOT’s User Guide). This crowded photometry was then compared to the standard photometry in the affected region with no significant difference between the two: we conclude that the use of crowded photometry parameters was unnecessary and that standard parameter photometry was as effective in all regions of GALFA Dw4. However, some of the stars from Dw4 may not be recovered successfully in either the standard or crowded photometry, and this will be further discussed in §5.2. ### 2.2 Other Observations Data from the Galaxy Evolution Explorer (GALEX; Martin & GALEX Team 2005) were also used to check for UV emission from GALFA Dw3, as this can be a strong indicator of recent star formation. Indeed, GALFA Dw3 shows substantial FUV and NUV emission, which we report alongside the HST data in Figure 3. These data were part of the All-Sky Imaging Survey; see Morrissey et al. (2007) for details. GALFA Dw4 is outside the GALEX footprint and therefore no conclusions can be drawn about its recent star formation with this dataset. We thus used UV images from the Neil Gehrels $Swift$ Observatory (Gehrels et al., 2004) and the Ultraviolet/Optical Telescope (UVOT; Roming et al., 2005), which were taken as part of proposal 1417202 (P.I. L. Hagen) in all 3 available UV filters (UVW1, UVM2, UVW2). There is no UV emission detected in these data, likely due to the high levels of extinction along the line of sight to Dw4. Supplemental H$\alpha$ narrow band imaging of GALFA Dw3 & Dw4 were obtained by our group with the WIYN 0.9-m telescope and the Half Degree Imager on 21 July 2017 (UT). These images are used to trace HII regions with active star formation within the last $\sim$10 Myrs (Calzetti, 2013) and can be seen in Figure 4. Figure 1: Color composite of F606W/F814W HST ACS imaging of the dwarf galaxies GALFA Dw3 (upper panel) and GALFA Dw4 (lower panel). The bright objects in the SW of Dw3 are background galaxies. Images are 1.2’x1.2’. North is up, east is left. ## 3 Tip of the Red Giant Branch Distances To determine distances to these resolved dwarf galaxies, we make use of the TRGB technique (e.g., Da Costa & Armandroff, 1990; Lee et al., 1993; Makarov et al., 2006; Rizzi et al., 2007; Freedman et al., 2020). The peak luminosity of the RGB is a standard candle in the red bands, because it is driven by core helium ignition and so it provides a useful distance estimate for galaxies with an old stellar component which are close enough that the RGB stars can be resolved. To determine TRGB magnitudes, we adopt the methodology described in Crnojević et al. (2019). Briefly, the photometry is first corrected to account for the color dependence of the TRGB (Jang & Lee, 2017); we also consider only RGB stars with colors in the range $0.85<(F606W-F814W)_{0}<1.35$, so as to exclude possible contamination from young red supergiant stars. The luminosity function for RGB stars is then computed (note that the field, background+foreground, contamination as derived from a dwarf-free region of the ACS field-of-view is not significant for the range of colors and magnitudes considered here), and a model luminosity function (convolved with the appropriate photometric uncertainty, bias and incompleteness function as derived from our ASTs) is fit to it with a non-linear least squares method. Using the HST data, we find a TRGB magnitudes of 25.37$\pm$0.08 mag and 23.42$\pm$0.12 mag for GALFA Dw3 and Dw4, this correspond to distance moduli of 29.41$\pm$0.08 and 27.46$\pm$0.12 mag, which translate to distances of 7.61${}_{-0.29}^{+0.28}$ Mpc and 3.10${}_{-0.17}^{+0.16}$ Mpc, respectively. We mark the position of the TRGB and its uncertainty in Figure 2, and tabulate our results in Table 1. Anand et al. (2019) used the same dataset presented here for GALFA Dw4 to study the peculiar velocities of galaxies at the edge of the Local Group, and reported a TRGB distance of 2.97$\pm$0.37 Mpc, which is consistent with the distance reported here. Figure 2: F606W/F814W CMD for the dwarf galaxies GALFA Dw3 (left panel) and GALFA Dw4 (right panel). Magnitudes are corrected for foreground extinction (see §2). Only point sources are shown (i.e., those sources with a DOLPHOT object type=1 or 2). Black dots are stars within the dwarfs, red dots are stars from an equal-area control field. In the left panel, the green crosses indicate those stars associated with the spatial position of the HII region in Dw3, see §5.1. In the right panel, the green crosses indicate those stars associated with the spatial position of the southeast HII region and the magenta crosses those associated with the northwest HII region, see §5.2. The black horizontal line indicates the best fit for the TRGB, and the dashed gray lines represent the 1$\sigma$ uncertainty. We display several Padova isochrones (Bressan et al., 2012), shown as solid lines of varying color, each line representing a stellar population of fixed age, shown in the legend of each panel. The red isochrone (RGB stars) is plotted at [Fe/H]=$-$1.6 for both dwarfs, while all other isochrones are at [Fe/H]=$-$1.0. Finally, the 50$\%$ completeness limit (black dashed line) and the photometric uncertainties are reported. Figure 3: The UV images of GALFA Dw3 from the GALEX All Sky Imaging Survey (AIS) alongside optical images from HST for illustrative purposes, see Figure 1. This clearly shows the elevated UV emission from Dw3. North is up, east is left. Each image is 1.1’x1.1’. The ellipses in this plot are illustrative. Left: HST Optical, Center: GALEX NUV, Right: GALEX FUV. Figure 4: The H$\alpha$ narrow band images (see §2) of GALFA Dw3 and Dw4 minus the continuum emission (right column), alongside optical images from HST for illustrative purposes (left column). We point out the elevated H$\alpha$ emission from the northeast corner of Dw3. GALFA Dw4 shows more H$\alpha$ emission within two clear regions, one at the southeast end of the dwarf and the other at the northwest end. These regions match with the blue regions seen in the HST imaging. North is up, east is left. Each image is 1.1’x1.1’. The ellipses in this plot are illustrative. ## 4 Structural Parameters Utilizing the HST imaging, we revisit the structural properties of these dwarf galaxies, previously reported in Sand et al. (2015). To constrain the structural parameters, we use the maximum-likelihood technique of Martin et al. (2008) using the implementation of Sand et al. (2009, 2012). First, we select the stars consistent with the RGB as seen in Figure 2. We fit a standard exponential profile plus constant background to the data, with the following free parameters: the central position (RA0, DEC0), position angle, ellipticity, half-light radius ($r_{h}$) and background surface density. Uncertainties on structural parameters are determined by bootstrap resampling the data 1000 times, from which 68% confidence limits are calculated. The resulting structural parameters are summarized in Table 1. Note that while the derived parameters describe the older stellar populations in our targets, both Dw3 and Dw4 host young populations that are highly irregular in appearance and are concentrated in the HII regions in the case of Dw4 (see Figure 5). We derive the absolute magnitude of the dwarfs via direct aperture photometry using an elliptical aperture with semi-major axis equal to the half-light radius. We estimate the flux within this aperture (after background correction), and multiply by a factor of two to account for the total flux of the dwarf, and then convert to a magnitude. After applying our measured distance modulus and correction for galactic extinction, we find M${}_{V}=-12.8\pm 0.3$ and $-11.8\pm 0.3$ for Dw3 and Dw4, respectively. Our results are consistent with the properties reported in Sand et al. (2015) within the uncertainties. We then estimate the present day stellar mass from the V band luminosity combined with the V-I color using the mass to light ratio formalism from Bell & de Jong (2001): $\log(M/L)_{V}=a_{V}+b_{V}\cdot(V-I)$ (1) where aV$=$$-$1.476 and bV$=$1.747 with an assumed solar luminosity of MV$=$4.77. This produces masses of 2.1$\times$106 M⊙ and 2.6$\times$106 M⊙ for Dw3 and Dw4 respectively. Our two targets broadly fit on the Local Group size-luminosity relations with slightly higher than typical surface brightness (see Figure 6). These properties are very similar to those found for Pisces A & B, two other gas- rich dwarf galaxies initially found in the GALFA survey of HI compact objects (Tollerud et al., 2015; Sand et al., 2015). Dw3 fits closer with the Local Group size-luminosity relation and has similar properties to many objects within the Local Group that are not satellites of the MW or M31. Dw4 appears to be higher surface brightness than many of these objects and is the most compact object at its magnitude (McConnachie et al., 2018), but has possible analogues at the edge of the Local Group such as GR8 (Dohm-Palmer et al., 1998; Tolstoy, 1999). This higher surface brightness when compared to Local Group satellites is likely explained by the recent star formation in both objects. These comparisons are discussed further in §6.2. Table 1: Properties of GALFA Dw3 & Dw4 \| GALFA Dw3 \| GALFA Dw4 ---\|---\|--- R.A. (J2000) \| 02h:58m:56s.5$\pm$0.6 \| 05h:45m:44s.7$\pm$0.5 Dec (J2000) \| +13∘:37${}^{{}^{\prime}}$:45${}^{{}^{\prime\prime}}$.4$\pm$0.5 \| +10∘:46${}^{{}^{\prime}}$:15${}^{{}^{\prime\prime}}$.7$\pm$0.3 l (deg) \| 164.15 \| 195.67 b (deg) \| $-$38.84 \| $-$24.70 GALFA ID \| 044.7+13.6+528 \| 086.4+10.8+611 Distance Modulus (mag) \| 29.41$\pm$0.08 \| 27.46$\pm$0.12 Distance (Mpc) \| 7.61${}_{-0.29}^{+0.28}$ \| 3.10${}_{-0.17}^{+0.16}$ mV (mag)aaVEGA Magnitude, derived from mF606W using the conversion from (Sahu et al., 2014) \| 16.6$\pm$0.2 \| 15.7$\pm$0.2 MV (mag)aaVEGA Magnitude, derived from mF606W using the conversion from (Sahu et al., 2014) \| $-$12.8$\pm$0.3 \| $-$11.8$\pm$0.3 V$-$I (mag) \| 0.44 \| 0.72 E(B$-$V)bbBased on the Schlafly & Finkbeiner (2011) dust maps \| 0.134 \| 0.531 AF606WbbBased on the Schlafly & Finkbeiner (2011) dust maps \| 0.322 \| 1.334 AF814WbbBased on the Schlafly & Finkbeiner (2011) dust maps \| 0.207 \| 0.811 rh (″) \| 12.62$\pm$1.2 \| 6.82$\pm$0.06 rh (pc) \| 466$\pm$46 \| 102$\pm$9 Ellipticity \| 0.54$\pm$0.03 \| 0.58$\pm$0.05 Position Angle (deg) \| 56.4$\pm$1.7 \| 100.4$\pm$1.8 fHα (erg s-1 cm-2) \| 0.514$\pm$0.051$\times$10-14 \| 5.221$\pm$0.110$\times$10-14 HI $v_{LSR}$ (km s-1)ccFrom the GALFA survey, see Saul et al. (2012), using the erratum values \| 528.59$\pm$18.90 \| 614.53$\pm$40.83 H$\alpha$ $v_{LSR}$ (km s-1)ddFrom Sand et al. (2015) \| 503$\pm$35 \| 607$\pm$35 Stot(Jy km s-1)ccFrom the GALFA survey, see Saul et al. (2012), using the erratum values \| 0.51 \| 0.53 M⋆ (M⊙) \| 2.1$\times$106 \| 2.6$\times$106 MHI (M⊙) \| 6.9$\times$106 \| 1.2$\times$106 SFRNUV (M⊙) \| 8.7$\pm$2.5$\times$10-3 \| – SFRFUV (M⊙) \| 8.7$\pm$0.6$\times$10-4 \| – SFRHα (M⊙) \| 3.77$\pm$0.47$\times$10-4 \| 1.37$\pm$0.15$\times$10-3 Figure 5: Spatial distribution of point sources consistent with stellar populations in GALFA Dw3 and Dw4. Point sources consistent with RGB stars are shown in red; these are selected via matching to the RGB isochrones seen in Figure 2. The blue points are those point sources consistent with a color of (F606W0-F814W0)$<$0.1, which are consistent with MS and blue helium burning stars. Only stars brighter than our 50% completeness limits are plotted. The approximate position and size of the HII regions in both dwarfs are shown by black outlines. The blue stars in Dw3 have a higher ellipticity than the RGB populations, but are generally spread throughout the dwarf. In Dw4 there is a concentration of blue stars around the HII region to the southeast, along with several associated with the HII region to the northwest, but few in the main body of the dwarf. Panels are 0.9′squares. North is up, East is left. Figure 6: Absolute V-band magnitude as a function of half-light radius for GALFA Dw3 and Dw4 (blue stars) as compared to satellites of the MW and M31 (Red Inverted Triangles) and other Local Group objects, i.e. those outside the virial radius of either the MW or M31 (Black Squares). Pisces A & B are shown for comparison (Cyan Triangles), along with Leo P (Green Circle). The lines of constant central surface brightness assume an exponential profile and range from 16 $\textrm{mag/arcsec}^{2}$ to 30 $\textrm{mag/arcsec}^{2}$ with a line every $\Delta$2 $\textrm{mag}/\textrm{arcsec}^{2}$. ### 4.1 HI mass The HI mass for GALFA Dw3 and Dw4 can be calculated using the HI flux and the distances derived in Section 3. This is done via the standard equation for an optically thin gas (Haynes & Giovanelli, 1984): $M_{HI}=2.356\times 10^{5}(D_{HI})^{2}S_{HI}M_{\odot}$ (2) where DHI is the distance in Mpc and SHI is the flux in Jy km s-1. These values are reported in Table 1. We use the HI fluxes from (Saul et al., 2012)222We use the revised flux values from the erratum. and the distances we derive here, along with the standard equation to derive HI masses for GALFA Dw3 and Dw4. We note that these fluxes are likely underestimated due to spatial and spectral smoothing procedures employed by Saul et al. (2012). An example of this underestimation is present in the discrepant fluxes for Pisces A and B, $\sim$1.2 and $\sim$1.6 Jy km s-1, respectively, found in Tollerud et al. (2015) compared to 0.445 and 0.957 Jy km s-1 from Saul et al. (2012). Nevertheless, for the purpose of this work, we carry on using the values from Saul et al. (2012) for Dw3 and Dw4. Given their optical luminosities, both GALFA dwarfs are relatively gas rich, with gas mass to light ratios of $\sim$0.6 M⊙/L⊙ for GALFA Dw3 and $\sim$0.3 M⊙/L⊙ for GALFA Dw4. These values are comparable to that of star forming objects within the Local Group with similar absolute magnitudes to those of the GALFA dwarfs (McConnachie, 2012). When we compare GALFA Dw3 and Dw4 to Pisces A and B, we find that the former have smaller gas mass to light ratios (Pisces A: $\sim$2.5 M⊙/L⊙, Pisces B: $\sim$2.7 M⊙/L⊙, Tollerud et al. 2016; Tollerud & Peek 2018; Beale et al. 2020), though this may be due to the underestimation of the HI fluxes discussed above. These gas masses are similar to other isolated field objects which are gas rich and star forming (Geha et al., 2012; Bradford et al., 2015; McQuinn et al., 2020). ## 5 Star Formation Histories It is immediately apparent from the HST images and the derived CMDs that GALFA Dw3 & Dw4 are nearby star-forming dwarf galaxies. They have well-resolved stellar populations, both show RGBs, asymptotic giant branch (AGB) stars, red helium burning stars, blue helium burning stars, MSs, an overall irregular morphology, and HII regions with young star clusters. We attempted to use the CMD-fitting code MATCH (Dolphin, 2002) to determine the SFHs of GALFA Dw3 and Dw4 similar to other works in the Local Volume (e.g. McQuinn et al., 2010; Weisz et al., 2011, 2014). However, the distance to these dwarfs and the shallow nature of the CMDs meant that the results did not provide meaningful constraints on the SFH of either dwarf, other than an indication of active star formation within the past 100 Myrs. Therefore we have qualitatively analyzed each dwarf’s possible SFH via comparison to the Padova isochrones (Bressan et al., 2012) and multi-wavelength observations, similar to other works with Local Volume low-mass dwarfs where more in depth analysis has not been possible (e.g. McQuinn et al., 2015a, 2020). ### 5.1 GALFA Dw3 #### 5.1.1 Isochrone Comparisons The CMD of GALFA Dw3 reveals a complex SFH, with both young and old stellar populations. We point the reader to the left panel of Figure 2 to guide this discussion, where we denote stars in the main body of GALFA Dw3, along with those associated with its HII region (see discussion below), and plot relevant isochrones of varying age and metallicity. There are several faint, blue stars (with 23 $\lesssim$F814W0$\lesssim$25 and (F606W0$-$F814W0) $<$ $-$0.1 mag) that are likely young MS stars, with an approximate age of $\sim$10 Myrs. Other young MS stars are apparent at fainter magnitudes. A sequence of stars spanning the same F814W0 range at slightly redder colors ((F606W0$-$F814W0) $\approx$ 0.0 mag) is likely a combination of slightly older MS stars and a blue helium burning sequence, to go hand in hand with the red helium burning sequence visible at 22 $\lesssim$F814W0$\lesssim$24.5 and 0.7 $\lesssim$(F606W0$-$F814W${}_{0})$ $\lesssim$ 1.0 mag. A RGB is apparent at faint magnitudes (see the TRGB at F814W0=25.4 mag), likely corresponding to an ancient and metal poor stellar population ($>$10–12 Gyr, [Fe/H]$\approx$$-$1.6). Stars immediately above the TRGB may be intermediate age AGB stars, or luminous helium burning stars. The separation of the helium burning branches is a strong indicator of metallicity, with a wider separation for more metal rich systems (Radburn- Smith et al., 2011), while the length and width of the branches is a good indicator of the age of the stars (McQuinn et al., 2011). Approximate properties of stellar populations can even be derived for systems with very few member stars (e.g. Sand et al., 2017). Using the approximate length of the red helium burning branch as a guide, we estimate a stellar population with ages between 25–100 Myrs. However, for stars older than this the red helium burning branch stars becomes hard to distinguish from AGB and RGB stars (McQuinn et al., 2015b). The blue helium burning branch shows stars with ages between 25–250 Myrs. The upper limit on the duration of this star formation is determined by the completeness of the HST data. Star formation may have happened before this estimated age, however deeper data would be required to determine this. The size and separation of the helium burning branches in Dw3 indicate a population with [Fe/H]$\approx-1.0$, based on an approximate match to isochrones. A metallicity of [Fe/H]$=-1.0$ is consistent with other galaxies of similar luminosity as Dw3 (MV=$-$12.8) based on the standard luminosity–metallicity relation for Local Volume galaxies (Berg et al., 2012). It is also consistent within 1$\sigma$ with the possible luminosity–metallicity relation for void galaxies (Pustilnik et al., 2016; Kniazev et al., 2018; McQuinn et al., 2020). Generally, dwarf irregulars form stars in bursts (Weisz et al., 2011), and this is also backed up by simulations (Wheeler et al., 2019). Deeper observations would be required to distinguish between continuous star formation and more episodic, bursty star formation in Dw3. Finally, isochrone fitting in the main body (excluding the HII region) shows a well populated young MS of stars below m${}_{F814W}\approx 25.5$. If this is the MS turnoff for the majority of the dwarf, it would show that star formation across most of the dwarf ceased $\sim$20 Myrs ago. #### 5.1.2 H$\alpha$ Imaging The H$\alpha$ imaging of GALFA Dw3 (see Section 2) reveals a single HII region located at the northeast edge of the dwarf, this image is shown alongside the HST image in Figure 4. This H$\alpha$ imaging shows a flux of 0.514$\pm$0.051$\times$10-14 erg s-1 cm-2, which combined with the distance, foreground extinction and the conversion factor from Kennicutt (1998) implies a star formation rate of 3.77$\pm$0.47$\times$10-4 M⊙ yr-1. If we limit the CMD to only those stars with a spatial position consistent with this HII region, we can see that the H$\alpha$ emission may be caused by a single MS O-star with a maximum age of 5 Myrs (see Figure 2 and 4). In this region we also see a population of lower-mass young MS stars as well as red and blue helium burning stars at higher density than across the main body of the dwarf. The RGB is at a similar density in the HII region when compared to the rest of the dwarf at a similar radius, indicating the overdensity of younger stars is not simply a result of higher overall stellar density in this region. We also find a point source (F814W0=23.2 and F606W0-F814W0=$-$0.25 ) consistent with an O-star, with an O5 spectral class (MV=$-$5.03; see the smoothed magnitudes in Table 1 of Wegner, 2000), outside of the HII region. As this star should be massive and young enough to drive H$\alpha$ emission, but we see no H$\alpha$ emission from its position, we can draw some conclusions. The first idea would be that this is a blended multiple star system (see the Leo P analysis in McQuinn et al. 2015a). If we assume equally massed component stars, then these components would be O8 (MV=$-$4.3) class stars, which would still be large enough to drive H$\alpha$ emission (even an equally massed triple star system would have components large enough to produce H$\alpha$). This source may be an evolved helium burning star that due to noise has been scattered into the region of the CMD equivalent to the MS. #### 5.1.3 GALEX As an additional method to determine the level and spatial position of recent star formation in GALFA Dw3, we checked the GALEX archive for the dwarf’s ultraviolet emission. Dw3’s position was observed by GALEX as part of the All- sky Imaging Survey (AIS, exposure time $\sim$270s). These GALEX images can be seen alongside the HST images in Figure 3. The GALEX data shows diffuse NUV and FUV emission across the body of Dw3, though slightly more concentrated toward the north. We see some concentration of FUV emission in the HII region found in the H$\alpha$ imaging, however the majority is spread across the dwarf. This significant NUV and FUV emission confirms the conclusion from the isochrone fitting that significant star formation has occurred across the dwarf within the last 100 Myrs (Calzetti, 2013). The detected level of NUV emission indicates that GALFA Dw3 has had recent star formation at a rate of 8.7$\pm$2.5$\times$10-3M⊙ yr-1, whereas the FUV emission indicates an order of magnitude lower star formation rate of 8.7$\pm$0.6$\times$10-4M⊙ yr-1. Both star formation rates were calculated using the relevant relations from Iglesias-Páramo et al. (2006). These relations have been shown to be potentially unreliable in low metallicity galaxies, like GALFA Dw3 (McQuinn et al., 2015a); in which case the star forming rate maybe up to $\sim$1.5 times higher than indicated, although this does not effect our overall results. The difference between the star formation rates drawn from the NUV and FUV emission may indicate that star formation in Dw3 has decreased significantly in the last $\sim$100 Myr. This is reinforced by the SFR derived from the H$\alpha$ imaging above (3.77$\pm$0.47$\times$10-4 M⊙ yr-1) which is comparable to the rate derived from the FUV emission but slightly lower. This difference in star formation rates between the tracers examined here can be explained by their differing sensitivity to different ages of star formation. As NUV is equally sensitive to all star formation across the last 100 Myrs, while FUV is most sensitive to stars formed in the last 10 Myrs (though there is some FUV sensitivity to populations up to 100 Myrs old, Calzetti, 2013), and H$\alpha$ is sensitive to only star formation within the last 10 Myrs. The UV emission coming from across the dwarf, along with the difference between the H$\alpha$, NUV and FUV, supports the conclusion drawn from the isochrone matching: that star formation was higher and more widespread in Dw3 in the recent past ($\lesssim$100 Myr), but has now quenched across most of the dwarf, and that there is ongoing star formation only in the single HII region (in the last $\sim$10 Myr). #### 5.1.4 Spatial structure Another diagnostic that we can use to analyze GALFA Dw3 is spatial maps, see Figure 5. When the stars are plotted on spatial maps we can see that the MS stars are concentrated in the central regions of the dwarf, have a more elliptical distribution and are preferentially found toward the northern end of the galaxy. This is true for all MS stars, aside from the very brightest which are only found in the HII region. This is in contrast to the RGB stars which are more evenly distributed throughout the galaxy. The helium burning stars are also more concentrated towards the center of the dwarf when compared to the RGB stars, however the concentration is less pronounced than it is for the MS stars. When we examine the star positions and compare them to the multi-wavelength observations, we find a strong match between the MS stars and the NUV emission. #### 5.1.5 Summary GALFA Dw3 shows an underlying old ($>$10–12 Gyr) metal poor ([Fe/H]$\approx$$-$1.6) stellar population across the body of the dwarf. There are also younger stellar populations. In the CMD we find well populated red and blue helium burning branches (20–100 Myr) across the body of the dwarf, this population can also be seen in the UV emission from Dw3 (see Figure 3). Finally we also find evidence in the CMD and H$\alpha$ emission for a very young population ($<$20 Myr) that is spatially limited to a single HII region in the northeast of the dwarf (see Figure 4). The differences in the spatial position and extent of the tracers of different ages of star formation can be used to reconstruct a qualitative SFH for GALFA Dw3: the star formation was at a higher level and distributed more evenly throughout the dwarf in the recent past, but is now restricted to a single HII region. This could indicate that GALFA Dw3 is concluding an episode of recent star formation that has now been quenched outside of the HII region. This interpretation appears to support the model that star formation in isolated dwarf galaxies is driven by a series of ‘bursts’ of intense star formation, interspersed with periods of quiescence (Weisz et al., 2011). In this model, galaxies go through intense bursts of active star formation which expels the HI gas through stellar feedback. This expulsion of the neutral gas causes the star formation to wane and the feedback to decrease. Without feedback, more HI gas falls onto the dwarf, producing a new episode of star formation (Wheeler et al., 2019). In this case, GALFA Dw3 would be in the concluding part of such a star forming episode with the last parts of star formation from an active burst. More detailed HI observations may be needed to determine the position and kinematic properties of the gas, as the existing HI information from the GALFA survey is low resolution (Saul et al., 2012). ### 5.2 GALFA Dw4 The position of GALFA Dw4 near the galactic plane complicates creating a comprehensive SFH due to the high extinction (particularly in the UV). #### 5.2.1 Isochrone Comparisons The CMD of GALFA Dw4 also reveals a complex SFH, with both young and old stellar populations, however there are substantial differences between Dw3 and Dw4. We point the reader to the right panel of Figure 2 to guide this discussion, where we denote stars in the main body of GALFA Dw4, along with those associated with both of its HII regions (see discussion below), and plot relevant isochrones of varying age and metallicity. Isochrone matching of the red and blue helium burning branches in GALFA Dw4 indicates a metallicity of [Fe/H]$\approx-$1.0 and ages of 50-500 Myrs, based on the branches’ length and separation. Similar to Dw3, the red helium burning branch shows stars with ages between 50-100 Myrs, with the blue helium burning branch showing stars from 100-500 Myrs. The upper age boundary is limited by the completeness of the CMD so star formation may have started even earlier than 500 Myrs ago, but this can not be determined without a deeper CMD. This metallicity is consistent within 1$\sigma$ with the luminosity–metallicity relationship for Local Volume dwarfs (Berg et al., 2012). Isochrone matching also shows Dw4 has an ancient ($>$10–12 Gyrs), low- metallicity ([Fe/H]$\approx-$1.6) RGB. We see some evidence for a limited metallicity spread in the RGB, with some stars being consistent with [Fe/H]$\approx-$1.0, or even slightly more metal-rich, and with most likely member stars being part of this ancient population. Isochrone matching of Dw4 indicates that there are relatively few young MS stars when compared to the stars of the helium burning branches. This could mean that the current star formation rate is at a lower level when compared to a few hundred Myrs ago when the stars that now make up the helium burning branches were formed. This could also be a function of the very low mass of Dw4, where even if there is active star formation very few high mass MS stars are formed and therefore the higher density of helium burning branch stars is caused by the initial stellar mass function, rather than differences in star formation rate over time. This would be similar to other isolated low mass dwarfs such as Leo P or Leoncino (McQuinn et al., 2015a, 2020). It is also possible that the young MS stars are being missed for some reason, and this possibility will be explored below. #### 5.2.2 H$\alpha$ Imaging The H$\alpha$ imaging (see §2) shows that Dw4 has two HII regions, one at each end of the galaxy, which match the blue regions seen in the HST imaging (see Figure 4). We find an H$\alpha$ flux of 4.184$\pm$0.097$\times$10-14 erg s-1 cm-2 for the southeast region and 1.037$\pm$0.052$\times$10-14 erg s-1 cm-2 for the northwest region, for a total H$\alpha$ flux of 5.221$\pm$0.110$\times$10-14 erg s-1 cm-2. Combined with the distance, foreground extinction and the conversion factor from Kennicutt (1998) this flux implies a star formation rate of 1.37$\pm$0.15$\times$10-3 M⊙ yr-1. When we examine these HII regions in the CMD (see the right panel of Figure 2), we find that there are no obvious O-stars to drive the H$\alpha$ emission. This could be caused by internal extinction within Dw4, which could cause the MS O-stars to appear as stars at the upper end of the blue helium burning branch. For this to be the case, the HII regions would have to be obscured by enough dust to cause extinction of AF606W$\approx$1.1 and AF814W$\approx$0.7. This level of extinction is substantially higher than the internal extinction reported for other dwarf galaxies (McQuinn et al., 2010) and is far larger than variations in the foreground extinction. Another possibility is that the stars which are driving H$\alpha$ emission are visible, but are not recovered in our point source photometry because they were culled at some stage in our reductions. To test this possibility, a CMD for Dw4 was constructed using the DOLPHOT catalog, but with the photometric quality cuts severely relaxed. This did not detect any sources with color and brightness consistent with MS O-stars across Dw4. We have also used the ASTs to confirm that artificial stars with properties similar to MS O-stars are successfully recovered by DOLPHOT in the HII regions of Dw4. We also tried a similar reduction in photometric quality cuts with the crowded photometry discussed in §2, and this yielded a few point sources consistent with MS O-stars of the spectral classes O7-O9. These could be the source of the H$\alpha$ emission, however these poorly recovered sources are generally too blue to be MS O-stars. We have considered that these objects may be O-stars with line contamination from the HII region sufficient to move it off the MS in the CMD, however this contamination would have to be larger than expected to have the observed effect. On the other hand, equivalent point sources are not found in the parallel field, indicating they are unique to the dwarf. ‘ Therefore, it is possible these are the MS O-stars, but they are in areas of the dwarf that preclude clean photometric recovery with the present data. It is also possible that a combination of the above scenarios are the reason we see no MS O-stars in Dw4 despite the presence of H$\alpha$ emission. In this case, internal extinction obscures and blurs the O-stars such that they are not recovered clearly by DOLPHOT. The two HII regions also contain most of the lower-mass MS stars seen in Dw4 (see Figure 5). This indicates that star formation is currently limited to these two regions. We also see overdensities of red and blue helium burning stars in the HII regions compared to the dwarf as a whole. RGB stars appear to be at a similar density in the HII regions when compared to other parts of the dwarf with similar radius, indicating the overdensities of young stars are genuine and not caused by general stellar overdensities in these regions. #### 5.2.3 SWIFT UVOT As stated in §2, GALFA Dw4 is outside of the GALEX footprint due to its proximity to the galactic plane. Therefore, to get UV information on this object, Swift UVOT (Roming et al., 2005) observations were required. These were taken as part of proposal 1417202 (P.I. L. Hagen) to observe the UV dust extinction properties in GALFA Dw4 (along with 4 other Local Volume dwarfs). Despite these Swift images with a reasonable total exposure time ($\sim$1100s), they show no detectable UV emission from Dw4 in any of the 3 filters examined (UVW1, UVM2, UVW2). This is likely due to the high levels of extinction around Dw4 (see Table 1). The H$\alpha$ emission from Dw4, combined with the presence of bright MS stars in the HST imaging, means it is likely that there is UV emission from Dw4, but that it is not observable with the present data due to the previously mentioned high levels of extinction. #### 5.2.4 Spatial structure In Dw4 the RGB stars are spread throughout the dwarf while the young MS and helium burning stars are largely confined to regions near the HII regions. These younger stellar populations are at higher relative density at either end of the dwarf near the HII regions, see Figure 5. We find that older helium burning stars are more evenly spread throughout the dwarf, though still more concentrated towards the current HII regions than the RGB stars. This may be the result of previous star formation being more evenly distributed, or a result of these older stars having had time to mix through the dwarf since they formed. #### 5.2.5 Summary GALFA Dw4 has an old ($\gtrsim$10–12 Gyrs) metal poor ([Fe/H]$\approx-$1.6) stellar population, with some evidence for a metallicity spread in the RGB. We also see younger stellar populations, with well populated red and blue helium burning sequences, and young MS stars. This is supported by H$\alpha$ imaging which shows emission concentrated in two regions at either end of the dwarf at the same position as the young stellar populations in the CMD. Therefore we conclude that star formation in Dw4 is limited to the HII regions at either end of the dwarf. We also find that star formation has been ongoing for $>$500 Myrs, and seems to be more concentrated in the HII regions. This can be seen by the concentration of young stars in these regions compared to the RGB stars, along with the H$\alpha$ emission. However, our conclusions here are less robust than for Dw3. This is due to the lack of UV information, and the lower total number of stars in Dw4 which makes it difficult to derive concrete information via examining stellar populations. ## 6 Discussion Having determined the distance (§3), structural properties (§4) and qualitative SFHs (§5) of both GALFA Dw3 and Dw4, we are in a position to discuss these galaxies in detail. ### 6.1 Environment We began exploring the environment around both GALFA Dw3 & Dw4 using their newly derived distances and a search of the NASA Extragalactic Database (NED)333http://ned.ipac.caltech.edu/. We searched for any catalogued objects within $\sim$5 degrees of angular separation and a relative velocity difference between $-$400 to +600 km s-1 (this range was chosen to avoid contamination by Galactic objects with velocities less than the MW escape velocity). This search showed that both GALFA Dw3 & Dw4 are extremely isolated, confirming the result from Sand et al. (2015). In addition, catalogs of known galaxies were searched for objects nearby to either galaxy and we found nothing within 1.5 Mpc of either dwarf (Karachentsev et al., 2013). The closest known object to GALFA Dw3 is NGC1156: NGC1156 has a distance consistent with GALFA Dw3 at 7.6$\pm$0.7 Mpc (Kim et al., 2012), however with a projected separation of 11.61 degrees (1.54 Mpc at the distance of Dw3/NGC1156) and a velocity separation of 155 km s-1 (Karachentsev et al., 2013), we consider direct association at the present time to be unlikely. GALFA Dw4 is projected near to the Orion Dwarf and A0554, however these objects are more distant at D$\sim$6.8 Mpc (Anand et al., 2019) and D$\sim$5.5 Mpc (Karachentsev & Musella, 1996) respectively, and therefore we consider association to be unlikely. The closest object to GALFA Dw4 is the HI source HIPASS J0630+08, with an angular separation of 11.2 degrees (a projected separation of 0.78 Mpc at the distance of Dw4) and a velocity difference of 240 km s-1 (Karachentsev et al., 2013). This is a HI source with no detected optical counterpart (Donley et al., 2005). We find that A0554 is the closest object with an optical counterpart, though this is extremely distant with a radial separation of 2.3 Mpc and a projected separation of 220 kpc. However, as GALFA Dw4 is in the ‘zone of avoidance’ around the Galactic plane, there have been relatively few deep wide field optical surveys done in the area, and therefore it can not be ruled out that there maybe other undetected galaxies closer than A0554. GALFA Dw4 is also unusual as it has large peculiar velocity $\sim$+350 km s-1, which is unexpected for isolated systems, which tend to move with the Hubble flow (Anand et al., 2019). The isolation of GALFA Dw3 & Dw4 can be seen in Figure 7, where the dwarfs are shown to be ‘below’ the supergalactic plane in very low density regions of the Local Volume. Therefore we conclude that both GALFA Dw3 & Dw4 are truly isolated with no other objects close enough to influence them at the current time or in the recent past. This isolation allows us to use them as probes into how star formation and galaxy evolution occur in isolated low-mass galaxies. Figure 7: The location of GALFA Dw3 and Dw4 in the Local Volume. GALFA Dw3 and Dw4 are shown as blue stars and labelled, as are Pisces A and B (Cyan Triangles), while the black dots are a 10 Mpc volume-limited sample of nearby galaxies (Karachentsev et al., 2013). The coordinates are supergalactic Cartesian with Earth at the center, oriented such that the x-axis points towards the origin and the z-axis points towards the Local Void (Lahav et al., 2000). ### 6.2 Local Volume Analogs We have examined other Local Volume dwarf galaxies to compare the properties of GALFA Dw3 and Dw4 with other low mass systems. GALFA Dw3 & Dw4 have very similar physical properties to Pisces A & B, which were also found in follow-up to the GALFA survey (Tollerud et al., 2015; Sand et al., 2015). All of these objects are very isolated, however Pisces A and B were theorised to be falling into local filamentary structure after spending most of cosmic time at the edge of the Local Void (Tollerud et al., 2016), which is speculated to have triggered recent star formation in Pisces A & B. The other object from Sand et al. (2015), ALFALFA Dw1 (also referred to as AGC 226067 or SECCO 1; Bellazzini et al., 2015) shows stellar populations that were found to be approximately consistent with a single burst of star formation with an age range of $\sim$7–50 Myr (Sand et al., 2017), with no accompanying old stellar population, the latter being typical in almost all known dwarf galaxies. Based on this and other results in the literature on this object (Bellazzini et al., 2015; Adams et al., 2015; Beccari et al., 2016), there is circumstantial evidence that ALFALFA Dw1 is a distant star- forming remnant of a ram pressure stripping event in the M86 subgroup, as recent simulations have predicted (Kapferer et al., 2009; Tonnesen & Bryan, 2012), and is therefore a very different class of object; it is possible that similar systems will be minor contaminants in field dwarf searches. In Figure 6, there are a number of Local Volume objects that have similar physical properties to GALFA Dw3 and Dw4. UGC9128 is an isolated Local Volume object (D$\sim$2.3 Mpc; Tully et al., 2013), and is a good analog for Dw3. It has very similar physical properties and a recent SFH that is comparable to Dw3, with recent star formation throughout the dwarf but current star formation limited to a few small regions (McQuinn et al., 2010). UGC9128 shows evidence of having had 3 bursts of star formation in the last $\sim$500 Myrs (McQuinn et al., 2010). GR8 (DDO155/UGC8091) is a star forming dwarf in the Local Volume with a distance of $\sim$2.2 Mpc (Tully et al., 2013). It has very similar physical properties to GALFA Dw4. In GR8, star formation is limited to HII complexes which seem to arise in associated regions approximately 100-200 pc in size, which last for $\sim$100 Myrs before star formation ceases and new regions begin to actively form stars (Dohm-Palmer et al., 1998; Tolstoy, 1999). The Survey of HI in Extremely Low-mass Dwarfs (SHIELD) galaxies (Cannon et al., 2011) are a selection of 12 galaxies initially detected in ALFALFA (Giovanelli et al., 2005; Haynes et al., 2018) data. These galaxies were selected based on low HI and stellar mass estimates. In terms of absolute magnitude and gas mass, the SHIELD galaxies are in the same range as GALFA Dw3 and Dw4. Examination of the SFHs of the SHIELD galaxies also shows a recent star formation rate consistent with that derived for Dw3 (see §5.1.3). The SHIELD galaxies are found in a number of different environments, with three (AGC 748778, AGC 174605, and AGC 74923) being isolated ($>$1 Mpc from their nearest neighbors, McQuinn et al., 2014, 2015b). These objects have very similar physical properties to Dw3, making them potentially good analogs, while Dw4 is fainter and physically smaller than the typical SHIELD galaxy. As previously mentioned Dw4 is one of the most compact objects at its luminosity detected. ## 7 Conclusions We have presented HST imaging of GALFA Dw3 and Dw4, two Local Volume dwarf galaxies which were initially discovered as optical counterparts to compact HI clouds in the GALFA survey. Both dwarfs resolve into stars, displaying complex stellar populations, including an old red giant branch, young helium burning sequences and main sequence stars. Each system also has young star clusters and HII regions which are evident in our H$\alpha$ imaging. In detail, the two dwarfs appear to have slightly different star formation histories based on a qualitative assessment of their CMDs and on the available UV data. GALFA Dw3 shows signs of a recently ceased episode of active star formation; although it is not well constrained, Dw4 seems to have a more consistent level of star formation within spatially limited HII regions at either end of the dwarf. Using the resolved CMDs, we measure the distance to each dwarf using the TRGB method, finding $D$=7.61${}_{-0.29}^{+0.28}$ Mpc and $D$=3.10${}_{-0.17}^{+0.16}$ Mpc for GALFA Dw3 and Dw4, respectively. With this information in hand, we found each dwarf to be extremely isolated, with no known neighbor within $\sim$1.5 Mpc, suggesting that neither galaxy has experienced a significant environmental influence. GALFA Dw3 and Dw4 are similar to other Local Volume dwarfs initially detected in wide field HI surveys (see §6.2 Cannon et al., 2011; Tollerud et al., 2015; Sand et al., 2015). The lack of detections of new gas-rich low-mass dwarf galaxies within the Local Group (similar to Leo P or Leo T Irwin et al., 2007; Rhode et al., 2013) in these surveys indicates that these ‘mini-halos’ are likely rare. The lack of new Local Group objects found in the GALFA survey has been used to examine a potential link between HI gas in dwarfs and the lower mass limit for reionization (Tollerud & Peek, 2018). It found that the lack of detections was very unlikely if these objects were common and this rarity could be used to determine the lower mass limit for reionization (see Tollerud & Peek, 2018, for more details). GALFA Dw3 and Dw4 (and other systems like them, such as Pisces A and B; Tollerud et al. 2016) present a unique opportunity to examine low-metallicity isolated dwarf galaxies analogous to the earliest galaxies in the Universe. Further work on GALFA Dw3 and Dw4, and related objects, will include gas phase metallicity measurements (e.g. Hirschauer et al., 2016; McQuinn et al., 2020) and high resolution HI mapping (e.g. Beale et al., 2020) to further understand the driving mechanisms of the structure and evolution of the faintest dwarf galaxies. Research by PB is supported by NASA through grant number HST-GO-14796.005-A from the Space Telescope Science Institute which is operated by AURA, Inc., under NASA contract NAS 5-26555. Research by DJS is supported by NSF grants AST-1821967 and AST-1813708. Research by DC is supported by NSF grant AST-1814208, and by NASA through grants number HST-GO-15426.007-A and HST- GO-15332.004-A from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. BMP is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2001663. EO is partially supported by NSF grant AST-1815767. JS acknowledges support from the Packard Foundation. This publication utilizes data from Galactic ALFA HI (GALFA HI) survey data set obtained with the Arecibo L-band Feed Array (ALFA) on the Arecibo 305m telescope. The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association. 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# Cubic spin-orbit coupling and anomalous Josephson effect in planar junctions Mohammad Alidoust Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Chenghao Shen University at Buffalo, State University of New York, Buffalo, NY 14260-1500, USA Igor Žutić University at Buffalo, State University of New York, Buffalo, NY 14260-1500, USA ###### Abstract Spin-orbit coupling in two-dimensional systems is usually characterized by Rashba and Dresselhaus spin-orbit coupling (SOC) linear in the wave vector. However, there is a growing class of materials which instead support dominant SOC cubic in the wave vector (cSOC), while their superconducting properties remain unexplored. By focusing on Josephson junctions in Zeeman field with superconductors separated by a normal cSOC region, we reveal a strongly anharmonic current-phase relation and complex spin structure. An experimental cSOC tunability enables both tunable anomalous phase shift and supercurrent, which flows even at the zero-phase difference in the junction. A fingerprint of cSOC in Josephson junctions is the $f$-wave spin-triplet superconducting correlations, important for superconducting spintronics and supporting Majorana bound states. Spin-orbit coupling (SOC) and its symmetry breaking provide versatile opportunities for materials design and brining relativistic phenomena to the fore of the condensed matter physics Kane2005:PRL ; Konig2007:S ; Wan2011:PRB ; Burkov2011:PRL ; Armitage2018:RMP ; Zutic2019:MT . While for decades SOC was primarily studied to elucidate and manipulate normal-state properties, including applications in spintronics and quantum computing Bychkov1984:PZETF ; DasSarma2001:SSC ; Winkler:2003 ; Zutic2004:RMP ; Hanson2007:RMP ; Fabian2007:APS ; Xiao2010:RMP ; Sinova2015:RMP ; Schliemann2017:RMP , there is a growing interest to examine its role on superconductivity Gorkov2001:PRL ; Samokhin2005:PRL ; Reynoso2008:PRL ; Buzdin2008:PRL ; Eschrig2015:RPP ; Smidman2017:RPP . Through the coexistence of SOC and Zeeman field, a conventional spin-singlet superconductivity can acquire spin-dependent long-range proximity effects Eschrig2015:RPP ; Martinez2020:PRA ; Jeon2020:PRX ; Gonzalez-Ruano2020:PRB as well as support topological superconductivity and host Majorana bound states, a building block for fault-tolerant quantum computing Lutchyn2010:PRL ; Oreg2010:PRL ; Aasen2016:PRX . In both cases, Josephson junctions (JJs) provide a desirable platform to acquire spin-triplet superconductivity through proximity effects Keizer2006:N ; Robinson2010:S ; Khaire2010:PRL ; Banerjee2014:NC ; Gingrich2016:NP ; Linder2015:NP ; Rokhinson2012:NP ; Fornieri2019:N ; Ren2019:N ; Desjardins2019:NM ; Mayer2019:P . In contrast, even seemingly well-established intrinsic spin-triplet superconductivity in Sr2RuO4 Mackenzie2003:RMP is now increasingly debated Pustogow2019:N ; Sharma2020:PNAS . Extensive normal-state studies of SOC in zinc-blende heterostructures usually distinguishing the resulting spin-orbit fields due to broken bulk inversion symmetry, Dresselhaus SOC, and surface inversion asymmetry, Rashba SOC, and focus on their dominant linear dependence in the wave vector, ${\bf k}$ Zutic2004:RMP ; Schliemann2017:RMP . In this linear regime, with a matching strengths of these SOC it is possible to strongly suppress the spin relaxation Schliemann2003:PRL and realize a persistent spin helix (PSH) Bernevig2006:PRL ; Koralek2009:N with a controllable spin precession over long distances Dettwiler2017:PRX ; Walser2012:NP ; Iizasa2020:PRB . Figure 1: (a) Spin-orbit fields in k-space for Rashba cubic spin-orbit coupling (cSOC) $(\alpha_{c}=-1$), Dresselhaus cSOC ($\beta_{c}=-1$, middle), and both ($\alpha_{c}=\beta_{c}=-1$, bottom). (b) Schematic of the Josephson junction. The middle region hosts cSOC and an effective Zeeman field, h, between the two $s$-wave superconductors (S). (c) Spin textures in the cSOC region resulting from the normal-incident electrons with in-plane spin orientations [see Fig. 1(b)] when S is at normal-state, the upper (lower) panel $\alpha_{c}=1$, $\beta_{c}=0$ ($\alpha_{c}=\beta_{c}=1$). The in-plane spin orientations of the incident electrons $\phi_{s}$ are from 0 (bottom row) to $\pi/2$ (top row). While typically k-cubic SOC contributions (cSOC) in heterostructures are neglected or considered just detrimental perturbations, for example, limiting the stability of PSH Dettwiler2017:PRX ; Walser2012:NP ; Iizasa2020:PRB , a more complex picture is emerging. Materials advances suggest that such cSOC, shown in Fig. 1(a), not only has to be included, but may also dominate the normal-state properties Winkler2002:PRB ; Krich2007:PRL ; Altmann2006:PRL ; Yoshizumi2016:APL ; Kammermeier2016:PRL ; Nakamura2012:PRL ; Moriya2014:PRL ; Cottier2020:PRB ; Liu2018:PRL ; Brosco2016:PRL . However, the role of cSOC in superconducting heterostructures is unexplored. It is unclear if cSOC is detrimental or desirable to key phenomena such as Josephson effect, spin- triplet superconductivity, or Majorana bounds states. To address this situation and motivate further cSOC studies of superconducting properties, we consider JJs depicted in Fig. 1(b), where $s$-wave superconductors (S) are separated by a normal region with cSOC which is consistent with the two-dimensional (2D) electron or hole gas, confined along the z-axis Winkler2002:PRB ; Nakamura2012:PRL . We find that the interplay between Zeeman field and cSOC results in an anomalous Josephson effect with a spontaneous supercurrent. While the commonly-expected current-phase relation (CPR) is $I(\varphi)=I_{c}\sin(\varphi+\varphi_{0})$ Buzdin2008:PRL ; Strambini2020:NN , where $I_{c}$ is the JJ critical current and $\varphi_{0}$ the anomalous phase ($\varphi_{0}\neq 0,\pi$), we reveal that CPR can be strongly anharmonic and host Majorana bound states. Instead of the $p$-wave superconducting correlations for linear SOC, their $f$-wave symmetry is the fingerprint of cSOC. To study cSOC, we consider an effective Hamiltonian $H=\frac{1}{2}\int d\mathbf{p}~{}\hat{\psi}^{{\dagger}}(\mathbf{p})H(\mathbf{p})\hat{\psi}(\mathbf{p}),$ (1) where $H(\mathbf{p})=\mathbf{p}^{2}/2m^{}+{\bm{\sigma}}\cdot\mathbf{h}+H_{\text{cSOC}}(\mathbf{p})$, with momentum, $\mathbf{p}=(p_{x},p_{y},0)$ [see Fig. 1(b)], effective mass, $m^{}$, Pauli matrices, ${{\bm{\sigma}}}$, effective Zeeman field, ${\bf h}$, realized from an externally applied magnetic field or through magnetic proximity effect Zutic2019:MT ; Takiguchi2019:NP , and cSOC term Winkler2002:PRB ; Krich2007:PRL ; Nakamura2012:PRL ; Moriya2014:PRL $H_{\text{cSOC}}(\mathbf{p})=\frac{i{\alpha_{\text{c}}}}{2\hbar^{3}}(p_{-}^{3}\sigma_{+}-p_{+}^{3}\sigma_{-})-\frac{{\beta_{\text{c}}}}{2\hbar^{3}}(p_{-}^{2}p_{+}\sigma_{+}+p_{+}^{2}p_{-}\sigma_{-}),$ (2) expressed using cSOC strengths $\alpha_{c}$ and $\beta_{c}$, for Rashba and Dresselhaus terms, where $p_{\pm}=p_{x}\pm ip_{y}$, and $\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$. The field operator in spin space is given by $\hat{\psi}(\mathbf{p})=[\psi_{\uparrow}(\mathbf{p}),\psi_{\downarrow}(\mathbf{p})]^{\mathrm{T}}$, with $\uparrow,\,\downarrow$ spin projections. To describe S regions in Fig. 1(b), we use an $s$-wave BCS model with a two- electron amplitude in spin-Nambu space $\Delta\langle\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\rangle+\text{H.c.}$, given by the effective Hamiltonian in particle-hole space ${\cal H}(\mathbf{p})=\left(\begin{array}[]{cc}H(\mathbf{p})-\mu\hat{1}&\hat{\Delta}\\\ \hat{\Delta}^{\dagger}&-H^{\dagger}(-\mathbf{p})+\mu\hat{1}\end{array}\right),$ (3) where $\mu$ is the chemical potential and $\hat{\Delta}$ is a $2\times 2$ gap matrix in spin space. The field operators in the rotated particle-hole and spin basis are $\hat{\psi}=(\psi_{\uparrow},\psi_{\downarrow},\psi_{\downarrow}^{{\dagger}},-\psi_{\uparrow}^{{\dagger}})^{\mathrm{T}}$. To calculate the charge current, we use its quantum definition where no charge sink or source is present. Therefore, the time variation of charge density vanishes, $\partial_{t}\rho_{\text{c}}\equiv 0=\lim\limits_{\mathbf{r}\rightarrow\mathbf{r}^{\prime}}\sum\limits_{\sigma\tau\sigma^{\prime}\tau^{\prime}}[\psi^{\dagger}_{\sigma\tau}(\mathbf{r}^{\prime}){\cal H}_{\sigma\tau\sigma^{\prime}\tau^{\prime}}(\mathbf{r})\psi_{\sigma^{\prime}\tau^{\prime}}(\mathbf{r})-\psi^{\dagger}_{\sigma\tau}(\mathbf{r}^{\prime}){\cal H}_{\sigma\tau\sigma^{\prime}\tau^{\prime}}^{\dagger}(\mathbf{r}^{\prime})\psi_{\sigma^{\prime}\tau^{\prime}}(\mathbf{r})]$. ${\cal H}_{\sigma\tau\sigma^{\prime}\tau^{\prime}}$ is the component form of ${\cal H}$, with spin (particle-hole) label $\sigma$ ($\tau$), and and $\mathbf{r}\equiv(x,y,0)$. From the current conservation, the charge current density is, $\mathbf{J}=\int d\mathbf{r}\\{\hat{\psi}^{\dagger}(\mathbf{r})\overrightarrow{{\cal H}}(\mathbf{r})\hat{\psi}(\mathbf{r})-\hat{\psi}^{\dagger}(\mathbf{r})\overleftarrow{{\cal H}}(\mathbf{r})\hat{\psi}(\mathbf{r})\\},$ where ${\cal H}(\mathbf{r})$ is obtained by substituting $\mathbf{p}\equiv-i\hbar(\partial_{x},\partial_{y},0)$. The arrow directions indicate the specific wavefunctions that the ${\cal H}$ operates on. By an exact diagonalization of ${\cal H}$, we obtain spinor wavefunctions $\hat{\psi}^{l,r,m}(\textbf{p})$ within the left ($x<0$) and right ($x>d$) S region and the middle normal region ($\smash{0}<x<d$) in Fig. 1(b). The wavefunctions and generalized velocity operators $v_{x}^{l,r,m}$ are continuous at the junctions, i.e., $\hat{\psi}^{l}$=$\hat{\psi}^{m}\|_{x=0}$, $\hat{\psi}^{m}$=$\hat{\psi}^{r}\|_{x=d}$, $v_{x}^{l}\hat{\psi}^{l}$=$v_{x}^{m}\hat{\psi}^{r}\|_{x=0}$, and $v_{x}^{m}\hat{\psi}^{m}$=$v_{x}^{r}\hat{\psi}^{r}\|_{x=d}$. The spinor wavefunctions are given in the Supplmental Material sm . Figure 2: (a) Josephson energy and (b) associated supercurrent evolution with the superconducting phase difference $\varphi$. Zeeman field values, $h_{x}$, are chosen near a $0$-$\pi$ transition. The other parameters are ${\alpha_{\text{c}}}=\pm 0.1$ and ${\beta_{\text{c}}}=0$, $\mu=\Delta$, $h_{y}=0$. The complexity of ${\cal H}$ precludes simple solutions and we evaluate the wavefunctions and supercurrent numerically. To reduce the edge effects, we consider Fig. 1(b) geometry with $W/d\gg 1$ Alidoust2015:JAP . This approach has been successfully used to study supercurrent in junctions with PSH, Weyl semimetals, phosphorene, and twisted bilayer graphene Alidoust2020:PRB0 ; Alidoust2020:PRB1 ; Alidoust2018:PRB1 ; Alidoust2020:PRB2 ; Alidoust2018:PRB2 ; Alidoust2018:PRB3 ; Alidoust2020:PRB4 . The calculated supercurrent is normalized by $I_{0}=2\|e\Delta\|/\hbar$, where $e$ is the electron charge, and $\Delta$ the energy gap in S. The energies are normalized by $\Delta$, lengths by $\xi_{\text{S}}=\hbar/\sqrt{2m^{}\Delta}$, cSOC strengths by $\Delta\xi_{\text{S}}^{3}$. The junction length is set at $d=0.3\xi_{\text{S}}$. To investigate the role of cSOC on the ground-state Josephson energy, $E_{\text{GS}}$, and the CPR obtained from the supercurrent $I(\varphi)\propto\partial E_{\text{GS}}/\partial\varphi$, we first consider a simple situation with only Rashba cSOC ($\alpha_{c}\neq 0$, $\beta_{c}=0$) and effective Zeeman field $h_{x}$ ($h_{y}=h_{z}=0$). The evolution of $E_{\text{GS}}$ with $\|h_{x}\|$, where its minima are denoted by dots in Fig. 2(a), shows a continuous transition from $\varphi=0$ to $\pi$ state (blue to green dot). For $\varphi_{0}\neq 0$, $E_{\text{GS}}$ minima come in pairs at $\pm\varphi_{0}$ Sickinger2012:PRL . The corresponding CPR reveals in Fig. 2(b) a competition between the standard, $\sin\varphi$, and the next harmonic, $\sin 2\varphi$, resulting in $I(-\varphi)=-I(\varphi)$. There is no spontaneous current expected in a Josephson junction with SOC, $I(\varphi=0)=0$, but only $I_{c}$ reversal with $h_{x}$. Such a scenario of a continuous and symmetric 0-$\pi$ transition is well studied without SOC in S/ferromagnet/S JJs due to the changes in the effective magnetization or a thickness of the magnetic region Kontos2002:PRL ; Ryazanov2001:PRL ; Bergeret2005:RMP ; Eschrig2003:PRL ; Halterman2015:PRB ; Wu2018:PRB ; Moen2020:PRB ; Yokoyama2014:PRB . Figure 3: (a) Josephson energy and (b) related supercurrent evolution with the superconducting phase difference $\varphi$ Zeeman field, $h_{y}$, at a fixed magnitude and varying Rashba cSOC strength ${\alpha_{\text{c}}}$ are considered. The other parameters are ${\beta_{\text{c}}}=0$, $\mu=\Delta$, $h_{x}=0$. (c) Three fits to the green curve in (b) using the generalized CPR from Eq. (4) with $N=1,2,3$ harmonics. While our previous results suggest no direct cSOC influence on CPR, a simple in-plane rotation of h, $h_{x}=0$, $h_{y}\neq 0$, drastically changes this behavior. This is shown in Figs. 3(b) where, at fixed $\|h_{y}\|=2.4\Delta$, we see a peculiar influence of a finite Rashba cSOC which is responsible for the anomalous Josephson effect with spontaneous current, $I(\varphi=0)\neq 0$, and strong anharmonic CPR that cannot be described by $I(\varphi)=I_{c}\sin(\varphi+\varphi_{0})$. Unlike in Fig. 3(a), a relative sign between ${\alpha_{\text{c}}}$ and $h$ alters the CPR and Josephson energy, where the ground states $\varphi_{0}$ appear at single points [green, red dots in Fig. 3(a)], consistent with $\varphi_{0}\propto\alpha_{c}h_{y}$. If instead of $\mu=\Delta$, we consider a regime $\mu\gg\Delta$, the evolution of Josephson energy from Fig. 2(a) changes. While 0-$\pi$ transitions with $\|h_{x}\|$ remain, there are no longer global minima with $\varphi\neq 0,\pi$ and the CPR reveals a stronger anharmonicity. In contrast, for $\mu\gg\Delta$, the anomalous Josephson effect from Fig. 3 remains robust and similar $\varphi_{0}$ states are accessible (see Ref. sm ). Simple harmonics used to describe anharmonic CPR in high-temperature superconductors Golubov2004:RMP ; Kashiwaya2000:RPP here are not very suitable. By generalizing a short-junction limit for CPR Yokoyama2014:PRB ; Golubov2004:RMP ; Hart2019:PRB , we identify a much more compact form where only a small number of terms gives an accurate description. To recognize the importance of SOC and two nondegenerate spin channels, $\sigma$, we write $I(\varphi)\approx\sum_{n=1}^{N}\sum_{\sigma=\pm}\frac{I_{n}^{\sigma}\sin(n\varphi+\varphi_{0n}^{\sigma})}{\sqrt{1-\tau_{n}^{\sigma}\sin^{2}(n\varphi/2+\varphi_{0n}^{\sigma}/2)}},$ (4) where $\tau_{n}^{\sigma}$ is the normal region transparency for spin channel $\sigma$. With only few lowest terms in this expansion ($N=1,2,3$), shown in Fig. 3(c) with the corresponding errors, it is possible to very accurately describe strong CPR anharmonicities for anomalous Josephson effect. To achieve the relative error from $N=3$ expansion in Eq. (4), in a standard $\\{\sin,\cos\\}$ expansion, with the corresponding phase shifts as extra fitting parameters, requires $N>20$ sm . Key insights into the CPR and an explicit functional dependence for the $\varphi_{0}$ state is obtained by a systematic $I(\varphi)$ symmetry analysis with respect to the cSOC ($\alpha_{c}$, $\beta_{c}$) and Zeeman field or, equivalently, magnetization ($h_{x,y,z}$) parameters sm . We find that $h_{z}$ plays no role in inducing the $\varphi_{0}$ state, it only produces $I(\varphi)$ reversals, explaining our focus on $h_{z}=0$ [Figs. 2 and 3]. These properties are expressed as an effective phase shift to the a sinusoidal CPR, $\sin(\varphi+\varphi_{0})$, extracted from Eq. (4). We again distinguish small- and large-$\mu$ regime ($\mu=\Delta$ v.s. $\mu=10\Delta$). In the first case, for the JJ geometry from Fig. 1, we obtain $\varphi_{0}\propto\Gamma_{y}\Big{(}\alpha_{\text{c}}^{2}+\Gamma_{1}\beta_{\text{c}}^{2}\Big{)}h_{x}{\beta_{\text{c}}}+\Gamma_{x}\Big{(}\alpha_{\text{c}}^{2}-\Gamma_{2}\beta_{\text{c}}^{2}\Big{)}h_{y}{\alpha_{\text{c}}},$ (5) where the parameters $\Gamma_{1,2,x,y}$ are introduced through their relations, $\Gamma_{2}>\Gamma_{1}$, $\Gamma_{1}<1$, $\Gamma_{2}>1$, $\Gamma_{y}(h_{y}=0)=\Gamma_{x}(h_{x}=0)=1$, $\Gamma_{y}(h_{y}\neq 0)<1$, $\Gamma_{x}(h_{x}\neq 0)<1$. These relations are modified as $\mu$ and $\mathbf{h}$ change. For $\mu\gg\Delta$, the functional dependence for the $\varphi_{0}$ state is simplified $\varphi_{0}\propto\Big{(}\alpha_{\text{c}}^{2}-\Gamma_{1}\beta_{\text{c}}^{2}\Big{)}h_{x}{\beta_{\text{c}}}+\Big{(}\alpha_{\text{c}}^{2}-\Gamma_{2}\beta_{\text{c}}^{2}\Big{)}h_{y}{\alpha_{\text{c}}},$ (6) where $\Gamma_{2}>\Gamma_{1}$ and $\Gamma_{1,2}>1$. Therefore, $\varphi_{0}$ state occurs when h shifts p $\bot$ to ${\bm{I}}(\varphi)$ and thus alters the SOC sm . Taken together, these results reveal that cSOC in JJ supports a large tunability of the Josephson energy, anharmonic CPR, and the anomalous phase, key to many applications, from post-CMOS logic, superconducting spintronics, quiet qubits, and topological quantum computing. Realizing $\pi$ states in JJs is desirable for improving rapid single flux quantum (RSFQ) logic, with operation $>100\,$GHz Likharev1991:IEEETAS ; Terzioglu1998:IEEETAS and enhancing coherence by decoupling superconducting qubits from the environment Yamashita2005:PRL . However, common approaches for $\pi$ states using JJs combining $s$\- and $d$-wave superconductors or JJs with ferromagnetic regions Golubov2004:RMP ; Kashiwaya2000:RPP pose various limitations. Instead, extensively studied gate-tunable SOC Zutic2004:RMP ; Dettwiler2017:PRX ; Nakamura2012:PRL ; Moriya2014:PRL ; Mayer2019:P ; Nitta1997:PRL , could allow not only a fast transformation between $0$ and $\pi$ states in JJs with cSOC, but also an arbitrary $\varphi_{0}$ state to tailor desirable CPR. An insight to the phase evolution and circuit operation of JJs with cSOC is provided by generalizing the classical model of resistively and capacitively shunted junction (RSCJ) Stewart1968:APL . The total current, $i$, is the sum of the displacement current across the capacitance, $C$, normal current characterized by the resistance, $R$, and $I(\varphi)$, $\frac{\phi_{0}}{2\pi}C\frac{d^{2}\varphi}{dt^{2}}+\frac{\phi_{0}}{2\pi R}\frac{d\varphi}{dt}+I(\varphi)=i,$ (7) where $\phi_{0}$ is the magnetic flux quantum and $I(\varphi)$ yields a generally anharmonic CPR, as shown from Eq. (4), which can support $0$, $\pi$, and turnable $\varphi_{0}$ states. As we have seen from Figs. 2 and 3, this CPR tunability is accompanied by the changes in Josephson energy, which in turn is responsible for the changes in effective values of $C$, $R$, and the nonlinear Josephson inductance. This JJ tunability complements using voltage or flux control Casparis2018:NN ; Krantz2019:APR . Figure 4: Normalized critical supercurrent as a function of cSOC strength ${\alpha_{\text{c}}}$ and ${\beta_{\text{c}}}$ for (a) $\mu=\Delta$ and (b) $\mu=10\Delta$. The Zeeman field is set to zero. In JJs with ferromagnetic regions, $I_{c}$ is the tunable $I_{c}$ by changing the underlying magnetic state Gingrich2016:NP ; Baek2014:NC ; Costa2017:PRB . In JJs with cSOC, tuning $I_{c}$ could be realized through gate control by changing the relative strengths of $\alpha_{c}$ and $\beta_{c}$, even at zero Zeeman field. This is shown in Fig. 4 by calculating $\text{Max}[I(\varphi)]$ with $\varphi\in[0,2\pi]$. In the low-$\mu$ regime, the maximum $I_{c}$ occurs at slightly curved region near the symmetry lines $\|\alpha_{c}\|=\|\beta_{c}\|$. For the high-$\mu$ regime, the region of maximum $I_{c}$ evolves into inclined symmetry lines, $\|\alpha_{c}\|={\cal A}\|\beta_{c}\|$, ${\cal A}<1$. Similar to linear SOC, in the diffusive regime for cSOC, one expects that the minimum in $I_{c}$ occurs near these symmetry lines because of the presence of long-range spin-triplet supercurrent Alidoust2015:NJP ; Alidoust2020:PRB1 . Figure 5: Real and imaginary parts of equal-spin superconducting correlations in the k-space, $\xi_{\text{S}}=\hbar/\sqrt{2m^{}\Delta}$ is the characteristic length. (a), (b) Linear Rashba, $\alpha=1$. (c), (d) cSOC, $\alpha_{c}=1$, $\beta_{c}=0$. (e), (f) cSOC, $\alpha_{c}=\beta_{c}=1$. The other parameters are the same for all panels. We expect that a hallmark of JJs with cSOC goes beyond CPR and will also influence the spin structure and symmetry properties of superconducting proximity effects. Linear SOC is responsible for mixed singlet-triplet superconducting pairing Gorkov2001:PRL , while with Zeeman or exchange field it is possible to favor spin-triplet proximity effects which can become long- range Eschrig2015:RPP ; Linder2015:NP or host Majorna bound states Lutchyn2010:PRL ; Oreg2010:PRL . To explore the proximity effects in the cSOC region, we calculate superconducting pair correlations using the Matsubara representation for the anomalous Green function, $F(\tau;\mathbf{r},\mathbf{r}^{\prime})$ Zagoskin:2014 , $F_{ss^{\prime}}(\tau;\mathbf{r},\mathbf{r}^{\prime})=+\langle T_{\tau}\psi_{s}(\tau,\mathbf{r})\psi_{s_{1}}(0,\mathbf{r}^{\prime})\rangle(-i\sigma^{y}_{s_{1}s^{\prime}}),$ (8) where $s,s^{\prime},s_{1}$ are spin indices, the summation is implied over $s_{1}$, $\tau$ is the imaginary time, $\psi_{s}$ is the field operator, and $T_{\tau}$ denotes time ordering of operators sm . For a translationally invariant SOC region, spin-triplet correlations in Fig. 5, obtained from Eq. (8), provide a striking difference between linear and cubic SOC. Unlike the $p$-wave symmetry for linear Rashba SOC [Figs. 5(a), 5(b)], we see that the $f$-wave symmetry is the fingerprint for cSOC, retained with only $\alpha_{c}\neq 0$ [Figs. 5(c), 5(d)] or both $\alpha_{c},\beta_{c}\neq 0$ [Figs. 5(e), 5(f)]. Remarkably, unlike the commonly-sought $p$-wave symmetry, we confirm that with a suitable orientation of the Zeeman field cSOC also supports Majorana flat bands sm . While we are not aware of any Josephson effect experiments in 2D systems dominated by cSOC, our studied parameters are within the range of already reported measurements. Choosing $m^{}$ of an electron mass, and $\Delta=0.2\,$meV, which is similar for both Al and proximity-induced superconductivity Mayer2019:P ; Mayer2020:NC , the characteristic length becomes $\xi_{\text{S}}\approx 14\,$nm. The resulting cSOC strength from Fig. 3(b) with $\alpha_{c}\Delta\xi_{\text{S}}^{3}\approx 50\,$eVÅ3 is compatible with the values in 2D electron and hole gases Cottier2020:PRB ; Liu2018:PRL . The Zeeman splitting $2.4\times 0.2\,$meV is available by applying magnetic field in large $g$-factor materials Zutic2004:RMP , or from magnetic proximity effects, measured in 2D systems to reach up to $\sim 20\,$meV Zutic2019:MT . Even though we have mostly focussed on the tunable Rashba SOC, the Dresselhaus SOC can also be gate tunable Dettwiler2017:PRX ; Iordanskii1994:JETPL , offering a further control of the anomalous Josephson effect. Our results reveal that the cSOC in JJs provides versatile opportunities to design superconducting response and test its unexplored manifestations. The anomalous Josephson effect could serve as a sensitive probe to quantify cSOC. While identifying the relevant form of SOC is a challenge even in the normal state Zutic2004:RMP ; Fabian2007:APS , in the superconducting state already a modest SOC can give a strong anisotropy in the transport properties Hogl2015:PRL ; Martinez2020:PRA ; Gonzalez-Ruano2020:PRB ; Vezin2020:PRB and enable extracting the resulting SOC. Identifying SOC, either intrinsic, or generated through magnetic textures, remains important for understanding which systems could host Majorana bound states Desjardins2019:NM ; Scharf2019:PRB ; Pakizer2020:P ; Fatin2016:PRL ; Matos-Abiague2017:SSC ; Ronetti2020:PRR ; Klinovaja2012:PRL ; Zhou2019:PRB ; Mohanta2019:PRA ; Turcotte2020:PRB ; Jiang2021:N ; Rex2020:PRB ; Kornich2020:PRB ; Mohanta2020:P ; Mohanta2018:PRB . With the advances in gate-tunable structures and novel materials systems Mayer2019:P ; Mayer2020:NC ; Nakamura2012:PRL ; Moriya2014:PRL ; Cottier2020:PRB ; Liu2018:PRL ; Assouline2019:NC , the functional dependence of the anomalous phase $\varphi_{0}$ and the $f$-wave superconducting correlations could also enable decoupling of the linear and cubic SOC contributions sm . For the feasibility of such decoupling, it would be useful to consider methods employed in the studies of the nonlinear Meissner effect Xu2015:PRB ; Bae2019:RSI ; Zhuravel2013:PRL ; Prozorov2006:SST ; Halterman2000:PRB ; Bhattacharya1999:PRL ; Zutic1997:PRB ; Zutic1998:PRB . Even small corrections to the supercurrent from the magnetic anisotropy of the nonlinear Meissner response offer a sensitive probe to distinguish different paring-state symmetries. ###### Acknowledgements. 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# Marginally bound circular orbits in the composed black-hole-ring system Shahar Hod The Ruppin Academic Center, Emeq Hefer 40250, Israel The Hadassah Institute, Jerusalem 91010, Israel ###### Abstract The physical and mathematical properties of the non-linearly coupled black- hole-orbiting-ring system are studied analytically to second order in the dimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of the black- hole horizon (here $M_{\text{ir}}$ is the irreducible mass of the slowly rotating central black hole). In particular, we determine analytically, to first order in the dimensionless ring-to-black-hole mass ratio $m/M_{\text{ir}}$, the shift $\Delta\Omega_{\text{mb}}/\Omega_{\text{mb}}$ in the orbital frequency of the marginally bound circular geodesic that characterizes the composed curved spacetime. Interestingly, our analytical results for the frequency shift $\Delta\Omega_{\text{mb}}$ in the composed black-hole-orbiting-ring toy model agree qualitatively with the recently published numerical results for the corresponding frequency shift in the physically related (and mathematically much more complex) black-hole-orbiting- particle system. In particular, the present analysis provides evidence that, at order $O(m/M_{\text{ir}})$, the recently observed positive shift in the angular frequency of the marginally bound circular orbit is directly related to the physically intriguing phenomenon of dragging of inertial frames by orbiting masses in general relativity. ## I Introduction Geodesic orbits of test particles in curved spacetimes are of central importance in black-hole physics Car ; Bar ; Chan ; Shap ; WT ; Willmb ; Gro ; Hodmb . They provide valuable information on the physical parameters (mass, charge, angular momentum) of the central black hole. In particular, the marginally bound circular orbit of a curved black-hole spacetime is of special importance in astrophysics and general relativity Car ; Bar ; Chan ; Shap ; WT ; Willmb ; Gro ; Hodmb . This physically interesting geodesic represents the innermost circular orbit of a massive particle which is energetically bound to the central black hole. For a test particle of proper mass $m$, the marginally bound circular geodesic is characterized by the simple energy relation Car ; Bar ; Chan ; Shap $E(r_{\text{mb}})=m\ ,$ (1) where $E$ is the energy of the particle as measured by asymptotic observers. Interestingly, the marginally bound circular geodesic (1) marks the boundary between bound orbits, which are characterized by the sub-critical energy relation $E<m$, and unbound circular orbits with $E>m$ which, given a small outward perturbation, can escape to infinity. In particular, as nicely demonstrated in Willmb ; WT , the critical (marginally bound) circular geodesic (1) plays a key role in the dynamics of star clusters around super- massive black holes in galactic nuclei. [The critical orbit (1) is sometimes referred to in the physics literature as the innermost bound spherical orbit (IBSO) Willmb ; Gro ]. An important gauge-invariant physical quantity that characterizes the motion of particles along the marginally bound circular geodesic is the orbital angular frequency $\Omega_{\text{mb}}$ of the particles as measured by asymptotic observers. For a test-particle moving in the spinless spherically symmetric Schwarzschild black-hole spacetime, this physically important orbital frequency is given by the simple dimensionless relation Bar ; Chan ; Shap $M_{\text{ir}}\Omega_{\text{mb}}={1\over 8}\ .$ (2) Here $M_{\text{ir}}$ is the irreducible mass Noteirr of the central black hole. In recent years, there is a growing physical interest in calculating the $O(m/M_{\text{ir}})$ corrections to the orbital frequency (1) of the marginally bound circular orbit in non-linearly coupled black-hole-particle systems (see the physically interesting work Barackmb and references therein). To this end, one should take into account the gravitational self- force corrections to the geodesic orbits of the particles Ori ; Poi ; Lou1 ; Det1 ; Bar1 ; Det2 ; Sag ; Kei ; Sha ; Dam ; Bar2 ; Fav2 ; Kol . The gravitational self-force has two distinct physical contributions to the dynamics of a composed black-hole-particle system: (1) It is responsible for non-conservative physical effects, like the emission of energy and angular momentum in the form of gravitational waves Ori . (2) The composed black-hole-particle system is also characterized by conservative gravitational self-force effects that preserve the total energy and angular momentum of the system but shift the orbital frequency of the marginally bound orbit. Computing the gravitational self-force (GSF) correction $\Delta\Omega_{\text{mb}}$ to the zeroth-order frequency (1) of the marginally bound circular orbit is a highly non-trivial task. Intriguingly, Barack at. al. Barackmb have recently used sophisticated numerical techniques in the composed Schwarzschild-black-hole-orbiting-particle system in order to compute the characteristic shift $\Delta\Omega_{\text{mb}}$ in the orbital frequency of the marginally bound orbit which is caused by the conservative part of the GSF. In particular, Barack at. al. Barackmb have found the (positive) dimensionless value ${{\Delta\Omega_{\text{mb}}}\over{\Omega_{\text{mb}}}}=c\cdot\eta+O(\eta^{2})\ \ \ \text{with}\ \ \ c\simeq 0.5536\ $ (3) for the shift in the orbital frequency of the marginally bound circular orbit, where $\eta\equiv{{m}\over{M_{\text{ir}}}}\ $ (4) is the dimensionless ratio between the mass of the orbiting particle and the mass of the central Schwarzschild black hole. The physical importance of the result (3) of Barackmb stems from the fact that it provides gauge-invariant information about the strong-gravity effects in the highly curved region ($r\simeq 4M_{\text{ir}}$) of the black-hole spacetime. The main goal of the present compact paper is to use analytical techniques in order to gain some physical insights on the intriguing $O(m/M_{\text{ir}})$ increase [see Eq. (3)] in the orbital frequency of the marginally bound circular orbit as recently observed numerically in the physically important work Barackmb . In particular, we shall analyze a simple black-hole-orbiting- ring toy model which, as we shall explicitly show below, captures some of the essential physical features of the (astrophysically more interesting and mathematically much more complex) black-hole-orbiting-particle system in general relativity. As nicely proved by Will Will , the composed black-hole- orbiting-ring toy model is amenable to a perturbative analytical treatment to second order in the dimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of the central slowly rotating black hole. ## II The orbital frequency of the marginally bound circular orbit in the composed black-hole-orbiting-ring spacetime In the present paper we would like to gain some analytical insights into the conservative part of the $O(m/M)$-shift in the orbital frequency $\Omega_{\text{mb}}$ of the marginally bound orbit that has recently been computed numerically in the highly interesting work Barackmb . To this end, we shall use the analytically solvable model of an axisymmetric ring of matter which orbits a central slowly spinning black hole Will . In particular, we shall use this simplified axisymmetric toy model (which, due to its symmetry, has no dissipative effects) in order to model the conservative part of the dynamics of the mathematically more complex black-hole-orbiting-particle system Notengw . We expect the composed axisymmetric black-hole-orbiting-ring system to capture, at least qualitatively, the essential physical features that characterize the conservative dynamics of the composed black-hole-orbiting- particle system. In particular, both the orbiting particle in the black-hole- particle system and the orbiting ring in the black-hole-ring system drag the generators of the central black-hole horizon Will . The physically intriguing general relativistic effect of dragging of inertial frames by an orbiting object is reflected, both in the black-hole-particle system and in the black-hole-ring system, by a non-linear spin-orbit interaction term of order $\omega_{\text{H}}\cdot j$ in the total gravitational energy of the composed systems (here $\omega_{\text{H}}$ is the angular velocity of the black-hole horizon and $j$ is the angular momentum per unit mass of the orbiting ring). Interestingly, and most importantly for our analysis, the main mathematical advantage of the black-hole-orbiting-ring system over the physically more interesting (but mathematically more complex) black-hole-orbiting-particle system stems from the fact that the spin-orbit interaction term in the black- hole-ring system is known in a closed analytical form to second order in the dimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of the central black hole Will [see Eq. (10) below]. In a very interesting work, Will Will has analyzed the total gravitational energy and the total angular momentum of a stationary physical system which is composed of an axisymmetric ring of particles of proper mass $m$ which orbits a central slowly rotating black hole of an irreducible mass $M_{\text{ir}}$. In particular, it has been proved in Will that the composed axisymmetric black-hole-orbiting-ring system is characterized by the total angular momentum $J_{\text{total}}(x)=mj+4M^{3}_{\text{ir}}\omega_{\text{H}}-8mjx^{3}\ ,$ (5) where $x\equiv{{M_{\text{ir}}}\over{R}}\ $ (6) is the dimensionless ratio between the irreducible mass of the black hole and the proper circumferential radius of the ring, $j(x)={{M_{\text{ir}}}\over{[x(1-3x)]^{1/2}}}\cdot[1+O(M_{\text{ir}}\omega_{\text{H}})]\ $ (7) is the angular momentum per unit mass of the orbiting ring, and $\omega_{\text{H}}$ is the angular velocity of the black-hole horizon. Since the first term on the r.h.s of (5) represents the angular momentum $J_{\text{ring}}$ of the orbiting ring of mass $m$, one concludes Will that the last two terms in (5) represent the angular momentum $J_{\text{H}}=4M^{3}_{\text{ir}}\omega_{\text{H}}-8mjx^{3}\ $ (8) which is contributed by the slowly spinning central black hole as measured by asymptotic observers. In particular, it is interesting to point out that, while the first term in (8) represents the usual relation between the angular momentum and the angular velocity of a slowly rotating Kerr black hole, the second term on the r.h.s of (8) is a direct consequence of the dragging of inertial frames caused by the orbiting ring Will . A simple inspection of the compact expression (8) reveals the physically important fact that, unlike vacuum Schwarzschild black holes, a zero angular momentum ($J_{\text{H}}=0$) black hole in the non-linearly coupled black-hole- orbiting-ring system is characterized by the non-zero horizon angular velocity $\omega_{\text{H}}(J_{\text{H}}=0)={{2x^{3}}\over{M^{3}_{\text{ir}}}}\cdot mj\ .$ (9) In addition, it has been explicitly proved in Will that, to second order in the angular velocity of the black-hole horizon, the composed axisymmetric black-hole-orbiting-ring system is characterized by the total gravitational energy $\displaystyle E_{\text{total}}(x)=m-m\Phi(x)+M_{\text{ir}}+2M^{3}_{\text{ir}}\omega^{2}_{\text{H}}-\omega_{\text{H}}mj\Psi(x)-{{m^{2}x}\over{2\pi M_{\text{ir}}}}\ln\Big{(}{{8M_{\text{ir}}\over{xr}}}\Big{)}\ $ (10) as measured by asymptotic observers. Here we have used the dimensionless radially dependent functions $\Phi(x)\equiv 1-{{1-2x}\over{(1-3x)^{1/2}}}\ \ \ ;\ \ \ \Psi(x)\equiv 12{{x^{3}-2x^{4}}\over{1-3x}}\ .$ (11) The various terms in the energy expression (10), which characterizes the composed black-hole-orbiting-ring system, have the following physical interpretations Will : * • The first term in the energy expression (10) represents the proper mass of the ring. * • In order to understand the physical meaning of the second term in the energy expression (10), it is worth pointing out that, in the small-$x$ regime (large ring radius, $R\gg M_{\text{ir}}$), this term can be approximated by the compact expression [see Eqs. (6), (10), and (11)] $-M_{\text{ir}}m/2R\cdot[1+O(M_{\text{ir}}/R)]$, which is simply the sum of the potential and rotational Newtonian energies of the ring in the background of the central compact object. Thus, this term represents the leading order (linear in the mass $m$ of the ring) interaction between the central black hole and the orbiting ring. * • In order to understand the physical meaning of the third and fourth terms in the energy expression (10), it is worth pointing out that a slowly spinning bare (isolated) Kerr black hole is characterized by the simple mass-angular- velocity relation $M_{\text{Kerr}}=M_{\text{ir}}+2M^{3}_{\text{ir}}\omega^{2}_{\text{H}}+O(M^{5}_{\text{ir}}\omega^{4}_{\text{H}})$. Thus, the third and fourth terms in (10) can be identified as the contribution of the slowly spinning central black hole to the total energy of the system. Interestingly, taking cognizance of Eq. (9) one learns that due to the general relativistic frame dragging effect, which is caused by the orbital motion of the ring, the fourth term in (10) contains a self-interaction contribution [of order $O(m^{2}/M_{\text{ir}})]$ to the total energy of the composed black- hole-orbiting-ring system. * • The fifth term in the energy expression (10) is a non-linear spin-orbit interaction between the slowly spinning central black hole and the orbiting ring. This energy term plays a key role in our composed black-hole-orbiting- ring toy model system since it is expected to mimic, at least qualitatively, the physically analogous non-linear spin-orbit interaction in the original black-hole-orbiting-particle system. Taking cognizance of Eq. (9) one learns that, due to the intriguing general relativistic phenomenon of frame dragging, the spin-orbit interaction term in (10) contains a non-linear contribution to the total energy of the composed black-hole-orbiting-ring system which is of order $O(m^{2}/M_{\text{ir}})$. * • The sixth term in the energy expression (10) is the gravitational self-energy of the ring Tho (not discussed in Will ), where $r\ll R$ is the half- thickness of the ring. This energy contribution represents the inner interactions between the many particles that compose the axisymmetric ring. Since our main goal in the present paper is to present a simple analytical toy-model for the physically more interesting (and mathematically more complex) two-body system in general relativity, which is characterized by a single orbiting particle, we shall not consider here this many-particle energy contribution. This approximation allows one to focus the physical attention on the general relativistic frame-dragging effect which characterizes both the black-hole-orbiting-particle system and the black-hole-orbiting-ring system. Taking cognizance of Eqs. (7), (9), (10), and (11), one finds the compact functional expression $\displaystyle E_{\text{total}}(x)=M_{\text{ir}}+m\cdot\Big{[}{{1-2x}\over{(1-3x)^{1/2}}}+{{8x^{5}(-2+3x)}\over{(1-3x)^{2}}}\cdot\eta+O(\eta^{2})\Big{]}\ $ (12) for the total gravitational energy of the non-linearly coupled black-hole- orbiting-ring system. In the decoupling $R/M_{\text{ir}}\to\infty$ limit, in which the ring is located at spatial infinity, the system is characterized by the presence of two non-interacting physical objects: (1) a bare (unperturbed) Schwarzschild black hole of mass $M=M_{\text{ir}}$ Notesmir , and (2) a ring of proper mass $m$. Thus, the total energy of the black-hole-ring system in the $R/M_{\text{ir}}\to\infty$ limit is given by the simple expression [see Eq. (12) with $x\to 0$] $\displaystyle E_{\text{total}}(R/M_{\text{ir}}\to\infty)=M+m\ \ \ \text{with}\ \ \ M=M_{\text{ir}}\ .$ (13) Energy conservation implies that the marginally bound orbit of the composed black-hole-orbiting-ring system is characterized by the same total gravitational energy Notemmir $\displaystyle E_{\text{total}}(x=x_{\text{mb}})=M_{\text{ir}}+m\ $ (14) as measured by asymptotic observers. Substituting the relation (14) into Eq. (12), one finds the simple expression $x_{\text{mb}}={1\over 4}\cdot\Big{[}1+{{5}\over{16}}\cdot\eta+O(\eta^{2})\Big{]}\ $ (15) for the $O(m/M_{\text{ir}})$-corrected location of the marginally bound circular orbit in the composed black-hole-orbiting-ring system. Substituting the dimensionless radial coordinate (15) of the marginally bound orbit into the functional expression Will $M_{\text{ir}}\Omega=x^{3/2}\cdot\Big{[}1-4x^{3/2}\cdot M_{\text{ir}}\omega_{\text{H}}+O[(M_{\text{ir}}\omega_{\text{H}})^{2}]\Big{]}\ $ (16) for the dimensionless orbital frequency of the axisymmetric orbiting ring and using Eqs. (7) and (9) Notesn , one obtains the $O(m/M_{\text{ir}})$-corrected expression $M_{\text{ir}}\Omega_{\text{mb}}={1\over 8}\cdot\Big{[}1+{{13}\over{32}}\cdot\eta+O(\eta^{2})\Big{]}\ $ (17) for the characteristic orbital frequency of the marginally bound circular geodesic in the composed black-hole-orbiting-ring system. ## III Summary We have analyzed the physical and mathematical properties of a composed black- hole-orbiting-ring system. In particular, we have proposed to use this analytically solvable conservative Notengw system as a simple toy model for the conserved dynamics of the astrophysically more interesting (and mathematically more complex) black-hole-orbiting-particle system in general relativity. Our main goal was to provide a simple qualitative analytical explanation for the increase in the orbital frequency of the marginally bound circular geodesic that has recently been observed numerically in the physically important work Barackmb . To this end, we have used the non-trivial spin-orbit interaction between the central black hole and the orbiting ring, which is known in a closed analytical form to second order in the dimensionless angular velocity $M_{\text{ir}}\omega_{\text{H}}$ of the black-hole horizon, in order to capture the essential physical features of a similar non-linear spin-orbit interaction which is expected to characterize the conservative dynamics of the black-hole-orbiting-particle system. Interestingly, the analytically derived expression [see Eqs. (2) and (17)] ${{\Delta\Omega_{\text{mb}}}\over{\Omega_{\text{mb}}}}={{13}\over{32}}\cdot\eta+O(\eta^{2})\ $ (18) for the dimensionless $O(m/M_{\text{ir}})$-shift in the orbital frequency of the marginally bound circular geodesic in the composed black-hole-orbiting- ring system provides the correct order of magnitude (with the correct sign) for the corresponding shift in the orbital frequency of the marginally bound circular geodesic of the physically more interesting black-hole-orbiting- particle system. This qualitative agreement suggests that the observed shift (3) in the characteristic orbital frequency of the marginally bound circular geodesic is mainly determined by the general relativistic effect of dragging of inertial frames by orbiting objects (the non-linear spin-orbit interaction between the orbiting object and the generators of the central black-hole horizon). ACKNOWLEDGMENTS This research is supported by the Carmel Science Foundation. I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions. ## References * (1) B. Carter, Phys. Rev. 174, 1559 (1968). * (2) J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astrophys. J. 178, 347 (1972). * (3) S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford University Press, New York, 1983). * (4) S. L. Shapiro and S. A. Teukolsky, Black holes, white dwarfs, and neutron stars: The physics of compact objects (Wiley, New York, 1983). * (5) D. Merritt, T. Alexander, S. Mikkola, and C. M. Will, Phys. Rev. D 84, 044024 (2011). * (6) C. M. Will, Class. Quantum Grav. 29, 217001 (2012). * (7) R. Grossman, J. Levin, and G. Perez-Giz, Phys. Rev. D 85, 023012 (2012). * (8) S. Hod, Phys. Rev. D 84, 104024 (2011) [arXiv:1201.0068]; S. Hod, Phys. Rev. D 84, 124030 (2011) [arXiv:1112.3286]; S. Hod, Phys. Lett. B 718, 1552 (2013) [arXiv:1210.2486]; S. Hod, Phys. Rev. D 87, 024036 (2013) [arXiv:1311.1281]; S. Hod, Phys. Rev. D 88, 087502 (2013) [arXiv:1707.05680]; S. Hod, Phys. Lett. B 726, 533 (2013) [arXiv:1312.4969]; S. Hod, The Euro. Phys. Jour. C 74, 2840 (2014) [arXiv:1404.1566]. * (9) The irreducible mass of a black hole is related to its surface area $A$ by the simple relation $M_{\text{ir}}=(A/16\pi)^{1/2}$. For a spherically symmetric vacuum Schwarzschild black hole, the irreducible mass coincides with the total ADM mass $M$ of the spacetime: $M_{\text{ir}}=M$. * (10) L. Barack, M. Colleoni, T. Damour, S. Isoyama, N. Sago, Phys. Rev. D 100, 124015 (2019). * (11) A. Ori and K. S. Thorne, Phys. Rev. D. 62, 124022 (2000); A. Buonanno and T. Damour, Phys. Rev. D 62, 064015 (2000). * (12) E. Poisson, Living Rev. Relativity 7, 6 (2004). * (13) C. O. Lousto, Class. and Quant. Grav. 22, S369 (2005). * (14) S. Detweiler, in Mass and Motion in General Relativity, edited by L. Blanchet, A. Spallicci, and B. Whiting (Springer, 2011). * (15) L. Barack, Class. and Quant. Grav. 26, 213001 (2009). * (16) S. Detweiler, Phys. Rev. D 77, 124026 (2008). * (17) N. Sago, L. Barack, and S. Detweiler, Phys. Rev. D 78, 124024 (2008). * (18) T. S. Keidl, A. G. Shah, J. L. Friedman, D. Kim, and L. R. Price, Phys. Rev. D 82, 124012 (2010). * (19) A. Shah, T. Keidl, J. Friedman, D. Kim, and L. Price, Phys. Rev. D 83, 064018 (2011). * (20) T. Damour, Phys. Rev. D 81, 024017 (2010). * (21) L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (2009); L. Barack and N. Sago, Phys. Rev. D 81, 084021 (2010); S. Akcay, L. Barack, T. Damour, and N. Sago, Phys. Rev. D 86, 104041 (2012). * (22) M. Favata, Phys. Rev. D 83, 024027 (2011); M. Favata, Phys. Rev. D 83, 024028 (2011). * (23) B. Kol, arXiv:1307.4064 . * (24) C. M. Will, The astrophysical Journal 191, 521 (1974); C. M. Will, The astrophysical Journal 196, 41 (1975). * (25) It is worth stressing the fact that the dynamics of the axisymmetric black-hole-orbiting-ring system is conservative in the sense that, due to its simple symmetry, it contains no gravitational waves. Likewise, the conservative part of the dynamics of the composed black-hole-orbiting-particle system ignores the emission of energy and angular momentum in the form of gravitational waves. * (26) K. S. Thorne, in Quasi-Stellar Sources and Gravitational Collapse (University of Chicago, 1965). * (27) Note that a bare Schwarzschild black hole is characterized by the simple relation $M_{\text{ir}}=(A/16\pi)^{1/2}=M$. * (28) We consider composed black-hole-orbiting-ring configurations which are characterized by a fixed value of the geometrically important quantity $M_{\text{ir}}$ (that is, a fixed value of the central black-hole surface area). * (29) Note that one finds from Eqs. (7) and (9) the dimensionless relation $M_{\text{ir}}\omega_{\text{H}}={1\over 8}\eta\cdot[1+O(\eta)]$ for the angular velocity of the black-hole horizon when the orbiting ring is in its marginally bound orbit (15).
revtex4-1Repair the float # Optimised Domain-engineered Crystals for Pure Telecom Photon Sources A. Pickston Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS F. Graffitti Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS P. Barrow Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS C. Morrison Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS J. Ho Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS A. M. Brańczyk Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5 A. Fedrizzi Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh, UK, EH14 4AS ###### Abstract The ideal photon-pair source for building up multi-qubit states needs to produce indistinguishable photons with high efficiency. Indistinguishability is crucial for minimising errors in two-photon interference, central to building larger states, while high heralding rates will be needed to overcome unfavourable loss scaling. Domain engineering in parametric down-conversion sources negates the need for lossy spectral filtering allowing one to satisfy these conditions inherently within the source design. Here, we present a telecom-wavelength parametric down-conversion photon source that operates on the achievable limit of domain engineering. We generate photons from independent sources which achieve two-photon interference visibilities of up to $98.6\pm 1.1\%$ without narrow-band filtering. As a consequence, we reach net heralding efficiencies of up to 67.5%, which corresponds to collection efficiencies exceeding $90\%$. Scalable photonic quantum technologies require pure photons created on demand. The simplicity of using photon sources based on spontaneous parametric down- conversion (PDC) means the process has been exploited widely and studied in great depth Christ et al. (2013); Slussarenko and Pryde (2019). Efforts have been made to achieve pseudo-deterministic operation via multiplexing Pittman et al. (2002); Migdall et al. (2002); Kaneda and Kwiat (2019); Collins et al. (2013); Kiyohara et al. (2016); Francis-Jones et al. (2016); Ma et al. (2011); Mendoza et al. (2016); Broome et al. (2011); Meyer-Scott et al. (2020), reach high heralding efficiencies, and generate indistinguishable photons—characteristics that all contribute towards an ideal source of photons. Whilst deterministic operation can be addressed separately, photon source engineering must focus on generating indistinguishable photons with high heralding efficiencies, since tasks such as measurement-based quantum computing Raussendorf et al. (2003); Walther et al. (2005), photonic Boson sampling Broome et al. (2013); van der Meer et al. (2020) and photonic quantum repeaters Azuma et al. (2015) are ultimately contingent on high visibility two-photon interference at high rates with minimal losses. Our work focuses on tailoring the phase matching function (PMF), modifying the PDC interaction to produce optimal photons. The quantum state resulting from PDC, when considering solely terms which describe emission of a single pair reads, $\ket{\psi}_{\text{pair}}=\iint d\omega_{s}d\omega_{i}f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a_{i}}^{\dagger}(\omega_{i})\ket{0}.$ (1) The state contains $f(\omega_{i},\omega_{s})$, which is determined by the pump envelope function (PEF) $\alpha(\omega_{s}+\omega_{i})$, and PMF $\phi(\omega_{s},\omega_{i})$, $f(\omega_{s},\omega_{i})=\phi(\omega_{s},\omega_{i})\hskip 1.0pt\alpha(\omega_{s}+\omega_{i}),$ (2) and is referred to as the Joint Spectral Amplitude (JSA). Under symmetric group-velocity matching—where the mean of the inverse signal-idler group velocities are matched to the inverse of the pump group velocity Grice et al. (2001); U’Ren et al. (2006); Mosley et al. (2008); Jin et al. (2014, 2013); Greganti et al. (2018)—the PMF and PEF are orthogonal. In this condition, signal and idler photons can be interfered interchangeably, unlike in Ref. Wang et al. (2016), as well as in heralded single-photon schemes. Achieving unit photon purities requires the bi-photon states to exist in a single spectral mode. In standard non-linear crystals, the PMF is a sinc- shaped function, which generates spectral correlations in the JSA, leading to bi-photon states that exist in a superposition of spectral modes Brańczyk et al. (2011); Graffitti et al. (2018a), illustrated in Figure 1 (a, b). These correlations reduce spectral photon purity and thus indistinguishability. Typically, tight filtering is used to suppress the spectral correlations, increasing purity and interference visibility. But filtering introduces optical loss, leading to a reduction in heralding efficiencies, source brightness and photon-number purity Brańczyk et al. (2010); Meyer-Scott et al. (2017). One can achieve a factorable JSA without tight filtering however, by engineering the properties of the crystal such that the PMF approximates a Gaussian function, shown in Figure 1 (c). First suggested by Brańczyk et al. in Ref. Brańczyk et al. (2011), several methods for obtaining a Gaussian PMF have been developed. Altering the poling duty cycle of the crystal domains Dixon et al. (2013); Chen et al. (2017, 2019); Cui et al. (2019), the orientation of the poling direction Dosseva et al. (2016), and tailoring both Graffitti et al. (2018b); Tambasco et al. (2016); Graffitti et al. (2017) can all generate the desired function. Using an optimal technique developed in Ref. Graffitti et al. (2017), Graffitti et al. demonstrated interference of photons generated from independent domain-engineered crystals in Ref. Graffitti et al. (2018b). Within that work, a symmetric heralding efficiency of 65% was achieved along with a source brightness of 4kHz/mW and lower-bound photon purity of 90.7$\pm$0.3%. While developed primarily for generating separable photons, domain engineering can also be exploited for tailoring high quality non-Gaussian PMFs, e.g. for efficient generation of time-frequency mode entanglement Graffitti et al. (2020a) and time-frequency hyper- entanglement Graffitti et al. (2020b). Here we present a PDC source based on domain-engineered crystals, operating on the achievable limits of this technique. Through the optimisation of parameters which trade off the non-trivial relationship between non-linearity and indistinguishability, we establish a lower bound on spectral purity of $98.0\pm 0.1\%$, achieve a maximal visibility of $98.6\pm 1.1\%$, a symmetric heralding efficiency of $67.5$% and a source brightness of $4.1$ kHz/mW. Figure 1: Theoretical Phase Matching Functions and Joint Spectral Amplitudes for periodically poled (a, b) and our Gaussian apodized crystals (c, d). The prevalent correlations in the joint spectrum for the periodically poled crystal are a result of the Sinc shaped phase matching function (a) and must be filtered out with narrow-band filters to achieve high spectral purity. These correlations are suppressed in the apodized crystals joint spectrum (compare (b) and (d)) by targeting a Gaussian phase matching function (c), increasing spectral purity, increasing source indistinguishability and removing the need for tight spectral filtering. PDC occurs in non-centrosymmetric optical materials, such as potassium titanyl phosphate (KTP). Quasi-phase-matching (QPM), a method commonly used to bypass non-critical phase-matching, is achieved by inverting the orientation of the crystal lattice structure with a period that prevents destructive interference of signal and idler fields. This allows photon generation along the crystallographic axes, thus avoiding birefringent walk-off effects and permits photon generation at desired wavelengths Fejer et al. (1992). The non-linear response of a uniformly periodically-poled crystal corresponds to a step function, which, in the Fourier domain, transforms to the detrimental sinc function seen in Figure 1 (a). Figure 2: Target function for Gaussian domain engineering. The top panel shows the target function of varying widths, with the red shaded areas indicating regions outside the boundary fixed by the crystal length $l$. A target function that is too wide, for example when $\sigma=l/2$, will result in side lobes in the PMF which is shown on the bottom panel. A narrow target function may produce minimal side lobes in the PMF, but will result in a lower effective non-linearity and therefore a lower source brightness. The blue dotted lines indicate the trade-off we chose for our implementation. To achieve high purities in the group-velocity matching (GVM) regime, the pump envelope function should be a transform-limited Gaussian envelope, whilst the phase-matching function should also be a Gaussian function Graffitti et al. (2018a); Quesada and Brańczyk (2018). Typical mode-locked lasers have a sech2-shaped PEF, which in our case contributes $\sim$1% in purity decrease. In this work we focus on PMF engineering only, but it’s also possible to reshape the pump field spectrum into a Gaussian PEF, as recently demonstrated by C. Chen et al. Chen et al. (2019). We define the PMF as: $\phi(\omega_{s},\omega_{i})=\int^{+\infty}_{-\infty}\hskip 1.0pt\text{g}(z)\text{e}^{i\Delta k(\omega_{s},\omega_{i})z}\text{d}z,$ (3) where $\Delta k(\omega_{s},\omega_{i})=k_{p}(\omega_{s}+\omega_{i})-k_{s}(\omega_{s})-k_{i}(\omega_{i})$ is the phase mismatch arising from material dispersion and $\text{g}(z)=\chi^{(2)}(z)/\chi^{(2)}_{0}$ is the normalised non-linear response of the material. We can modify this function by aperiodically altering the width and orientation of each domain while tracking a predefined target function $\text{g}(z)_{\text{target}}$ Graffitti et al. (2017). This target function produces a target PMF amplitude which is scaled to possess the maximum gradient achievable—that is $\frac{\pi}{2}$ Boyd (2008)—to ensure that the non-linear response along the longitudinal direction is maximised. The resulting PMF amplitude Graffitti et al. (2017); Tambasco et al. (2016) is given by: $\text{PMF}(\Delta k_{0})=\sqrt{\frac{\pi}{2}}\left(\text{erf}\left(\frac{l}{2\sqrt{2}\sigma}\right)+\text{erf}\left(\frac{z-\frac{l}{2}}{\sqrt{2}\sigma}\right)\right).$ (4) A crucial parameter is the choice of $\sigma$, the width of the Gaussian function. This parameter balances source brightness with spectral purity. In order to obtain high brightness the function must be wide, but to avoid correlations the function must be narrow. Thus we choose a width that both avoids a large step in non-linearity—avoiding spectral correlations—whilst wide enough to obtain a reasonably high effective nonlinearity and thus brightness. This trade-off is illustrated in Figure 2. With a $\sigma=l/4.7$, where $l$ is the crystal length, the generation of side lobes is minimal whilst not adversely reducing generation rates, see Figure 1 (c) for our apodized crystals theoretical PMF. The crystal length is $l=30$ mm, resulting in $\sigma=6.38$ mm. Figure 3: Experimental Layout. (a) A Ti:Saphh laser pumps a standard ppKTP, or domain-engineered aKTP crystal, at a repetition rate of 80.9MHz. The down- converted photon pairs are collected after reflection from a dichroic mirror and separated by a PBS. Individual photons from two sources are temporally synchronised with an adjustable free-space gap before they are superposed in an in-fibre BS. Photons are then detected by Superconducting Nano-wire Single Photon Detectors (SNSPDs), with photon arrival times being time-tagged and processed. (b) Two $\sim 20$km fibre spools of telecommunication grade fibre are used for dispersive spectroscopy, exploiting chromatic dispersion allowing us to reconstruct the joint photon spectrum Avenhaus et al. (2009). We collect the photon pairs in the same manner as above, however the collected photons are subjected to the fibre delay. Using a mode-locked Ti:Sapphire laser with a non-ideal $\text{sech}^{2}$-shaped spectral envelope we pump our crystals at a wavelength of 774.9 nm, down-converting to 1549.8 nm. The pulse duration can be tuned between 1.3 ps and 1.4 ps to be matched to different crystal and filter combinations. Operating at just below $1550$ nm was necessary to ensure temperature stabilisation, enabled by keeping our crystals sufficiently far from room temperature for degenerate photon generation. We focus into the centre of the crystal with a 40 cm lens, generating a slightly elliptical spot with a waist of $\sim 124\mu$m in the horizontal and $\sim 119\mu$m in the vertical axis. This focusing condition was chosen as an optimal trade-off between brightness and heralding efficiency Bennink (2010); Grice et al. (2011); Dixon et al. (2014). To collect the down-converted modes we separate the emitted photon pairs on a polarising beam splitter, with an initial dichroic mirror removing pump photons. Signal and idler photons are collected into single-mode fibres after long-pass filtering to reduce any residual pump photons further. We introduce some gentle filtering around the central spectral lobe of our down-converted photons via a filter with a transmission profile of $\text{exp}[-\frac{(\omega-\omega_{0})^{4}}{2\sigma^{4}}]$, a FWHM of 7.4 nm and is $\sim$5 times wider than the generated photon bandwidth which minimally impacts heralding efficiencies. Down-converted photons then pass through optical interference or spectroscopy setups before being collected by Superconducting Nano-wire Single Photon Detectors (SNSPDs) operating at $\sim 80\%$ detection efficiencies. See Figure 3 (a) for the experimental layout. We investigated two-photon interference visibilities for different configurations of crystals—a 22 mm periodically-poled KTP crystal and a 30 mm custom-poled KTP crystal—and filters. We interfered photons generated from separate, but identical (manufactured from the same lithographic mask) crystals. In order to obtain a lower bound on the implied photon purity and to generate the data in Figure 4 (a), the two sources were pumped with the same amount of pump power and at least five independent two-photon interference scans were run consecutively. The data acquisition time for each of these scans was sufficient to obtain at least 1000 four-photon coincidence events outside of the dip position. From this data set we fit a linear function and extrapolate the expected visibility at zero pump power. This technique eliminates visibility degradation due to photon-number impurity (see the Appendix of Ref. Graffitti et al. (2018b)) and serves to lower bound photon purity. The performance of all results are summarised in Table 1. The different crystal photon generation rates, in terms of number of coincident photon counts per second, are shown in Figure 4 (c). Importantly, the generation rates and heralding efficiencies are quoted with consistent focusing conditions in the same optical setup and provide a comparison and not an upper limit on crystal performances. Different pump focusing conditions as well as different collection optics will result in different values for source brightness, heralding efficiencies and can also impact photon purity Bennink (2010). Crystal \| \| Interference --- Visibility (%) \| Mean --- Heralding Efficiency (%) \| Collection --- Efficiency (%) \| Mean --- Brightness (cc/mW/s) \| Experimental --- $\sqrt{\text{JSI}}$ Purity (%) \| Theoretical --- JSA Purity (%) aKTP \| 98.0 $\pm$ 0.1 \| 67.5 \| $91.8$ \| 3900 \| 91.17 $\pm$ 0.02 \| 98.7 ppKTP \| 95.3 $\pm$ 0.1 \| 57.2 \| $77.4$ \| 4900 \| 94.43 $\pm$ 0.03 \| 98.4 Table 1: A summary of results comparing our custom aKTP crystal with loose spectral filters to a ppKTP crystal with tight spectral filters. The interference visibilities are quoted at zero pump power. The mean heralding efficiencies and brightness respectively for each crystal result from an analysis of the performance of each source as a function of pump power. The collection efficiencies are calculated with respect to the upper limit detection efficiency of our detectors (80%) as well as other known optical losses (7.9%). Finally we also include the purities calculated from our experimental JSI reconstructions, as well as the theoretical purities. We use the $\sqrt{\text{JSI}}$ to calculate purities as it represents a better approximation of the JSA compared to calculating the purity of the JSI Graffitti et al. (2018a). Figure 4: Experimental results from interfering photons generated from two independent sources. (a) Visibility dependence on the squeezing parameter, $\gamma$. Each data point represents the average visibility from five interference measurements for each value of pump power (or, equivalently, for each value of $\gamma$). From this data set we can infer a minimum spectral purity of $98.0\pm 0.1\%$ and compare the performance of our aKTP crystals with loose spectral filtering against a ppKTP crystal with narrow-band, tight spectral filtering. (b) A two-photon interference measurement between photons generated from separate sources. At a pump power of 10 mW we achieve an interference visibility of $98.6\pm 1.1\%$, with a four-photon coincidence rate of around 5 Hz. (c) Photon pair generation rates of our previous crystal, a filtered ppKTP crystal and our aKTP crystal. (d) Theoretical amplitude of the phase matching function along the longitudinal direction (z axis) of the crystal at $\Delta k_{0}$. We observe an improvement in both interference visibility and generation rates upon Ref. Graffitti et al. (2018b), a result of altering the width of the Gaussian target function tracked by our algorithm from $\sigma=l/4$ to $\sigma=l/4.7$. Ref. Graffitti et al. (2018b) reported a lower bound purity of $92.7\pm 0.2\%$. This data was obtained using a delayed two-photon interference technique, interfering photons generated from the same source. Instead of this technique, we perform interference measurements on photons from independent crystals, representing a true proxy for source scalability. Our new apodized crystals have a lower-bound purity, under the same gentle filtering, of $98.0\pm 0.1\%$. Without any filtering we obtain a lower-bound purity of $95.3\pm 0.1\%$ and the respective data contributes to a full plot of all results found in the Appendix. Rather than expressing results in terms of pumping power, we show the main results in terms of $\gamma$, the effective squeezing of the two-mode squeezed vacuum, which encompasses the pump power and focusing conditions applied to the crystal. In the photon number basis, $n$, the PDC process can be expressed as $(1-\gamma^{2})^{1/2}\sum^{\infty}_{\text{n}=0}\gamma^{\text{n}}\ket{\text{n},\text{n}}_{s,i}$, with $\gamma$ defined as: $\gamma=(\tau\text{p})^{1/2}$, where p is the pump power and $\tau$ is a constant quantifying the non-linear interaction of the medium Jin et al. (2015). In this work, we evaluate $\gamma$ from the measured coincidence rates, single rates and the clock rate of the pulsed laser in a similar manner as in Ref. Jin et al. (2015). With knowledge of $\gamma$, the photon pair rate and multi-photon pair rates can be determined. This forms a more representative analysis of crystal performance as the variety of experimental conditions distinct to our analysis are gathered into this one term. Figure 4 (a) therefore, compares the interference visibility of our aKTP crystals performance with a ppKTP crystal as a function the squeezing, $\gamma$. With apodization, the need for tight filtering is removed, resulting in significantly higher heralding efficiencies, seen in Table 1. This higher efficiency means that when both sources are generating photons at the same raw rate, the source with higher heralding efficiencies will lead to higher rates of detector clicks. Factoring out known optical losses and detection efficiencies (taken as the quoted operational upper bound of 80%), overall collection efficiencies are lower bounded to 91.8%. Optical losses were determined by measuring the transmission properties of each optical element between pair production and the housing of our detectors, this accounted for a loss of 7.9%. Anti-reflection coated optics were used where possible to minimise any losses, including on the end facets of all the KTP crystals used in this investigation. Figure 5: (a) and (b) Experimental reconstruction of the JSI and marginal photon spectrum. Using a dispersive spectroscopy technique, we construct the full joint spectrum spanning a whole repetition of our lasers cycle, for our aKTP crystal (a) and ppKTP crystal (b). The reconstruction reveals all spectral correlations which are then either suppressed by filtering, or already suppressed through modification of the PMF. These figures are plotted with a logarithmic scale in order to highlight any correlations. (c) Normalised heralding and purities of the crystals we have analysed in this manuscript as a function of the photon bandwidth or the filtered photons bandwidth. The solid data points represent the normalised heralding, the filled data points are purity values and the solid (dashed) lines are the simulated results of the heralding (purity) for the ppKTP crystal. Another means of quantifying source performance is to analyse a reconstruction of the joint photon spectrum. Reconstruction of the JSA is experimentally demanding since it requires a spectrally resolved amplitude and phase measurement, which can be achieved for example via phase-sensitive stimulated emission tomography Jizan et al. (2016). Constructing the joint spectral intensity (JSI), equivalent to $\|{\text{JSA}}\|^{2}$, can be achieved with comparative ease and is therefore commonly shown, although one has to be careful what conclusions to draw in the absence of phase information normally contained in the JSA Graffitti et al. (2018a). With 20 km of standard telecommunication fibre optic we can exploit chromatic dispersion to map photon arrival time to the associated spectral component of the JSI, as performed in Graffitti et al. (2020a). The experimental arrangement is depicted in Figure 3 (b). Collection of at least $10^{6}$ photons detected by SNSPD’s operating with $<50$ ps jitter, $<25$ ns reset time and processed via a Picoquant HydraHarp with 1 ps timing resolution, enabled the construction of the respective JSI for combinations of filter and crystal. The spectral window of our results span 12.5 ns and the achievable timing resolution of this spectrometer translates to a spatial resolution of $\sim 0$.0028 nm. Figure 5 (a) and (b) shows the respective experimental JSIs of un-filtered aKTP and un-filtered ppKTP on a logarithmic scale. Any spectral correlations that exist along the main diagonal are visually highlighted. These correlation are clearly prevalent for ppKTP but almost non-existent for unfiltered aKTP, a result of non-zero contributions from the PMF. Along the diagonal, from bottom left to the top right, as well parallel to the x and y axes, through the central lobe of the joint spectra, we see a constant background signal arising from dark counts. An additional PDC source was used as a trigger, to measure the arrival of signal and idler photons. A dark count detected in the trigger channel, as opposed to an actual photon, corresponds to a displacement of the central lobe along the diagonal, resulting in temporally correlated background noise. If, either the signal or idler photon is lost, but a dark count is detected in that channel along with the trigger and remaining signal/idler photon, the central lobe is shifted in the parallel to the x or y axes depending on which of signal and idler photons are lost. The probability of this is smaller, proportional to the pair emission probability. To produce estimates for both the JSI and $\sqrt{\textrm{JSI}}$ purities, we reconstructed the JSI across increasingly long measurement intervals. Each estimation is calculated using $50\times 50~{}\textrm{ps}$ bins; doing so reduces the sparsity of the raw data and provides a more accurate and reliable Singular Value Decomposition (SVD). The SVD is used to numerically implement the Schmidt decomposition, used to quantify the non-separability of the JSA Law et al. (2000). By observing the value the estimation converges towards, we truncate the total measurement time to avoid instability. These estimation of purities are contained in Table 1. Neither the JSI nor the $\sqrt{\text{JSI}}$ truly reveal photon purity due to lack of phase information, something two- photon interference can incorporate Graffitti et al. (2018a). Thus, two-photon interference results represent a more faithful estimate of photon purity. Discrepancies between the lower-bound purities determined by two-photon interference results, and inferred purities from experimental JSIs could be caused by a combination of different factors, such as drifts in the laser central frequency and pulse duration, as well as non-negligible jitter in the detection system. Visually noticeable elongation of central lobe along the diagonal suggests pump pulse durations that are shorter than the crystal is optimised for, which in turn would result in a lower purity for experimental JSI analysis. From simulations we estimate that, pulse durations that are $\pm 0.4$ ps away from the ideal value can result in purities dropping by 6%, see the Appendix for more details. The importance of achieving the photon source characteristics displayed in this work was recently highlighted in Ref. van der Meer et al. (2020), which concludes that quantum supremacy in a Boson sampling task with non-multiplexed single-photons from PDC sources can only be achieved with Gaussian-profile aKTP crystals due to the higher collection efficiencies. Notably, photonic quantum supremacy has just been demonstrated in Gaussian Boson Sampling (GBS), in an experiment which created 50 photons from 25 Gaussian apodized crystals using a duty-cycle poling technique Zhong et al. (2020). Using our improved poling algorithm and considering the trade off between non-linearity and photon purity highlighted in this manuscript, an optimal $\sigma$ could enable higher purities and heralding efficiencies. This, in turn, would culminate in a greater advantage and scalability of the scheme. The discrepancy in brightness between our aKTP source and the ppKTP source highlighted within Table 1 can be balanced by adjusting the relative pump powers to achieve the same squeezing $\gamma$. At a fixed value of $\gamma$, the single and multi-pair production probability for aKTP and ppKTP are the same, independent of the pump power, as the different pumping powers act to equate the probabilities of generating $n$ photon pairs. A hard limit on available pump power for multiple bulk PDC sources could restrict one’s ability to maximise brightness. Future scalable architectures however are likely to be based on waveguides, which typically require only $\mu$W of pumping power. Gaussian PMFs can also be achieved in waveguide sources either through domain engineering, or via inter-modal Spontaneous Four Wave Mixing (SFWM) in a multi-mode waveguide Paesani et al. (2020). Future improvements will target higher interference visibilities by modifying the PEF. In this work, the PEF is a $1.3~{}\textrm{ps}$ long sech2 pulse, imposing a theoretical limit on the maximum visibility of 98.7%. However, it is possible to achieve up to 99.9% visibility directly with our crystals by engineering the PEF Graffitti et al. (2018a). Modification of the PEF can be achieved using pump-shaping techniques Chen et al. (2019). Additionally, further improvements may be obtained by exploring the interplay of spatial and spectral modes generated in a non-linearity engineered crystal Bruno et al. (2014); Guerreiro et al. (2013). ## References * Christ et al. (2013) A. Christ, A. Fedrizzi, H. Hübel, T. Jennewein, and C. 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We used the same optical setup for each analysis. We also determined the interference visibilities for signal-idler interference. As mentioned in the main text, being able to interchangeably interfere daughter photons from PDC offers some additional capabilities when it comes to building multi-photon states. Under the same conditions as the results obtained with idler-idler interference we obtain a lower-bound purity of $97.0\pm 0.1\%$, a $1\%$ decrease. Figure S1: Visibility power dependence. At each pump power setting, we measured five two-photon interferograms to obtain the average visibility shown here as solid data points. Error bars are taken as one standard deviation from these measurements. We then infer a minimum for spectral purity for a combination of crystals under certain filtering condition by fitting a linear function (dashed lines) to extract visibility under 0 pump power. Our efforts to consider why we witnessed lower purities in our experimental JSI analysis led to simulations into how pulse duration affects photon purity, the results of which are shown in Figure S2. Maximum purities are achieved when the width of the PEF and PMF are matched. From the JSI reconstruction results, the elongation along the diagonal could have been caused by instability of our pulsed laser source, a reasonable argument as scans were run for hours at a time. Any drifting in pulse duration from ideal leads to a reduction in purity. Figure S2: Theoretical simulations of photon purity as a function of varying pulse duration. A non-ideal pulse duration affects photon purity as the bandwidths of the PEF and PMF are not matched for pulse durations not 1.3 ps. (a), (b) and (c) depict the $\sqrt{\text{JSI}}$ from a range of pulse durations. Shorter pulse durations contribute towards spectral correlations along the diagonal, something visible in our reconstructed $\sqrt{\text{JSI}}$s. The red dash lines represent the width of the PEF corresponding to the pulse duration under investigation. (d) The effects of non-ideal pulse durations on photon purity. Analysing the range of purities derived from $\sqrt{\text{JSI}}$ and JSI as a function of pump pulse duration.
# Accumulation of chiral hinge modes and its interplay with Weyl physics in a three-dimensional periodically driven lattice system Biao Huang<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01069 Dresden, Germany Viktor Novičenko<EMAIL_ADDRESS>Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10257 Vilnius, Lithuania André Eckardt<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01069 Dresden, Germany Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany Gediminas Juzeliūnas<EMAIL_ADDRESS>Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10257 Vilnius, Lithuania ###### Abstract We demonstrate that a three dimensional time-periodically driven lattice system can exhibit a second-order chiral skin effect and describe its interplay with Weyl physics. This Floquet skin-effect manifests itself, when considering open rather than periodic boundary conditions for the system. Then an extensive number of bulk modes is transformed into chiral modes that are bound to the hinges (being second-order boundaries) of our system, while other bulk modes form Fermi arc surface states connecting a pair of Weyl points. At a fine tuned point, eventually all boundary states become hinge modes and the Weyl points disappear. The accumulation of an extensive number of modes at the hinges of the system resembles the non-Hermitian skin effect, with one noticeable difference being the localization of the Floquet hinge modes at increasing distances from the hinges in our system. We intuitively explain the emergence of hinge modes in terms of repeated backreflections between two hinge-sharing faces and relate their chiral transport properties to chiral Goos-Hänchen-like shifts associated with these reflections. Moreover, we formulate a topological theory of the second-order Floquet skin effect based on the quasi-energy winding around the Floquet-Brillouin zone for the family of hinge states. The implementation of a model featuring both the second-order Floquet skin effect and the Weyl physics is straightforward with ultracold atoms in optical superlattices. ## I Introduction In recent years, researches have demonstrated that time-periodically driven systems can show intriguing and unique effects that find no counterparts in non-driven systems. Examples include anomalous Floquet topological insulators featuring robust chiral edge modes for vanishing Chern numbers Kitagawa _et al._ (2010); Rudner _et al._ (2013); Roy and Harper (2017a); Yao _et al._ (2017a); Mukherjee _et al._ (2017a); Peng _et al._ (2016); Maczewsky _et al._ (2016); von Keyserlingk and Sondhi (2016); Else and Nayak (2016); Potirniche _et al._ (2017); Wintersperger _et al._ (2020) and discrete time crystals Sacha (2015); Khemani _et al._ (2016); Else _et al._ (2016); Yao _et al._ (2017b); Zhang _et al._ (2017); Choi _et al._ (2017); Rovny _et al._ (2018); Ho _et al._ (2017); Huang _et al._ (2018); Sacha and Zakrzewski (2017); Yao and Nayak (2018); Sacha (2020). The periodic driving shifts the fundamental theoretical framework from focusing on Hamiltonian eigen problems to the unitary evolution operators genuinely depending on time, resulting in a plethora of new concepts and methods such as spacetime winding numbers Rudner _et al._ (2013) and spectral pairing von Keyserlingk _et al._ (2016). In this context, it appears natural and tantalizing to explore possible new classes of periodically driven systems that go beyond descriptions by traditional theories. In this paper, we show that time-periodic driving of a three dimensional (3D) lattice can give rise to the coexistence of Weyl physics Armitage _et al._ (2018); Anderson _et al._ (2012, 2013); Dubček _et al._ (2015); Sun _et al._ (2018); Higashikawa _et al._ (2019); Lu _et al._ (2020); Wang _et al._ ; Zhu _et al._ with a new type of hinge (i.e. second-order boundary) states. These states are chiral in the sense that they transport particles in a unidirectional fashion along the hinge. As an intriguing effect, we find a macroscopic accumulation of these chiral Floquet hinge states, which is associated with a complete reorganization of the system’s quasienergy spectrum in response to shifting from periodic to open boundary conditions. At a fine- tuned point, even all states of the system become hinge states. Tuning away from that point a pair of Weyl points is created at quasienergy $\pi$, leading to Fermi-arc surface (i.e. first-order boundary) states that coexist with the hinge modes. The localization of an extensive number of hinge modes at the boundaries of the system resembles the non-Hermitian skin effect Yao and Wang (2018); Bergholtz _et al._ (2020); Ashida _et al._ (2020); Kawabata _et al._ (2020), with a notable difference being the localization of the modes at increasing distances from hinges (higher-order boundaries) in such a way that the hinge modes cover the whole lattice. Different from the case of higher- order topological phases Benalcazar _et al._ (2017a, b); Khalaf (2018), the hinge states are buried deeply inside the bulk spectrum. Their existence and robustness is, therefore, not captured by the theory of higher-order topological insulators/semimetals which rely on the existence of bulk energy gaps. Instead, the chiral hinge modes can be understood as resulting from the repeated backreflection from two hinge-sharing surface planes and their chiral motion as the result of chiral Goos-Hänchen-like shift associated with the reflection at a boundary face. Furthermore, different from one-dimensional (1D) periodically driven lattices Budich _et al._ (2017), the modes with opposite chirality residing at opposite hinges are well spatially separated, so the scattering due to a local perturbation, such as a local disorder, does not affect the transport chirality at individual hinges. The model system proposed and studied here consists of a simple, stepwise modulation of tunnelling matrix elements involving six steps. It generalizes to three dimensions a two-dimensional lattice model introduced by Rudner et. al. Rudner _et al._ (2013) for studying anomalous Floquet topolgical insulators (see also Ref. Kitagawa _et al._ (2010)). The latter 2D tunnel modulation has been recently applied to ultracold atoms for the realization of anomalous topological band structures Wintersperger _et al._ (2020). Our 3D model can equally be implemented with ultracold atoms using such a stepwise tunnel modulation, now in 3D optical superlattices. Furthermore, besides giving rise to a new phenomenon, the unconventional chiral second-order Floquet skin effect, the model proposed here also provides a simple recipe for the robust implementation of Weyl physics by means of time-periodic driving, which should be easier to realize compared to previous proposals Higashikawa _et al._ (2019). This paper is structured as follows. In the next Section II a 3D periodically driven lattice is defined. Subsequently the characteristic features of the bulk and hinge physics are considered in Secs. III and IV. In particular, in Sec IV.3 a topological theory based on the quasi-energy winding around the frequency-Brillouin zone is formulated for the family of localized hinge states enforced by reflections from open-boundaries. The experimental implementation of our model, using ultracold atoms in modulated superlattices, is discussed in Sec. V, before the Concluding Section VI. Some technical details are presented in three Appendices A, B and C. (a) Driving (b) Bulk dynamics (c) Reflection by open boundary surfaces at $x=1$ and $y=1$ accompaned by sublattice changes $B\rightarrow A$ and $A\rightarrow B$, respectively. (d) Hinge dynamics Figure 1: (a) Bonds connected during the driving steps 1 to 6. (b) Bulk trajectories within a Floquet cycle at the fine tunned point $\phi=\pi/2$. Depending on the starting sublattice, particles will travel in opposite directions along the cubic diagonal $\mathbf{d}=(1,-1,1)$. (c) Trajectories at the same $\phi=\pi/2$ but with a surface termination (open boundary). Particles starting from sublattice $A$ (or $B$) near the $y=1$ (or $x=1$) surface would have their dynamics impeded by the open boundary at a certain driving step. That results in a switch of sublattice after a Floquet cycle, and therefore the direction of the trajectory is reversed after the reflection from the surface. (d) The hinge formed by two intersecting terminating surfaces renders uni-directional modes. The figure shows two of the modes closest to the hinge starting at a site of $B$ (lower plot) or $A$ (upper plot) sublattices directly at the hinge. Each color for the arrow denotes one Floquet cycle. ## II Model We consider a bipartite cubic lattice with alternating A-B sublattices in all three Cartesian directions. The lattice is described by a time-periodic Hamiltonian $H(t+T)=H(t)$, with the driving period $T$ divided into 6 steps. In each step tunnelling matrix elements $-J$ between sites $\mathbf{r}_{A}$ of sublattice $A$ and neighboring sites $\mathbf{r}_{A}\pm a\mathbf{e}_{\mu}$ of sublattice $B$ are switched on, with $\mu=x,y,z$. During the driving period $T$ the tunneling steps appear in a sequence $\mu\pm=x+,\,y+,\,z+,\,x-,\,y-,\,z-$, as illustrated in Fig. 1(a) 111Without including the tunneling along the $z$ direction described by third and the sixth driving steps, the dynamics reduces to a 2D motion in a periodically driven square lattice considered in refs. Rudner _et al._ (2013); Mukherjee _et al._ (2017a); Wintersperger _et al._ (2020).. Within each step the evolution is determined by a coordinate-space Hamiltonian $H_{\pm\mu}=-J\sum_{\bm{r}_{A}}(\|\bm{r}_{A}\rangle\langle\bm{r}_{A}\pm a\bm{e}_{\mu}\|+\|\bm{r}_{A}\pm a\bm{e}_{\mu}\rangle\langle\bm{r}_{A}\|)$, where $J$ is the tunneling matrix element, $\bm{r}_{A}$ specifies the location of sublattices $A$, and $a$ is the lattice spacing such that $\bm{r}_{A}\pm a\bm{e}_{\mu}$ denotes the locations of sites in sublattice $B$ neighboring to the sublattice $A$ site $\bm{r}_{A}$. The tunnelling processes occurring in each of the driving steps are characterized by a single dimensionless parameter, the phase $\phi=-\frac{JT}{6\hbar}.$ (1) The one-cycle evolution operator (or Floquet operator), $U_{F}={\cal T}e^{-(i/\hbar)\int_{0}^{T}dtH(t)},$ (2) whose repeated application describes the time-evolution in stroboscopic steps of the driving period $T$, can be decomposed into terms corresponding to the six driving stages, $U_{F}=U_{z-}U_{y-}U_{x-}U_{z+}U_{y+}U_{x+}.$ (3) When dealing with the bulk dynamics we impose periodic boundary conditions in all three spatial directions. The evolution operators for the individual driving steps can then be represented as: $U_{\mu\pm}=U\left(\pm k_{\mu}\right)=e^{-\frac{i}{6}H_{\mu\pm}}=e^{-i\phi(\tau_{1}\cos k_{\mu}\pm\tau_{2}\sin k_{\mu})}\,,$ (4) where $\tau_{1,2,3}$ are Pauli matrices associated with the sublattice states $A$ and $B$ and where $k_{\mu}$ with $\mu=x,y,z$ denotes the Cartesian components of the quasimomentum vector $\mathbf{k}$. Here and in the following, we will use a dimensionless description, where time, energy, length and quasimomentum are given in units of $T,\hbar/T$, $a$, and $\hbar/a$, respectively. The quasienergies $E_{n,\bm{k}}$ and the Floquet modes $\left\|n,\bm{k}\right\rangle$ are defined via the eigenvalue equation $U_{F}\|u_{n,\bm{k}}\rangle=\exp(-iE_{n,\bm{k}})\|u_{n,\bm{k}}\rangle.$ (5) We first note that the only global symmetry satisfied by the Floquet operator (3)-(4) is a particle-hole flip $\Gamma=CK$, where $Ki=-iK$ is complex conjugation and $C=\tau_{3}$ the third Pauli matrix. Thus, the system belongs to class D in Altland-Zirnbauer notation Altland and Zirnbauer (1997); Chiu _et al._ (2016). The Floquet operator satisfies $CU_{F}(\bm{k})C^{-1}=U_{F}^{}(-\bm{k})$ Roy and Harper (2017b); Yao _et al._ (2017a), and therefore the quasienergies must appear in pairs $E_{1,\bm{k}}=-E_{2,-\bm{k}}$. Meanwhile, the system obeys the inversion symmetry $PU_{F}(\bm{k})P^{-1}=U_{F}(-\bm{k})$, with $P=\tau_{1}$, enforcing that for each band one has $E_{n,\bm{k}}=E_{n,-\bm{k}}$. Together, we see that the Floquet spectrum has pairs of states with quasi-energies $E_{1,\bm{k}}=-E_{2,\bm{k}}$. This means possible gaps or nodal points/lines can, modulo $2\pi$, appear only at quasienergy $0$ or $\pi$. At $k_{x}=k_{y}=k_{z}=\pm\frac{\pi}{2}\,(\text{mod }\pi)$ Eqs. (3) and (4) yield $U_{F}=1$, so that $E_{n,\bm{k}}=0$. Therefore the quasienergy spectrum is always gapless at quasienergy $0$ (modulo $2\pi$), regardless of the driving strength $\phi$. Thus a single band spectrum could only possibly open up a gap at quasienergy equal to $\pi$ (modulo $2\pi$). To draw a complete phase diagram, we first note that flipping the sign of $\phi$ amounts to a particle-hole transformation $U_{F}\|_{-\phi}=CU_{F}\|_{\phi}C^{-1}$, and from the previous analysis we see that such a flip does not change the spectrum. Furthermore, from $e^{-i\phi\hat{n}\cdot\bm{\tau}}=\cos\phi-i\hat{n}\cdot\bm{\tau}\sin\phi$, the periodicity of the Floquet operator with respect to the parameter $\phi$ is clearly seen: $U_{F}\|_{\phi}=U_{F}\|_{\pi+\phi}$. In this way, the irreducible parameter range is $\phi\in[0,\pi/2]$ as illustrated in Fig. 2. $\phi$ \| (1) PBC $x,y,z$ \| (2) PBC $z$, OBC $x,y$ \| (3) OBC $x,y,z$ ---\|---\|---\|--- $\frac{\pi}{2}$ \| \| 4 eigenstates \| 2 eigenstates $\frac{\pi}{3}$ \| \| 3 eigenstates \| 2 eigenstates $\frac{\pi}{8}$ \| \| 1 eigenstate \| 1 eigenstate Figure 2: Table showing energy spectra and Floquet modes. The phases of the system are indicated in the vertical axis on the left. The table’s rows refer to the driving parameters $\phi=\pi/8,\pi/3,\pi/2$, corresponding to the metallic phase, the Weyl semimetal/hinge phase, and the fine-tuned point, respectively. The columns represent periodic/open boundary conditions (PBC/OBC) along the specified directions. The system sizes are $L_{x}=L_{y}=40$ in column (1), 20 in (2), and 16 in (3). Columns (1) and (2) contain energy spectra projected onto $k_{z}$. Although the spectra for fully periodic and open boundary conditions in $x$ and $z$ direction are almost identical in the metallic phase ($\phi=\pi/8$), they are very different in the WSM/hinge phase ($\phi=\pi/3$) and at fine tuning ($\phi=\pi/2$). The difference is a result of the formation of chiral hinge modes for open boundary conditions in $x$ and $z$ directions (column 2). This can be inferred also from the dot size reflecting the inverse participation ratio with respect to the site basis as a measure for localization, as well as from the color code indicating the mean distance to two opposite hinges and interpolating from blue for one hinge over black near the center of the system to red for the other hinge. The green dots mark the Weyl points. We also plot the real- space densities of various Floquet modes for open boundary conditions along $x$ and $y$ (column 2) or all (column 3) directions. ## III Phase diagram for periodic boundary conditions $\phi$ \| Weyl points ---\|--- $\frac{\pi}{2}$ \| $\frac{\pi}{3}$ \| $\frac{\pi}{6}$ \| $\frac{\pi}{8}$ \| (a) Positions of Weyl points. (b) $\phi=\pi/3$, PBC $x,y,z$ (c) $\phi=\pi/3$, PBC $x,y$ (d) $\phi=\pi/6$, PBC $x,y,z$ Figure 3: (a) The different phases can be distinguished by the presence or absence of Weyl points at quasienergy $\pi$. For $\phi=\pi/8$, in the metallic phase ($\phi<\pi/6$), no Weyl points are present. At the transition, $\phi=\pi/6$, the band touching point appears at $\bm{k}=\bm{0}$ (green circle). For $\pi/6<\phi<\pi/2$ the band touching splits into two Weyl points of opposite charge (green circles with $\pm$ sign) that separate along the diagonal $k_{x}=-k_{y}=k_{z}$, as shown for $\phi=\pi/3$. At fine tuning $\phi=\pi/2$ the dispersion becomes flat in two directions and the Weyl points disappear to reappear for $\phi>\pi/2$ with reversed charges (signs in green dots correspond to the limit $\phi=\pi/2-0)$. (b-d) Quasienergy spectra versus $k_{x},k_{y}$ for periodic boundary conditions in $x$ and $y$ and either periodic (b,d) or open (c) boundary conditions in $z$ direction. A surface Fermi arc can be observed for $\phi=\pi/3$ by comparing the spectra with periodic (b) and open (c) boundary conditions the $z$ direction. The orange surface denotes the contour formed by quasi-energies closest to $E=\pi$ at each $(k_{x},k_{y})$. For $\phi=\pi/6$ (d) a pair of Weyl points reduces to a single band touching point at the quasi-energy $E=\pi$. (a) Fine tuned point (b) Slight deviation Figure 4: (a) Anisotropic 1D-like dispersion along the corrdinate $k_{x}-k_{y}+k_{z}$ at the fine-tuned point $\phi=\pi/2$. (b) For a small detuning, $\phi=\pi/2-0.1$, the dispersion is no longer completely flat in other two directions, and a pair of non-equivalent Weyl points is formed along the diagonal at $\mathbf{k}=\pm k_{0}(1,-1,1)$. Yet the dispersion remains highly anisotropic. Before investigating the system with open boundary conditions and the emergence of chiral hinge modes, let us first discuss the phase diagram considering the case of a translation invariant system with periodic boundary conditions. Let us begin with the topologically trivial high-frequency (weak driving) limit corresponding to $\phi\ll 1$. In that case one can retain only the lowest order terms in $\phi$ when expanding the stroboscopic evolution operator $U_{F}$ of Eq. (3), resulting in $U_{F}\|_{\phi\rightarrow 0}\simeq\cos^{6}\phi\tau_{0}-2i\sin\phi\cos\phi(\cos k_{x}+\cos k_{y}+\cos k_{z})\tau_{1}+O(\sin^{2}\phi)\simeq e^{-i\phi 2(\cos k_{x}+\cos k_{y}+\cos k_{z})\tau_{1}}$, where $\tau_{0}$ is the identity matrix. The spectrum $\pm 2\phi\sum_{\mu=x,y,z}\cos k_{\mu}$ corresponds to that of a static simple cubic lattice artificially described by 2 sublattices, where the bands are folded as the Brillouin zone size is halved. While the system remains gapless at quasienergy $0$ for arbitrary $\phi$, a characteristic feature of the high- frequency (weak driving) regime is a finite energy gap at quasienergy $\pi$, resulting from the fact that the band width is proportional to $\phi$ and thus is small compared to the dimensionless driving energy $\hbar\omega=2\pi$. This behaviour can be observed in the spectrum for $\phi=\pi/8$ shown in Fig. 2 at the bottom of column (1). Increasing the driving strength $\phi$, the band width grows relative to $2\pi$. When $\phi=\pi/6$ the single Floquet band touches itself at quasienergy $\pi$ and quasimomentum $\mathbf{k}=0$ (see Fig. 3 (d)), so that the quasienergy spectrum becomes gapless. We refer to this as Floquet autogamy. Going to the regime $\phi\in(\pi/6,\pi/2)$, the band touching point transforms into a pair of Weyl points forming at quasienergy $\pi$ with topological charges $\pm 1$, as shown in Fig. 3 (a). They are located at the quasimomenta $\mathbf{k}=\pm k_{0}\mathbf{d}$ along the diagonal vector $\mathbf{d}=(1,-1,1),$ (6) with $k_{0}=\left(1/2\right)\arccos\left[\left(1/2-\sin^{2}\phi\right)/\sin^{2}\phi\right]$ modulo $\pi$ (see Appendix A), so that $k_{0}\to\pi/3$ as $\phi\to\pi/2$. We observe the emergence of surface Fermi arc states connecting the Weyl points, when comparing the spectrum with full periodic boundary conditions to that with open boundary conditions along z-direction, as illustrated in Fig. 3 (b) and (c) respectively. As the driving strength approaches the fine tuned point, $\phi=\pi/2-\varepsilon$ with $\varepsilon\ll 1$, the Weyl dispersion acquires a highly anisotropic form shown in Fig. 4 (b). The dispersion remains steep along the diagonal coordinate $\bm{k}\cdot\mathbf{d}=k_{x}-k_{y}+k_{z}$, but becomes increasingly flat in other two directions. Exactly at fine tuning, $\phi=\pi/2$, the constituent evolution operators (4) reduce to $U_{\mu\pm}\left(\mathbf{k}\right)=-i(\cos k_{\mu}\tau_{1}\pm\sin k_{\mu}\tau_{2})$, and the Floquet stroboscopic operator takes the form $U_{F}=-e^{-i2\tau_{3}\bm{k}\cdot\mathbf{d}}$. This provides the quasi- energies $E_{\bm{k},\pm}=\pm 2\bm{k}\cdot\mathbf{d}+(2m+1)\pi$, where $m\in\mathbb{Z}$ labels the Floquet bands, and where the upper and lower branches labeled by $\pm$ now directly correspond to sublattices A and B, i.e. $\pm\rightarrow\tau_{3}$. In that case the Weyl points disappear and the dispersion $E_{\bm{k},\pm}$ is completely flat for the momentum plane normal to $\mathbf{d}$, as one can see in Fig. 4 (a). Hence a particle can only propagate along the diagonal $\mathbf{d}$ with a dimensionless velocity $\bm{v_{\pm}}=\pm 2\bm{d}$, depending on whether the particle occupies a site on the sublattice $A$ or $B$ at the beginning of a driving cycle. It is noteworthy that for fine tuned driving, the effective Floquet Hamiltonian, $H_{F}\equiv-i\ln U_{F}=2\tau_{3}\mathbf{d}\cdot\mathbf{k}+\pi\quad\text{for}\quad\phi=\pi/2,$ (7) is periodic in the momentum space only by taking into account the periodicity in quasi-energies. Such an effective Hamiltonian does not have a static counterpart, and can only be produced in periodically driven systems. The fine-tuned dispersion can be understood by considering the dynamics in real space. For $\phi=\pi/2$ in each step $\mu\pm$ the particle is fully transferred from a sublattice A site positioned at $\mathbf{r}_{A}$ to a neighboring site B situated at $\mathbf{r}_{B}=\mathbf{r}_{A}\pm\mathbf{e}_{\mu}$ or vice versa. During the six steps composing the driving period, the particle follows the trajectory shown in Fig. 1 (b). Thus, after completing each period the particle located on a site of sublattice $A$ ($B$) is transferred by $+2\mathbf{d}$ ($-2\mathbf{d}$) to an equivalent site of the same sublattice, giving rise to stroboscopic motion along the diagonal directions $\pm\mathbf{d}$ at the velocity $\bm{v}_{\pm}$. ## IV Accumulation of chiral hinge modes for open boundary conditions Let us now consider the case of open boundary conditions with the faces of the boundary planes oriented along the directions $\pm\mu$, either with $\mu=x,y,z$ (fully open boundary conditions) or with $\mu=x,y$, keeping periodic boundary conditions along the $z$ direction in the latter case. As an intriguing effect, we find that, when the system is subjected to open boundary conditions with at least two properly chosen boundary planes, an extensive number of chiral Floquet hinge modes is formed. This phenomenon is best understood by considering the real-space propagation of the particle at the fine-tuned parameter $\phi=\pi/2$ in stroboscopic steps of the driving period, as it will be done next. For this purpose, we will label the lattice sites by vectors ${\bm{r}}=(x,y,z)$, with coordinates $x,y,z$ taking integer values between $1$ and $L_{x,y,z}$. The lattice sites are considered to belong to sublattice $A$ (or $B$) if $s=(-1)^{x+y+z}$ equals $+1$ (or $-1$). Thus we will use $s=1=A$ and $s=-1=B$ to label the sublattice. ### IV.1 Fine-tuned driving Let us consider a hinge along the $z$ axis confined by the $-x$ and $-y$ surface planes. In that case the particle is restricted to the lattice sites with $x\geq 1$ and $y\geq 1$. Starting from a site of, say, sublattice $B$, a particle will propagate on this sublattice at constant velocity in steps $-2\bm{d}$ in diagonal direction, until it reaches the $-x$ boundary face positioned at $x=1$. At the boundary, tunnelling between the lattice sites $B$ and $A$ cannot occur during one of the six steps of the driving cycle. As a result, the particle changes the sublattice and starts to propagate on the $A$ sublattice in opposite direction. The microscopic processes leading to such a reflection are illustrated in Fig. 1 (c). The left plot shows the two possible ways of how a change of sublattice can occur at the boundary face oriented in the $-x$ direction. The two processes are distinguished by whether they start on a $B$ lattice site directly at the boundary with $x=1$ (dark grey arrow) or one site away from the boundary with $x=2$ (yellow arrow). A tunnelling event is impeded in the first (dark grey) or the fifths (yellow) driving step, respectively, as marked by small planes. After changing the sublattice at the $-x$ surface, the particle travels in reversed direction in steps of $2\bm{d}$ until it eventually reaches the $-y$ surface and is again backreflected, this time with the sublattice change $A\rightarrow B$. The microscopic details of such a reflection at the $-y$ surface are depicted on the right hand side of Fig. 1 (c). In this way, the particle will move back and forth between the $-x$ and the $-y$ surfaces sharing the hinge. Such a dynamics is illustrated in Figs. 5 and 6, showing the path of a particle projected onto the $xy$-plane. Interestingly, within this plane, the particle returns to the same transverse position $(x,y)$ only after having travelled twice between both faces, as one can see in Figs. 5 and 6. It is noteworthy that the change in the sublattice $B\rightarrow A$ or $A\rightarrow B$ during the backward reflection from the corresponding hinge planes is accompanied by a lateral Goos-Hänchen-like (GH) shift of particle’s trajectory. This is similar to changing a track for a train before sending it backwards. Importantly the back reflected particle propagates in a trajectory situated closer the hinge or further away from it for the $B\rightarrow A$ or $A\rightarrow B$ reflections, respectively. Because of such chiral GH shifts, the particle visits a larger number of $B$ sites than $A$ sites when travelling between the two surface planes. Since the ballistic motion along the $B$ ($A$) sites is the accompanied by a spatial shift in $-z$ ($z$) direction, one arrives at an overall steady advance in $-z$ direction, i.e. along the hinge, during the forward and backward motion of the particle between the hinge-sharing surfaces oriented in $-x$ and $-y$ direction. More precisely, as demonstrated in Appendix B, the particle traces out two slightly mismatched “loops” shown in Figs. 5 and 6 that involve four reflections by the hinge surfaces before it comes back to exactly the same point in the $xy$ plane, but shifted by $-2$ in $z$-direction at an equivalent (i.e. B-type) lattice site. In this way the particle’s trajectory roughly covers a two- dimensional strip along the z-direction, whose width is about twice its distance from the hinge. Although such a picture applies to a particle situated further away from the hinge, the advance in the $-z$ direction takes place also for trajectories situated very close to the hinge where the particle is reflected simultaneously from both hinge planes $x=1$ and $y=1$, as illustrated in Fig. 1(d). Figure 5: An example of a fine-tunned stroboscopic motion of a particle at the lower-left hinge. The picture shows the projection of the particle’s trajectory in the $xy$ plane. The sites of the $B$ and $A$ sublattices are marked in blue and red, respectively. The particle is initially at the site of the sublattice $B$ characterized by the coordinates $x=M+1$ and $y=1$, with even $M=4$. This corresponds to the lower row ($y=1$) and the fifth column ($x=5$). Subsequently the particle undergoes the stroboscopic evolution described by Eqs. (35)-(41) in Appendix B. Dashed lines with arrows show stroboscopic reflections from the planes $x=1$ or $y=1$. Bulk ballistic trajectories over one/two driving periods are indicated by thin/thick solid arrows. The particle returns to its initial site after $2M+1=9$ periods. Figure 6: Like in Fig. 5 the particle is initially at the site of the sublattice $B$ with $x=M+1$ and $y=1$, but now with odd $M=5$. The particle returns to the initial site after $2M+1=11$ periods. Suppose the particle intially occupies a site of the sublattice $B$ with transverse coordinates $y=1$ and $x=M+1$, so the particle is situated at the hinge surface oriented in $-y$ direction, and is $M\geq 0$ sites away from another $-x$ hinge surface. In that case, it takes $\left(2M+1\right)$ driving periods for the particle to come back to the initial position $\left(M+1,1\right)$ in the $xy$ plane, while having shifted by $-2$ lattice units in $z$ direction, i.e. $\left(U_{F}\right)^{2M+1}\left\|B,M+1,1,z\right\rangle=-\left\|B,M+1,1,z-2\right\rangle\,.$ (8) After averaging over such a full reflection cycle, the particle travels with an $M$-dependent mean velocity of $v_{M-}=-2/\left(2M+1\right)$ along the $z$ (hinge) axis (see Appendix B for more details). Here $\|s,x,y,z\rangle$ describes a particle located at site $(x,y,z)$ belonging to sublattice $s=B=-1$ or $s=A=1$ with $s=(-1)^{x+y+z}$. Let us now consider periodic boundary conditions in $z$ direction with $L_{z}=2N_{z}$, i.e. $\left\|s,x,y,z+2N_{z}\right\rangle=\left\|s,x,y,z\right\rangle$, while keeping open boundary conditions in the $x$ and $y$ . It is then convenient to introduce mixed basis states $\left\|{s,x,y,k_{z}}\right\rangle^{\prime}=\frac{1}{\sqrt{N_{z}}}\sum_{z}e^{ik_{z}z}\left\|s,x,y,z\right\rangle.$ (9) They are characterized by quasimomenta $k_{z}$, which are defined modulo $\pi$ corresponding to the lattice period of 2 when moving in $z$ direction, and obey $\left(U_{F}\right)^{2M+1}\left\|{B,M+1,1,k_{z}}\right\rangle^{\prime}=-\left\|{B,M+1,1,k_{z}}\right\rangle^{\prime}e^{2ik_{z}}\,.$ (10) In a similar manner, the system returns to any state of the stroboscopic sequence $\left\|M,k_{z},p\right\rangle=\left(U_{F}\right)^{p}\left\|{B,M+1,1,k_{z}}\right\rangle^{\prime}$ (11) after $2M+1$ driving periods: $\left(U_{F}\right)^{2M+1}\left\|M,k_{z},p\right\rangle=-\left\|M,k_{z},p\right\rangle e^{2ik_{z}}\,,$ (12) with $p=0,1,\ldots 2M$. By superimposing the subharmonic hinge states $\left\|M,k_{z},p\right\rangle$, we can now construct Floquet hinge states: $\overline{\left\|M,k_{z},q\right\rangle}=\frac{1}{\sqrt{2M+1}}\sum_{p=0}^{2M}\left\|M,k_{z},p\right\rangle\exp\left(\mathrm{i}\frac{2\pi q-2k_{z}+\pi}{2M+1}s\right)\,,$ (13) where the index $q=0,1,\ldots,2M$ labels these modes. The corresponding quasienergies are given by $E_{M,k_{z},q}=\left(\frac{2\pi q-2k_{z}+\pi}{2M+1}\right)\textrm{ mod }2\pi\,.$ (14) An analogous dispersion but with an opposite slope (opposite sign) is obtained for the states formed at the opposite hinge confined by the planes at $x=y=L$ facing the $+x$ and $+y$ directions. The dispersion at both hinges reproduces the spectrum for the beam geometry shown in the row 1 and column (2) of Fig. 2. Such a spectrum of the system with open boundary conditions in $x$ and $y$ directions looks completely different from the one for full periodic boundary conditions shown in column (1) of Fig. 2 or Fig. 4 (a), where all the modes have the same positive or negative dispersion slope (group velocity) $v_{z}=\pm 2$. In contrast, for the beam geometry [column (1) of Fig. 2] the spectrum due to the chiral hinge modes is now organized in linear branches given by Eq. (14) and the analogous dispersion with inverted sign for the opposite hinge. Each branch is characterized by a different group velocity $v_{M\pm}=\pm\frac{2}{2M+1}$ (15) decreasing with the distance from the hinge $M$, where the lower and upper sign in $\pm$ correspond to the states located around opposite hinges $x=y=1$ and $x=y=L$. The red / blue colors in Fig. 2 indicate the mean distance of each mode from the two relevant hinges. The dark red (blue) mode associated with $M=0$ is localized directly at the hinge $x=y=1$ ($x=y=L$) and propagates at the largest velocity in negative (positive) $z$ direction. Modes with a smaller slope have larger $M$ and thus are located further away from the particular hinge, as indicated by the color. The real-space density of four different hinge modes at $\phi=\pi/2$ is illustrated in the real-space plot shown in row 1 and column (2) of Fig. 2. We can see that the modes located further away from the hinge, having smaller chiral group velocities, are less localized than the fastest modes located directly at the hinge. A measure for the degree of localization of a mode $\|\psi\rangle$ is the inverse participation ratio $\text{IPR}=\sum_{j}\|\langle j\|\psi\rangle\|^{4}$, with real-space site states $\|j\rangle$ and the Floquet eigenstates $\|\psi\rangle$. It is shown in the spectra of Fig. 2 via the dot size roughly indicating the inverse of the number of sites a mode is spread over. Thus 1D-like modes localized at the hinges have larger IPR than those that are spread over a 2D-like ribbon further away from the hinges. All in all, for the given geometry, all modes are hinge modes at fine tuning. This effect resembles an extensive accumulation of boundary modes featured in the non-Hermitian skin effect Yao and Wang (2018); Bergholtz _et al._ (2020); Ashida _et al._ (2020); Kawabata _et al._ (2020). It is noteworthy that the modes of the present periodically driven lattice are localized at the second- order boundaries, viz. at the hinges rather than directly at the boundary faces. Therefore, the formation of an extensive number of hinge modes might be called _chiral second-order Floquet skin effect_ , in analogy to the terminology used for non-Hermitian systems Kawabata _et al._ (2020). An important difference is that in non-Hermitian systems the skin modes are localized directly at the boundaries Yao and Wang (2018); Bergholtz _et al._ (2020); Ashida _et al._ (2020); Kawabata _et al._ (2020), whereas in the present periodically driven system the hinge modes are localized at various distances from the hinges to which they are bound to in such a way that the hinge modes cover the whole lattice. This is because the eigenstates of a unitary Floquet evolution operator are orthogonal to each other (like those of a Hermitian operator) implying that there can be at most one Floquet eigenstate per lattice site on average, so an accumulation of boundary states is not possible directly at the boundary. To put in another way, the non- Hermitian skin effect is associated with the exceptional points of the non- Hermitian Hamiltonian when the boundaries are introduced Bergholtz _et al._ (2020); Ashida _et al._ (2020), whereas no exceptional points are formed for periodically driven systems described by the unitary Floquet evolution operators. More details on these issues are available in Appendix C. ### IV.2 Beyond fine-tuned driving (a) without defect (b) with defect (c1) $t=3$ (c2) $t=5$ (c3) $t=20$ Figure 7: The dynamics of a particle initially localized at the site $(x,y,z)=(1,1,16)$ for a system of $16\times 16\times 16$ sites with full open boundary conditions and fine tuned $\phi=\pi/2$. The squared wave function at different times is reflected in the opacity of the plotted dots. Different colors indicate time. (a) Without defect. (b) In the presence of a potential defect of energy $\Delta=3\pi$ at the two sites marked by the green tube. Additionally, we plot the squared wave function of one hinge mode. (c) Snapshots of the time evolution in the presence of the defect at different times. (a) without defect (b) with defect (c1) $t=3$ (c2) $t=5$ (c3) $t=20$ Figure 8: As Fig. 7, but for $\phi=0.9\times\pi/2$, away from fine tuning. Although the above discussion is based on ballistic trajectories at the fine- tuned driving parameter $\phi=\pi/2$, we expect the chirality of the hinge modes to be robust also against perturabtions and tuning away from $\phi=\pi/2$. This applies especially the hinge states with larger chiral velocities, which are situated closer to the hinge than those with smaller chiral velocities, and are spatially well separated from counter-propagating modes at the opposite hinge. The chiral hinge modes persist for a rather wide range of $\phi$ beyond $\phi=\pi/2$. This can be observed from the example of $\phi=\pi/3$ ($33.3\%$ detuning, half-way across the Weyl phase transition) displayed in Fig. 2 column (2) around $k_{z}=\pi/2$ and $E=0$. We can see that the hinge modes at smaller distances $M$ from the hinge still preserve their chirality. In turn, hinge modes with larger $M$ that are closer to the sample center, are gradually mixed with modes of opposite chirality and become bulk modes when $\phi$ deviates away from $\phi=\pi/2$. We have also considered the exemplary eigenstates for a cube geometry with full open boundary conditions along all Cartesian axes $x$, $y$ and $z$ presented in column (3) of Fig. 2 showing that for $\phi=\pi/2$ and $\phi=\pi/3$ the chiral modes at different hinges are joined to form a closed loop respecting inversion symmetry of the system [see also Fig. 7 (a)]. The six hinges not participating in this closed loop do not carry hinge modes, since their two boundary planes are not connected along the diagonal direction $\bm{d}$. Meanwhile, non-hinge modes, representing the bulk dynamics all center along the cubic diagonal [column (3) of Fig. 2]. To further confirm the robustness of the hinge states, in Figs. 7 and 8 we simulate the dynamics of a particle in the presence of a defect for a system with open boundary conditions in all three directions, corresponding to column (3) of Fig. 2. Figure 7 illustrates the dynamics of a particle initially located at a corner $(x,y,z)=(1,1,16)$ of the system, where two transporting hinges intersect each other, (a) for the fine tuned situation without defect and (b) in the presence of a strong potential offset of $\Delta=3\pi$ on two neighboring hinge sites (indicated by a green tube) at $(x,y,z)=(1,1,8),(1,1,9)$, respectively. The corresponding plots for non-fine- tuned driving with $\phi=0.9(\pi/2)$ are presented in Fig. 8. We find that despite this strong defect the chiral nature of the hinge modes ensures that no backscattering occurs at the defect and the majority of the wave-packets continues to follow chiral trajectories along the hinges. In Figs. 7 (b) and 8 (b) we also plot representative eigenstates of the system with the defect. The eigenstates remain delocalized along the hinge, with only a small distortion compared to the situation without defect shown in column (3) of Fig. 2. The fact that the defect does not induce scattering away from the hinge (modes) can be seen also from Fig. 9. It shows the time-evolved state after 100 driving cycles for the non-fine-tuned system ($\phi=0.9\times\pi/2$) both without defect (a) and with defect (b). Very similar distributions are also found after even longer evolution, e.g. over 1000 driving cycles; the densities are, thus, representative for late-time states in the limit $t\to\infty$. (a) $\Delta=0$ (b) $\Delta=3\pi$ Figure 9: Density distribution of a particle initially localized at the corner site $(x,y,z)=(1,1,16)$ after an evolution over $100$ driving cycles for the non-fine-tuned parameter $\phi=0.9\times\pi/2$, without defect (a) and with a defect (b) [corresponding to the parameters of Fig. 8 (b)]. The densities are representative for late-time states in the limit $t\to\infty$; similar distributions are found also after 1000 driving cycles. ### IV.3 Topological origin It is an interesting question, whether the robust chiral hinge states are a consequence of topological properties of the driven system. However, as we see from column (2) of Fig. 2, the quasi-energies of hinge states are fully mixed with the bulk spectrum, and therefore no traditional topological band theories for gapped or semimetallic systems apply. Here, it is the collaboration of boundary geometry and Floquet driving that generates such topologically protected states. In section IV.1 we have obtained equation (12) describing the evolution over $2M+1$ driving periods of the $M$th hinge state $\left\|M,k_{z},p\right\rangle$ at the fine tunned point. Using this equation one can define the the quasienergy winding number Kitagawa _et al._ (2010) for the $M$th hinge state via the Floquet evolution operator over the $2M+1$ driving periods: $W_{M}=\frac{1}{2\pi i}\int_{0}^{\pi}dk_{z}U_{k_{z},2M+1}^{-1}\partial_{k_{z}}U_{k_{z},2M+1}=1\,,$ (16) where $U_{k_{z},2M+1}=\left\langle M,k_{z},p\right\|\left(U_{F}\right)^{2M+1}\left\|M,k_{z},p\right\rangle=-e^{2ik_{z}}\,$ (17) and $p=0,1,\ldots 2M$. In Eq. (16) the integration over $k_{z}$ extends over one Brillouin zone of width $\pi$, as the distance between two non-equivalent lattice sites equals to $2$ in $z$ direction. A similar procedure can be applied to the opposite hinge at $(x,y)=(L,L)$, where the hinge modes shown in blue in column (2) of Fig. 2 are characterzed by the opposite group velocity and thus the opposite winding number $W_{M}=-1$. The rigorous quantization of the topological invariant $W_{M}$ is associated with fine tuning, $\phi=\pi/2$. However, the spatial separation between hinge modes of opposite chirality allows to preserve their chiral character also away from fine tuning point $\phi=\pi/2$, as one can see in column (2) of Fig. 2. Thus, the formation of bulk states via the mixing of hinge modes of opposite chirality happens mostly in the center of the system, where hinge modes corresponding to large $M$ and small chiral velocities lie close by to their counter propagating partners associated with the opposite hinge. In turn, the states with the largest chiral velocity, which are situated close to the hinge and far away from counter-propagating modes of the opposite hinge, are much less affected by a small detuning. In this way, tuning away from $\phi=\pi/2$ we find a crossover (rather than a topological transition) in which the chiral hinge modes are gradually destroyed, as can be observed in the real-space plots in columns (2) and (3) of Fig. 2. Note that already a small deviation from the fine tuned point $\phi=\pi/2$ destroys the chiral hinge states in a narrow region near $k_{z}=0$ and $E=\pi$, as can be seen from the spectrum shown for $\phi=\pi/3$ in column (2) of Fig. 2. In this spectral surface Fermi arc states are formed, which equally provide definite chiral transport, yet around the surfaces rather than the hinges. Thus a fraction of the chiral hinge states is transformed into Fermi arc surface states in the vicinity of the Weyl points. The latter states extend to a larger and larger spectral area as the detuning increases. steps \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 ---\|---\|---\|---\|---\|---\|--- $\alpha_{00}$ \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 $\alpha_{10}$ \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 $\alpha_{01}$ \| 1 \| 1 \| 0 \| 0 \| 0 \| 1 $\alpha_{11}$ \| 0 \| 1 \| 1 \| 1 \| 0 \| 0 $None$ $None$ $None$ $None$ $None$ $None$ Figure 10: The stepwise modulation of the dimensionless superlattice amplitudes $\alpha_{ab}$ according to the protocol given in the table, gives rise to different dimerizations of the cubic lattice in each driving step, that enables tunneling along the desired bonds. ## V Experimental Realization with ultracold atoms in optical lattices ### V.1 Engineering of the driven lattice Above, we have shown that the proposed modulation of tunnelling gives rise to a variety of phenomena, including the robust creation of a pair of Weyl points, unidirectional bulk transport, chiral Goos-Hänchen-like shifts, and the macroscopic accumulation of chiral hinge modes for open boundary conditions corresponding to a chiral second-order Floquet skin effect. The model itself is, nevertheless, rather simple and its implementation with ultracold atoms in optical lattices can be accomplished using standard experimental techniques. All what is needed is a static cubic host lattice potential of equal depth $V_{0}$ in each Cartesian direction and a superlattice potential, whose amplitudes along various diagonal lattice directions are modulated in a stepwise fashion in time in order to suppress/allow tunneling along the six different bonds specified by our protocol. This can be achieved using the following optical lattice potential: $\displaystyle V(\bm{r})=V_{0}\sum_{\mu=x,y,z}\cos^{2}(2k_{L}r_{\mu})$ $\displaystyle+V_{1}\sum_{a,b=0,1}\alpha_{ab}(t)\cos^{2}k_{L}(x+(-1)^{a}y+(-1)^{b}z)\,,$ (18) where only two of the four modulating lasers $\alpha_{ab}$ with $a,b=0,1$ are turned on in each driving step, as shown in Fig. 10. Such a modulation provides the required six-stage driving of the cubic lattice. Note that a similar modulation has recently been implemented in two dimensions Wintersperger _et al._ (2020). ### V.2 Detection of hinge dynamics To observe the dynamics associated with the hinge modes, one can apply the boxed potential achieved in recent experiments Gaunt _et al._ (2013); Navon _et al._ (2016); Mukherjee _et al._ (2017b); Lopes _et al._ (2017a, b); Eigen _et al._ (2017); Ville _et al._ (2018). There, thin sheets of laser beams penetrate through the quantum gases creating a steep potential barrier. Three pairs of such beams are imposed in a three-dimensional system, creating the sharp “walls” for the box potential while leaving the central part of the gases homogeneous. Essentially, such a potential combined with our lattice driving scheme immediately leads to the particle dynamics described in Fig. 7 and Fig. 8. To take into account realistic experimental situations, two modifications are adopted in our following simulations. First, we consider the effect of a relatively “softer” wall for the box potential with $\displaystyle V_{\text{box}}(\bm{r})=\frac{V_{b}}{2}\sum_{\mu=x,y,z}\left(2+\tanh\frac{r_{\mu}^{(1)}-r_{\mu}}{\xi}+\tanh\frac{r_{\mu}-r_{\mu}^{(2)}}{\xi}\right),$ (19) where the potential ramps up over a finite distance of roughly $4\xi$ near the boundaries $r_{\mu}^{(1,2)}$, see Fig. 11 for instance. The second modification we adopt is that the initial state is not taken to be localized on a single lattice site but described by a gaussian wave packet of finite width, $\displaystyle\psi_{i=(x,y,z)}(t=0)=e^{-[(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}]/2s_{0}^{2}}.$ (20) (a) $\|\psi_{i}(t=0)\|$ (b) Projection to $x$-$y$ (left) and $x$-$z$ (right) (c1) Ideal boundary (c2) $\|\psi_{i}(t=5)\|$ (c3) $\|\psi_{i}(t=10)\|$ (d1) Softer boundary (d2) $\|\psi_{i}(t=5)\|$ (d3) $\|\psi(t=10)\|$ Figure 11: The dynamics of particles in a box potential with different softness of the boundary. The initial density distribution takes a Gaussian profile spreading over several lattice sites. The parameters are $\phi=\pi/2,V_{b}T/6\hbar=7.5\pi$. The initial Gaussian profile has the center $(x_{0},y_{0},z_{0})=(4,4,12)$ and width $s_{0}=0.75$. The results of the dynamics are presented in Fig. 11 (c1)–(c3) and (d1)–(d3), for the ideal sharp boundary (as in Fig. 7) and the realistic softer boundaries in experimental setting respectively. We see that the chiral motion snapshots for the ideal/softer boundaries exhibit qualitatively the same characters, signalling that a softer boundary does not cause significant changes. This is in consistent with the previous simulation showing the robustness of hinge states and their resulting chiral dynamics against local defects in Fig. 7 and Fig. 8. The major difference from previous cases, then, derives from the initial state that overlaps with more than one set of eigenstates near the hinge, each with different group velocities as given in Eq. (15). Finally, we mention that some portion of initial particle distribution would reside within the region with significant changes in $V_{\text{box}}$. That portion of the particles could be permanently confined to the initial hinge due to a mechanism similar to Wannier-Stark localization. However, the majority of the particles are still traveling into the connecting hinges, as shown in Fig. 11 (c3) and (d3). In cold atom experiments, the density profiles are usually detected by taking a certain projection plane, where the integrated (column-averaged) densities are observed. To this end, we point out that the hinge dynamics can be confirmed by observing the density profiles in two perpendicular planes. A schematic plot is given in Fig. 11 (b), corresponding to the dynamics along the hinge $x=y\sim 1$. The density profile taken at $x-y$ plane (i.e. the “top” view) would show a localized distribution at the corner, verifying the particles only locate at $x=y\sim 1$. Meanwhile, the profile at $x-z$ plane (i.e. “side” view) indicates the movement/spreading along $z$. In a more general situation, i.e. at long time limit with all 6 hinges populated as in Fig. 9, additional image projection planes could be exploited. We also mention that a simultaneous implementation of multiple imaging planes have been applied in experiments Lu _et al._ (2020); Wang _et al._ . ### V.3 Detection of Floquet Weyl points Weyl physics has been explored in recent cold atom experiments and theoretical proposals Lu _et al._ (2020); Wang _et al._ ; Zhang _et al._ (2015); Dubček _et al._ (2015); He _et al._ (2016), and also extensively in solid state systems Armitage _et al._ (2018). Here, we discuss a scheme closely related to a recent experiment Ünal _et al._ (2019); Wintersperger _et al._ (2020) detecting the spacetime singularities in anomalous Floquet insulators. (a1) $\Delta^{(0)}$ at $\phi=\pi/8$ (a1) $\Delta^{(0)}$ at $\phi=\pi/3$ (b) Exemplary gap at $\bm{k}_{0}=k_{0}(1,-1,1)$ with $k_{0}=\pi/3-0.2$. Figure 12: Simulation of gap measurements using Stückelberg interferometry. (a1) and (a2) Contours for quasi-energy gap at $E=0$, for $\phi=\pi/8$ and $\phi=\pi/3$ respectively. (b) The gaps at $E\sim 0$ and $E\sim\pi$, and the measured gap which takes the smaller one of the two. First, the band touching at Weyl points can be verified using the Stückelberg interferometry Zenesini _et al._ (2010); Kling _et al._ (2010); Shevchenko _et al._ (2010); Wintersperger _et al._ (2020). Such a method measures the smaller gap for the two bands at $E\sim 0$ and $\pi$, see Ref. Wintersperger _et al._ (2020) for details. Compared with the experiments for insulators, a difference here is that the two bands are, overall, always gapless at $E=0$. That means if a global gap is measured, it will prevent us from gaining information about the gaps or band touching at quasienergy $E=\pi$. But fortunately, there exists a finite region neighboring to $\bm{k}_{0}=\pi/3(1,-1,1)$ where the bands are always gapped at $E=0$ for all $\phi$, see Fig. 12 (a1), (a2) for example. Then a local gap closure can be measured near $\bm{k}_{0}$. The specific measurement for our case can be performed in the following way. An example for $\bm{k}_{0}=(\pi/3-0.2)(1,-1,1)$ is presented in Fig. 12 (b). Let us denote the quasi-energy of the two Floquet bands at $\bm{k}_{0}$ with $E_{\pm}(\bm{k}_{0})=\pm E_{0}$. Here, we use the branch cut along $\pi$ in taking the logarithm of Floquet eigenvalues $e^{-iE_{\pm}(\bm{k}_{0})}$. They have the same magnitudes and opposite signs due to particle-hole and inversion symmetry as explained in Sec. II. Then, the local gap around $E\sim 0$ is $\Delta^{(0)}\equiv 2E_{0}$, while the other gap around the Floquet Brillouin zone boundary $E\sim\pi$ is $\Delta^{(\pi)}\equiv 2\pi-2E_{0}$. Therefore, $\Delta^{(0)}=\Delta^{(\pi)}$ can only occur at $0,\pi$ mod $2\pi$. In experiments, one can start from the high frequency limit ($\phi\rightarrow 0$) where the band width is small compared to $2\pi$ and therefore the measured gap always corresponds to $\Delta^{(0)}$. Slowing down the driving, the gap $\Delta^{(\pi)}$ shrinks while the other gap $\Delta^{(0)}$ expands. At some point, the two gaps coincide with their magnitudes, as shown in Fig. 12 (b). Since it is always the smaller one of $\Delta^{(0)}$ and $\Delta^{(\pi)}$ that will show up in experimental measurement, one will observe a cusp shape of the measured gap, i.e. near $\phi\approx 0.41$ in Fig. 12 (b). One could then imply from the occurrence of the cusp that for $\phi>0.41$, the experimental data starts to reveal $\Delta^{(\pi)}$, whose vanishing at $\phi\approx 0.87$ shows the existence of the Weyl point around $E\sim\pi$. Similar measurements can be performed for $\bm{k}$ slightly deviating from $\bm{k}_{0}$, which will show that at $\phi=0.87$, $\Delta^{(\pi)}$ remains finite, proving that the band closure around $E\sim\pi$ is a point contact. When the designated $\phi$ is slowly approached, one can perform a measurement of the gap at a certain $\bm{k}$. A shortcut for our system is that focusing on momenta along the diagonal $\bm{k}_{0}=k_{0}(1,-1,1)$ is sufficient to determine the Weyl point, as discussed in Sec.III. With the Weyl points determined, one could further apply band tomography Hauke _et al._ (2014); Fläschner _et al._ (2016) method for momentum states surrounding a certain Weyl point in order to determine its charge. Note that one does not need the eigenstate information throughout the whole Brillouin zone as the two bands are gapless in certain regions, except for just an arbitrarily small surface wrapping a Weyl point $\bm{k}^{\text{(Weyl)}}$ determined previously. As shown before, near the Weyl points in our model, there exists a finite region where the two bands are fully gapped in both $E\sim 0$ and $\pi$, which allows for populating eigenstates with bosons at a certain band Fläschner _et al._ (2016); Wintersperger _et al._ (2020). As an example, in Fig. 13 (a) we illustrate the surfaces formed by 6 faces $q_{x,y,z}=\pm 0.5$ of a cube, where $\bm{q}=\bm{k}-\bm{k}^{(\text{Weyl})}$, with $\bm{k}^{(\text{Weyl})}\approx 0.955\times(1,-1,1)$ for $\phi=\pi/3$, as in Fig. 2. From the full information of the Floquet eigenstates $\|u_{n,\bm{k}}\rangle$ given by Eq. (5), the Berry curvature penetrating out of a plane normal to the unit vector $\bm{e}_{\mu}$ can be computed as $\Omega_{\mu}(\bm{k})=\pm i\sum_{\nu\rho}\varepsilon_{\mu\nu\rho}\left(\langle\partial_{k_{\nu}}u_{n,\bm{k}}\|\partial_{k_{\rho}}u_{n,\bm{k}}\rangle\right)$, where $\varepsilon_{\mu\nu\rho}$ is the Levi-Civita symbol, and $\pm$ sign denotes that the unit vector penetrating out of the cube is along $\pm\bm{e}_{\mu}$ directions. Figure 13 shows momentum resolved Berry curvatures in each wrapping surface and their net fluxes $\int_{\text{surf}}d\bm{k}\Omega_{\mu}(\bm{k})$ in that plane. (a) The surfaces wrapping a Weyl point (b1) $\Omega_{z+}$, net 1.284 (b2) $\Omega_{z-}$, net 0.866 (b3) $\Omega_{x+}$, net 1.284 (b4) $\Omega_{x-}$, net 0.866 (b5) $\Omega_{y+}$, net 1.469 (b6) $\Omega_{y-}$, net 0.513 Figure 13: The Berry curvatures for the surfaces wrapping a Weyl point. Adding the net Berry curvatures up we have $2\pi$. ## VI Conclusion In this paper, we have shown that three-dimensional periodically driven lattice systems can show a macroscopic accumulation of chiral hinge modes, when subjected to open boundary conditions. This corresponds to a chiral second-order Floquet skin effect. An intuitive understanding of this effect was given by considering the system at a fine-tuned point of the periodic driving, where the bulk motion can only take place forwards or backwards along a single diagonal direction. As a consequence, for open boundary conditions, particles are reflected back and forth between hinge-sharing surface planes, with a drift along the direction of the hinge being accomplished by chiral Goos-Hänchen-like shifts associated with these reflections. This effect is different from higher-order Floquet topological insulators (HOFTI) not only regarding the underlying mechanism, but also because no macroscopic accumulation of hinge modes takes place in HOFTI. The effect resembles, however, the accumulation boundary modes in non-Hermitian systems, even though here noticeable differences are also found. Namely in the non-Hermitian case the accumulation of boundary modes occurs close to the hinge. This is not possible in our system, since different from the eigenmodes of a non-Hermitian Hamiltonian, the Floquet hinge modes are orthogonal to each other, as they are eigenstates of the unitary Floquet evolution operator $U_{F}$. Thus we find hinge bound modes also at larger distances from the hinge. Another interesting aspect is the competition or interplay between the hinge modes and the emergence of robust Wely points in our system, so the hinge states can co- exist with the Fermi arc surface states. The implementation of the model featuring both the second-order Floquet skin effect and the Weyl physics is straightforward with ultracold atoms in optical superlattices. ## VII Acknowledgment The authors thank E. Anisimovas and F. Nur Ünal for helpful discussions. We acknowledge funding by the European Social Fund under grant No. 09.3.3-LMT-K-712-01-0051 and the Deutsche Forschungsgemeinschaft (DFG) via the Research Unit FOR 2414 under Project No. 277974659. ## Appendix A Evolution operator and quasienergies along the diagonal ### A.1 Evolution operator In the bulk the stroboscopic evolution operator $U_{F}$ is generally given by Eqs. (3)-(4) in the main text. Let us consider the operator $U_{F}$ for wave- vectors $\mathbf{k}$ along the cubic diagonal direction $\bm{d}=(1,-1,1)$, for which $\bm{k}=\pm k_{0}\bm{d}\quad\text{and, thus,}\quad\|\bm{k}\|=\sqrt{3}k_{0}\,.$ (21) with $k_{0}>0$. In that case Eqs. (3)-(4) simplify to $U_{F}=\left[U\left(\mp k_{0}\right)U\left(\pm k_{0}\right)\right]^{3}\,,$ (22) where $U\left(\mp k_{0}\right)U\left(\pm k_{0}\right)=\left(\cos\phi-\mathrm{i}\tau_{\mp k_{0}}\sin\phi\right)\left(\cos\phi-\mathrm{i}\tau_{\pm k_{0}}\sin\phi\right)\,,$ (23) with $\tau_{\pm k_{0}}=\tau_{1}\cos{k_{0}}\pm\tau_{2}\sin{k_{0}}$ and Pauli matrices $\tau_{1,2,3}$ for the sublattice freedom. Explicitly one, thus, has $U\left(\mp k_{0}\right)U\left(\pm k_{0}\right)=\left[\cos^{2}\phi-\sin^{2}\phi\cos\left(2k_{0}\right)\right]-\mathrm{i}d\,,$ (24) with $d=\sin^{2}\phi\cos\left(2k_{0}\right)\tau_{3}-2\mathrm{i}\cos\phi\sin\phi\cos k_{0}\tau_{1}.$ (25) ### A.2 Quasi-energies The evolution operator $U_{F}=e^{-\mathrm{i}H_{F}}$ defines the quasienergies representing the eigenvalues of the of tthe Floquet Hamiltonian $H_{F}$, which describes the stroboscopic time evolution in multiples of the driving period $T=1$. Using Eqs. (22) and (24) for the evolution operator, one arrives at the following equation for the quasi-energies $E_{\mathbf{k}}$ $\cos\left(E_{\mathbf{k}}/3\right)=\cos^{2}\phi-\sin^{2}\phi\cos\left(2k_{0}\right)\,.$ (26) This provides the dispersion (modulo $2\pi$) along the diagonal $k_{x}=-k_{y}=k_{z}=k_{0}$ $E_{k_{0}\bm{d},\gamma}=3\gamma\arccos\left[\cos^{2}\phi-\sin^{2}\phi\cos\left(2k_{0}\right)\right],\,\,\mathrm{with}\,\,\gamma=\pm 1\,.$ (27) In particular, quasienergies $E_{k_{0}\bm{d},\gamma}=\pi$ (modulo $2\pi$) correspond to $\cos^{2}\phi-\sin^{2}\phi\cos\left(2k_{0}\right)=1/2\,,$ (28) and thus $\cos\left(2k_{0}\right)=\frac{1/2-\sin^{2}\phi}{\sin^{2}\phi}\,,$ (29) giving (modulo $\pi$) $k_{0}=\left(1/2\right)\arccos\left[\left(1/2-\sin^{2}\phi\right)/\sin^{2}\phi\right]\,.$ (30) At the fine tunned point ($\phi=\pi/2$) the condition Eq.(28) reduces to $\cos\left(2k_{0}\right)=-1/2\,,\quad\mathrm{giving}\quad k_{0}=\pi/3\,.$ (31) On the other hand, at $\phi=\pi/6$ one has $\sin^{2}\phi=1/4$, so that $\cos\left(2k_{0}\right)=1,\,\quad\mathrm{giving}\quad k_{0}=0\,.$ (32) In this way, two band touching points are formed at quasienergy $\pi$ for $\pi/6<\phi<\pi/2$, as well as for $\pi/2<\phi<5\pi/6$ (beyond the fine tuning point at $\phi=\pi/2$). By taking $\phi<\pi/6$ or $\phi>5\pi/6$, Eq.(28) can no longer be fulfilled, so a band gap is formed at quasienergy $\pi$. ## Appendix B Stroboscopic hinge motion at fine tuning In this Appendix we give a detailed description of the stroboscopic real-space dynamics of the system at fine tuning, $\phi=\pi/2$, giving rise to chiral hinge-bound Floquet modes. We will consider the hinge that is shared by the two surface planes oriented in the $-x$ and $-y$ direction, which is parallel to the $z$-axis. The projection of the particle’s trajectory in the $xy$ plane is illustrated in Figs. 5 and 6. A particle of sublattice $s=+1,-1\equiv A,B$ is translated by $2s\bm{d}$ during each driving cycle, provided $x+2s\geq 1$ and $y-2s\geq 1$ to ensure it does not hit any the boundary plane. In that case the state-vector $\left\|s,x,y,z\right\rangle$ transforms according to the following rule after a single driving period: $U_{F}\left\|s,x,y,z\right\rangle=-\left\|s,x+2s,y-2s,z+2s\right\rangle\,.$ (33) The particle thus propagates with a stroboscopic velocity $\mathbf{v}=2s\left(1,-1,1\right)$ in opposite directions $s=\pm 1$ for different sublattices $A$ and $B$. Suppose initially the particle occupies a site of the sublattice $B$ at the boundary $y=1$ situated $M$ sites away from the hinge ($x=M+1$) with odd $M+z$, so that $s=B=-1$. The corresponding initial state vector is $\left\|s,M+1,1,z\right\rangle\equiv\left\|B,M+1,1,z\right\rangle$. The subsequent stroboscopic trajectory projected to the $xy$ plane is shown in Fig. 5 for $M=4$ and in Fig. 6 for $M=5$. Generally it takes $\left(2M+1\right)$ driving periods for the system to return to its initial state $\left\|M+1,1,z\right\rangle$. To see this, consider the stroboscopic evolution of the particle with an even $M>2$ and odd $z$. The stroboscopic motion of the particle then splits into four bulk and four boundary segments illustrated in Fig. 5 for $M=4$. During the first $M/2$ driving periods the particle undergoes the bulk ballistic motion along the sites of the $B$ sublattice, and the state vector transforms as $\left\|B,M+1,1,z\right\rangle\rightarrow\left\|B,1,M+1,z-M\right\rangle$. Subsequently the particle is reflected from the plane $x=1$ to a site of the sublattice $A$ situated closer to the hinge, $\left\|B,1,M+1,z-M\right\rangle\rightarrow\left\|A,2,M-1,z+2-M\right\rangle$, as shown in Fig. 1(c) of the main text. During the next $M/2-1$ driving periods the particle propagates ballistically along the sites of the $A$ sublattices, giving $\left\|A,2,M-1,z+2-M\right\rangle\rightarrow\left\|A,M,1,z\right\rangle$. The subsequent reflection from the plane $y=1$ brings the particle to a site of the $B$ sublattice situated further away to from the hinge, $\left\|A,M,1,z\right\rangle\rightarrow\left\|B,M,2,z-2\right\rangle$. The evolution takes place in the similar way during final four segments. Explicitly the full stroboscopic dynamics is given by: $\displaystyle\left(U_{F}\right)^{M/2}\left\|B,M+1,1,z\right\rangle=\left(-1\right)^{M/2}\left\|B,1,M+1,z-M\right\rangle\,,$ (34) $\displaystyle U_{F}\left\|B,1,M+1,z-M\right\rangle=-\mathrm{i}\left\|A,2,M-1,z+2-M\right\rangle\,,$ (35) $\displaystyle\left(U_{F}\right)^{M/2-1}(-\mathrm{i})\left\|A,2,M-1,z+2-M\right\rangle$ $\displaystyle=\mathrm{i}\left(-1\right)^{M/2}\left\|A,M,1,z\right\rangle\,,$ (36) $\displaystyle U_{F}\mathrm{i}\left\|A,M,1,z\right\rangle=\left\|B,M,2,z-2\right\rangle\,,$ (37) $\displaystyle\left(U_{F}\right)^{M/2-1}\left\|B,M,2,z-2\right\rangle=-\left(-1\right)^{M/2}\left\|B,2,M,z-M\right\rangle\,,$ (38) $\displaystyle U_{F}(-1)\left\|B,2,M,z-M\right\rangle=\mathrm{i}\left\|A,1,M,z-M\right\rangle\,,$ (39) $\displaystyle\left(U_{F}\right)^{M/2-1}\mathrm{i}\left\|A,1,M,z-M\right\rangle$ $\displaystyle=(-\mathrm{i})\left(-1\right)^{M/2}\left\|A,M-1,2,z-2\right\rangle\,,$ (40) $\displaystyle U_{F}(-\mathrm{i})\left\|A,M-1,2,z-2\right\rangle=-\left\|B,M+1,1,z-2\right\rangle\,,$ (41) In this way, after $\left(2M+1\right)$ driving periods the particle returns back to the intial position $\left(M+1,1\right)$ in the $xy$ plane and is shifted by $2$ lattice units to an equivalent point of the sublattice $B$ in the direction opposite to the $z$ axis. The same holds for the initial state vector $\left\|B,M+1,1,z\right\rangle$ characterized by an odd $M$ and even $z$ (see Fig. 6 for $M=5$), as well as for a particle situated closer to the hinge ($0\leq M\leq 3$) where the reflections can take place simultaneously from both planes $x=1$ and $y=1$, as illustrated in Fig. 1(d) in the main text. Thus one can write for any distance $M\geq 0$ from the hinge: $\left(U_{F}\right)^{2M+1}\left\|B,M+1,1,z\right\rangle=-\left\|B,M+1,1,z-2\right\rangle\,,$ (42) This means the particle propagates along the hinge in the $-z$ direction with the stroboscopic velocity equal to $-2/\left(2M+1\right)$. The relations analogous to Eq.(42) hold for all $2M+1$ states of the stroboscopic sequence featured in Eqs. (35)-(41) The origin of such chiral hinge states can be explained as follows. The particle in the sublattice $B$ is reflected to a site of the $A$ sublattice situated closer to the hinge, whereas the particle in the sublattice A is reflected to a site of the sublattice $B$ situated further away from the hinge, as one can see in Figs. 5 and 6, as well as in Eqs. (35), (37), (39), (41). Consequently the number of $B$ sites visited over all $2M+1$ driving periods ($M+1$) exceeds the corresponding number of $A$ sites ($M$). The four reflections do not yield any total shift of the particle in the $z$ direction. On the other hand, the ballistic motion between sites the same sublattice $B$ ($A$) is accompanied by a shift by $2$ lattice sites in the $z$ ($-z$) direction for each driving period. This leads to the overall shift of the particle to an equivalent site in the $-z$ direction is due to the difference in the number of the visited $B$ and A sites after $2M+1$ driving periods. ## Appendix C Non-Hermitian Hamiltonian corresponding to stroboscopic operator Recently it was suggested Bessho and Sato (2020) to associate a non-Hermitian Hamiltonian $H_{NH}\left(\mathbf{k}\right)$ to the momentum space stroboscopic evolution operator $\mathrm{i}U_{F}\left(\mathbf{k}\right)$. Let us consider such a non-Hermitian Hamiltonian for our 3D periodically driven lattice $H_{NH}=\mathrm{i}U_{F}\,.$ (43) For the fine tunned driving ($\phi=\pi/2$) the bulk stroboscopic evolution operator corresponds to a non-Hermitian Hamiltonian describing a unidirectional transfer between the lattice sites along the diagonal $\mathbf{d}=(1,-1,1)$ and in the opposite direction $-\mathbf{d}$ for the sublattices $A$ and $B$, respectively: $H_{NH}^{bulk}=-\mathrm{i}\sum_{\mathbf{r}_{A}}\left\|A,\mathbf{r}_{A}+2\mathbf{d}\right\rangle\left\langle A,\mathbf{r}_{A}\right\|-\mathrm{i}\sum_{\mathbf{r}_{B}}\left\|B,\mathbf{r}_{B}-2\mathbf{d},\right\rangle\left\langle B,\mathbf{r}_{B}\right\|\,.$ (44) The open boundary conditions for the hinge corresponding to $x\geq 1$ and $y\geq 1$ are obtained by imposing a constraint on the state-vectors entering the real space non-Hermitian Hamiltonian (44) $\left\|s,\mathbf{r}_{s}\right\rangle=0\quad\mathrm{for}\quad\mathbf{r}_{s}\cdot\mathrm{e}_{x,y}\leq 0,\,\,\mathrm{with}\,\,s=A,B\,.$ (45) The bulk non-Hermitian Hamiltonian (44) supplied with the open boundary conditions (45) describes a unidirectional coupling between unconnected linear chain of the $A$ or $B$ sites terminating at the hinge planes. The eigenstates of each linear chain represent non-Hermitian skin modes which are localized at one end of the chain depending on the direction of asymmetric hopping Bergholtz _et al._ (2020); Ashida _et al._ (2020). In the present situation such skin modes would be trivially localised on different planes of the hinge for the chains comprising different sublattice sites $A$ or $B$, and no chiral motion is obtained along the hinge. Yet the open boundary conditions (45) are not sufficient to properly represent the boundary behavior of a particle in our periodically driven lattice. In fact, bulk non-Hermitian Hamiltonian (44) supplied with the boundary conditions (45) is no longer a unitary operator. Thus one can not associate such an non-Hermitian operator with the evolution operator, in contradiction with Eq. (43). 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# The buildup of the intracluster light of Abell 85 as seen by Subaru’s Hyper Suprime-Cam Mireia Montes School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Sarah Brough School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Matt S. Owers Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia Astronomy, Astrophysics and Astrophotonics Research Centre, Macquarie University, Sydney, NSW 2109, Australia Giulia Santucci School of Physics, University of New South Wales, Sydney, NSW 2052, Australia ###### Abstract The study of low surface brightness light in large, deep imaging surveys is still uncharted territory as automated data reduction pipelines over-subtract or eliminate this light. Using archival data of the Abell 85 cluster of galaxies taken with Hyper Suprime-Cam on the Subaru Telescope, we show that using careful data processing can unveil the diffuse light within the cluster, the intracluster light. We reach surface brightness limits of $\mu_{g}^{limit}$(3$\sigma$, $10\arcsec\times 10\arcsec$) = 30.9 mag/arcsec2, and $\mu_{i}^{limit}$(3$\sigma$, $10\arcsec\times 10\arcsec$) = 29.7 mag/arcsec2. We measured the radial surface brightness profiles of the brightest cluster galaxy out to the intracluster light (radius $\sim 215$ kpc), for the $g$ and $i$ bands. We found that both the surface brightness and the color profiles become shallower beyond $\sim 75$ kpc suggesting that a distinct component, the intracluster light, starts to dominate at that radius. The color of the profile at $\sim 100$ kpc suggests that the buildup of the intracluster light of Abell 85 occurs by the stripping of massive ($\sim 10^{10}M_{\odot}$) satellites. The measured fraction of this light ranges from $8\%$ to $30\%$ in $g$, depending on the definition of intracluster light chosen. galaxies: clusters: individual (A85) — galaxies: elliptical and lenticular, cD — galaxies: halos — galaxies: evolution — techniques: image processing ††journal: ApJ††facilities: Subaru Telescope††software: Astropy (The Astropy Collaboration et al., 2018), SExtractor v2.19.5 (Bertin & Arnouts, 1996), SWarp v2.38.0 (Bertin et al., 2002), SCAMP v2.0.4 (Bertin, 2006), photutils v0.7.2 (Bradley et al., 2019), pillow (van Kemenade et al., 2020), ellipse (Jedrzejewski, 1987), GALFIT (Peng et al., 2002) ## 1 Introduction Deep observations of galaxy clusters have revealed the existence of a diffuse glow produced by stars not bound to any individual galaxy; the intracluster light (ICL, see Mihos, 2016; Montes, 2019, for reviews). As the by-product of galaxy interactions, the ICL forms a fossil record of all the dynamical processes the system has undergone and provides a holistic view of the interaction history of the cluster (e.g., Merritt, 1984). The ICL is key to understanding how brightest cluster galaxies (BCGs) grow with time. Their formation and evolution have been predicted to be rather different than satellite galaxies (e.g., De Lucia & Blaizot, 2007). The innermost regions of these massive galaxies appear to have formed the majority of their stars at high redshift and on short timescales (e.g., Thomas et al., 2005) whereas their outer parts are likely assembled as a consequence of multiple minor merging more recently (e.g., Trujillo et al., 2011). As the ICL is often found to be more concentrated around the BCG (e.g., Mihos et al., 2005), it implies that the growth of both components, BCG and ICL, are connected. In addition, simulations of the growth rate of BCGs better agree with observations if formation of ICL is included (e.g., Conroy et al., 2007; Contini et al., 2019; Spavone et al., 2020). A useful tool to characterize the ICL is the study of its stellar populations, as they reflect the properties of the galaxies from which the ICL accreted its stars. Knowing the stellar populations of the ICL in clusters allow us to infer the mechanisms at play in the formation of this component, and therefore how (and when) the assembly history of these clusters occurred. Observations show clear radial gradients in colors indicating radial gradients in metallicity (e.g., Zibetti et al., 2005; Iodice et al., 2017; Mihos et al., 2017; DeMaio et al., 2015, 2018) and, in some cases, age (e.g., Montes & Trujillo, 2014; Morishita et al., 2017; Montes & Trujillo, 2018). These studies point to the tidal stripping of massive satellites (a few $\times 10^{10}M_{\odot}$; Montes & Trujillo, 2014, 2018) as the dominant process of ICL formation for massive clusters ($\sim 10^{15}M_{\odot}$)111Diffuse light has also been detected and studied in groups of galaxies (e.g., Da Rocha & Mendes de Oliveira, 2005; DeMaio et al., 2018; Iodice et al., 2020) but the main mechanism of the formation of this intragroup light appears to differ from that for clusters (e.g., Spavone et al., 2020). A clear limitation in the study of the ICL is the lack of statistically significant samples with the required depth ($\mu_{V}>26.5$ mag/arcsec2; Rudick et al., 2006). Unfortunately, the long exposure times required for these observations mean that very few clusters have been studied, so far. To date, studies have only analysed small samples (1-20 clusters; Krick & Bernstein, 2007; Montes & Trujillo, 2014, 2018; Burke et al., 2015; Jiménez- Teja et al., 2018) or employed stacking of many clusters to obtain a coarse measurement (e.g. Zibetti et al., 2005; Zhang et al., 2019). This is changing with the next generation of surveys using state-of-the-art cameras that will be able to reach unprecedented depths over large areas in the sky. An example is the Hyper Suprime-Cam (HSC; Miyazaki et al., 2018) on the 8.2-meter Subaru Telescope. This camera is well suited to not only provide the wide field-of-view necessary to observe nearby clusters but also the time efficiency of a large telescope being able to reach ICL depths in short exposure times. The HSC is currently carrying out the HSC Subaru Strategic Program (HSC-SSP), a survey of 1400 deg2 in five different bands ($grizy$) plus four narrow filters. The depth and area of this survey will provide the large numbers of galaxy clusters necessary to _deepen_ our knowledge of the formation of the ICL (Aihara et al., 2019). However, ICL studies need very accurate data processing. The data reduction of HSC data is undertaken with the HSC pipeline (Bosch et al., 2018), a custom version of the LSST222The Vera C. Rubin Observatory Legacy Survey for Space and Time. pipeline. The sky subtraction algorithm in the HSC-SSP data release 1 over-subtracts extended halos of bright objects making it almost impossible to study nearby or very extended objects (Aihara et al., 2018)333This issue was improved in the data release 2 (Aihara et al., 2019) but not completely resolved.. In addition, ICL studies are susceptible to biases due to flat- field inaccuracies and the scattered light from bright stars. In this work, we use archival HSC images of the cluster Abell 85 (A85) to test a dedicated data processing technique for low surface brightness science and study the diffuse light of this cluster out to $\sim 215$ kpc. The main properties of A85 are listed in Table 1. A85 is a rich cluster of galaxies ($\sim 800$ spectrocopically confirmed galaxies within $2R_{200}$, Owers et al., 2017; Habas et al., 2018) hosting a massive BCG (M${}_{}\sim 3\times 10^{12}M_{\odot}$, Mehrgan et al. 2019). Many studies have shown that this cluster is slowly accreting material through several ongoing mergers with, at least, two subclusters or groups of galaxies (Bravo-Alfaro et al., 2009; Ichinohe et al., 2015; Owers et al., 2017). In addition, models of the X-ray temperature across the cluster support the picture that A85 has undergone several small mergers in the past few billion years (Durret et al., 2005; Ichinohe et al., 2015). This cluster provides an ideal target for a pilot study of the ICL using HSC and dedicated data processing techniques for low surface brightness science. Studying the properties of the ICL in this cluster will inform us of the ongoing processes shaping this cluster, and its BCG. Throughout this work, we adopt a standard cosmological model with the following parameters: $H_{0}=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$. All magnitudes are in the AB magnitude system. Table 1: Main properties of A85. Redshift, mass and radius are taken from Owers et al. (2017). Name \| RA \| DEC \| z \| Distance \| Angular scale \| Virial M200 \| R200 ---\|---\|---\|---\|---\|---\|---\|--- \| [deg] \| [deg] \| \| [Mpc] \| [kpc/arcsec] \| [10${}^{14}M_{\odot}$] \| [Mpc] Abell 85 \| 10.458750 \| -9.301944 \| 0.0549 \| 245 \| 1.068 \| 17.0$\pm$1.3 \| $2.42$ ## 2 Data HSC is a 1.77 $\deg^{2}$ imaging camera on the Subaru Telescope operated by the National Astronomical Observatory of Japan (NAOJ) on the summit of Maunakea in Hawaii. It consists of 116 CCDs (104 science CCDs, 4 guide and 8 focus sensors) with a resolution of $0\farcs 168$/pixel. For this work, we have used archival data. A85 was observed on the 2014-09-24 (Proposal ID: o14171). The science data consist of 9 frames in both the HSC-$G$ ($g$) and the HSC-$I$ ($i$) bands. The exposure times for each frame are 200s and 240s, respectively. The observational strategy consisted of a dithering pattern of 9 positions around the center of the cluster. The offsets of $1\farcm 367$ are enough to fill the gaps of the camera mosaic. All the data used in this work was downloaded from the Subaru-Mitaka-Okayama-Kiso Archive (SMOKA; Baba et al., 2002)444https://smoka.nao.ac.jp/. ### 2.1 Custom-made processing Exploring the ICL of clusters of galaxies is difficult as it is not only faint, but also extended. This means that in order to avoid biases when measuring the ICL caused by inhomogeneities in the images such as gradients and oversubtraction, see Mihos 2019 for a detailed description of the possible biases in low surface brightness imaging), the images must have a flat background and the background subtraction should be performed carefully so as not to eliminate this light. At the time we started this project, the data reduced with the HSC pipeline (Bosch et al., 2018) for the DR1 of the HSC-SSP survey (Aihara et al., 2018), showed significant oversubtraction around bright sources caused by a background estimation using a relatively small mesh size ($128\times 128$ pix${}^{2}=\,21\arcsec\times 21\arcsec$). Because the cluster of interest is at low redshift (i.e, extended in the sky, $R_{200}=2.42$ Mpc $=0\fdg 63$; Owers et al. 2017), this oversubtraction would likely eliminate the ICL. For this reason, we developed a custom-made process in order to reduce the data, preserving low surface brightness light, i.e. the extended and faint ICL. The code is mainly written in Python and uses Astropy (The Astropy Collaboration et al., 2018) and astronomical software such as SExtractor, SWarp, and SCAMP (Bertin & Arnouts, 1996; Bertin et al., 2002; Bertin, 2006). The steps followed here to reduce the HSC images, after the individual CCD processing, are similar to those performed in Trujillo & Fliri (2016). For this work, as the images are dithered around the BCG of the cluster, we focus only on the innermost $40$ CCDs of the camera to reduce inaccuracies due to the non-uniform illumination of the CCDs. This corresponds to a radius of $\sim 0\fdg 42$ ($1.6$ Mpc) around the BCG. These are the main steps we conduct to process the data: 1. 1. Derivation of the calibration files (bias and dark) 2. 2. Individual CCD processing and assembly 3. 3. Flat-field derivation using science images and correction 4. 4. Camera mosaic with a careful determination of the sky background 5. 5. Mosaic coaddition and final image calibration. In the following sections, we describe in detail how these steps are performed. The HSC CCDs are full-depletion Hamamatsu CCDs (Miyazaki et al., 2012). The individual raw images are $2144\times 4241$, divided into four science channels of $512\times 4096$ pixels along with pre-scan, overscan and non- science regions of each of those channel.555As described in: https://hsc.mtk.nao.ac.jp/pipedoc/pipedoc_4_e/e_hsc/index.html#e-hsc Therefore, the next steps to calibrate each CCD have to be performed in each channel separately before the CCD is assembled. #### 2.1.1 Calibration files Bias frames were taken the same night as part of the observing program of A85. They consist of $15$ bias frames per CCD. However, there were only $2$ dark frames taken the same night. In order to derive a robust master dark for the images, we also downloaded the darks taken on adjacent nights; the 2014-09-22, 2014-09-23, and 2014-09-25, to a total of $10$ dark frames per CCD. The master bias and dark frames were created as the sigma-clipped ($3\sigma$) median for each of the channels of each of the CCDs. #### 2.1.2 Individual CCD processing In this step, we perform the processing and assembly of each CCD for each of the frames to produce a calibrated image. Each of the CCDs is processed independently. For each channel, we compute a median overscan using the corresponding overscan regions, and correct for overscan, dark and bias (as derived in Sec. 2.1.1). We also correct each channel for nonlinearity as done in the HSC pipeline (Bosch et al., 2018) by applying a polynomial with coefficients determined per amplifier. Before assembling the final CCD image, we applied the gains for each of the channels (provided in the headers). The final size of the assembled CCD is $2048\times 4176$ pixels. #### 2.1.3 Flat-field correction An accurate estimation of the flat-field correction is crucial to achieving the goals of this study. Dome flats are not suitable for our goals due to inhomogeneities in the dome illumination that can result in gradients across the image (e.g., Trujillo & Fliri, 2016). Consequently, our flat-field correction should be based on science exposures and, ideally, they should be the same science exposures used in this work. However, the images of the cluster are not appropriate for two reasons: 1) there are only 9 different exposures meaning that the resulting flats will be very noisy and 2) the offsets of the dithering pattern are not large enough for this purpose, so the galaxies of the cluster occupy roughly the same physical area of the CCD in all the exposures. The latter means that there will not be enough pixels to average in those regions in the resulting flats. To address this, we downloaded images of random fields from the HSC-SSP Wide survey taken on adjacent nights to the A85 observations in order to derive the most reliable flat-field correction possible. Using the SSP-Wide survey reduces the probability of an extended object in the same physical space of the CCD in all exposures. For the $g$ band the images were taken on 2014-10-01 (31), 2014-11-18 (9), 2014-11-25 (9), a total of 49 frames per CCD. The exposure times are $150$s per frame. For the $i$ band, taking the images from the adjacent nights resulted in substructure remaining after the flat-field correction. This was found to be due to differences in the rotation angle of the instrument in the different set of images (see Appendix A for more details). Therefore, the final images used were taken on 2014-03-27, 2014-09-17, 2015-01-22, 2015-07-11, 2015-07-20, 2014-09-22, 40 images per CCD in total. The exposure times are $200$s for each frame. The assembled CCDs show a steep gradient across the detector that can cause detection algorithms to mistakenly detect and mask regions of the image that do not correspond to any source. To account for this, the construction of the flats was done in two steps. We first derived a rough flat or _preflat_. These were derived by stacking the HSC-SSP science images using a median of the normalized images for each CCD, without any masking, to make a CCD flat. Each of the images that went into the flats was visually inspected to eliminate those presenting very bright, saturated stars and extended objects that might introduce errors in the derived flat-field frames. First, we normalized each CCD image to one, using a region of $1000\times 1000$ pixels located at the same position in the middle of each CCD. The _preflats_ were created as the sigma-clipped ($3\sigma$) median of the normalized images. Once these _preflats_ are derived, we use them to correct the assembled CCD images. We use these _preflat_ -corrected CCD images to build an object mask with SExtractor (Bertin & Arnouts, 1996). The settings used for the detection are optimized for faint object detection so to better mask faint sources. Again, for each CCD the masked and normalized images are combined to create the final flats. Finally, each CCD is divided by the corresponding final flat. In Appendix B, we show a region of our $i$ band images where the improvement of using the flats with the same rotation as the science images can be seen. #### 2.1.4 Astrometric calibration and frame assembly Before combining the CCDs into frames, we need to refine the rough astrometry that the HSC camera provides. To do that, we use SCAMP (Bertin, 2006) to put the science images into a common astrometric solution. SCAMP reads SExtractor catalogs and computes astrometric solutions for each individual CCD. The reference used is the stars of the SDSS DR9 catalogue (Ahn et al., 2012) in our field of view. The number of stars used in each mosaic frame (40 CCDs) for our astrometric solution is typically around a couple of hundred. After computing the accurate astrometry for each CCD in each frame, we need to make sure the CCDs are levelled before building the frame, i.e. all CCDs in the frame have the same sky counts. For each CCD, we run SExtractor again. We build a mask by using the segmentation map obtained, further expanding the detected objects by 10 pixels. In addition, we masked all bright sources in the CCDs. This includes bright stars to minimize the contamination of their extended halos, large galaxies and $\sim 700\arcsec$ in radius around the BCG. This constant correction is computed as the $3\sigma$-clipped median of the remaining pixels and subtracted from the respective CCDs. After levelling each CCD, we use SWarp (Bertin et al., 2002) to put together the $40$ CCDs from each exposure into single mosaic frames. SWarp resamples the CCDs putting them into a common grid using a LANCZOS3 interpolation function. The result is $9$ mosaic frames for both $g$ and $i$ bands. Figure 1: Image of the cluster A85 in the $g$-band. The area displayed is $52\arcmin\times 52\arcmin$ around the centre of the cluster (RA = 00h42m01.2s, DEC = -09d18m18.9s). Two regions of the cluster are highlighted. Zoom-in A ($390\arcsec\times 350\arcsec$, purple) shows an RGB image of the BCG of A85 where the ICL can be seen. Zoom-in B ($220\arcsec\times 200\arcsec$, light green) shows a satellite galaxy of the cluster. Tidal streams and other signs of interaction are easily seen when the images are zoomed-in on. Three of these features are marked with a blue arrow (one in the zoom-in B). The RGB images are a combination of the $g$, and $i$ bands whereas a black and white $g$ image is used for the main image. North is up and East is left. #### 2.1.5 Sky subtraction Sky subtraction is one of the most important steps for reducing low surface brightness data as if done incorrectly it can introduce unwanted gradients or remove partially or entirely the object we want to study. The sky determination and subtraction is done for each of the mosaic frames individually before the final co-addition step. We first masked all sources in the individual mosaics using the segmentation maps provided by SExtractor and further dilated each object by 20 pixels to minimize contamination of the fainter regions of objects that are not included in SExtractor’s segmentation map. Separately, we generously masked all bright sources (stars and galaxies) as well as the gaps between CCDs and created an additional mask to cover the centre of the cluster to avoid contamination of the outer parts of the BCG (as done in Sec. 2.1.4 but now for the full mosaic). Once the mosaic is masked, we distributed $50,000$ boxes of $100\times 100$ pixels randomly through the image and computed the 3-$\sigma$ clipped median of the counts. We subtract this constant sky value from the respective mosaic. In addition, we also fitted a first degree 2D polynomial to the masked mosaics. As the size of the mosaics is larger than the physical extent of the ICL in the images, this ensures the correction of any remaining gradients in the image while preserving the diffuse light in this cluster. This 2D polynomial is then subtracted from the entire mosaic. #### 2.1.6 Image co-addition Once the science mosaics are sky-subtracted and in a common astrometric solution, we use SWarp to co-add the mosaics into a final image. SWarp projects the input images into the output frame and co-adds them in an optimum way. The method used for the geometric resampling is LANCZOS3. The final output is created as the median of the $9$ mosaic frames. Finally, we computed and subtracted a constant sky value from the final co-added images. The final exposure times of the images are 1800s (30 mins) for the $g$ band and 2160s (36 mins) for the $i$ band. The final $g$ band mosaic is shown in Fig. 1. The field of view is $52\arcmin\times 52\arcmin$. In Fig. 1, we also show RGB zoom-in images of two regions of the cluster. Region A shows a postage stamp of $390\arcsec\times 350\arcsec$ around the BCG of A85 (framed in purple) and region B shows a $220\arcsec\times 200\arcsec$ region around a massive galaxy belonging to one of the subclusters that are merging into A85 (Ichinohe et al. 2015; Owers et al. 2017; framed in green). The astrometric calibration is not accurate at the corners of our field of view, likely due to the lack of stars available to perform accurate astrometry there. #### 2.1.7 Photometric calibration The photometric calibration of our images is based on the photometry of non- saturated stars in our field of view in common with the SDSS DR12 catalogue (Alam et al., 2015). For each band, we chose stars within the magnitude range (SDSS ‘psfMag’) 18 to 21 mag, to avoid saturated stars in our mosaics, as seen in Fig. 2, and very faint and noisy sources in SDSS. For our images, we used ‘MAG_PETRO’ which provides an estimate of the total flux of the star. We matched the SDSS DR12 photometric catalogue to ours, multiplying the frames by a factor to make the photometry in both catalogues equal. The typical number of stars that are used for photometric calibration within each individual mosaic image is $\sim 700$ stars. The average dispersion in the photometry for each band is $\sim 0.1$ mag, for both the $g$ and $i$ bands. ### 2.2 Modeling and subtraction of stars The careful modeling and removal of stars in deep images is now a common technique in low surface brightness science (e.g., Slater et al., 2009; Trujillo & Fliri, 2016; Román et al., 2020). This is important in order to minimize the contamination by light scattered by the stars, especially bright stars, in our photometry of the faint ICL. Figure 2: Magnitude as a function of the half-light radius, in pixels, for all detected sources in the image of A85. The selection boxes for the stars used for the core (light green) and intermediate (blue) parts of the PSF are drawn and the selected stars are highlighted, for the $g$ (left panel) and $i$ (right panel) bands. #### 2.2.1 Point spread function derivation A robust and extended characterization of the point spread function (PSF) of the image is crucial to remove the stars in the field of view, in particular bright stars close to the object of interest. For example, Uson et al. (1991) showed that the total amount of diffuse light measured around the BCG of Abell 2029 would be in excess without removing nearby stars (their figure 5). In order to correct for this, we first construct the PSF of our images. Generally, to derive PSFs, we need to use stars with a wide range of brightnesses. The bright, saturated stars are used to characterize the outer parts of the PSF, or wings of the PSF, while fainter stars are used to characterize the core and intermediate parts. The bright stars in Fig. 1 show asymmetries due to internal reflections in the telescope and the non-uniform illumination through it. These asymmetries become more significant further away from the centre of the camera. Given the limited amount of very bright stars in our image (N$\approx 10$), we cannot build a PSF in every position of the camera. Luckily, the object of interest (BCG + ICL) is very close to the centre of the camera, therefore deriving a symmetric PSF to subtract nearby stars is a good approximation in this case. #### 2.2.2 Core and intermediate part of the PSF In order to build the inner parts of the PSF, we followed a similar approach to the one in PSFEx (Bertin, 2011). We first obtain a source catalog using SExtractor. The SExtractor catalog provides the half-light radius (‘FLUX_RADIUS’) and the magnitude (‘MAG_AUTO’) of the detected sources. It also provides the stellarity index ‘CLASS_STAR’ for discerning between stars and galaxies. A ‘CLASS_STAR’ close to 0 means that the object is very likely a galaxy, and 1 that it is a star. We select the objects of the catalog with ‘CLASS_STAR’ greater than 0.98. To minimize the asymmetries that can smear the structure of the PSF, we selected stars only in the inner $40\arcmin\times 40\arcmin$ of the image. Their magnitude and half-light radius distribution is shown in Fig. 2. We selected non-saturated stars (light green box) to derive the core, while brighter and saturated stars (blue box) are used to derive the intermediate parts of the PSF. We obtained the core and intermediate parts of the PSF by stacking the corresponding stars following these steps. First, we cut postage stamps around the selected stars of size $100$ and $500$ pixels2 for the core and intermediate parts, respectively. In order to stack the stars, we need to accurately estimate their center. To do that, we need to mask all sources other than the selected star in the postage stamp. We use SExtractor’s segmentation map for this. Then, we fitted a 2D Moffat model to the masked postage stamp of the star. Once the center of the Moffat is obtained, we re- centered the postage stamp. Second, we normalized each star by measuring a 1-pixel width ring at a radial distance of 15 pixels, avoiding the noisier outer parts (for the core stars) and the central saturated parts (for the intermediate stars). We also subtracted the sky around the stars in a 5 pixel-width ring at a radius of $13\arcsec$ for the core stars and $75\arcsec$ for the intermediate stars666These radii were defined to reach the background in each of the postage stamps, i.e., to not include flux from the star, at SNR$\sim 1$.. Finally, we stacked the normalized stars using a 3-$\sigma$ clipped median. The number of stars that were used for the stacking are 51 and 41, for the core, and 29 and 73, for the intermediate parts for the $g$ and $i$ bands, respectively. #### 2.2.3 Outer parts of the PSF As discussed above, we want a model PSF that is extended enough that we can subtract the wings of stars close to the BCG + ICL system. However, in our field of view there are not enough bright stars to properly derive the outer parts of the PSF. This is also limited by the asymmetries that are more evident as we move away from the center of the image. For that reason, we selected a few very bright stars that are in our field of view and derived their radial profiles. The profiles of these stars look very similar despite the asymmetries, therefore we decided to use the radial profile of the closest bright star ($m_{i}\approx 11$ mag, although saturated) to the center to build the outer part of the PSF. We adopted this methodology as a profile is more resistant to masking residuals or other artifacts that could bias the resulting PSF. It also means that this PSF will be symmetrical (i.e., we lose the spatial information). Note that the center parts of these very bright stars are strongly saturated causing bleeding in the detector, seen as spikes in Figure 1. We do not model these spikes. We followed the same steps as for the core and intermediate parts. First, we cut a postage stamp of $2000$ pix $\times$ $2000$ pix around the star. We masked all sources that are not the selected stars using the segmentation map. In addition, to mask sources that are in the star’s halo, we run SExtractor on an unsharp-masked image (Sofue, 1993) of the postage stamp. The unsharp-masked image was obtained smoothing the stamp by a Gaussian with $\sigma=30$ pix, which was then subtracted from the original. We combined both segmentation maps, from the original and the unsharp-masked image, to create the final mask. We re-centered the postage stamp by fitting a Moffat2D model and shifting it to the new center given by the fit. In this case, the sky is subtracted at a distance of $325\arcsec$ to avoid contamination from the star flux (SNR$\sim 1$). Then, we measured the radial profile of the star. After deriving the radial profile of the star, we build the 2D outer PSF by assigning the value of each point of the profile to its corresponding radial distance ring around the centre. We then convolved the whole stamp with a $\sigma=1$ pix Gaussian to smooth the abrupt changes at each given radius. This smoothing does not change the shape of the profile of the star. Finally, we extend this outer part with a power-law in a similar way to Montes & Trujillo (2018). This last step is to minimise any sky subtraction issues in the outer parts of the star. We fit a power-law to the PSF image between $95\arcsec$ to $141\arcsec$, for the $g$ band, and $221\arcsec$ to $289\arcsec$, in the case of the $i$ band777The bending in the profile of the $i$ band at $200\arcsec$ is seen in the bright stars’ postage stamps as well as in their profiles and is not a consequence of the sky subtraction.. This power-law fit was used to extrapolate the outer regions of the PSF to a radius of $420\arcsec$. Figure 3: Radial PSF profile in the HSC $g$ band in black. The different shaded regions correspond to the four different parts derived in Sec. 2.2.2 and 2.2.3 from which the final PSF was constructed. The colored lines are the individual radial profiles of the four different parts. The vertical lines that divide the shaded regions indicate the radii at which these different parts were joined. Figure 4: Same as Fig. 3 but for the HSC $i$ band. #### 2.2.4 Connecting the different parts of the PSF Once we derived the four different parts described above, we constructed our final PSF. We follow a similar approach to Infante-Sainz et al. (2020). We use the radial profile of the bright star derived above as a reference for the connection and multiply the other profiles by a factor so they match the profile of the bright star at a given radius. The radius at which these connections are made change depending on the band. Fig. 3 and Fig. 4 show the final PSF profiles (black thick line) for the $g$ and $i$ bands, respectively. The shaded regions indicate the four different parts used to construct the final PSF derived in Sec. 2.2.2 and 2.2.3. The profile of the bright star, which was used for building the outer part of the PSF is labelled as _Outer 1_ in orange. The power-law extrapolation to the bright star profile is _Outer 2_ , in magenta. The core and intermediate parts are in teal and blue, respectively. We also show the different individual profiles used to construct the final PSF, in their respective colors. The radii where the connections were made for each band and each of the different parts are indicated by the vertical lines in the plots, in teal (connection between core and intermediate part), orange (between intermediate and the bright star profile) and magenta (between the bright star profile and the power-law extension). The total flux of the final PSFs ($g$ and $i$) is normalized to 1. #### 2.2.5 Star subtraction Figure 5: Example of the star subtraction and masking process in a $500\arcsec\times 500\arcsec$ region of the image of A85 for the $i$ band. The images shown have the same contrast. To subtract the stars in our images, we follow similar steps to Román et al. (2020). We started by building a catalogue of the positions of visually- selected bright stars. There are two key aspects when fitting these stars: to obtain an accurate centre of the star and to perform the flux calibration. We produced postage stamps for each of the stars of $500\times 500$ pixels. Then, we masked all sources that are not the central star to avoid contamination that could affect the flux calibration and centering. This masking is done in two steps: 1) a first run to detect all sources with and 2) a second run where the detection image is an unsharp-masked image, with a Gaussian smoothing of $\sigma=20$ pix. This second step allows us to mask sources that are covered by the halo of the star and not properly detected in the first run. In both cases, the detection was done with SExtractor. To accurately center the star, we calculated the centroids for each star by fitting a 2D Gaussian to the 2D flux distribution of the star using centroid_sources in photutils. centroid_sources allows us to define a box for fitting the Gaussian, useful in cases where the centre of the star is strongly saturated. Once the star is masked and centered, we performed the flux calibration. We first derived radial profiles for both star and the PSF. By using the profiles rather than the stamps we are minimizing contamination due to masking residuals or other artifacts. To fit each star, we selected a range in radius for the calibration. The radial range is from $0.1$ times the saturation level of the image to $4$ times the value of the background of each postage stamp. This background was calculated as the standard deviation of the postage stamp with all the sources masked (including the star). We scaled the PSF profile to match the star profile, using ratio between star and PSF values derived from the profiles. Once the PSF is centered and calibrated we subtracted it from the image. We repeated the same process for each of the stars in the catalogue for both $g$ and $i$ bands. Fig. 5 shows a region of our image of A85 in the $i$ band. The original image is seen in the left panel while the middle panel shows the same region with the stars subtracted. As mentioned above, the stars in HSC images show asymmetries that become more evident further away from the center of the image. However, we have built a symmetric PSF. As the object of interest, the BCG, is centered in the image, nearby stars that could affect our photometry are not going to present significant asymmetries. However, we note that this is a potential source of error for this study. ### 2.3 Masking The study of the ICL in clusters of galaxies requires a very careful masking of foreground and background sources to reduce contamination that can affect the determination of the color of this light. In the case of deep images, this masking must be optimized not only for faint and small background objects but also for those that are closer and large. As a single setup for the detection and masking of both types of sources is unfeasible, we used a two-step approach like Montes & Trujillo (2018); a “hot+cold” mode (e.g., Rix et al., 2004). The “cold” mode will detect the extended bright galaxies from the cluster while the “hot” mode is optimized to detect the faint and small sources. We use this approach on a deep combined $g+i$ image, after star subtraction. In the case of the “hot” mode, we unsharp-masked the original image, to enhance the contrast, particularly in the central parts of the BCG. To create the unsharp-masked image, we convolved the image with a box filter with a side of $25$ pixels and then subtracted it from the original. The threshold for detection is $1.1\sigma$ above the background. The “cold” mask was further expanded 10 pixels while the “hot” was expanded 5 pixels. Both masks were combined to create the final mask for our images. Before this, we unmasked the BCG on the “cold” mask. The bleeding spikes were manually masked as well as the residuals from the subtraction of stars and their asymmetries. We created two masks for the cluster. In the first mask, all the objects of the image are masked except for the members of the cluster contained in our field of view and the diffuse ICL. For that, we use the spectroscopic membership information obtained in Owers et al. (2017). The morphological information obtained from SExtractor’s “cold” mask run is used to unmask the members of the cluster. For the second mask, all the objects are masked except for the BCG and ICL. As SExtractor does a poor job detecting low surface brightness outskirts, we manually extended the masks for the remaining objects after visual inspection. The final mask was again visually inspected to manually mask any remaining light that was missed by the process described above. In the right panel of Fig. 5, we show an example of the mask in one region of our image. ### 2.4 Surface brightness limits Our goal is to study the low surface brightness features in A85 down to the faintest surface brightness possible. For this reason, we need to know how deep our images are by estimating the surface brightness limits that they reach. To obtain these limits, we calculated the r.m.s of the final masked images by randomly placing $20000$ boxes of $10\times 10$ arcsec2 ($\sim 10\times 10$ kpc2) across the images. In this case, we also masked the BCG and ICL by adding an ellipse of semi-major axis of $672\arcsec$ centered in the image. The $3\sigma$ surface brightness limits are: $\mu_{g}^{limit}$(3$\sigma$, $10\arcsec\times 10\arcsec$) = 30.9 mag/arcsec2, and $\mu_{i}^{limit}$(3$\sigma$, $10\arcsec\times 10\arcsec$) = 29.7 mag/arcsec2. These limits are calculated following Appendix A in Román et al. (2020). ## 3 The intracluster light of A85 ### 3.1 Radial surface brightness profiles Figure 6: The left panel shows the inner $700\arcsec\times 700\arcsec$ of the A85 image with the isophotes from ellipse. The middle panel presents the surface brightness profiles as a function of the semi-major axis for the BCG + ICL of A85 to a radius of $250\arcsec$ (258 kpc), for the $g$ (green) and $i$ (purple) bands. The profiles are $k$-corrected and corrected for the extinction of the Milky Way and surface brightness dimming. The right panel shows the ellipticity (top) and position angle (PA, bottom) profiles as a function of the semi-major axis for the $g$ and $i$ bands. The vertical gray lines indicate the FWHM of point sources in the $g$ (solid) and $i$ (dashed) images. The goal of this paper is to study the diffuse light in HSC images of A85. To that end, we derived the radial profiles for the $g$ and $i$ bands using the software ellipse in IRAF. ellipse fits elliptical isophotes to the 2-D images of galaxies using the method described in Jedrzejewski (1987). It provides the mean intensity, ellipticity, position angle and harmonic amplitudes (deviations from perfect ellipticity) for each fitted isophote. By deriving the 1-D profiles this way, we are not assuming any particular model or models to describe the BCG+ICL, as they might be sensitive to the choice of the particular model and prone to degeneracies between the different parameters. ellipse was run on the star-subtracted, masked images. We first run the task allowing all parameters to vary freely. In the second run, we fixed the centers to the median centers of the isophotes returned by ellipse in the first iteration.We adopted the median setup in ellipse. The surface brightness profiles reach a signal-to-noise ratio of $2.8$ ($3.0$) at $27.1$ ($25.7$) mag/arcsec2 in $g$ ($i$), which corresponds to a radius of $200\arcsec$ or $213$ kpc. Fig. 6 shows the output of ellipse for A85. The left panel shows the $700\arcsec\times 700\arcsec$ region around the BCG888Known as Holm 15A. with the fitted ellipses. The 1-D radial surface brightness profiles as a function of semi-major axis (SMA) for the $g$ (green) and $i$ (purple) bands are shown in the middle panel, up to $250\arcsec$. These surface brightness profiles are corrected for the absorption of our galaxy (E(B-V) $=0.034$; Schlafly & Finkbeiner, 2011) and surface brightness dimming. The profiles are also $k$-corrected (Chilingarian et al., 2010; Chilingarian & Zolotukhin, 2012). The shaded regions represent the errors of the profiles computed as the r.m.s scatter of each isophote. The vertical gray lines in all the panels indicate the full-width-at-half- maximum (FWHM) of the $g$ (solid; $1\farcs 07$) and $i$ (dashed; $0\farcs 78$) bands. The FWHM of each image is given by twice the average ‘FLUX_RADIUS’ (the half-light radius) of stars obtained from SExtractor (see Fig. 2). These lines define the regions where the isophotal fits are not reliable. The right panel shows the ellipticity (top) and position angle (PA; bottom) with SMA for both bands. The surface brightness profiles derived here show a flattening in the central regions of the BCG ($\lesssim 10\arcsec$, $11$ kpc). This flattening in the inner $\sim 10\arcsec$ has already been reported (e.g., López-Cruz et al., 2014; Madrid & Donzelli, 2016). In fact, the BCG of A85 is known to host one of the largest cores measured to date (López-Cruz et al., 2014). Beyond the core, the surface brightness radial profiles roughly follow a Sérsic (1968) profile. However, in the middle panel of Fig. 6, there appears to be a break in the profile at a radius of $\sim 70\arcsec$ ($\sim 75$ kpc), where the profiles become shallower. In order to explore whether there is a break in the surface brightness radial profiles, we fit a single Sérsic profile to both bands, excluding the inner $10\arcsec$. These fits are performed using a least squares fitting method as suggested in Seigar et al. (2007). The best fit to the profiles are shown in Fig. 15 and the parameters are listed in Appendix C. We show the residuals of subtracting the best Sérsic fit from the surface brightness profiles in the top panel of Fig. 7. The figure shows that at a radius of $\sim 70\arcsec$ ($\sim 75$ kpc) there is an excess of light with respect to the Sérsic fit.999Note that the goal of this fit is to locate the break, not to describe the light profile. This indicates that there is an extra component over the galaxy; the ICL. The position of the break found here is consistent with Zibetti et al. (2005), where a similar flattening is found at a radius of $\sim 80$ kpc, in their stacked profiles of multiple clusters. The ellipticity of the diffuse light of A85 increases with radius up to a value of $\epsilon\sim 0.55$ for both bands at a radius of $\sim 200\arcsec$ ($\sim 213$ kpc), as shown in the top right panel of Fig. 6. Kluge et al. (2020b) also observed an increase in ellipticity for A85. However, at a radius of $\sim 250$ kpc, their ellipticity profile drops sharply to a value of $0.1$. We do not see any evidence of such a decrease in our profiles101010Our ellipticity profile remains constant at $\sim 0.5$ to a radius of $320$ kpc, although the signal-to-noise at that radius is $\lesssim 1$. In contrast, the PA does not show any significant change with radius. Figure 7: The top panel shows the residuals from a Sérsic fit to the surface brightness radial profiles in the $g$ (green) and $i$ (purple) bands. The bottom panel shows the B4 coefficient (4$th$ Fourier harmonic) as a function of semi-major axis for both bands. The vertical grey line at $70\arcsec$ ($\sim 75$ kpc), tentatively marks the radius where an extra component starts to dominate and the isophotes become boxier. Departures from perfect elliptical isophotes can be described as Fourier harmonic perturbations (Jedrzejewski, 1987). The coefficients of these harmonic series carry physical meaning. For example, B4, the 4$th$ Fourier amplitude, indicates the boxyness/diskyness of the isophotes. In the bottom panel of Fig. 7, we show the B4 coefficient as a function of SMA. The radius where the break of the surface brightness profile is located, $70\arcsec$, also corresponds to where the B4 becomes negative, i.e. the ellipses start showing a boxy shape. This radius is indicated in both panels of Fig. 7 by a gray vertical line. This is a confirmation of the boxyness visible in the outer parts of the BCG (inset A in Fig. 1). Boxyness has been found to be related to galaxy interactions (e.g., Nieto & Bender, 1989). ### 3.2 Color profile of the BCG+ICL Radial color gradients provide valuable constraints in the formation processes of galaxies and, consequently, the BCG and ICL (e.g., Montes & Trujillo, 2014, 2018). The radial color profile was measured in $55$ logarithmic spaced bins from 0 to $200\arcsec$. The distance to each pixel on the images is computed as the elliptical distance to the BCG, where the morphological parameters (ellipticity and PA) are the median values from the ellipse isophotes excluding the inner $10\arcsec$: $0.37$ for the ellipticity and $56\deg$ for the PA. For each radial bin, the surface brightness in each band was obtained by averaging the pixel values. The errors are drawn from jackknife resampling, i.e. repeating the photometry in a sub-sample of the data for each bin. The number of sub-samples per bin was 100. Fig. 8 shows the $g-i$ color profile for the BCG+ICL of A85 down to $200\arcsec$ ($213$ kpc; light blue line). The color profile is $k$-corrected and corrected for the extinction of the Galaxy. The error in the color profile, represented as the light blue area, is the sum of the errors in the individual surface brightness radial profiles. We have also plotted the $g-i$ color of the satellite galaxies in the cluster as reported by Owers et al. (2017). The color profile of the BCG + ICL shows three distinct regions: i) a flat region out to $10\arcsec$ indicative of the core of the galaxy, ii) a negative color gradient from $10$ to $\sim 70\arcsec$ and iii) a region from $\sim 70\arcsec$ to $\sim 200\arcsec$ where the color gradient of the diffuse light becomes shallower. To see if there is a difference, we calculated the gradients of each region as a linear fit to the color profile $g-i$ vs. log R ($\Delta gi$). The fits are shown in Fig. 8 as the dark blue lines. The gradients for the different regions are: i) $-0.01\pm 0.01$ (dashed line), ii) $-0.24\pm 0.01$ (dotted line) and iii) $-0.06\pm 0.04$ (dash-dotted line). Figure 8: The $g-i$ color profile of BCG + ICL of A85 in blue. The errors are indicated by the light blue shaded area. The red spirals indicate the average color of member galaxies of the cluster as derived in Owers et al. (2017). The color profile presents three different regions: a flattening at $<10\arcsec$, a color gradient from $10\arcsec$ to $75\arcsec$ and a region from $70\arcsec$ to $200\arcsec$ where the gradient shallows. The linear fits to the color profiles for each region are shown as the dark blue lines: dashed for $<10\arcsec$, dotted for $10\arcsec$ to $75\arcsec$ and dash-dotted for $75\arcsec$ to $242\arcsec$. The flat color profile at SMA $<10\arcsec$ ($<11$ kpc) coincides with the size of the core of the galaxy as seen by López-Cruz et al. (2014). This is consistent with a mixing of the stellar populations in the centre of the galaxy. The region between $10\arcsec$ to $\sim 75\arcsec$ ($11$ to $\sim 80$ kpc) presents a negative gradient in $g-i$ color from $1.45$ to $\sim 1.25$ ($\Delta gi=-0.24\pm 0.01$). It is well known that massive early-type galaxies have negative optical color gradients indicating gradients in their stellar populations, generally metallicity (e.g., Peletier et al., 1990; Davies et al., 1993; La Barbera et al., 2012; Huang et al., 2018; Santucci et al., 2020). Beyond $\sim 75\arcsec$ ($\sim 80$) kpc, the color profile becomes significantly shallower ($\Delta gi=-0.06\pm 0.04$) with a median color of $g-i=1.25$. The observed behaviour of the color profile of A85 is consistent with the color profile in Zibetti et al. (2005) (also, Coccato et al., 2010; Montes et al., 2014; Spavone et al., 2020). Zibetti et al. (2005) explored the $g-r$ color profile of stacked clusters in SDSS. Their color profile also shows a gradient down to $\sim 80$ kpc where it shallows. Figure 9: The left panel shows the inner $700\arcsec\times 700\arcsec$ of A85 where the different sections are shaded in different colors: North (N, orange), East (E, green), South (S, purple) and West (W, magenta). In the right panel we show the corresponding $g-i$ color profiles to a radius of $200\arcsec$. The Northern, Southern and Western profiles flatten at different radius possibly indicating the presence of accreted material at those distances. There are some nearby bright stars both East and West of the BCG. Inaccuracies in the star subtraction process could bias the colors that we obtain, particularly the colors of the faintest regions of the ICL. In order to assess that potential issue, we derive the color profiles in 4 different directions of the BCG: North, East, South and West. The four different profiles were derived by masking the BCG+ICL except for $90\deg$-wide sections as shown in the left panel of Fig. 9, labelled as North (orange, N), East (green, E), South (purple, S) and West (magenta, W). The profiles are derived in the same way as the overall color profile. The right panel in Fig. 9 shows the color profiles color-coded by their respective section. The color profiles behave similarly up to $\sim 50\arcsec$, where the Southern profile flattens (purple line, $g-i\approx 1.3$) to a radius of $\sim 100\arcsec$ ($107$ kpc). Similarly, the Northern (orange) and Western (magenta) profiles show flattening, and even reddening (North profile), between $80\arcsec$ to $130\arcsec$. While for the Southern profile there is not a clear origin for the observed flattening, the shape of the Northern profile could be affected by the presence of a large satellite (at a projected radius of $\sim 75\arcsec$, zoomed-in in Fig. 11). This is also the case for the Western profile as there are some galaxies at $\sim 145\arcsec$. We will discuss this in detail in the following Section. Given that the closest bright stars are only located East and West of the BCG, these color profiles confirm that the change in gradient is not caused by the presence of these stars but rather caused by the presence of diffuse light associated with ongoing mergers. ### 3.3 Fraction of light in the ICL Studying the amount of light in the ICL can provide information on the efficiency of the interactions that form the ICL. This is given by the ICL fraction, defined as the ratio between the ICL and the total (BCG + galaxies + ICL) flux or luminosity of the cluster. This ICL fraction is an ill-defined quantity in photometric-only studies as separating between BCG and ICL is not obvious. To overcome this problem, astronomers have been using different ways of defining the ICL component in deep photometry. In the following, we describe two of the most widely used definitions. We derived the ICL fraction for A85 using both of them, for ease of comparison with other studies. #### 3.3.1 ICL fraction from surface brightness cuts The most widely used definition is to apply a cut in surface brightness and assume that the light fainter than a certain surface brightness limit is the ICL (typically $\mu_{V}>26.5$ mag/arcsec2, e.g., Feldmeier et al., 2004; Rudick et al., 2011). To derive the ICL fraction for the $g$ and $i$ bands, we followed similar steps to Montes & Trujillo (2018). First, we applied the mask where all the members of the cluster are unmasked, derived in Sec. 2.3, to each of the images. In each of the bands, we summed all the pixels fainter than a given ICL threshold. The fractions given have a fainter limit of $\mu<29.5$ mag/arcsec2 in order to minimize the contamination from inhomogeinities in the background. The ICL fractions are derived applying 3 different surface brightness cuts: $\mu>$ 26, 26.5 and 27 mag/arcsec2. We provide the ICL fractions for both the $g$ and $i$ bands in Table 2.. The ICL fractions calculated this way account not only for the diffuse light associated with the BCG but also with other galaxies in the cluster. Note that defining the ICL this way means that the measured fractions are a lower limit of the true value; we are missing light in both the brighter (e.g., in projection) and fainter limit. Table 2: ICL fraction ($\%$) for A85 Surface brightness cuts --- \| $26<\mu<29.5$ \| $26.5<\mu<29.5$ \| $27<\mu<29.5$ \| $27.5<\mu<29.5$ [mag/arcsec2] $f_{\mathrm{ICL}}(g)$ \| $8.8\pm 0.5$ \| $6.2\pm 0.7$ \| $4.0\pm 0.9$ \| $2.4\pm 0.9$ $f_{\mathrm{ICL}}(i)$ \| $3.1\pm 0.7$ \| $1.9\pm 0.7$ \| $1.1\pm 0.7$ \| $0.6\pm 0.7$ 2D fit \| $g$ \| $i$ $f_{\mathrm{ICL}}$ \| $11.0\pm 1.0$ \| $11.5\pm 1.0$ $f_{\mathrm{BCG+ICL}}$ \| $16.7\pm 2.0$ \| $18.0\pm 2.0$ $f_{\mathrm{ICL}/\mathrm{BCG+ICL}}$ \| $66.1\pm 2.2$ \| $63.7\pm 2.2$ #### 3.3.2 ICL fraction assuming a functional form Despite its simplicity, one of the limitations of the above definition is that it does not account for the amount of ICL in projection on top of the BCG. Another common approach is using functional forms to describe both BCG and ICL (e.g., Gonzalez et al., 2005; Seigar et al., 2007; Spavone et al., 2018, to name a few). In our case, we use GALFIT (Peng et al., 2002) to simultaneously fit two two-dimensional Sérsic profiles: one to describe the BCG and one for the ICL. The parameters for the two fitted Sérsic components are given in Table 4 in Appendix C. Although the fits seem to describe well the overall profile, they are not able to reproduce the inner core of the galaxy (as in the case of the single Sérsic fit, Fig. C). Contrary to the single Sérsic fit performed in Sec. 3.1, we now find that the inner component is an exponential, similar to the outer component ($n_{1}\sim 1$ and $n_{2}\sim 2.15)$. This difference between the single and double Sérsic fits is probably caused by the single component trying to fit the outer parts of the BCG+ICL profile. As expected from the ellipse 1-D profiles in Sec. 3.1, the more extended component (the ICL) has a higher ellipticity than the inner component (the BCG, see also Kluge et al., 2020a). However, the PA in both models are not significantly different ($\Delta\mathrm{PA}\sim 4^{\circ}$). The 1-D surface brightness profiles obtained with ellipse for the double Sérsic fits are shown in Fig. 10. As in Fig. 6, the observed surface brightness profiles of the $g$ and $i$ bands are shown in mint green and purple, respectively. The two different Sérsic models (inner and outer) are shown with the dashed grey lines while the sum of both models is the solid black line. As in Fig. 7, it can be seen that the outer component, the ICL, dominates at around $\sim$ 60-70$\arcsec$. The ICL fraction obtained using the outer Sérsic model is given in Table 2. We have also derived the fraction of BCG+ICL with respect to the total and the ratio between ICL and BCG+ICL. Figure 10: ellipse 1-D profiles of the $g$ (top) and $i$ (bottom) bands of the BCG + ICL of A85. The grey dashed lines correspond to the profiles of the two Sérsic components fitted with GALFIT. The solid black line is the sum of both components. A double Sérsic fit reproduces the light profile of A85 in both bands and the outer component, the ICL, dominates at around $\sim$ 60-70$\arcsec$. The ICL fractions derived from assuming a double Sérsic to describe BCG+ICL are higher than those from surface brightness cuts. This is expected because we are extrapolating the contribution of the diffuse light in the line of sight of the BCG which results in adding more ICL that surface brightness cuts cannot account for. Fig. 10 shows that while the extended Sérsic component begins to dominate at r $=70$ kpc, a surface brightness limit of $\mu_{g}>26$ mag/arcsec2 will measure all the light beyond $110$ kpc as ICL. Note that a surface brightness cut also accounts for diffuse light that is associated with the other galaxies in the cluster and might give a more complete picture of the formation of ICL in clusters. The fractions calculated in this Section include all the member galaxies of the cluster in our images. That is r = 0$\fdg$42 = 0.67$\times R_{200}$ (where R${}_{200}=$0$\fdg$63 = 2.42 Mpc, Owers et al., 2017). Other studies measure the ICL fraction in smaller radius, tipically $R_{500}$ (e.g., Gonzalez et al., 2007). This means that, in comparison, we are including more galaxies and therefore deriving a higher total luminosity for the cluster111111While not adding almost any ICL.. That yields to lower ICL fractions than in other studies with more limited field of views (e.g., Burke et al., 2015; DeMaio et al., 2020). For this reason, we have also calculated the fractions within $R_{500}=1.2$ Mpc = $18\farcm 7$ (Ichinohe et al., 2015). The fractions within $R_{500}$ can be found in Table 5 in Appendix F. ## 4 Discussion In this work, we have used archival HSC data to explore the radial surface brightness and color profile of the BCG of A85 to a radius of $200\arcsec$ ($213$ kpc). We found that both the surface brightness and color profile become shallower beyond $70\arcsec$ ($75$ kpc), indicating that an extra component, the ICL, starts to dominate. In the following, we will discuss the implications of our results. ### 4.1 The fraction of light in the ICL The ICL is a product of the interactions between galaxies within the cluster (Rudick et al., 2009), therefore its fraction can provide information of the efficiency of those interactions, while the evolution of this component with time gives an estimation of their timescales. However, measuring the ICL fraction is difficult as the transition between BCG and ICL happens smoothly, making it hard to separate both components. In addition, studies use different bands and definitions for the ICL complicating direct comparison. In general, the ICL fractions derived here using surface brightness cuts are in agreement with those in the literature for clusters at similar redshifts (although in different, adjacent, bands and surface brightness limits, e.g., Krick & Bernstein 2007). Our ICL fraction at $\mu_{g}>26$ mag/arcsec2 is $\sim 9.8\pm 0.5\%$ (Table 5). This is in agreement with the median ICL fraction (using the same band and surface brightness cut) in Kluge et al. (2020a): $13\pm 13\%$. It is also in agreement with the $\sim 11\%$ at $\mu_{V}>26.5$ mag/arcsec2 derived in the simulations of Rudick et al. (2011). Simulations show that $70\%$ of the stellar mass of the BCG is accreted (Qu et al., 2017; Pillepich et al., 2018). This means that most of the BCG is formed in a similar way to the ICL, and therefore they should be studied in order to understand the growth of BCGs. For this reason, we also measured the fraction of BCG+ICL over the total luminosity of the cluster, $f_{BCG+ICL}$. This fraction is $\sim 46\%$ at $r<R_{500}$ in agreement with Gonzalez et al. (2007, 2013) for clusters at similar redshifts. The fraction of ICL over the BCG+ICL component, fICL/BCG+ICL ($\sim 64\%$) indicates that most of the total light in the BCG+ICL system in A85 is in the ICL121212More specifically, it is the stellar halo or envelope (bound to the BCG) + ICL (bound to the cluster) as we cannot distinguish between both components using imaging alone.. This result agrees with the fractions from previous observations and simulations (e.g., Gonzalez et al., 2005; Zibetti et al., 2005; Seigar et al., 2007; Cañas et al., 2020; Kluge et al., 2020a). In the simulations of Conroy et al. (2007), similar fractions are achieved if all the stars from disrupted satellites end up in the ICL (their Fig. 4). These results from simulations, coupled with the observed mild evolution in mass of BCGs (e.g., Whiley et al., 2008; Collins et al., 2009; Lidman et al., 2012; Oliva-Altamirano et al., 2014; Bellstedt et al., 2016), suggests indicate that a significant fraction of the mass of infalling satellites goes to the stellar halo + ICL instead of adding a significant fraction of mass to the BCG (e.g., Laporte et al., 2013; Contini et al., 2018). ### 4.2 Stellar populations of the BCG Studying the colors of the ICL in clusters allows us to infer the properties of the progenitor galaxies from which the ICL accreted its stars and, consequently, the mechanisms at play in the formation of this component. In Section 3.1, we presented the surface brightness radial profiles of the BCG + ICL for the $g$ and $i$ bands. Both surface brightness profiles show a flat region in the inner $10\arcsec$ ($11$ kpc), denoting the presence of a core (e.g., López-Cruz et al., 2014; Madrid & Donzelli, 2016). In the same way, the measured color profile is flat in the inner $10\arcsec$, indicating that the stellar populations in this region are well mixed. Mehrgan et al. (2019) used MUSE data to infer the stellar kinematics of this BCG finding that this central region hosts a supermassive black hole with a mass of $4.0\pm 0.8\times 10^{10}M_{\odot}$. They concluded that the BCG of A85 is a result of the merger of two cored early-type galaxies. Beyond $10\arcsec$, the surface brightness profiles follow a Sérsic (1968) profile down to $\sim 70\arcsec$ ($\sim 75$ kpc). At the same time, the color profile shows a negative gradient from $g-i=1.45$ to $g-i\approx 1.25$. The central flattening and subsequent gradient in color is also observed in the integral field spectroscopy observations of A85 in Edwards et al. (2020). They find that out to $30$ kpc the metallicity of the galaxy shows the same behaviour as our color profile: a flattening in the inner $\sim 10$ kpc ($\sim 10\arcsec$), followed by a decrease to $\sim 30$ kpc ($28\arcsec$). At $\sim 70\arcsec$, the surface brightness profiles depart from the Sérsic fit (top panel in Fig. 7). This corresponds to where the isophotes show a boxy shape (indicated by the gray vertical line in Fig. 7). Simulations suggest that boxyness is the result of a past dry merger event (e.g., Naab et al., 2006). In addition, at this radius, the color profile becomes shallower. These pieces of evidence point to an extra component originating from accreted stars: the ICL131313ICL + stellar halo.. It is not possible to disentangle between stellar age and metallicity using only one color. Previous deep observations of clusters of galaxies show clear radial gradients in colors (e.g., Williams et al., 2007; Montes & Trujillo, 2014, 2018; DeMaio et al., 2015, 2018; Mihos et al., 2017; Iodice et al., 2017) indicating radial gradients in metallicity while the ages of the ICL in nearby systems are old ($>10$ Gyr, e.g., Williams et al., 2007; Coccato et al., 2010). This is consistent with Edwards et al. (2020), who only found a very mild decrease in age to $30$ kpc for A85, from $\sim 15$ to $10$ Gyr. Therefore at $<30$ kpc, the color profiles likely mostly trace changes in metallicity. However, we cannot test here whether the decrease in age becomes significant beyond $30$ kpc. The shape of the color profile is reminiscent of the three different regions found in the metallicity profile of M87 in Montes et al. (2014) (see also Coccato et al. 2010). In M87, the metallicity gradient becomes shallower in the outer parts of the galaxy. This is the consequence of the mixing of the stellar populations of the accreted galaxies. This also appears to be the case for A85 and is supported by the change in the slope of the surface brightness profiles, where the outer parts of the galaxy (the ICL) are built via the accretion of satellite galaxies. Figure 11: Inner $700\arcsec\times 700\arcsec$ RGB image of A85. The ellipse model obtained in Sec. 3.1 has been subtracted. The teal cross marks the centre of the BCG while the green arrow marks a collection of galaxies and diffuse light to the South of the BCG. The colored arcs mark the area where there is flattening in the color profiles derived in different directions as seen in Fig. 9. The inset shows a zoom-in into a galaxy that presents a faint tail towards the North as indicated by the purple arrow. The North (orange) and West (magenta) arcs are highlighting diffuse light associated with galaxies interacting with the BCG. In Section 3.2, we derived color profiles of the BCG+ICL in four different 90 deg-wide sections, finding that the Southern color profile between $50\arcsec$ to $100\arcsec$ ($53$ to $106$ kpc) is redder than the other profiles ($\sim 1.3$, Fig. 9). Similarly, the North and West profiles become flat between $80\arcsec$ to $130\arcsec$. To explore whether there is any evidence of infalling material that might be causing the flattening of the profiles, we subtracted the ellipse models from the image for both bands to enhance any signs of interactions or asymmetries. In Fig. 11, we show the inner $700\arcsec\times 700\arcsec$ region of A85 with the model generated from the ellipse fits subtracted. We have drawn arcs to demarcate the areas in the image corresponding to the flattening of the color profiles, color-coded according to the direction of the corresponding profile in Fig. 9. Towards the North, we found a faint tail associated with a large satellite galaxy (zoomed-in in Fig. 11 and marked with a purple arrow). The presence of this faint tail might explain the sudden reddening of the Northern color profile at $\sim 130\arcsec$ in Fig. 9. As there are no other signs of disturbance, this galaxy is probably just starting to interact with the BCG. To the West, there is some diffuse light associated with two galaxies that are likely interacting with the BCG. Even with our careful and conservative masking, the diffuse light associated with these interactions might be contaminating our profiles. In addition, there is a collection of galaxies between $\sim 75\arcsec$ and $\sim 160\arcsec$ ($80$ to $170$ kpc) South of the BCG with associated diffuse light. This structure is marked with a green arrow in Fig. 11. These galaxies appear to be interacting with each other rather than with the BCG. The color of the diffuse light of the region, with the galaxies masked, is $g-i=1.24\pm 0.15$. However, this structure is not associated to any signature in the Southern color profile (lies outside the purple arcs in Fig. 11). To investigate whether these diffuse structures could be biasing our results, we repeated the ellipse fitting and color profile derivation but, in this case, generously masking all of these structures (North, South and West). The resulting color profile and the $4th$ Fourier harmonic, B4, are shown in Fig. 16 in Appendix D. We also plotted the original color profile in dark blue for reference. The new color profile is compatible with the original. The gradient between $70\arcsec$ to $200\arcsec$ ($75$ to $213$ kpc) is now $-0.07\pm 0.04$, slightly steeper but compatible within errors with the gradient derived in Sec. 3.2 ($-0.06\pm 0.04$). The boxyness is still present. Therefore, the diffuse light associated with these interactions is not producing the change of gradient nor the boxyness observed. The lack of any obvious tidal feature related to the color flattening towards the South means that any tidal feature has had time to smear, only emerging here in the color profile. Given its preferential position to the South, it did not have enough time to mix with the rest of the galaxy. The orbital period for a star around this BCG in the radial range from $50\arcsec$ to $100\arcsec$ ($53$ to $106$ kpc, the approximate range of the flattening in the Southern color profile) is between $1.5$ to $3.7$ Gyr. The calculations of the orbital period are described in Appendix E. In the simulations of Rudick et al. (2009), streams found in the core of the cluster are quickly destroyed by the strong, evolving tidal field of the cluster with timescales of $\lesssim 1$ Gyr. Therefore, for a star not to have orbited the galaxy but any stream to be smeared, we suggest that this interaction likely happened a few Gyrs ago. #### 4.2.1 The ellipticity profile and the ICL as a tracer of dark matter A significant anisotropy in the orientation of the orbits of the progenitors of the ICL will produce an elongation in the ICL distribution. This elongation, i.e. ellipticity, is expected to increase with increasing radius up to the value of the original distribution, i.e. the cluster distribution. The ellipticity of the diffuse light of A85 increases with radius up to a value of $\sim 0.55$ at $\sim 200\arcsec$ ($\sim 213$ kpc), for $g$ and $i$ (Fig. 6). However, the PA does not change significantly with radius, i.e., inner and outer components are aligned. This increase in ellipticity with radius was also observed in this cluster by Kluge et al. (2020b). The same trend with radius has been measured in other massive galaxies and clusters (e.g., Gonzalez et al., 2005; Tal & van Dokkum, 2011; Huang et al., 2018; Kluge et al., 2020b, a). The ellipticities of the diffuse light in these systems tend to the typical values for the distribution of galaxies within clusters (Shin et al., 2018). When fitting a double Sérsic model to the 2-D distribution of light of the BCG+ICL, we also find that the ellipticity of the outer component, the ICL, has a higher ellipticity ($\sim 0.5$) than the inner component, the BCG ($\sim 0.2$). The value of the ellipticity at large radii derived here is consistent with the axis ratio measured for A85 using weak-lensing modeling by Cypriano et al. (2004). That is, the ICL has the ellipticity of the dark matter halo of the cluster. These results agree with the picture proposed in Montes & Trujillo (2019) that the ICL is a good luminous tracer of the dark matter distribution in clusters of galaxies. ### 4.3 The buildup of the ICL of A85 The change in slope of the surface brightness profile of the BCG, the boxyness of the isophotes and the change in the slope of the color gradient at a radius of $\sim 70\arcsec$ ($\sim 75$ kpc) suggests strongly that the BCG and ICL can be considered as distinct stellar components with different assembly histories, and that the accreted component (ICL) starts to dominate at that radius. Integrated light spectroscopy (e.g., Dressler, 1979) and planetary nebulae kinematics (e.g., Arnaboldi et al., 1996) of nearby clusters, show that the radial velocity dispersion increases with radius to reach the value of the velocity dispersion of the galaxies in the cluster (Longobardi et al., 2018). That means that the stars forming the ICL are following the potential of the cluster rather than the potential of the BCG. We can conclude that the radius where the potential of the A85 cluster begins to dominate is $\sim 70\arcsec$ ($\sim 75$ kpc, see Fig. 10). Previous works have also shown that, in massive clusters ($10^{14-15}M_{\odot}$), BCGs tend to show this break radius at around $60-80$ kpc (e.g., Zibetti et al., 2005; Gonzalez et al., 2005; Seigar et al., 2007; Iodice et al., 2016). We can calculate an approximate mass of the progenitor of the merger using the color of the Southern profile. If the average color of the reddening in the Southern profile is around $g-i=1.3$ (Fig. 9), and assuming an age of $10$ Gyr, the metallicity of the progenitor would be [Z/H] $=-0.013$ (using the models of Vazdekis et al., 2016), i.e. slightly subsolar metallicity. Using the mass-metallicity relation from Gallazzi et al. (2005), this corresponds to a galaxy of $\sim 3\times 10^{10}M_{\odot}$. The galaxies towards the North and West that are interacting with the BCG have masses of the order of $\sim 7\times 10^{10}M_{\odot}$ (Owers et al., 2017). This is in agreement with observations in other clusters (Montes & Trujillo, 2014, 2018; Morishita et al., 2017; DeMaio et al., 2018) and with simulations (Purcell et al., 2007; Cui et al., 2014; Contini et al., 2014, 2019). These studies conclude that galaxies of $\sim 10^{10}M_{\odot}$ are the main contributors to the ICL in massive clusters of galaxies. ## 5 Conclusions In this work, we have presented deep observations in the $g$ and $i$ bands of the central $52\arcmin\times 52\arcmin$ of the cluster Abell 85, obtained with Hyper Suprime-Cam on the Subaru Telescope. The surface brightness limits reached are $30.1$ and $29.7$ mag/arcsec2 ($3\sigma$, $10\times 10$ arcsec2), for the $g$ and $i$ bands, respectively. Taking advantage of the depth of these images, we are able to study the diffuse light of this cluster down to $200\arcsec$ ($213$ kpc) from the BCG. At $\sim 70\arcsec$ ($\sim 75$ kpc), the surface brightness profiles become shallower and the isophotes show a boxy shape, strongly indicating the presence of an accreted component: the ICL. In addition, at the same radius the color profile becomes shallower, a consequence of the mixing of the stellar populations of the accreted galaxies. Furthermore, the color profile towards the North, West and South of the BCG show a redder color compared to the other profiles as if there is remaining material in that direction from a merger that happened a few Gyrs ago. This work shows that even short exposure times ($\sim 30$ mins) on large telescopes can unveil the assembly history of clusters of galaxies. The results presented in this work show the extraordinary power of ground- based observatories to address the origin and evolution of the ICL. In the future, the LSST will be able to provide deep multi-wavelength observations of the southern sky allowing the study of the ICL in a range of cluster masses and redshifts (Montes, 2019; Brough et al., 2020). However, as demonstrated in this work, careful data processing techniques are crucial in order to take the maximum benefit from the upcoming data. We thank the referee for constructive comments that helped to improve the original manuscript. MM would like to thank Raúl Infante-Sáinz and Nacho Trujillo for their very useful comments on the data reduction and Alexandre Vazdekis for his comments on stellar populations. SB acknowledges funding support from the Australian Research Council through a Future Fellowship (FT140101166). Based on data collected at Subaru Telescope and obtained from the SMOKA, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan. This research includes computations using the computational cluster Katana supported by Research Technology Services at UNSW Sydney. ## Appendix A Effect of instrument rotation on the flat-fields In Sec. 2.1.3, we discussed that we used HSC-SSP images of adjacent nights to the observations of A85 in order to derive a sky flat. However, in the case of the $i$ band, using the images of adjacent nights resulted in significant background substructure in the individual CCDs, and consequently, a global structure in the individual frames. The source of this structure seems to be related to the rotation angle of the instrument (‘INR-STR’ in the header) being considerably different from that of the observed images ($<$INR-STR$>$ = $-5$ in the HSC-SSP Wide images compared to $<$INR-STR$>$ = $124$ for A85). Figure 12: Effect of rotation angles of the HSC (‘INR-STR’) on the flat-fields derived for CCD = 80 in the $i$ band. Left panel: Flat derived using the images taken from adjacent nights to the images of A85 ($<$INR-STR$>$ = $-5$). Middle panel: Flat derived using the images with similar rotation angles as the A85 science images ($<$INR-STR$>$ = $114$). Right panel: Ratio of the two flats, Flat/Flatr, where a gradient of $\sim 1\%$ across the CCD can be seen. For reference $<$INR-STR$>$ of the A85 images is $124$. To test this hypothesis, we downloaded images from the SMOKA archive where ‘INR-STR’ was close to the angle of the A85 $i$ band images. As it was difficult to find the same rotation angles as the A85 images, we downloaded images with angles between 100 and 140. The average rotation angle of these images are $<$INR-STR$>$ = 114. The dates when those images were taken are listed in Sec. 2.1.3. In Fig. 12, we show a comparison of the two different flats derived for CCD number 80141414Map of the CCD arrangement here: https://hsc.mtk.nao.ac.jp/pipedoc/pipedoc_4_e/_images/CCDPosition_20140811-1.png. The flat derived from the images from adjacent nights is shown in the left panel, labelled as _Flat_. The middle panel shows the flat derived from the images with a median instrument rotation close to the A85 images, labelled as _Flat r_. The right panel of Fig. 12 is the ratio of the two flats; _Flat_ divided by _Flat r_. The presence of a significant gradient across the CCD can be seen. This gradient is of the order of $\sim 1\%$. In Fig. 13, we show the comparison of the final co-added images for the $i$ band using the flats derived using science images of adjacent nights (labelled: with Flat, left panel) and using the flats obtained with the science images with the same rotation as the A85 image (the final image used in this work labelled: with Flatr, right panel). In the left image we can see inhomogeneities caused by the poor flat-field correction to the individual CCDs. Figure 13: Final $i$ band co-added image using the flat-field frames from science images of adjacent nights to the A85 observations (‘with Flat’, left panel) and using the flat-field frames from science images with the same rotation as A85 (‘with Flatr’, right panel). The left image shows inhomogeneities caused by the inaccuracy of the flat-field correction. ## Appendix B Example of the custom-made flat-field An accurate flat-field correction is crucial to minimise errors in low surface brightness science, especially in extended and diffuse objects such as the ICL. For this reason, we derived the flats from science observations instead of using the HSC master flats as inhomogeneities in the illumination can introduce gradients in our images. During this work, we found that the HSC master flat from CCD 75 does not contain a feature that is present in the data (indicated by purple lines in Fig. 14). We do not know the reason for this discrepancy. However, the master flat does seem to reproduce all other features seen in the CCD image. Figure 14: Left panel: Image from CCD 75 processed before flat-field correction. Middle panel: Master flat-field for CCD 75 from 2014-09-17. Right panel: Custom flat derived in this work. The purple lines highlight the structure present in both the image and custom flat, but not in the HSC master flat. ## Appendix C Sérsic fits 1D and 2D In Fig. 15, we show the single Sérsic fits to the radial surface brightness profiles of A85, for the $g$ (green) and $i$ (purple) bands. The best fit effective radius ($r_{eff}$) and Sérsic index (n) parameters are listed in Table 3, for both bands. Figure 15: 1-D single Sérsic fits to the BCG+ICL profile of A85. Table 3: Parameters of the single Sérsic fits for the surface brightness profiles of A85 Band \| $r_{\mathrm{eff}}$ [arcsec] \| n ---\|---\|--- g \| $39\pm 6$ \| $4.0\pm 0.7$ i \| $36\pm 4$ \| $4.9\pm 0.8$ Table 4: Parameters from the double Sérsic fit from GALFIT Band \| m1 \| $r_{\mathrm{eff},1}$ \| n1 \| $\varepsilon_{1}$ \| PA1 \| m2 \| $r_{\mathrm{eff},2}$ \| n2 \| $\varepsilon_{2}$ \| PA2 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| [mag] \| [arcsec] \| \| \| \| [mag] \| [arcsec] \| \| \| g \| $14.37\pm 0.01$ \| $15.02\pm 0.01$ \| $1.08\pm 0.01$ \| $0.20\pm 0.01$ \| $54.40\pm 0.02$ \| $13.49\pm 0.01$ \| $173.0\pm 0.2$ \| $2.14\pm 0.01$ \| $0.49\pm 0.01$ \| $58.63\pm 0.02$ i \| $12.96\pm 0.01$ \| $14.29\pm 0.01$ \| $1.04\pm 0.01$ \| $0.19\pm 0.01$ \| $54.16\pm 0.02$ \| $12.20\pm 0.01$ \| $172.6\pm 0.2$ \| $2.18\pm 0.01$ \| $0.53\pm 0.01$ \| $58.82\pm 0.02$ ## Appendix D Color profiles of A85 after masking diffuse structures In Sec. 4.2, we discussed the existence of diffuse structures North, South and West of the BCG with associated diffuse light that might be causing the flattening of the color profile. In Fig. 16, we show the $g-i$ color profile (top panel) and the B4 coefficient (bottom panel) with these structures masked. We do not find any significant change within errors. Figure 16: Color profile and B4 coefficients as a function of SMA after masking the collection of galaxies to the South of the BCG. We do not find any significant difference within errors with respect to the results from Sec. 3.2 (in dark blue). ## Appendix E Calculation of the orbital period around A85 To estimate the orbital period ($T$) of a star at a given radius ($R$) around A85 we used Kepler’s third law: $T=2\pi\sqrt{\frac{R^{3}}{GM_{R}}}$ (E1) where $M_{R}$ is the mass of the BCG of A85 inside a radius $R$ and $G$ is the gravitational constant (Binney & Tremaine, 1987). In order to calculate $M_{R}$ for A85, we followed the prescription in Bell et al. (2003) to calculate the mass-to-light ratio ($M/L$) as a function of the $g-i$ color, assuming a Salpeter 1955 IMF. The radius is computed as the elliptical distance to the BCG, where the morphological parameters (ellipticity and PA) are the median values from the ellipse isophotes excluding the inner $10\arcsec$: 0.37 for the ellipticity and 56 deg for the PA. The expression we have used to estimate the $M/L$ in the $g$ band is: $\log(M/L_{g})=-0.379+0.914\times(g-i)$ (E2) from Table 7 in Bell et al. (2003). The color used is the median $g-i$ color inside a radius $R$, $k$-corrected and corrected by the extinction of the Milky Way. ## Appendix F ICL fraction at R$<$R500 In Table 5, we list the ICL fractions inside $R_{500}$ (= 1.2 Mpc = 18$\farcm$7). We also list the BCG+ICL fraction with respect to the total luminosity of the cluster and the ICL fraction with respect to the BCG+ICL. Table 5: ICL fraction (%) for A85 within r$<R_{500}$ Surface brightness cuts --- \| $26<\mu<29.5$ \| $26.5<\mu<29.5$ \| $27<\mu<29.5$ \| $27.5<\mu<29.5$ [mag/arcsec2] $f_{\mathrm{ICL}}(g)$ \| $9.8\pm 0.5$ \| $7.0\pm 0.8$ \| $4.6\pm 1.0$ \| $2.7\pm 1.0$ $f_{\mathrm{ICL}}(i)$ \| $3.6\pm 0.8$ \| $2.2\pm 0.8$ \| $1.4\pm 0.8$ \| $0.7\pm 0.8$ 2D fit \| $g$ \| $i$ $f_{\mathrm{ICL}}$ \| $29.9\pm 1.0$ \| $30.3\pm 1.0$ $f_{\mathrm{BCG+ICL}}$ \| $45.4\pm 2.0$ \| $47.7\pm 2.0$ $f_{\mathrm{ICL}/\mathrm{BCG+ICL}}$ \| $66.1\pm 2.2$ \| $63.7\pm 2.2$ ## References Ahn et al. (2012) Ahn, C. P., Alexandroff, R., Allende Prieto, C., et al. 2012, ApJS, 203, 21, doi: 10.1088/0067-0049/203/2/21 * Aihara et al. (2018) Aihara, H., Armstrong, R., Bickerton, S., et al. 2018, PASJ, 70, S8, doi: 10.1093/pasj/psx081 * Aihara et al. (2019) Aihara, H., AlSayyad, Y., Ando, M., et al. 2019, PASJ, 71, 114, doi: 10.1093/pasj/psz103 * Alam et al. (2015) Alam, S., Albareti, F. 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# A Spatially-Resolved Survey of Distant Quasar Host Galaxies: II. Photoionization and Kinematics of the ISM Andrey Vayner Department of Physics, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA Center for Astrophysics & Space Sciences, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA Department of Physics and Astronomy, Johns Hopkins University, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA Shelley A. Wright Department of Physics, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA Center for Astrophysics & Space Sciences, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA Norman Murray Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada Canada Research Chair in Theoretical Astrophysics Lee Armus Spitzer Science Center, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 USA Anna Boehle ETH Zürich Wolfgang-Pauli-Str. 27 8093 Zürich, Switzerland Maren Cosens Department of Physics, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA Center for Astrophysics & Space Sciences, University of California San Diego, 9500 Gilman Drive La Jolla, CA 92093 USA James E. Larkin Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 USA Etsuko Mieda National Astronomical Observatory of Japan, Subaru Telescope, National Institutes of Natural Sciences, Hilo, HI 96720, USA Gregory Walth Observatories of the Carnegie Institution for Science 813 Santa Barbara Street Pasadena, CA 91101 USA (Accepted January 15, 2021) ###### Abstract We present detailed observations of photoionization conditions and galaxy kinematics in eleven z$=1.39-2.59$ radio-loud quasar host galaxies. Data was taken with OSIRIS integral field spectrograph (IFS) and the adaptive optics system at the W.M. Keck Observatory that targeted nebular emission lines (H$\beta$ ,[OIII] ,H$\alpha$ ,[NII] ) redshifted into the near-infrared (1-2.4 µm). We detect extended ionized emission on scales ranging from 1-30 kpc photoionized by stars, shocks, and active galactic nuclei (AGN). Spatially resolved emission-line ratios indicate that our systems reside off the star formation and AGN-mixing sequence on the Baldwin, Phillips $\&$ Terlevich (BPT) diagram at low redshift. The dominant cause of the difference between line ratios of low redshift galaxies and our sample is due to lower gas-phase metallicities, which are 2-5$\times$ less compared to galaxies with AGN in the nearby Universe. Using gas velocity dispersion as a proxy to stellar velocity dispersion and dynamical mass measurement through inclined disk modeling we find that the quasar host galaxies are under-massive relative to their central supermassive black hole (SMBH) mass, with all systems residing off the local scaling ($M_{\bullet}-\sigma~{}$,$M_{\bullet}-M_{}~{}$) relationship. These quasar host galaxies require substantial growth, up to an order of magnitude in stellar mass, to grow into present-day massive elliptical galaxies. Combining these results with part I of our sample paper (Vayner et al., 2021) we find evidence for winds capable of causing feedback before the AGN host galaxies land on the local scaling relation between black hole and galaxy stellar mass, and before the enrichment of the ISM to a level observed in local galaxies with AGN. ††journal: ApJ††software: OSIRIS DRP (Larkin et al., 2013), Matplotlib (Hunter, 2007), SciPy (Virtanen et al., 2020), NumPy (Harris et al., 2020), Astropy (Astropy Collaboration et al., 2018), MAPPINGS (Alarie & Morisset, 2019), emcee (Foreman-Mackey et al., 2013) ## 1 Introduction Today, feedback from supermassive black holes (SMBH) is an integral part of galaxy evolution models. It is commonly used to explain the lack of observed baryons in local massive galaxies (Behroozi et al., 2010), the enrichment of the circumgalactic medium with metals (Prochaska et al., 2014) and the observed local scaling relation between the mass of the galaxy/bulge and the SMBH (Ferrarese & Merritt, 2000; Gebhardt et al., 2000; McConnell & Ma, 2013). The latest observational and theoretical results point to a critical question; at what points does the AGN drive an outflow powerful enough to clear the galaxy of its gas into the surrounding CGM? (King & Pounds, 2015) According to theoretical work, this typically happens once the galaxy reaches the $M_{\bullet}-\sigma~{}$relationship (Zubovas & King, 2014). However, there has been growing evidence for galaxies with massive SMBH and powerful outflows that are offset from the local scaling relationship (Vayner et al., 2017). The origin and evolution of the local scaling relationships with redshift have been an active debate topic over the last decade. When are the local scaling relations established? Are the local scaling relationships the end product of galaxy evolution? Meaning, as galaxies form and evolve, do they fall in and out of the relationships due to rapid growth or feedback process? Do galaxies eventually end up on the local scaling relations once the galaxy or SMBH catch up and finish growing (Volonteri, 2012)? Alternatively, is there an inherent evolution in the scaling relationship with redshift and a symbiosis between the galaxy and SMBH growth? (i.e., evolution in slope, offset, and scatter). Finally, there is still an open question regarding the role of quasar feedback in establishing the relationship and whether the merging of galaxies can produce the $M_{\bullet}-\sigma~{}$relation following the central limit theorem (Jahnke & Macciò, 2011). From a sample of AGN in the COSMOS field (Merloni et al., 2010) finds an offset in the local scaling relationship between redshift 0 and 2. These authors use SED decomposition with numerous spectral bands to measure the stellar mass of the AGN host galaxy in the redshift range of $1<\rm z<2.2$. From a sample of lensed quasars at $1<z<4.5$ and broadband HST imaging, Peng et al. (2006) finds an offset in the local scaling relationship. While Sun et al. (2015) using multi-band SED fitting of galaxies in the COSMOS field finds that z$\sim 0.2-2$ galaxies are consistent with being on the local scaling relationship. Schramm & Silverman (2013) using HST imaging in the Chandra Extended Deep Field also finds that galaxies at z$\sim 0.6-1$ are also consistent with being on the local scaling relationship. In the nearby Universe, there is tentative evidence that all of the most massive black holes ($>10^{9}$ M⊙ ) are systematically more massive relative to their host galaxies (Martín-Navarro et al., 2018). Fields such as COSMOS or the Extended Chandra Deep Field-South are relatively small in the sky; hence, the number of luminous quasars with massive SMBH is small. Studies that explored the evolution of the local scaling relationships have generally focused on lower-mass black holes with masses $<10^{9}$ M⊙ . Furthermore, a large fraction of these studies used broadband HST imaging to study the host galaxies of their quasars/AGN. It is often difficult to disentangle the bright AGN emission from the host galaxy at smaller angular separations ($<$0.5″). These studies have a limited number of filters to measure the stellar population’s age and the mass to light ratio. Alternatively, mm-interferometry observations have become an essential tool in measuring the dynamical masses of quasar host galaxies across different redshift ranges. At the highest redshifts (z$>4$), the [CII] 158µm line has been the most commonly used tracer of the dynamics of the ISM. There is growing evidence that the most massive ($>10^{9}$ M⊙ ) SMBH in the highest redshift quasars known to date (z$>6$) appear to be over massive for the mass of their host galaxies (Wang et al., 2013; Venemans et al., 2016; Decarli et al., 2018), indicating that the most massive SMBHs residing in high redshift quasars grow first relative to their host galaxies. At more intermediate redshifts 1$<$z$<$3, some systems also appear to have overly massive SMBH relative to their stellar/dynamical mass (Shields et al., 2006; Trakhtenbrot et al., 2015; Vayner et al., 2017). While a significant fraction of galaxies with lower SMBH $<10^{9}$ M⊙ appear closer or within the scatter of the local scaling relations, galaxies with the luminous quasars and massive SMBH appear to be under massive relative to the mass of their SMBH. As outlined by (Lauer et al., 2007; Schulze & Wisotzki, 2014), the offset from the local scaling relations for the systems with more massive black holes is biased due to the steep decline in the galaxy mass function at the massive end. Integral field spectroscopy (IFS) behind adaptive optics is another method with which it is possible to disentangle the bright quasar emission from the extended emission of the host galaxy. A point spread function can be constructed using data channels confined to the broad emission line of the quasar. After the point spread function is normalized, it is subtracted from the rest of the data channels in the cube. This technique was first shown to be able to resolve host galaxies of low redshift ($z<0.2$) luminous type-1 quasars in seeing limited observations (Jahnke et al., 2004) and extended Ly$\alpha$ emission around high redshift quasars (Christensen et al., 2006). Later, when the first near-infrared IFS came online along with their own adaptive optics system, this technique was expanded to samples of higher redshift quasars in search for their host galaxies (Inskip et al., 2011; Vayner et al., 2016) and has shown to work on all the 8-10m class near- infrared IFS (e.g., SINFONI, NIFS, and OSIRIS). This technique has shown continued success in seeing limited optical IFS data as well (Herenz et al., 2015; Arrigoni Battaia et al., 2019). This PSF subtraction routine provides better contrast at smaller angular separations than HST, with an inner working angle of 0.1-0.2″, compared to $\sim$ 0.5″for HST (Vayner et al., 2016). Although today’s near-infrared IFSs are not sensitive enough to detect the stellar continuum from the quasar/AGN host galaxies, they can still detect extended ionized emission, enabling us to extract the dynamical properties of the galaxy (Inskip et al., 2011; Vayner et al., 2017) and compare systems to the local scaling relation. However, today, the largest fraction of quasar host galaxy masses still come from HST and mm-interferometric observations. Most likely, selection effects play an important role in determining whether there is a systematic offset from the local scaling relations among the different studies. Besides measuring the host galaxies and SMBH masses, there are vital open questions regarding the gas phase properties. Galaxies exhibit a correlation between the stellar mass and metallicity across a wide redshift range (Erb et al., 2006a; Sanders et al., 2015). It is often difficult to place galaxies with bright AGN on the mass-metallicity relationship due to limited contrast and the fact that the AGN has a strong impact on the ISM’s ionization state. What are the metallicities of the gas in quasar hosts? How does the metallicity in quasar host galaxies evolve with redshift? What is the dominant source of ionization in quasar hosts? What are the star formation rates? One of the best ways to measure the ionization properties of the gas in galaxies is through the BPT (Baldwin, Phillips $\&$ Terlevich) diagram (Baldwin et al., 1981; Veilleux & Osterbrock, 1987). The traditional BPT diagram plots the ratio of log([OIII]/H$\beta$ ) vs. log([NII]/H$\alpha$) and contains two clearly defined sequences: the star-forming sequence and the mixing sequence. The star-forming sequence provides information about the metallicity of HII regions, the stellar ionizing radiation field as well as information on the gas condition in star-forming regions. On the other hand, the mixing sequence consists of gas photoionized by hot stars, AGN, and shocks. It can potentially provide information on the hardness of the AGN ionizing radiation and the metallicity of the gas photoionized by the quasar/AGN (Groves et al., 2006) and shocks. Studies of high redshift star-forming galaxies have shown evidence for elevated line ratios relative to low redshift galaxies. At z$\sim$2, the observed elevated line ratios have been attributed to denser ISM conditions (Sanders et al., 2016) and harder ionizing radiation fields at fixed N/O and O/H abundances relative to typical z=0 galaxies (Strom et al., 2017). Evolutionary BPT models by Kewley et al. (2013a) are consistent with these observations. The evolutionary BPT models also provide a prediction on the evolution of the mixing sequence between z=0 and 3. The location of the mixing sequence moves to lower log([NII]/H$\alpha$) value at a relatively fixed log([OIII]/H$\beta$ ) value, primarily due to lower on average gas-phase metallicity at higher redshift (Groves et al., 2006; Kewley et al., 2013a). There is tentative evidence that gas photoionized by AGN is consistent with this picture, as there are several galaxies with AGN, which have emission line ratios offset from the local mixing sequence (Juneau et al., 2014; Coil et al., 2015; Strom et al., 2017; Nesvadba et al., 2017b; Law et al., 2018). Given the presence of the AGN, young stars and shocks in quasar host galaxies, it is crucial to spatially resolve the quasar host galaxy to understand the various contributions to gas ionization. In the distant Universe, this generally requires observations with an IFS and adaptive optics. Resolved BPT diagnostics in both nearby and distant AGN/quasar host galaxies have found regions with distinct photoionization mechanisms (Davies et al., 2014; Williams et al., 2017; Vayner et al., 2017). The question remains whether the ISM condition in the most luminous high redshift quasar host galaxies is different from local AGN and where they lie relative to the mass metallicity relationship. We have begun a survey to study the host galaxies of z$=1.4-2.6$ radio-loud quasars, which are likely to evolve into the most massive elliptical galaxies in the nearby Universe. The sample consists of eleven objects, selected to have young-jets with sizes up to 20 kpc in order to study their impact on galactic scales at early times. The observations consist of near-infrared IFS observation behind laser-guide-star adaptive optics (LGS-AO) at the W.M. Keck Observatory with the OSIRIS instrument. The survey aims to understand the gas phase conditions and ionization mechanisms in high redshift quasar host galaxies and search for evidence of quasar feedback and weighing the masses of the quasar hosts. The observations target nebular emission lines (H$\beta$ ,[OIII] ,H$\alpha$ ,[NII] ,[SII] ) redshifted in the near-infrared bands ($1-2.4$ µm), at the distance of our sample, the angular resolution of the OSIRIS/LGS-AO mode corresponds to approximately 1.4 kpc in projection. This paper is part two of two papers focusing on understanding the photoionization mechanisms of gas in radio-loud quasar host galaxies and weigh the mass of the galaxy and SMBH to compare them to the local scaling relations. Refer to paper I (Vayner et al., 2021) for details on the sample selection, properties, and data reduction. Details on archival HST imaging data set are presented in §2. Blackhole masses are presented in §3, we describe how we identify spatially-resolved dynamically quiescent regions in each quasar host galaxy in §4.1, resolved BPT diagrams and our interpretation of the line ratios are present in §5, dynamical masses of the quasar host galaxies and their place relative to the local scaling relations is presented in §7 & §6, we discuss our results in broader context of massive galaxy evolution in §8 and present our conclusions in §9. Notes on individual sources are presented in §9. Throughout the paper we assume a $\Lambda$-dominated cosmology (Planck Collaboration et al., 2014) with $\Omega_{M}$=0.308, $\Omega_{\Lambda}$=0.692, and Ho=67.8 km s-1 Mpc-1. All magnitudes are on the AB scale unless otherwise stated. ## 2 Archival HST imaging The sources within our sample have a rich set of multi-wavelength space and ground-based data sets. To assist in our analysis and interpretation of distinct regions within these quasar host galaxies, we utilize high angular resolution images from the Hubble Space Telescope. We download fully-reduced data from the Barbara A. Mikulski Archive for Space Telescopes (MAST). Table 1 list the archival HST observations used in this study. Table 1: Archival HST imaging Object \| Proposal ID \| Instrument \| Filter \| Exposure time ---\|---\|---\|---\|--- \| \| \| \| (s) 3C446 \| 12975 \| ACS-WFC \| F814W \| 2200 3C298 \| 13023 \| WFC3-UV \| F606W \| 1100 3C268.4 \| 13023 \| WFC3-UV \| F606W \| 1100 4C09.17 \| 5393 \| WFPC2 \| F555W \| 2100 3C9 \| 13945 \| ACS-WFC \| F814W \| 2040 We construct a model of the PSF using stars in the vicinity of the quasar within the FOV of each instrument. Images centered on each star are extracted in a box region of roughly 5″x 5″. We then subtract the local background for each star and median combine the stellar images into a normalized “master” PSF. This PSF is then re-scaled to the quasar’s peak flux and subtracted out at the spatial location of the quasar. In cases where the quasar was saturated, we scale the flux in the diffraction pattern of the PSF. ## 3 Black hole mass measurement Blackhole masses are calculated using the broad-H$\alpha$ line width and luminosity using the scaling relation from Greene & Ho (2005) for a single epoch SMBH mass estimate. We describe the details of the nuclear spectrum fitting in Vayner et al. (2021), which comprises of multi Gaussian models with a broad component for the BLR emission, a narrow Gaussian for the narrow-line region, and an intermediate width Gaussian for the case where there is an outflow. We use the flux and width of the broadest Gaussian to compute the black hole mass. For 3C9, 3C298, there are strong telluric/filter transmission issues that prevent accurate measurement of the FWHM for the emission line. For these targets, we use the Mg II single epoch black hole mass estimate from Shen et al. (2011). The black hole masses are provided in Table 2. We assume an uncertainty of 0.4 dex on the SMBH masses. Table 2: QUART Sample properties Name \| RA \| DEC \| z \| Lbol \| L178MHz \| MBH ---\|---\|---\|---\|---\|---\|--- \| J2000 \| J2000 \| \| ($10^{46}$ erg s-1 ) \| ($10^{44}$ erg s-1 ) \| M⊙ 3C 9 \| 00:20:25.22 \| +15:40:54.77 \| 2.0199 \| 8.17$\pm$0.31 \| 9.0 \| 9.87 4C 09.17 \| 04:48:21.74 \| +09:50:51.46 \| 2.1170 \| 2.88$\pm$0.14 \| 2.6 \| 9.11 3C 268.4 \| 12:09:13.61 \| +43:39:20.92 \| 1.3987 \| 3.57$\pm$0.14 \| 2.3 \| 9.56 7C 1354+2552 \| 13:57:06.54 \| +25:37:24.49 \| 2.0068 \| 2.75$\pm$0.11 \| 1.4 \| 9.86 3C 298 \| 14:19:08.18 \| +06:28:34.76 \| 1.439 \| 7.80$\pm$0.30 \| 12 \| 9.51 3C 318 \| 15:20:05.48 \| +20:16:05.49 \| 1.5723 \| 0.79$\pm$0.04 \| 4.0 \| 9.30 4C 57.29 \| 16:59:45.85 \| +57:31:31.77 \| 2.1759 \| 2.1$\pm 0.1$ \| 1.9 \| 9.10 4C 22.44 \| 17:09:55.01 \| +22:36:55.66 \| 1.5492 \| 0.491$\pm$0.019 \| 0.6 \| 9.64 4C 05.84 \| 22:25:14.70 \| +05:27:09.06 \| 2.320 \| 20.3$\pm$1.00 \| 4.5 \| 9.75 3C 446 \| 22:25:47.26 \| -04:57:01.39 \| 1.4040 \| 7.76 \| 4.4 \| 8.87 4C 04.81 \| 23:40:57.98 \| +04:31:15.59 \| 2.5883 \| 0.62$\pm$0.02 \| 9.3 \| 9.58 ## 4 Distinct regions within each quasar host galaxy In this section we outline how we define various regions within the data cube of each individual object. ### 4.1 Spatially-Resolved Dynamically “Quiescent” Regions In the first survey paper, we outline our methodology for fitting the emission lines in individual spaxels of our data cubes. From these fits, we derive integrated intensity, velocity, and velocity dispersion maps. The errors on the radial velocity and dispersion maps come directly from the Least-squares Gaussian model fit. The flux map’s errors come directly from integrating a noise spectrum in quadrature over the same wavelength range where the emission line is integrated. The fits are presented in the appendix of (Vayner et al., 2021). Here we utilize the radial velocity and dispersion maps to select regions with low-velocity dispersion to search for gas in gravitational motion and search for regions where star formation may have recently happened. We define a dynamically “quiescent” region of our data set that contains gas with a velocity dispersion ($V_{\sigma}$) less than 250 km s-1 . A quiescent region that belongs to the host galaxy of the quasar must have a radial velocity $<400$ km s-1 as we expect the maximum rotational velocity for a given host galaxy to be at most 400 km s-1 . The maximum rotational velocity found for the most massive galaxies studied with IFS at z$\sim$2 is about 400 km s-1 (Förster Schreiber et al., 2018). We define gas with $V_{r}>\|400\|$ km s-1 and $V_{\sigma}<$ 250 km s-1 belonging to a merger system. A system is defined as a merger if there are components with $V_{r}>\|400\|$ km s-1 or more than one distinct kinematic component. For example, in the 3C298 system, two galactic disks are found to be offset by less than 400 km s-1 . All radial velocity and velocity dispersion measurements are relative to the redshift of the quasar. The redshifts for the individual quasars are calculated in Vayner et al. (2021) and are taken from the fit to the narrow-line region. For sources with no spatially unresolved narrow emission, we use the redshift of the broad-line region. We label quiescent regions in the following manner: source name + direction + component A or B where A = component associated with the quasar, B = component associated with the galaxy merging with the quasar host galaxy. We follow these with a one or two-word comment about the region. Examples of description words are clump, diffuse, or tidal feature. Where clump referrers to a typical few kpc in size compact ionized emission typically seen in high redshift star-forming galaxies. Diffuse referrers to gas that has a surface density of less than typical clumpy star-forming regions. A tidal feature refers to ionized gas associated with a tidal tail in a merging system, containing both diffuse and clumpy ionized gas morphology. For each dynamically quiescent region, we construct a 1D spectrum by integrating over its spaxels. We show an example of this for 4C09.17 in Figure 1, spectra of distinct regions for the rest of the sources are presented in the appendix (Figures 11-18). The emission lines in each spectrum are fit with multi-Gaussian profiles. In these plots, we also present the outflow regions from (Vayner et al., 2021), to illustrate the location of dynamically quiescent regions relative to turbulent regions in these quasar hosts. From these fits, we derive integrated intensity and velocity dispersion that are presented in Tables 3 and 5. Figure 1: On the left, we present the spectra of distinct regions and fits to individual emission lines for the 4C09.17 system. On the right, we present the three-color composite where H$\alpha$ is color-coded to red, [OIII] to green and [NII] to blue. The contour outlines the spatial location of the region. The bar on the right in each stamp represents 1″or approximately 8.6 kpc at the system’s redshift. Table 3: Fluxes of distinct “dynamically quiescent” regions in individual sources Source \| Region \| $\rm F_{[OIII]}$ \| $\rm F_{H\alpha}$ \| $\rm F_{[NII]}$ ---\|---\|---\|---\|--- \| \| 10-17 erg s-1 cm-2 \| 10-17 erg s-1 cm-2 \| 10-17 erg s-1 cm-2 3C9 \| SE-SW component A \| 199$\pm$20 \| 65$\pm$7 \| 21$\pm$2 \| N component B \| 127$\pm$13 \| 40$\pm$4 \| 15$\pm$1 4C09.17 \| SW component A \| 9.55$\pm$0.98 \| 3.37$\pm$0.35 \| 1.32$\pm$0.2 \| W component B clumps \| 26$\pm$3 \| 10$\pm$1 \| 0.77$\pm$0.13 \| W component B diffuse \| 92$\pm$9 \| 25$\pm$2 \| 3.5$\pm$0.4 3C268.4 \| SW component B \| 245$\pm$25 \| 51$\pm$5 \| 9$\pm$1 7C 1354+2552 \| component A \| 46$\pm$1 \| 12$\pm$1 \| – \| E component B \| 6.2$\pm$0.6 \| 4.7$\pm$0.5 \| $<$0.7 3C298 \| SE component B ENLR \| 649$\pm$65 \| 188$\pm$20 \| 65$\pm$7 \| SE component B tidal feature \| 55$\pm$5 \| 20$\pm$2 \| 3.6$\pm$0.5 4C57.29 \| NE component A \| 26$\pm$3 \| – \| – \| N component A/B(?) \| 12$\pm$1 \| – \| – 4C22.44 \| N,S component A \| 54$\pm$5 \| 25$\pm$2 \| 3.5$\pm$0.3 4C05.84 \| SW component A clump \| 7.7$\pm$0.8 \| 3.3$\pm$0.3 \| 0.48$\pm$0.05 3C446 \| NW component A tidal feature \| 11$\pm$1 \| 5.9$\pm$0.6 \| $<$0.15 \| E-W component B \| 132$\pm$10 \| 48$\pm$4 \| 6.9$\pm$1.0 ### 4.2 Spatially unresolved narrow-line regions We search for narrow spatially unresolved emission in each object. To do so, we first subtracted a model of the extended emission from our fits to each emission line in individual spaxels. We then perform aperture photometry on the spatially unresolved emission and extract a spectrum. The emission lines are fit with multiple Gaussian profiles. The fluxes of the narrow emission ($\sigma<250$ km s-1 ) lines from unresolved regions are presented in Table 4. For sources where no unresolved narrow emission line is detected, we place a 1 sigma upper limit on the line flux. Based on the average angular resolution of about 0.1″, the unresolved narrow line emitting regions’ sizes are $<$ 1 kpc. Table 4: Fluxes of spatially unresovled narrow emission line regions in individual sources Source \| $\rm F_{[OIII]}$ \| $\rm F_{H\alpha}$ \| $\rm F_{[NII]}$ ---\|---\|---\|--- \| 10-17 erg s-1 cm-2 \| 10-17 erg s-1 cm-2 \| 10-17 erg s-1 cm-2 4C09.17 \| 52$\pm$5 \| 30$\pm$3 \| 102$\pm$1 3C268.4 \| 649$\pm$70 \| 239$\pm$20 \| 76$\pm$8 4C22.44 \| 1521$\pm$200 \| 102$\pm$10 \| 3.5$\pm$0.3 3C318 \| 35$\pm$4 \| 66 $\pm$7 \| 5$\pm$ 1 3C446 \| 11$\pm$1 \| 5.9$\pm$0.6 \| $<$0.15 7C1354 \| 65$\pm$10 \| $<$4 \| $<$4 4C57.29 \| $<12$ \| $<$5 \| $<$ 10 4C04.81 \| $<3$ \| $<2$ \| $<2$ 4C05.84 \| $<0.9$ \| $<1$ \| $<1$ ## 5 Nebular Emission Line Diagnostics and Sources of Gas Excitation In this section, we explore the photoionization mechanism in all distinct regions of each quasar host galaxy. The Baldwin, Phillips $\&$ Terlevich (BPT) diagram is used to differentiate between different gas photoionization sources (Baldwin et al., 1981). Here, we use the log([OIII]/H$\beta$ ) and log([NII]/H$\alpha$) line flux ratios to distinguish heating from young stars, AGN, and shocks. To construct the BPT diagram for our sources, we integrated each emission line over the same velocity width ($\Delta$V) and velocity offset relative to the redshift derived from the [OIII] emission line at each spaxel. We integrate the maps relative to [OIII] since it is typically the brightest emission line in any given spaxel. The higher signal-to-noise [OIII] emission line leads to a smaller spaxel-spaxel variation in the radial velocity and dispersion maps, creating a more consistent log([OIII]/H$\beta$ ) and the log([NII]/H$\alpha$) ratio between neighboring spaxels. We find that for the entire sample, the standard deviation on the log([OIII]/H$\beta$ ) ratio decreases by 0.2 dex compared to when integrating the cubes relative to the H$\alpha$ line. A resolved BPT diagram allows us to investigate the source of ionization throughout each quasar host galaxy. Due to sensitivity and, in some cases, wavelength coverage, we cannot create an integrated emission-line map for H$\beta$ on a similar scale to H$\alpha$ , [OIII] , or [NII] maps. For our BPT diagrams, we construct our H$\beta$ map by assuming case B recombination (H$\beta$ =H$\alpha$ /2.86) with a gas temperature of $10^{4}$ K and an electron density of $10^{2}$ cm-3. Assuming other recombination cases and ISM conditions with reasonable temperatures and densities would not change our results by a significant amount as the ratios between H$\beta$ and H$\alpha$ would only change at most by a factor of $\sim$1.3 (Osterbrock & Ferland, 2006). For sources with the brightest extended emission and wavelength coverage of both H$\alpha$ and H$\beta$ we find a maximum V band extinction of 1 mag, however in most cases, line ratios consistent with case B recombination. In regions where gas extinction is present, the log([OIII]/H$\beta$ ) ratios are preferentially lower. Only spaxels where at least H$\alpha$ and [OIII] were detected are analyzed and presented here. Typically [NII] is detected in far fewer spaxels compared to H$\alpha$ and [OIII] . For spaxels where only H$\alpha$ and [OIII] are detected, we calculate a limit on [NII] by integrating a sky spectrum over the same velocity width as [OIII] at the expected wavelength location of [NII] . In Figure 2 we plot the ratios from each spaxel. Diamonds are regions where [NII] , H$\alpha$ , and [OIII] were detected, and triangles are regions where only H$\alpha$ and [OIII] were detected with a limit on the [NII] flux. A total of 3160 spaxels are plotted corresponding to 21 distinct galactic regions. For each distinct regions identified in section 4.1 and from (Vayner et al., 2021) we over plot their line ratios and label them with a star. Individual spaxels typically have high uncertainties in their ratios but tend to cluster together on the BPT diagram. Integrating over distinct regions and re- calculating the ratios from a high SNR spectrum confirms that region’s true line ratio. To conserve space, we do not over-plot the error bars on points from individual spaxels in Figure 2, we only show the error bars of ratios computed for integrated values of the distinct regions. In Figure 3, we plot points of individual spaxels along with the error bars. Figure 2: Line ratio diagnostics of individual resolved distinct regions. In grey, we plot the line ratios of individual spaxels where at least [OIII] and H$\alpha$ was detected at an SNR$>$3\. Uncertainties on these line ratios are generally large; hence, we also integrate over all spaxels in individual regions to increase the SNR and lower the uncertainties on the line ratios. We show region-integrated line ratios with the colored symbols where each object has the same symbol, and each region has a different color. The names of the distinct region are present in the lower-left corner, and these match the names given in Table 3. We present the evolutionary models of the mixing and star-forming sequence with red and green curves from Kewley et al. (2013a). We show the upper limit of a sequence with a straight line and the lower boundary of each sequence with a dashed curve. Teal curves represent the bounds of the two sequences where the majority of the line ratio in low redshift galaxies fall. Our line ratios are consistent with a model where gas photoionized by the quasar is denser, has lower metallicity, and experiencing harder ionization compared to the gas photoionized by AGN in nearby galaxies. Figure 3: We present line ratio diagnostics for spaxels in each source where at least [OIII] and H$\alpha$ were detected at SNR great than 3. We show the uncertainties on the line ratio, which were omitted from figure 2 to conserve space. The dashed red line in each panel shows the theoretical separation between gas photoionized by star formation and AGN or shocks from Kewley et al. (2013a). Points above the line are photoionized by the quasar, while regions below are photoionized by O and B stars. Solid black mesh represents the location of radiative shocks from the astrophysical plasma modeling code MAPPINGS V (Alarie & Morisset, 2019). The shock model uses solar abundances from Gutkin et al. (2016). Either shocks or the quasar photoionizes the majority of the gas within our systems. ### 5.1 Ionization Diagnostic Models We find that a large portion of our line ratios values lies outside the two typical sequences of the BPT diagram (Figure 2). At a fixed log([NII]/H$\alpha$) nearly, all values are above both the local mixing and star-forming sequence. At a fixed log([OIII]/H$\beta$ ) value, nearly all values are outside the local mixing sequence. A large portion of points falls between the star-forming and mixing sequence, with relatively high log([OIII]/H$\beta$ ) values. Metallicity, electron density, the hardness of the ionization field, and the ionization parameter determines the location of the galaxy/region on a given sequence. With changing conditions in the ISM between z=0 and the median redshift of our sample, the locus of both the star- forming and mixing sequence can change locations (Kewley et al., 2013a). Galaxies at a fixed stellar mass have lower metallicities at high redshift compared to galaxies today (Yuan et al., 2013). Near the peak of the star formation rate density at z$\sim 1-3$, the ISM conditions and star formation histories of star-forming galaxies may differ from local systems. Star formation appears to happen in denser environments in the distant Universe with higher electron densities (Sanders et al., 2016), akin to conditions seen in local ULIRGs. According to Steidel et al. (2014); Strom et al. (2017) ISM in high redshift galaxies experiences harder ionization at a fixed N/O and O/H abundances than z=0 star-forming galaxies. On the other hand, galaxies at higher redshift have elevated N/O rations (Shapley et al., 2015). Taken together, Kewley et al. (2013a) shows that such changes in ISM conditions can alter the location of the star formation sequence between z=0 and z=2.5. Notably, the combination of harder ionization, electron density, and ionization parameter can shift the locus of the star-forming sequence to higher log([NII]/H$\alpha$) and log([OIII]/H$\beta$ ) values. It appears that UV/emission line selected galaxy samples tend to show a more significant shift from the SDSS star formation locus, as evident in a large sample of 377 star- forming galaxies explored by Strom et al. (2017). Nearly all their galaxies have a higher log([OIII]/H$\beta$ ) value at a fixed log([NII]/H$\alpha$) compared to local galaxies studied in SDSS. Various galaxy selection techniques may lead to samples of galaxies with inherently different ionization properties. However, the overall conclusion from studying star- forming galaxies in the distant Universe is that the line ratios of these systems lie on different star formation locus compared to the local Universe. Changes in the ISM conditions of distant galaxies may also lead to changes in the location of the mixing sequence. Kewley et al. (2013a) and Groves et al. (2006) show that for galaxies with lower metallicities, the mixing sequences shifts to lower log([NII]/H$\alpha$) values with relatively small changes in the log([OIII]/H$\beta$ ) value. We explore the various evolutionary models of the star-forming and mixing sequence with redshift and ISM conditions proposed by Kewley et al. (2013a). The best fit model to our sample is the one where the ISM of high redshift galaxies have more extreme condition (higher electron density, harder ionization field, and larger ionization parameters) and the metallicity of the gas photoionized by the quasar is at a lower metallicity compared to the gas ionized by local AGN in the SDSS sample. The median log([NII]/H$\alpha$) value is about 1.0 dex lower than that of the mixing sequence at z=0. If the primary source in the shift of the mixing sequence from z=0 to z=1.5-2.5 is a change in the gas phase metallicity, then the gas photoionized by the quasar in our sample has a metallicity a 1/2-1/5 of that in narrow line regions of z=0 AGN on the Kewley & Dopita (2002) metallicity scale. One of the consequences of the shift in the mixing sequence is that it becomes harder to distinguish between gas photoionized by AGN vs. star formation, especially in systems with potentially multiple ionization sources. Changes in the photoionization condition likely also play a role in the offset from the local mixing sequence. In a sample of local type-1 quasars, (Stern & Laor, 2013) found that systems with higher bolometric luminosities and higher Eddington ratios are systematically offset to lower log([NII]/H$\alpha$) ratios. Most of the gas in our quasar host galaxies lies in the mixing sequence where the gas is photoionized by a combination of quasar ionization and radiative shocks. In Figure 3, a significant fraction of the points in individual objects match the predicted location of radiative shocks on the BPT diagram. The radiative shock models assume solar metallicity and a preshock density of 215 cm-3 . With the present data, it is difficult to distinguish the percentage of photoionization from shocks vs. AGN. However, given the overlap of both line ratios and kinematics with shock models, we cannot rule them out to contribute to gas photoionization. A number of our distinct regions appear to have low log([NII]/H$\alpha$) values ($<$0.5) with low-velocity dispersion gas (V${}_{\sigma}<250$ km s-1 ). Morphologically these regions appear to be clumpy in their H$\alpha$ maps reminiscent of typical star-forming regions in galaxies at $z>1$. The line ratios of these points do not coincide with regions of fast or slow shocks photoionization on the BPT diagram (Allen et al., 2008; Newman et al., 2014). Archival HST data of 3C9, 3C298, 4C09.17, 3C268.4, 3C446 all showcase that the dynamically “quiescent” regions in these galaxies have clumpy morphology in rest-frame UV continuum data, similar to those of star-forming galaxies at these redshifts. In Figure 4, we overlay the H$\alpha$ emission from dynamically quiescent regions onto archival HST observations at rest-frame UV wavelength. Combining these clues suggests that the quasar does not entirely photoionize the gas in these regions. The elevated log([OIII]/H$\beta$ ) in these regions compared to local and distant star-forming regions may be from the mixing of photoionization from massive stars and the quasar. There is some evidence for this based on the morphology of the ionized gas and their respective log([OIII]/H$\beta$ ) ratios. For example, in 4C09.17, we see that more diffuse emission with low-velocity dispersion tends to have a higher log([OIII]/H$\beta$ ) value compared to clumpier regions where there is evidence for recent star formation activity and potentially more shielding from the quasar radiation field. Using the empirical star formation rate H$\alpha$ luminosity relationship from Kennicutt (1998), we convert the H$\alpha$ luminosity of the distinct extended quiescent regions to star formation rates. Most likely, the majority of these regions are photoionized by a combination of AGN and star formation, hence the derived star formation rates are upper limits. Regions “3C298 SE component B Tidal feature” and “4C09.17 W component B clumps” have line rations most consistent with photoionization by O/B stars, the star formation rates derived in these regions are closer to their actual value. To partially address the contribution from AGN to photoionization in dynamically quiescent regions, we also derive star formation rates only using spaxles that fall within the line ratio error inside the star-forming region of the BPT diagram based on the Kewley et al. (2013b) maximum separation between star formation and AGN photoionization. Generally, we find lower (1/2 - 1/10) star formation rates when using the BPT diagram criteria. We also measure the metallicities of these regions using the Pettini & Pagel (2004) empirical gas-phase metallicity - log([NII]/H$\alpha$) relationship. Given that log([NII]/H$\alpha$) is elevated in the presence of an AGN/quasar ionization field, the metallicities for the majority of the regions are also upper limits. We also calculate the metallicity using theoretically derived chemical abundance calibration for narrow-line regions of AGN (Storchi-Bergmann et al., 1998). We present quantitative values of these regions in Table 5, where we show the H$\alpha$ luminosity of each quiescent region, along with the star formation rate, metallicities, and velocity dispersion. Since nearly all of the unresolved narrow line regions are consistent with quasar photoionization, we do not include them in Table 5 with the exception of 3C318. In this object, the line ratios are consistent with star formation, indicating a nuclear starburst on scales $<$1 kpc. For cases where we do not detect any unresolved narrow emission, we place an average upper limit on the star formation rate. We do so if there is an ongoing nuclear starburst with a star formation rate below the sensitivity of OSIRIS at our given exposure times. We find an average star formation rate limit of 9 M⊙ yr-1 for the four objects (4C05.84, 4C04.81, 4C57.29, and 7C1354) where no strong narrow nuclear emission was detected. Table 5: Star formation rates and metallicities of distinct dynamically quiescent regions Source \| Region \| LHα \| SFR aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| L$\rm{}_{H\alpha_{BPT}}$ \| SFR BPTbbStar formation rate derived using spaxels that fall within the star formation sequence on the BPT diagram. \| 12+log(O/H) \| 12+log(O/H) \| $\sigma_{gas}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| 1043erg s-1 \| M⊙ yr-1 \| 1043erg s-1 \| M⊙ yr-1 \| PP04 \| SB98 \| km s-1 3C9 \| SE-SW A \| 2$\pm$0.2 \| 160$\pm$16 \| 0.20$\pm$0.02 \| 15$\pm$2 \| 8.6 \| 8.6 \| 173.1$\pm$25.7 \| N B \| 1.2$\pm$0.1 \| 99$\pm$10 \| 0.06$\pm$0.006 \| 5$\pm$1 \| 8.6 \| 8.5 \| 200.5$\pm$5 4C09.17 \| SW A \| 0.12$\pm$0.01 \| 9$\pm$1 \| 0.04$\pm$0.01 \| 3$\pm$1 \| 8.6 \| 8.5 \| 126.6$\pm$3 \| W B clumps \| 0.35$\pm$0.04 \| 28$\pm$3 \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| 8.2 \| 8.4 \| 136$\pm$4.7 \| W B diffuse \| 0.86$\pm$0.09 \| 68$\pm$7 \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| 8.4 \| 8.5 \| 146.2$\pm$7 3C268.4 \| SW B \| 0.64$\pm$0.06 \| 51$\pm$5 \| 0.20$\pm$0.02 \| 18$\pm$2 \| 8.5 \| 8.5 \| 144.6$\pm$5 7C 1354+2552 \| A \| 0.37$\pm$0.04 \| 29$\pm$3 \| 0.4$\pm$0.04 \| 33$\pm$3 \| $<$8.5 \| 8.4 \| 182.16$\pm$38.2 3C298 \| SE B TTddTidal tail \| 0.3$\pm$0.03 \| 22$\pm$2 \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| 8.5 \| 8.4 \| 109.6$\pm$5.5 3C318 \| Nuclear \| 1.1$\pm$0.1 \| 88$\pm$9 \| bbStar formation rate derived using spaxels that fall within the star formation sequence on the BPT diagram. \| aaStar formation rate derived using the H$\alpha$ luminosity of the entire distinct quiescent region \| $<$8.3 \| $<$8.5 \| 179.8$\pm$7.4 4C22.44 \| N,S A \| 0.40$\pm$0.04 \| 32$\pm$3 \| 0.2$\pm$0.02 \| 20$\pm$2 \| 8.4 \| 8.4 \| 184.6$\pm$6.5 4C05.84 \| SW A clump \| 0.14$\pm$0.01 \| 11$\pm$1 \| 0.05$\pm$0.01 \| 5$\pm$0.5 \| 8.5 \| 8.4 \| 198.7$\pm$16 3C446 \| NW A TTddTidal tail \| 0.07$\pm$0.001 \| 6 $\pm$1 \| 0.06$\pm$0.01 \| 5$\pm$1 \| $<$8.4 \| $<$8.4 \| 167.9$\pm$0.7 \| E-W B \| 0.48$\pm$0.05 \| 38$\pm$4 \| 0.15$\pm$0.02 \| 12$\pm$2 \| 8.5 \| 8.5 \| 204.3 $\pm$ 15 ccfootnotetext: Value indistinguishable from the integrated value over the entire dynamically quiescent region Figure 4: Detection of dynamically quiescent regions in archival Hubble Space Telescope observation. In the background, we show PSF-subtracted images of rest-frame UV emission in the quasar host galaxy. Overlaid in contours is the extended H$\alpha$ emission of the dynamically quiescent regions detected with OSIRIS. Note the similarities in both morphology and extent, indicating massive young stars photoionize at least a portion of the gas. The bar represents a spatial scale of 1″ or about 8.5 kpc. ## 6 SMBH-galaxy scaling relationships In this section, we place our galaxies on the velocity dispersion and galaxy mass vs. SMBH mass plots, comparing their locations to the local scaling relations ($M_{\bullet}-\sigma~{}$and $M_{\bullet}-M_{}~{}$). We calculate the SMBH masses from the broad H$\alpha$ luminosity and line width using the methodology presented in Greene & Ho (2005). The SMBH masses span a range of $10^{8.87-9.87}$ M⊙ . The velocity dispersions are taken from dynamically quiescent regions, while the galaxy masses are calculated from the virial equation and from modeling the radial velocity of targets with rotating disks and extracting a dynamical mass. ### 6.1 Host Galaxy Velocity Dispersion We identify several dynamically quiescent regions within most of the quasar host galaxies in our sample. These regions show lower log([NII]/H$\alpha$) line ratios and typically have clumpy morphology, reminiscent of the general star-forming regions seen in nebular emission and UV continuum in high redshift galaxies. In most galaxies, these regions lie away from any galactic- scale outflows. Hence their observed dynamics could be a probe of the galactic gravitational potential. These regions can be used to measure the velocity dispersion of our quasar host galaxies. In combination with the measured black hole masses, we can compare them to the local scaling relation between the SMBH mass and the velocity dispersion of the galaxy/bulge. In Figure 5, we plot the mass of the SMBH presented in Table 2 against the velocity dispersion of distinct quiescent regions measured with the H$\alpha$ line. Also, we include the velocity dispersion measured from CO (3-2) emission for 3C 298 from Vayner et al. (2017). We find a significant offset from the local scaling relation between the SMBH mass and the velocity dispersion of the galaxy/bulge ($M_{\bullet}-\sigma~{}$) (Gültekin et al., 2009; McConnell & Ma, 2013). To address the issue that the velocity dispersion may be systematically lower in dynamically quiescent regions offset from the quasar (3C446) or regions where the surface area of the narrow emission is significantly lower than the outflow (4C09.17, 3C298), we recalculate the velocity dispersion in a larger aperture that includes outflows and narrow emission. We see no strong systematic difference in the velocity dispersion of the narrow gas. The source integrated narrow velocity dispersion for 3C298, 3C446 and 4C09.17 are 100.7 $\pm$ 16.6, 187.5 $\pm$ 1.0 and 144.0$\pm$10.0 km s-1 , respectively. In the nearby universe, the velocity dispersion is typically measured inside the effective radius of the galactic bulge. The difference within our observations is that we do not know the bulges’ true sizes for our galaxies as we have no way to measure them with current data and telescope/instrument technology. However, the extents of the dynamically quiescent regions are in the range of the effective radii for bulges in massive ($10^{10.5-11.5}$ M⊙ ) galaxies studied in the CANDELS survey (Bruce et al., 2014). Figure 5: The location of our galaxies on the velocity dispersion vs. SMBH mass plot compared to the local $M_{\bullet}-\sigma~{}$relationship. We use the narrow H$\alpha$ emission line velocity dispersion of dynamically quiescent regions as a proxy for the stellar velocity dispersion. Red stars are the measurements from our sample, where we measure the velocity dispersion from the narrow H$\alpha$ emission line. We measure the black hole masses using the broad H$\alpha$ line from the broad-line-region discussed in section 3. The two blue stars represent the velocity dispersion measured in the disk of the host galaxy of 3C 298 and the tidal tail feature 21 kpc away from the quasar (Vayner et al., 2017). Blue circles are quasars from the Shields et al. (2006) sample, where they measure the velocity dispersion from CO emission lines. The yellow points are from quasars at z$>6$, where they measure the velocity dispersion from the 158 µm [CII] emission line (Decarli et al., 2018). Green points represent the local sample of all the bright cluster galaxies with SMBH greater than $10^{9}$ M⊙ taken from McConnell & Ma (2013). The blue curve represents the best fit to the entire galaxy sample from McConnell & Ma (2013) with the blue shaded region represents the intrinsic scatter on the $M_{\bullet}-\sigma~{}$relationship from the fit while the green curve is the fit to the sample studied in Gültekin et al. (2009). We find a significant offset between the galaxies in our sample and local BCG and the general local $M_{\bullet}-\sigma~{}$relationship. This large offset indicates that the host galaxies appear to be under-massive for their SMBHs. Figure 6: We present the location of individual galaxies compared to the local scaling relation between the mass of the SMBH and mass of the galaxy/bulge shown with a blue curve. Blue points represent systems with virial dynamical masses. Red points represent systems where we calculate the dynamical mass by modeling the radial velocity maps with an inclined disk model. Gray points show the location of galaxies at z$>0.5$, with lower SMBH masses and lower AGN luminosity compared to our sample. The blue curve represents the local scaling relationship as measured in McConnell & Ma (2013), with the shaded region representing the intrinsic scatter. We find the majority of our points are offset from the local scaling relationship, outside the observed scatter. There have seen numerous discussions in the literature about whether the velocity dispersion measured from gas traces the stellar velocity dispersion. The gas and stars might not have the same uniform distribution, and winds can broaden the nebular emission lines. Furthermore, the line of sight absorption and emission lines from which the velocity dispersion is calculated are luminosity weighted subject to galactic dust extinction. Because of the different light distribution between stars and gas, the measured velocity dispersion can differ. These differences can lead to increased scattering in any correlation between $\sigma_{}$ and $\sigma_{gas}$ . Data-sets that spatially resolve the gas and stellar components and have enough resolving power to separate multi-component emission from different regions (e.g., outflowing/in-flowing gas vs. galactic disk rotation) are important when making a comparison between $\sigma_{}$ and $\sigma_{gas}$ . In Bennert et al. (2018), for a large sample of local AGN, they find when fitting a single Gaussian component to the [OIII] emission line, they can overestimate the stellar velocity dispersion by about 50-100$\%$. Only by fitting multiple Gaussian components to account for both the narrow core and the broader wings of the [OIII] line profile can they adequately match the velocity dispersion from the narrow component of the [OIII] line to that of the stellar velocity dispersion. For their entire sample, the average ratio between the velocity dispersion of narrow Gaussian component and the stellar velocity dispersion is $\sim$1\. The 1 $\sigma$ scatter on the ratio between $\sigma_{[OIII],narrow}$ and $\sigma_{}$ is about 0.32 with a maximum measured ratio of about a factor of 2 which translates to a scatter in $\Delta\sigma=\sigma_{[OIII]}-\sigma_{}$ of 43.22 km s-1 with a maximum difference of about $\pm$100 km s-1 . However, only a few sources show such drastic velocity differences ($\sim 2.5\%$ of the entire sample, 82$\%$ of the sources show $\|\sigma_{[OIII]}-\sigma_{}\|<$ 50 km s-1 ). When fitting for the $M_{\bullet}-\sigma~{}$relationship with the narrow [OIII] emission as a proxy for stellar velocity dispersion, the resultant fit agrees with that of quiescent galaxies reverberation-mapped AGNs. These results indicate that for the sample as a whole Bennert et al. (2018) finds that both the stars and gas follow the same gravitation potential. Given the Bennert et al. (2018) results that demonstrates that the gas velocity dispersion can be used as a proxy for the stellar velocity dispersion, we follow a similar analysis using our IFS data sets to explore the location of our galaxies relative to the $M_{\bullet}-\sigma~{}$relation at high redshift. We attempted to the best of our ability to separate regions that contain galactic scales winds from those with more quiescent kinematics both spectrally and spatially with OSIRIS. Hence similar to Bennert et al. (2018) we think that the measured velocity dispersions in quiescent regions are good tracers of the galactic potential on average. Throughout the paper we use the narrow velocity dispersion of [OIII] and H$\alpha$ emission lines of dynamically quiescent regions as a proxy for the stellar velocity dispersion. We still find a significant offset for our sample after applying the observed scatter in the difference between $\sigma_{}$ and $\sigma_{gas}$ . This is also true when applied to the more distant quasar host galaxies studied with 158 µm [CII] emission. In nearby galaxies, there is a dependence on the velocity dispersion with the radius from the galaxy center (Bennert et al., 2018; Ene et al., 2019). However, based on the local galaxy observations, the velocity dispersion is unlikely to increase by $\sim$ 200 km s-1 that is necessary to bring the galaxies onto the local scaling relations. Using N-body smoothed-particle hydrodynamics simulations Stickley & Canalizo (2014) examines how the stellar velocity dispersion evolves in a binary galaxy merger. At various stages in the merger (e.g., a close passage, nucleus coalescence), they measure the stellar velocity dispersion along $10^{3}$ random lines of sight. Near each close passage and during coalescence, they find that the scatter on the velocity dispersion significantly increases from $\sim 5-11$ km s-1 to about 60 km s-1 with the average velocity dispersion a factor of $\sim$1.7 higher than after the galaxies have finished merging. For several sources in our sample (3C9, 3C298, and 3C446), the measured velocity dispersion might be higher than what it will be once the galactic merger is complete adding uncertainty due to projection effects. Following the simulations results, we add in quadrature an additional uncertainty on the velocity dispersion of 60 km s-1 given that the majority of our mergers are near coalescence or a close passage ($\Delta R<10$ kpc). It should be noted that this is near the maximum scatter seen in the simulations on $\sigma$. These simulations also find that for merging galaxies at their maximum separation, the measured velocity could be a factor of $\sim 1.7$ times smaller compared to the final system. They find that for a 1:1 merger, the maximum separation after the first passage is 10-100 kpc, which is much larger than any separation that we find in our systems from observed projected separations and measured relative velocities. No obvious merging companions are found for 3C318, 4C22.44, or 4C05.84 hence for these systems, the mergers might be past their coalescence stage where the measured velocity dispersion is close to its final value, and the scatter due to the line of sight effects is minimal ($\sim$ a few km s-1 ). However, we still apply an additional 60 km s-1 uncertainty in these regions. Even after these corrections are made to approximate the stellar velocity dispersion from the [OIII] emission lines in our sample, we still find that all of our systems are offset from the local scaling relation between the mass of the SMBH and the velocity dispersion of the bulge/galaxy. Given that we are dealing with relatively small sample size, we performed statistical tests to confirm the offset between the local scaling relation and our sample. We measure the offset between the observed and predicted velocity dispersion for the SMBH mass of our systems for each object. We use the local scaling relation fit from (McConnell & Ma, 2013), and H$\alpha$ measured SMBH masses. We construct a data set consisting of velocity differences. From bootstrap re- sampling of the velocity difference data set, we find that the average offset of 188.7 km s-1 is significant at the 3.25$\sigma$ level. Using Jackknife re- sampling similarly, we find that the offset is significant at the 3.3$\sigma$ level with the 95$\%$ confidence intervals of 154.4 km s-1 to 223.0 km s-1 on the velocity dispersion offset. Performing similar statistical tests on the Decarli et al. (2018) sample, we find an average offset of 178.8 km s-1 with a significance of the shift at 2.7$\sigma$ and 2.8$\sigma$ for Jackknife and bootstrap re-sampling, respectively from the local relationship. We also measure the offsets of massive BCGs in the local Universe from the $M_{\bullet}-\sigma~{}$relationship. Using a two-sided Kolmogorov-Smirnov test, we can ask if the observed offsets of the local and high redshift data sets are drawn from the same continuous distribution. We find a p-value of 5.7$\times 10^{-9}$, indicating that the two populations are not drawn from the same distribution. Applying the Kolmogorov-Smirnov test to the velocity dispersion offsets from our sample and in the higher redshift quasars, we find a p-value of 0.84, indicating that these two data sets could be drawn from the same continuous distribution. We find similar results by comparing the Shields et al. (2006) sample at $z\sim 2$ to our own and that of Decarli et al. (2018). ## 7 Dynamical mass measurements We can also test whether these systems lie off the local scaling relationship between the SMBH mass and the dynamical mass of the bulge/galaxy. First by using a virial estimator for the dynamical mass of the galaxy $\rm M_{virial}=\frac{C\sigma^{2}r}{G}$ where C=5 for a uniform rotating sphere (Erb et al., 2006b). We assume 7 kpc for the radius, which is the median effective radius of massive quiescent galaxies in the local Universe (Ene et al., 2019). Here $\sigma$ is derived from a Gaussian fit to the integrated spectra over the distinct region. For galaxies with multiple distinct regions, we derive two or more dynamical masses as there may be a dependence on the velocity dispersion as a function of position with the galaxy. For systems in a clear merger, the galactic component belonging to the quasar is used to estimate the dynamical mass since we are interested in the correlation between the SMBH and the velocity dispersion of the quasar host galaxy. For systems with velocity shear in the 2D radial velocity map, we fit a 2D inclined disk model to the kinematics data to measure the dynamical mass. The model is a 2D arctan function $V(r)=\frac{2}{\pi}V_{max}\arctan\Big{(}\frac{r}{r_{dyn}}\Big{)},$ (1) where V(r) is rotation velocity at radius r from the dynamical center, $V_{max}$, is the asymptotic velocity, and $r_{dyn}$ is the radius at which the arc-tangent function transitions from increasing to flat velocity. The measured line-of-sight velocity from our observations relates to V(r) as $V=V_{0}+\sin i\cos\theta V(r),$ (2) where $\cos\theta=\frac{(\sin\phi(x_{0}-x))+(\cos\phi(y_{0}-y))}{r}.$ (3) Radial distance from the dynamical center to each spaxel is given by $r=\sqrt{(x-x_{0})^{2}+\Big{(}\frac{y-y_{0}}{\cos i}\Big{)}^{2}},$ (4) where $x_{0},y_{0}$ is spaxel location of the dynamical center, we quote the value relative to the centroid of the quasar, $V_{0}$ is velocity offset at the dynamical center relative to the redshift of the quasar, $\phi$ is position angle in spaxel space, and $i$ is the inclination of the disk. $V_{max}$ is not the true “plateau” velocity of the galaxy’s disk. $V_{max}$ can have arbitrarily large numbers, especially when $r_{dyn}$ is very small (Courteau, 1997). To fit the data we use the MCMC code emcee. We construct the model in a grid with a smaller plate scale than the observed data, which gets convolved with a 2D Gaussian PSF with an FWHM measured from the quasar PSF image. The image is then re-sized to the plate scale of the data. We construct the priors on each of the seven free parameters. The prior on $V_{max}$ is $300<V_{max}<1000$ km s-1 the prior on both $x_{0},y_{0}$ is the boundary of the FOV of the imaged area, the prior on the position angle is $0<\phi<2\pi$, the prior on the inclination angle is $0<i<\pi/2$, the prior on the radius is $0.5<r_{dyn}<10$ pixels and the prior on $V_{0}$ is $-100<V_{0}<100$ km s-1 . We then sample this distribution with emcee. We initialize 1000 walkers for each free parameter using the best fit values from leastsquares fitting as the starting point, with a small random perturbation in each walker. We run MCMC for 500 steps starting from the perturbed initial value. The best-fit parameters, along with their confidence intervals, are presented in 6 for the quasar host galaxies of 7C 1354+2552, 3C9. For 3C 298 we do not see the disk in the ionized emission with the OSIRIS data, it is solely detected in CO (3-2) observations from ALMA, here we present the best fit values from Vayner et al. (2017). Also, we present $\Delta v_{obs}/2$, the average between the maximum and the minimum velocity along the kinematic major axis as determined by the position angle ($\phi$). We also present the intrinsic velocity dispersion ($\sigma_{0}$), measured along the kinematic major axis, towards the outskirts, away from the steep velocity gradient near the center of the disk. Table 6: Best fit values for each inclined disk model parameter Parameters \| 7C 1354+2552 \| 3C9 \| 3C298 ---\|---\|---\|--- $V_{max}$ [km s-1 ] \| 449.67${}^{+0.24}_{-0.64}$ \| 442.0${}^{+23.9}_{-5.7}$ \| 392${}^{+65}_{-65}$ $x_{0}$ [kpc] \| -2.37${}^{+0.04}_{-0.03}$ \| 0.5${}^{+2}_{-1}$ \| 0.43${}^{+0.1}_{-0.1}$ $y_{0}$ [kpc] \| -0.93${}^{+0.08}_{-0.08}$ \| -4.8${}^{+1.22}_{-1.5}$ \| 0 ${}^{+0.1}_{-0.1}$ $\phi$ [∘] \| 75.68${}^{+0.47}_{-0.48}$ \| 74.10${}^{+3.5}_{-35.4}$ \| 5.3${}^{+1.28}_{-1.28}$ $i$ [∘] \| 47.6${}^{+0.8}_{-0.8}$ \| 47.1${}^{+5.0}_{-3.7}$ \| 54.37${}^{+6.4}_{-6.4}$ $r$ [kpc] \| $<$0.017 \| 0.26${}^{+0.49}_{-0.14}$ \| 2.1${}^{+0.9}_{-0.9}$ $V_{0}$ [km s-1 ] \| -93.9${}^{+1.2}_{-1.7}$ \| -9.22${}^{+30.45}_{-86.46}$ \| -13.0${}^{+3.15}_{-3.15}$ $\Delta v_{obs}/2$ [km s-1 ] \| 309.84$\pm$20.47 \| 370.84$\pm$45.4 \| 150.0 $\pm$23.7 $\sigma_{0}$ [km s-1 ] \| 61.3$\pm$7.9 \| 186.9$\pm$32.7 \| 42.35 $\pm$ 12.68 In addition, we measure $V_{rot}/\sigma_{0}$ to gauge whether these systems are dynamically supported by rotation or dispersion. We measure a value of 6.8$\pm$1, 2.7$\pm$0.6, and 4.4$\pm$1.5 for 7C 1354, 3C9, and 3C298, respectively. In all systems, rotation dominates over the velocity dispersion for the dynamical support according to the criteria outlined by Förster Schreiber et al. (2018), and henceforth the systems can be classified as true disks. Assuming a spherically symmetric system, we can compute the total enclosed mass using the following formula: $M(R)=2.33\times 10^{5}rV_{r}^{2}/\sin(i)^{2}$ (5) Where $V_{r}$ is the radial velocity, $i$ is the inclination angle from the disk fit. For the radial velocity we use $\Delta v_{obs}/2$. Similarly, we assume a radius that is the median value of nearby BCGs (7.1 kpc). The selected radius should give us an absolute upper limit on the dynamical mass of the galaxy/bulge as this radius is much larger than the typical size of a galactic bulge at this redshift and is larger than the observed extent of the galactic disks. The reason for choosing a larger radius is to address the case where the quasar host galaxy extends to a larger radius and is not captured in our OSIRIS observations because they are not sensitive enough to low surface brightness emission at larger separation from the quasar. Virial and dynamical masses are presented in 7. However, it is not guaranteed that the extent of the ionized gas will match the stellar. We attempted to measure the size of the stellar continuum from the HST observations but were unsuccessful. Using the Galfit package, we were unable to constrain the radius due to the sources’ complex morphologies and the increased inner working angle due to the quasars’ brightness and saturated counts in the HST observations. Due to the limited sensitivity of OSIRIS to lower surface brightness emission, we are missing an accurate measurement of the plateau velocity for the galactic disks at large separations from the quasar. Hence, our fitting routine is unable to constrain $V_{max}$ for 3C9 and 7C 1354. Also, it appears that the turn over radius is very small for these two systems, smaller than the resolution element of our observations. For this reason, we are unable to constrain the turn over radius, and we only provide a limit. For 7C 1354, there is a degeneracy between the maximum velocity, turn over radius, and inclination; thus, the values that we provide are those that give the smallest velocity residual. Figure 7: Fitting an inclined disk model to the radial velocity map of the 3C9 quasar host galaxy. Far-left we plot the isolated radial velocity structure belonging to the quasar host galaxy of 3C9, middle left shows the best fit model overlaid as contours on top of the radial velocity map, middle right is the best fit model. On the right, we plot the residuals. Larger blue-shifted residuals at $-1$″ south from the quasar are from the outflow (3C9 SE component A outflow A). Figure 8: Fitting an inclined disk model to the radial velocity map of the 7C1354 quasar host galaxy. Far-left we plot the isolated radial velocity structure belonging to the quasar host galaxy of 7C1354, middle left shows the best fit model overlaid as contours on top of the radial velocity map, middle right is the best fit model, and on the right, we plot the residuals. Figure 9: Fitting an inclined disk model to the radial velocity map of the 3C298 quasar host galaxy. Far-left we plot the isolated radial velocity structure belonging to the quasar host galaxy of 3C298, middle left shows the best fit model overlaid as contours on top of the radial velocity map, middle right is the best fit model, and on the right, we plot the residuals. Table 7: Virial and dynamical mass values. Source \| Virial Mass \| Disk-fit Dynamical mass ---\|---\|--- \| $\times 10^{11}$ M⊙ \| $\times 10^{11}$ M⊙ 3C9 \| 2.5$\pm$0.7 \| 4.3$\pm$0.8 4C09.17 \| 1.3$\pm$0.1 \| – 3C268.4 \| 1.4$\pm$0.1 \| – 7C1354+2552 \| 1.5$\pm$0.3 \| 3.0$\pm$0.4 3C298 \| 0.45$\pm$0.13aacomputed from CO 3-2 velocity dispersion \| 0.6$\pm$0.1 3C318 \| 2.9$\pm$0.5 \| – 4C57.29 \| 3.3$\pm$0.3 \| – 4C22.44 \| 2.8$\pm$0.1 \| – 4C05.84 \| 3.3$\pm$0.1 \| – 3C446 \| 2.3$\pm$0.1 \| – Using the measured virial and disk fit dynamical masses and the SMBH masses, we can now compare our galaxies to the local $M_{\bullet}-M_{}~{}$relationship. Not only are these galaxies offset from the local $M_{\bullet}-\sigma~{}$relationship, but we also find that these galaxies are on an average offset from the local $M_{\bullet}-M_{}~{}$relationship. The galaxies need about an order of magnitude of stellar growth if they are to evolve into the present-day massive elliptical galaxies. We note that we have used two different methods for exploring the scaling relationship for galaxy mass vs. SMBH. Both the gas velocity dispersion method and dynamical measurement imply that the SMBH is over-massive compared to their host galaxies when exploring the local scaling relationship. It will be important to further compare these methods with larger samples, as well as future observations with the James Webb Space Telescope that will be able to directly measure the stellar velocity dispersion. ## 8 Discussion Our survey aimed to study host galaxies of redshift 1.4 - 2.6 radio-loud quasars through rest frame nebular emission lines redshifted into the near- infrared. We place distinct regions of each quasar host galaxy on the traditional BPT diagram (log([OIII]/H$\beta$ ) vs. log([NII]/H$\alpha$) ). The majority of the points for our sources lie outside the two local sequences (the mixing and star-forming sequence). In section 5, we introduce evolutionary BPT models from Kewley et al. (2013a) that indicate changes in the photoionization and metallicity conditions of the gas can shift both of the star-forming and mixing sequences. We fit these models to our data and find that the best- fitting model is the one where the gas in our quasar host galaxies is at least two to five times less metal-rich compared to the narrow line regions of nearby (z$<$0.2) AGN. The best fit model also indicates that the gas is ten times denser compared to nearby galaxies. In Figure 2, we show all of our points on the BPT diagram along with the best fit model. Kewley et al. (2013b) studied a sample of star-forming galaxies and galaxies with AGN in the redshift range of 0.8$<$z$<$2.5. They also find that galaxies at z$>$2 show elevated line ratios on average outside the local star formation and mixing sequences. They find that normal ISM conditions similar to the SDSS sample transition to the more extreme conditions with elevated line ratios somewhere between redshift z=1.5 and z=2. This is an agreement with our results as the majority of our targets are at $z>1.5$. High redshift radio galaxies also appear to show ISM conditions with metallicities that are lower compared to local AGN. In a study of a large sample of distant radio galaxies, Nesvadba et al. (2017a) finds that their gas-phase metallicities are at least half of that seen in local AGN. Nesvadba et al. (2017a) finds the same best-fitting model from Kewley et al. (2013a) as we do for our sample to explain their observed nebular line ratios. The average log([NII]/H$\alpha$) value of our sample seems to be lower than that of Nesvadba et al. (2017a); this could be due to the lower metallicity of our sample. On the other hand, a different approach to how we compute our line ratios can cause the discrepancy. Nesvadba et al. (2017a) only presents source integrated line ratios, while we explore ratios of distinct regions because we typically have a factor of 5-10 better angular resolution due to adaptive optics and hence can resolve the different ionized/kinematics structures of our galaxies. In the majority of our sources, we see significant variations in log([NII]/H$\alpha$) and log([OIII]/H$\beta$ ) values across each system, hence why we explore distinct regions. Line ratios from integrated spectra that include regions with various ionization sources and from multiple components of a merger system may shift towards higher log([NII]/H$\alpha$) , and log([OIII]/H$\beta$ ) values as the regions photoionized by the quasar/AGN tend to be brighter. Line ratios of galaxies with lower luminosity AGN compared to quasars/radio galaxies studied in Strom et al. (2017) are nearly all outside the local mixing sequence. These points overlap with the location of our line ratios and that of the radio galaxy sample. The MOSDEF survey finds similar results for their AGN sample at a range of bolometric luminosities (Coil et al., 2015). The ubiquity of elevated line ratios in host galaxies of AGN, meaning they are typically above the local mixing or star- forming sequence on the traditional BPT diagram (log([OIII]/H$\beta$ ) vs. log([NII]/H$\alpha$) ), indicates that regardless of the active galaxy population selected at z$\sim$2 the conditions of the gas that is photoionized by an AGN are different from those in the local Universe. Overall, this suggests that the ISM conditions in high redshift galaxies with AGN at a range of bolometric luminosities are different from those in local systems. The ISM conditions appear to be far more extreme with gas-phase metallicity lower than that of local AGN, suggesting an evolution in the ISM gas that is photoionized by AGN from z=0 to z=2.5. ### 8.1 Star formation and dynamically “quiescent” regions in the host galaxies In 9/11 quasar host galaxies within our sample, we see the morphology of clumpy star-forming regions seen in other galaxies at these redshifts. These regions also typically show lower velocity dispersion and lower log([NII]/H$\alpha$) values. We described them in more detail in section 4.1. These regions lie 1 - 21 kpc from the quasar and generally do not coincide with the location of galactic outflows. For sources with available HST imaging of rest-frame UV continuum, these regions appear bright and clumpy (see Figure 4). Taking these two results together indicates that O and B stars could photoionize a non-negligible fraction of the gas in these clumpy regions. In section 5, we derive an upper limit on their star formation rates and gas- phase metallicities. Taking this together, there is evidence for very recent star formation activity in 9/11 quasars within our sample. We find an average star formation rate of 50 M⊙ yr-1 for the star-forming regions within our sample. The average dynamical mass of our quasar host galaxies of $\sim 10^{11}$ M⊙ , indicates that the galaxies sit near the galaxy star formation rate - stellar-mass sequence at z$\sim$ 2 (Tomczak et al., 2016). Using the average metallicity of 8.5 measured in dynamically quiescent regions and the average stellar mass of our sample indicates that our galaxies sit on the mass-metallicity relationship at z$\sim$2 (Sanders et al., 2015). Quasars at z$\sim$ 2 are found to reside in galaxies with a broad range of star formation rates, spanning from quiescent to star-bursting galaxies. However, our sample preferentially contains quasar host galaxies in a star- burst phase. High specific accretion rate AGN are more likely to be found in star-bursting galaxies with rates on or above the star formation rate - stellar-mass sequence in the distant Universe (Aird et al., 2019). We selected to observe compact steep spectrum radio-loud quasars, this class of objects tend to contain younger AGN. One of the mechanisms to trigger a luminous AGN is through a massive gas-rich galaxy merger (Treister et al., 2012). During the ongoing merger, the loss of angular momentum feeds gas into the centers of galaxies, providing fuel for both star formation and SMBH growth. Since we selected AGN that may have recently triggered, they are more likely to be in an ongoing merger, where star formation activity is enhanced. Indeed, about 7/11 of the quasar host galaxies in our sample are mergers. This can explain why our sample preferentially contains galaxies with active or recent star formation and rapid accretion onto the SMBH. The measured star formation rates within our sample are significantly lower than those measured through dust emission in the far-infrared by the Herschel Space Observatory (Podigachoski et al., 2015; Barthel et al., 2017) for 4C04.81, 4C09.17, 3C318, and 3C298. The most likely explanation is that the quasar itself could partially heat the dust, H$\alpha$ misses a significant fraction of the obscured star formation, or the dust traces a different star formation history. Interestingly for 3C298 and 3C318, where both high spatial resolution imaging of the dust and H$\alpha$ emission is present, there is a significant misalignment between the maps. In places where we see evidence for recent star formation based on nebular emission-line ratios in 3C 298 and 3C 318, Barthel et al. (2018); Barthel & Versteeg (2019) does not see any dust emission. For the case of 3C 298 in the location where we see recent star formation traced by H$\alpha$ , we also detect a molecular reservoir; however, no dust emission is present there. Furthermore, in the places where dust emission exists in the case of 3C 298, the molecular gas at that location is stable against gravitational collapse and has been on a time scale longer than the propagation of the observed outflow. For the case of 4C09.17 and 4C04.81, no high-resolution dust maps are available. The dust emission could originate at any location within the $\sim$ 17″ Herschel SPIRE beam, which translates to a physical scale of about 150 kpc. Future high spatial resolution dust and molecular gas emission maps are necessary for proper comparison between the obscured and unobscured star formation traces and the molecular gas dynamics. ### 8.2 Offset from local scaling relations The majority of our systems appear to be offset from both local scaling relationships between the mass of the SMBH and mass and the velocity dispersion of the bulge (see Figures 5, 6). To explain the large offset from the local $M_{\bullet}-\sigma~{}$and $M_{\bullet}-M_{}~{}$relationship, we could invoke a significant error in the estimated SMBH masses. The bolometric luminosities of some of our quasars are far greater than those used for reverberation mapping in the nearby Universe, which is used in calibrating the single epoch SMBH mass (Greene & Ho, 2005). The SMBH masses would have to be off by 2-3 orders of magnitude to explain the observed offsets. By assuming that the SMBH grows primarily through gas accretion, we can use the Eddington luminosity formula to estimate the SMBH mass. Given that our quasars are most likely not all accreting at or close to the Eddington limit, this derived mass is effectively a lower limit. $\rm M_{SMBH,min}=\frac{L_{Eddington}}{1.26\times 10^{38}}M_{\odot}$ (6) For the derived bolometric luminosities in Table 2 we find a range of minimum SMBH of 107.5-9M⊙ , consistent with what we measure from single epoch SMBH masses using the H$\alpha$ emission line. Hence, ther is likely no significant error in our measured black hole masses. In Figure 10, we plot the offset from the local scaling relation against the redshift of each object from our sample, the local galaxies sample with SMBH $>10^{9}$ M⊙ and higher redshift quasars. Quasars with SMBH $>10^{9}$ M⊙ appear to be offset from the local scaling relationship, which indicates that SMBH growth appears to outpace that of stars in these systems. The SMBHs may grow rapidly up to a mass of several times $10^{9}$ M⊙ as early as 690 Myr after the Big Bang (Bañados et al., 2018), matching in mass to some of the most massive SMBH seen today. Some galaxies with lower luminosity AGN and lower mass SMBH also appear to be offset from the local scaling relation at z$>1$ (Merloni et al., 2010; Bennert et al., 2011). Given the typically large uncertainty on the measured values and generally small sample sizes, it is difficult today to say whether a different population of AGN/galaxies are offset differently from the local scaling relationships at z$>1$. Figure 10: Measured offset of galaxies from the local $M_{\bullet}-\sigma~{}$scaling relationship (McConnell & Ma (2013), $\log_{10}(M_{BH}/M_{\odot})=8.32+5.64\log_{10}(\sigma/200\rm\enspace km\enspace s^{-1})$). On the y-axis, we quantify the offset as the difference between the observed and predicted velocity dispersion from the local scaling relation based on the observed SMBH mass. We plot the observed offset from the local scaling relation against the redshift for individual targets. The labels are similar to 5. The shaded blue region represents the intrinsic scatter in the $M_{\bullet}-\sigma~{}$relationship for black holes with a mass of $10^{9.5}$ M⊙ . There is an overall offset for galaxies with massive SMBH at z$>$1 from the local $M_{\bullet}-\sigma~{}$relationship. We find no statistically significant difference in the offset between any of the high redshift samples, while there is a statistically significant offset from the local BCG points (green). Under the assumption that SMBH primarily grows through Eddington-limited gas accretion, the growth is expected to be exponential. The e-folding or “Salpeter” time scale is about 50-300 Myr, depending on the SMBH spin. At the mean redshift of our sample (z=1.87), the SMBHs are expected to experience 30-200 e-folds in mass growth. However, for a duty cycle of around $10\%$ (Wang et al., 2006) the expected number of e-folds drops down to about 3-20. Furthermore, the quasars in our sample are not accreting near the Eddington limit and can eventually switch from high to low accretion-rate mode, further decreasing the Eddington ratio. Hence, the SMBHs in our sample have nearly finished forming and will only further grow by a factor of 1.2-7 under the assumption of an Eddington ratio of 10$\%$, and a duty cycle of 10$\%$. If these galaxies are to assemble onto the local scaling relation and to evolve into the most massive early-type galaxies that we see today, then the rapid SMBH growths at early times in the Universe must be followed by significant stellar growth. On average, the galaxies within our sample need to grow the stellar mass within a radius of 7 kpc at a constant rate of 100 M⊙ yr-1 from z=2 to z=0. In the host galaxy of 3C 298, there is currently insufficient molecular gas for the galaxy to grow in stellar mass to match the mass predicted by the local scaling relationship. Furthermore, the quasar 3C 298 does not appears to live in an over-dense environment based on the number count of galaxies seen with the Spitzer space telescope imaging data (Ghaffari et al., 2017). The open question is, how do these galaxies obtain the stellar mass necessary to grow into the massive galaxies we see today? Are minor mergers responsible for growing these galaxies? Alternatively, is the accretion of cool gas from the CGM responsible for providing the fuel necessary for future star formation? The results we find for the host galaxy of 3C 298 favor the scenario where cold accretion flows from the CGM will supply most of the fuel necessary for future star formation. Another scenario could be that the Spitzer observations are too shallow to see lower mass galaxies. If these systems are gas-rich, they can supply future fuel for star formation from merging the gas in their CGM and ISM with the quasar’s host. Indeed in recent hydrodynamical simulation (Anglés-Alcázar et al., 2017) found that for dark matter halos with masses $>10^{12.5}$ M⊙ majority of the mass build up happens from gas accreted from the CGM and transfer/exchange of gas from CGM and ISM of cannibalized low mass galaxies. These simulations also find that stellar build-up from dry mergers and just accretion of stars from merging galaxies is not significant to grow the stellar mass of galaxies in massive halos. If this is the case for the majority of our galaxies, it implies that they have enormous amounts of gas inside their CGM. Our results can be in stark contrast to the predicted evolutionary paths of massive galaxies. In today’s theoretical framework (Di Matteo et al., 2005; Hopkins et al., 2008; Zubovas & King, 2012, 2014), feedback from the SMBH is predicted to happen once the galaxy reaches the local $M_{\bullet}-\sigma~{}$relationship. However, our systems are showcasing outflows that are capable of causing major feedback when the mass of the galaxies is a fraction of their predicted final mass from the local scaling relations. Also, the gas-phase metallicities are far lower than those observed in nearby AGN. The kinetic luminosities for half of the outflows in our sample are far lower than the values predicted in simulations for the bolometric luminosities of our quasars (Vayner et al., 2021). Ionized outflows in other samples show similar results, where about half the objects lie below the predicted minimum energy-coupling between the quasar and the outflow of 0.1$\%$ at z$\sim 2$ (Vayner et al., 2021). If all these systems are offset from the local scaling relationship, it would be easier to launch the outflows because their masses are smaller compared to if they were on the local scaling relations. This could lead to lower energy coupling efficiency. On the other hand, we might be missing a significant fraction of the gas within the outflows because a large portion of the gas could be in either a molecular or neutral phase. In the quasar host galaxy of 3C298, we find the majority of the gas in the outflow is in a molecular state, and once combined with the ionized kinetic luminosity we find values closer to those predicted in simulations. The kinetic luminosity in 3C298 is close to 1$\%$ of the quasar’s bolometric luminosity. Regardless if we are accounting all the gas in the outflow, outflows capable of causing feedback are occurring before the galaxies are on the $M_{\bullet}-\sigma~{}$relationship. We might need to reconsider our theoretical framework for massive galaxy formation, where the gas is not cleared from the galaxy in a single “burst” of feedback once the galaxies reach the $M_{\bullet}-\sigma~{}$relationship. Instead, the SMBH grows first in massive dark matter haloes, followed by a delayed growth of the host galaxy with regulatory feedback from the SMBH and near-continuous accretion of gas from the CGM and nearby satellite galaxies. In such a scenario, the coupling efficiency might be lower per outflow event, compared to a single burst model where a single outflow-event clears all the gas. At later times, maintenance mode feedback from jets can heat the CGM, preventing gas from cooling and accreting onto the galaxy, keeping the galaxies “quenched”. ## 9 Conclusions We have conducted a near diffraction-limited survey of 11 quasar host galaxies to study the distribution, kinematics, and dynamics of the ionized ISM using the OSIRIS IFS at the W.M. Keck Observatory. This survey paper aimed to understand the source of gas ionization, the physical and chemical conditions of the ISM and to estimate the masses of radio-loud quasar host galaxies at z$\sim$2\. We detected extended emission in all objects on scales from 1-30 kpc and found that: • The AGN photoionizes the majority of the extended gas. A significant fraction of emission-line ratios are found to reside between the two sequences on the traditional BPT diagram. By applying evolutionary models of the mixing and star-forming sequence from z=0 to z=2.5, we find that the gas within our systems is denser and has lower metallicity compared to the gas photoionized in local AGN. * • In 9 objects, we find dynamically quiescent regions, with lower average log([OIII]/H$\beta$ ) ratios. For systems where Hubble Space Telescope imaging is available, their morphologies are consistent with clumpy star-forming regions commonly observed in the distant Universe, indicating the presence of recent star formation. We find these systems to be forming stars at a maximum rate of 9-160 M⊙ yr-1 based on the H$\alpha$ luminosity. * • For nine objects, we are able to measure the mass of the SMBH, the stellar velocity dispersion using the narrow component of H$\alpha$ emission line as a proxy, and galaxy mass. We compare these nine objects to the local scaling relation between the mass of the SMBH and the mass or velocity dispersion of the galaxy. Our systems are both offset from the $M_{\bullet}-\sigma~{}$and $M_{\bullet}-M_{}~{}$relationship. Substantial growth is still necessary if these systems are to evolve into the present-day massive elliptical galaxies. Gas accretion from the CGM and gas-rich minor mergers are necessary to grow the stellar mass and increase the metallicity of the ISM. On average, the galaxies need to grow by at least an order of magnitude in stellar mass if they are to assemble onto the local scaling relations. A near-constant mass growth rate of $\sim$100 M⊙ yr-1 is necessary within a radius of 10 kpc from the quasar from z$\sim 2$ to 0. • Combining the results of this paper with (Vayner et al., 2021) we find evidence for the onset of feedback before the galaxies are on the local $M_{\bullet}-\sigma~{}$relationship. Luminous type-1 quasars are not the end phase of massive galaxy formation. The authors wish to thanks Jim Lyke, Randy Campbell, and other SAs with their assistance at the telescope to acquire the Keck OSIRIS data sets. We want to thank the anonymous referee for their constructive comments that helped improve the manuscript. The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Figure 11: Spectra of distinct regions along with fits to individual emission lines for the 3C 9 system. Figure 12: Spectra of distinct regions along with fits to individual emission lines for the 3C268.4 system. Figure 13: Spectra of distinct regions along with fits to individual emission lines for the 7C 1354+2552 system. Figure 14: Spectra of distinct regions along with fits to individual emission lines for the 3C 298 system. Figure 15: Spectra of distinct regions along with fits to individual emission lines for the 3C 446 system. Figure 16: Spectra of distinct regions along with fits to individual emission lines for the 4C 57.29 system. Figure 17: Spectra of distinct regions along with fits to individual emission lines for the 4C 22.44 system. Figure 18: Spectra of distinct regions along with fits to individual emission lines for the 4C 05.84 system. ## Appendix A 3C 9 3C9 is a luminous quasar at z = 2.019922 with a prominent blue rest-frame UV continuum. For this source, we identify three distinct regions. “SE-SW component A” is a region with a ring-like morphology associated with the 3C9 quasar host galaxy. We measure a velocity dispersion from a Gaussian fit to the nebular emission lines of 407.6$\pm$12.9km s-1 and the kinematics resembling a rotating disk. “SE component A” is classified as an outflow region with a very high emission line FWHM of 1362.7$\pm$60.5 km s-1 and elevated log([OIII]/H$\beta$ ) and log([NII]/H$\alpha$) ratios relative to the rest of the system. “N component B” is the merging galaxy in the 3C9 system showcasing a line FWHM of 472.15$\pm$11.8 km s-1 and a velocity offset of $\sim$200 km s-1 from the quasar. The projected spatial separation between the two apparent nuclei is 9 kpc. The quasar lies in the galaxy with a ring-like morphology showing the kinematic structure of a disk. Archival HST imaging of rest-frame UV continuum shows the ring morphology as well (see Figure 4), indicating very recent star formation activity in the ring. The merging galaxy “N component B” appears to be a dispersion dominated system with active star formation and appears in rest-frame UV emission. The 3C9 system best resembles the local galaxy merger system Arp 148 (z=0.036), also known as Mayall’s Object. The outflow in this system appears to be emanating from the ring of the galaxy with the quasar. ## Appendix B 4C 09.17 4C 09.17 is a luminous quasar at z=2.117 with a blue UV continuum. For this source, we identify four distinct regions. “SW component A” is a star-forming clump associated with the quasar host galaxy. The spectrum of this region shows a single narrow emission line with an FWHM of 312.0$\pm$7 km s-1 . “S/E component A” is an outflow region driven by the quasar, the nebular emission lines for this region have an FWHM of 887.2$\pm$22.4 km s-1 . A second narrow component is required for a good fit for each emission line in this region, with a line FWHM of 290.4$\pm$29.9 km s-1 . “W component B clumps” is a region part of the merging galaxy within the 4C09.17 system. The region consists of clumpy emission selected by isolating spaxels with an H$\alpha$ line surface density $>6\times 10^{-16}$ erg s-1 cm-2 arcsec-2. “W component B diffuse” is emission associated with “diffuse” ionized emission in the merging galaxy selected by isolating spaxels with an H$\alpha$ spatial line surface density $<6\times 10^{-16}$ erg s-1 cm-2 arcsec-2. The diffuse region shows higher log([OIII]/H$\beta$ ) and log([NII]/H$\alpha$) line ratios associated with both AGN and star formation photoionization while the clumpy regions of the merging galaxy showcase lower ionization levels consistent with photoionization by star formation. This region is associated with bright UV emission in HST imaging of this object (Lehnert et al., 1999). “S/E component A outflow” shows high log([NII]/H$\alpha$) and log([OIII]/H$\beta$ ) values relative to the rest of the system, indicating this region is predominantly photoionized by the quasar. The 4C09.17 system is a merger of two galaxies with velocity offsets of $\sim$1000 km s-1 and a projected separation of $\sim$ 4 kpc. HST imaging of rest-frame UV continuum (see Figure 4) shows evidence for a population of young hot stars indicating recent star formation activity. The majority of the star formation activity is confined to the merging galaxy. ## Appendix C 3C 268.4 3C 268.4 is a luminous quasar at z=1.39, with a slightly reddened UV continuum compared to the average type-1 quasar. For this target, we identified two distinct regions. “SW component A” is an outflow driven by the quasar. The FWHM of the emission lines is 2075$\pm$354 km s-1 as measured from the Gaussian fit to the [OIII] line. The spectrum extracted over this region also shows a narrow component with an FWHM of 603.7$\pm$54.9 km s-1 , most likely signaling emission from an extended narrow-line region close to the quasar. Because of issues with miss-assignment of flux in the OSIRIS pipeline (Lockhart et al., 2019), the rows below and above the centroid of the quasar do not have properly extracted spectra in the H band observations of this object. Hence we do not have a good spectrum of the extended emission in a 0.2-0.3″ radius around the quasar in the H band, which covers the H$\alpha$ and [NII] emission lines of the ionized outflow. “SW component B” is a region associated with the merging galaxy, showcasing clumpy morphology in ionized gas emission. The emission lines have an FWHM of 367.7 $\pm$ 3.9 km s-1 and an offset of $-300$ km s-1 relative to the redshift of the quasar. The log([OIII]/H$\beta$ ) line ratios are lower for this region compared to the rest of the system, consistent with a mixture of AGN and star formation photoionization. This region is also associated with bright rest-frame UV continuum emission, seen with HST observations of this target Hilbert et al. (2016). ## Appendix D 7C 1354+2552 7C 1354+2552 is a luminous quasar at z=2.0064 with a blue UV continuum. For this target, we identify two distinct regions. “Component A” is the extended emission associated with the quasar host galaxy. The kinematics show a smooth velocity gradient, indicating the presence of a galactic disk. The size, morphology, and kinematics of the disk are similar to that of star-forming galaxies on the more massive end of the star formation main sequence at $z\sim$2 (Förster Schreiber et al., 2018). We measure an emission line FWHM of 357.2$\pm$2.0 km s-1 on the redshifted side of the disk and 497.7$\pm$6.5 km s-1 on the blue-shifted side of the disk. Although this region only has a single label (“component A”), in Figure 13 rows one and two show the fits to the red and blue-shifted sides of the disk that are part of this region. This region is selected based on the location where H$\alpha$ emission is detected. This is done to boost the SNR in the H$\alpha$ line as it appears to be clumpier, more compact, and less extended than [OIII] . In Table 3 we provide values integrated over the entire galactic disk. “E component B ” is a region associated with the merging galaxy at a projected separation of 6-7 kpc. The kinematics are consistent with a dispersion dominated galaxy. The entire “component A” is consistent with quasar photoionization. The gas in “E component B” is photoionized by star formation. ## Appendix E 3C 298 3C298 is a luminous quasar at z=1.439 with a slightly reddened UV continuum. For this target, we identify five distinct regions. We present a detailed discussion of each region in Vayner et al. (2017). “W/E component A” are outflow regions with a bi-conical morphology, where the western (W) is the redshifted receding cone, and the eastern (E) is the approaching cone. In Vayner et al. (2017), they are referred to as the red(blue) shifted outflow region. The emission lines over the outflows are very broad, with FWHM up to $\sim$1500 km s-1 . A combination of shocks and quasar activity is likely responsible for photoionizing the gas. “SE component B outflow” is an outflow region belonging to a merging galaxy. “SE component B ENLR” is an extended narrow-line region belonging to the disk of the merging galaxy, with gas photoionized by the quasar or secondary AGN. “SE component B Tidal feature” is a region of the merging galaxy with active/recent star formation as evident by lower log([NII]/H$\alpha$) and log([OIII]/H$\beta$ ) values compared to the rest of the regions. ## Appendix F 3C318 3C318 is a luminous quasar at z=1.5723 with a reddened UV continuum. There is evidence for a spatially unresolved nuclear star-burst with an upper limit on the star formation rate of 88$\pm$9M⊙ yr-1 . This star formation rate is far lower than the far infrared derived rate of 580M⊙ yr-1 . The extinction towards the nuclear region measured from Willott et al. (2000) alone cannot explain the mismatch between the H$\alpha$ and far-infrared derived SFR. Either a larger fraction of the far-infrared emission is from dust that is being heated by the AGN, or the far-infrared emission traces a different star formation history than H$\alpha$ (Calzetti, 2013). No narrow extended emission is detected in this object. The merger status of this object is unclear. Two nearby galaxies to the north and west of the quasar are visible in archival HST imaging (Willott et al., 2000). We do not detect the western object that is 2″ away from the quasar in our OSIRIS observations in any emission line. Willott et al. (2007) studied this object with PdBI through CO 2-1 emission at a fairly coarse ($\sim$8 arcseconds) resolution. There appears to be CO emission that could plausibly be associated with the western object. We have recently obtained a much higher angular resolution CO 3-2 spectroscopy of this target that will be discussed in detail in a forthcoming paper. We confirm the existence of CO 3-2 emission associated with the CO 2-1 emission. We resolve the molecular emission into multiple components. However, the CO 3-2 emission is not associated with either one of the galaxies seen in the HST data. We obtained a wide field of view IFS observations of this target with KCWI aimed at attempting to measure the redshifts of the nearby galaxies and to confirm the merger scenario of this object. We detect both the northern and western objects in the continuum. We confirm that the northern target is at a different redshift than the quasar from the detection of [OII] emission, while for the western object, a reliable redshift is challenging to determine with the current data set. Hence no clear evidence of a companion galaxy that is part of a merger is detected for this quasar associated with the brightest galaxies seen in optical imaging within a few arcseconds from the quasar. ## Appendix G 4C 57.29 4C 57.29 is a luminous quasar at z=2.1759 with a blue UV continuum. For this target, we identify two regions. Region “NE component A” belongs to the host galaxy of the quasar. The relatively high log([OIII]/H$\beta$ ) value indicates that this region is consistent with being photoionized by the quasar. The 500.7 nm [OIII] is the only emission line detected for this region. The region is marginally resolved, making it hard to measure the kinematic structure. We require a double Gaussian fit to the [OIII] emission in this region to obtain a good fit, and we measure an FWHM of 474.3 and 502.5 km s-1 with offsets of 35.0 km s-1 and -1050.1 km s-1 relative to the redshift of the quasar. We identify a second region north of the quasar. It is unclear if it belongs to a merging galaxy or the quasar host galaxy. There is a $\sim 100$ km s-1 offset from the quasar, and the line has an FWHM of 550.13$\sim$19 km s-1 . This region is also only detected in [OIII] . The SNR is too low to measure any kinematics structure. ## Appendix H 4C 22.44 4C22.44 is a luminous quasar at z=1.5492 with a reddened UV continuum. Similar to 3C318, we do not detect any evidence for a merging galaxy for this system. For this target, we identify a single region, “N, S component A”. The kinematics of this region may be consistent with a galactic disk belonging to the quasar host galaxy. We see evidence for a smooth gradient in the radial velocity map, however, the region is marginally resolved. We measure an emission line FWHM of 434.8 km s-1 . The region is consistent with being ionized by star formation with some contribution from quasar photoionization. ## Appendix I 4C 05.84 4C05.84 is a luminous quasar at z=2.323 with a slightly reddened UV continuum. For this target, we identify three distinct regions. Regions “S component A” and “NE component A” are the blue(red) shifted outflow regions resembling a bi-conical outflow. They showcase broad extended emission with a line FWHM of $\sim$800 km s-1 . The quasar photoionizes these regions. Region “SW component A clump”, shows a line FWHM of 467.9$\pm$3.0 km s-1 and is photoionized by a combination of star formation and the quasar. This clump is also detected in NIRC2 imaging of this object studied by Krogager et al. (2016), where they consider this clump to be associated with a damped Ly$\alpha$ system. However, here we confirm that this objected is part of the quasar host galaxy. We find no evidence for a merging galaxy within our OSIRIS observations. ## Appendix J 3C 446 3C446 is a quasar at z = 1.404. For this target, we identify two regions, “N component A tidal feature” is a region belonging to the quasar host galaxy, resembling a tidal feature that is most likely induced by the merger. We measure an FWHM of 395.14$\pm$2.0 km s-1 for this region. “E-W component B” belongs to the merging galaxy, a portion of it resembles a tidal feature, counter to the tidal arm of “N component A tidal feature.” For this region, we measure a line FWHM of 558.5$\pm$63 km s-1 however, it appears to be a blend of two velocity components. It is unclear where the nucleus of the merging galaxies resides. It could be that it has already merged with that of the quasar host galaxy. 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# Run-Time Safety Monitoring of Neural-Network-Enabled Dynamical Systems Weiming Xiang Weiming Xiang is with the School of Computer and Cyber Sciences, Augusta University, Augusta, GA, 30912 USA e-mail<EMAIL_ADDRESS> ###### Abstract Complex dynamical systems rely on the correct deployment and operation of numerous components, with state-of-the-art methods relying on learning-enabled components in various stages of modeling, sensing, and control at both offline and online levels. This paper addresses the run-time safety monitoring problem of dynamical systems embedded with neural network components. A run-time safety state estimator in the form of an interval observer is developed to construct lower-bound and upper-bound of system state trajectories in run time. The developed run-time safety state estimator consists of two auxiliary neural networks derived from the neural network embedded in dynamical systems, and observer gains to ensure the positivity, namely the ability of estimator to bound the system state in run time, and the convergence of the corresponding error dynamics. The design procedure is formulated in terms of a family of linear programming feasibility problems. The developed method is illustrated by a numerical example and is validated with evaluations on an adaptive cruise control system. ###### Index Terms: Dynamical systems, interval observer, neural networks, run-time monitoring. ## I Introduction Complex dynamical systems, for instance, medical robotic systems, autonomous vehicles and a variety of cyber-physical systems (CPS), have been increasingly benefiting from the recent rapid development of machine learning (ML) and artificial intelligence (AI) techniques in various aspects ranging from modeling to control, for instance stabilizing neural network controllers and state observers [1, 2, 3], adaptive neural network controllers [4, 5] and a variety of neural network controllers [6]. However, because of the well-known vulnerability of neural networks, those systems equipped with neural networks which are also called neural-network-enabled systems are only restricted to scenarios with the lowest levels of the requirement of safety. As often observed, a slight perturbation that is imposed onto the input of a well- trained neural network would lead to a completely incorrect and unpredictable result [7]. When neural network components are involved in dynamical system models such as neural network controllers applied in feedback channels, there inevitably exist noises and disturbances in output measurements of the system that are fed into the neural network controllers. These undesired but inevitable noises and disturbances may bring significant safety issues to dynamical systems in run-time operation. Moreover, with advanced adversarial machine learning techniques recently developed which can easily attack learning-enabled systems in run time, the safety issue of such systems only becomes worse. Therefore, for the purpose of safety assurance of dynamical systems equipped with neural network components, there is a need to develop safety monitoring techniques that are able to provide us the online information regarding safety properties for neural-network-enabled dynamical systems. To assure the safety property of neural networks, there are a few safety verification methods developed recently. These approaches are mostly designed in the framework of offline computation and usually represent high computational complexities and require huge computation resources to conduct safety verification. For instance, the verification problem of a class of neural networks with rectified linear unit (ReLU) activation functions can be formalized as a variety of sophisticated computational problems. One geometric computational approach based on the manipulation of polytopes is proposed in [8, 9] which is able to compute the exact output set of an ReLU neural network. In their latest work [10, 11], a novel Star set is developed to significantly improve the scalability. Optimization-based methods are also developed for verification of ReLU neural networks such as mixed-integer linear programming (MILP) approach [12, 13], linear programming (LP) based approach [14], and Reluplex algorithm proposed in [15] which is stemmed from classical Simplex algorithm. For neural networks with general activation function, a simulation-based approach is introduced in [16] inspired by the maximal sensitivity concept proposed in [17]. The output reachable set estimation for feedforward neural networks with general activation functions is formulated in terms of a chain of convex optimization problems, and an improved version of the simulation-based approach is developed in the framework of interval arithmetic [18, 19]. These optimization and geometric methods require a substantial computational ability to verify even a simple property of a neural network. For example, some properties in the proposed ACAS Xu neural network in [15] need even more than 100 hours to complete the verification, which does not meet the real-time requirement of run-time safety monitoring for dynamical systems. One way to resolve the real-time challenge of run-time monitoring is to develop more computational efficient verification methods that can be executed sufficiently fast to satisfy the run-time requirement such as the specification-guide method and Star set method do in [19, 10, 11]. However, these offline methods are essentially with an open-loop computation structure and there always exist computational limitations for these offline algorithms implemented online. On the other hand, inspired by observer design techniques in classical control theory, another way is to design a closed-loop structure of run-time monitoring using the instantaneous measurement of the system, which is illustrated in Figure 1. Recently, interval observer design techniques have been developed to provide lower- and upper-bounds of state trajectories during the system’s operation which can be used to conduct run- time monitoring for dynamical systems [20, 21, 22, 23, 24, 25, 26]. Inspired by the idea of interval observer methods developed in the framework of positive systems [27, 28, 29, 30], a novel run-time safety state estimator is developed for neural-network-enabled dynamical systems. The run-time state estimator design consists of two essential elements, the auxiliary neural networks and observer gains. Briefly speaking, the auxiliary neural networks stemmed from the neural network in the original system are designed to deal with neural network components and observer gains are handling system dynamics, ensuring positivity of error states and convergence. The design process can be formulated in terms of a family of LP feasibility problems. Notably, if the neural network component is driven by measurement instead of system state, the design process is independent with the neural network which makes the developed method applicable for neural-network-enabled systems regardless of the scalability concern for the size of neural networks. Figure 1: The generic structure of run-time safety monitoring of neural- network-enabled dynamical systems considered in this paper. The rest of this paper is organized as follows. In Section II, some preliminaries and problem formulation are introduced. The main result, run- time monitoring design, is proposed in Section III. Two auxiliary neural networks are derived from the weights and biases of the neural network of original systems. Interval observers are designed in the framework of LP problems and furthermore, the convergence of the error system is discussed. In Section IV, the developed approach is applied to an adaptive cruise control (ACC) system. Conclusions and future remarks are given in Section V. _Notations:_ $\mathbb{R}$ and $\mathbb{R}_{+}$ stand for the sets of real numbers and nonnegative real numbers respectively, and $\mathbb{R}^{n}$ denotes the vector space of all $n$-tuples of real numbers, $\mathbb{R}^{n\times n}$ is the space of $n\times n$ matrices with real entries. We denote $I_{n\times n}\in\mathbb{R}^{n\times n}$ as an $n$-dimensional identity matrix and $\mathbf{1}_{n\times 1}=[1,\ldots,1]^{\top}\in\mathbb{R}^{n\times 1}$. Matrix $A\in\mathbb{R}^{n\times n}$ is a Metzler matrix if its off-diagonal entries are nonnegative, and $\mathbb{M}_{n}$ denotes the set of the Metzler matrices of the size $n$. For $x\in\mathbb{R}^{n}$, $x_{i}$ denotes the $i$th component of $x$, and the notation $x>0$ means $x_{i}>0$ for $1\leq i\leq n$. $\mathbb{R}_{+}^{n}=\\{x\in{\mathbb{R}^{n}}:x>0\\}$ denotes the nonnegative orthant in $\mathbb{R}^{n}$, $\mathbb{R}_{+}^{n\times m}$ denotes the set of $n\times m$ real non-negative matrices. For $x\in\mathbb{R}^{n}$, its 1-norm is $\left\\|x\right\\|=\sum\nolimits_{k=1}^{n}{\left\|{{x_{k}}}\right\|}$. Similarly, for an $A\in\mathbb{R}^{n\times m}$, $a_{ij}$ denotes the element in the $(i,j)$ position of $A$, and $A>0$ means that $a_{ij}>0$ for $1\leq i,j\leq n$. $A>B$ means that $A-B>0$. $\left\|A\right\|$ means $\left\|a_{ij}\right\|$ for $1\leq i,j\leq n$ and $A^{\top}$ is the transpose of $A$. ## II System Description and Problem Formulation ### II-A Neural-Network-Enabled Dynamical Systems In this work, we consider an $L$-layer feedforward neural network $\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$ defined by the following recursive equations in the form of $\displaystyle\begin{cases}\eta_{\ell}=\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell}),~{}\ell=1,\ldots,L\\\ \Phi(\eta_{0})=\eta_{L}\end{cases}$ (1) where $\eta_{\ell}$ denotes the output of the $\ell$-th layer of the neural network, and in particular $\eta_{0}\in\mathbb{R}^{n_{0}}$ is the input to the neural network and $\eta_{L}\in\mathbb{R}^{n_{L}}$ is the output produced by the neural network, respectively. $W_{\ell}\in\mathbb{R}^{n_{\ell}\times n_{\ell-1}}$ and $b_{\ell}\in\mathbb{R}^{n_{\ell}}$ are weight matrices and bias vectors for the $\ell$-th layer. $\phi_{\ell}=[\psi_{\ell},\cdots,\psi_{\ell}]$ is the concatenation of activation functions of the $\ell$-th layer in which $\psi_{\ell}:\mathbb{R}\to\mathbb{R}$ is the activation function. The following assumptions which are related to activation functions are proposed. ###### Assumption 1 Assume that the following properties holds for activation functions $\psi_{\ell}$, $\ell=1,\ldots,L$: 1. 1. Given any two scalar $x_{1}$ and $x_{2}$, there exists an $\alpha>0$ such that $\left\|\psi_{\ell}(x_{1})-\psi_{\ell}(x_{2})\right\|\leq\alpha\left\|x_{1}-x_{2}\right\|,~{}\forall\ell=1,\ldots,L.$ (2) 2. 2. Given any two scalars $x_{1}\leq x_{2}$, the following inequality holds $\psi_{\ell}(x_{1})\leq\psi_{\ell}(x_{2}),~{}\forall\ell=1,\ldots,L.$ (3) ###### Remark 1 The above two assumptions hold for most popular activation functions such as ReLU, sigmoid, tanh, for instance. The maximum Lipschitz constant of all $\psi_{\ell}$ can be chosen as the $\alpha$ for condition (2). In addition, those popular activation functions are monotonically increasing so that condition (3) is explicitly satisfied. Neural-network-enabled dynamical systems are dynamical systems driven by neural network components such as neural network feedback controllers. In general, neural-network-enabled dynamical systems are in the form of $\displaystyle\begin{cases}\dot{x}(t)=f(x(t),u(t),\Phi(x(t),u(t)))\\\ y(t)=g(x(t),u(t))\end{cases}$ (4) where $x(t)\in\mathbb{R}^{n_{x}}$ is the state vector, $u(t)\in\mathbb{R}^{n_{u}}$ is the system input and $y(t)\in\mathbb{R}^{n_{y}}$ is the measurement of the system, respectively. $f:\mathbb{R}^{n_{x}+n_{u}}\to\mathbb{R}^{n_{x}}$ and $g:\mathbb{R}^{n_{x}+n_{u}}\to\mathbb{R}^{n_{y}}$ are nonlinear functions. $\Phi:\mathbb{R}^{n_{x}+n_{u}}\to\mathbb{R}^{n_{x}}$ is the neural network component embedded in the system dynamics. In the rest of this paper, the time index $t$ in some variables may be omitted for brevity if no ambiguity is introduced. In the work, we focus on a class of neural-network-enabled systems with system dynamics in the form of Lipschitz nonlinear model described as $\displaystyle\mathfrak{L}:\begin{cases}\dot{x}=Ax+f(x)+\Phi(x,u)\\\ y=Cx\end{cases}$ (5) where $A\in\mathbb{R}^{n_{x}\times x_{x}}$, $C\in\mathbb{R}^{n_{y}\times n_{x}}$, and $f(x,u)$ is a Lipschitz nonlinearity satisfying the Lipschitz inequality $\displaystyle\left\\|f(x_{1})-f(x_{2})\right\\|\leq\beta\left\\|x_{1}-x_{2}\right\\|,~{}\beta>0.$ (6) ###### Remark 2 It is worthwhile mentioning that any nonlinear system in the form of $\dot{x}=f(x)+\Phi(x,u)$ can be expressed in the form of (5), as long as $f(x)$ is differentiable with respect to $x$. The neural network $\Phi(x,u)$ is an interval component affecting the system behavior. For example, if $\Phi(x,u)$ is trained as the neural network feedback controller, model (5) represents a state-feedback closed-loop system. Finally, the nonlinearity $f(x)$ is assumed to have the following property. ###### Assumption 2 It is assumed that there exist functions $\underline{f},\overline{f}:\mathbb{R}^{2n_{x}}\to\mathbb{R}^{n_{x}}$ such that $\displaystyle\underline{f}(\underline{x},\overline{x})\leq f(x)\leq\overline{f}(\underline{x},\overline{x})$ (7) holds for any $\underline{x}\leq x\leq\overline{x}$. ### II-B Problem Statement The run-time safety monitoring problem considered in this paper is to design a run-time safety state estimator $\mathfrak{E}$ which is able to estimate the lower- and upper-bounds of the instantaneous value of $x(t)$ for the purpose of safety monitoring. The information of system $\mathfrak{L}$ that is available for estimator $\mathfrak{E}$ includes: System matrices $A$, $C$, nonlinearity $f$ and neural network $\Phi$, namely the weight matrices $\\{W_{\ell}\\}_{\ell=1}^{L}$ and bias vectors $\\{b_{\ell}\\}_{\ell=1}^{L}$, and the $\underline{u}$ and $\overline{u}$ such that input $u(t)$ satisfies $\underline{u}\leq u(t)\leq\overline{u}$, $\forall t\geq 0$, and the instantaneous value of measurement $y(t)$ in running time. The run-time safety monitoring problem for neural-network-enabled dynamical system (5) is summarized as follows. ###### Problem 1 Given a neural-network-enabled dynamical system $\mathfrak{L}$ in the form of (5) with input $u(t)$ satisfying $\underline{u}\leq u(t)\leq\overline{u}$, $\forall t\geq 0$, how does one design a run-time safety state estimator $\mathfrak{E}$ to reconstruct two instantaneous values $\underline{x}(t)$ and $\overline{x}(t)$ such that $\underline{x}(t)\leq x(t)\leq\overline{x}(t)$, $\forall t\geq 0$? Inspired by interval observers proposed in [20, 21, 22, 23, 24, 25, 26], the run-time safety state estimator $\mathfrak{E}$ is developed in the following Luenberger observer form $\displaystyle\mathfrak{E}:\begin{cases}\dot{\underline{x}}=A\underline{x}+\underline{f}(\underline{x},\overline{x})+\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})+\underline{L}(y-C\underline{x})\\\ \dot{\overline{x}}=A\overline{x}+\overline{f}(\underline{x},\overline{x})+\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})+\overline{L}(y-C\overline{x})\end{cases}$ (8) where initial states satisfy $\underline{x}(t_{0})\leq x(t_{0})\leq\overline{x}(t_{0})$ and $\underline{f}$, $\overline{f}$ are functions satisfying Assumption 2. Neural networks $\underline{\Phi}$, $\overline{\Phi}$ and observer gains $\underline{L}$, $\overline{L}$ are to be determined. Furthermore, letting the error states $\underline{e}(t)=x(t)-\underline{x}(t)$ and $\overline{e}(t)=\overline{x}(t)-x(t)$, the error dynamics can be obtained as follows: $\displaystyle\begin{cases}\dot{\underline{e}}=(A-\underline{L}C)\underline{e}+f(x)-\underline{f}(\underline{x},\overline{x})+\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})\\\ \dot{\overline{e}}=(A-\overline{L}C)\overline{e}+\overline{f}(\underline{x},\overline{x})-f(x)+\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})-\Phi(x,u)\end{cases}$ (9) with initial states $\underline{e}(t_{0})\geq 0$ and $\overline{e}(t_{0})\geq 0$. The problem of ensuring the run-time value of $x(t)$ satisfying $\underline{x}(t)\leq x(t)\leq\overline{x}(t)$, $\forall t\geq 0$ is equivalent to the one that the run-time values of error states $\underline{e}(t)$ and $\overline{e}(t)$ are required to be always positive, that is to say, $\underline{e}(t)\geq 0$ and $\overline{e}(t)\geq 0$, $\forall t\geq 0$. Thus, with the run-time safety state estimator in the form of (8), the run-time safety monitoring problem for system (5) can be restated as follows. ###### Problem 2 Given a neural-network-enabled dynamical system $\mathfrak{L}$ in the form of (5) with input $u(t)$ satisfying $\underline{u}\leq u(t)\leq\overline{u}$, $\forall t\geq 0$, how does one construct proper neural networks $\underline{\Phi}$, $\overline{\Phi}$ and observer gains $\underline{L}$, $\overline{L}$ such that the error states $\underline{e}(t)$, $\overline{e}(t)$ governed by (9) satisfy $\underline{e}(t)\geq 0$ and $\overline{e}(t)\geq 0$, $\forall t\geq 0$? As stated in Problem 2, the run-time safety monitoring consists of two essential design tasks, neural network design and observer gain design. Then, the result about positivity of dynamical systems is recalled by the following lemma. ###### Lemma 1 [21] Consider a system $\dot{z}=Mz+p(t)$, $z\in\mathbb{R}^{n}$, where $M\in\mathbb{M}_{n}$ and $p:\mathbb{R}_{+}\to\mathbb{R}^{n}_{+}$, the system is called cooperative and the solutions of the system satisfy $z(t)\geq 0$, $\forall t\geq 0$ if $z(0)\geq 0$. Based on Lemma 1 and owing to Assumption 2 implying $f(x)-\underline{f}(\underline{x},\overline{x})\in\mathbb{R}_{+}^{n_{x}}$ and $\overline{f}(\underline{x},\overline{x})-f(x)\in\mathbb{R}_{+}^{n_{x}}$, the run-time safety monitoring problem can be resolved if observer gains $\underline{L}$, $\overline{L}$ and nerual networks $\underline{\Phi}$, $\overline{\Phi}$ satisfy the conditions proposed in the following proposition. ###### Proposition 1 The run-time safety monitoring Problem 1 is solvable if there exist observer gains $\underline{L}$, $\overline{L}$ and neural networks $\underline{\Phi}$ and $\overline{\Phi}$ such that 1. 1. $A-\underline{L}C\in\mathbb{M}_{n_{x}}$, $A-\overline{L}C\in\mathbb{M}_{n_{x}}$. 2. 2. $\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})\in\mathbb{R}_{+}^{n_{x}}$, $\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})-\Phi(x,u)\in\mathbb{R}^{n_{x}}_{+}$. Proof. Due to Assumption 2, it implies $f(x)-\underline{f}(\underline{x},\overline{x})\in\mathbb{R}_{x}^{n_{x}}$. Then, owing to $\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})\in\mathbb{R}_{+}^{n_{x}}$, one can obtain $f(x)-\underline{f}(\underline{x},\overline{x})+\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})\in\mathbb{R}_{+}^{n_{x}}$. Together with $A-\underline{L}C\in\mathbb{M}_{n_{x}}$, it leads to $\underline{e}(t)\geq 0$, $\forall t\geq 0$ according to Lemma 1. Same guidelines can be applied to ensure $\overline{e}(t)\geq 0$. The proof is complete. $\hfill\hfill\square$ The observer gains $\underline{L}$, $\overline{L}$ and neural networks $\underline{\Phi}$, $\overline{\Phi}$ satisfying conditions in Proposition 1 can ensure the system state $x(t)$ to be bounded by the estimator states $\underline{x}(t)$ and $\overline{x}(t)$, but there is no guarantee on the boundedness and convergence of error state $\underline{e}(t)$ and $\overline{e}(t)$. The values of $\underline{e}(t)$ and $\overline{e}(t)$ may diverge, namely $\lim_{t\to\infty}\underline{e}(t)=\infty$ and $\lim_{t\to\infty}\overline{e}(t)=\infty$, thus make no sense in terms of safety monitoring in practice. The following notion of practical stability concerned with boundedness of system state is introduced. ###### Definition 1 [31] Given $(\epsilon,\delta)$ with $0<\epsilon\leq\delta$. Let $x(t,x(t_{0}))$, $t\geq t_{0}$, be a solution of the system $\dot{x}(t)=f(x(t),u(t))$, then the trivial solution $x=0$ of the system is said to be practically stable with respect to $(\epsilon,\delta)$ if $\left\\|x(t_{0})\right\\|\leq\epsilon$ implies $\left\\|x(t)\right\\|\leq\delta$, $\forall t\geq t_{0}$. Furthermore, if there is a $T=T(t_{0},\epsilon,\delta)>0$ such that $\left\\|x(t_{0})\right\\|\leq\epsilon$ implies $\left\\|x(t)\right\\|\leq\delta$ for any $t\geq t_{0}+T$, then the system is practically asymptotically stable. ###### Remark 3 Practical stability ensures the boundedness of state trajectories of a dynamical system. If the inequality $\displaystyle\left\\|x(t)\right\\|\leq Ce^{-\lambda(t-t_{0})}\left\\|x(t_{0})\right\\|+r$ (10) holds for any $x(t_{0})\in\mathbb{R}^{n_{x}}$, any $t\geq t_{0}$ and constants $C>0$, $r\leq 0$, it implies that the state trajectories of a dynamical system are bounded in terms of $\left\\|x(t)\right\\|\leq C\left\\|x(t_{0})\right\\|+r$, $\forall t\geq t_{0}$, and moreover, the state trajectories also converge to ball $\mathcal{B}_{r}=\\{x\in\mathbb{R}^{n_{x}}\mid\left\\|x\right\\|\geq r,r\geq 0\\}$ exponentially at a decay rate of $\lambda$. In particular, we call this system is globally practically uniformly exponentially stable. ## III Run-Time Safety Monitoring Design This section aims to design neural networks $\underline{\Phi}$, $\overline{\Phi}$ and observer gains $\underline{L}$, $\overline{L}$ satisfying conditions in Proposition 1. Furthermore, the convergence of error state is also analyzed and assured. First, we design neural networks $\underline{\Phi}$ and $\overline{\Phi}$ based on the weight matrices $W_{\ell}$ and bias vectors $b_{\ell}$ of neural network $\Phi$ in system (5). Given a neural network $\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$ in the form of (1) with weight matrices $\displaystyle W_{\ell}=[w_{\ell}^{i,j}]=\begin{bmatrix}w_{\ell}^{1,1}&w_{\ell}^{1,2}&\cdots&w_{\ell}^{1,n_{\ell-1}}\\\ w_{\ell}^{2,1}&w_{\ell}^{2,2}&\cdots&w_{\ell}^{2,n_{\ell-1}}\\\ \vdots&\vdots&\ddots&\vdots\\\ w_{\ell}^{n_{\ell},1}&w_{\ell}^{n_{\ell},2}&\cdots&w_{\ell}^{n_{\ell},n_{\ell-1}}\end{bmatrix}$ (11) where $w_{\ell}^{i,j}$ denotes the element in $i$-th row and $j$-th column, we define two auxiliary weight matrices as below: $\displaystyle\underline{W}_{\ell}$ $\displaystyle=[\underline{w}_{\ell}^{i,j}],~{}\underline{w}_{\ell}^{i,j}=\begin{cases}w_{\ell}^{i,j}&w_{\ell}^{i,j}<0\\\ 0&w_{\ell}^{i,j}\geq 0\end{cases}$ (12) $\displaystyle\overline{W}_{\ell}$ $\displaystyle=[\overline{w}_{\ell}^{i,j}],~{}\overline{w}_{\ell}^{i,j}=\begin{cases}w_{\ell}^{i,j}&w_{\ell}^{i,j}\geq 0\\\ 0&w_{\ell}^{i,j}<0\end{cases}$ (13) for which it is explicit that we have $W_{\ell}=\underline{W}_{\ell}+\overline{W}_{\ell}$. Then, we construct two auxiliary neural networks $\underline{\Phi}:\mathbb{R}^{2n_{0}}\to\mathbb{R}^{n_{L}}$, $\overline{\Phi}:\mathbb{R}^{2n_{0}}\to\mathbb{R}^{n_{L}}$ with inputs $\underline{\eta}_{0},\overline{\eta}_{0}\in\mathbb{R}^{n_{0}}$ in the following form: $\displaystyle\begin{cases}\underline{\eta}_{\ell}=\phi_{\ell}(\underline{W}_{\ell}\overline{\eta}_{\ell-1}+\overline{W}_{\ell}\underline{\eta}_{\ell-1}+b_{\ell}),~{}\ell=1,\ldots,L\\\ \underline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})=\underline{\eta}_{L}\end{cases}$ (14) $\displaystyle\begin{cases}\overline{\eta}_{\ell}=\phi_{\ell}(\underline{W}_{\ell}\underline{\eta}_{\ell-1}+\overline{W}_{\ell}\overline{\eta}_{\ell-1}+b_{\ell}),~{}\ell=1,\ldots,L\\\ \overline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})=\overline{\eta}_{L}\end{cases}$ (15) Given $\underline{\eta_{0}}\leq\eta_{0}\leq\overline{\eta_{0}}$, the following theorem can be derived with auxiliary neural networks in the form of (14) and (15), which implies the positivity of $\Phi(\eta_{0})-\underline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})$ and $\overline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})-\Phi(\eta_{0})$. ###### Theorem 1 Given neural networks $\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$ and its two auxiliary neural networks $\underline{\Phi}:\mathbb{R}^{2n_{0}}\to\mathbb{R}^{n_{L}}$, $\overline{\Phi}:\mathbb{R}^{2n_{0}}\to\mathbb{R}^{n_{L}}$ defined by (14) and (15), the following condition $\displaystyle\begin{bmatrix}\Phi(\eta_{0})-\underline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})\\\ \overline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})-\Phi(\eta_{0})\end{bmatrix}\in\mathbb{R}_{+}^{2n_{L}}$ (16) holds for any $\underline{\eta}_{0}\leq\eta_{0}\leq\overline{\eta}_{0}$. Proof. Let us consider the $\ell$-th layer. For any $\underline{\eta}_{\ell-1}\leq\eta_{\ell-1}\leq\overline{\eta}_{\ell-1}$, it can be obtained from (12), (13) such that $\displaystyle\underline{w}_{\ell}^{i,j}\overline{\eta}_{\ell-1}^{j}+\overline{w}_{\ell}^{i,j}\underline{\eta}_{\ell-1}^{j}\leq w^{i,j}_{\ell}\eta_{\ell-1}^{j}\leq\underline{w}_{\ell}^{i,j}\underline{\eta}_{\ell-1}^{j}+\overline{w}_{\ell}^{i,j}\overline{\eta}_{\ell-1}^{j}$ which implies that $\displaystyle W_{\ell}\eta_{\ell-1}+b_{\ell}-(\underline{W}_{\ell}\overline{\eta}_{\ell-1}+\overline{W}_{\ell}\underline{\eta}_{\ell-1}+b_{\ell})\geq 0$ $\displaystyle\underline{W}_{\ell}\underline{\eta}_{\ell-1}+\overline{W}_{\ell}\overline{\eta}_{\ell-1}+b_{\ell}-(W_{\ell}\eta_{\ell-1}+b_{\ell})\geq 0.$ Under Assumption 1, the monotonic property (3) of activation function $\phi_{\ell}$ leads to $\displaystyle\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell})-\phi_{\ell}(\underline{W}_{\ell}\overline{\eta}_{\ell-1}+\overline{W}_{\ell}\underline{\eta}_{\ell-1}+b_{\ell})\geq 0$ $\displaystyle\phi_{\ell}(\underline{W}_{\ell}\underline{\eta}_{\ell-1}+\overline{W}_{\ell}\overline{\eta}_{\ell-1}+b_{\ell})-\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell})\geq 0.$ Using the definitions of neural networks $\Phi$, $\underline{\Phi}$ and $\overline{\Phi}$ described by (1), (14) and (15), namely $\eta_{\ell}=\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell})$, $\underline{\eta}_{\ell}=\phi_{\ell}(\underline{W}_{\ell}\overline{\eta}_{\ell-1}+\overline{W}_{\ell}\underline{\eta}_{\ell-1}+b_{\ell})$ and $\overline{\eta}_{\ell}=\phi_{\ell}(\underline{W}_{\ell}\underline{\eta}_{\ell-1}+\overline{W}_{\ell}\overline{\eta}_{\ell-1}+b_{\ell})$, the above derivation implies that $\displaystyle\begin{bmatrix}\eta_{\ell-1}-\underline{\eta}_{\ell-1}\\\ \overline{\eta}_{\ell-1}-\eta_{\ell-1}\end{bmatrix}\in\mathbb{R}_{+}^{2n_{\ell-1}}\Rightarrow\begin{bmatrix}\eta_{\ell}-\underline{\eta}_{\ell}\\\ \overline{\eta}_{\ell}-\eta_{\ell}\end{bmatrix}\in\mathbb{R}_{+}^{2n_{\ell}}.$ (17) Thus, given any $\underline{\eta}_{0}\leq\eta_{0}\leq\overline{\eta}_{0}$, one can obtain $\displaystyle\begin{bmatrix}\eta_{L}-\underline{\eta}_{L}\\\ \overline{\eta}_{L}-\eta_{L}\end{bmatrix}=\begin{bmatrix}\Phi(\eta_{0})-\underline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})\\\ \overline{\Phi}(\underline{\eta_{0}},\overline{\eta}_{0})-\Phi(\eta_{0})\end{bmatrix}\in\mathbb{R}_{+}^{2n_{L}}.$ (18) The proof is complete. $\hfill\hfill\square$ With the two auxiliary neural networks $\underline{\Phi}$, $\overline{\Phi}$, we are ready to design observer gains $\underline{L}$ and $\overline{L}$ to construct run-time safety state estimator $\mathfrak{E}$ in the form of (8) via the following theorem. ###### Theorem 2 The safety monitoring Problem 1 is solvable if the following conditions hold for observer gains $\underline{L}$, $\overline{L}$ and neural networks $\underline{\Phi}$ and $\overline{\Phi}$: 1. 1. There exist $a\in\mathbb{R}$, $\underline{L},\overline{L}\in\mathbb{R}^{n_{x}\times n_{y}}$ such that $\displaystyle A-\underline{L}C$ $\displaystyle\geq aI_{n_{x}\times n_{x}}$ (19) $\displaystyle A-\overline{L}C$ $\displaystyle\geq aI_{n_{x}\times n_{x}}.$ (20) 2. 2. $\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})$, $\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})$ are in the form of (14) and (15) with $\underline{\eta}_{0}=[\underline{x}^{\top},\underline{u}^{\top}]^{\top}$ and $\overline{\eta}_{0}=[\overline{x}^{\top},\overline{u}^{\top}]^{\top}$. Proof. First note that (19) and (20) imply that $A-\underline{L}C\in\mathbb{M}_{n_{x}}$, $A-\overline{L}C\in\mathbb{M}_{n_{x}}$. Then, from Theorem 1, it leads to the fact of $\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})\in\mathbb{R}_{+}^{n_{x}}$, $\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})-\Phi(x,u)\in\mathbb{R}^{n_{x}}_{+}$. Based on Proposition 1, the error $e(t)$ will be bounded as $\underline{e}(t)\leq e(t)\leq\overline{e}(t)$, $\forall t\geq 0$, thus the safety monitoring problem is solvable. The proof is complete. $\hfill\hfill\square$ Theorem 2 provides us a method to design run-time safety state estimator in the interval observer form of (8). The observer gains $\underline{L}$ and $\overline{L}$ can be obtained by solving the linear inequalities (19), (20), and neural networks $\underline{\Phi}$ and $\overline{\Phi}$ are determined by (14) and (15) with weight matrices $\underline{W}_{\ell}$, $\overline{W}_{\ell}$ defined by (12), (13). The boundedness of $\underline{x}(t)\leq x(t)\leq\overline{x}(t)$, $\forall t\geq 0$ can be established during the system’s operation, however, the boundedness and convergence of error states $\underline{e}(t)$ and $\overline{e}(t)$ are not guaranteed, which means the error dynamics (9) could be unstable. In this case, the estimated bounds $\underline{x}(t)$ and $\overline{x}(t)$ will diverge from system state $x(t)$ to infinite values, and consequently, the run-time safety monitoring does not make sense in practice. In the following, the convergence of run-time estimation bounds is discussed in the framework of practical stability proposed in Definition 1. First, the following assumption is proposed for nonlinearity $f(x)$ and $\underline{f}(\underline{x},\overline{x})$, $\overline{f}(\underline{x},\overline{x})$ mentioned in Assumption 2. ###### Assumption 3 It is assumed that there exist scalars $\underline{\gamma}_{1}$, $\overline{\gamma}_{1}$, $\underline{\gamma}_{2}$, $\overline{\gamma}_{2}\in\mathbb{R}_{+}$ and vector $\underline{\rho},\overline{\rho}\in\mathbb{R}^{n_{x}}_{+}$ such that $\displaystyle f(x)-\underline{f}(\underline{x},\overline{x})\leq\underline{\gamma}_{1}(x-\underline{x})+\underline{\gamma}_{2}(\overline{x}-x)+\underline{\rho}$ (21) $\displaystyle\overline{f}(\underline{x},\overline{x})-f(x)\leq\overline{\gamma}_{1}(x-\underline{x})+\overline{\gamma}_{2}(\overline{x}-x)+\overline{\rho}$ (22) holds for $f(x)$, $\underline{f}(\underline{x},\overline{x})$, $\overline{f}(\underline{x},\overline{x})$. ###### Remark 4 These parameters $\underline{\gamma}_{1}$, $\overline{\gamma}_{1}$, $\underline{\gamma}_{2}$, $\overline{\gamma}_{2}$, $\underline{\rho}$, and $\overline{\rho}$ in Assumption 3 can be estimated under Lipschitz condition (6), using the results in [32], i.e., Lemma 6 in [32]. The following lemma is developed for neural network $\Phi$ and its auxiliary neural networks $\underline{\Phi}$ and $\overline{\Phi}$. ###### Lemma 2 Given a feedforward neural network $\Phi:\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}}$, there always exist a series of matrices $\underline{S}_{\ell},\overline{S}_{\ell}\in\mathbb{R}^{n_{L}\times n_{\ell}}_{+}$, $\ell=0,\ldots,L$, with $\underline{S}_{L}=\overline{S}_{L}=I_{n_{L}\times{n_{L}}}$ such that $\displaystyle\alpha\begin{bmatrix}\underline{S}_{\ell}\overline{W}_{\ell}-\overline{S}_{\ell}\underline{W}_{\ell}\\\ \overline{S}_{\ell}\overline{W}_{\ell}-\underline{S}_{\ell}\underline{W}_{\ell}\end{bmatrix}$ $\displaystyle\leq\begin{bmatrix}\underline{S}_{\ell-1}\\\ \overline{S}_{\ell-1}\end{bmatrix},~{}\ell=1,\ldots,L$ (23) $\displaystyle\overline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})-\underline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})$ $\displaystyle\leq\underline{S}_{0}(\eta_{0}-\underline{\eta}_{0})+\overline{S}_{0}(\overline{\eta}_{0}-\eta_{0})$ (24) hold for any $\underline{\eta}_{0}\leq\eta_{0}\leq\overline{\eta}_{0}$, where $\alpha$ is the Lipschitz constant of activation functions given in (2). Proof. Starting from $\underline{S}_{L}=\overline{S}_{L}=I_{n_{L}\times{n_{L}}}$, we can recursively define $\displaystyle\underline{S}_{\ell-1}=\alpha(\underline{S}_{\ell}\overline{W}_{\ell}-\overline{S}_{\ell}\underline{W}_{\ell})+\epsilon\mathbf{1}_{n_{L}\times 1}\mathbf{1}_{n_{\ell-1}\times 1}^{\top}$ (25) $\displaystyle\overline{S}_{\ell-1}=\alpha(\overline{S}_{\ell}\overline{W}_{\ell}-\underline{S}_{\ell}\underline{W}_{\ell})+\epsilon\mathbf{1}_{n_{L}\times 1}\mathbf{1}_{n_{\ell-1}\times 1}^{\top}$ (26) where $\epsilon>0$ could be any positive value. Thus, there always exist $\underline{S}_{\ell}$, $\overline{S}_{\ell}$, $\ell=0,\ldots,L$ such that (23) holds. Then, we are going to establish (24). We consider the $\ell$-th layer $\eta_{\ell}=\phi_{\ell}(W_{\ell}\eta_{\ell-1}+b_{\ell})$. Under Assumption 1, it implies $\displaystyle\eta_{\ell}-\underline{\eta}_{\ell}=$ $\displaystyle\phi_{\ell}({W}_{\ell}\eta_{\ell-1}+b_{\ell})-\phi_{\ell}(\underline{W}_{\ell}\overline{\eta}_{\ell-1}+\overline{W}_{\ell}\underline{\eta}_{\ell-1}+b_{\ell})$ $\displaystyle\leq$ $\displaystyle\alpha\left\|{W}_{\ell}\eta_{\ell-1}+b_{\ell}-\underline{W}_{\ell}\overline{\eta}_{\ell-1}-\overline{W}_{\ell}\underline{\eta}_{\ell-1}-b_{\ell}\right\|$ (27) Following the same guideline in the proof of Theorem 1 one obtains $\displaystyle{W}_{\ell}\eta_{\ell-1}+b_{\ell}-\underline{W}_{\ell}\overline{\eta}_{\ell-1}-\overline{W}_{\ell}\underline{\eta}_{\ell-1}-b_{\ell}\geq 0$ (28) and using the fact of $W_{\ell}=\underline{W}_{\ell}+\overline{W}_{\ell}$, inequality (27) equals $\displaystyle\eta_{\ell}-\underline{\eta}_{\ell}\leq$ $\displaystyle\alpha\overline{W}_{\ell}(\eta_{\ell-1}-\underline{\eta}_{\ell-1})-\alpha\underline{W}_{\ell}(\overline{\eta}_{\ell-1}-\eta_{\ell-1})$ $\displaystyle=$ $\displaystyle\begin{bmatrix}\alpha\overline{W}_{\ell}&-\alpha\underline{W}_{\ell}\end{bmatrix}\begin{bmatrix}\eta_{\ell-1}-\underline{\eta}_{\ell-1}\\\ \overline{\eta}_{\ell-1}-\eta_{\ell-1}\end{bmatrix}.$ (29) Similarly, one can obtain $\displaystyle\overline{\eta}_{\ell}-\eta_{\ell}\leq\begin{bmatrix}-\alpha\underline{W}_{\ell}&\alpha\overline{W}_{\ell}\end{bmatrix}\begin{bmatrix}\eta_{\ell-1}-\underline{\eta}_{\ell-1}\\\ \overline{\eta}_{\ell-1}-\eta_{\ell-1}\end{bmatrix}.$ (30) Based on inequalities (29) and (30), the following inequality can be established $\displaystyle\underline{S}_{\ell}(\eta_{\ell}-\underline{\eta}_{\ell})+\overline{S}_{\ell}(\overline{\eta}_{\ell}-\eta_{\ell})$ $\displaystyle\leq\alpha\begin{bmatrix}\underline{S}_{\ell}\overline{W}_{\ell}-\overline{S}_{\ell}\underline{W}_{\ell}&\overline{S}_{\ell}\overline{W}_{\ell}-\underline{S}_{\ell}\underline{W}_{\ell}\end{bmatrix}\begin{bmatrix}\eta_{\ell-1}-\underline{\eta}_{\ell-1}\\\ \overline{\eta}_{\ell-1}-\eta_{\ell-1}\end{bmatrix}$ with any $\underline{S}_{\ell},\overline{S}_{\ell}\in\mathbb{R}^{n_{L}\times n_{\ell}}_{+}$. Due to (24) which always holds with existence of $\underline{S}_{\ell}$, $\overline{S}_{\ell}$, $\ell=0,\ldots,L$ as proved by (25) and (26), the above inequality ensures $\displaystyle\underline{S}_{\ell}(\eta_{\ell}-\underline{\eta}_{\ell})+\overline{S}_{\ell}(\overline{\eta}_{\ell}-\eta_{\ell})$ $\displaystyle\leq$ $\displaystyle\begin{bmatrix}\underline{S}_{\ell-1}&\overline{S}_{\ell-1}\end{bmatrix}\begin{bmatrix}\eta_{\ell-1}-\underline{\eta}_{\ell-1}\\\ \overline{\eta}_{\ell-1}-\eta_{\ell-1}\end{bmatrix}$ $\displaystyle=$ $\displaystyle\underline{S}_{\ell-1}(\eta_{\ell-1}-\underline{\eta}_{\ell-1})+\overline{S}_{\ell-1}(\overline{\eta}_{\ell-1}-\eta_{\ell-1})$ which can be iterated to yield $\displaystyle\underline{S}_{L}(\eta_{L}-\underline{\eta}_{L})+\overline{S}_{L}(\overline{\eta}_{L}-\eta_{L})\leq\underline{S}_{0}(\eta_{0}-\underline{\eta}_{0})+\overline{S}_{0}(\overline{\eta}_{0}-\eta_{0}).$ Owing to the fact of $\eta_{L}=\Phi(\eta_{0})$, $\underline{\eta}_{L}=\underline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})$, $\overline{\eta}_{L}=\overline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})$ and $\underline{S}_{L}=\overline{S}_{L}=I_{n_{L}\times n_{L}}$, the following inequality can be established $\overline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})-\underline{\Phi}(\underline{\eta}_{0},\overline{\eta}_{0})\leq\underline{S}_{0}(\eta_{0}-\underline{\eta}_{0})+\overline{S}_{0}(\overline{\eta}_{0}-\eta_{0})$ (31) for any $\underline{\eta}_{0}\leq\eta_{0}\leq\overline{\eta}_{0}$. The proof is complete. $\hfill\hfill\square$ ###### Remark 5 Lemma 2 ensures the existence of $\underline{S}_{\ell}$, $\overline{S}_{\ell}$, $\ell=0,\ldots,L$ such that the input-output relationship in the description of (24) holds for auxiliary neural networks $\underline{\Phi}$ and $\overline{\Phi}$. It also provides a method to compute $\underline{S}_{\ell}$, $\overline{S}_{\ell}$, $\ell=0,\ldots,L$, that is solving linear inequality (23) with initialized $\underline{S}_{L}=\overline{S}_{L}=I_{n_{L}\times{n_{L}}}$. In practice, optimal solution of $\underline{S}_{0}$, $\overline{S}_{0}$ such as $\min~{}\mathrm{trace(diag}\\{\underline{S}_{0},\overline{S}_{0}\\})$ is of interest. With respect to objective functions of interest, optimization problems such as linear programming (LP) problems can be formulated to compute $\underline{S}_{0}=\overline{S}_{0}$. For instance, the following LP problem can be formulated $\displaystyle\min~{}\mathrm{trace(diag}\\{\underline{S}_{0},\overline{S}_{0}\\})$ $\displaystyle\mathrm{s.t.~{}}$ $\displaystyle\alpha\begin{bmatrix}\underline{S}_{\ell}\overline{W}_{\ell}-\overline{S}_{\ell}\underline{W}_{\ell}\\\ \overline{S}_{\ell}\overline{W}_{\ell}-\underline{S}_{\ell}\underline{W}_{\ell}\end{bmatrix}\leq\begin{bmatrix}\underline{S}_{\ell-1}\\\ \overline{S}_{\ell-1}\end{bmatrix},~{}\ell=1,\ldots,L$ $\displaystyle\underline{S}_{L}=\overline{S}_{L}=I_{n_{L}\times{n_{L}}}.$ (32) Based on Lemma 2, we are ready to derive the following result to ensure the boundedness and convergence of run-time error states, namely the practical stability of error system (9). Before presenting the result, we assume that the input vector of neural network component $\Phi(\eta_{0})$ is $\eta_{0}=[x,u]\in\mathbb{R}^{n_{x}+n_{u}}$. ###### Theorem 3 Consider error system (9), if there exist a diagonal matrix $X\in\mathbb{R}^{n_{x}\times n_{x}}$, matrices $\underline{Y},\overline{Y}\in\mathbb{R}^{n_{x}\times n_{y}}$ and a scalar $a\in\mathbb{R}$ such that $\displaystyle X\mathbf{1}_{n_{x}\times 1}$ $\displaystyle>0$ (33) $\displaystyle XA-\underline{Y}C$ $\displaystyle>aI_{n_{x}\times n_{x}}$ (34) $\displaystyle XA-\overline{Y}C$ $\displaystyle>aI_{n_{x}\times n_{x}}$ (35) $\displaystyle\begin{bmatrix}A^{\top}X-C^{\top}\underline{Y}^{\top}+\underline{\gamma}X+\underline{U}^{\top}X\\\ A^{\top}X-C^{\top}\overline{Y}^{\top}+\overline{\gamma}X+\overline{U}^{\top}X\end{bmatrix}\mathbf{1}_{n_{x}\times 1}$ $\displaystyle<0$ (36) where $\underline{\gamma}=\underline{\gamma}_{1}+\underline{\gamma}_{2}$, $\overline{\gamma}=\overline{\gamma}_{1}+\overline{\gamma}_{2}$ and $\underline{U}$, $\overline{U}$ are defined by $\underline{S}_{0}=[\underline{U},~{}\underline{V}]$ and $\overline{S}_{0}=[\overline{U},~{}\overline{V}]$ where $\underline{U},\overline{U}\in\mathbb{R}^{n_{x}\times n_{x}}$, $\underline{V},\overline{V}\in\mathbb{R}^{n_{x}\times n_{u}}$, and $\underline{S}_{0},\overline{S}_{0}\in\mathbb{R}^{n_{x}\times(n_{x}+n_{u})}$ are the solution of the following conditions: $\displaystyle\alpha\begin{bmatrix}\underline{S}_{\ell}\overline{W}_{\ell}-\overline{S}_{\ell}\underline{W}_{\ell}\\\ \overline{S}_{\ell}\overline{W}_{\ell}-\underline{S}_{\ell}\underline{W}_{\ell}\end{bmatrix}$ $\displaystyle\leq\begin{bmatrix}\underline{S}_{\ell-1}\\\ \overline{S}_{\ell-1}\end{bmatrix},~{}\ell=1,\ldots,L$ (37) with $\underline{S}_{L}=\overline{S}_{L}=I_{n_{x}\times n_{x}}$, then the error system (9) is globally practically uniformly exponentially stable with observer gains $\underline{L}=X^{-1}\underline{Y}$ and $\overline{L}=X^{-1}\overline{Y}$. Proof. First, we construct the co-positive Lyapunov function candidate $V(\underline{e},\overline{e})=\underline{V}(\underline{e})+\overline{V}(\overline{e})$, where $\underline{V}(\underline{e})=\underline{e}^{\top}v$, $\overline{V}(\overline{e})=\overline{e}^{\top}v$ with $v=X\mathbf{1}_{n_{x}\times 1}\in\mathbb{R}_{+}^{n_{x}}$. Considering $\underline{V}(\underline{e})$, one can obtain $\displaystyle\dot{\underline{V}}(\underline{e})=\underline{e}^{\top}(A^{\top}-C^{\top}\underline{L}^{\top})v+\underline{\mathcal{F}}^{\top}v+\underline{\mathcal{G}}^{\top}v$ (38) where $\underline{\mathcal{F}}=f(x)-\underline{f}(\underline{x},\overline{x})$, $\underline{\mathcal{G}}=\Phi(x,u)-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})$. Under Assumption 3, it implies that $\displaystyle\underline{\mathcal{F}}^{\top}v\leq\underline{\gamma}_{1}\underline{e}^{\top}v+\underline{\gamma}_{2}\overline{e}^{\top}v+\underline{\rho}^{\top}v.$ (39) Thus, we have $\displaystyle\dot{\underline{V}}(\underline{e})\leq\underline{e}^{\top}\underline{\Theta}v+\underline{\rho}^{\top}v+\underline{\mathcal{G}}^{\top}v$ (40) where $\underline{\Theta}=A^{\top}-C^{\top}\underline{L}^{\top}+(\underline{\gamma}_{1}+\underline{\gamma}_{2})I_{n_{x}\times n_{x}}$. Similarly, the following inequality can be obtained for ${\overline{V}(\overline{e})}$: $\displaystyle\dot{\overline{V}}(\overline{e})\leq\overline{e}^{\top}\overline{\Theta}v+\overline{\rho}^{\top}v+\overline{\mathcal{G}}^{\top}v$ (41) where $\overline{\Theta}=A^{\top}-C^{\top}\overline{L}^{\top}+(\overline{\gamma}_{1}+\overline{\gamma}_{2})I_{n_{x}\times n_{x}}$ and $\overline{\mathcal{G}}=\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})-\Phi(x,u)$. From (40) and (41), we have $\displaystyle\dot{V}(\underline{e},\overline{e})\leq\begin{bmatrix}\underline{e}^{\top}&\overline{e}^{\top}\end{bmatrix}\begin{bmatrix}\underline{\Theta}\\\ \overline{\Theta}\end{bmatrix}v+(\underline{\rho}^{\top}+\overline{\rho}^{\top})v+\mathcal{G}^{\top}v$ (42) where $\mathcal{G}=\underline{\mathcal{G}}+\overline{\mathcal{G}}$. Due to $\mathcal{G}=\underline{\mathcal{G}}+\overline{\mathcal{G}}=\overline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})-\underline{\Phi}(\underline{x},\overline{x},\underline{u},\overline{u})$ and using (37) based on Lemma 2, it leads to $\displaystyle\mathcal{G}\leq$ $\displaystyle\begin{bmatrix}\underline{U}&\underline{V}\end{bmatrix}\left(\begin{bmatrix}x\\\ u\end{bmatrix}-\begin{bmatrix}\underline{x}\\\ \underline{u}\end{bmatrix}\right)+\begin{bmatrix}\overline{U}&\overline{V}\end{bmatrix}\left(\begin{bmatrix}\overline{x}\\\ \overline{u}\end{bmatrix}-\begin{bmatrix}{x}\\\ {u}\end{bmatrix}\right)$ $\displaystyle=$ $\displaystyle\underline{U}\underline{e}+\underline{V}(u-\underline{u})+\overline{U}\overline{e}+\overline{V}(\overline{u}-u)$ $\displaystyle\leq$ $\displaystyle\begin{bmatrix}\underline{U}&\overline{U}\end{bmatrix}\begin{bmatrix}\underline{e}\\\ \overline{e}\end{bmatrix}+(\underline{V}+\overline{V})(\overline{u}-\underline{u}).$ (43) Therefore, one has $\displaystyle\dot{V}(\underline{e},\overline{e})\leq\begin{bmatrix}\underline{e}^{\top}&\overline{e}^{\top}\end{bmatrix}\begin{bmatrix}\underline{\Theta}+\underline{U}^{\top}\\\ \overline{\Theta}+\overline{U}^{\top}\end{bmatrix}v+\theta$ (44) where $\theta=(\overline{u}^{\top}-\underline{u}^{\top})(\underline{V}^{\top}+\overline{V}^{\top})v+(\underline{\rho}^{\top}+\overline{\rho}^{\top})v$. Due to (36) and $v=X\mathbf{1}_{n_{x}\times 1}$, it leads to $\displaystyle\begin{bmatrix}\underline{\Theta}+\underline{U}^{\top}\\\ \overline{\Theta}+\overline{U}^{\top}\end{bmatrix}v<0$ (45) which implies that there always exists a sufficient small $\lambda>0$ such that $\displaystyle\begin{bmatrix}\underline{\Theta}+\underline{U}^{\top}\\\ \overline{\Theta}+\overline{U}^{\top}\end{bmatrix}v<-\lambda\begin{bmatrix}I_{n_{x}\times n_{x}}\\\ I_{n_{x}\times n_{x}}\end{bmatrix}v$ (46) and that implies $\displaystyle\dot{V}(\underline{e},\overline{e})\leq-\lambda\underline{e}^{\top}v-\lambda\overline{e}^{\top}v+\theta=-\lambda V(\underline{e},\overline{e})+\theta.$ (47) Defining $\underline{v}$ and $\overline{v}$ the minimal and maximal element in $v$, (47) implies that $\displaystyle\underline{v}\left\\|\xi\right\\|\leq e^{-\lambda(t-t_{0})}\overline{v}\left\\|\xi_{0}\right\\|+\frac{\theta}{\lambda}\Rightarrow\left\\|\xi\right\\|\leq Ce^{-\lambda(t-t_{0})}\left\\|\xi_{0}\right\\|+r$ where $\xi^{\top}=[\underline{e}^{\top},\overline{e}]^{\top}$, $C={\overline{v}}/{\underline{v}}$ and $r={\theta}/{\lambda\underline{v}}$. Therefore, the error system is globally practically uniformly exponentially stable, the error state converges to ball $\mathcal{B}_{r}=\\{x\in\mathbb{R}^{n_{x}}\mid\left\\|x\right\\|\geq r,r\geq 0\\}$ exponentially at a decay rate of $\lambda$. The proof is complete. $\hfill\hfill\square$ The design process of observer gains $\underline{L}$ and $\overline{L}$ allows one to use coordinate transformation techniques used in several works such as [21, 33, 25] to relax the conditions of both Mezleter and Hurwitz conditions being satisfied. Based on Theorem 3, an design algorithm is proposed in Algorithm 1. The outputs of the algorithm, the auxiliary neural networks $\underline{\Phi}$, $\overline{\Phi}$ and observer gains $\underline{L}$, $\overline{L}$, are able to ensure the run-time boundedness as well as the convergence of the error state in terms of practical stability. 1 Compute $\underline{W}_{\ell}$, $\overline{W}_{\ell}$, $\ell=1,\ldots,L$ by (12), (13) and obtain neural networks $\underline{\Phi}$, $\overline{\Phi}$; 2 Solve LP problem (32) to obtain $\underline{S}_{0}$ and $\overline{S}_{0}$ ; 3 Compute $\underline{U}$, $\overline{U}$ by $\underline{S}_{0}=[\underline{U},~{}\underline{V}]$, $\overline{S}_{0}=[\overline{U},~{}\overline{V}]$ where $\underline{U},\overline{U}\in\mathbb{R}^{n_{x}\times n_{x}}$, $\underline{V},\overline{V}\in\mathbb{R}^{n_{x}\times n_{u}}$ ; 4 Solve LP problem (33)-(36) to obtain $X$, $\underline{Y}$, $\overline{Y}$ ; Compute observer gains $\underline{L}$, $\overline{L}$. Algorithm 1 Run-Time Safety State Estimator Design A numerical example is proposed to illustrate the design process of Algorithm 1. ###### Example 1 Consider a neural-network-enabled system in the form of $\dot{x}=Ax+\Phi(x,u)$, $y=Cx$, where system matrices are $\displaystyle A=\begin{bmatrix}-2&1\\\ 3&-5\end{bmatrix},~{}C=\begin{bmatrix}0&1\end{bmatrix}$ and neural network $\Phi$ is determined by $\displaystyle W_{1}=\begin{bmatrix}0.6266&0.8433&0.3241\\\ -0.2485&-1.5838&-0.5620\\\ 0.5243&-1.4939&1.1992\\\ -0.4300&-1.4659&0.1102\\\ 0.2629&0.6789&-1.2695\end{bmatrix},~{}b_{1}=\begin{bmatrix}-1.0191\\\ -1.3852\\\ 0.9549\\\ -0.6011\\\ -1.1719\end{bmatrix}$ $\displaystyle W_{2}^{\top}=\begin{bmatrix}-0.4617&-0.6691\\\ 0.6824&0.3819\\\ 0.2419&0.3326\\\ 0.0344&-0.7591\\\ 0.4333&-0.6569\end{bmatrix},~{}b_{2}=\begin{bmatrix}-1.0719\\\ -1.0741\end{bmatrix}$ and activation functions are tanh and purelin. _Step 1. Design Auxiliary Neural Networks:_ By (12) and (13), matrices $\underline{W}_{\ell}$, $\overline{W}_{\ell}$, $\ell=1,2$ are as follows: $\displaystyle\underline{W}_{1}=\begin{bmatrix}0&0&0\\\ -0.2485&-1.5838&-0.5620\\\ 0&-1.4939&0\\\ -0.4300&-1.4659&0\\\ 0&0&-1.2695\end{bmatrix}$ $\displaystyle\overline{W}_{1}=\begin{bmatrix}0.6266&0.8433&0.3241\\\ 0&0&0\\\ 0.5243&01.1992\\\ 0&0&0.1102\\\ 0.2629&0.6789&0\end{bmatrix}$ $\displaystyle\underline{W}_{2}=\begin{bmatrix}-0.4617&0&0&0&0\\\ -0.6691&0&0&-0.7591&-0.6569\end{bmatrix}$ $\displaystyle\overline{W}_{2}=\begin{bmatrix}0&0.6824&0.2419&0.0344&0.4333\\\ 0&0.3819&0.3326&0&0\end{bmatrix}.$ _Step 2: Design Observer Gains:_ By (33)–(37), the observer gains are computed as $\displaystyle\underline{L}=\begin{bmatrix}0\\\ 12.0394\end{bmatrix},~{}\overline{L}=\begin{bmatrix}1\\\ 8.0044\end{bmatrix}.$ Assuming input $u=10\sin(5t)$, thus we have $\underline{u}=-10$ and $\overline{u}=10$. The initial state is assumed to be bounded in $[-1,1]$. The run-time safety monitoring of system state $x_{1}(t)$ and $x_{2}(t)$ is illustrated in Figure 2. The run-time state trajectories $x_{1}(t)$, $x_{2}(t)$ are bounded in run-time estimated states $\underline{x}_{i}(t)$, $\overline{x}_{i}(t)$, $i=1,2$, in other words, the safety of system state $x(t)$ can be monitored by $\underline{x}_{i}(t)$, $\overline{x}_{i}(t)$, $i=1,2$ in run time. Figure 2: State response of $x(t)$ (solid lines) and run-time safety monitoring of $\underline{x}(t)$ and $\overline{x}(t)$ (dashed lines). State response $x(t)$ is bounded between states $\underline{x}(t)$, $\overline{x}(t)$ of state estimator in run time. As one of the most common neural-network-enabled dynamical systems, the neural network is driven by the measurement of the system, which means the input of the neural network is the measurement of the system $y(t)$ instead of system state $x(t)$. For instance, neural network control systems use the measurement $y(t)$ to compute the control input instead of system state $x(t)$ since system state $x(t)$ may not be measurable. This class of systems with neural network $\Phi(y,u)$ are in the following description of $\displaystyle\mathfrak{L}_{y}:\begin{cases}\dot{x}=Ax+f(x)+\Phi(y,u)\\\ y=Cx\end{cases}$ (48) where neural network $\Phi(y,u)$ is measured output driven. Since the output $y(t)$ is measurable in run time, it can be employed in the safety monitoring. The run-time safety state estimator is developed in the form of $\displaystyle\mathfrak{E}_{y}:\begin{cases}\dot{\underline{x}}=A\underline{x}+\underline{f}(\underline{x},\overline{x})+\underline{\Phi}(y,\underline{u},\overline{u})+\underline{L}(y-C\underline{x})\\\ \dot{\overline{x}}=A\overline{x}+\overline{f}(\underline{x},\overline{x})+\overline{\Phi}(y,\underline{u},\overline{u})+\overline{L}(y-C\overline{x})\end{cases}$ (49) where $\underline{\Phi}$ and $\overline{\Phi}$ are defined by (14) and (15) with $\underline{y}=\overline{y}=y$, respectively. Consequently, the error dynamics is in the form of $\displaystyle\begin{cases}\dot{\underline{e}}=(A-\underline{L}C)\underline{e}+f(x)-\underline{f}(\underline{x},\overline{x})+\Phi(y,u)-\underline{\Phi}(y,\underline{u},\overline{u})\\\ \dot{\overline{e}}=(A-\overline{L}C)\overline{e}+\overline{f}(\underline{x},\overline{x})-f(x)+\overline{\Phi}(y,\underline{u},\overline{u})-\Phi(y,u)\end{cases}$ (50) The following result represents the observer gain design process in (49). ###### Corollary 1 Consider error system (50), if there exist a diagonal matrix $X\in\mathbb{R}^{n_{x}\times n_{x}}$, matrices $\underline{Y},\overline{Y}\in\mathbb{R}^{n_{x}\times n_{y}}$ and a scalar $a\in\mathbb{R}$ such that $\displaystyle X\mathbf{1}_{n_{x}\times 1}$ $\displaystyle>0$ (51) $\displaystyle XA-\underline{Y}C$ $\displaystyle>aI_{n_{x}\times n_{x}}$ (52) $\displaystyle XA-\overline{Y}C$ $\displaystyle>aI_{n_{x}\times n_{x}}$ (53) $\displaystyle\begin{bmatrix}A^{\top}X-C^{\top}\underline{Y}^{\top}+\underline{\gamma}X\\\ A^{\top}X-C^{\top}\overline{Y}^{\top}+\overline{\gamma}X\end{bmatrix}\mathbf{1}_{n_{x}\times 1}$ $\displaystyle<0$ (54) where $\underline{\gamma}=\underline{\gamma}_{1}+\underline{\gamma}_{2}$, $\overline{\gamma}=\overline{\gamma}_{1}+\overline{\gamma}_{2}$, then the error system (50) is globally practically uniformly exponentially stable with observer gains $\underline{L}=X^{-1}\underline{Y}$ and $\overline{L}=X^{-1}\overline{Y}$. Proof. Construct a co-positive Lyapunov function candidate $V(\underline{e},\overline{e})=\underline{e}^{\top}v+\overline{e}^{\top}v$ where $v=X\mathbf{1}_{n_{x}\times 1}\in\mathbb{R}^{n_{x}}_{+}$. Following the same guideline in Theorem 3, the following inequality can be obtained $\displaystyle\dot{V}(\underline{e},\overline{e})\leq\begin{bmatrix}\underline{e}^{\top}&\overline{e}^{\top}\end{bmatrix}\begin{bmatrix}\underline{\Theta}\\\ \overline{\Theta}\end{bmatrix}v+(\underline{\rho}^{\top}+\overline{\rho}^{\top})v+\mathcal{G}^{\top}v$ (55) where $\overline{\Theta}=A^{\top}-C^{\top}\overline{L}^{\top}+(\overline{\gamma}_{1}+\overline{\gamma}_{2})I_{n_{x}\times n_{x}}$, $\underline{\Theta}=A^{\top}-C^{\top}\underline{L}^{\top}+(\underline{\gamma}_{1}+\underline{\gamma}_{2})I_{n_{x}\times n_{x}}$, and $\mathcal{G}=\overline{\Phi}(y,\underline{u},\overline{u})-\underline{\Phi}(y,\underline{u},\overline{u})$. Based on Lemma 2 and using the fact of $\underline{y}=\overline{y}=y$ in $\underline{\Phi}$ and $\overline{\Phi}$, it leads to $\displaystyle\mathcal{G}\leq$ $\displaystyle\begin{bmatrix}\underline{U}&\underline{V}\end{bmatrix}\left(\begin{bmatrix}y\\\ u\end{bmatrix}-\begin{bmatrix}y\\\ \underline{u}\end{bmatrix}\right)+\begin{bmatrix}\overline{U}&\overline{V}\end{bmatrix}\left(\begin{bmatrix}y\\\ \overline{u}\end{bmatrix}-\begin{bmatrix}y\\\ {u}\end{bmatrix}\right)$ $\displaystyle\leq$ $\displaystyle\begin{bmatrix}\underline{U}&\overline{U}\end{bmatrix}\begin{bmatrix}0\\\ 0\end{bmatrix}+(\underline{V}+\overline{V})(\overline{u}-\underline{u})$ $\displaystyle=$ $\displaystyle(\underline{V}+\overline{V})(\overline{u}-\underline{u})$ (56) which is irrelevant to $\underline{U}$ and $\overline{U}$. Then, we have $\displaystyle\dot{V}(\underline{e},\overline{e})\leq\begin{bmatrix}\underline{e}^{\top}&\overline{e}^{\top}\end{bmatrix}\begin{bmatrix}\underline{\Theta}\\\ \overline{\Theta}\end{bmatrix}v+\theta$ (57) where $\theta=(\overline{u}^{\top}-\underline{u}^{\top})(\underline{V}^{\top}+\overline{V}^{\top})v+(\underline{\rho}^{\top}+\overline{\rho}^{\top})v$. Due to (54) and following the same guidelines in Theorem 1, the following inequality can be derived $\displaystyle\dot{V}(\underline{e},\overline{e})\leq-\lambda\underline{e}^{\top}v-\lambda\overline{e}^{\top}v+\theta=-\lambda V(\underline{e},\overline{e})+\theta$ (58) which ensure the practical stability of error system. The proof is complete. $\hfill\hfill\square$ ###### Remark 6 As Corollary 1 indicates, the design process of observer gain computation has nothing to do with the neural network. The observer gains $\underline{L}$ and $\overline{L}$ are obtained by solving an LP problem in terms of (51)–(54) which is dependent upon system dynamics without considering neural network components. This is because the measurable output $y$ makes the portion of the output of neural network $\Phi$ driven by $y$ completely compensated by the outputs of auxiliary neural networks $\underline{\Phi}$, $\overline{\Phi}$ which are also driven by same values of measurement $y$. This promising feature of irrelevance to neural networks leads this developed methods to be able to deal with dynamical systems with large-scale neural network components such as deep neural network controllers regardless of the size of neural networks. ## IV Application to Adaptive Cruise Control Systems Figure 3: Illustration of adaptive cruise control systems and simulink block diagram of the closed-loop system. In this section, the developed run-time safety monitoring approach will be evaluated by an adaptive cruise control (ACC) system which is under control of a neural network controller as depicted in Figure 3. Two cars are involved in the ACC system, an ego car with ACC module and a lead car. A radar sensor is equipped on the car to measure the distance to the lead car in run time. The run-time measured distance is denoted by $d_{\mathrm{rel}}$. Moreover, the relative velocity against the lead car is also measured in run time which is denoted by $v_{\mathrm{rel}}$. There are two system operating modes including speed control and spacing control. Two control modes are operating in run time. In speed control mode, the ego car travels at a speed of $v_{\mathrm{set}}$ set by the driver. In spacing control mode, the ego car has to maintain a safe distance from the lead car denoted by $d_{\mathrm{safe}}$. The system dynamics of ACC is expressed in the following form of $\displaystyle\left\\{{\begin{array}[]{{20}l}\dot{x}_{l}(t)=v_{l}(t)\\\ \dot{v}_{l}(t)=\gamma_{l}(t)\\\ \dot{\gamma}_{l}(t)=-2\gamma_{l}(t)+2\alpha_{l}(t)-\mu v^{2}_{l}(t)\\\ \dot{x}_{e}=v_{e}(t)\\\ \dot{v}_{e}(t)=\gamma_{e}(t)\\\ \dot{\gamma}_{e}(t)=-2\gamma_{e}(t)+2\alpha_{e}(t)-\mu v^{2}_{e}(t)\end{array}}\right.$ (65) where $x_{l}$, $v_{l}$ and $\gamma_{l}$ are the position, velocity and actual acceleration of the lead car, and $x_{e}$, $v_{e}$ and $\gamma_{e}$ are the position, velocity and actual acceleration of the ego car, respectively. $\alpha_{l}$ and $\alpha_{e}$ is the acceleration control inputs applied to the lead and ego car. $\mu=0.0001$ is the friction parameter. A $2\times 20$ feedforward neural network controller with tanh and purelin is trained for the ACC system. Specifically, the measurement of the ACC system which also performs as the inputs to the neural network ACC control module are listed in Table I. TABLE I: Measured Outputs of ACC system and Inputs to Neural Network Controller Driver-set velocity \| $v_{\mathrm{set}}$ ---\|--- Time gap \| $t_{\mathrm{gap}}$ Velocity of the ego car \| $v_{e}$ Relative distance to the lead car \| $d_{\mathrm{rel}}=x_{l}-x_{e}$ Relative velocity to the lead car \| $v_{\mathrm{rel}}=v_{l}-v_{e}$ The output for the neural network ACC controller is the acceleration of the ego car, namely $\alpha_{e}$. In summary, the neural network controller for the acceleration control of the ego car is in the form of $\displaystyle\alpha_{e}(t)=\Phi(d_{\mathrm{rel}}(t),v_{\mathrm{rel}}(t),v_{e}(t),v_{\mathrm{set}}(t),t_{\mathrm{gap}}).$ (66) Letting $x=[x_{l},v_{l},\gamma_{l},x_{e},v_{e},\gamma_{e}]^{\top}$, $u=[\alpha_{l},v_{\mathrm{set}},t_{\mathrm{gap}}]^{\top}$ and $y=[v_{e},d_{\mathrm{rel}},v_{\mathrm{rel}}]^{\top}$, the ACC system can be rewritten in the following neural-network-enabled system $\displaystyle\begin{cases}\dot{x}=Ax+f(y)+\tilde{\Phi}(y,u)\\\ y=Cx\end{cases}$ (67) where system matrices $A$, $C$, nonlinearity $f(y)$ and neural network component $\Phi(y,u)$ are defined as below: $\displaystyle A$ $\displaystyle=\begin{bmatrix}0&1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&-2&0&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&0&-2\end{bmatrix}$ $\displaystyle C$ $\displaystyle=\begin{bmatrix}0&0&0&0&1&0\\\ 1&0&0&-1&0&0\\\ 0&1&0&0&-1&0\end{bmatrix}$ $\displaystyle f(y)$ $\displaystyle=\begin{bmatrix}0&0&-0.0001(y_{1}+y_{3})^{2}&0&-0.0001y_{1}^{2}\end{bmatrix}^{\top}$ $\displaystyle\tilde{\Phi}(y,u)$ $\displaystyle=\begin{bmatrix}0&0&2u_{1}&0&0&\Phi(y,u_{2},u_{3})\end{bmatrix}^{\top}$ where $\Phi(y,u_{2},u_{3})$ is the neural network controller (66). In addition, considering the physical limitations of the vehicle dynamics, the acceleration is constrained to the range $[-3,2]$ ($m/s^{2}$), thus input $u_{1}\in[-3,2]$. Since the nonlinearity of $f(y)$ can be obtained with the measurement of $y$, we can let $f(y)=\underline{f}(\underline{x},\overline{x})=\overline{f}(\underline{x},\overline{x})$ in state estimator (49), which is thus constructed as follows $\displaystyle\begin{cases}\dot{\underline{x}}=A\underline{x}+f(y)+\underline{\Phi}(y,\underline{u},\overline{u})+\underline{L}(y-C\underline{x})\\\ \dot{\overline{x}}=A\overline{x}+f(y)+\overline{\Phi}(y,\underline{u},\overline{u})+\overline{L}(y-C\overline{x})\end{cases}.$ (68) The auxiliary neural networks $\underline{\Phi}$ and $\overline{\Phi}$ are designed based on $\Phi$ according to (14) and (15). The observer gains $\underline{L}$ and $\overline{L}$ can be computed by Corollary 1 via solving a collection of LP problems. Figure 4: Run-time safety monitoring of positions $x_{l}(t)$ and $x_{e}(t)$. The state trajectories $x(t)$ (solid lines) are bounded within the estimated bounds $\underline{x}(t)$, $\overline{x}(t)$ (dashed lines). Magnified time windows are used for clear clarification. Figure 5: Run-time safety monitoring of velocities $v_{l}(t)$ and $v_{e}(t)$. The state trajectories $v(t)$ (solid lines) are bounded within the estimated bounds $\underline{v}(t)$, $\overline{v}(t)$ (dashed lines). Magnified time windows are used for clear clarification. Figure 6: Run-time safety monitoring of accelerations $\gamma_{l}(t)$ and $\gamma_{e}(t)$. The state trajectories $\gamma(t)$ (solid lines) are bounded within the estimated bounds $\underline{\gamma}(t)$, $\overline{\gamma}(t)$ (dashed lines). Magnified time windows are used for clear clarification. The run-time boundary estimations of state trajectories of positions $\\{x_{l}(t),x_{e}(t)\\}$, velocities $\\{v_{l}(t),v_{e}(t)\\}$ and accelerations $\\{\gamma_{l}(t),\gamma_{e}(t)\\}$ during ACC system evolves in time interval $[0,100]$ are shown in Figures 4, 5 and 6. As shown in these simulation results, the state trajectories are always bounded within the lower and upper-bounds of observers which can be used as a run-time safety monitoring state for system state during operation. ## V Conclusions The run-time safety monitoring problem of neural-network-enabled dynamical systems is addressed in this paper. The online lower- and upper-bounds of state trajectories can be provided by the run-time safety estimator in the form of interval Luenberger observer form. The design process includes two essential parts, namely two auxiliary neural networks and observer gains. In summary, two auxiliary neural networks are derived from the neural network component embedded in the original dynamical system and observer gains are computed by a family of LP problems. Regarding neural networks driven by measurements of the system, it is noted that the design process is independent with the neural network so that there is no scalability concern for the size of neural networks. An application to ACC system is presented to validate the developed method. Further applications to complex dynamical systems such as systems with switching behaviors [34, 35, 36, 37, 38] should be considered in the future. Beyond the run-time safety monitoring approach developed in this paper, the future work should be the run-time correction of neural networks once the unsafe behavior is detected. 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# Text Line Segmentation for Challenging Handwritten Document Images Using Fully Convolutional Network Berat Barakat, Ahmad Droby, Majeed Kassis and Jihad El-Sana The Department of Computer Science Ben-Gurion University of the Negev Email: {berat, drobya, majeek<EMAIL_ADDRESS> ###### Abstract This paper presents a method for text line segmentation of challenging historical manuscript images. These manuscript images contain narrow interline spaces with touching components, interpenetrating vowel signs and inconsistent font types and sizes. In addition, they contain curved, multi-skewed and multi-directed side note lines within a complex page layout. Therefore, bounding polygon labeling would be very difficult and time consuming. Instead we rely on line masks that connect the components on the same text line. Then these line masks are predicted using a Fully Convolutional Network (FCN). In the literature, FCN has been successfully used for text line segmentation of regular handwritten document images. The present paper shows that FCN is useful with challenging manuscript images as well. Using a new evaluation metric that is sensitive to over segmentation as well as under segmentation, testing results on a publicly available challenging handwritten dataset are comparable with the results of a previous work on the same dataset. ## I Introduction Historical handwritten documents are valuable since they connect past and present. Commonly they are converted into digital form for being easily available to scholars worldwide. However, digital historical documents pose real challenges for automatic writer identification, keyword searching, and indexing. Text line segmentation of document images is an essential pre- processing operation for these automatizing problems. The problems for text line segmentation of handwritten documents consist of touching, overlapping and crowded characters and vowel signs among consecutive text lines besides narrow interline spacing (Figure 1). In addition to the problems of handwritten documents, challenging handwritten documents contain various writing styles with inconsistent font types and font sizes through multi-skewed, multi-directed and curved text lines (Figure 2). Several text line extraction methods for handwritten documents have been proposed. Projection method was initially used for printed documents [1, 2] then modified for skewed [3, 4] and multi-skewed documents [5]. Smearing method [6] which fills the space between consecutive foreground pixels can be used on skewed documents [7] as well. Grouping method aggregates pixels or connected components in a bottom up strategy and is superior in case of skewed and curved text lines [8, 9]. Machine learning algorithms, a type of grouping method, handle text line segmentation as a pixel classification problem. Pixel classification can be done in a sliding window manner [10, 11] which is not desirable due to redundant and expensive computation of overlapping areas in the sliding windows. On the other hand, dense prediction does not suffer from redundant computation and has been successfully used for text line segmentation of handwritten documents [12, 13]. Figure 1: Text line segmentation problems with regular handwritten documents Figure 2: Additional text line segmentation problems with challenging handwritten documents. Various writing styles are also noticeable. However, text line extraction of challenging documents has not been extensively studied. This paper provides a dataset (https://www.cs.bgu.ac.il/ vml/) of challenging documents with multi-skewed, multi-directed and curved handwritten text lines. It then describes text line segmentation of this dataset using Fully Convolutional Network (FCN). We also propose a new evaluation metric that is sensitive to both, over and under segmentation of lines in contrast to the available metrics. Using the new evaluation metric we show that FCN based method is comparable to Cohen et al.’s method [9]. In the rest of the paper we describe our method and the new evaluation metric in detail, and present the challenging dataset and report experimental results. After comparing results we conclude and discuss the future directions. ## II Method Fully Convolutional Network has made great improvements in object segmentation field [14]. It is an end to end semantic segmentation framework that extracts the features and learns the classifier function simultaneously. FCN inputs the original images and their pixel level annotations for learning the hypothesis function that can predict whether a pixel belongs to a text line label or not. So the crucial question is how to annotate the text lines. Baseline labeling is not applicable to all the alphabets whereas bounding polygon is applicable but very cumbersome for crowded documents [15]. Instead of baseline or bounding polygon, we used line mask labeling that connects the characters in the same line (Figure 4). A line mask disregards diacritics and touching components between lines. ### II-A FCN architecture The FCN architecture (Figure 3) we used is based on the FCN proposed for semantic segmentation [14]. First five blocks, encoder part, follow the design of VGG 16-layer network [16] except the discarded final layer. The encoder consists of five convolutional blocks. Each convolutional block contains a number of convolutional layers followed by a max pooling layer. Through the encoder, input image is downsampled, and the filters can see coarser information with larger receptive field. Then the decoder part of FCN upsamples coarse outputs to dense pixels. Upsampling is done by transpose convolution by applying a convolution filter with a stride equal to $\frac{1}{f}$, for upsampling by a factor $f$. FCN has two types, FCN8 and FCN32, according to the upsampling factor used in the last layer. FCN32 upsamples the last convolutional layer by $f=32$ at one time. However, particularly FCN8 architecture was selected because it has been successful in page layout analysis of a similar dataset [17]. FCN8 adds final layer of encoder to the lower layers with finer information, then upsamples the combined layer back to the input size. Default input size of VGG is $224\times 224$, which does not cover more than 2 main text lines and 3 side text lines. To include more context we changed the input size to $320\times 320$ pixels. We also changed the output channel to 2 which is the number of classes, text line or background. Figure 3: The FCN architecture. Pooling and prediction layers are shown as grids that show relative coarseness. Convolutional layers are shown as vertical lines. FCN8 4 times upsamples the final layer, 2 times upsamples the pool4 layer and combine them with pool3 layer finally to upsample to input size. ### II-B Pre-processing We binarize the 30 document images, each with an approximate size of $3000\times 4000$, by applying an adaptive binarization method for historical documents [18]. Binarized images were inverted before inputting them to the FCN. Then we manually annotated the line masks on the document images. A sequence of original, binarized and labeled document images is demonstrated in Figure 4. Finally a total of $50.000$ and $6.000$ random patches of size $320\times 320$ were generated for training and validation sets of each fold respectively. Figure 4: A sequence of original, binarized and labeled document images. Random patches for training are generated from the binarized and labeled images. ### II-C Training and testing We applied 6 fold cross validation scheme for the experiments. Each fold was split into train, validation and test sets. In each fold, training continued for 80 epochs and the model with the least validation loss value was saved. The best model was then used to predict the corresponding test set. This training procedure ensures generalizability of the proposed model. The FCN was trained by a batch size of 16, using Stochastic Gradient Descent (SGD) with momentum equals to $0.9$ and learning rate equals to $0.001$. VGG was initialized with its publicly available pre-trained weights. During the testing, a sliding window of size $320\times 320$ was used for prediction, but only the inner window of size $100\times 100$ was considered. Page was padded with black pixels at its right and bottom sides if its size is not an integer multiple of the sliding window size, in addition to padding it at 4 sides for considering only the central part of the sliding window. ### II-D Post-processing Occasionally predicted line masks were disconnected. Thus, we needed to post- process the FCN output. Given a predicted line mask image, firstly the orientation of each connected component was computed. Then the image was split into $N$ layers where each layer contains the connected components with same orientation. Later a directional morphological operation was applied on each layer. Resulting layers at the end were combined back using a pixel-wise OR operation. Let $C=\\{c_{1},c_{2},...,c_{M}\\}$ is the set of connected components in the predicted line mask image. $C$ is further divided into $N$ intersecting subsets $B_{1},B_{2},...,B_{N}\subseteq C$ such that: $B_{i}=\\{c_{i}:\alpha(c_{i})^{2}\|v_{j}^{T}\cdot\theta(c_{i})\|<\epsilon\\}$ (1) $i=1,2,\dots M,j=1,2,\dots N$ $v_{j}=(\cos(j\frac{\pi}{N}),\sin(j\frac{\pi}{N}))$ (2) $\alpha(c)=\frac{R_{maj}}{R_{maj}+R_{min}}$ (3) where $v_{j}\in[0,\pi]$ is a particular orientation and $\epsilon\in[0,1]$ is the threshold for selecting the connected components perpendicular to this particular orientation. $R_{maj}$ and $R_{min}$ are the major and minor axes of the fitted ellipse to the connected component $c$ respectively. $\alpha(c)\in[0.5,1]$ indicates how sure are we about the orientation of the component $c$. $\theta(c)$ is the unit vector that represents the orientation of the fitted ellipse to the connected component $c$. Ellipse fitting was done using the algorithm described in [19]. Eventually for each subset $B_{i}$ a morphological operation with a narrow kernel in the orientation of this subset was applied. Figure 5 shows the result of post-processing on a sample predicted line mask image. Figure 5: Post processing phases: (a) Predicted line mask may have disconnected components. (b) For each component an ellipse (red) is fitted and its orientation vector $\theta(c)$ (blue) is computed. (c) Morphological dilation is applied to each component with a narrow kernel in the direction of its fitted ellipse. ### II-E Connectivity Component Based Line Extraction Accuracy Metric Available evaluation methods for text line segmentation either use a pixel- wise matching mostly normalized by line length or maximum overlap according to a certain threshold between the extracted and annotated lines. These methods have their short-comings. Thus, we present a different evaluation method that provides a better picture of the results. The theoretical basis is as follows. A line extraction algorithm succeeds in extracting a complete text line if it has succeeded in finding all the connected components of this line. That is if the algorithm labels all the connected components of a line with the same label, then it has successfully extracted this line without any errors. This is in contrast to having multiple labels, over segmentation, or extracting part of the connected components, under segmentation, along the same text line. To describe the new metric, we define the term connectivity component. A connectivity component is the connection between two consecutive components with the same label. The number of connectivity components in a line is equal to the number of connectivity components between every two consequent connected components and in addition to it a beginning of line connectivity component. The extra connectivity component handles cases where a line contains one connected component only. _C_ omplete extraction of a line with several connectivity components is extracting all its connectivity components and assigning them the same label. To quantify the new metric we define recall and precision for calculating F-measure. Recall is the number of connectivity components extracted by the algorithm in a line, out of all connectivity components found in the corresponding line in ground truth. Precision is the number of correct connectivity components extracted by the algorithm in a line out of all connectivity components extracted by the algorithm. Note that some connectivity components extracted by the algorithm are not found in the ground truth, and some connectivity components are found in the ground truth but not extracted by the algorithm. First type of error is quantified in the precision part of the metric, while the latter type of error is quantified in the recall part of the metric. Let $G=\\{g_{1},g_{2},g_{3},\dots g_{m}\\}$ is the set of connected components of a line in the ground truth, $E_{i}\in\\{E_{1},E_{2},E_{3},\dots E_{n}\\}$ is the set of extracted lines such that $E_{i}\cap G\neq\emptyset$, then for this line in the ground truth, recall ($R$) and precision ($P$) is: $R=\sum\limits_{i}{\frac{\|E_{i}\cap G\|-1}{\|G\|-1}}$ (4) $P=\frac{\sum\limits_{i}{\|E_{i}\cap G\|-1}}{\sum\limits_{i}{\|E_{i}\|-1}}$ (5) The recall definition penalizes over segmentation of a line where an extraction algorithm assigns multiple labels to the components of a single line. In contrast, the precision definition penalizes under segmentation where an extraction algorithm incorrectly assigns a line label to the components that are not in the ground truth of this line (Figure 6). Figure 6: Connectivity component based metric penalizes under segmentation by its precision definition and over segmentation by its recall definition. ## III Dataset Although several benchmark datasets [20, 21, 22] of handwritten document images are available, a challenging document dataset is absent. We collected a set of challenging document images from the Islamic Heritage Project (IHP), Harvard. This dataset is publicly available (https://www.cs.bgu.ac.il/ vml/). The challenging dataset contains 30 pages from two different manuscripts. It is written in Arabic language and contains 2732 text lines where a considerable amount of them are multi-directed, multi-skewed or curved. Ground truth where text lines were labeled manually by line masks is also available in the dataset. ## IV Results We tested the proposed system on the new challenging handwritten document dataset. In each fold we trained FCN on 50.000 patches randomly cropped from 20 pages, validated on 6.000 patches randomly cropped from 5 pages and tested on 5 whole pages using a sliding window. Predicted line mask images were then post-processed with $N=10$ and $\epsilon=0.2$. Extracted text lines were evaluated using the new metric to calculate the F-measure. Entire training took around 9 days. Visualization of the first convolutional layer filters shows that network have learned and filters have converged (Figure 7). The model achieved $89\%$ training accuracy and $88\%$ validation accuracy on average. Two characteristics of the dataset lead the model lacking to overfit to the training set. First it contains two manuscripts with 6 and 24 pages. The manuscript with 6 pages caused most of the errors. Second, although dataset contains considerable amount of multi-skewed, multi-directed and curved lines, they spatially cover smaller area due to smaller font size. This lead to less number of random patches with skewed or curved lines in relative to the number of random patches with regular lines. Figure 7: Visualization of the filters in the first convolutional layer. Table I shows the performance of our method compared with the method of Cohen et al.[9]. Their approach achieved outstanding results on ICDAR2009 [20] and ICDAR2013 [21] datasets. We run their publicly available code (http://www.cs.bgu.ac.il/ rafico/LineExtraction.zip) on the challenging handwritten dataset. TABLE I: Comparison with the method of Cohen et al. Method \| Recall \| Precision \| F-measure ---\|---\|---\|--- Proposed \| 0.82 \| 0.78 \| 0.80 Cohen et al.[9] \| 0.74 \| 0.60 \| 0.66 Figure 8: Sample image of ground truth and corresponding outputs of Cohen et al. [9] and FCN. Lower precision values show that both method tend to under segment. Most errors of FCN method occur at curved areas whereas most errors of method of Cohen et al. occur at the main text areas. Our method outperforms the method of Cohen et al. in terms of both recall and precision. Both methods have lower precision values than recall values. This demonstrates that most of their errors are due to wrongly connected lines in their output. Therefore both method tend to under segment more than over segment. We have noticed that in the output of our method, wrongly connected lines mostly crop up at the curved areas in contrast to the output of Cohen et al where the wrongly connected lines are mostly crop up at the main text areas. The former was a result of small number of training patches with curved lines. Curved lines can be long but their curved part covers relatively a small spatial area which is one or two corner parts of a page. The latter was a result of small number of main text lines in relative to the number of side text lines in a page, where the average height of text lines converges to the height of side text lines. Therefore method of Cohen et al., which runs according to the average height of text lines, has most errors in main text areas. Figure 8 shows some qualitative results for the latter and the former types of errors on the challenging dataset. ## V Conclusion This paper introduces challenging handwritten documents, presents a dataset of challenging handwritten documents and its text line segmentation using FCN. Line mask labeling is less cumbersome for challenging handwritten documents and is a proper way for FCN training. We have also defined a new evaluation metric with the concept of connectivity component. This metric is sensitive to both over and under segmentation. New metric is used to validate the proposed method on the challenging handwritten dataset. 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# Aesthetics, Personalization and Recommendation: A survey on Deep Learning in Fashion Wei Gong<EMAIL_ADDRESS>University of Science and Technology of ChinaNo.96, JinZhai Road Baohe DistrictHefeiAnhuiChina230026 and Laila Khalid <EMAIL_ADDRESS>University of Science and Technology of ChinaNo.96, JinZhai Road Baohe DistrictHefeiAnhuiChina230026 (2020) ###### Abstract. Machine learning is completely changing the trends in the fashion industry. From big to small every brand is using machine learning techniques in order to improve their revenue, increase customers and stay ahead of the trend. People are into fashion and they want to know what looks best and how they can improve their style and elevate their personality. And since systems are already monitoring every sale and coming trends , why not utilize their power in getting a recommendation regarding outfit. Using Deep learning technology and infusing it with Computer Vision techniques one can do so by utilizing Brain-inspired Deep Networks, and engaging into Neuroaesthetics, working with GAN’s and Training them, playing around with Unstructured Data,and infusing the transformer architecture are just some highlights which can be touched with the Fashion domain. It’s all about designing a system that can tell us information regarding the fashion aspect that can come in handy with the ever growing demand. Personalization is a big factor that impacts the spending choices of customers.The survey also shows remarkable approaches that encroach the subject of achieving that by divulging deep into how visual data can be interpreted and leveraged into different models and approaches. Aesthetics play a vital role in clothing recommendation as users’ decision depends largely on whether the clothing is in line with their aesthetics, however the conventional image features cannot portray this directly. For that the survey also highlights remarkable models like tensor factorization model, conditional random field model among others to cater the need to acknowledge aesthetics as an important factor in Apparel recommendation.These AI inspired deep models can pinpoint exactly which certain style resonates best with their customers and they can have an understanding of how the new designs will set in with the community. With AI and machine learning your businesses can stay ahead of the fashion trends and deliver exactly what your customers want and when they want it. Deep Learning, neural networks, Fashion, Aesthetics ,Recommendation, Personalization ††copyright: acmcopyright††journalyear: 2020††doi: not available yet††journal: JACM††journalvolume: 00††journalnumber: 0††article: 111††publicationmonth: 0††ccs: Computing methodologies Computer vision††ccs: Applied computing Online shopping††ccs: Computing methodologies Neural networks ## 1\. Introduction If we go over the past decade and see how deep learning has achieved significant success in many popular Industries and areas. We observe how perception tasks, including visual object recognition and text understanding and speech recognition, have revolutionized different regions. There is no comparison as to how successful deep learning has been. Still, suppose we want to discuss deep learning in the real terms of the fashion industry. In that case, we see a lot of opportunities and research areas that are still available to work on. As we all know, fashion is an ever-evolving industry. There are new trends that are setting in every second that is passing by. Although clothing design is like one of the most creative realms in the Contemporary World (Insight, [n.d.]), whether it’s because of the considerable creative part of the design process or equivocal information about clothing, the fact remains to be. Internet shopping has also grown incredibly in the last few years, and fashion has created immense opportunities. Exciting applications for image understanding , retrieval and tagging are surfacing, and there are loads of different application areas that they can be used on. For example, text analysis, image analysis, and similarity retrieval can be utilized in fashion. So deep learning is an aspect that we can use to train our computer to perform human-like tasks such as recognizing speech, identifying images or making predictions. For example, the results described in the apparel design and fashion industry allow users to translate the image into the text that might as well be interpreted as a description of the garment based on its sketch. We also know that images are an essential aspect because they display content and convey emotions like sadness, excitement, anger, etc. So useful image classification is beneficial, and obviously, it’s been used in computer vision in multimedia. Still, if you find research regarding fashion and specifically in terms of aesthetic features or personalization, you will find only a few specific directions. Discussing one of them that is available to describe images inspired by art theories, which are, you know, intuitive, discriminative, and easily understandable. So we know that the effective image classification based on these features can achieve high accuracy compared with the state-of-the-art. For that, we take an example in the paper (Wang, 2013) where they develop an Emotion guided image gallery to demonstrate the proposed feature collection. So the authors achieve mining the interpretable visual features directly affecting human emotional perception from the viewpoint of art theories. Another example in another paper (Borràs et al., 2003) is where they discussed that content-based image retrieval is done in terms of people’s appearance. It’s a two-stage process that is composed of image segmentation and region- based interpretation. The modelling of an image is due to an attributed graph and a hybrid method that follows a split and merge strategy. There are a lot of different stuff that is being worked on in this field of computer vision specifically, and image retrieval from databases is usually a formula, in terms of descriptions that combine the Salient features such as colour, texture, shapes etc. Today, more and more retail companies are trying to understand, to stay ahead of the trend curve and because they want to reshape their business to stay ahead while implementing tech forward approaches and solutions. And data analysis brings diverse opportunities to companies, which allows them to reach their customer goal and offer a smarter experience to them. But the thing is that the lack of profound insights based on reliable statistics is the major challenge of fashion retailers that they face. So for that, computer vision technologies and deep learning can come in very handy. And as we all know, computer vision is still an evolving technology, so we can speak about specific optimization and cost reduction techniques that can come in handy, for example, like how the information regarding what people wear, how customers kind of match their garments and what or which or who influences their taste is essential for fashion retailers. As we can see the Instagram influencers, we see that many people follow them and try to copy their trends and how they are inspiring a lot of followers. Image recognition technology also helps business owners collect data, process it, and gain an actionable insight for Effective Trend forecasting. For that, in this particular article (ELEKS, [n.d.]b), we see that the dashboard they developed allows seeing how frequently one specific type of garment appears a day. Like what type of apparel is popular within a particular age range or how people sort of match their attire. Like for example, how a specific jacket is trending or why is it popular among teenagers? Or why is a scarf popular amongst the elders. They developed the graph that shows how certain prevalent types of garments would be over the next season’s you know, which could broadly impact the new upcoming trend for the fashion. This kind of analysis also aims to help fashion retailers and brands plan sales and learn to avoid any surplus. The author suggests that in the visual search domain with a focus mainly on image similarity for like, e-commerce and Online shops and understanding images of clothing, it means a lot more than just classifying them into different categories. Because if you don’t get a meaningful description of the whole image you classify, then you are losing a lot of information that could come in handy. In this way, one can gain reliable and timely insights into fashion trends across any location. What defines those trends is people’s unique choices, like how they choose something and what goes with their personality. The element of personalization is one of the biggest game-changers in this apparel recommendation. By targeting this factor, businesses can attract more customers. The thing that I like about this deep learning aspect is that it penetrates the industry and, you know, activities where human creativity has traditionally dominated. It adds a futuristic touch to fashion, art , architecture and music so on. Another paper’s (ELEKS, [n.d.]a) key finding is that the representation of content and style in the convolutional neural networks are separable. That is, you know if we can manipulate both representations independently to produce new and perceptually meaningful images. If you look, fashion is an entirely new direction for machine learning. So to design clothes one should you know, basically have an understanding of the mechanism of technique, like how certain styles go famous, what things they are having that are attracting millions of followers around and what causes the spread, you know the spread of the Fashion trends and principles and evolution of patterns, so the task of designing or predicting trends can be simplified. The paper under discussion where the author suggests that now designing or predicting Trends can be simplified, thanks to a new class of neural networks. These networks basically can automatically allocate shapes, elements, and types of clothing and further combine them. This allows a whole fresh feel of how you can manipulate the patterns and see which patterns can influence more influence than the others. Now aesthetics play a vital role in the user’s pick, and even though personalization is tricky to play with, aesthetics are not. Because everyone appreciates eye-pleasing items and if we can manipulate the role of aesthetics in our fashion recommendation, we can hit the jackpot. So there are many various aspects of fashion in which deep learning can enhance and help us out. There are multiple domains for improving the current elements and how we can help predict and revolutionize this industry. This survey is organized in the following sections. Sec. 2 reviews the fashion recommendation systems and approaches that come out on top and are the basis for future work. Sec. 3 illustrates the positions for aesthetics in fashion, all it’s analysis containing various approaches. Sec. 4 provides an overview of personalization in fashion , different top approaches that have tasks comprising Deep Neural Networks, GAN’s, and handling unstructured data. Sec. 5 demonstrates selected applications and future horizons that can be worked on. Last but not least, concluding remarks are given in Sec. 6. ## 2\. Recommendation Systems Well, if you indulge in object recognition, you will find that fashion sense is a bit more subtle and sophisticated, you know, which can require specific domain expertise in outfit composition. So, for example, if you refer to an outfit as a set of clothes working together kind of typically for a desired specific style or to find a good Outfit composition, what we need is not only to follow the appropriate dressing course, but it can also have a creative aspect in balancing the contrast of colours and different styles. And although we have seen a relative number of researches that are mainly based on clothes retrieval and recommendation but what we have seen is that none of them consider the problem of fashion outfit composition. On the one hand, you know a fashion concept is often subtle and subjective and is non-trivial to get you to know consensus from ordinary labellers if they are not Fashion experts. On the other hand, there may be a large number of attributes for describing fashion. ### 2.1. End-to-End Deep Learning Approach on Set Data. It is challenging to obtain exhaustive labels for training. So, as a result, most of the existing studies are kind of, you know, limited to the simple scenario of retrieving similar clothes or choosing individual clothes for a given event. So the paper (Li et al., 2016) that is being reviewed proposes a data-driven approach to train a model that can automatically compose a suitable fashion outfit. This approach is motivated by the surge of the increasing online fashion trends, including Pinterest and YouTube, and how teenagers have been addicted to creating every new culture trend on these sites. #### 2.1.1. Background So basically what they have done is that they have developed a full automatic composition system that is based upon a scorer by iteratively evaluating all the possible outfit candidates. But this model had some challenges in which they had to look out for possible solutions. For example, one of the challenges that they Encountered was that complicated visual contents of the fashion images? So, you know, there are potentially many kinds of different attributes like color, textures, categories and spectrum’s etc and it is impossible to label or even list all possible attributes. So there is this hindrance and second one would be the rich context of fashion outfit for example, clothing outfits can kind of sort of reflect current personality and interest. So if one style is acceptable to a specific group or culture. It may be offensive to the others. So to infer such information they have taken into account not only the pixel information but also the context information in the fashion output. #### 2.1.2. Proposed Approach So basically for These challenges they proposed different solutions like for the first challenge they have proposed an end-to-end system of encoding visual features through a deep convolutional network which sort of, you know, takes a fashion outfit as an input and processes it and then predicts the user engagement levels. And for the Second Challenge what happens is that a multimode Deep learning framework, which sort of leverages the context information from the image itself and the experiment that they did through that was that the multi-modal approach significantly outperforms the single model. And provides the suitable and more reliable solution for the fashion outfit for scoring tasks and thus the full composition tasks. So these are the contributions that they are enlisting and they are basically proposing an end- to-end trainable system to fuse signals from multi-level hybrid modalities that includes image and metadata of the fashion items and they also collected a large scale of database that are for the fashion outfit related research. Lastly they propose a fashion outfit composition to the solution based on a reliable sort of outfit quality predictor and predicting fashion is never easy, but it is something that they have put forward because many interleaving factors visible or hidden contribute to the process the combinatorial nature of the problem also makes it very interesting and it’s a test tone for the state-of-the-art machine learning systems. Figure 1. The proposed fashion outfit scoring model ### 2.2. Implicit Feedback Based As we know that the fashion domain has quite a lot of several intriguing properties that can be personalized and which make personalization recommendations even far more difficult than the traditional domains. So in order to sort of avoid potential bias, like when using explicit user ratings, which are also pretty much expensive to obtain. #### 2.2.1. Background So this paper (Nguyen et al., 2014) basically suggests the work that approaches fashion recommendations by sort of analyzing the implicit feedback from users in an app. Basically the design criteria is that the system shall be completely unobstructive and thus the recommendation system cannot , you know, rely explicitly on the ratings rather It will be based on the rich history and the interaction between the user and the app. In simple words it relies on the implicit feedback that is you know, the user preference is to be automatically inferred from the behavior. Though there are still some challenges that can be gathered in this approach that is the most notable interaction a user has with an item is a sign of interest ,so the system therefore never receives a negative feedback and of course, you know an item can be both clicked and loved so it is also multi- faceted and then the different types of feedback will have to be combined into a single numerical value as defined for an experiment. Set a preference score for the recommendation algorithms. It is difficult to evaluate such a system compared to explicit-rating-systems, because the system does not have a target rating to compare its predictions to. So all in all the success basically relies on the implicit feedback system that has a well-defined strategy for inferring user preference from implicit feedback data and combining even types into implicit scores and then evaluating these scores and recommendations by using a suitable metric. #### 2.2.2. Proposed Approach So basically the authors in order to build this recommendation system took the first step and that was to generate implicit preference scores and to you know translate data that is being captured by a user’s interaction with an item into a specific number that can be called employees implicit preference score and that can be also later used to rank it with the other items so that most important factor in this was when they created such numbers to understand the data available and their implications for user preference. So once you can have the data analyzed suitable generalizations can then be furthermore chosen. And then the second step was for defining the penalisation functions. Important properties in the fashion domain that must be captured by the system include seasons and trends, price sensitivity and popularity. In general, when a user $u$ triggers an event $e,$ e.g. Clicks, we have a range of possible scores to give this event. We use $S_{e}$ to denote this score, and let $m_{e}$ and $M_{e}$ denote the minimum and maximum score possible for event $e,$ respectively. We then use a penalisation function $p_{u}(x)$ taking a feature value $x$ (e.g., related to an item’s price), and returns a number between zero and one to adjust the score inside the possible range. $S_{e}=M_{e}-\left(M_{e}-m_{e}\right)\cdot p_{u}(x)$ So as mentioned that you know fashion is all about the trend and timing, so the recentness of an event is a natural feature for having the events importance and therefore, penalise the items that the user has not , you know considered recently. So for that they had a look at the number of days since the user did the event in question. Let’s say we can denote that event by $x$. And then compare this to the old event that the user has in the database and that can be, you know denoted by $F_{u}.$ So this can be known later on, forced to create a linear penalization letting $p_{u}(x)=x/F_{u},$ , but it wasn’t fitting well with the idea of Seasons. So as an example what they did was that even if a store may be selling the warm clothes from November to March , they wanted to focus on the recommendations on summer clothes when the season changes so for that they had to, kind of duplicate this behavior and choose a sigmoid function that you know, considers the recentness in a way that could obscure the preference of users that have been absent from the app for some time. So they used linear penalization because you know, it could ensure that the difference in penalization between the two most recent items is equal to the difference between the two old ones. Figure 2. Screenshots from the application. From the left to right: editorial information, product details, and a “love list”. So for the price what they did was that, you know different users have different price range because they tend to be price sensitive and if an item’s price should also be taken into account then what they did was that the users typical price range was used and that was that created a personalized score and penalized that were not in the price layer range preferred by the user. So this procedure was basically done in simple two steps. In the first step what they did was they found the average of all the price items related to a user and on second base they pretty much calculated the difference that was found in the price of an item $i$ that triggered the event $e$ and the average and then used that to penalize that item. Rregarding the third aspect that they used was popularity. So for the popularity expect what they did was that they considered popularity as a feature by, you know, having a comparison with users Behavior to the rest of the population. So, you know, we can tell it like that that if a user’s activities conform to the common standards that are likely to be his or her taste then it is more unique giving significant clues about the items to recommend. So basically they judged each user’s behavior by looking at the overall popularity of the items. They pretty much interacted with them and they use a linear pair punishment for items with different popularities. And lastly what they did was they combined all these different penalisation and came over a sum of all models this sort of required setting different weights for different factors. So simply what they did was in order to validate their approach that there were scores built using features and that was you know, Event for the fashion domain and secondly, they distributed the scores over a full range of valid scores and had an average confirming the hypothesis. ### 2.3. Based on Weak Appearance Feature As we know that the technology regarding online shopping has been developing rapidly and that online fitting and other clothing intelligent equipment have been introduced in the fashion industry. A lot of different Advanced algorithms have been developed and there are many more currently in the process. #### 2.3.1. Background For example the CRESA (Melo et al., 2015) combined textual attributes, visual features and human visual attention to compose the clothes profile in the recommendation. Recommendation that is based on the content is usually applicable for multiple regions. So for new projects, let’s say if the user has according to the individual browsing records, they can recommend results have been proven to be explicit and accessible but the content-based recommendation usually is improper when you kind of apply it in the industry. And obviously this means that the new users that sign up would not be getting any recommendations based on the browsing record. #### 2.3.2. Proposed Approach Basically what this paper (Zhang et al., 2017) proposes is that the classification process usually needs to consider the quarter sales clothing styles and other factors. So as a result, they basically divided this into four categories where the fashion level is a subjective method that usually needs subjective evaluation on image characters through the expert group. So knowledge background and psychological motivation of the edge experts is involved. And as for the researchers of visual psychological characteristics, there wasn’t a quantitative description method by which the objective evaluated results can represent the subjective evaluation results. So what this aims to find out is to have a set of objective indexes, which can be used to access the fashion level. This was done by considering all the factors that usually affect the evaluation of personal scoring. So this paper basically regards the weak appearance feature as an important index that can influence the fashion level. So there are many, as you know weak appearance features related to the individual fashion level. But the three major categories that can be known namely if we want to go over are makeup ,accessories and hair colors. So this could include the blush, the lip color, eyebrow color ,hat, any accessories on hand and neck etc. By utilizing all these features what they do is that the SVM classification method is leveraged in this and they evaluate based on whether the human body has weak appearance features. So there is no effective way to sort of establish a fashion level database. But the one established in this paper is a basis of the follow-up studies that can be taken up by the future researchers. So basically the image database is of a pretty much very important significance in all this training and testing of algorithms. Figure 3. Fashion level classification framework based on weak appearance feature. For the extraction of weak feature index, the current face detection methods usually have sort of two categories in which knowledge based ones and statistics based ones are available. So in order to extract the weak facial feature, they find the facial feature points and then they use the facial recognition. This paper basically adopts the Adaptive boosting method for facial feature positioning. So the idea behind is that they have to endure large amounts of unsuccessful training samples making the algorithm learning focus on the difficult training samples in the subsequent study and finally they weight and add the number of weak classifiers selected by the algorithms to Strong classifier. Table 1. Customer fashion level classification. Fashion level \| Description classification ---\|--- First level \| Wonderful Second level \| Great Third level \| Good Fourth level \| Common Table 2. Weak appearance features catalogue. Category \| Weak feature index ---\|--- Make-up \| Eyebrow, blush, lips, eye shadow Accessories \| Neck accessories, hand accessories, brooch, nail, hat Hair color \| Red, yellow, green, blue, brown, black, gray, white So all in all what the paper does is that it uses the appearance week feature to sort of characterize consumers’ fashion level and what it does is that it draws the conclusion by, you know, comparing the science experiment and expert evaluation. So both categories of evaluation are involved in this study. Basically the fashion level of the users is what they determine which is based on their makeup ,the accessories they are wearing and the hair color they have. So if a person is into red hair color or you know, having a lot of makeup on they can you know access their level that oh, okay so this person is more into fashion. So based on their level they kind of you know just recommend them the things that they like so for example, let’s say if a certain person is into dark eye shades and dark lip color and you know, they are having some sort of streaks in their hair and stuff like that. So these May indicate a level that is higher in the fashion aspect and they will obviously recommend the products accordingly. ### 2.4. Semantic Attribute Region Guided Approach A lot of multiple semantic attributes built up a fashion product for example sleeves, collars etc. So while making you know these decisions regarding the clothes, a lot of preferences for different semantics attributes, like v neck collar ,deep neck or pointed toes shoes, high heels etc, are looked over. Semantic attributes can not only let you know how one generates a comprehensive representation of products, but they can also help us make an understanding of how the user preferences work. But unfortunately, there aren’t any unique challenges that can be inherited in designing efficient solutions in order to integrate semantic attribute information for the fact that we want fashion recommendation. #### 2.4.1. Previous Methods It is quite difficult to obtain semantic attribute features without the manual attribute annotation and especially in large scale e-commerce. On the other hand if the user preferences are basically classy or sophisticated while traditional methods usually have to transform the item image into a vector directly. So these two aspects make it very very difficult to explain recommendations with current recommendation models. It is very hard on the other hand with these aspects to generate an explainable recommendation with the current recommendation models (Wu et al., 2019; Kang et al., 2017; Xu Chen, 2018) that are currently being used in the industry. #### 2.4.2. Proposed Approach So for that this the paper (Hou et al., 2019) basically proposes a novel semantic attribute explainable recommendation system as a fine-grained interpretable space name semantic attribute space is introduced in which each Dimension corresponds to a semantic attribute. So basically they project the users and items into this space. The users’ fine-grained preferences are being able to generate explainable recommendations specifically if they first develop a semantic extraction Network that can be used to extract the region specific attribute representations. Then by this each item is then projected to the semantic attribute space and then you can easily capture the diversity of semantic attribute. The design aspects contain a fine-grained preferences attention FPA module which basically does that it automatically matches and the user preferences for each given attribute in the space and aggregate all these attributes with different weights. So now each attribute has a weight of it’s own so in the end what happens is that finally they optimize the SAERS models in Bayesian personalized rank BPR framework, which not only significantly improves and out performs several base lines on the visual recommendation task, but it also sort of provides interpretable insights by highlighting attribute semantics in a personalized manner. Basically, what they have done is that previously as we know that these attempts were made to capture users’ visual preferences, but in order to make institutional explanations for the recommendations, they were pretty much very limited on item level. So the paper basically takes a further step to discuss the user preferences on Visual attribute level. Table 3. List of semantic attributes used in method Category \| Attribute: Class ---\|--- Top \| high neck: ruffle semi-high, turtle,… \| collar: rib collar, puritan collar,… \| lapel: notched, shawl, collarless,… \| neckline: V, square, round,… \| sleeves length: sleeveless, cap, short,… \| body length: high waist, long, regular,… Bottom \| skirt length: short, knee, midi, ankle,… \| trousers length: short, mid, 3/4, cropped,… Shoes \| heel height: flat, 1 in-7/4 in, under 1 in,… \| boots height: ankle, knee-high, mid-calf,… \| closure: lace-up, slip-on, zipper,… \| toe style: round, pointed, peep, open,… With their semantic attribute explainable recommendations system. They basically bridge the gap and utilize a new semantic attribute visual space in which each Dimension represents an attribute that corresponds to the region that basically different regions of the clothing items are usually split into several semantics attributes via the extraction Network and then they are later projected into the visual space. So later the users are projected according to the Fine graded preferences for clothing attributes. So this all makes it easily for them to obtain the fashion item projection in the semantic feature space. And from there they can use the FPA to project users into the same semantic feature space. Here FPA is the Fine grain preferences attention where they jointly learned the item representation in both Global visual space and semantic attribute visual space under a pairwise learning framework. And with this they are able to generate the explainable recommendations. ### 2.5. Complementary Recommendations Using Adversarial Feature Transformer Traditional procedures for complementary product hints depend on behavioral and non-visible facts along with consumer co-perspectives or co-buys. However, positive domain names along with style are often visible. Recommendation algorithms are important to many business applications, specially for online shopping. In domain names along with style, clients are seeking out apparel hints that visually supplement their modern outfits, styles, and wardrobe. Which the conventional strategies do now no longer cater to. #### 2.5.1. Previous Methods Now we have seen that there are traditional content-based and collaborative recommendation algorithms (Adomavicius and Tuzhilin, 2005; Lew et al., 2006). But among these collaborative filtering approaches (Koren and Bell, 2015; Melville et al., 2002) are the common ones that primarily rely on behavioral and historical data such as you know, Co purchases , the views and past purchases to suggest new items to customers. So this work basically on providing complimentary item recommendations for a given query item based on visual cues. #### 2.5.2. Proposed Approach So basically what this paper (Huynh et al., 2018) does is that it proposes a framework in which they harness visual clues in an unsupervised manner in order to learn the distribution that exists between co-occurring complimentary items in real world images. The model runs are nonlinear transformations between two manifolds of source and Target complimentary item categories, for example, a top and a bottom in an outfit. And training it on a large data set they train generative Transformer Network directly on the feature representation space by just casting it as an Adversarial Optimization problem. Now such a conditional generative model can produce multiple novel samples of complimentary items in the feature space for a given query item.Now for that they develop an unsupervised learning approach for complementary recommendation using adversarial feature transform CRAFT by learning the co- occurrence of item pairs in real images. So the Assumption here is that the co-occurrence frequency of item pairs is sort of a strong indication of likelihood of their complementary relationship. So the paper advises a defined and adversarial process to train a conditional generative Transformer Network which can then learn the joint distribution of item pairs by observing samples from the real distribution. Now their approach is quite novel and unique in a certain way that they utilize generative adversarial training with several advantages over traditional generative adversarial network (GAN) (Goodfellow et al., 2014) based image generation. Well, we know that the quality of visual image generation using GANs has improved a lot but it still lacks the realism required for many real world applications and fashion apparel is one of them. And more importantly if we see that their goal of recommendation systems in certain types of application is often not to generate synthetic images, but they have to recommend real images from a catalog of items. Now we know that an approach that generates synthetic images will still need to perform a search and that will be typically done by searching in the feature space in order to find the most visually similar image in the catalog. Now CRAFT directly generates these features of the recommended items and bypasses the need to generate synthetic images and enable a simpler and more efficient algorithm. So by working in a feature space, what they do is that they can use a simpler Network architecture that improves stability during the training time and avoid common pitfalls such as model collapse (Che et al., 2016). Figure 4. Generating recommendations using the proposed CRAFT network. #### 2.5.3. Network Architecture The network architecture basically comprises several steps and first is the selection of appropriate visual representations for the source and Target images. Then what they do is that the encoding which are fixed feature representations are generally derived from pre-trained CNN’s. Typically it is advisable to use application specific feature representations, for example, apparel feature embeddings for clothing recommendations, but a general representation such as one trained on ImageNet (Deng et al., 2009) or MS-COCO (Lin et al., 2014) offer nice efficient alternatives. So as shown in figure, what basically is happening, is that the source and the target feature encoders $E_{s}$ and $E_{t},$ respectively are fixed and are used to generate feature vectors for training and inference. Now, the architecture resembles traditional Grand designs with two main components , a conditional feature transformer and a discriminator. The role of the feature transformer is to transform the source feature $s_{q}$ into a complementary target feature $\hat{t}_{q}.$ The input to the transformer also consists of a random noise vector $z$ sampled uniformly from a unit sphere in a $d_{z}$ -dimensional space. By design, the transformer is generative since it is able to sample various features in the target domain. As the transformer consists of several fully-connected layers in which each is followed by batch normalization (Ioffe and Szegedy, 2015) and leaky ReLU (Maas, 2013) activation layers. The discriminator is commensurate to the transformer in capacity, consisting of the same number of layers. This helps balance the power between the transformer and the discriminator in the two- player game, leading to stable training and convergence. From a query image, the query feature $f$ is extracted by the source encoder, $E_{s},$ and multiple samples of transformed features $\left\\{\hat{t}_{i}\right\\}$ are generated by sampling random vectors $\left\\{z_{i}\right\\}.$ Now basically what it does is that it allows them to generate a diverse set of complementary recommendations by sampling the underlying conditional probability distribution function. And when they performed a nearest neighbor search within a set of pre-indexed target features extracted using the same target encoder, $E_{t},$ used during training. Actual recommendation images were retrieved by a reverse lookup that maps the selected features to the original target images. Figure 5. Complementary recommendation for a common query item (dark jeans) #### 2.5.4. Performance evaluation The feature transformer in CRAFT samples from a conditional distribution to generate diverse and relevant item recommendations for a given query. The recommendations generated by CRAFT are preferred by the domain experts over those produced by competing approaches. ### 2.6. Neural Compatibility Modeling It’s easy these days where fashion communities are online and we can experience that a lot of fashion experts are publicly sharing their own fashion tips by showing how their outfit compositions work , where each item a top or a bottom usually has an image and context metadata title and category. With such Rich information, fashion data offers us an opportunity to investigate the code in clothing matching. Now we know that the colors, materials and shape are some aspects that affect the compatibility of fashion items and also each fashion item involves multiple modalities and also if we notice that the composition relation between fashion items is rather sparse. Now this makes Matrix factorization methods not applicable. Figure 6. Illustration of the proposed scheme. They employed a dual autoencoder network to learn the latent compatibility space, where they jointly model the coherent relation between visual and contextual modalities and the implicit preference among items via the Bayesian personalized ranking. #### 2.6.1. Previous Methods The recent advancement in these Fashion aspects has been done, but the previous models (Iwata et al., 2011; Hu et al., 2015; McAuley et al., 2015; Liu et al., 2012) proposed were lacking in terms of how they wanted to approach this subject. #### 2.6.2. Proposed Approach So what this paper (Song et al., 2017) proposes is a content-based neural scheme that models the compatibility between fashion items based on the Bayesian personalized ranking BPR framework. Now this scheme jointly models the coherent relation between modalities of items and their implicit matching preference.So basically they propose focusing on modeling the sophisticated compatibility between fashion items by seeking the nonlinear latent compatibility space with neural networks. And they also were able to aggregate the multimodal data of fashion items and exploit the inherent relationship that basically exists between different modalities to comprehensively model the compatibility between fashion items. Now we know that it is not correct to directly measure the compatibility between fashion items from a distinct space due to their heterogeneity. So for that the author’s they assume that there exists a little compatibility space that is able to bridge the gap between heterogeneous fashion items where highly compatible fashion items share the similar style material which can show high similarity or functionality should also show high similarity. For example a wide casual T-shirt goes really well with black jeans, but it does not go with a black suit while a pair of high boots prefer skinny jeans rather than flared pants. So they further go along and assume that the subtle compatibility factors lie in a highly nonlinear space that can be learned by the advanced neural network models. So they employ the auto encoders networks to learn the latent space which has been proven to be effective in the latent space learning.(Wang et al., 2016) To fully take advantage of the implicit relation between tops and bottoms, basically what they did was that they naturally adopt the BPR framework and assumed that bottoms from the positive set $\mathcal{B}_{i}^{+}$ are more favorable to top $t_{i}$ than those unobserved neutral bottoms. According to BPR, built a training set: $\mathcal{D}_{S}:=\left\\{(i,j,k)\mid t_{i}\in\mathcal{T},b_{j}\in\mathcal{B}_{i}^{+}\wedge b_{k}\in\mathcal{B}\backslash\mathcal{B}_{i}^{+}\right\\}$ where the triple $(i,j,k)$ indicates that bottom $b_{j}$ is more compatible than bottom $b_{k}$ with top $t_{i}$ Then according to(Rendle et al., 2012) , they got the following objective function, $\mathcal{L}_{bpr}=\sum_{(i,j,k)\in\mathcal{D}_{S}}-\ln\left(\sigma\left(m_{ij}-m_{ik}\right)\right)$ Taking the modality consistency into consideration, they got the following objective function: $\mathcal{L}=\mathcal{L}_{bpr}+\gamma\mathcal{L}_{mod}+\mu\mathcal{L}_{rec}+\frac{\lambda}{2}\\|\Theta\\|_{F}^{2}$ where $\mathcal{L}_{\text{rec}}=\mathcal{L}_{\text{rec}}^{v}+\mathcal{L}_{\text{rec}}^{c}$ with $\mathcal{L}_{\text{rec}}^{v}=\Sigma_{(i,j,k)\in\mathcal{D}_{S}}\left(l\left(\mathbf{v}_{i}^{t}\right)+l\left(\mathbf{v}_{j}^{b}\right)+\right.\left.l\left(\mathbf{v}_{k}^{b}\right)\right)$ and $\mathcal{L}_{rec}^{c}=\Sigma_{(i,j,k)\in\mathcal{D}_{S}}\left(l\left(\mathbf{c}_{i}^{t}\right)+l\left(\mathbf{c}_{j}^{b}\right)+l\left(\mathbf{c}_{k}^{b}\right)\right)\cdot\mu,\gamma,\lambda$ are non-negative trade-off hyperparameters. $\Theta$ refers to the set of network parameters (i.e., $\mathbf{W}_{k}$ and $\left.\hat{\mathbf{W}}_{k}\right)$. The last regularizer term is designed to avoid overfitting. ## 3\. Aesthetics and Fashion The word aesthetic (of Philosophy, 2009) was basically introduced in the 18th century where it has come to be used to designate among other things a kind of object, a kind of judgment, a kind of attitude or experience and a kind of value. Where aesthetic comes the concept of aesthetic descends usually from the concept of taste. So in the 18th century, the theory of taste emerged in part as a corrective to the rise of rationalism particularly as applied to Beauty and the rise of egoism particularly as applied to virtue. ### 3.1. Mapping Association So how do people usually describe clothing ,so there are words like informal, casual ,formal ,party, where they are usually used. But the recent focus on recognizing or extraction of the features that are available visually in clothing is pretty much different. #### 3.1.1. Background To accurately guess that, the authors in the paper (Jia et al., 2016) describe a way to bridge the gap between visual features and the aesthetic words. So what they basically do is that they formulate a novel three-level framework visual features (VF) - image-scale space (ISS) - aesthetic words space (AWS) and then they leverage the Art field image scale space which serves as an intermediate layer. So firstly they proposed a stacked diagnosing auto encoder Guided by correlative labels SDAEGCL, to map the visual features to the image scale space and then with that accordingly what they do is that the semantic distance is computed by the Wordnet similarity (Pedersen et al., 2004). They map the most often using static words available and being used by people in the online clothing shops to the image scale space. Now, what they do is that they employ the upper body menswear images that they have downloaded from several different online shops as their experimental data and they proposed a 3-level framework that can help to capture the relationship that is standing between visual features and aesthetic words. It is quite important for people to wear aesthetically and properly and specifically given a user input occasion wedding ,shopping or dating ,a system should be able to suggest the most suitable clothing that is from the user’s own clothing available. So another paper (Liu et al., 2012) similar idea was mentioned where the two criterion’s are explicitly considered for the system where it is paid heed to wear properly and to wear aesthetically like for example that red T shirt matches better with white pants than green pants and to basically narrow down the semantic Gap that is between the low-level features of clothing and the high-level occasion categories. From where these clothing attributes are treated as latent variables in the support Vector machine based recommendation model. But nevertheless the matching rules cannot reveal the aesthetic effects holistically and lacked Interpretability. Figure 7. Examples of clothing images and their corresponding aesthetic words. #### 3.1.2. Proposed Approach So the paper (Jia et al., 2016) basically aims to bridge the gap between visual features and aesthetic words of clothing where in order to capture the intrinsic and holistic relationship between them they sort of introduce a middle layer ,intermediate layer and form a novel three-level framework, which is based on the proposed Theory by Kobayashi (Kobayashi, [n.d.]). Where two dimensional space warm cool and hard soft aspects are applied in the art design. Basically the contribution of the papers is that they build an association between clothing images and aesthetic words by proposing a three-level framework. It basically does a novel notation of using the 2D continuous image scale space as a layer that is intermediate with a very strong ability of description thus it facilitates the deep and high-level understanding of aesthetic effects. And secondly what it does is that the paper proposes a stacked denoising auto-encoder Guided by correlative labels SDAEGCL to implement mapping of visual features to the image scale space and that can amend the random error existing in initial input and make full use of the information of both labeled and unlabeled data and moreover we can also find that the stack methods improve the representation capability of model by adding more hidden layers. So basically Kobayashi proposed 180 keywords into different 16 categories of Aesthetics and defined their coordinate values in the image scale space. But as in fashion, there are some words that are unrelated like alert ,robust, sad, happy. These are not something that we usually use to describe clothing. So first the authors sort of removed manually all these not often used words and established a static word space $Y$ for clothing containing 527 words. Now in order to illustrate how to map the aesthetic words $y_{i}\left(\forall y_{i}\in Y\right)$ to the image-scale space $D.$ To determine the coordinate value $D_{y_{i}}\left(wc_{y_{i}},hs_{y_{i}}\right)$ of an aesthetic word $y_{i}\in Y,$ the authors basically first define the 180 keywords as keyword ${}_{j}(j=1,2,\cdots,180)$ and calculate the semantic distances between $y_{i}$ and each keyword j using WordNet::Similarity . Then what they do is that they basically pick 3 keywords with the shortest distances $d_{i_{1}},d_{i_{2}}$ and $d_{i_{3}},$ marking the coordinate values of these 3 keywords as $D_{i_{1}}\left(wc_{i_{1}},hs_{i_{1}}\right),D_{i_{2}}\left(wc_{i_{2}},hs_{i_{2}}\right)$ $D_{i_{3}}\left(wc_{i_{3}},hs_{i_{3}}\right).$ Afer that they take the reciprocals of distances $rec_{i_{1}}$ rec ${}_{i_{2}},$ rec ${}_{i_{3}}$ as weights (e.g. rec ${}_{i_{1}}=\frac{1}{d_{i_{1}}}$ ), the weighted arithmetic mean 1 of $D_{i_{1}},D_{i_{2}}$ and $D_{i_{3}}$ can also be regarded as the coordinate value $D_{y_{i}}\left(wc_{y_{i}},hs_{y_{i}}\right)$ of $y_{i}.$ The formula is shown as follows: $wc_{y_{i}}=\frac{\sum_{k=1}^{3}wc_{i_{k}}\cdot rec_{i_{k}}}{\sum_{k=1}^{3}rec_{i_{k}}},hs_{y_{i}}=\frac{\sum_{k=1}^{3}hs_{i_{k}}\cdot rec_{i_{k}}}{\sum_{k=1}^{3}rec_{i_{k}}}$ So by this way what they do is that for each $y_{i}\in Y,$ they basically calculate its coordinate value $D_{y_{i}}$ in the image-scale space as $\left(wc_{yi},hs_{yi}\right).$ To label an input clothing image $v$ with an aesthetic word, they use the proposed SDAE-GCL to predict its coordinate value $D_{v}\left(wc_{v},hs_{v}\right)$ in $D.$ Then, after that they find a word $y_{v}\in Y$ whose corresponding coordinate value $D_{yv}$ has the shortest Euclidean distance to the $D_{v}$. Thus, $y_{v}$ can be regarded as the aesthetic word of image $v$ ### 3.2. Brain-inspired Deep Network Now we know that most existing methods sort of rely on conventional features in order to represent an image. Such features that can be extracted by convolutional neural networks are the scale-invariant feature, transform algorithm, color histogram and so on but one important type of feature is the aesthetic feature and as we have already discussed it before it plays an important role in clothing and specially in clothing recommendation since users largely depend on whether the clothing is in line with their aesthetics or not. #### 3.2.1. Previous Methods Now we have seen in some papers (Han et al., 2017; Hsiao and Grauman, 2017; McAuley et al., 2015; Vasileva et al., 2018) in which there was a recommendation for different fashion garments for an unfinished outfit. But their goal was different from the one mentioned in this paper. That is basically that they focused on clean per-garment catalog photos and the recommendations were mostly restricted to retrieve garments from a specific data set. Now the only feature in those recommendation systems was that they were adding to the Garment. Most prior fashion work addresses recognition problems, like matching street-to shop (Kalantidis et al., 2013; Kiapour et al., 2015; Yan, 2012; Vittayakorn et al., 2015) But in this case, what they are doing is that they are saying that some problems demand going beyond seeking an existing garment and adding to it and for that, they said that there are garments which are detrimental and it should be taken off. You know like cuff the jeans above the ankle or how to adjust the presentation and detail of them within a complete outfit to improve its style. #### 3.2.2. Background So in order to bridge the gap there are a lot of different methods but we are going to discuss another one (Yu et al., 2018) which introduces the intense static information. Which is highly relevant with user’s preference into the clothing recommendation system. So what they basically do, is that the aesthetic feature extracted by the pre-training on network, which is a brain inspired deep structured trained for the assessment task of Aesthetics. So for that they consider the aesthetic preference which varies significantly from user to user as different people have different sorts of reference in Aesthetics. So they proposed a new tensor factorization model that incorporates the static features in a very personalized manner. So what they do is that they conduct different experiments and demonstrate that the approach they are putting forward captures the static preference of the user. It significantly outperforms the already available state-of-the-art recommendation methods.What happens is that usually when we are shopping for clothing on the web. We used to look through product images before making a certain decision before buying that thing and product images usually provide a lot of information including design, color schemes ,patterns structure and so on. We can get an estimation of the thickness and quality of a product from its images. As such product images play a lot of key roles in the clothing recommendation task. So what the authors in this paper do is that they leverage this information and enhance the performance of the existing clothing recommendation systems. Figure 8. Brain-inspired Deep Network (BDN) architecture. However, an important factor regarding aesthetics is that it has been considered not much in previous researchers’ research. So basically what happens is that while most user’s concern regarding clothing is that the product should be good looking. What happens is that the author’s use the static Network to extract relevant features that is between an aesthetic network and a CNN. That are demonstrated and they proposed a brain inspired deep Network, which is a deep structure trained for image aesthetic assessment that inputs several raw features that are indicative of aesthetic feelings like hue, saturation, color, duotones ,complementary color etc. And what is it that extracts high-level aesthetics from these barely barely raw features. #### 3.2.3. Proposed Approach So the paper works on BDN that is utilized to strike the holistic feature in order to represent the static elements of a clothing. And as different people prefer different aesthetic tastes. So to capture the diversity of the aesthetic preference among different consumers and over different times. They exploit tensor factorization as a basic model. Now, there are several ways to decompose a tensor however, there are certain drawbacks in existing models (Kolda and Bader, 2009; Rendle and Schmidt-Thieme, 2010; Sidiropoulos et al., 2016) . So what they do is that they address the clothing recommendation task better and propose a dynamic collaborative filtering DCF model that is trained with coupled matrices to mitigate the sparsity problem. And then afterwards they combined the models with Bayesian personalized ranking optimization criteria and evaluated the proper performance on an Amazon clothing dataset. So basically what they are doing is that they are proposing an novel DCF model to portray the purchase events in three dimensions: user, items, and time and then incorporate the aesthetic features into DCF and train it. And of course, they are leveraging the novel aesthetic features in recommendation to capture consumers specific aesthetic preference and they compare the effect with several conventional features to demonstrate the necessity of the aesthetic features. So in order to illustrate the hybrid model that integrates image features into the basic model the DCFA. They first introduced the basic tensor factorization model DCF. So the basic model is the impact of time on aesthetic preference. So what they do is that they proposed a context-aware model as the basic model to account for the temporal factor. What they do is that they use P × Q × R tensor a to indicate the purchase events among the users clothes and time dimensions. So if a user P purchase an item Q in the time interval R Then ${\mathrm{A}}_{pqr}$ is 1 otherwise, it will be 0. so for that the tensor factorization has been widely used to predict all the missing entries 0 elements in $A$ which can be used for recommendation. So as the previous models have some limitations what they do is that they proposed a new tensor factorization method in which a user makes a purchase by deciding a product and there are two primary factors. So the first one is that if the product fits the users preference and the appearance is good looking or appealing to that specific user. And if the time is correct that if it’s in the season and fashionable, for example, of course winter clothing cannot be recommended or aesthetically fine if it’s being recommended in the summer season, so for user $p$, clothing $q$, and time interval $r$, they use the scores $S_{1}$ and $S_{2}$ to indicate how the user likes the clothing and how the clothing fits the time respertively. $S_{1}=1$ when the user likes the clothing and $S_{1}=0$ otherwise. Similarly, $S_{2}=1$ if the clothing fits the time and $S_{2}=0$ otherwise. The consumer will buy the clothing only if $S_{1}=1$ and $S_{2}=1,\mathrm{so},\hat{\mathrm{A}}_{pqr}=S_{1}\&S_{2}.$ To make the formula differentiable, they approximately formulated it as $\hat{\mathrm{A}}_{pqr}=S_{1}\cdot S_{2}.$ And the presented $S_{1}$ and $S_{2}$ in the form of matrix factorization: $\begin{array}[]{l}S_{1}=\sum_{i=1}^{K_{1}}\mathbf{U}_{ip}\mathbf{V}_{iq}\\\ S_{2}=\sum_{j=1}^{K_{2}}\mathbf{T}_{jr}\mathbf{W}_{jq}\end{array}$ where $\mathrm{U}\in\mathbb{R}^{K_{1}\times P},\mathrm{V}\in\mathbb{R}^{K_{1}\times Q},\mathrm{T}\in\mathbb{R}^{K_{2}\times R},$ and $\mathrm{W}\in\mathbb{R}^{K_{2}\times Q}.$ The prediction is then given by: $\hat{\mathrm{A}}_{pqr}=\left(\mathrm{U}_{p}^{\mathrm{T}}\mathrm{V}_{\cdot q}\right)\left(\mathrm{T}_{r}^{\mathrm{T}}\mathrm{W}_{q}\right)$ We can see that in Equation that the latent features relating users and clothes are independent with those relating clothes and time. Though $K_{1}$ -dimensional vector $\mathrm{V}_{q}$ and $K_{2}$ -dimensional vector $\mathrm{W}_{}q$ are all latent features of clothing $q,\mathrm{V}_{q}$ captures the information about users” preference intuitively whereas $\mathrm{W}_{}q$ captures the temporal information of the clothing. The model is more expressive in capturing. The underline related patterns in purchases. Moreover this model is efficient and easy to train compared with the Tucker decomposition. ### 3.3. Minimalistic Approach We know that the physical attributes of a product are very much influencing the buying behavior. (Streamoid, [n.d.]) We also know that the aesthetic calls intuitively while we shop. so it may not even be you know, the person might not even be aware of making multiple decisions on every product, for example, you know like the style but not the color of the product. Various aspects of our life influence the style of how we dress. Every look that we wear tells a different story about us. So basically it communicates a certain image representation which is you know decoded by others within their own cultural context. So it is sort of possible that the Aesthetics of a garment is similar for all in a particular society. #### 3.3.1. Background So when we look into a garment, what are the main things that we should or we usually look into. Queries like. so can I wear it? , What occasion it would suit and how does it make me feel? And also another precise preference is , you know included in this aspect and how does it reflect their own personality. So these are just a few of the questions that we usually, ask ourselves when we are out shopping and when we want to wear clothes that are aesthetically pleasing. But as we have seen in this new modern era that minimalism is getting into every aspect of life and people are tending to move towards simpler versions, but aesthetically pleasing ones. As Coco Chanel has said: “before you leave the house look in the mirror and take one thing off” So minimal outfit edits in an already used outfit they can use to change the existing outfit and improve its fashionability. Whether it can be removing an accessory selecting a blouse with a higher neckline or you know, just tucking your shirt in or simply, you know, changing the pants to a darker color. So these all small adjustments are accountable for a more stylish outfit that is more aesthetically pleasing to a large group of people or to your own self as well. #### 3.3.2. Proposed Approach So motivated by these observations which made the authors of this particular paper (Hsiao et al., 2019) go for the minimal edits for fashion outfit improvement. So minimally editing an outfit and getting an algorithm must impose alternations to the garments and accessories that are slight, yet visibly improve the overall fashionability. So basically what they’re doing is that a minimal edit need not strictly minimize the out amount of change rather it incrementally adjust in an outfit as opposed to starting from scratch. So basically, it can be a recommendation regarding which garment you need to you know, replace or take off or you know to swap out or simply, you know, just wear the same garment in a better way. And also it is well known that clothing fashion is sort of just intuitive and often a habitual trend in the style in which you know, an individual usually dresses but it is sort of not clear which visual stimulus places higher or lower significance or influence on the updation of clothing and fashion trends. So another paper (Zou et al., 2016) that we have seen in which they have employed machine learning techniques in order to analyze the influence that the visual stimuli of different clothing fashion are having on the fashion trends and specifically classification-based model was proposed by them that quantified the influence of different visual stimuli in which each stimuli influenced was quantified by, you know, it’s a corresponding accuracy in fashion classification. So experimental results also, demonstrated that if they were quantifying style color and texture so out of those three on clothing fashion updates the style holds a higher influence than the color. And the color holds a higher influence than the texture. So all of these are very important in determining the Aesthetics as well. Figure 9. Overview of our Fashion $++$ framework. We first obtain latent features from texture and shape encoders $E_{t}$ and $E_{s}$. Our editing module $F^{++}$ operates on the latent texture feature $t$ and shape feature s. After an edit, the shape generator $G_{s}$ first decodes the updated shape feature $s^{++}$ back to a $2\mathrm{D}$ segmentation mask $\mathrm{m}^{++},$ and then we use it to region-wise broadcast the updated texture feature $\mathrm{t}^{++}$ into a $2\mathrm{D}$ feature $\operatorname{map}\mathbf{u}^{++}.$ This feature map and the updated segmentation mask are passed to the texture generator $G_{t}$ to generate the final updated outfit $x^{++}$. So basically the main idea and approach for this model. Is that the activation maximization method. That works on localized encodings from a deep image generation Network. So what they basically do is that you give them an original outfit and they map it’s composing pieces for example, you know, the bag, boots, jeans. blouse to their respective codes. And then what they do is that they use a discriminative fashionability model for the editing in which it gradually updates the encodings in the direction that maximizes the outfit score so when they do this, they are hence improving its style. And also the update trajectory offers various ranges of edits starting from you know, the least changed and going towards the item that is most fashionable from you know, which users can choose a preferred endpoint. The approach basically says that it provides its outputs in two formats: 1. (1) Retrieved garments from an inventory that would best achieve its recommendation. 2. (2) And the second one is rendering of the same person in the newly adjusted look generated from the edited outfits encoding #### 3.3.3. System Working So basically, what they do is that they present an image generation framework, which is comprised of outfit images into their garment regions and factorizes shape/fit and texture in support of the later objectives. So the framework is basically about coordination of all composing pieces defines and outfits look. What they do is that they can control which parts like the pants or the skirts or you know shirts and then aspects like the length of their sleeve, color, the pattern and neckline to change and sort of, you know, keep the identity and fashion irrelevant factors unchanged. So what they want to do is they want to explicitly model their spatial locality and to perform minimal edits. So what they needed to do was to control the piece’s textures as well as their shapes. So basically what textures comprise in outfits is for example, like in denim with solid patterns gives more casual look or like leather with red colors, give more street style look. So with the same material color and pattern of garment and how they are worn, you know, like tucked in or pulled out and skinny or baggy pants and you know, what sort of cut they have v-neck or turtleneck or you know boatneck. So the Garment will compliment a person’s silhouette in different ways. So what they do is that they account for all of these factors and devise an image generation framework that gives control over individual pieces accessories body parts and also factorize the shapes from the texture. For computing an edit the main steps are: calculating the desired edit, and generating the edited image. For calculation of an edit, they basically took an activation maximization approach where they iteratively alter the outfit’s feature such that it increases the activation of the fashionable label according to $f$. Formally, let $\mathbf{z}^{(0)}:=\left\\{\mathbf{t}_{0},\mathbf{s}_{0},\ldots,\mathbf{t}_{n-1},\mathbf{s}_{n-1}\right\\}$ be the set of all features in an outfit, and $\tilde{\mathbf{z}}^{(0)}\subseteq\mathbf{z}^{(0)}$ be a subset of features corresponding to the target regions or aspects that are being edited ( $e.g.,$ shirt region, shape of skirt, texture of pants). The updated outfit’s representation is as follows: $\tilde{\mathbf{z}}^{(k+1)}:=\tilde{\mathbf{z}}^{(k)}+\lambda\frac{\partial p_{f}\left(y=1\mid\mathbf{z}^{(k)}\right)}{\partial\tilde{\mathbf{z}}^{(k)}},k=0,\ldots,K-1$ where $\tilde{\mathbf{z}}^{(k)}$ denotes the features after $k$ updates, $\mathbf{z}^{(k)}$ denotes substituting only the target features in $\mathbf{z}^{(0)}$ with $\tilde{\mathbf{z}}^{(k)}$ while keeping other features unchanged, $p_{f}\left(y=1\mid\mathbf{z}^{(k)}\right)$ denotes the probability of fashionability according to classifier $f$, and $\lambda$ denotes the update step size. Each gradient step yields an incremental adjustment to the input outfit. #### 3.3.4. Performance evaluation This Approach makes slight yet noticeable improvements better than baseline methods in both quantitative evaluation and user studies and it effectively communicates to users through image generation and supports all possible edits from swapping, adding, removing garments to adjusting outfit presentations through qualitative examples. ### 3.4. Neuroaesthetics Mark Twain has said that the “Finest Clothing made is a person skin”, but of course society demands something more than this. Now, we know that fashion has a tremendous impact on our society and clothing is basically something that reflects the person’s social status and thus puts pressure on how they are to dress to, you know, fit a particular occasion. For this the authors of this particular paper (Simo-Serra et al., 2015) analyze the fashion of clothing of a large social website in which their main aim is to learn and predict how fashionable a person looks on a photograph and suggest subtle improvements that they can make in order to improve their image and appeal. #### 3.4.1. Previous Methods Now the approach these authors have suggested is also somewhat related to recent approaches (Dhar et al., 2011; Gygli et al., 2013; Isola et al., 2013; Khosla et al., 2014) that were aimed at modeling the human perception of what beauty actually is. So in papers these authors basically address the questions of what makes a particular image memorable and interesting or you know popular to viewers. So this line of work usually contains mining of large image data sets in order to you know, find a relation of visual cues to popularity scores. But in this paper what they do is that they tackle the problem of predicting fashionability. So they are going a step further from the previous work by identifying High-level semantic properties that cause a particular aesthetic score which can be then conveyed to the user so that they can improve their outfit or their look. And this work is very much closest to (Khosla et al., 2013) which was able to infer whether our faces are memorable or not and then upon that results modify it such that it becomes. Although this is quite different as their domain is different and it is also different in formulation. #### 3.4.2. Proposed Approach So they are modeling the perception of fashionability. And for that what they have done is that they have proposed a conditional random field model that jointly reasons about several fashionability factors such as the type of outfit and garments that an individual is wearing and the type of user and the photograph setting for example, the scenery and fashionability score. And based on that they give the recommendation to user in which they convey which garments or scenery the individual should change in order to improve fashionability. This paper predicts how fashionable a person looks on a particular photograph. So the fashionability is then affected by the clothes the subject is wearing and also by a large number of other factors such as how appealing they are in a scene that is containing that person and how that image was taken and how appealing visually the person is ,their age and also the garment itself being fashionable is not a perfect indicator of someone’s fashionability as people typically judge how well the garments aligned with someone’s look, body, characteristic or even personality. So the model proposed exploit several domain inspired features which include beauty, age and mood inferred from the image. And the scene and the type of photograph and if available metadata in the form of where the user is from, how many online followers he/she has the and the sentiment of comments by other users. For this they have to create their own data set from different online sources. And if we see our daily lives we can see how much of an impact fashion has in it. So this also proves the growing interest in clothing related applications in Vision community. Early work (Jammalamadaka et al., 2013; Simo-Serra et al., 2014; Yamaguchi et al., 2013, 2012; Yang et al., 2015) that was focused was mainly on clothing parsing in terms of diverse set of garments types.The paper’s objective was basically to be able to predict fashionability of a given post, but they also wanted to build a model that can understand fashion at a higher level. So for that purpose what they did was they made a Conditional Random Field (CRF) to learn the different outfits , types of peoples and settings. Now here the word setting is basically something that describes the location where the picture is taken and both at a scenic and geographic level. They use their own fashion data set fashion144k Images and metadata to produce accurate predictions of how fashionable a certain person is. More formally, let $u\in\left\\{1,\cdots,N_{U}\right\\}$ be a random variable capturing the type of user, $o\in\left\\{1,\cdots,N_{O}\right\\}$ the type of outfit, and $s\in\left\\{1,\cdots,N_{S}\right\\}$ the setting. Further, we denote $f\in\\{1,\cdots,10\\}$ as the fashionability of a post $\mathbf{x}$. They represented the energy of the CRF as a sum of energies encoding unaries for each variable as well as non-parametric pairwise potentials which reflected the correlations between the different random variables. It is defined as: $\displaystyle E(u,o,s,f)$ $\displaystyle=E_{user}(u)+E_{out}(o)+E_{set}(s)+E_{fash}(f)$ $\displaystyle+E_{np}^{uf}(u,f)+E_{np}^{of}(o,f)+E_{np}^{sf}(s,f)$ $\displaystyle+E_{np}^{uo}(u,o)+E_{np}^{so}(s,o)+E_{np}^{us}(u,s)$ Figure 10. An overview of the CRF model and the features used by each of the nodes. #### 3.4.3. Performance Output An exciting property of this specific model was that it could be used for outfit recommendation.What they basically did was they used to take a post as an input and estimated the outfit that maximizes the fashionability while the kept the other variables fixed. So basically what was happening was that they were predicting what the user should be wearing in order to increase their looks instead of their current outfit. And this can be just one example of the flexibility of the approach. They proposed other thoughts such as what would be the low fitting outfit and what would be the best place to go with the current outfit or you know, what type of users this outfit fits the most, this can be done with this same model. ## 4\. Personalisation in Fashion One of the key aspects in fashion is personalization. So personalization is basically something that is intended for a certain individual based on their likes and dislikes and what they cater as good for them. And we know that fashion industry included e-commerce worldwide is supposed to hit the 35 billion dollar Mark by 2020 this year and there’s a need for applications which can help the user in making Intelligent Decisions on their day-to-day purchases or a system that can recommend them a model or something that is personalized to their liking. ### 4.1. Personalized Outfit Recommendation with Deep Neural Network So for this purpose the use of deep neural networks for this challenge is needed and we are going to discuss one of a system that is dubbed as FashionNet (He and Hu, 2018) that consists of basically two components: a feature Network for the feature extraction function and a matching Network for the compatibility computation. The former one is achieved through a deep convolutional Network and the second one for that they adopt a multi-layered fully connected Network structure and design, and compare the three alternative architectures for FashionNet and to achieve personalized recommendations, what they do is that they develop a two stage training strategy, which uses the fine-tuning technique to sort of transfer a general compatibility model to the model that embeds personal preference. #### 4.1.1. Previous Methods Now we know that existing recommender systems are heavily dependent on the collaborative filtering techniques CF which basically uses historical ratings given to the item by users as the sole source of information for their learning expect and the performance is very much sensitive to the sparsity level of user item metrics. The recent progress of deep neural networks provides promising solution to the representation problem of image content(Lecun et al., 1998; Krizhevsky et al., 2012; Chatfield et al., 2014; Szegedy et al., 2015) . #### 4.1.2. Background This specific paper explores the Deep use of neural networks for outfit recommendation and specifically for the personalized outfit recommendation. Now for this they encounter two key problems. The first one was modeling of the compatibility among multiple fashion items and obviously the second one was capturing users personal interest. So for that the former one was solved by first mapping the item images to a latent semantic space with convolutional neural network and for the second one they adopt a multi-layer fully-connected network structure. And they also studied alternative architectures that combine feature learning and compatibility modeling. Different ways for the other problem. What they do is that they encode user-specific information in terms of parameters of the network. Although we know that each user may have his own unique personal taste and they follow some general rules for making outfits. But besides that the usual small number of training samples for individual users makes it very much important to borrow training data from other users that share similar tastes. So with these observations in mind, what they do is that they adopt a two-stage strategy for the training of their model network; the first stage basically learns a general compatibility model from outfits of users. And in the later stage, what they do is that they fine-tune the general model with the specific data that they get from the user in fine-tuning. It is an important technique for training deep neural networks for applications that have limited number of training samples. #### 4.1.3. Proposed Approach So in their approach they basically assume that heterogeneous fashion items can be grouped into n categories. Let’s take an example where the three most social categories for fashion are usually shoes, tops and bottoms and outfit is a collection of fashion items which are usually coming from different categories. So an outfit can consist of a bottom, top and a pair of shoes. So given some historical data what they did was that for any user outfit pair they pretty much assigned a rating score as the score kind of reflected the level of affection the user has for the outfit. So the higher the score then obviously the more appealing the outfit is for the users and those outfits that had the highest score were recommended to the users. So basically the rating system was used and the rating $s$ for a user outfit pair is determined by how well the items in the outfit go with each other. So if you know a pair of red shirts and you know, let’s say black slacks or tight jeans and maybe they go well instead of, you know, something with a yellow skirt and red shirt. So we basically see the author’s design appropriate deep neural network structure to model the interactions among these items and they achieve Personalization by developing a two-stage training strategy and embed the user specific preferences in the parameter of the network. So what they basically do is that they explore three different network architectures and naming them as fashionet A ,B and C and without the loss of generality. They assume an outfit consists of three items: top, bottom and pair of shoes. So in fashionNet A the images of the items are first concatenated to create a new image with nine color channels, and the compounded images are then forwarded to a widely used CNN model VGGNet. The output layer is a fully connected layer with softmax function as its activation function. So in this architecture the components of representation learning and compatibility measure are fully integrated. The two steps are carried out simultaneously right from the first convolution layer. Figure 11. Network architectures Now in fashionNet B we see that they apply representation learning and compatibility measures sequentially and the images are first of all mapped to a feature representation through a feature Network. So the same CNN model is used for items from different categories. To model the compatibility they concatenate the features of all items and feed them to three fully connected layers. So in this work what they show that this network structure also has the capacity for approximating the underlying compatibility among multiple features. Now for fashionNet C , what they do is that both FashionNet A and B try to directly model the compatibility among multiple items. They sort of come across difficulties when trying to capture the High order relationships and the data is significantly expanded when we concatenate all the items. Due to the dimensionality issue a huge number of training samples may be required for a good model to be learned and we know that users on the internet have contributed so many outfit ideas. It is still minor compared to the number of all possible outfits. So in order to overcome this problem what the authors propose is that a prior restraint in fashionNet C. They assume that the compatibility of a set of items is mainly determined by how well a pair of these items go with each other. Then all the outfits from the final layers regarding the probabilities that the item pairs are matched while are added together to get a final score as for the whole outfit. The learning task is formulated as a learn to rank problem.A training sample contains two outfits, e.g. $\left\\{I_{t}^{+},I_{b}^{+},I_{s}^{+}\right\\}$ and $\left\\{I_{t}^{-},I_{b}^{-},I_{s}^{-}\right\\},$ where the former is preferable to the latter. A two-tower structure to train the networks and rank loss is used to minimize this following equation. $L=\frac{1}{M}\sum_{i=1}^{M}\log\left(1+\exp\left(-\left(s_{i}^{+}-s_{i}^{-}\right)\right)\right)$ In the training expect what happens is that for an individual user they usually have a small number of training outfits. And furthermore, although each user may have their own preference. There are some rules that should be followed by most people for making an outfit. For example t-shirts and jeans are usually paired up. With these observations. What they do is that they design a two stage procedure to train the deep network for personalized outfit recommendation. So the first stage is basically that they learn a general model for compatibility. Here they discard the information of the user and mix the outfit created by different users all together. And then they create a new neutral outfit by mixing randomly selected fashion items. Now, this is reasonable in order to assume that items in a user created outfit are more compatible than those in neutral outfit. So for that ,training samples can be made by pairing a user-generated outfit with a neutral one. So they initialize the parameters in VGGNet that would be trained on imagenet and initialize the other layers with random numbers drawn from gaussian distribution. Then furthermore these are optimized for the whole network using the mixed data set and in the second stage we see that the authors train using the specific model for personalized recommendations so we can say that for each user what they did was they first initialize the network with the certain parameters that were obtained by the previous general training and then they use each user’s own personal data to fine grain or fine tune the parameters. We know that fine-tuning is very important in this aspect. It sort of helps the data insufficiency problem in a lot of different applications. So for fashionNet A they saw that they fine-tune the whole network in this stage and for fashionNet B and C. There were two strategies used. The first one was to fine-tune the whole network. So both the feature Network and the matching network will have personalized parameters. Now this one resulted in different feature representations of each item for different users. The second method was to freeze the feature Network and only fine-tune the matching Network. So the features will keep the same and the user-specific information will be carried only by the matching Network and this will save a lot of computation during testing and which is quite a favorable aspect in terms of practice. #### 4.1.4. Performance evaluation In the end they found that the performance of FashionNet A was inferior to the other two architectures namely FashionNet B and C. When all the possible reasons for fashionNet B and C to obtain such an advantage was that the representation learning incompatibility modeling was performed in them separately so that they were able to use different network structures in order to achieve different functionalities. So these kinds of networks are easier to design and optimize in this case. ### 4.2. Generative Adversarial Training For personalization another approach is the generative adversarial training. So for that we go over another paper (Yu et al., 2019) in which they propose an approach in which a convolutional network is first used to map the query image into a latent Vector presentation. Now this latent representation all together with another Vector which characterizes users style preference as an input are taken into the generator Network in order to generate the target image item. #### 4.2.1. Previous Methods Although there are few works (Hu et al., 2015; Xu Chen, 2018) that have shown the personalized model is more capable of picking outfits that suit or a model to generate new items images for some category for a user that was personalized. But no query item was provided in their settings. They did not consider the compatibility between items. Figure 12. Network architecture for personalized fashion design. It contains one generator and two discriminators. The generator uses an encoder-decoder architecture. One of the discriminators is for real/fake supervision. And the other one is for compatibility prediction #### 4.2.2. Proposed Method Now, discriminator networks are built to guide the generation process. One of them is the classic real fake discriminator. And the other is a matching Network which simultaneously models the compatibility between fashion items and also learns the preference representations.When the given inventory is limited. It’s a possibility there. There are no good items enough to complement the query and when we have the inventory that is too large then generating the recommendation may face some efficiency problems. So this paper basically suggests that existing items can be synthesized images of new items that are compatible to a given one. So basically this solves the deficit problem for small inventories and for large inventory when targeting real items is necessary. We can adjust search items that are similar with the synthesized ones. Which is pretty much more efficient in terms than the exhaustive compatibility valuations, since similarity search can be very fast with techniques like hashing. Now aside from General compatibility they are also considering the personal issue. Personalization comes in here, which is an important trend as we have already discussed. Now given the same query item different persons would like to choose different items which goes with their own personal style. So while personalized recommendations have been prevalent in areas, like movies, songs and book recommendations, but for fashion, they are still not user-specific. So basically what this paper suggests is that the proposed system is personalized using the generative adversarial training framework GAN’s. Generative adversity networks have pretty much achieved a great success in synthesizing realistic images for different applications. So they apply this technique and they first use an encoder Network to map the query image into a latent Vector representation. And then this representation together with another vector that characterizes user style preference is taken into the input as for the generator Network that generates the target item. So basically the approach goes like this: the task of personalized fashion design is basically to develop a fashion item for a specific individual given an input query item. So there are two general requirements for this design that they have: the first one is the realness requirement which practically means that the design item should look realistic. And then the second thing comes is the compatibility requirement that is basically that the design item should be compatible with the query item. ### 4.3. Personalization in Unstructured Data Now we know that a lot of challenges in e-commerce usually come up from the fact that new products are continuously being added to the catalog. So the challenge invoked is properly personalizing the customers experience forecasting demand and planning the product range. #### 4.3.1. Background The paper (Ângelo Cardoso et al., 2018) in discussion is about a global e-commerce company that creates and curates clothing and beauty products for fashion lovers. So over the years they have a lot of products and this amounts to more than 1 million unique Styles. So for each product different divisions within the company produce and consume different product attributes, so mostly the attributes are manually curated and there could be cases in which information is sometimes missing or wrongly labeled. However, sometimes incomplete information still carries a lot of potential value for the business; the ability to have a systematic and quantitative characterization of a product is basically one of the key aspects for the company to make data- driven decisions that can be used across a set of problems including personalization. So the paper basically shows how to predict a consistent and complete set of product attributes that will illustrate how this enables them to personalize the customer experience by providing more relevant products. Figure 13. Schematic view of the multi-task attribute prediction network #### 4.3.2. Proposed Approach So basically the model that they proposed attracts attribute values from product images and textual descriptions. In terms of image processing what they do is that fashion is predominantly a visual business and visual features are at the core of many data science products. They use image features for many of their applications. So in order to minimize the computational cost what they did was they implemented a centralized visual feature generation pipeline. That uses a pre-trained convolutional neural network to extract product representation from images. Now for the text processing what they did was that the CNN’s were originally applied to images which are treated as matrices of pixel color values. And it’s a possibility to apply these convolutions to other types of matrices as well and in particular paragraphs of text. So similarly, they process images to produce product representations they also used the same technique for text descriptions. In multi modal Fusion, they say that the image and the text representations simply are concatenated together within a neural network, which is trained to predict the product attributes. This is pretty much straight forward because it’s a common way to fuse the different inputs. That works well in practice. Now the primary focus of the paper design was to find a solution that deals with missing labels at scale. Because in the paper, they also argue that the foundational piece to solve all of the problems is having consistent and detailed information about each product, which is rarely available. So they show this by having a quantitative understanding of the products. Can be used to improve recommendations in a Hybrid recommender system approach.They say that they could have chosen to build a separate model for each attribute, but then they would have to maintain multiple models in production. And in terms of independent models would also be oblivious to the correlations between attribute values and they would also only work well for common attributes, where there must be enough training data. Alternatively they said that they could have built a single model to predict all attributes at once also, but however few products are fully annotated and there would have not been enough data to train such a model. So because of these reasons what they did was they chose to cast attribute prediction as a multitask learning problem. This means training a neural network for each attribute but sharing most of the parameters between Networks. #### 4.3.3. Hybrid Approach The hybrid approach incorporates several state-of-the-art advances in recommender systems and not only incorporates new products, but also enhances the recommendations that customers receive overall. Their approach creates an embedding of products, i.e. a representation of all the products in their catalogue in a high-dimensional vector space. In this vector space, products with similar styles and attributes will be closer than unrelated ones. When producing personalised recommendations, the algorithm also assigns a vector to every customer. The items with the highest inner product with the customer vector are the recommended ones. The position of products and customers in this space is determined not only by the customer-product interactions, but also by the augmented product attributes. This ensures that newly added products are positioned correctly in the space and can be recommended to the right customers. ### 4.4. POG: Personalized Outfit Generation Another paper (Chen et al., 2019) proposes a personalized outfit generation POG model. Basically what happens in this model is that they connect the user preferences regarding individual items and then the outfits with transformer architecture. So the extensive offline and online experiments they did provided them with strong quantitative evidence that the method they proposed found alternative methods regarding port compatibility and personalization metrics. So basically what happens is that they can generate compatible and personalized outfits based on user recent behavior. So specifically for this they use a Transformer encoder decoder architecture that models both signals from user preference and outfit compatibility. And this is interestingly one of the first study to generate personalized outfits based on user historical Behavior within encoder decoder framework. They also developed a platform named IDA where POG. has been deployed in order to help out without regeneration and recommendation at a very large scale application Ifashion. #### 4.4.1. Previous Methods There are several methods for generating a fashion outfit that is likeable by the user and usually these methods fall into basically two types. So the first type is basically the one in which they focus on calculating a pairwise compatibility metric (McAuley et al., 2015; Song et al., 2018; Veit et al., 2015) . And the second type is in which they present modeling and outfit as a set or an ordered sequence. And then there are models (Li et al., 2016) in which they classify a given outfit as popular or unpopular or train a bi- directional LSTM model (Han et al., 2017) sequentially generate outfits. Now we can see that all these methods generally use a simple pooling of item vectors in order to represent an outfit and they have to rely on the order of the outfits item. So this is noted that these methods belonging to either category hardly considers all the interactions between the items in an outfit. And it is quite unreasonable to consider an outfit as an ordered sequence because you know shuffling of items in the outfit itself should make no difference on its compatibility. #### 4.4.2. Proposed Approach So what they are trying to say is that they want to explicitly incorporate this into their modeling architecture by which they require that each item should have a different interaction weight with respect to other item in one outfit and they have given example like a red shirt should have a higher interaction with you know, blue jeans or black jeans, but a smaller weight with a pair of white gloves. So the model they propose in this is basically what they do, is that they build a three-step process in which the first step has the items that are to be embedded and in the second they build FOM which learns compatibilities of items within an outfit and lastly the third stage once their training is completed. They use the result to pretrained FOM to initialize POG Transformer architecture. Representing these items using a multi model embedding model. So for every fashion item f they compute a non linear feature embedding f . The concept of fashion basically relies on Visual and textual information So basically in previous models (Li et al., 2016; Han et al., 2017) what they did was the authors used the image and text to learn the multimodal embeddings. But in this scenario, what they do is that they use a multi-modal embedding model that takes the following input for every item 1. (1) Dense vector encoding the white background picture of the item from a CNN model, 2. (2) Dense vector encoding the title of the item obtained from a TextCNN network, which has been pre-trained to predict an item’s leaf category based on its title 3. (3) D ense vector encoding a collaborative filtering signal for the item using Alibaba’s proprietary Behemoth Graph embedding platform. So this platform is used for generating item embeddings based on the co-occurrence statistics of items in recorded user click sessions in the taobao application. Figure 14. The architecture of POG, which is an encoder-decoder architecture with a Per network and a Gen network. The outfit item is generated step by step according to the user preference signal from the Per network and the compatibility signal from the Gen network. So the generation model works like this, it generates personalized and compatible outfit by introducing user preference signals. Taking the advantage of encoder-decoder structure, it translates an user’s historical behaviors to a personalized outfit. Let $\mathcal{U}$ denote the set of all users and $\mathcal{F}$ be the set of all outfits. They have used a sequence of user behaviors $U=\left\\{u_{1},\ldots,u_{i},\ldots,u_{m}\right\\}$ to characterize an user, where $u_{i}$ are the clicked items by the user. $F=\left\\{f_{1},\ldots,f_{t},\ldots,f_{n}\right\\}$ is the clicked outfit from the same user, where $f_{t}$ are the items in the outfit. At each time step, it predicts the next outfit item given previous outfit items and user’s click sequence on items $U.$ Thus for pair $(U,F)$ the objective function of $\mathrm{POG}$ can be written as: $\mathcal{L}_{(U,F)}=-\frac{1}{n}\sum_{t=1}^{n}\log\operatorname{Pr}\left(f_{t+1}\mid f_{1},\ldots,f_{t},U;\Theta_{(U,F)}\right)$ where $\Theta_{(U,F)}$ denotes the model parameters. $\operatorname{Pr}(\cdot)$ is the probability of seeing $f_{t+1}$ conditioned on both previous outfit items and user clicked items. In POG the encoder basically what it does is that it takes the user clicked input items and then it gives a special token like [start]. And then the decoder generates an outfit one item at a time. So at each step what happens is that the model is basically autoregressively consuming the previously generated items as input.The generation basically stops when a special token [end] appears. So basically what happens is that there in the end an outfit is given that is generated by composing the output items. So in the figure, you can also see that the encoder is termed as PER Network and then the decoder is as Gen Network. So the PER’s natural is basically that it provides a user preference in terms of signal and then the Gen Network what it does is that it generates outfits based on both personalization signal and compatibility signal. So basically the general network is initialized using the aforementioned pre trained FOM. ### 4.5. Item-to-Set Metric Learning Approach Social media has been a great source for fashion recommendation and fashion promotion. It provides us with an open and new data source for personalized fashion analysis. #### 4.5.1. Background So this paper (Zheng et al., 2020) basically studies the problem of personalized fashion recommendation by gathering the data from different social media. That is they recommend new outfits to social media users that fit their fashion preferences. They present an item to set metric learning framework that basically learns to compute similarity that exists between a set of historical fashion items of a user to a new fashion item. For extracting features from a multi model street view fashion item the author basically proposes an embedding module that performs multi-modality feature extraction and cross Modality gated fusion. By studying the problem of personalized fashion recommendation with social media data that they are seeking to recommend new fashion outfits based on the activities that are being carried by the social network users. #### 4.5.2. Previous Methods A lot of different studies (Kiapour et al., 2015; Hu et al., 2015; Huang et al., 2015; Iwata et al., 2011; Jagadeesh et al., 2014) are done for clothing retrieval and recommendation. But leveraging the user’s interaction on social media for data for fashion recommendation is very much still challenging and is quite less explored. And usually what we can gather from social media is online activities like a street view selfie with additional word description. So this gives that the granularity of such data is much coarser than you know, that is unexplored. And most models (Li et al., 2016; Tangseng et al., 2018) are not directly applicable to the task due to their lack of supervision. #### 4.5.3. Proposed Approach So paper basically proposes a self supervisor approach for effective and personalized fashion recommendation in which they divide into two categories the pictures in which the selfie posts of users a set that reveals their personal fashion preferences or outfit items that are to be recommended items. So they proposed that to learn an item to set metric that measures similarities between a set and items for personalized recommendation. They minimize the item to set distance for the set and items of a user and while making sure they maximize such distances for certain items of different users. And benefiting from this framework they are able to perform personalized recommendations without requiring any sort of additional supervision. Now we know that metric learning is well studied in literature and learning such an item to set metric is previously unexplored. And therefore pose new challenges because we know that the user can have interest in more than one fashion style and not the one that is being depicted in their picture. So the item to set similarity cannot be captured by an over simplified average of multiple items by similarities. Which therefore states that the nearest neighbor item to set metric is difficult to learn as it is susceptible to noise and outliers. So in highlight what their contribution is that they present a fashion recommendation system built on personal social media data and their system recommends personalized outfits for using few constraint street view selfie post of the users. They also proposed a self supervise scheme in which they enable the training of the system. The approach is based on a novel item to set a metric learning framework that basically needs only the user selfie pose as the supervision. For this they design a multi model embedding module that better fuses the social media data for obstruction of fashion features. Built upon the item-wise measurement $d\left(f_{i},f_{j}\right),$ they propose an item-to-set similarity metric $D(S,f),$ which measures how dissimilar an item $f$ is to a set of items $S=\left\\{f_{1},\cdots,f_{K}\right\\}.$ The itemto-set metric aims to predict how similar a outfit candidate is to a set of user selfies for personalized fashion recommendation. To design a metric that better captures the multiple interests of a user while facilitating robust training, the paper proposes a generalized item-to-set distance. Specifically, given a set $S$ and a query $f$, they first assign an importance weight $w_{i}$ to each item $f_{i}\in S$ before feature averaging and distance computation. The importance weight is computed using an importance estimator $w_{i}=K\left(f_{i};f,S\right)$ Such a item-to-set distance is defined by: $\displaystyle D(S,\boldsymbol{f})$ $\displaystyle=d\left(\sum_{i=1}^{K}\alpha_{i}f_{i},\boldsymbol{f}\right)$ $\displaystyle\alpha_{i}$ $\displaystyle=\frac{\exp\left(w_{i}\right)}{\sum_{j}\exp\left(w_{j}\right)}$ To reduce the influences of noise and outliers when computing the distance, basically what they did was that they further considered an intra-set importance weight: $v\left(f_{i};S\right)=\operatorname{MLP}_{v}\left(\left[f_{i},\operatorname{stat}(S)\right]\right)$ where $\mathrm{MLP}_{v}$ outputs a scalar from an input vector, and $\operatorname{stat}(S)$ is a vector that captures the statistics of the set $S$ along all feature dimensionalities ${}^{2}.$ In this way, we compare each item $f_{i}$ with the set $S$ to eliminate the outliers from the sets. Now as we know that there are different individuals that focus on different particular aspects of fashion items and the item to set metric itself should be user specific. So for that issue what they did was that for the minimalist fashion style users the items that distance was made more sensitive to the amount of colors that are used but for users of the artsy style the item to set distance should focus more on unusual prints and the complexity of accessories. So they extended the similarity metric equation to a user specific metric in which they performed a user specific space transformation before the distance computation. In particular, given the set $S$, we compute a scaling vector $t(S)$ which indicates the scaling factor at each feature dimension: $\boldsymbol{t}(S)=\operatorname{softmax}\left(\operatorname{MLP}_{t}(\operatorname{stat}(S))\right)$ Using the space transformation, they extended the item-to-set metric to a set- specific metric. Specifically, they defined a user-specific item-to-set metric: $D_{us}(S,f)=d\left(t(S)\odot\left(\sum_{i=1}^{K}\alpha_{i}f_{i}\right),t(S)\odot f\right)$ where $\odot$ represents vector elementwise multiplication. It filters out the feature dimensions that a user focuses less on before the distance computation. This procedure helps the recommendation system to be more user- specific. ## 5\. Future Research In the post coronavirus era one of the industries that is obviously undoubtedly incorporating advanced technologies at much faster speed than ever before is fashion. And thanks to AI and computer vision power tools, new and engaging experiences are being born for both retailers and consumers. The e-commerce customer experience is completely incorporated with AI Solutions like online site navigation, search, retrieval ,target marketing, labeling, personalized offers ,size fitting, recommendations and online fitting rooms and also style recommendation analytics and much more. So by using computer vision and AI the image pixels are automatically taken and then they generate semantic data from them, which is very crucial for the e-commerce stores. One of the things that is the basic thing is the discovery of the products that the visual search should be easy enough for the Shoppers to find what they are looking for and should also be benefiting the retailers as well so that they can take the advantage of users behavior and then show them the recommendations and can get more profit from this aspect as the stores are getting more online this post covid era. So the AI technology enables fashion brands to sort of gain insight as to which product features their customers would like to prefer. Now an interesting aspect (Countants, 2020) is that we can see that the fashion industry is at over 3 trillion dollars that contributes to the healthy portion of the global GDP and in the 21st century, we can see that AI or machine learning or specifically deep learning in the fashion industry is changing every expectation of this forward-looking business. So the use of AI let alone in the fashion industry of 2020 has so entrenched that 44 percent of the fashion retailers that are not using AI today are facing bankruptcy. So you can take this as an example and as a result of this Global spending on AI Technologies by fashion and retail industry is expected to reach 7.3 billion each year by the year 2022 and that’s just in two years. AI powered fashion designing can be based to get the preferred customer color textures and other style preferences and then they can be used ahead in order to design the apparel, the textile itself. Regarding The factoring process, what they can do is that they can use AI tools to identify the Super fast changing trends and supply the latest fashion accessories to the retail shelves, which is pretty much faster than the traditional retailing system. And a lot of leading fashion brands like Zara, Topshop and Achieve and are already using this and they are pretty much quicker in providing instant gratification to retail customers by recognizing seasonal demand and Manufacturing the right supply of the latest clothing and obviously virtual merchandising is something that has enabled technologies like augmented reality and virtual reality and now are closing the gap that is between online and in-store shopping. So this is also really popular regarding this system. And this is something that can be worked in the recommendation systems. As a lot of people would like to experience the virtual reality and augmented reality aspect in terms of the clothes fitting and checking out the online buying experience and making it more human-like. ## 6\. Conclusion As the advancements in deep learning, CV and AI are getting stronger day by day their usage in the fashion industry has also become a very popular topic. From product personalization or better designing there are multiple ways in which AI and machine learning Technologies are impacting the global fashion industry and they are increasing the investment by Leading fashion brands in these Technologies are a proof of their immense potential. They provide enhanced customer service, Virtual merchandising, smart manufacturing process and improved inventory management and need less Manpower through Automation and provide reduction in returned products which also improves customer satisfaction. And one of the biggest things is personalization, which is pretty much the key of business success and thanks to deep learning Technologies like AI and ML along with business analytics is enabling fashion business to keep track of fashion trends and purchasing behavior of individual customers. So now it may be a trend or it may be a season prediction. You can do anything with these powerful tools and the fashion industry is magnified. And this is a field that has the potential to grow and ever expand, so any future research in this line that will be done would be something that paves way ahead for more jaw dropping phenomenon. ## References (1) * Adomavicius and Tuzhilin (2005) Gediminas Adomavicius and Alexander Tuzhilin. 2005. Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions. _Knowledge and Data Engineering, IEEE Transactions on_ 17 (07 2005), 734–749. https://doi.org/10.1109/TKDE.2005.99 * Borràs et al. 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Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs À La Apéry Robert DOUGHERTY-BLISS, Christoph KOUTSCHAN, and Doron ZEILBERGER In honor of our irrational guru Wadim Zudilin, on his $\lfloor 50\,\zeta(5)\rfloor$-th birthday [Actual] Historical Introduction: How Beukers’ Proofs Were ACTUALLY found Hilbert’s 0-th problem Before David Hilbert [H] stated his famous 23 problems, he mentioned two problems that he probably believed to be yet much harder, and indeed, are still wide open today. One of them was to prove that there are infinitely many prime numbers of the form $2^{n}+1$, and the other one was to prove that the Euler-Mascheroni constant is irrational. Two paragraphs later he stated his optimistic belief that “in mathematics there is no ignorabimus.” As we all know, he was proven wrong by Gödel and Turing in general, but even for such concrete problems, like the irrationality of a specific, natural, constant, like the Euler-Mascheroni constant (that may be defined in terms of the definite integral $\quad-\int_{0}^{\infty}e^{-x}\log x$) , that is most probably decidable in the logical sense, (i.e. there probably exists a (rigorous) proof), we lowly humans did not yet find it, (and may never will!). While the Euler-Mascheroni constant (and any other, natural, explicitly- defined, constant that is not obviously rational) is surely irrational, in the everyday sense of the word sure (like death and taxes), giving a proof, in the mathematical sense of ‘proof’ is a different matter. While $e$ was proved irrational a long time ago (trivial exercise), and $\pi$ was proved irrational by Lambert around 1750, we have no clue how to prove that $e+\pi$ is irrational. Ditto for $e\cdot\pi$. Exercise: Prove that at least one of them is irrational. Apéry’s Miracle As Lindemann first proved in 1882, the number $\pi$ is more than just irrational, it is transcendental, hence it follows that $\zeta(n)$ is irrational for all even arguments, since Euler proved that $\zeta(2n)$ is a multiple of $\pi^{2n}$ by a rational number. But proving that $\zeta(3)$, $\zeta(5)$, $\dots$ are irrational remained wide open. Since such problems are so hard, it was breaking news, back in 1978, when 64-years-old Roger Apéry announced and sketched a proof that $\zeta(3):=\sum_{n=1}^{\infty}{1\over n^{3}}$ is irrational. This was beautifully narrated in a classic expository paper by Alf van der Poorten [vdP], aided by details filled-in by Henri Cohen and Don Zagier. While beautiful in our eyes, most people found the proof ad-hoc and too complicated, and they did not like the heavy reliance on recurrence relations. To those people, who found Apéry’s original proof too magical, ad-hoc, and computational, another proof, by a 24-year-old PhD student by the name of Frits Beukers [B] was a breath of fresh air. It was a marvelous gem in human- generated mathematics, and could be easily followed by a first-year student, using partial fractions and very easy estimates of a certain triple integral, namely $\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{(x(1-x)y(1-y)z(1-z))^{n}\over(1-z+xyz)^{n+1}}\,dx\,dy\,dz\quad.$ The general approach of Apéry of finding concrete sequences of integers $a_{n},b_{n}$ such that $\|\zeta(3)-{a_{n}\over b_{n}}\|\,<\,{CONST\over b_{n}^{1+\delta}}\quad,$ (see below) for a positive $\delta$ was still followed, but the details were much more palatable and elegant to the average mathematician in the street. As a warmup, Beukers, like Apéry before him, gave a new proof of the already proved fact that $\zeta(2)={\pi^{2}\over 6}$ is irrational, using the double integral $\int_{0}^{1}\,\int_{0}^{1}\,{(x(1-x)y(1-y))^{n}\over(1-xy)^{n+1}}\,dx\,dy\quad.$ Ironically, we will follow Beukers’ lead, but heavily using recurrence relations, that will be the engine of our approach. Thus we will abandon the original raison d’être of Beukers’ proof of getting rid of recurrences, and bring them back with a vengeance. [Alternative World] Historical Introduction: How Beukers’s Proofs Could (and Should!) have been Discovered Once upon a time, there was a precocious teenager, who was also a computer whiz, let’s call him/her/it/they Alex. Alex just got a new laptop that had Maple, as a birthday present. Alex typed, for no particular reason, int(int(1/(1-xy),x=0..1),y=0..1); and immediately got the answer: ${\pi^{2}\over 6}$. Then Alex was wondering about the sequence $I(n):=\int_{0}^{1}\,\int_{0}^{1}\,{(x(1-x)y(1-y))^{n}\over(1-xy)^{n+1}}\,dx\,dy\quad.$ (why not, isn’t it a natural thing to try out for a curious teenager?), and typed I1:=n->int(int(1/(1-xy)(x(1-x)y(1-y)/(1-xy))n,x=0..1),y=0..1); (I is reserved in Maple for $\sqrt{-1}$, so Alex needed to use I1), and looked at the first ten values by typing: L:=[seq(I1(i),i=1..10)]; , getting after a few seconds $[5-{\pi^{2}\over 2},-{{125\over 4}}+{{19\,{\pi}^{2}\over 6}},{{8705\over 36}}-{{49\,{\pi}^{2}\over 2}},-{{32925\over 16}}+{{417\,{\pi}^{2}\over 2}},$ ${{13327519\over 720}}-{{3751\,{\pi}^{2}\over 2}},-{{124308457\over 720}}+{{104959\,{\pi}^{2}\over 6}},$ ${{19427741063\over 11760}}-{{334769\,{\pi}^{2}\over 2}},-{{2273486234953\over 141120}}+{{9793891\,{\pi}^{2}\over 6}},$ ${{202482451324891\over 1270080}}-{{32306251\,{\pi}^{2}\over 2}},-{{2758128511985\over 1728}}+{{323445423\,{\pi}^{2}\over 2}}]\quad.$ Alex immediately noticed that, at least for $n\leq 10$, $I(n)=a_{n}-b_{n}{\pi^{2}\over 6}\quad,$ for some integers $b_{n}$ and some rational numbers $a_{n}$. By taking evalf(L), Alex also noticed that $I(n)$ get smaller and smaller. Knowing that Maple could not be trusted with floating point calculations (unless you change the value of Digits from its default, to something higher, say, in this case Digits:=30), that they get smaller and smaller. Typing ‘evalf(L,30);’, Alex got: $[0.06519779945532069058275450006,0.0037472701163022929758881663,$ $0.000247728866269394110526059,0.00001762713127202699137347,$ $0.0000013124634659314676853,0.000000100776323486001254,$ $0.00000000791212964371946,0.0000000006317437711206,$ ${5.1111100706\times 10^{-11}},{4.17922459\times 10^{-12}}]\quad.$ Alex realized that $I(n)$ seems to go to $zero$ fairly fast, and since $I(10)/I(9)$ and $I(9)/I(8)$ were pretty close, Alex conjectured that the limit of $I(n)/I(n-1)$ tends to a certain constant. But ten data points do not suffice! When Alex tried to find the first $2000$ terms, Maple got slower and slower. Then Alex asked Alexa, the famous robot, Alexa: how do I compute many terms of the sequence $I(n)$ given by that double-integral? and Alexa replied: Go to Doron Zeilberger’s web-site and download the amazing program https://sites.math.rutgers.edu/~zeilberg/tokhniot/MultiAlmkvistZeilberger.txt , that accompanied the article [ApaZ]. Typing MAZ(1,1/(1-xy),x(1-x)y(1-y)/(1-xy),[x,y],n,N, $\\{\\}$)[1]; immediately gave a recurrence satisfied by $I(n)$ $I(n)=-{{\left(11\,{n}^{2}-11\,n+3\right)\over{n}^{2}}}\cdot{\it I}\left(n-1\right)+{{\left(n-1\right)^{2}\over{n}^{2}}}\cdot{\it I}\left(n-2\right)\quad.$ Using this recurrence, Alex easily computed the first $2000$ terms, using the following Maple one-liner (calling the sequence defined by the recurrence I2(n)): I2:=proc(n) option remember: if n=0 then Pi2/6 elif n=1 then 5-Pi2/2 else -(11n2-11n+3)/n*2I2(n-1)+(n-1)2/n2*I2(n-2):fi: end: and found out that indeed $I(n)/I(n-1)$ tends to a limit, about $0.09016994$. Writing $I(n)=a_{n}-b_{n}{\pi^{2}\over 6}\quad$ and realizing that $I(n)$ is small, Alex found terrific rational approximations to ${\pi^{2}\over 6}$, $a_{n}/b_{n}$, that after clearing denominators can be written as $a^{\prime}_{n}/b^{\prime}_{n}$ where now both numerator $a^{\prime}_{n}$ and denominator $b^{\prime}_{n}$ are integers. ${\pi^{2}\over 6}\approx{a^{\prime}_{n}\over b^{\prime}_{n}}\quad.$ Alex also noticed that for all $n$ up to $2000$, for some constant $C$, $\|{\pi^{2}\over 6}-{a^{\prime}_{n}\over b^{\prime}_{n}}\|\leq{C\over(b^{\prime}_{n})^{1+\delta}}\quad,$ where $\delta$ is roughly $0.09215925467$. Then Alex concluded that this proves that ${\pi^{2}\over 6}$ is irrational, since if it were rational the left side would have been $\geq{C_{1}\over b^{\prime}_{n}}$, for some constant $C_{1}$. Of course, some details would still need to be filled-in, but that was not too hard. The General Strategy Let’s follow Alex’s lead. (Of course our fictional Alex owes a lot to the real Beukers and also to Alladi and Robinson [AR]). Start with a constant, let’s call it $C$, given by an explicit integral $\int_{0}^{1}K(x)\,dx\quad,$ for some integrand $K(x)$, or, more generally, a $d$-dimensional integral $\int_{0}^{1}\dots\int_{0}^{1}K(x_{1},\dots,x_{k})\,dx_{1}\dots dx_{k}\quad.$ Our goal in life is to prove that $C$ is irrational. Of course $C$ may turn out to be rational (that happens!), or more likely, an algebraic number, or expressible in terms of a logarithm of an algebraic number, for which, there already exist irrationality proofs (albeit not always effective ones). But who knows? Maybe this constant has never been proved irrational, and if it will happen to be famous (e.g. Catalan’s constant, or $\zeta(5)$, or the Euler- Mascheroni constant mentioned above), we will be famous too. But even if it is a nameless constant, it is still extremely interesting, if it is the first irrationality proof, since these proofs are so hard, witness that, in spite of great efforts by experts like Wadim Zudilin, the proofs of these are still wide open. In this article we will present numerous candidates. Our proofs of irrationality are modulo a ‘divisibility lemma’ (see below), that we are sure that someone like Wadim Zudilin, to whom this paper is dedicated, can fill-in. Our only doubts are whether these constants are not already proved to be irrational because they happen to be algebraic (probably not, since Maple was unable to identify them), or more complicated numbers (like logarithms of algebraic numbers). Recall that Maple’s identify can’t (yet) identify everything that God can. Following Beukers and Alladi-Robinson, we introduce a sequence of integrals, parameterized by a non-negative integer $n$ $I(n)=\int_{0}^{1}K(x)\,(x(1-x)K(x))^{n}\,dx\quad,$ and analogously for multiple integrals, or more generally $I(n)=\int_{0}^{1}K(x)\,(x(1-x)S(x))^{n}\,dx\quad,$ for another function $S(x)$. Of course $I(0)=C$, our constant that we want to prove irrational. It so happens that for a wide class of functions $K(x)$, $S(x)$, (for single or multivariable $x$) using the Holonomic ansatz [Ze1], and implemented (for the single-variable case) in [AlZ], and for the multi-variable case in [ApZ], and much more efficiently in [K], there exists a linear recurrence equation with polynomial coefficients, that can be actually computed (always in theory, but also often in practice, unless the dimension is high). In other words we can find a positive integer $L$, the order of the recurrence, and polynomials $p_{0}(n),p_{1}(n),\dots,p_{L}(n)$, such that $p_{0}(n)I(n)+p_{1}(n)I(n+1)+\dots+p_{L}(n)I(n+L)\,=\,0\quad.$ If we are lucky (and all the cases in this paper fall into this case) the order $L$ is $2$. Furthermore, it would be evident in all the examples in this paper that $p_{0}(n),p_{1}(n),p_{2}(n)$ can be taken to have integer coefficients. Another ‘miracle’ that happens in all the examples in this paper is that $I(0)$ and $I(1)$ are rationally-related, i.e. there exist integers $c_{0},c_{1},c_{2}$ such that $c_{0}I(0)+c_{1}I(1)=c_{2}\quad,$ that our computers can easily find. It then follows, by induction, that one can write $I(n)=b_{n}C-a_{n}\quad,$ for some sequences of rational numbers $\\{a_{n}\\}$ and $\\{b_{n}\\}$ that both satisfy the same recurrence as $I(n)$. Either using trivial bounds on the integral, or using the so-called Poincaré lemma (see, e.g. [vdP], [ZeZu1],[ZeZu2]) it turns out that $a_{n}\,=\,\Omega(\alpha^{n})\quad,\quad b_{n}\,=\,\Omega(\alpha^{n})\quad,$ for some constant $\alpha>1$, and $\|I(n)\|=\Omega(\,{1\over\beta^{n}}\,)\quad,$ for some constant $\beta>1$. [Please note that we use $\Omega$ in a looser-than-usual sense, for us $x(n)=\Omega(\alpha^{n})$ means that $\lim_{n\rightarrow\infty}\,{\log x(n)\over n}=\alpha$.] In the tweaks of Beukers’ integrals for $\zeta(2)$ and $\zeta(3)$ coming up later, $\alpha$ and $\beta$ are equal, but in the tweaks of the Alladi- Robinson integrals, $\alpha$ is usually different than $\beta$. It follows that $\|C-{a_{n}\over b_{n}}\|=\Omega({1\over(\alpha\beta)^{n}})\quad.$ Note that $a_{n}$, and $b_{n}$ are, usually, not integers, but rather rational numbers (In the original Beukers/Apéry cases, the $b_{n}$ were integers, but the $a_{n}$ were not, in the more general cases in this article, usually neither of them are integers). It so happens, in all the cases that we discovered, that there exists another sequence of rational numbers $E(n)$ such that $a^{\prime}_{n}:=a_{n}\,E(n)\quad,\quad b^{\prime}_{n}:=b_{n}\,E(n)\quad,$ are always integers, and, of course $gcd(a^{\prime}_{n}\,,\,b^{\prime}_{n})=1$. We call $E(n)$ the integer-ating factor. In some cases we were able to conjecture $E(n)$ exactly, in terms of products of primes satisfying certain conditions (see below), but in other cases we can only conjecture that such an explicitly-describable sequence exists. In either case there exists a real number, that sometimes can be described exactly, and other times only estimated, let’s call it $\nu$, such that $\lim_{n\rightarrow\infty}{\log E(n)\over n}\,=\,\nu\quad,$ or, in our notation, $E(n)=\Omega(\,e^{n\nu}\,)$ . Since we have $\|C-{a^{\prime}_{n}\over b^{\prime}_{n}}\|=\Omega({1\over(\alpha\beta)^{n}})\quad,$ where $b^{\prime}_{n}=\Omega(e^{\nu\,n}\alpha^{n})$. We need a positive $\delta$ such that $(e^{\nu\,n}\alpha^{n})^{1+\delta}=(\alpha\beta)^{n}\quad.$ Taking $\log$ (and dividing by $n$) we have $(\nu+\log\alpha)(1+\delta)=\log\alpha+\log\beta\quad,$ giving $\delta={\log\beta-\nu\over\log\alpha+\nu}\quad.$ If we are lucky, and $\log\beta>\nu$, then we have $\delta>0$, and an irrationality proof!, Yea! We also, at the same time, determined an irrationality measure (see [vdP]) $1+{1\over\delta}\,=\,{\log\alpha+\log\beta\over\log\beta-\nu}\quad.$ If we are unlucky, and $\delta<0$, it is still an exponentially fast way to compute our constant $C$ to any desired accuracy. Summarizing: For each specific constant defined by a definite integral, we need to exhibit $\bullet$ A second-oder recurrence equation for the numerator and denominator sequence $a_{n}$ and $b_{n}$ that feature in $I(n)=b_{n}C-a_{n}$. $\bullet$ The initial conditions $a_{0},a_{1}$, $b_{0},b_{1}$ enabling a very fast computation of many terms of $a_{n},b_{n}$. $\bullet$ The constants $\alpha$ and $\beta$ $\bullet$ Exhibit a conjectured integer-ating factor $E(n)$, or else conjecture that one exists, and find, or estimate (respectively), $\nu:=\,\lim_{n\rightarrow\infty}{\log E(n)\over n}$ . $\bullet$ Verify that $\beta>e^{\nu}$ and get (potentially) famous. The Three Classical Cases ${\bf\log 2}$ ([AR]) $C\,=\,\int_{0}^{1}\,{1\over 1+x}\,dx\,=\,\log 2\quad.$ $I(n)=\int_{0}^{1}{(x(1-x))^{n}\over(1+x)^{n+1}}\,dx\quad.$ Recurrence: $\left(n+1\right)X\left(n\right)+\left(-6\,n-9\right)X\left(n+1\right)+\left(n+2\right)X\left(n+2\right)\,=0\,\quad.$ $\alpha\,=\,\beta=3+2\sqrt{2}\quad.$ Initial conditions $a_{0}=0\,,\,a_{1}=2\quad;\quad b_{0}=1\,,\,b_{1}=3\quad.$ Integer-ating factor $E(n)=lcm(1\dots n)$, $\nu=1$. $\delta={\log\beta-\nu\over\log\alpha+\nu}={\log\beta-1\over\log\alpha+1}={\log(3+2\sqrt{2})-1\over\log(3+2\sqrt{2})+1}=0.276082871862633587\quad.$ Implied irrationality measure: $1+1/\delta=4.622100832454231334\dots$. $\bf{\zeta(2)}$ ([B]) $C=\int_{0}^{1}\,\int_{0}^{1}\,{1\over 1-xy}\,dx\,dy\,=\,\zeta(2)\quad.$ $I(n)=\int_{0}^{1}\,\int_{0}^{1}\,{(x(1-x)y(1-y))^{n}\over(1-xy)^{n+1}}\,dx\,dy\quad.$ Recurrence: $-\left(1+n\right)^{2}X\left(n\right)+\left(11\,{n}^{2}+33\,n+25\right)X\left(n+1\right)+\left(2+n\right)^{2}X\left(n+2\right)\,=0\quad.$ $\alpha\,=\,\beta={11\over 2}+{5\sqrt{5}\over 2}\quad.$ Initial conditions $a_{0}=0\,,\,a_{1}=-5\quad;\quad b_{0}=1\,,\,b_{1}=-3\quad.$ Integer-ating factor $E(n)=lcm(1\dots n)^{2}$, $\nu=2$. $\delta={\log\beta-\nu\over\log\alpha+\nu}={\log\beta-2\over\log\alpha+2}={\log(11/2+5\sqrt{5}/2)-2\over\log(11/2+5\sqrt{5}/2)+2}=0.09215925473323\dots\quad.$ Implied irrationality measure: $1+1/\delta=11.8507821910523426959528\dots$. $\bf{\zeta(3)}$ ([B]) $C=\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{1\over 1-z+xyz}\,dx\,dy\,\,dz=\,\zeta(3)\quad.$ $I(n)=\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{(x(1-x)y(1-y)z(1-z))^{n}\over(1-z+xyz)^{n+1}}\,dx\,dy\,dz\quad.$ Recurrence: $\left(1+n\right)^{3}X\left(n\right)-\left(2\,n+3\right)\left(17\,{n}^{2}+51\,n+39\right)X\left(n+1\right)+\left(n+2\right)^{3}X\left(n+2\right)\,=0\quad.$ $\alpha\,=\,\beta=17+12\,\sqrt{2}\quad.$ Initial conditions $a_{0}=0\,,\,a_{1}=12\quad;\quad b_{0}=1\,,\,b_{1}=5\quad.$ Integer-ating factor $E(n)=lcm(1\dots n)^{3}$, $\nu=3$. $\delta={\log\beta-\nu\over\log\alpha+\nu}={\log\beta-3\over\log\alpha+3}={\log(17+12\,\sqrt{2})-3\over\log(17+12\,\sqrt{2})+3}=0.080529431189061685186\dots\quad.$ Implied irrationality measure: $1+1/\delta=13.41782023335376578458\dots$. Accompanying Maple packages This article is accompanied by three Maple packages, GenBeukersLog.txt, GenBeukersZeta2.txt, GenBeukersZeta3.txt all freely available from the front of this masterpiece https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/beukers.html , where one can find ample sample input and output files, that readers are welcome to extend. Zudilin’s Tweak of the Beukers $\zeta(2)$ integral to get the Catalan constant The inspiration for our tweaks came from Wadim Zudilin’s brilliant discovery [Zu1] that the famous Catalan constant, that may be defined by the innocent- looking alternating series of the reciprocals of the odd perfect-squares $C:=1-{1\over 3^{2}}+{1\over 5^{2}}-{1\over 7^{2}}+\dots=\sum_{n=0}^{\infty}{(-1)^{n}\over(2n+1)^{2}}\quad,$ can be written as the double integral ${1\over 8}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{-{1\over 2}}(1-y)^{-{1\over 2}}\over 1-xy}\,dx\,dy\quad.$ This lead him to consider the sequence of Beukers-type double-integrals $I(n)\,=\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{-{1\over 2}}(1-y)^{-{1\over 2}}\over 1-xy}\cdot\left({x(1-x)y(1-y)\over 1-xy}\right)^{n}\,dx\,dy\quad.$ Using the Zeilberger algorithm, Zudilin derived a three term recurrence for $I(n)$ leading to good diophantine approximations to the Catalan constant, alas not good enough to prove irrationality. This was elaborated and extended by Yu. V. Nesterenko [N]. See also [Zu2]. Using the multivariable Almkvist-Zeilberger algorithm we can derive the recurrence much faster. Using Koutschan’s package [K], it is yet faster. Our Tweaks Inspired by Zudilin’s Beukers-like integral for the Catalan constant, we decided to use our efficient tools for quickly manufacturing recurrences. We systematically investigated the following families. Generalizing the Alladi-Robinson-Like Integral for $\log 2$ Alladi and Robinson [AR] gave a Beukers-style new proof of the irrationality of $\log 2$ using the elementary fact that $\log 2\,=\,\int_{0}^{1}\,{1\over 1+x}\,dx\quad,$ and more generally, ${1\over c}\,\log(1+c)\,=\,\int_{0}^{1}\,{1\over 1+cx}\,dx\quad.$ They used the sequence of integrals $I(n):=\int_{0}^{1}\,{1\over 1+cx}\left({x(1-x)\over 1+cx}\right)^{n}\,dx\quad,$ and proved that for a wide range of choices of rational $c$, this leads to irrationality proofs and irrationality measures (see also [ZeZu1]). Our generalized version is the three-parameter family of constants $I_{1}(a,b,c):={1\over B(1+a,1+b)}\,\int_{0}^{1}\,{x^{a}(1-x)^{b}\over 1+cx}\,dx$ that is easily seen to equal ${}_{2}F_{1}(1,a+1;a+b+2;-c)$. We use the sequence of integrals $I_{1}(a,b,c)(n):=\,{1\over B(1+a,1+b)}\,\int_{0}^{1}\,{x^{a}(1-x)^{b}\over 1+cx}\cdot\left({x(1-x)\over 1+cx}\right)^{n}\,dx\quad.$ Using the (original!) Almkvist-Zeilberger algorithm [AlZ], implemented in the Maple package https://sites.math.rutgers.edu/~zeilberg/tokhniot/EKHAD.txt , we immediately get a second-order recurrence that can be gotten by typing ‘OpeL(a,b,c,n,N);’ in the Maple package https://sites.math.rutgers.edu/~zeilberg/tokhniot/GenBeukersLog.txt . This enabled us to conduct a systematic search, and we found many cases of ${}_{2}F_{1}$ evaluations that lead to irrationality proofs, i.e. for which the $\delta$ mentioned above is positive. Many of them turned out to be (conjecturally) expressible in terms of algebraic numbers and/or logarithms of rational numbers, hence proving them irrational is not that exciting, but we have quite a few not-yet-identified (and inequivalent) cases. See the output file https://sites.math.rutgers.edu/~zeilberg/tokhniot/oGenBeukersLog1.txt , for many examples. Whenever Maple was able to (conjecturally) identify the constants explicitly, it is mentioned. If nothing is mentioned then these are potentially explicit constants, expressible as a hypergeometric series ${}_{2}F_{1}$, for which this would be the first irrationality proof, once the details are filled-in. We also considered the four-parameter family of constants $I^{\prime}_{1}(a,b,c,d):={\int_{0}^{1}\,{x^{a}(1-x)^{b}\over(1+cx)^{d+1}}\,dx\over\int_{0}^{1}\,{x^{a}(1-x)^{b}\over(1+cx)^{d}}\,dx}\quad,$ and, using the more general recurrence, also obtained using the Almkvist- Zeilberger algorithm (to see it type ‘OpeLg(a,b,c,d,n,Sn);’ in GenBeukersLog.txt), found many candidates for irrationality proofs that Maple was unable to identify. See the output file https://sites.math.rutgers.edu/~zeilberg/tokhniot/oGenBeukersLog2.txt . Generalizing the Beukers Integral for $\zeta(2)$ Define $I_{2}(a_{1},a_{2},b_{1},b_{2})(n)\,=\,{1\over B(1-a_{1},1-a_{2})B(1-b_{1},1-b_{2})}\cdot$ $\int_{0}^{1}\,\int_{0}^{1}\,{x^{-a_{1}}(1-x)^{-a_{2}}y^{-b_{1}}(1-y)^{-b_{2}}\over 1-xy}\cdot\left({x(1-x)y(1-y)\over 1-xy}\right)^{n}\,dx\,dy\quad,$ that happens to satisfy a linear-recurrence equation of second order, yielding Diophantine approximations to the constant $I_{2}(a_{1},a_{2},b_{1},b_{2})(0)$, let’s call it $C_{2}(a_{1},a_{2},b_{1},b_{2})$ $C_{2}(a_{1},a_{2},b_{1},b_{2})\,=\,{1\over B(1-a_{1},1-a_{2})B(1-b_{1},1-b_{2})}\cdot\int_{0}^{1}\,\int_{0}^{1}\,{x^{-a_{1}}(1-x)^{-a_{2}}y^{-b_{1}}(1-y)^{-b2}\over 1-xy}\,dx\,dy\quad.$ It is readily seen that $C_{2}(a_{1},a_{2},b_{1},b_{2})\,=\,{}_{3}F_{2}\left({{1\,,\,1-a_{1}\,,\,-b_{1}+1}\atop{2-a_{1}-a_{2}\,,\,2-b_{1}-b_{2}}}\,;1\,\right)\quad.$ Most choices of random $a_{1},a_{2},b_{1},b_{2}$ yield disappointing, negative $\delta$’s, just like $C_{2}({1\over 2},0,0,{1\over 2})$ (alias $8$ times the Catalan constant), but a systematic search yielded several hundred candidates that produce positive $\delta$’s and hence would produce irrationality proofs. Alas, many of them were conjecturally equivalent to each other via a fractional-linear transformation with integer coefficients, $C\rightarrow{a+bC\over c+dC}$, with $a,b,c,d$ integers, hence the facts that they are irrational are equivalent. Nevertheless we found quite a few that are (conjecturally) not equivalent to each other. Modulo filling-in some details, they lead to irrationality proofs. Amongst them some were (conjecturally) identified by Maple to be either algebraic, or logarithms of rational numbers, for which irrationality proofs exist for thousands of years (in case of $\sqrt{2}$ and $\sqrt{3}$ etc.), or a few hundred years (in case of $\log 2$, etc.). But some of them Maple was unable to identify, so potentially our (sketches) of proofs would be the first irrationality proofs. Beukers $\zeta(2)$ Tweaks That produced Irrationality Proofs with Identified Constants Denominator 2 We first searched for $C_{2}(a_{1},a_{2},b_{1},b_{2})$ where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $2$, there were quite a few of them, but they were all conjecturally equivalent to each other. Here is one of them: $\bullet$ $C_{2}(0,0,{1\over 2},0)={}_{3}F_{2}(1,1,1/2;2,3/2;1)$, alias $2\log 2$. Denominator 3 There were also quite a few where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $3$, but again they were all equivalent to each other, featuring $\pi\sqrt{3}$. Here is one of them. $\bullet$ $C_{2}(0,0,{1\over 3},-{2\over 3})={}_{3}F_{2}(1,1,2/3;2,7/3;1)$, alias (conjecturally) $-6+4\pi\sqrt{3}/3$. Denominator 4 There were also quite a few where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $4$, but again they were all equivalent to each other, featuring $\sqrt{2}$, yielding a new proof of the irrationality of $\sqrt{2}$ (for what it is worth). Here is one of them. $\bullet$ $C_{2}(-{3\over 4},-{3\over 4},-{1\over 4},-{3\over 4})={}_{3}F_{2}(1,7/4,5/4;7/2,3;1)$, alias (conjecturally) $-240\,+\,{512\over 3}\,\sqrt{2}$. Denominator 5 There were also quite a few where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $5$, but again they were all equivalent to each other, featuring $\sqrt{5}$, yielding a new proof of the irrationality of $\sqrt{5}$ (for what it is worth). Here is one of them. $\bullet$ $C_{2}(-{4\over 5},-{4\over 5},-{2\over 5},-{3\over 5})={}_{3}F_{2}(1,9/5,7/5;18/5,3;1)$, alias (conjecturally) $-{845\over 2}\,+\,{2275\over 12}\,\sqrt{5}$ Denominator 6 with identified constants We found two equivalence classes where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $6$, for which the constants were identified. Here are one from each class. $\bullet$ $C_{2}(-5/6,-5/6,-1/2,-1/2)={}_{3}F_{2}(1,11/6,3/2;11/3,3;1)$, alias (conjecturally) $-{{1344\over 5}}+{{16384\,\sqrt{3}\over 105}}$ $\bullet$ $C_{2}(-5/6,-5/6,-1/3,-2/3)={}_{3}F_{2}(1,11/6,4/3;11/3,3;1)$, alias (conjecturally) ${{972\,{2}^{2/3}\over 5}}-{{1536\over 5}}$ denominator 7 with identified constants We found two cases where the parameters $a_{1},a_{2},b_{1},b_{2}$ have denominator $7$, for which the constants were identified. $\bullet$ $C_{2}(-6/7,-6/7,-4/7,-3/7)={}_{3}F_{2}(1,13/7,11/7;26/7,3;1)$, alias (conjecturally) the positive root of $13824\,{x}^{3}-2757888\,{x}^{2}-10737789048\,x+16108505539=0$ . $\bullet$ $C_{2}(-6/7,-1/7,4/7,2/7)={}_{3}F_{2}(1,13/7,3/7;3,8/7;1)$, alias (conjecturally) the positive root of $2299968\,{x}^{3}+7074144\,{x}^{2}-11234916\,x-12663217=0$ Beukers $\zeta(2)$ Tweaks That produced Irrationality Proofs with Not-Yet- Identified Constants (and Hence Candidates for First Irrationality Proofs) For the following constants, Maple was unable to identify, and we have potentially the first irrationality proofs of these constants. Denominator 6 with not yet identified constants We found two cases (up to equivalence): $\bullet$ $C_{2}(0,-1/2,1/6,-1/2)={}_{3}F_{2}(1,1,5/6;5/2,7/3;1)$ While Maple was unable to identify this constant, Mathematica came up with $-24\,-\,{81\sqrt{\pi}\Gamma(7/3)\over\Gamma(-1/6)}$. $\bullet$ $C_{2}(-2/3,-1/2,1/2,-1/2)={}_{3}F_{2}(1,5/3,1/2;19/6,2;1)$ While Maple was unable to identify this constant, Mathematica came up with ${13\over 2}\,-\,{6\Gamma(19/6)\over\sqrt{\pi}\Gamma(8/3)}$. Denominator 7 with not yet identified constants We found six cases (up to equivalence): $\bullet$ $C_{2}(-6/7,-6/7,-4/7,-5/7)={}_{3}F_{2}(1,13/7,11/7;26/7,23/7;1)$ $\bullet$ $C_{2}(-6/7,-5/7,-3/7,-5/7)={}_{3}F_{2}(1,13/7,10/7;25/7,22/7;1)$ $\bullet$ $C_{2}(-6/7,-5/7,-2/7,-1/7)={}_{3}F_{2}(1,13/7,9/7;25/7,17/7;1)$ $\bullet$ $C_{2}(-6/7,-4/7,-1/7,-1/7)={}_{3}F_{2}(1,13/7,8/7;24/7,16/7;1)$ $\bullet$ $C_{2}(-6/7,-3/7,-5/7,-3/7)={}_{3}F_{2}(1,13/7,12/7;23/7,22/7;1)$ $\bullet$ $C_{2}(-5/7,-3/7,-4/7,-2/7)={}_{3}F_{2}(1,12/7,11/7;22/7,20/7;1)$ For each of them, to get the corresponding theorem and proof, use procedure TheoremZ2 in the Maple pacgage GenBeukersZeta2.txt. To get a statement and full proof (modulo a divisibility lemma) type , in GenBeukersZeta2.txt TheoremZ2(a1,a2,b1,b2,K,0): with K at least $2000$. For example, for the last constant in the above list ${}_{3}F_{2}(1,12/7,11/7;22/7,20/7;1)$, type TheoremZ2( -5/7, -3/7, -4/7, -2/7 ,3000,0): For more details (the recurrences, the estimated irrationality measures, the initial conditions) see the output file https://sites.math.rutgers.edu/~zeilberg/tokhniot/oGenBeukersZeta2g.txt . Generalizing the Beukers Integral for $\zeta(3)$ The natural extension would be the six-parameter family (but now we make the exponents positive) ${1\over B(1+a_{1},1+a_{2})B(1+b_{1},1+b_{2})B(1+c_{1},1+c_{2})}\cdot$ $\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{a_{1}}(1-x)^{a_{2}}y^{b_{1}}(1-y)^{b_{2}}z^{c_{1}}(1-z)^{c_{2}}\over 1-z+xyz}\cdot\left({x(1-x)y(1-y)z(1-z)\over 1-z+xyz}\right)^{n}\,dx\,dy\,dz\quad.$ However, for arbitrary $a_{1},a_{2},b_{1},b_{2},c_{1},c_{2}$ the recurrence is third order. (Wadim Zudilin pointed out that this may be related to the work of Rhin and Viola in [RV]). Also, empirically, we did not find many promising cases. Instead, let’s define $J_{3}(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2};e)(n)$ $\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{a_{1}}(1-x)^{a_{2}}y^{b_{1}}(1-y)^{b_{2}}z^{c_{1}}(1-z)^{c_{2}}\over(1-z+xyz)^{e}}\cdot\left({x(1-x)y(1-y)z(1-z)\over 1-z+xyz}\right)^{n}\,dx\,dy\,dz\quad.$ and $I_{3}(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2};e)(n):={J_{3}(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2};e+1)(n)\over J_{3}(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2};e)(0)}$ The family of constants that we hope to prove irrationality is the five- parameter: $I_{3}(a_{1},a_{2},b_{1},b_{2},c_{1},c_{2};e)(0)\quad.$ $={\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{a_{1}}(1-x)^{a_{2}}y^{b_{1}}(1-y)^{b_{2}}z^{c_{1}}(1-z)^{c_{2}}\over(1-z+xyz)^{e+1}}\,dx\,dy\,dz\over\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{a_{1}}(1-x)^{a_{2}}y^{b_{1}}(1-y)^{b_{2}}z^{c_{1}}(1-z)^{c_{2}}\over(1-z+xyz)^{e}}\,dx\,dy\,dz}\quad.$ Of course, for this more general, $7$-parameter, family, there is no second- order recurrence, but rather a third-order one. But to our delight, we found a five-parameter family, let’s call it $K(a,b,c,d,e)(n):=I_{3}(b,c,e,a,a,c,d)(n)\quad.$ Spelled-out, our five-parameter family of constants is $K(a,b,c,d,e)(0)=$ ${\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{b}(1-x)^{c}y^{e}(1-y)^{a}z^{a}(1-z)^{c}\over(1-z+xyz)^{d+1}}\,dx\,dy\,dz\over\int_{0}^{1}\,\int_{0}^{1}\,\int_{0}^{1}\,{x^{b}(1-x)^{c}y^{e}(1-y)^{a}z^{a}(1-z)^{c}\over(1-z+xyz)^{d}}\,dx\,dy\,dz}\quad.$ Now we found (see the section on finding recurrences below) a general second- order recurrence, that is too complicated to display here in full generality, but can be seen by typing OPEZ3(a,b,c,d,e,n,Sn); In the Maple package GenBeukersZeta3.txt. This enabled us, for each specific, numeric specialization of the parameters $a,b,c,d,e$ to quickly find the relevant recurrence, and systematically search for those that give positive $\delta$. Once again, many of them turned out to be (conjecturally) equivalent to each other. Denominator 2: We only found one class, up to equivalence, all related to $\log 2$. One of them is $K(0,0,0,1/2,1/2)=I_{3}(0,0,1/2,0,0,0,1/2)\quad,$ that is not that exciting since it is (conjecturally) equal to $-{{2-4\,\log\left(2\right)\over 3-4\,\log\left(2\right)}}$. For details, type TheoremZ3(0,0,0,1/2,1/2,3000,0); in GenBeukersZeta3.txt . Denominator 3: We found three inequivalent classes, none of them Maple was able to identify. $K(0,0,0,1/3,2/3)=I_{3}(0,0,2/3,0,0,0,1/3)\quad,$ for details, type TheoremZ3(0,0,0,1/3,2/3,3000,0); in GenBeukersZeta3.txt. $K(0,0,0,2/3,1/3)=I_{3}(0,0,1/3,0,0,0,2/3)\quad,$ for details, type TheoremZ3(0,0,0,2/3,1/3,3000,0); in GenBeukersZeta3.txt. $K(0,1/3,2/3,1/3,2/3)=I_{3}(0,0,1/3,0,0,0,2/3)\quad,$ for details, type TheoremZ3(0,1/3,2/3,1/3,2/3,3000,0); in GenBeukersZeta3.txt, These three constants are candidates for ‘first-ever-irrationality proof’. Denominator 4: We only found one family, all expressible in terms of $\log 2$. Here is one of them. For example $K(0,1/2,0,1/4,3/4)=I_{3}(1/2,0,3/4,0,0,0,1/4)\quad,$ that, conjecturally equals $-{{-30+45\,\log\left(2\right)\over-11+15\,\log\left(2\right)}}$. For details, type TheoremZ3(0,1/2,0,1/4,3/4,3000,0); in GenBeukersZeta3.txt. Denominator 5: We only found one family, up to equivalence, but Maple was unable to identify the constant. So it is potentially the first irrationality proof of that constant $K(0,1/5,0,3/5,2/5)=I_{3}(1/5,0,2/5,0,0,0,3/5)\quad.$ For details, type TheoremZ3(0,1/5,0,3/5,2/5,3000,0); in GenBeukersZeta3.txt. Denominator 6: We found three families, up to equivalence, none of which Maple was able to identify. Once again, these are candidates for first-ever irrationality proofs for these constants. $K(0,1/2,1/2,1/3,1/6)=I_{3}(1/2,1/2,1/6,0,0,1/2,1/3)\quad.$ For details, type TheoremZ3(0,1/2,1/2,1/3,1/6,3000,0); in GenBeukersZeta3.txt. $K(0,1/2,1/2,1/6,1/3)=I_{3}(1/2,1/2,1/3,0,0,1/2,1/6)\quad.$ For details, type TheoremZ3(0,1/2,1/2,1/6,1/3,3000,0); in GenBeukersZeta3.txt. $K(1/3,0,2/3,1/2,5/6)=I_{3}(0,2/3,5/6,1/3,1/3,2/3,1/2)\quad.$ For details, type TheoremZ3(1/3,0,2/3,1/2,5/6,3000,0); in GenBeukersZeta3.txt. Denominator 7: We found five families, up to equivalence, none of which Maple was able to identify. Once again, these are candidates for first-ever irrationality proofs for these constants. $K(1/7,0,2/7,3/7,4/7)=I_{3}(0,2/7,4/7,1/7,1/7,2/7,3/7)\quad.$ For details, type TheoremZ3(1/7,0,2/7,3/7,4/7,3000,0); in GenBeukersZeta3.txt. $K(1/7,0,2/7,5/7,3/7)=I_{3}(0,2/7,3/7,1/7,1/7,2/7,5/7)\quad.$ For details, type TheoremZ3(1/7,0,2/7,5/7,3/7,3000,0); in GenBeukersZeta3.txt. $K(1/7,0,3/7,4/7,5/7)=I_{3}(0,3/7,5/7,1/7,1/7,3/7,4/7)\quad.$ For details, type TheoremZ3(1/7,0,3/7,4/7,5/7,3000,0); in GenBeukersZeta3.txt. $K(1/7,0,4/7,2/7,5/7)=I_{3}(0,4/7,5/7,1/7,1/7,4/7,2/7)\quad.$ For details, type TheoremZ3(1/7,0,4/7,2/7,5/7,3000,0); in GenBeukersZeta3.txt. $K(2/7,0,3/7,4/7,5/7)=I_{3}(0,3/7,5/7,2/7,2/7,3/7,4/7)\quad.$ For details, type TheoremZ3(2/7,0,3/7,4/7,5/7,3000,0); in GenBeukersZeta3.txt. If you don’t have Maple, you can look at the output file https://sites.math.rutgers.edu/~zeilberg/tokhniot/oGenBeukersZeta3All.txt , that gives detailed sketches of irrationality proofs of all the above constants, some with conjectured integer-ating factors. Guessing an INTEGER-ating factor In the original Beukers cases the integer-ating factor was easy to conjecture, and even to prove. For $\zeta(2)$ it was $lcm(1\dots n)^{2}$, and for $\zeta(3)$ it was $lcm(1\dots n)^{3}$. For the Alladi-Robinson case of $\log 2$ it was even simpler, $lcm(1\dots n)$. But in other cases it is much more complicated. A natural ‘atomic’ object is, given a modulo M, a subset C of $\\{0,...,M-1\\}$, rational numbers $e_{1}$, $e_{2}$ between $0$ and $1$, rational numbers $e_{3},e_{4}$, the following quantity, for positive integers $n$ $Pp(e_{1},e_{2},e_{3},e_{4},C,M;n):=\prod_{p}p\quad,$ where $p$ ranges over all primes such that (let $\\{a\\}$ be the fractional part of $a$, i.e. $a-\lfloor a\rfloor$) $\bullet$ $e_{1}<\\{n/p\\}<e_{2}$ $\bullet$ $e_{3}<p/n<e_{4}$ $\bullet$ $p\,\,mod\,\,M\in C$ Using the prime number theorem, it follows (see e.g. [Zu2]) that $\lim_{n\rightarrow\infty}{\log Pp(e_{1},e_{2},e_{3},e_{4},C,M;n)\over n}\quad,$ can be evaluated exactly, in terms of the function $\Psi(x)={\Gamma^{\prime}(x)\over\Gamma(x)}$ (see procedure PpGlimit in the Maple packages) thereby giving an exact value for the quantity $\delta$ whose positivity implies irrationality. Of course, one still needs to rigorously prove that the conjectured integer- ating factor is indeed correct. Looking under the hood: On Recurrence Equations For ‘secrets from the kitchen’ on how we found the second-order, four- parameter recurrence operator OPEZ2(a1,a2,b1,b2,n,N) in the Maple package GenBeukersZeta2.txt, that was the engine driving the $\zeta(2)$ tweaks, and more impressively, the five-parameter second-order recurrence operator OPEZ3(a,b,c,d,e,n,N) in the Maple package GenBeukersZeta3.txt, that was the engine driving the $\zeta(3)$ tweaks, the reader is referred to the stand- alone appendix available from the following url: https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/beukersAppendix.pdf . Other Variations on Apéry’s theme Other attempts to use Apéry’s brilliant insight are [Ze2][Ze3][ZeZu1]. Recently Marc Chamberland and Armin Straub [CS] explored other fascinating aspects of the Apéry numbers, not related to irrationality. Conclusion and Future Work We believe that symbolic computational methods have great potential in irrationality proofs, in particular, and number theory in general. In this article we confined attention to approximating sequences that arise from second-order recurrences. The problem with higher order recurrences is that one gets linear combinations with rational coefficients of several constants, but if you can get two different such sequences coming from third-order recurrences, both featuring the same two constants, then the present method may be applicable. More generally if you have a $k$-th order recurrences, you need $k-1$ different integrals. The general methodology of this article can be called Combinatorial Number Theory, but not in the usual sense, but rather as an analog of Combinatorial Chemistry, where one tries out many potential chemical compounds, most of them useless, but since computers are so fast, we can afford to generate lots of cases and pick the wheat from the chaff. Encore: Hypergeometric challenges As a tangent, we (or rather Maple) discovered many exact ${}_{3}F_{2}(1)$ evaluations. Recall that the Zeilberger algorithm can prove hypergemoetric identities only if there is at least one free parameter. For a specific ${}_{3}F_{2}(a_{1}\,a_{2}\,a_{3}\,;b_{1}\,b_{2};1)$, with numeric parameters, it is useless. Of course, it is sometimes possible to introduce such a parameter in order to conjecture a general identity, valid for ‘infinitely’ many $n$, and then specialize $n$ to a specific value, but this remains an art rather than a science. The output file https://sites.math.rutgers.edu/~zeilberg/tokhniot/oGenBeukersZeta2f.txt contains many such conjectured evaluations, (very possibly many of them are equivalent via a hypergeometric transformation rule) and we challenge Wadim Zudilin, the birthday boy, or anyone else, to prove them. References [AR] Krishna Alladi and Michael L. Robinson, Legendre polynomials and irrationality, J. Reine Angew. Math. 318 (1980), 137-155. [AlZ] Gert Almkvist and Doron Zeilberger, The method of differentiating under the integral sign, J. Symbolic Computation 10, 571-591 (1990). https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/duis.html . [ApaZ] Moa Apagodu and Doron Zeilberger, Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger Theory , Adv. Appl. Math. 37 (2006)(Special Regev issue), 139-152. https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/multiZ.html . [Ape] Roger Apéry, “Interpolation de fractions continues et irrationalité de certaine constantes” Bulletin de la section des sciences du C.T.H.S. #3 p. 37-53, 1981. [B] Frits Beukers, A note on the irrationality of $\zeta(2)$ and $\zeta(3)$, Bull. London Math. Soc. 11 (1979), 268-272. [CS] Marc Chamberland and Armin Straub, Apéry limits: Experiments and Proofs, arxiv:2001.034400v1, 6 Nov 2020. https://arxiv.org/abs/2011.03400 . [H] Professor David Hilbert, Mathematical Problems [Lecture delivered before the International Congress of Mathematicians at Paris in 1900], translated by Dr. Mary Winston Newson, Bulletin of the American Mathematica Society 8 (1902), 437-479. https://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00881-8/S0273-0979-00-00881-8.pdf . [K] Christoph Koutschan, Advanced applications of the holonomic systems approach, PhD thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009. http://www.koutschan.de/publ/Koutschan09/thesisKoutschan.pdf, http://www.risc.jku.at/research/combinat/software/HolonomicFunctions/ . [N] Yu. V. Nesterenko, On Catalan’s constant, Proceedings of the Steklov Institute of Mathematics 292 (2016), 153-170. [vdP] Alf van der Poorten, A proof that Euler missed… Apéry’s proof of the irrationality of $\zeta(3)$, Math. Intelligencer 1 (1979), 195-203. [RV] Georges Rhin and Carlo Viola, The group structure of $\zeta(3)$, Acta Arithmetica, 97(2001), 269-293. [Ze1] Doron Zeilberger, A Holonomic systems approach to special functions identities, J. of Computational and Applied Math. 32, 321-368 (1990). https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/holonomic.html . [Ze2] Doron Zeilberger, Computerized deconstruction, Adv. Applied Math. 30 (2003), 633-654. https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/derrida.html . [Ze3] Doron Zeilberger, Searching for Apéry-style miracles [using, inter-alia, the amazing Almkvist-Zeilberger algorithm], Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/apery.html . [ZeZu1] Doron Zeilberger, and Wadim Zudilin, Automatic discovery of irrationality proofs and irrationality measures, International Journal of Number Theory , published on-line before print, volume and page tbd. Also to appear in a book dedicated to Bruce Berndt. https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/gat.html . [ZeZu2] Doron Zeilberger, and Wadim Zudilin, The irrationality measure of Pi is at most 7.103205334137…, Moscow J. of Combinatorics and Number Theory 9 (2020), 407-419. https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pimeas.html . [Zu1] Wadim Zudilin, Apéry-like difference equations for Catalan’s constant https://arxiv.org/abs/math/0201024 . [Zu2] Wadim Zudilin, Arithmetic of linear forms involving odd zeta values, J. Théorie Nombres Bordeaux 16 (2004), 251-291. https://arxiv.org/abs/math/0206176 . Robert Dougherty-Bliss, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. Email: Robert.w.Bliss at gmail dot com . Christoph Koutschan, Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Strasse 69, A-4040 Linz, Austria Email: christoph.koutschan at ricam dot oeaw dot ac dot at . Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA. Email: DoronZeil at gmail dot com .
# Chest X-ray lung and heart segmentation based on minimal training sets Balázs Maga Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, H-1117 Hungary, email<EMAIL_ADDRESS> ###### Abstract As the COVID-19 pandemic aggravated the excessive workload of doctors globally, the demand for computer aided methods in medical imaging analysis increased even further. Such tools can result in more robust diagnostic pipelines which are less prone to human errors. In our paper, we present a deep neural network to which we refer to as Attention BCDU-Net, and apply it to the task of lung and heart segmentation from chest X-ray (CXR) images, a basic but ardous step in the diagnostic pipeline, for instance for the detection of cardiomegaly. We show that the fine-tuned model exceeds previous state-of-the-art results, reaching $98.1\pm 0.1\%$ Dice score and $95.2\pm 0.1\%$ IoU score on the dataset of Japanese Society of Radiological Technology (JSRT). Besides that, we demonstrate the relative simplicity of the task by attaining surprisingly strong results with training sets of size 10 and 20: in terms of Dice score, $97.0\pm 0.8\%$ and $97.3\pm 0.5$, respectively, while in terms of IoU score, $92.2\pm 1.2\%$ and $93.3\pm 0.4\%$, respectively. To achieve these scores, we capitalize on the mixup augmentation technique, which yields a remarkable gain above $4\%$ IoU score in the size 10 setup. ## 1 Introduction All around the world, a plethora of radiographic examinations are performed day to day, producing images using different imaging modalities such as X-ray, computed tomography (CT), diagnostic ultrasound and magnetic resonance imaging (MRI). According to the publicly available, official data of the National Health Service ([13]), in the period from February 2017 to February 2018, the count of imaging activity was about 41 million only in England. The thorough examination of the vast quantity of these images imposes a huge workload on radiologists, which increases the number of the avoidable human mistakes. Consequently, automated methods aiding the diagnostic processes are sought- after. The examination of medical images customarily includes various segmentation tasks, in which detecting and pixelwise annotating different tissues and certain anomalies are vital. Common examples include lung nodule segmentation in the diagnosis of lung cancer, lung and heart segmentation in the diagnosis of cardiomegaly, or plaque segmentation in the diagnosis of thrombosis. Even in the case of 2-dimensional modalities, such segmentation tasks can be extremely time-demanding, and the situation gets even worse in three dimension. Taking into consideration that these tasks are easier to formalize as a standard computer vision exercise than the identification of a particular disease, it is not surprising that they sparked much activity in the field of automated medical imaging analysis. Semantic segmentation – that is assigning a pre-defined class to each pixel of an image – requires a high level of visual understanding, in which state-of- the-art performance is attained by methods utilizing Fully Convolutional Networks (FCN) [4]. An additional challenge of the field is posed by the strongly limited quantity of training data on which one train machine learning models, as annotating medical imaging requires specialists in contrast to “real-life” images. To overcome this difficulty, the so-called U-Net architecture was proposed: its capability to being efficiently trained on small datasets has been demonstrated in [5]. Over the past few years several modifications and improvements have been proposed on the original architecture, some of which involved different attention mechanisms designed to help the network to detect the important parts of the images. In the present paper we introduce a new network primarily based on the ideas of [12] and [8], to which we refer to as Attention BCDU-Net. We optimize its performance through hyperparameter tests on the depth of the architecture and the loss function, and we demonstrate the superiority of the resulting model compared to the state-of-the-art network presented in [15] in the task of lung and heart segmentation on chest X-rays. Besides that, we will also give an insight into two interesting phenomena arising during our research which might be interesting for the AI medical imaging community: one of them is the very small data requirement of this particular task, while the other one is the peculiar evolution of the loss curves over the training. ## 2 Deep learning approach ### 2.1 Related work As already mentioned in Section 1, [5] introducing U-Nets is of paramount importance in the field. Since then U-Nets have been used to cope with diverse medical segmentation tasks, and numerous papers aimed to design U-Net variants and mechanisms such that the resulting models tackles better the problem considered. Some of these paid primary attention to the structure of the encoder and the decoder – that is the downsampling and the upsampling path – of the original architecture. For example in [18], the authors developed a network (CoLe-CNN) with multiple decoder branches and Inception-v4 inspired encoder to achieve state-of-the-art results in 2-dimensional lung nodule segmentation. In [10] and [14], the authors introduced U-Net++, a network equipped with intermediate upsampling paths and additional convolutional layers, leading to essentially an efficient ensemble of U-Nets of varying depths, and demonstrated its superiority compared to the standard U-Net in many image domains. Other works put emphasis on the design of skip connections and the way the higher resolution semantic information joins the features coming through the upsampling branch. In [12], the authors proposed the architecture BCDU-Net, in which instead of the simple concatenation of the corresponding filters, the features of different levels are fused using a bidirectional ConvLSTM layer, which introduces nonlinearity into the model at this point and makes more precise segmentations available. In [8] it has been shown that for medical image analysis tasks the integration of so-called Attention Gates (AGs) improved the accuracy of the segmentation models, while preserving computational efficiency. In [15], this network was enhanced by a critic network in a GAN-like scheme following [9], and achieved state-of-the- art results in the task of lung and heart segmentation. Other attention mechanisms were introduced in [17] and in [16]. ### 2.2 Our proposal The network architecture Attention BCDU-Net we propose is a modification of the Attention U-Net, shown at Figure 1. Figure 1: Schematic architecture of Attention U-Net [8]. Figure 2: Schematic figure of the attention gate used in Attention U-Net [8], the tensor addition to alter is highlighted by an arrow. In [12], the authors demonstrated that it is beneficial to use bidirectional ConvLSTM layers to introduce nonlinearity in the step of merging semantic information gained through skip connections and the features arriving through the decoder. This inspired us to modify the attention gates (see Figure 2) in a similar manner, in which these pieces of information are merged via tensor addition, that is a linear operation as well. This addition is replaced by a bidirectional ConvLSTM layer, to which the output of $W_{g}$ and $W_{x}$ – the processed features and the semantic information, respectively – is fed in this order. We note that to our best knowledge, there is a slight ambiguity about the structure of the resampling steps in the attention gate: while the official implementation is in accordance with the figure, there are widely utilized implementations in which the output of $W_{g}$ is upsampled instead of downsampling the output of $W_{x}$ in order to fit their shape. We tested both solutions and did not experience a measurable difference in the performance. We also experimented with the usage of additional spatial and channel attention layers as proposed by [17], however, we found that it does not improve the performance of our model. The depth of the network is to be determined by hyperparameter testing. Our tests confirmed that four downsampling steps results in the best performance, however, the differences are minuscule. ### 2.3 Loss function A standard score to compare segmentations is the Intersection over Union (IoU): given two sets of pixels $X,Y$, their IoU is $IoU(X,Y)=\frac{\|X\cap Y\|}{\|X\cup Y\|}.$ In the field of medical imaging, Dice Score Coefficient (DSC) is probably the most widespread and simple way to measure the overlap ratio of the masks and the ground truth, and hence to compare and evaluate segmentations. It is a slight modification of IoU: given two sets of pixels $X,Y$, their DSC is $DSC(X,Y)=\frac{2\|X\cap Y\|}{\|X\|+\|Y\|}.$ If $Y$ is in fact the result of a test about which pixels are in $X$, we can rewrite it with the usual notation true/false positive (TP/FP), false negative (FN) to be $DSC(X,Y)=\frac{2TP}{2TP+FN+FP}.$ We would like to use this concept in our setup. The class $c$ we would like to segment corresponds to a set, but it is more appropriate to consider its indicator function $g$, that is $g_{i,c}\in\\{0,1\\}$ equals 1 if and only if the $i$th pixel belongs to the object. On the other hand, our prediction is a probability for each pixel denoted by $p_{i,c}\in[0,1]$. Then the Dice Score of the prediction in the spirit of the above description is defined to be $DSC=\frac{\sum_{i=1}^{N}p_{i,c}g_{i,c}+\varepsilon}{\sum_{i=1}^{N}\left(p_{i,c}+g_{i,c}\right)+\varepsilon},$ where $N$ is the total number of pixels, and $\varepsilon$ is introduced for the sake of numerical stability and to avoid divison by 0. The IoU of the prediction can be calculated in a similar manner. The linear Dice Loss (DL) of the multiclass prediction is then $DL=\sum_{c}\left(1-DSC_{c}\right).$ A deficiency of Dice Loss is that it penalizes false negative and false positive predictions equally, which results in high precision but low recall. For example practice shows that if the region of interests (ROI) are small, false negative pixels need to have a higher weight than false positive ones. Mathematically this obstacle is easily overcome by introducing weights $\alpha,\beta$ as tuneable parameters, resulting in the definition of Tversky similarity index [1]: $TI_{c}=\frac{\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,c}+\varepsilon}{\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,c}+\alpha\displaystyle\sum_{i=1}^{N}p_{i,\overline{c}}g_{i,c}+\beta\displaystyle\sum_{i=1}^{N}p_{i,c}g_{i,\overline{c}}+\varepsilon},$ where $p_{i,\overline{c}}=1-p_{i,c}$ and $g_{i,\overline{c}}=1-g_{i,c}$, that is the overline simply stands for describing the complement of the class. Tversky Loss is obtained from Tversky index as Dice Loss was obtained from Dice Score Coefficient: $TL=\sum_{c}\left(1-TI_{c}\right).$ Another issue with the Dice Loss is that it struggles to segment small ROIs as they do not contribute to the loss significantly. This difficulty was addressed in [11], where the authors introduced the quantity Focal Tversky Loss in order to improve the performance of their lesion segmentation model: $FTL=\sum_{c}\left(1-TI_{c}\right)^{\gamma^{-1}},$ where $\gamma\in[1,3]$. In practice, if a pixel with is misclassified with a high Tversky index, the Focal Tversky Loss is unaffected. However, if the Tversky index is small and the pixel is misclassified, the Focal Tversky Loss will decrease significantly. In our work we use multiclass DSC and IoU to evaluate segmentation performance. As our initial tests demonstrated that training our network with Focal Tversky loss results in better scores, we will use this loss function. The optimal $\alpha,\beta,\gamma$ parameters should be determined by extensive hyperparameter testing and grid search. We worked below with $\alpha=0.6,\beta=0.4,\frac{1}{\gamma}=0.675$. ### 2.4 Dataset and preprocessing For training- and validation data, we used the public Japanese Society of Radiological Technology (JSRT) dataset ([3]), available at [2]. The JSRT dataset contains a total of 247 images, all of them are in $2048\times 2048$ resolution, and have 12-bit grayscale levels. Both lung and heart segmentation masks are available for this dataset. In terms of preprocessing, similarly to [15], the images were resized to the resolution $512\times 512$ first. As X-rays are grayscale images with typically low contrast, which makes their analysis a difficult task. This obstacle might be overcome by using some sort of histogram equalization technique. The idea of standard histogram equalization is spreading out the the most frequent intensity values to a higher range of the intensity domain by modifying the intensities so that their cumulative distribution function (CDF) on the complete modified image is as close to the CDF of the uniform distribution as possible. Improvements might be made by using adaptive histogram equalization, in which the above method is not utilized globally, but separately on pieces of the image, in order to enhance local contrasts. However, this technique might overamplify noise in near-constant regions, hence our choice was to use Contrast Limited Adaptive Histogram Equalization (CLAHE), which counteracts this effect by clipping the histogram at a predefined value before calculating the CDF, and redistribute this part of the image equally among all the histogram bins. ### 2.5 Data augmentation Concerning data augmentation, we follow [7], in which the method mixup was used to improve glioma segmentation on brain MRI’s. This slightly counter- intuitive augmentation technique was introduced by [6]: training data samples are obtained by taking random convex combinations of original image-mask pairs. That is, for $(x_{1},y_{1})$ and $(x_{2},y_{2})$ image-mask pairs, we create a random mixed up pair $x=\lambda x_{1}+(1-\lambda)x_{2}$, $y=\lambda y_{1}+(1-\lambda)y_{2}$, where $\lambda$ is chosen from the beta distribution $B(\delta,\delta)$ for some $\delta\in(0,\infty)$. In each epoch, the original samples are paired randomly, hence during the course of the training, a multitude of training samples are fed to the network. (From the mathematical point of view, as the coefficient $\lambda$ is chosen independently in each case from a continuous probability distribution, the network will encounter pairwise distinct mixed up training samples with probability 1, modulo floating point inaccuracy.) In [6], the authors argue that generating training samples via this method encourages the network to behave linearly in-between training examples, which reduces the amount of undesirable oscillations when predicting outside the training examples. The choice of $\delta$ should be determined by hyperparameter testing for any network and task considered. In [6], $\delta\in[0.1,0.4]$ is proposed, while in [7] $\delta=0.4$ is applied. ## 3 Experiments ### 3.1 Training schedule In our main tests, the JSRT dataset was randomly split so that 85% of it was used for training and the rest for validation and testing. This split was carried out independently in each case, enhancing the robustness of our results. Besides that, we also experimented with small dataset training, in which rather modest sets of 10 and 20 X-rays was utilized as training set. (The test set remained the same.) It enabled us to measure the benefits of mixup more transparently. In each of these cases, we trained our network with Adam optimizer: in the former case, for 50 epochs, while in the latter cases for 1000 and 500 epochs, respectively. As these epoch numbers are approximately inversely proportional to the size of the training sets, these choices correspond to each other in terms of training steps. ### 3.2 Results Table 1 summarizes the numerical results we obtained during the testing of Attention BCDU-Net with different train sizes and choices of $\delta$, while Figure 3-5 display visual results. Note that the highest DSC scores slightly exceed the ones attained by the state-of-the-art, adversarially enhanced Attention U-Net introduced in [15] ($97.6\pm 0.5\%$) and admit higher stability. The effect of augmentation is the most striking in the case of training on an X-ray set of size 10, when the choice $\delta=0.2$ results in a 5% increase of IoU compared to the no mixup case. In general, we found this case particularly interesting: it was surprising that we could achieve IoU and DSC scores of this magnitude using such a small training set. Nevertheless the predictions have some imperfections, displayed by Figure 3: the contours of the segmentation are less clear and both the heart and the lung segmentation tend to contain small spots far from the ground truth. However, such conspicuous faults are unlikely to occur in the case of the best models for 20 train X-rays (Figure 4), which is still remarkable. The sufficiency of such small training sets is probably due to the relative simplicity of the task. Notably, lung and heart regions admit large similarity across a set of chest X-rays, and they are strongly correlated with simple intensity thresholds. Consequently, even small datasets have high representing potential. We note that as $\delta$ gets smaller, the probability density function of $B(\delta,\delta)$ gets more strongly skewed towards the endpoints of the interval $[0,1]$, which results in mixed up samples being closer to original samples in general. The perceived optimality of $\delta=0.2$ in the small dataset cases show that a considerable augmentation is beneficial and desirable, yet it is unadvised to use too wildly modified samples. The benevolent effect of mixup gets more obscure as we increase the size of the training set. Notably, the results of different augmentation setups are almost indistinguishable from each other. We interpret this phenomena as another consequence of the similarity of masks from different samples, which inherently drives the network towards simpler representations in the case of a sufficiently broad training set, even without using mixup. We also note that in the case of 10 training samples, while the IoU differences between the no mixup and the mixup regime are striking, the gain in DSC is less remarkable. It hints that it is unadvised to rely merely on DSC when evaluating and comparing segmentation models. Figure 3: Ground truth (left) compared to the prediction of the Attention BCDU-Net (right), train size: 10. Figure 4: Ground truth (left) compared to the prediction of the Attention BCDU-Net (right), train size: 20. Figure 5: Ground truth (left) compared to the prediction of the Attention BCDU-Net (right), train size: complete. \| Train size: 10 \| Train set: 20 \| Train size: 209 (Complete) ---\|---\|---\|--- \| IoU \| DSC \| IoU \| DSC \| IoU \| DSC No mixup \| $87.2\pm 1.9$% \| $94.8\pm 1.1$% \| $91.9\pm 0.6$% \| $96.9\pm 0.5$% \| $94.9\pm 0.2$% \| $98.0\pm 0.1$% $\delta=0.1$ \| $91.9\pm 1.3$% \| $96.8\pm 0.9$% \| $92.5\pm 0.5$% \| $97.1\pm 0.5$% \| $95.2\pm 0.2$% \| $98.1\pm 0.1$% $\delta=0.2$ \| $92.2\pm 1.2$% \| $97.0\pm 0.8$% \| $93.3\pm 0.4$% \| $97.3\pm 0.5$% \| $95.0\pm 0.1$% \| $98.0\pm 0.1$% $\delta=0.3$ \| $91.3\pm 1.2$% \| $96.5\pm 1.0$% \| $92.9\pm 0.5$% \| $97.2\pm 0.5$% \| $94.9\pm 0.1$% \| $98.0\pm 0.1$% $\delta=0.4$ \| $91.3\pm 1.4$% \| $96.4\pm 1.0$% \| $93.0\pm 0.5$% \| $97.2\pm 0.4$% \| $94.8\pm 0.1$% \| $97.9\pm 0.1$% Table 1: Dice scores and IoU scores of Attention BCDU-Net with different mixup parameters We would also like to draw attention to the peculiar loss curves we primarily encountered during the small dataset trainings, as displayed in Figure 6. Notably, the curve of the validation DSC flattens far below the also flattening curve of the train DSC, strongly inciting the usage of early stopping. (Train DSC reaches essentially 1 in fact, which is unsurprising with such a small training set.) However, in the later stages the validation DSC catches up, even though the train DSC does not have any room for further improvement. We were especially puzzled by this behaviour in the 10-sized training setup, in which both the train and validation DSC seems completely stabilized after from epoch 50 to epoch 400, yet validation DSC skyrockets in the later stages in a very short amount of time. The same behaviour was experienced during each test run. We have yet to give the intuitive or theoretical explanation for this phenomenon that how the generalizing ability of the model can improve further when it seems to be in a perfect state from the training perspective. We note that these observations naturally led us to experiment with even longer trainings, but to no avail. Figure 6: From left to right: the evolution of the train DSC (blue) and the validation DSC (orange) with 10 training samples, 20 training samples, and the complete training dataset, respectively. The IoU curves admit similar patterns. ## 4 Conclusion In the present work, we addressed the problem of automated lung and heart segmentation on chest X-rays. We introduced a new model, Attention BCDU-Net, a variant of Attention U-Net equipped with modified attention gates, and surpassed previous state-of-the-art results. We also demonstrated its ability to attain surprisingly reasonable results with strongly limited training sets. Performance in these cases was enhanced using the mixup augmentation technique, resulting in highly notable contribution in the IoU score. Concerning future work, a natural extension of this work would be adding a structure correcting adversarial network to the training scheme, similarly to [9] and [15], and measuring its effect on the performance, especially in the setup of limited training sets. We would also like to give some kind of explanation to the phenomenon of peculiar loss curves. ## Acknowledgements The project was supported by the grant EFOP-3.6.3-VEKOP-16-2017-00002. ## References * [1] Amos Tversky “Features of similarity.” In _Psychological review_ 84.4 American Psychological Association, 1977, pp. 327 * [2] , http://db.jsrt.or.jp/eng.php, 2000 * [3] Junji Shiraishi et al. “Development of a digital image database for chest radiographs with and without a lung nodule: receiver operating characteristic analysis of radiologists’ detection of pulmonary nodules” In _American Journal of Roentgenology_ 174.1 Am Roentgen Ray Soc, 2000, pp. 71–74 * [4] Jonathan Long, Evan Shelhamer and Trevor Darrell “Fully convolutional networks for semantic segmentation” In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2015, pp. 3431–3440 * [5] Olaf Ronneberger, Philipp Fischer and Thomas Brox “U-Net: Convolutional Networks for Biomedical Image Segmentation” In _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III_ Springer International Publishing, 2015, pp. 234–241 DOI: 10.1007/978-3-319-24574-4˙28 * [6] Hongyi Zhang, Moustapha Cisse, Yann Dauphin and David Lopez-Paz “mixup: Beyond Empirical Risk Minimization”, 2017 * [7] Zach Eaton-Rosen, Felix J. 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Partitions of an Integer into Powers Matthieu Latapy liafa, Université Paris 7, 2 place Jussieu, 75005 Paris. <EMAIL_ADDRESS> ###### Abstract In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer. ## 1 Introduction We study here the problem of writing a non-negative integer $n$ as the sum of powers of another positive integer $b$: $n=p_{0}b^{0}+p_{1}b^{1}+\dots+p_{k-1}b^{k-1}$ with $p_{k-1}\not=0$ and $p_{i}\in\mathbb{N}$ for all $i$. Following [Rod69], we call the $k$-tuple $(p_{0},p_{1},\dots,p_{k-1})$ a _$b$ -ary partition_ of $n$. The integers $p_{i}$ are called the _parts_ of the partition and $k$ is the _length_ of the partition. A $b$-ary partition of $n$ can be viewed as a representation of $n$ in the basis $b$, with digits in $\mathbb{N}$. Conversely, given a $k$-tuple $(p_{0},\dots,p_{k-1})$ and a basis $b$, we will denote by $v_{b}(p_{0},\dots,p_{k-1})$ the integer $p_{0}b^{0}+p_{1}b^{1}+\dots+p_{k-1}b^{k-1}$. There is a unique $b$-ary partition such that $p_{i}<b$ for all $i$, and it is the usual (canonical) representation of $n$ in the basis $b$. Here, we consider the problem without any restriction over the parts: $p_{i}\in\mathbb{N}$, which is actually equivalent to say that $p_{i}\in\\{0,1,\dots,n\\}$ for all $i$. We will mainly be concerned with the enumeration and counting of the $b$-ary partitions of $n$, for given integers $n$ and $b$. This natural combinatorial problem has been introduced by Mahler [Mah40], who showed that the logarithm of the number of $b$-ary partitions of $n$ grows as $\frac{(\log n)^{2}}{2\log b}$. This asymptotic approximation was later improved by de Bruijn [dB48] and Pennington [Pen53]. Knuth [Knu66] studied the special case where $b=2$. In this case, the function counting the $b$-ary partitions for a given $n$ is called the _binary partition function_. This function has been widely studied. Euler and Tanturri [Eul50, Tan18a, Tan18b] studied its exact computation and Churchhouse [Chu69, Chu71] studied its congruence properties, while Fröberg [Fro77] gave a final solution to its asymptotical approximation. Later, Rödseth [Rod69] generalized some of these results to $b$-ary partitions for any $b$. Finally, Pfaltz [Pfa95] studied the subcase of the binary partitions of integers which are powers of two. We are concerned here with the exact computation of the number of $b$-ary partitions of a given integer $n$, for any $b$. We will use a powerful technique we developped in [LP99] and [LMMP98]: incremental construction of the set of $b$-ary partitions of $n$, infinite extension and coding by an infinite tree. This method gives a deep understanding of the structure of the set of $b$-ary partitions of $n$. We will obtain this way a tree structure which permits the enumeration of all the $b$-ary partitions of $n$ in linear time with respect to their number. We will also order these partitions in a natural way which gives the distributive lattice structure to this set. We recall that a lattice is a partially ordered set such that any two elements $a$ and $b$ have a least upper bound (called supremum of $a$ and $b$ and denoted by $a\vee b$) and a greatest lower bound (called infimum of $a$ and $b$ and denoted by $a\wedge b$). The element $a\vee b$ is the smallest element among the elements greater than both $a$ and $b$. The element $a\wedge b$ is defined dually. A lattice is _distributive_ if for all $a$, $b$ and $c$: $(a\vee b)\wedge(a\vee c)=a\vee(b\wedge c)$ and $(a\wedge b)\vee(a\wedge c)=a\wedge(b\vee c)$. A distributive lattice is a strongly structured set, and many general results, for example efficient coding and algorithms, are known about such sets. For more details, see for example [DP90]. Notice that if we consider $b=1$ and restrict the problem to partitions of length at most $n$, then we obtain the _compositions_ of $n$, i.e. the series of at most $n$ integers, the sum of which equals $n$. Many studies already deal with this special case. In particular, the (infinite) distributive lattice $R_{1}(\infty)$ which we will introduce in Section 4 is isomorphic to the well known Young lattice [Ber71]. Therefore, we will suppose $b>1$ in the following. Notice however that some of the results we present here are already known in this special case (for example the distributive lattice structure), therefore they can be seen as an extension of the existing ones. ## 2 The lattice structure In this section, we define a simple dynamical model which generates _all_ the $b$-ary partitions of an integer. We will show that the set of $b$-ary partitions, ordered by the reflexive and transitive closure of the successor relation, has the distributive lattice structure. Let us consider a $b$-ary partition $p=(p_{0},p_{1},\dots,p_{k-1})$ of $n$, and let us define the following transition (or rewriting) rule: $p\stackrel{{\scriptstyle i}}{{\longrightarrow}}q$ if and only if for all $j\not\in\\{i,i+1\\}$, $q_{j}=p_{j}$, $p_{i}\geq b$, $q_{i}=p_{i}-b$ and $q_{i+1}=p_{i+1}+1$ (with the assumption that $p_{k}=0$). In other words, if $p_{i}$ is at least equal to $b$ then $q$ is obtained from $p$ by removing $b$ units from $p_{i}$ and adding one unit to $p_{i+1}$. We call this operation _firing $i$_. The important point is to notice that $q$ is then a $b$-ary partition of $n$. We call $q$ a _successor_ 111Notice that the term _successor_ can have many different meanings. We follow here the standard usage in discrete dynamical models, but in order theory the term has another meaning, and one may also consider that a _successor_ of an integer $n$ should be the integer $n+1$, which is not the case here. of $p$, and we denote by $Succ_{b}(p)$ the set of all the successors of $p$, with respect to the rule. We denote by $R_{b}(n)$ the set of $b$-ary partitions of $n$ reachable from $(n)$ by iterating the evolution rule, ordered by the reflexive and transitive closure of the successor relation. Notice that the successor relation is the covering relation of the order, since it is defined as the transitive and reflexive closure of the successor relation, and one can easily verify that this relation has no reflexive ($x\longrightarrow x$) and no transitive ($x\longrightarrow z$ with $x\longrightarrow y$ and $y\longrightarrow z$) edge. See Figure 1 for some examples. Figure 1: From left to right, the sets $R_{2}(9)$, $R_{3}(9)$, $R_{3}(10)$, $R_{3}(11)$, $R_{3}(12)$ and $R_{3}(15)$. From Theorem 1, both of these sets is a distributive lattice. Given a sequence $f$ of firings, we denote by $\|f\|_{i}$ the number of firings of $i$ during $f$. Now, consider an element $p$ of $R_{b}(n)$, and two sequences $f$ and $f^{\prime}$ of firings which transform $(n)$ into $p$. Then, $p_{i}=\|f\|_{i-1}-b\cdot\|f\|_{i}=\|f^{\prime}\|_{i-1}-b\cdot\|f^{\prime}\|_{i}$. Suppose that there exists an integer $i$ such that $\|f\|_{i}\not=\|f^{\prime}\|_{i}$, and let $i$ be the smallest such integer. Then, $\|f\|_{i-1}=\|f^{\prime}\|_{i-1}$ and the equality $\|f\|_{i-1}-b\cdot\|f\|_{i}=\|f^{\prime}\|_{i-1}-b\cdot\|f^{\prime}\|_{i}$ is impossible. Therefore, we have $\|f\|_{i}=\|f^{\prime}\|_{i}$ for all $i$. This leads to the definition of the _shot vector_ $s(p)$: $s(p)_{i}$ is the number of times one have to fire $i$ in order to obtain $p$ from $(n)$. Now we can prove: ###### Lemma 1 For all $p$ and $q$ in $R_{b}(n)$, $p\leq q$ if and only if for all $i$, $s(p)_{i}\geq s(q)_{i}$. Proof : If $p\leq q$, i.e. $p$ is reachable from $q$ then it is clear that for all $i$, $s(p)_{i}\geq s(q)_{i}$. Conversely, if there exists $i$ such that $s(p)_{i}>s(q)_{i}$, then let $j$ be the smallest such integer. Therefore, $q_{j}>p_{j}+b$ and so $q$ can be fired at $j$. By iterating this process, we finally obtain $p$, and so $p\leq q$. ###### Theorem 1 For all integers $b$ and $n$, the order $R_{b}(n)$ is a _distributive lattice_ which contains _all_ the $b$-ary partitions of $n$, with the infimum and supremum of any two elements $p$ and $q$ defined by: $s(p\vee q)_{i}=\min(s(p)_{i},s(q)_{i})\mbox{ and }s(p\wedge q)_{i}=\max(s(p)_{i},s(q)_{i}).$ Proof : We first show that $R_{b}(n)$ contains all the $b$-ary partitions of $n$. Consider $p$ a $b$-ary partition of $n$. If $p=(n)$, then $p\in R_{b}(n)$, so we suppose that $p\not=(n)$. Therefore, there must be an integer $i>0$ such that $p_{i}>0$. Let us define $q$ such that $q_{j}=p_{j}$ for all $j\not\in\\{i-1,i\\}$, $q_{i-1}=p_{i-1}+b$ and $q_{i}=p_{i}-1$. It is clear that $q$ is a $b$-ary partition of $n$, and that if $q\in R_{b}(n)$ then $p\in R_{b}(n)$ since $q\stackrel{{\scriptstyle i-1}}{{\longrightarrow}}p$. It is also obvious that, if we iterate this process, we go back to $(n)$, and so $p\in R_{b}(n)$. We now prove the formula for the infimum and the supremum. Let $p$ and $q$ be in $R_{b}(n)$, and $r$ such that $s(r)_{i}=\min(s(p)_{i},s(q)_{i})$. From Lemma 1, $p$ and $q$ are reachable from $r$. Moreover, if $p$ and $q$ are reachable from $t\in R_{b}(n)$, then, from Lemma 1, $r$ is reachable from $t$ since we must have $s(t)_{i}\leq\min(s(p)_{i},s(q)_{i})$ (else one can not transform $t$ into $p$ or $q$). Therefore, $r$ is the supremum of $p$ and $q$, as claimed in the theorem. The argument for the infimum is symmetric. Finally, to prove that the lattice is _distributive_ , we only have to check that the formulae satisfy the distributivity laws. We will now show that the dynamical model defined here can be viewed as a special Chip Firing Game (CFG). A CFG [BLS91, BL92] is defined over a directed multigraph. A configuration of the game is a repartition of a number of chips over the vertices of the graph, and it obeys the following evolution rule: if a vertex $\nu$ contains as many chips as its outgoing degree $d$, then one can transfer one chip along each of its outgoing edges. In other words, the number of chips at $\nu$ is decreased by $d$ and, for each vertex $v\not=\nu$, the number of chips at $v$ is increased by the number of edges from $\nu$ to $v$. This model is very general and has been introduced in various contexts, such as physics, computer science, economics, and others. It is in particular very close to the famous Abelian Sandpile Model [LP00]. It is known that the set of reachable configurations of such a game, ordered with the reflexive and transitive closure of the transition rule, is a Lower Locally Distributive (LLD) lattice (see [Mon90] for a definition and properties), but it is not distributive in general [BL92, LP00, MPV01]. However, if a lattice is LLD and its dual, i.e. the lattice obtained by reversing the order relation, also is LLD, then the lattice is distributive. Therefore, we can give another proof of the fact that $R_{b}(n)$ is a distributive lattice by showing that it is the set of reachable configurations of a CFG, and that its dual too 222This idea is due to Clémence Magnien, who introduced this new way to prove that a set is a distributive lattice using two Chip Firing Games.. Given two integers $n$ and $b$, let us consider the following multigraph $G=(V,E)$ defined by: $V=\\{0,\dots,n\\}$ and there are $b^{i+1}$ edges from the $i$-th vertex to the $(i+1)$-th, for all $n<i\leq 0$. Now, let us consider the CFG $C$ defined over $G$ by the initial configuration where the vertex $0$ contains $n$ chips, the other ones being empty. Now, given a configuration $c$ of the CFG, where $c_{i}$ denotes the number of chips in the vertex number $i$, let us denote by $\bar{c}$ the vector such that $\bar{c}_{i}=\frac{c_{i}}{b^{i}}$. Then, if the CFG is in the configuration $c$, an application of the rule to the vertex number $i$ gives the configuration $c^{\prime}$ such that $c^{\prime}_{i}=c_{i}-b^{i+1}$, $c^{\prime}_{i+1}=c_{i+1}+b^{i+1}$ and $c^{\prime}_{j}=c_{j}$ for all $j\not\in\\{i,i+1\\}$. Notice that this means exactly that $\bar{c}_{i}$ is decreased by $b$ and that $\bar{c}_{i+1}$ is increased by $1$, therefore an application of the CFG rule corresponds exactly to an application of the evolution rule we defined above, and so the set of reachable configurations of the CFG is isomorphic to $R_{b}(n)$. This leads to the fact that $R_{b}(n)$ is a LLD lattice. Conversely, let $G^{\prime}$ be the multigraph obtained from $G$ by reversing each edge, and let us consider the CFG $C^{\prime}$ over $G^{\prime}$ such that the initial configuration of $C^{\prime}$ is the final configuration of $C$. Then it is clear that the set of reachable configurations of $C^{\prime}$ is nothing but the dual of the one of $C$, therefore it is isomorphic to the dual of $R_{b}(n)$. This leads to the fact that the dual of $R_{b}(n)$ is a LLD lattice, which allows us to conclude that $R_{b}(n)$ is a distributive lattice. ## 3 From $R_{b}(n)$ to $R_{b}(n+1)$ In this section, we give a method to construct the transitive reduction (i.e. the successor relation) of $R_{b}(n+1)$ from the one of $R_{b}(n)$. In the following, we will simply call this the _construction of $R_{b}(n+1)$ from $R_{b}(n)$_. This will show the self-similarity of these sets, and give a new way, purely structural, to obtain a recursive formula for $\|R_{b}(n)\|$, which is previously known from [Rod69] (the special case where $b=2$ is due to Euler [Eul50]). This construction will also show the special role played by certain $b$-ary partitions, which will be widely used in the rest of the paper. Therefore, we introduce a few notations about them. We denote by $P_{i}(b,n)$ the set of the partitions $p$ in $R_{b}(n)$ such that $p_{0}=p_{1}=\dots=p_{i-1}=b-1$. Notice that for all $i$ we have $P_{i}(b,n)\subseteq P_{i+1}(b,n)$ and that $P_{0}(b,n)=R_{b}(n)$. If $p=(p_{0},\dots,p_{k-1})$ is in $P_{i}(b,n)$, we denote by $p^{\hookrightarrow_{i}}$ the $k$-uple $(0,\dots,0,p_{i}+1,p_{i+1},\dots,p_{k-1})$. In other words, $p^{\hookrightarrow_{i}}$ is obtained from $p$ by switching all the $i$ first components of $p$ from $b-1$ to $0$ and adding one unit to its $i$-th componend 333This operator is known in numeration studies as an odometer. See [PJG95] for more precisions.. Notice that the $k$-uple $p^{\hookrightarrow_{0}}$, which is simply obtained from $p$ by adding one unit to its first component, is always a $b$-ary partition of $n+1$. If $S$ is a subset of $P_{i}(b,n)$, we denote by $S^{\hookrightarrow_{i}}$ the set $\\{p^{\hookrightarrow_{i}}\|\ p\in S\\}$. Notice that, if $p\stackrel{{\scriptstyle i}}{{\longrightarrow}}q$ in $R_{b}(n)$, then $p^{\hookrightarrow_{0}}\stackrel{{\scriptstyle i}}{{\longrightarrow}}q^{\hookrightarrow_{0}}$ in $R_{b}(n+1)$. This remark makes it possible to construct $R_{b}(n+1)$ from $R_{b}(n)$: the construction procedure starts with the lattice $R_{b}(n)^{\hookrightarrow_{0}}$ given by its diagram. Then, we look for those elements in $R_{b}(n)^{\hookrightarrow_{0}}$ that have a successor out of $R_{b}(n)^{\hookrightarrow_{0}}$. The set of these elements will be denoted by $I_{0}$, with $I_{0}\subseteq R_{b}(n)^{\hookrightarrow_{0}}$. At this point, we add all the missing successors of the elements of $I_{0}$. The set of these new elements will be denoted by $C_{0}$. Now, we look for the elements in $C_{0}$ that have a successor out of the constructed set. The set of these elements is denoted by $I_{1}$. More generally, at the $i$-th step of the procedure we look for the elements in $C_{i-1}$ with missing successors and call $I_{i}$ the set of these elements. We add the new successors of the elements of $I_{i}$ and call the set of these new elements $C_{i}$. At each step, when we add a new element, we also add its covering relations. Since $R_{b}(n+1)$ is a finite set, this procedure terminates. At the end, we obtain the whole set $R_{b}(n+1)$. In the rest of this section, we study more precisely this construction process. ###### Lemma 2 Let $p$ be a $b$-ary partition in $P_{i}(b,n)$. If $p_{i}\not=b-1$ then $Succ_{b}(p^{\hookrightarrow_{i}})={Succ_{b}(p)}^{\hookrightarrow_{i}}$. Else, $Succ_{b}(p^{\hookrightarrow_{i}})={Succ_{b}(p)}^{\hookrightarrow_{i}}\cup\\{p^{\hookrightarrow_{i+1}}\\}$. Proof : If a transition $p\stackrel{{\scriptstyle j}}{{\longrightarrow}}q$ is possible, then $p^{\hookrightarrow_{i}}\stackrel{{\scriptstyle j}}{{\longrightarrow}}q^{\hookrightarrow_{i}}$ is obviously possible. Moreover, an additional transition is possible from $p^{\hookrightarrow_{i}}$ if and only if $p_{i}=b-1$. In this case, $p^{\hookrightarrow_{i}}\stackrel{{\scriptstyle i}}{{\longrightarrow}}p^{\hookrightarrow_{i+1}}$. ###### Lemma 3 For all integer $b$, $n$ and $i$, we define the function $r_{i}:P_{i}(b,n)\rightarrow R_{b}(\frac{n+1}{b^{i}}-1)$ by: $r_{i}(p)$ is obtained from $p\in P_{i}(b,n)$ by removing its $i$ first components (which are equal to $b-1$). Then, $r_{i}$ is a bijection. Proof : Let us consider $p$ in $P_{i}(b,n)$: $p=(b-1,b-1,\dots,b-1,p_{i},\dots,p_{k})$. Then, it is clear that $r_{i}(p)=(p_{i},\dots,p_{k})$ is in $R_{b}(\frac{n-(b-1)-(b-1)b-\dots-(b-1)b^{i-1}}{b^{i}})=R_{b}(\frac{n+1-b^{i}}{b^{i}})=R_{b}(\frac{n+1}{b^{i+1}}-1)$. Conversely, if we consider $p$ in $R_{b}(\frac{n+1}{b^{i}}-1)$, then $r_{i}^{-1}(p)=(b-1,b-1,\dots,b-1,p_{0},p_{1},\dots,p_{k})$ is a $b$-ary partition of $m=(b-1)+(b-1)b+\dots+(b-1)b^{i-1}+\frac{n+1-b^{i}}{b^{i}}$, which is nothing but $n$. Therefore, $r_{i}^{-1}(p)$ is in $R_{b}(n)$. ###### Lemma 4 For all integer $b$, $n$ and $i$, we have $I_{i}=P_{i+1}(b,n)^{\hookrightarrow_{i}}$ and $C_{i}=P_{i+1}(b,n)^{\hookrightarrow_{{i+1}}}$. Proof : By induction over $i$. For $i=0$, it is clear from Lemma 2 that the set of elements in $R_{b}(n)^{\hookrightarrow_{0}}$ with a missing successor, namely $I_{0}$, is exactly $P_{1}(b,n)^{\hookrightarrow_{0}}$. Moreover, the set of these missing successors, namely $C_{0}$, is clearly $P_{1}(b,n)^{\hookrightarrow_{1}}$. Now, let us suppose that the claim is proved for $i$ and let us prove it for $i+1$. The set $I_{i+1}$ is the set of elements in $C_{i}$ with one missing successor. By induction hypothesis, we have $C_{i}=P_{i+1}(b,n)^{\hookrightarrow_{{i+1}}}$ and so, from Lemma 2, $I_{i+1}=P_{i+2}(b,n)^{\hookrightarrow_{{i+1}}}$. Then, by application of the evolution rule, it is clear that the set $C_{i+1}$ of the missing successor is $P_{i+2}(b,n)^{\hookrightarrow_{{i+2}}}$, which proves the claim. ###### Theorem 2 For any positive integer $b$ and $n$, we have: $R_{b}(n)=\bigsqcup_{i\geq 0}r_{i}^{-1}\left(R_{b}\left(\frac{n}{b^{i}}-1\right)\right)^{\hookrightarrow_{i}}$ $\|R_{b}(n)\|=\sum_{i=0}^{\lfloor n/b\rfloor}\begin{tabular}[]{\|c\|}$R_{b}(\frac{i}{b})$\end{tabular}$ where $\bigsqcup$ denotes the disjoint union, where $R_{b}(n)$ is taken as $\emptyset$ when $n$ is not a positive integer, and with $R_{b}(0)=\\{0\\}$. Proof : From the construction procedure described above, we have $R_{b}(n)=R_{b}(n-1)^{\hookrightarrow_{0}}\sqcup\bigsqcup_{i\geq 0}C_{i}$. From Lemma 4, we obtain $R_{b}(n)=R_{b}(n-1)^{\hookrightarrow_{0}}\sqcup\bigsqcup_{i\geq 0}P_{i+1}(b,n)^{\hookrightarrow_{{i+1}}}$. Moreover, since $R_{b}(n-1)^{\hookrightarrow_{0}}$ is nothing but $P_{0}(b,n)^{\hookrightarrow_{0}}$, this is equivalent to $R_{b}(n)=\bigsqcup_{i\geq 0}P_{i}(b,n)^{\hookrightarrow_{i}}$. Finally, from Lemma 3, we obtain the announced formula. From this formula, we have $R_{b}(\frac{n}{b})=\bigsqcup_{i\geq 0}r^{-1}(R_{b}(\frac{n}{b^{i+1}}-1)^{\hookrightarrow_{i}})$. Therefore, $\|R_{b}(n)\|=\sum_{i\geq 0}\|R_{b}(\frac{n}{b^{i}}-1)\|=\|R_{b}(n-1)\|+\sum_{i\geq 0}\|R_{b}(\frac{n}{b^{i+1}}-1)\|=\|R_{b}(n-1)\|+\|R_{b}(\frac{n}{b})\|$. We obtain the claim by iterating this last formula. The first formula given in this theorem can be used to compute the sets $R_{b}(n)$ efficiently since it only involves _disjoint_ unions. We will give in Section 5 another method to compute $R_{b}(n)$ which is much simplier, as it gives $R_{b}(n)$ a tree structure. However, the formula is interesting since it points out the self-similar structure of the set (see Figure 4). The second formula is previouly known from [Rod69], and from [Eul50] in the special case where $b=2$. Notice that this does not give a way to compute $\|R_{b}(n)\|$ in linear time with respect to $n$, which is an unsolved problem in the general case, but it gives a very simple way to compute recursively $\|R_{b}(n)\|$. ## 4 Infinite extension $R_{b}(n)$ is the lattice of the $b$-ary partitions of $n$ reachable from $(n)$ by iteration of the evolution rule. We now define $R_{b}(\infty)$ as the set of all $b$-ary partitions reachable from $(\infty)$. The order on $R_{b}(\infty)$ is the reflexive and transitive closure of the successor relation. For $b=2$, the first $b$-ary partitions in $R_{b}(\infty)$ are given in Figure 2 along with their covering relation (the first component, which is always infinity, is not represented on this diagram). Notice that it is still possible to define the shot vector $s(p)$ of an element $p$ of $R_{b}(\infty)$ by: $s(p)_{i}$ is the number of times one has to fire $i$ in order to obtain $p$ from $(\infty)$. Figure 2: The first $b$-ary partitions obtained in $R_{b}(\infty)$ when $b=2$. Two parts isomorphic to $R_{2}(4)$ are distinguished, as well as two parts isomorphic to $R_{2}(7)$. ###### Theorem 3 The set $R_{b}(\infty)$ is a distributive lattice with: $s(p\vee q)_{i}=\min(s(p)_{i},s(q)_{i})\mbox{ and }s(p\wedge q)_{i}=\max(s(p)_{i},s(q)_{i})$ for all $p$ and $q$ in $R_{b}(\infty)$. Moreover, for all $n$ the functions $\pi:s=(s_{1},s_{2},\dotsi,s_{k})\longrightarrow\pi(s)=(\infty,s_{2},\dots,s_{k})$ and $\tau:s=(s_{1},s_{2},\dotsi,s_{k})\longrightarrow\tau(s)=(\infty,s_{1},s_{2},\dots,s_{k})$ are lattice embeddings of $R_{b}(n)$ into $R_{b}(\infty)$. Proof : The proof for the distributive lattice structure and for the formulae of the infimum and supremum is very similar to the proof of Theorem 1. Therefore, it is left to the reader. Given $p$ and $q$ in $R_{b}(n)$, we now prove that $\pi(p)\vee\pi(q)=\pi(p\vee q)$. From Theorem 1, we have $s(p\vee q)_{i}=\min(s(p)_{i},s(q)_{i})$. Moreover, it is clear that $s(\pi(x))_{i}=s(x)_{i}$ for all $x$ in $R_{b}(n)$. Therefore, $s(\pi(p\vee q))_{i}=\min(s(\pi(p))_{i},s(\pi(q))_{i}))$, which shows that $\pi$ preserves the supremum. The proof of $\pi(p)\wedge\pi(q)=\pi(p\wedge q)$ is symmetric. Therefore, $\pi$ is a lattice embedding. The proof for $\tau$ is very similar when one has noticed that the shot vector of $\tau(s)$ is obtained from the one of $s$ by adding a new first component equal to $n$. With similar arguments, one can easily show that $\pi(R_{b}(n))$ is a sublattice of $\pi(R_{b}(n+1))$, and so we have an infinite chain of distributive lattices: $\pi(R_{b}(0))\leq\pi(R_{b}(1))\leq\dots\leq\pi(R_{b}(n))\leq\pi(R_{b}(n+1))\leq\dots\leq R_{b}(\infty),$ where $\leq$ denotes the sublattice relation. Moreover, one can use the self- similarity estalished here to construct filters of $R_{b}(\infty)$ (a _filter_ of a poset is an upper closed part of the poset). Indeed, if one defines $R_{b}(\leq n)$ as the sub-order of $R_{b}(\infty)$ over $\cup_{i\leq n}R_{b}(i)$, then one can construct efficiently $R_{b}(\leq n+1)$ from $R_{b}(\leq n)$ by extracting from $R_{b}(\leq n)$ a part isomorphic to $R_{b}(n+1)$ and pasting it to $R_{b}(\leq n)$. See Figures 2 and 4. Notice that, for all integer $b$, $R_{b}(\infty)$ contains exactly all the finite sequences of integers, since any such sequence can be viewed as a $b$-ary partition of an integer $n$. Therefore, we provide infinitely many ways to give the set of finite sequences of integers the distributive lattice structure. ## 5 Infinite tree As shown in our construction of $R_{b}(n+1)$ from $R_{b}(n)$, each $b$-ary partition $p$ in $R_{b}(n+1)$ is obtained from another one $p^{\prime}\in R_{b}(n)$ by application of the ↪ operator: $p=\mbox{$p^{\prime}$}^{\hookrightarrow_{i}}$ with $i$ an integer between $0$ and $l(p^{\prime})$, where $l(p^{\prime})$ denotes the number of $b-1$ at the beginning of $p^{\prime}$. Thus, we can define an infinite tree $T_{b}(\infty)$ whose nodes are the elements of $\bigsqcup_{n\geq 0}{R_{b}(n)}$ and in which the fatherhood relation is defined by: $q\mbox{ is the $(i+1)$-th son of $p$ if and only if }q=p^{\hookrightarrow_{i}}\mbox{ for some }i,\ 0\leq i\leq l(p).$ The root of this tree is $(0)$ and each node $p$ of $T_{b}(\infty)$ has $l(p)+1$ sons. The first levels of $T_{b}(\infty)$ when $b=2$ are shown in Figure 3 (we call the set of elements of depth $n$ the “level $n$” of the tree). Figure 3: The first levels of $T_{b}(\infty)$ when $b=2$. We distinguished some special subtrees, which will play an important role in the following. ###### Proposition 1 The level $n$ of $T_{b}(\infty)$ contains exactly the elements of $R_{b}(n)$. Proof : Straightforward from the construction of $R_{b}(n+1)$ from $R_{b}(n)$ given above and the definition of the tree. If we define $\overline{R_{b}(n)}$ as $\\{(s_{2},\dots,s_{k})\ \|\ (s_{1},s_{2},\dots,s_{k})\in R_{b}(n)\\}$, then: ###### Proposition 2 For all integer $n$, the elements of $\overline{R_{b}(n)}$ are exactly the elements of the $\lfloor\frac{n}{b}\rfloor$ first levels of $T_{b}(\infty)$. Proof : Let us first prove that the elements of $R_{b}(n)$ are the nodes of a subtree of $T_{b}(\infty)$ that contains its root. This is obviously true for $n=0$. The general case follows by induction, since by construction the elements of $\overline{R_{b}(n+1)}\setminus\overline{R_{b}(n)}$ are sons of elements of $\overline{R_{b}(n)}$. Now, let us consider an element $e$ of the $l$-th level of $T_{b}(\infty)$. If there is a $b$-ary partition $p$ of $n$ such that $\overline{p}=e$, then clearly $p_{i}=e_{i-1}$ for all $i>0$ and $p_{0}=n-b\cdot l$. Therefore, if $e$ is in $\overline{R_{b}(n)}$ then all the elements of the $l$-th level are in $\overline{R_{b}(n)}$, and this is clearly the case exactly when $0\leq l<\lfloor\frac{n}{b}\rfloor$. This ends the proof. Notice that this proposition gives a simple way to enumerate the elements of $R_{b}(n)$ for any $n$ in linear time with respect to their number, since it gives this set a tree structure. Algorithm 1 acheives this. Input: An integer $n$ and a basis $b$ Output: The elements of $R_{b}(n)$ begin $\mbox{Resu}\leftarrow\\{(n)\\}$; $\mbox{CurrentLevel}\leftarrow\leftarrow\\{()\\}$; $\mbox{OldLevel}\leftarrow\emptyset$; $l\leftarrow 0$; while _$l <\lfloor\frac{n}{b}\rfloor$_ do $\mbox{OldLevel}\leftarrow\mbox{CurrentLevel}$; $\mbox{CurrentLevel}\leftarrow\emptyset$; $l\leftarrow l+1$; foreach _$p$ in OldLevel_ do $i\leftarrow 0$; repeat Add $p^{\hookrightarrow_{i}}$ to CurrentLevel; $i\leftarrow i+1$; until _$p_{i-1}\not=b-1$_ ; end foreach foreach _$e$ in CurrentLevel_ do Create $p$ such that $p_{i}=e_{i-1}$ for all $i>0$ and $p_{0}=n-b\cdot l$; Add $p$ to Resu; end foreach end while Return(Resu); end Algorithm 1 Efficient enumeration of the elements of $R_{b}(n)$. We will now show that $T_{b}(\infty)$ can be described recursively, which allows us to give a new recursive formula for $\|R_{b}(n)\|$. In order to do this, we will use a series known as the $b$-ary carry sequence [Slo73]: $c_{b}(n)=k$ if $b^{k}$ divides $n$ but $b^{k+1}$ does not. Notice that this function is defined only for $n>0$ (or one can consider that $c_{b}(0)=\infty$). These series appear in many contexts, and have many equivalent definitions 444 For example, if one defines the series $C_{b,0}=0$ and $C_{b,i}=C_{b,i-1},\stackrel{{\scriptstyle\mbox{$b-1$ times}}}{{\overbrace{i,C_{b,i-1}}}}$, then $c_{b}(i)$ is nothing but the $i$-th integer of the series $C_{b,i}$. The ten first values for $c_{2}(i)$ are $0,1,0,2,0,1,0,3,0,1$ and the ten first ones for $c_{3}(i)$ are $0,0,1,0,0,1,0,0,2,0$.. Here, we will mainly use the fact that the first $n$ such that $c_{b}(n)=k$ is $n=b^{k}$, and the fact that $c_{b}(n)$ is nothing but the number of components equal to $b-1$ at the begining of the canonical representation of $n-1$ in the basis $b$. ###### Definition 1 Let $p\in T_{b}(\infty)$. Let us consider the rightmost branch of $T_{b}(\infty)$ rooted at $p$ ($p$ is considered as the first node of the branch). We say that $p$ is the root of a _$X_{b,k}$ subtree (of $T_{b}(\infty)$)_ if this rightmost branch is as follows: for $i\leq b^{k-1}$, the $i$-th node on the branch has $j=c_{b}(i)+1$ sons, and the $l$-th ($1\leq l<j$) of these sons is the root of a $X_{b,l}$ subtree. Moreover, the $(b^{k-1}+1)$-th node of the branch is itself the root of a $X_{b,k}$ subtree. For example, we show in Figure 3 a $X_{2,2}$ subtree of $T_{2}(\infty)$, composed of a $X_{2,1}$ subtree and another $X_{2,2}$ subtree. Notice that a $X_{b,1}$ subtree is simply a chain. ###### Proposition 3 Let $p=(0,0,\dots,0,p_{k},\dots)$ in $T_{b}(\infty)$ with $p_{k}>b-1$. Then, $p$ is the root of a $X_{b,k+1}$ subtree of $T_{b}(\infty)$. Proof : The proof is by induction over $k$ and the depth of $p$. Let us consider the rightmost branch rooted at $p$. Since, for all $q$ in $T_{b}(\infty)$, the rightmost son of $q$ is $q^{\hookrightarrow_{i}}$ with $i$ the number of $b-1$ at the beginning of $q$, it is clear that the $j$-th node of this branch for $j\leq b^{k}$ is $q=(q_{0},\dots,q_{k-1},p_{k},\dots)$ where $(q_{0},\dots,q_{k-1})$ is the canonical representation of $j-1$ in the basis $b$. Therefore, $q$ begins with $c_{b}(j)$ components equal to $b-1$, and so, for $l=1,\dots,c_{b}(j)$, the $l$-th son of $q$ starts with $l-1$ zeroes followed by a component equal to $b>b-1$. By induction hypothesis, we then have that the sons of $q$ are the roots of $X_{b,l}$ subtrees. Moreover, the $(b^{k}+1)$-th node on the rightmost branch begins with exactly $k$ zeroes followed by a component greater than $b-1$, and so it is the root of a $X_{b,k+1}$ subtree by induction hypothesis. ###### Theorem 4 The infinite tree $T_{b}(\infty)$ is a $X_{b,\infty}$ tree: it is a chain (its rightmost branch) such that its $i$-th node has $c_{b}(i)$ sons and the $j$-th of these sons, $1\leq j\leq c_{b}(i)$, is the root of a $X_{b,j}$ subtree. Moreover, the $i$-th node of the chain is the canonical representation of $i-1$ in the basis $b$. Proof : Since the rightmost son of $p\in T_{b}(\infty)$ is $p^{\hookrightarrow_{i}}$, where $i$ is the number of $b-1$ at the beginning of $p$, and since the root of $T_{b}(\infty)$ is nothing but the canonical representation of $0$, it is clear by induction that the $i$-th node of the rightmost branch of $T_{b}(\infty)$ is the canonical representation of $i-1$ in the basis $b$. Then, the theorem follows from Proposition 3. We now have a recursive description of $T_{b}(\infty)$, which allows us to give recursive formula for the cardinal of some special sets. Let us denote by $\pi_{b}(l,k)$ the number of paths of length exactly $l$ starting from the root of a $X_{b,k}$ subtree of $T_{b}(\infty)$. We have: ###### Theorem 5 $\pi_{b}(l,k)=\left\\{\begin{array}[]{ll}1&\mbox{if $0\leq l<b$}\\\ 1+\sum_{i=1}^{l}\sum_{j=1}^{c_{b}(i)}\pi_{b}(l-i,j)&\mbox{if $b\leq l\leq b^{k-1}$}\\\ \pi_{b}(l-b^{k-1},k)+\sum_{i=1}^{b^{k-1}}\sum_{j=1}^{c_{b}(i)}\pi_{b}(l-i,j)&\mbox{otherwise ($l>b^{k-1}$)}\end{array}\right.$ Moreover, $\|R_{b}(n)\|=\pi_{b}(n,n)$ and the number of $b$-ary partitions of $n$ into exactly $l$ parts is $\pi_{b}(n-(b-1)^{l},l)$. Proof : The formula for $\pi_{b}(l,k)$ is directly deduced from the definition of the $X_{b,k}$ subtrees. The other formulae derive from Theorem 4 and from the fact that all the $b$-ary partitions of length $l$ are in a $X_{b,l}$ subtree of $T_{b}(\infty)$which is rooted at the $(b-1)^{l}$-th node of the righmost branch of $T_{b}(\infty)$. ## 6 Perspectives The results presented in this paper mainly point out the strong self- similarity and the structure of the sets $R_{b}(n)$. As already noticed, it is an open question to compute the cardinal of $R_{b}(n)$ in linear time with respect to $n$, and one may expect to obtain a solution using these results. Another interesting direction is to investigate how one can extend the dynamics we study. A first idea is to consider non-integer basis, in particular complex basis or Fibonnacci basis. For example, if we consider the complex basis $b=i-1$ then we can obtain all the ways to write an integer $n$ as the sum of powers of $b$ by iterating the following evolution rule from $(n)$: $q$ is a successor of $p$ if $p-q=(0,\dots,0,2,0,-1,-1,0\dots,0)$. In other words, we can decrease by two the $j$-th component of $p$ and increase by one its $(j+2)$-th and its $(j+3)$-th components for some integer $j$. This gives to the set of representations of $n$ in the complex basis $b=i-1$ the lattice structure, since this can be encoded by a Chip Firing Game [LP00] (notice however that in this case the lattice is no longer distributive). Another interesting case is when $b=1$. As already noticed, we obtain the Young lattice, or equivalently the lattice of the compositions of $n$. ## 7 Acknowledgments I thank Christiane Frougny and Clémence Magnien for many useful comments on preliminary versions, which deeply improved the manuscript. ## References * [Ber71] Claude Berge. Principles of Combinatorics, volume 72 of Mathematics in science and engineering. Academic Press, 1971. * [BL92] Anders Björner and László Lovász. Chip-firing games on directed graphs. J. Algebraic Comb., 1(4):305–328, December 1992. * [BLS91] A. Bjorner, L. Lovász, and W. 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Latapy and H.D. Phan. The lattice of integer partitions and its infinite extension. 1999\. To appear in DMTCS, special issue, proceedings of ORDAL’99. Preprint ava ilable at `http://www.liafa.jussieu.fr/~latapy/`. * [LP00] M. Latapy and H.D. Phan. The lattice structure of chip firing games. 2000\. To appear in Physica D. Preprint available at `http://www.liafa.jus sieu.fr/~latapy/`. * [Mah40] Kurt Mahler. On a special functional equation. J. London Math. Soc, 15:115–123, 1940. * [Mon90] Bernard Monjardet. The Consequences of Dilworth’s Work on Lattices with Unique Irreducible Decompositions, pages 192–199. Birkhäuser Boston, Boston, MA, 1990. * [MPV01] Clémence Magnien, Ha Duong Phan, and Laurent Vuillon. Characterization of lattices induced by (extended) chip firing games. In Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001, volume AA of DMTCS Proceedings, pages 229–244, 2001\. * [Pen53] W.B. Pennington. On Mahler’s partition problem. 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# On-state commutativity of measurements and joint distributions of their outcomes Jan Czajkowski<EMAIL_ADDRESS>QuSoft, University of Amsterdam Alex B. Grilo<EMAIL_ADDRESS>Sorbonne Université, CNRS, LIP6 ###### Abstract In this note, we analyze joint probability distributions that arise from outcomes of sequences of quantum measurements performed on sets of quantum states. First, we identify some properties of these distributions that need to be fulfilled to get a classical behavior. Secondly, we prove that a joint distribution exists iff measurement operators “on-state” permute (permutability is the commutativity of more than two operators). By “on-state” we mean properties of operators that hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that the property of partial on- state permutation implies full on-state permutation. We disprove this conjecture with a counterexample where pairwise “on-state” commutativity does not imply on-state permutability, unlike in the case of commutativity for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hastings (Communications in Mathematical Physics, 2019) for general operators. ## 1 Introduction In this work we propose a basic formalism for studying classical distributions that come from joint measurements on quantum states. Our initial motivation comes from studying a conjecture proposed in a recent paper by Carstens, Ebrahimi, Tabia, and Unruh [CETU18]. Their result on quantum indifferentiability111Indifferentiability is a strong security notion capturing security of cryptographic constructions such as hash functions, where we require that any polynomial-time adversary cannot distinguish if it has access to a cryptographic hash function or an ideal random function, even if she has access to some internal auxiliary functions used to construct the hash function. The quantum version assumes the adversary makes quantum queries. relies on a conjecture proposed by them, which informally states that commutation of projectors with respect to a fixed quantum state implies a classical joint distribution of their measurement outcomes. More concretely, they conjecture the following.222See Conjecture 2 for the formal statement. ###### Conjecture 1 (Informal). If we have a set of $N$ measurements $P_{1},\dots,P_{N}$ that commute on a quantum state $\|\psi\rangle,$333Informally, two operators $A$ and $B$ commute on $\|\psi\rangle$ if $[A,B]\|\psi\rangle=0$. then there exist random variables $X_{1},\dots,X_{N}$ drawn from a distribution $D$ such that for any $t>1$, $\forall_{1},\dots,i_{t}$, the marginals of this distribution on $X_{i_{1}},\dots,X_{i_{t}}$ correspond to measuring $\|\psi\rangle$ with measurements $P_{i_{1}},\dots,P_{i_{t}}$. Motivated by this conjecture, our goal is to study the behavior of $N$ random variables $X_{1},X_{2},\dots,X_{N}$ corresponding to the outcomes of a sequence of quantum measurements that commute on a set of quantum states $\mathcal{F}\subseteq\mathcal{D}(\mathcal{H})$. Surprisingly, such results have only been studied for $\mathcal{F}=\mathcal{D}(\mathcal{H})$, i.e. measurements commuting on all quantum states. The focal point of this note is to study which are the necessary and sufficient properties of the quantum setup, so that such a probability distribution is well-defined. With this in hand, we then have two applications. First, we disprove 1. Secondly, we show a simpler proof for a variant of the result by Hastings [Has09] on operators that almost-commute on specific states. To be able to explain our contributions in more details, we will first start with a detour to very basic properties of probability distributions that arise from classical processes. Then, we discuss how these properties could be defined in the quantum setting (but, unfortunately, they do not hold for general quantum setups), and finally we state our results and discuss related works. ### 1.1 Classical Distributions We discuss here properties of classical distributions that may be obvious at first but are crucial and not trivial in the quantum world. In the following, we let $A,B,C$ be events that come from a classical experiment. We denote the event corresponding to $A$ not happening as $\overline{A}$, the probability that $A$ and $B$ both happen as $\mathbb{P}[A,B]$, and the probability that $A$ happens conditioned on the fact that event B happens as $\mathbb{P}[A\|B]=\frac{\mathbb{P}[A,B]}{\mathbb{P}[B]}$ (assuming $\mathbb{P}[B]\neq 0$). The first property that we want to recall on classical distributions is that we can compute the marginals of the distribution when given the joint distribution: ###### Property 1 (Classical Marginals). $\mathbb{P}[A\mid C]=\mathbb{P}[A,B\mid C]+\mathbb{P}[A,\overline{B}\mid C]$. A second property that we want to recall is that the probability that $A$ and $\overline{A}$ occur is $0$, even when considering other events: ###### Property 2 (Classical Disjointness). $\mathbb{P}[A,B\mid\bar{A}]\mathbb{P}[\overline{A}]=\mathbb{P}[A,B,\overline{A}]=0$. Another property that we have classically is _reducibility_ , which says that the probability of events $A$ and $A$ both happening is the same as the probability of $A$. ###### Property 3 (Classical Reducibility). $\mathbb{P}[A,B\mid A]\mathbb{P}[A]=\mathbb{P}[A,B,A]=\mathbb{P}[A,B]$. Finally, the last property we study is _sequential independence_ 444Sequential independence has been originally defined in [GN01] in the context of quantum measurements. of random variables. Roughly, this property just says that the probability that event $A$ happens and that event $B$ happens is the same as the probability that event $B$ happens and that event $A$ happens. ###### Property 4 (Classical Sequential Independence). $\mathbb{P}[A\mid B]\mathbb{P}[B]=\mathbb{P}[A,B]=\mathbb{P}[B\mid A]\mathbb{P}[A]$. We stress that these properties hold trivially for all classical distributions and all events such that $\mathbb{P}[A]\neq 0$, $\mathbb{P}[\overline{A}]\neq 0$, $\mathbb{P}[B]\neq 0$, and $\mathbb{P}[C]\neq 0$. ### 1.2 Quantum Distributions and their Properties Our goal is to find necessary conditions for the existence of a classical description of the experiment where we perform a sequence of $N$ general measurements, irrespective of the order. More concretely, we aim to find the properties of measurement operators $Q_{1},...,Q_{N}$ on specific subsets of quantum states $\mathcal{F}$ so that there exists a joint distribution of $X_{1},X_{2},\dots,X_{N}$ such that all marginals of this distribution on $X_{i_{1}},\dots,X_{i_{t}}$ correspond to measuring a state $\|\psi\rangle\in\mathcal{F}$ with measurements $Q_{i_{1}},\dots,Q_{i_{t}}$. In this case, we call it a _quantum distribution_. The main obstacle in this task is the fact that quantum measurements do not necessarily commute, unlike in the classical world: the chosen order for performing the measurements influences the final probability distribution of the joint measurement outcomes. Because of that, we will consider the quantum analog of Properties 1 to 4, and study when such properties hold in the quantum case, and their implication for having such a joint distribution. Our connections closely follow [ME84], where they show that the existence of a joint distribution for two arbitrary quantum observables (Hermitian operators) on every quantum state is equivalent to their commutation. In this work, we show how to extend their analysis in two ways: we are interested in multiple observables and we consider specific sets of quantum states. In order to carry out this analysis, we extend the properties described in Section 1.1 to quantum measurements and study their relations to each other. We leave the formal definitions of the quantum analogs of these classical properties to Section 3.1. ### 1.3 Our Results Using the formalism described in the previous section, we prove the following connections between joint quantum distributions and the measurement operators. First, we show that quantumly, the marginal property also implies the sequential independence one. ###### Result 1 (Informal statement of 1). If a joint distribution has the quantum marginal property, then it also has the quantum sequential independence property. Then, we show that in the on-state case, we have that there is a quantum joint distribution iff all operators permute555Informally, a set of operators permutes on $\|\psi\rangle$ if applied in any order they yield the same state: $A_{1}\cdots A_{N}\|\psi\rangle=A_{\sigma(1)}\cdots A_{\sigma(N)}\|\psi\rangle$, where $\sigma$ is a permutation.. This result is a generalization of the classic results from [Nel67, Fin73, Fin82, ME84] to the on-state case. ###### Result 2 (Informal statement of 4). Fix a set of quantum states $\mathcal{F}$. A set of measurements yield a quantum joint distribution on each state in $\mathcal{F}$ iff these operators permute on every state in $\mathcal{F}$. Then, we show that pairwise on-state commutation does not imply full on-state permutation, unlike in the case of permutation on all states. This fact—that we prove via a numerical example—together with Result 2 implies that 1 is false. ###### Result 3. 1 is false. Finally, our last result is a simpler proof for a restricted version of Theorem 1 in [Has09], which states that if two operators $A$ and $B$ almost- commute, we can find commuting operators $A^{\prime}$ and $B^{\prime}$ that are close to $A$ and $B$, respectively. In our case, we consider on-state commutation instead of the regular one, and unlike in [Has09], our proof works only for projectors. ###### Result 4 (Making almost commuting projectors commute). Given any two projectors $P_{1}$ and $P_{2}$ and a state $\|\psi\rangle$ we have that if $\left\\|(P_{1}P_{2}-P_{2}P_{1})\|\psi\rangle\right\\|=\epsilon$ then there is a projector $P_{2}^{\prime}$ that is close to the original projector on the state $\left\\|(P_{2}^{\prime}-P_{2})\|\psi\rangle\right\\|\leq\sqrt{2}\epsilon$ and $[P_{1},P_{2}^{\prime}]=0$. ### 1.4 Related Work A prominent result in the literature is that a joint distribution for a set of measurements exists iff all the operators pairwise commute. Different versions of this result were previously proven: In [Nel67] the author considers the case of continuous variables and $N$ observables. A similar result but without specifying the Hilbert space is achieved with different mathematical tools in [Fin82]. In the specific case where we have only two observables, we mention three works; In [Fin73] and [ME84] the authors prove the classic problem in a way similar to each other, but using different mathematical tools. All but the first work mentioned here focus on the joint distribution as a functional from the space of states. An approach using $$-algebras was presented by Hans Maassen in [Maa06, Maa10]. The authors of [GN02] analyze the case of general measurements but prove that the measurement operators pairwise commute iff the square-root operators permute (Corollaries 3 and 6 in [GN02]), in the sense of our Definition 3 (for all states in $\mathcal{H}$). In general the problem of conditional probabilities in Quantum Mechanics was discussed by Cassinelli and Zanghi in [CZ83]. The related problems of incompatible devices measurement and joint measurability of quantum effects are covered in [HMZ16] and [BN18] respectively. In [Lin97, FR96] the authors prove that for any two Hermitian matrices if their commutator has small norm, then there are operators close to the originals that fully commute. In [Has09] Hastings proves how close the new operators are in terms of the norm of the commutator. ### Organization In Section 2, we provide some preliminaries. Then in Section 3, we discuss quantum distributions and their properties. Finally, in Section 4, we discuss the almost-commuting case. ### Acknowledgements JC thanks Dominique Unruh and Christian Schaffner for helpful discussions. JC was supported by a NWO VIDI grant (Project No. 639.022.519). Most of this work was done while AG was affiliated to CWI and QuSoft. ## 2 Preliminaries ### 2.1 Notation In this work, we are going to use calligraphic letters ($\mathcal{S},\mathcal{R},...$) to denote sets. We denote $[N]:=\\{1,2,\dots,N\\}$. For $\mathcal{S}\subseteq[N]$, we denote by $\mathcal{S}(i)$ the $i$-th element of the set $\mathcal{S}$ in ascending order. For some fixed sets $\mathcal{X}_{1},...,\mathcal{X}_{N}$, we denote by $\vec{x}$ an element of $\mathcal{X}_{1}\times\cdots\mathcal{X}_{N}$ and for $\mathcal{S}\subseteq[N]$ we have $\vec{x}_{\mathcal{S}}:=(x_{\mathcal{S}(1)},\dots,x_{\mathcal{S}(\|\mathcal{S}\|)})$. We denote the set of all $t$-element permutations by $\Sigma_{t}$. For some complex number $c=a+b\textnormal{i}$, we define $\mathfrak{Re}(c)=a$ as its real part. ### 2.2 Quantum Measurements We briefly review some concepts in quantum computation/information and we refer to [NC11] for an more detailed introduction to these topics. Quantum states are represented by positive semi-definite operators with unit trace, i.e., $\rho\succeq 0,\textnormal{Tr}(\rho)=1$. We denote the set of all density operators on some Hilbert space $\mathcal{H}$ by $\mathcal{D}(\mathcal{H})$. To describe general measurements, we use the notion of Positive Operator Valued Measure (POVM). The only requirement of POVMs is that they consist of positive operators and sum up to the identity operator. More formally, a POVM with set of outcomes $\mathcal{X}$ is described by a set of operators $\mathcal{M}=\\{Q^{x}\\}_{x\in\mathcal{X}}$, where $\forall x\in\mathcal{X}:Q^{x}\succeq 0$, and $\sum_{x\in\mathcal{X}}Q^{x}=\mathbbm{1}$. We denote the probability of getting the outcome $x$ when measuring $\rho$ with the measurement $\mathcal{M}$ by $\mathbb{P}[x\leftarrow\mathcal{M}(\rho)]:=\textnormal{Tr}(Q^{x}\rho)$. To describe the post-measurement state, we can write down operators of $\mathcal{M}$ as products of linear operators on $\mathcal{H}$ (denoted by $\mathcal{L}(\mathcal{H})$), $Q^{x}=A^{x\dagger}A^{x}$, where $A^{x}\in\mathcal{L}(\mathcal{H})$ (such a decomposition is always possible since $Q^{x}\succeq 0$). The post-measurement state when the outcome of $\mathcal{M}$ on $\rho$ is $x$ is given by $\rho_{x}:=\frac{A^{x}\rho A^{x\dagger}}{\textnormal{Tr}(Q^{x}\rho)}.$ (1) The operator $A^{x}$ is called the _square root_ of $Q^{x}$. ## 3 Quantum Distributions In this section, we study the description of the statistics of outcomes derived from a sequence of measurements. Our approach is to consider the quantum version of the classical properties described in Section 1.1. Since quantum measurements do not commute in general, these quantum properties do not always hold. We then study the connection between properties of the measurements and the properties of their outcome distribution. The structure of our proofs follows [ME84], where they show that for two Hermitian observables there is a joint distribution for the outcomes of their joint measurement iff they commute. We stress that the result in [ME84] only works for measurements that commute on every quantum state and our result extends it to the case of joint distributions defined on a limited set of states. In the following, we denote the observables with $Q$ and their square-roots with $R$. In Section 3.1, we define the quantum analogues of the classical properties of distributions defined in Section 1.1. Then, in Section 3.2, we state and prove the main result of this section, where we show a connection between existence of a distribution and permutability—a generalization of commutativity—of the corresponding measurement operators. ### 3.1 Quantum Distributions We analyse a functional from the set of density operators and finite sets of $N$ random variables $X_{1},X_{2},\dots,X_{N}$ to reals $\mathbf{W}_{[N]}:\mathcal{D}(\mathcal{H})\times(\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{N})\to[0,1]$. We define this functional666Note that the superscript of $\mathbf{W}^{\rho}_{[N]}(\vec{x})$ denotes the first input to the functional, so we have $\mathbf{W}_{[N]}(\rho,\vec{x})$. as $\mathbf{W}_{[N]}^{\rho}(\vec{x}):=\textnormal{Tr}\left(Q_{[N]}^{\vec{x}}\rho\right),$ (2) where $Q_{[N]}^{\vec{x}}$ is a positive semidefinite operator corresponding to the outcome $\vec{x}\in\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{N}$. The subscript $[N]$ of $\mathbf{W}$ denotes the set of indices of random variables that the distribution is defined on. This definition is similar to the one proposed by [ME84]. The starting point of our discussion is that every random variable $X_{i}$ corresponds to the measurement $\mathcal{M}_{i}=\\{Q_{i}^{x_{i}}\\}$. So we have to keep in mind that these operators are fixed throughout this note. Given the definition of $\mathbf{W}$, we can state conditions so that it can be seen as a joint quantum distribution: • Normalization: $\sum_{\vec{x}}Q^{\vec{x}}_{[N]}=\mathbbm{1}$, which implies that for all $\rho\in\mathcal{D}(\mathcal{H}),\;\sum_{\vec{x}}\mathbf{W}^{\rho}(\vec{x})=1$. * • Linearity: for every $\vec{x}\in\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{N}$, $\rho_{1},\rho_{2}\in\mathcal{D}(\mathcal{H})$ and $\lambda_{1},\lambda_{2}\in[0,1]$, we have that $\mathbf{W}^{\lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}}_{[N]}(\vec{x})=\lambda_{1}\mathbf{W}^{\rho_{1}}_{[N]}(\vec{x})+\lambda_{2}\mathbf{W}^{\rho_{2}}_{[N]}(\vec{x}).$ * • Non-negativity: for every $\vec{x}\in\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{N}$ and $\rho\in\mathcal{D}(\mathcal{H})$, we have $\mathbf{W}^{\rho}_{[N]}(\vec{x})\geq 0$. We describe the quantum analogues of the properties described in Section 1.1. #### Marginals. We start with a sequence of general measurements (i.e. POVMs): $\\{\mathcal{M}_{i}\\}_{i\in[N]}$, where $\mathcal{M}_{i}:=\\{Q_{i}^{x}\\}_{x\in\mathcal{X}_{i}}$ for all $i$ and all $Q_{i}^{x}\succeq 0$ and $\sum_{x\in\mathcal{X}_{i}}Q^{x}_{i}=\mathbbm{1}$. Moreover, for $\mathcal{S}\subseteq[N]$ we define $Q^{\vec{y}}_{\mathcal{S}}$ to be measurement operators where $\vec{y}\in\mathcal{X}_{\mathcal{S}(1)}\times\cdots\times\mathcal{X}_{\mathcal{S}(\|\mathcal{S}\|)}$. Given $Q^{\vec{y}}_{\mathcal{S}}$ and their corresponding square-roots $R^{\vec{y}}_{\mathcal{S}}$ and $\rho$, we have that if $\textnormal{Tr}\left(R^{\vec{y}}_{\mathcal{S}}\rho R^{\vec{y}\dagger}_{\mathcal{S}}\right)\neq 0$, then we define the conditional distribution for any sequence $\vec{x}$ as $\mathbf{W}_{[N]}^{\rho}(\vec{x}\mid\vec{y}):=\mathbf{W}_{[N]}^{R^{\vec{y}}_{\mathcal{S}}\rho R^{\vec{y}\dagger}_{\mathcal{S}}}(\vec{x})/\textnormal{Tr}R^{\vec{y}}_{\mathcal{S}}\rho R^{\vec{y}\dagger}_{\mathcal{S}}.$ (3) For all those measurements, for a set $\mathcal{F}\subseteq\mathcal{D}(\mathcal{H})$, and for $\mathcal{U}\subseteq[N]$, we define the “orbit” of the post-measurement states. For any $\mathcal{T}\subseteq[N]$ of size $t$, we take $s\leq t$ sets $\mathcal{S}_{1},...,\mathcal{S}_{s}$ that are a partition of $\mathcal{T}$. We then consider the post-measurement states generated by sequences of measurements corresponding to $\mathcal{S}_{i}$: $\displaystyle\mathcal{G}_{\mathcal{U}}(\mathcal{F}):=\left\\{R^{\vec{y}_{s}}_{\mathcal{S}_{s}}\cdots R^{\vec{y}_{1}}_{\mathcal{S}_{1}}\psi R^{\vec{y}_{1}\dagger}_{\mathcal{S}_{1}}\cdots R^{\vec{y}_{s}\dagger}_{\mathcal{S}_{s}}/\textnormal{Tr}\left(R^{\vec{y}_{s}}_{\mathcal{S}_{s}}\cdots R^{\vec{y}_{1}}_{\mathcal{S}_{1}}\psi R^{\vec{y}_{1}\dagger}_{\mathcal{S}_{1}}\cdots R^{\vec{y}_{s}\dagger}_{\mathcal{S}_{s}}\right):\psi\in\mathcal{F},\mathcal{T}\subseteq\mathcal{U},\right.$ $\displaystyle\left.s\leq\left\|\mathcal{T}\right\|,\;\mathcal{S}_{1},...,\mathcal{S}_{s}\subseteq\mathcal{T},\bigcup_{i=1}^{s}\mathcal{S}_{i}=\mathcal{T},\forall i\neq j\,\mathcal{S}_{i}\cap\mathcal{S}_{j}=\emptyset,\vec{y}_{i}\in\mathcal{X}_{\mathcal{S}_{i}(1)}\times\cdots\times\mathcal{X}_{\mathcal{S}_{i}(\|\mathcal{S}_{i}\|)}\right\\},$ (4) where $R^{\vec{y}_{i}}_{\mathcal{S}_{i}}$ are the square-root operators of $Q^{\vec{y}_{i}}_{\mathcal{S}_{i}}=R^{\vec{y}_{i}\dagger}_{\mathcal{S}_{i}}R^{\vec{y}_{i}}_{\mathcal{S}_{i}}$. The subscript of $\mathcal{G}$ denotes the set we take the subsets of, usually it is $[N]$ but later we also consider $[N]\setminus\mathcal{S}$ for some $\mathcal{S}$. With our quantum marginals property, we require that the operator we get after we sum over a subset of variables is still a valid measurement operator. ###### Property 5 (Quantum Marginals). We say that the joint distribution $\mathbf{W}$ has the _quantum marginals property on set $\mathcal{F}$_ if for every $\mathcal{S}\subseteq[N]$, there is a measurement $\mathcal{M}_{\mathcal{S}}=\\{Q^{\vec{y}}\\}_{\vec{y}\in(\mathcal{X}_{i})_{i\in\mathcal{S}}}$ such that for every value $\vec{x}\in\mathcal{X}_{1}\times\mathcal{X}_{2}\times\cdots\times\mathcal{X}_{N}$, denoting $\vec{x}:=(x_{1},x_{2},\dots,x_{N})$ and for every density operator $\rho\in\mathcal{G}_{[N]}(\mathcal{F})$ defined as in Equation (3.1) we have that $\displaystyle\mathbf{W}^{\rho}_{\mathcal{S}}(\vec{x}_{\mathcal{S}}):=\textnormal{Tr}\left(Q^{\vec{x}_{\mathcal{S}}}_{\mathcal{S}}\rho\right)=\sum_{x_{i}\in\mathcal{X}_{i},i\in[N]\setminus\mathcal{S}}\mathbf{W}^{\rho}_{[N]}(\vec{x}).$ (5) Additionally for $\|\mathcal{S}\|=1$ the operators $Q^{x_{i}}_{i}$ are the operators from $\mathcal{M}_{i}$. #### Disjointness. It follows from the definition of POVMs that the quantum measurement operators need not be orthogonal, and this implies that the disjointness property (Property 2 does not hold in generality quantumly. Disjointness is a property that concerns a post-measurement state of a set $\mathcal{S}$ of variables. To ensure the existence of a measurement operator corresponding to $\mathcal{S}$, we need to assume Property 5. ###### Property 6 (Quantum Disjointness). Let $\mathbf{W}$ be a joint distribution for which Property 5 holds. We say that $\mathbf{W}$ has the _quantum disjointness property on set $\mathcal{F}$_ if for every subset $\mathcal{S}\subseteq[N]$, for every density operator $\rho\in\mathcal{G}_{[N]\setminus\mathcal{S}}(\mathcal{F})$, and for every value $\vec{x}\in\mathcal{X}_{1}\times\mathcal{X}_{2}\times\cdots\times\mathcal{X}_{N}$ and $\vec{y}\in\prod_{i\in\mathcal{S}}\mathcal{X}_{i}$, we have that if $\vec{y}\neq\vec{x}_{\mathcal{S}}$, then $\displaystyle\mathbf{W}^{\rho}_{[N]}(\vec{x}\mid\vec{y})\mathbf{W}^{\rho}_{[N]}(\vec{y})=\textnormal{Tr}\left(Q^{\vec{x}}_{[N]}R^{\vec{y}}_{\mathcal{S}}\rho R^{\vec{y}\dagger}_{\mathcal{S}}\right)=0.$ (6) #### Reducibility. Reducibility (Property 3) is a similar property to disjointness but with the key difference that we condition on the same event: ###### Property 7 (Quantum Reducibility). Let $\mathbf{W}$ be a joint distribution for which Property 5 holds. We say that $\mathbf{W}$ has the _quantum reducibility property on set $\mathcal{F}$_ if for every subset $\mathcal{S}\subset[N]$, every density operator $\rho\in\mathcal{G}_{[N]\setminus\mathcal{S}}(\mathcal{F})$, and value $\vec{x}\in\mathcal{X}_{1}\times\mathcal{X}_{2}\times\cdots\times\mathcal{X}_{N}$, we have that $\displaystyle\mathbf{W}^{\rho}_{[N]}(\vec{x}\mid\vec{x}_{\mathcal{S}})\mathbf{W}^{\rho}_{[N]}(\vec{x}_{\mathcal{S}})=\textnormal{Tr}\left(Q^{\vec{x}}_{[N]}R^{\vec{x}_{\mathcal{S}}}_{\mathcal{S}}\rho R^{\vec{x}_{\mathcal{S}}\dagger}_{\mathcal{S}}\right)=\textnormal{Tr}\left(Q^{\vec{x}}_{[N]}\rho\right).$ (7) new line Note that the last two properties together allow us to conclude that the operators are (morally) _on-state projections_ : Property 6 plays the role of different projectors being orthogonal and Property 7 that projecting twice to the same space does not change the resulting state. More concretely, we say that the $R_{i}$’s are on-state projectors on $\psi\in\mathcal{F}$ if for all $\mathcal{S}\subseteq[N]$, all $\vec{x}\in\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{N}$, all $\vec{y}\in\prod_{i\in\mathcal{S}}\mathcal{X}_{i}$, and for $R_{\mathcal{S}}^{\vec{y}}:=R^{y_{1}}_{\mathcal{S}(1)}R^{y_{2}}_{\mathcal{S}(2)}\cdots R^{y_{t}}_{\mathcal{S}(t)}$ and $Q_{\mathcal{S}}^{\vec{y}}:=R_{\mathcal{S}}^{\vec{y}\dagger}R_{\mathcal{S}}^{\vec{y}}$ (similarly for $[N]$), we have $\displaystyle\textnormal{Tr}\left(R^{\vec{y}\dagger}_{\mathcal{S}}Q^{\vec{x}}_{[N]}R^{\vec{y}}_{\mathcal{S}}\psi\right)=\delta_{\vec{y},\vec{x}_{\mathcal{S}}}\textnormal{Tr}\left(Q^{\vec{x}}_{[N]}\psi\right).$ (8) #### Sequential Independence As previously discussed, the notion of time order in the quantum setting is much more delicate as the probabilistic events no longer commute. Let us go back to the example of the simple sequence $(A,B)$ from Section 1.1 but now consider $A$ and $B$ as quantum observables measured on the state $\rho$. Let us assume for simplicity that $A$ and $B$ are projections. The probability of measuring $a$ with $A$ is $\textnormal{Tr}\left(A\rho\right)$ and the state after this measurement is $\rho_{a}:=\frac{A\rho A}{\textnormal{Tr}\left(A\rho\right)}$ so the probability of measuring the sequence $(a,b)$ equals $\mathbb{P}[b\leftarrow B(\rho_{a})]\mathbb{P}[a\leftarrow A(\rho)]=\textnormal{Tr}\left(B\frac{A\rho A}{\textnormal{Tr}(A\rho)}\right)\textnormal{Tr}\left(A\rho\right)=\textnormal{Tr}\left(ABA\rho\right).$ (9) On the other hand the probability of measuring the sequence $(b,a)$ equals $\textnormal{Tr}\left(BAB\rho\right)$ which is in general different than Equation (9). This simple example shows that sequential independence is not attained by all quantum joint probabilities. More formally, the notion of sequential independence from [GN02] for quantum joint probabilities can be stated as follows. ###### Property 8 (Quantum Sequential Independence). Let $\mathbf{W}$ be a joint distribution for which Property 5 holds. We say that $\mathbf{W}$ has the _quantum sequential independence property on set $\mathcal{F}$_ if for every density operator $\psi\in\mathcal{F}$, for any $\mathcal{T}\subseteq[N]$ of size $t$, for all $s\leq t$, partition $\mathcal{S}_{1},...,\mathcal{S}_{s}$ of $\mathcal{T}$, permutation $\sigma\in\Sigma_{s}$, and $\vec{x}\in\mathcal{X}_{1}\times\mathcal{X}_{2}\times\cdots\times\mathcal{X}_{N}$ such that $\vec{x}:=(x_{1},x_{2},\dots,x_{N})$ and $\vec{y}_{i}:=\left(x_{\mathcal{S}_{i}(1)},x_{\mathcal{S}_{i}(2)},\dots,x_{\mathcal{S}_{i}(\left\|\mathcal{S}_{i}\right\|)}\right)$, we have that Tr $\displaystyle\left(Q^{\vec{y}_{s}}_{\mathcal{S}_{s}}R^{\vec{y}_{s-1}}_{\mathcal{S}_{s-1}}\cdots R^{\vec{y}_{1}}_{\mathcal{S}_{1}}\psi R^{\vec{y}_{1}\dagger}_{\mathcal{S}_{1}}R^{\vec{y}_{2}\dagger}_{\mathcal{S}_{2}}\cdots R^{\vec{y}_{s-1}\dagger}_{\mathcal{S}_{s-1}}\right)$ $\displaystyle=\textnormal{Tr}\left(Q^{\vec{y}_{\sigma(s)}}_{\mathcal{S}_{\sigma(s)}}R^{\vec{y}_{\sigma(s-1)}}_{\mathcal{S}_{\sigma(s-1)}}\cdots R^{\vec{y}_{\sigma(1)}}_{\mathcal{S}_{\sigma(1)}}\psi R^{\vec{y}_{\sigma(1)}\dagger}_{\mathcal{S}_{\sigma(1)}}R^{\vec{y}_{\sigma(2)}\dagger}_{\mathcal{S}_{\sigma(2)}}\cdots R^{\vec{y}_{\sigma(s-1)}\dagger}_{\mathcal{S}_{\sigma(s-1)}}\right).$ (10) ###### Theorem 1 (Marginals imply Sequential Independence). If a joint distribution $\mathbf{W}$ has Property 5, 6, and 7, then it also has Property 8. ###### Proof. First note that Properties 5, 6, and 7 directly imply Equation (8). We now prove the statement for two sets of indices and then argue how to extend it for $s$ sets. In our restricted case, we have that $\displaystyle\textnormal{Tr}\left(Q^{\vec{y}_{1}}_{\mathcal{S}_{1}}R^{\vec{y}_{2}}_{\mathcal{S}_{2}}\psi R^{\vec{y}_{2}\dagger}_{\mathcal{S}_{2}}\right)=\textnormal{Tr}\left(\sum_{\vec{y}^{\prime}}Q^{\vec{y}^{\prime}}_{\mathcal{S}_{1}\cup\mathcal{S}_{2}}R^{\vec{y}_{2}}_{\mathcal{S}_{2}}\psi R^{\vec{y}_{2}\dagger}_{\mathcal{S}_{2}}\right)$ (11) $\displaystyle=\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{S}_{1}\cup\mathcal{S}_{2}}R^{\vec{y}_{2}}_{\mathcal{S}_{2}}\psi R^{\vec{y}_{2}\dagger}_{\mathcal{S}_{2}}\right)$ (12) $\displaystyle=\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{S}_{1}\cup\mathcal{S}_{2}}\psi\right),$ (13) where the first equality comes from the marginals property, the second equality comes from the disjointness property if we take $\vec{y}$ that agrees with $\vec{y}_{2}$ on $\mathcal{S}_{2}$, and the last equality comes from the reducibility property. The above derivation can be repeated for any other two sets $\mathcal{S}^{\prime}_{1}$ and $\mathcal{S}^{\prime}_{2}$ such that $\mathcal{S}^{\prime}_{1}\cup\mathcal{S}^{\prime}_{2}=\mathcal{S}_{1}\cup\mathcal{S}_{2}$. Any pair like this will yield $\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{S}_{1}\cup\mathcal{S}_{2}}\psi\right)$ that equals $\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{S}^{\prime}_{1}\cup\mathcal{S}^{\prime}_{2}}\psi\right)$ for all other sets $\mathcal{S}^{\prime}_{1}$ and $\mathcal{S}^{\prime}_{2}$, hence we have proven Property 8 for any two subsets. In general, we prove a slightly stronger statement than Property 8. We prove that not only different sequences give the same probabilities but also that these probabilities equal $\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{T}}\psi\right)$ for some operator $Q^{\vec{y}}_{\mathcal{T}}$. The general case holds by taking $\rho\in\mathcal{G}_{[N]\setminus\mathcal{S}_{s}}(\mathcal{F})$ instead of $\psi$ and use it in the above calculation. To prove Property 8 for any $s$ sets we consider $\rho=R^{\vec{y}_{s-1}}_{\mathcal{S}_{s-1}}\cdots R^{\vec{y}_{1}}_{\mathcal{S}_{1}}\psi R^{\vec{y}_{1}\dagger}_{\mathcal{S}_{1}}R^{\vec{y}_{2}\dagger}_{\mathcal{S}_{2}}\cdots R^{\vec{y}_{s-1}\dagger}_{\mathcal{S}_{s-1}}$ and $\textnormal{Tr}\left(Q^{\vec{y}_{s}}_{\mathcal{S}_{s}}\rho\right)$. We shave off operators from $\rho$ one by one using Equation (13). After $s$ steps we have that $\textnormal{Tr}\left(Q^{\vec{y}_{s}}_{\mathcal{S}_{s}}\rho\right)=\textnormal{Tr}\left(Q^{\vec{x}_{\mathcal{T}}}_{\mathcal{T}}\psi\right)$. Again, repeating this procedure for a different $\rho^{\prime}$ with sets that sum to the same $\mathcal{T}$ yields the claimed result. ∎ ### 3.2 Joint Distributions and Permutability In this section, we state the definition of a joint distribution that describes a sequence of quantum measurements done on states from a small set. We also define a generalization of commutativity and prove that a joint distribution exists if and only if the measurement operators are permutable. ###### Definition 2 (Quantum Joint Distribution On State). A joint distribution of $N$ random variables $X_{1},\dots,X_{N}$ that describe outcomes of general quantum measurements of states $\psi\in\mathcal{F}$ is defined as a positive, normalized, and linear functional $\displaystyle\mathbf{W}_{[N]}:\mathcal{D}(\mathcal{H})\times(\mathcal{X}_{1}\times\mathcal{X}_{2}\times\cdots\times\mathcal{X}_{N})\to[0,1],$ (14) for which 1. (1) Quantum Marginals on states in $\mathcal{F}$, Property 5, holds, 2. (2) Quantum Disjointness on states in $\mathcal{F}$, Property 6, holds, 3. (3) Quantum Reducibility on states in $\mathcal{F}$, Property 7, holds, 4. (4) Quantum Sequential Independence on states in $\mathcal{F}$, Property 8, holds. Next we show the connection between the existence of joint distributions and requirements on the measurement operators. But first let us define the notion of on-state permutability. ###### Definition 3 (On-state permutator, (fully) permutable operators). For any $s$ operators $R_{i}$ and a permutation of the $s$-element set $\sigma\in\Sigma_{s}$ the permutator on $\psi$ is defined as $\displaystyle[R_{1},$ $\displaystyle R_{2},\dots,R_{s}]_{\psi}(\sigma):=$ $\displaystyle\textnormal{Tr}\left(R_{s}R_{s-1}\cdots R_{1}\psi R^{\dagger}_{1}R^{\dagger}_{2}\cdots R^{\dagger}_{s}\right)-\textnormal{Tr}\left(R_{\sigma(s)}R_{\sigma(s-1)}\cdots R_{\sigma(1)}\psi R^{\dagger}_{\sigma(1)}R^{\dagger}_{\sigma(2)}\cdots R^{\dagger}_{\sigma(s)}\right).$ (15) We say that the operators $R_{1},\dots,R_{s}$ are _permutable_ if $[R_{1},\dots,R_{s}]_{\psi}(\sigma)=0$ for all $\sigma\in\Sigma_{s}$. Moreover, we call a set of measurements $\\{\mathcal{M}_{i}\\}_{i}$ _fully permutable on $\mathcal{F}$_ if all square-root operators $R^{\vec{y}}_{i}$ of these measurements $\mathcal{M}_{i}$ permute on $\psi\in\mathcal{F}$ for all $\sigma\in\Sigma_{s}$. Now we state the theorem connecting existence of joint distributions with permutability of the measurement operators. This statement extends Theorem $3.2$ of [ME84], where they prove that if the joint distribution satisfies the marginals property (they use the name “nondisturbance”), then the operators pairwise commute. ###### Theorem 4 (Quantum Joint Distribution and Permutability). There is a quantum joint distribution on states $\psi\in\mathcal{F}$ describing measurement outcomes of $N$ observables $X_{1},\dots,X_{N}$ if and only if all square roots $R^{x}_{i}$ of $Q^{x}_{i}$ operators of measurements $\mathcal{M}_{i}$ permute on $\psi\in\mathcal{F}$ and are on-state projectors according to Equation (8). ###### Proof. $(\Rightarrow)$ Permutability follows from Property 8 for $N$ single-element sets $\mathcal{S}_{i}=\\{i\\}$. Being on-state projectors (Equation (8)) follows from Property 6 and Property 7. $(\Leftarrow)$ The other direction of the proof follows by setting the measurement operators to $Q_{\mathcal{S}}^{\vec{y}}:=R^{y_{t}\dagger}_{\mathcal{S}(t)}R^{y_{t-1}\dagger}_{\mathcal{S}(t-1)}\cdots R^{y_{1}\dagger}_{\mathcal{S}(1)}R^{y_{1}}_{\mathcal{S}(1)}R^{y_{2}}_{\mathcal{S}(2)}\cdots R^{y_{t}}_{\mathcal{S}(t)}$, for every set $\mathcal{S}\subseteq[N]$ with $\|\mathcal{S}\|=t$. Similarly we define $R_{\mathcal{S}}^{\vec{y}}:=R^{y_{1}}_{\mathcal{S}(1)}R^{y_{2}}_{\mathcal{S}(2)}\cdots R^{y_{t}}_{\mathcal{S}(t)}$. The marginals property follows from the fact that $\sum_{x_{i}\in\mathcal{X}_{i}}Q_{i}^{x_{i}}=\mathbbm{1}$: $\displaystyle\sum_{x_{i}\in\mathcal{X}_{i},i\in[N]\setminus\mathcal{S}}\textnormal{Tr}\left(Q^{x_{i_{i}}}_{i_{1}}R^{\vec{y}}_{[N]\setminus\\{i_{1}\\}}\rho R^{\vec{y}\dagger}_{[N]\setminus\\{i_{1}\\}}\right)$ $\displaystyle=\sum_{x_{i}\in\mathcal{X}_{i},i\in[N]\setminus(\mathcal{S}\cup\\{i_{1}\\})}\textnormal{Tr}\left(Q^{x_{i_{2}}}_{i_{2}}R^{\vec{y}}_{[N]\setminus\\{i_{1},i_{2}\\}}\rho R^{\vec{y}\dagger}_{[N]\setminus\\{i_{1},i_{2}\\}}\right)=\cdots=\textnormal{Tr}\left(Q^{\vec{y}}_{\mathcal{S}}\rho\right).$ (16) Properties 6 and 7 are natural consequences of Equation (8). The only difference between the properties and on-state projections is the set $\mathcal{G}_{[N]\setminus\mathcal{S}}(\mathcal{F})$ versus just $\mathcal{F}$. Nonetheless, with our definition of $R_{\mathcal{S}}^{\vec{y}}$, Equation (8) implies disjointness and reducibility. In Theorem 1 we have already proved that marginals together with disjointness and reducibility imply sequential independence, which concludes our proof. ∎ ### 3.3 Pairwise on-state commutation does not imply full commutation We now investigate whether full permutability is the weakest assumption we can have for joint distributions to exist. When we consider this question for the full Hilbert space $\mathcal{F}=\mathcal{D}(\mathcal{H})$, this problem has been considered by a number of works in the literature [Nel67, Fin73, Fin82, ME84] and it is well- known that it suffices for the measurement operators to pairwise commute, i.e., pairwise commutation on all possible quantum states implies permutability of the operators. Our goal in this section is to consider the case where $\mathcal{F}\subsetneq\mathcal{D}(\mathcal{H})$. In particular, in [CETU18], in order to connect perfect quantum indifferentiability to classical indifferentiability with stateless simulators777 Roughly, we say that $A$ is classical (quantum) indifferentiable from $B$ iff we can map classical (resp. quantum) attacks on $A$ to classical (resp. quantum) attacks on $B$ using simulators. Moreover, we say that the simulator is stateless if it does not store any internal state. , they rely on the following conjecture. ###### Conjecture 2 (Conjecture 2 from [CETU18]). Consider $N$ binary measurements described by projectors $P_{1},\dots,P_{N}$, and a quantum state $\|\Psi\rangle$. Assume that any $t$ out of the $N$ measurements permute on state $\|\Psi\rangle$. That is, for any $I$ with $\|I\|=t$, if $P^{\prime}_{1},\dots,P^{\prime}_{t}$ and $P^{\prime\prime}_{1},\dots,P^{\prime\prime}_{t}$ are the projectors $\\{P_{i}\\}_{i\in I}$ (possibly in different order), then $P^{\prime}_{t},\dots,P^{\prime}_{1}\|\Psi\rangle=P^{\prime\prime}_{t},\dots,P^{\prime\prime}_{1}\|\Psi\rangle$. Then there exist random variables $X_{1},\dots,X_{N}$ with a joint distribution $D$ such that for any $I=\\{i_{1},\dots,i_{t}\\}$ the joint distribution of $X_{i_{1}},\dots,X_{i_{t}}$ is the distribution of the outcomes when we measure $\|\Psi\rangle$ with measurements $P_{i_{1}},\dots,P_{i_{t}}$. Conjecture 2 states that if any $t$ measurement operators permute on state $\|\Psi\rangle$ then there is a joint distribution. From Theorem 4, we know that if there is a joint distribution then the operators fully permute. Hence, the key point of the conjecture is that if any $t$ operators permute on a state, then they fully permute on it. However, we show here that 2 is not true, in general. ###### Theorem 5. There is a set of four projectors $\\{P_{1},P_{2},P_{3},P_{4}\\}$ and a state $\|\phi\rangle\in\mathbb{C}^{8}$ such that the projectors are 2-permutable (they pairwise commute) on state $\|\phi\rangle$ and they are _not_ 4-permutable on $\|\phi\rangle$. ###### Proof. To prove the statement we found an example of such operators and a state numerically by random constrained search. We consider 4 projectors $P_{i}$ and a state $\|\phi\rangle$ of dimension 8. The constraints we impose are $\displaystyle\forall i,j\neq i\;[P_{i},P_{j}]\|\phi\rangle=0,$ (17) moreover operators $P_{i}$ are projectors: $\forall iP_{i}^{2}=P_{i}$ and $\|\phi\rangle$ is a unit-norm complex vector. We look for an example to the statement that 2-permutability (commutativity) does not imply 4-permutability, so that $\displaystyle(P_{1}P_{2}P_{3}P_{4}-P_{3}P_{4}P_{1}P_{2})\|\phi\rangle\neq 0.$ (18) To find such example we used software for symbolic computing to define the problem and maximize $\\|(P_{1}P_{2}P_{3}P_{4}-P_{3}P_{4}P_{1}P_{2})\|\phi\rangle\\|$, where we maximize over operators $P_{i}$ and the state $\|\phi\rangle$. The result of our optimization can be found in Appendix A. Note that however we consider vector equalities—instead of just traces like in Theorem 4—our example provides a stronger argument for the necessity of full permutability. ∎ One can notice by looking at the optimization problem that it is not a semidefinite problem, nor that it has any other structure that is easy to exploit. For that reason finding larger instances is computationally very expensive. We notice that Theorem 5 actually disproves a slightly stronger version of Conjecture 2. In the use of Conjecture 2 in [CETU18], they implicitly assume that we can replace $P_{i}$ by $\mathbbm{1}-P_{i}$ and the permutation still holds. While this modification gives a slightly stronger assumption, our counterexample in Theorem 5 works just as well. For the joint distribution in Conjecture 2 to exist, we know from Theorem 4 that all operators must on-state permute. An important observation is that Conjecture 2 regards vector equalities and Theorem 4 regards measurement outcomes. The theorem is “easier” than the former and our counterexample works with vector equalities, hence we indeed disprove Conjecture 2. In [Ebr18] the author uses a different conjecture and different reasoning to prove the existence of the joint distribution. Our counterexample does not disprove this other approach and we refer interested readers to [Ebr18]. ## 4 Almost On-State Commutation In the last part of the paper we discuss almost commutativity in the on-state case. In particular, we show here that if we have two projectors that almost commute on a state then we can define a projector that fully commutes with one of the original operators and is on-state close to the second one. The main tool that we need to prove this result is the Jordan’s lemma. ###### Lemma 6 (Jordan’s lemma [Jor75]). Let $P_{1}$ and $P_{2}$ be two projectors with rank $r_{i}:=\textnormal{rank}(P_{i})$ for $i\in\\{1,2\\}$. Then both projectors can be decomposed simultaneously in the form $P_{i}=\bigoplus_{k=1}^{r_{i}}P_{i}^{k}$, where $P_{i}^{k}$ denote rank-1 projectors acting on one- or two-dimensional subspaces. We denote the one- and two-dimensional subspaces by $S_{1},\dots,S_{l}$ and subspaces by $T_{1},\dots,T_{l^{\prime}}$, respectively. The eigenvectors $\|v_{k,1}\rangle$ and $\|v_{k,2}\rangle$ of $P_{1}^{k}$ and $P_{2}^{k}$ respectively are related by: $\displaystyle\|v_{k,2}\rangle=\cos\theta_{k}\|v_{k,1}\rangle+\sin\theta_{k}\|v^{\perp}_{k,1}\rangle,\|v_{k,1}\rangle=\cos\theta_{k}\|v_{k,2}\rangle-\sin\theta_{k}\|v^{\perp}_{k,2}\rangle.$ (19) We can now prove our result. ###### Theorem 7 (Making almost commuting projectors commute). Given any two projectors $P_{1}$ and $P_{2}$ and a state $\|\psi\rangle$ we have that if $\left\\|(P_{1}P_{2}-P_{2}P_{1})\|\psi\rangle\right\\|=\epsilon$ then there is a projector $P_{2}^{\prime}$ that is close to the original projector on the state $\left\\|(P_{2}^{\prime}-P_{2})\|\psi\rangle\right\\|\leq\sqrt{2}\epsilon$ and $[P_{1},P_{2}^{\prime}]=0$. ###### Proof. By Jordan’s lemma (Lemma 6), there exist bits $\lambda_{i,1},\lambda_{i,2}\in\\{0,1\\}$ and vectors $\|u_{1}\rangle,...,\|u_{m}\rangle$ and $\|v_{1,1}\rangle,\|v_{1,2}\rangle,...,\|v_{\ell,1}\rangle,\|v_{\ell,2}\rangle$, such that 1. 1. $P_{1}=\sum_{i\in[m]}\lambda_{i,1}\|u_{i}\rangle\langle u_{i}\|+\sum_{i\in[\ell]}\|v_{i,1}\rangle\langle v_{i,1}\|$ and $P_{2}=\sum_{i\in[m]}\lambda_{i,2}\|u_{i}\rangle\langle u_{i}\|+\sum_{i\in[\ell]}\|v_{i,2}\rangle\langle v_{i,2}\|$; 2. 2. $\langle u_{i}\|u_{k}\rangle=0$ and $\langle u_{i}\|v_{j,b}\rangle=0$ for all $b,i,j$ and $k\neq i$; 3. 3. $\langle v_{j,b^{\prime}}\|v_{i,b}\rangle=0$ for $i\neq j$ and any $b,b^{\prime}$; 4. 4. $0<\langle v_{i,1}\|v_{i,2}\rangle<1$. Let $\theta_{i}$ be the angle between $\|v_{i,1}\rangle$ and $\|v_{i,2}\rangle$ (i.e. $\cos\theta_{i}=\langle v_{i,1}\|v_{i,2}\rangle$), and $\|v_{i,1}^{\perp}\rangle$ be the state orthogonal to $\|v_{i,1}\rangle$ in the subspace spanned by these two vectors. Since the non-commuting part of $P_{1}$ and $P_{2}$ must come from the pairs $\|v_{i,1}\rangle,\|v_{i,2}\rangle$, we will define $P_{2}^{\prime}$ by removing the non-commuting part of $P_{2}$, shifting the vector $\|v_{i,2}\rangle$, to either $\|v_{i,1}\rangle$ or $\|v_{i,1}^{\perp}\rangle$: $P_{2}^{\prime}=\sum_{i\in[m]}\lambda_{i,2}\|u_{i}\rangle\langle u_{i}\|+\sum_{i\in[\ell]:\theta_{i}\leq\frac{\pi}{4}}\|v_{i,1}\rangle\langle v_{i,1}\|+\sum_{i\in[\ell]:\theta_{i}>\frac{\pi}{4}}\|v_{i,1}^{\perp}\rangle\langle v_{i,1}^{\perp}\|.$ We have clearly that $[P_{1},P_{2}^{\prime}]=0$ since the two projectors are simultaneously diagonalizable and we now want to prove that $\displaystyle\left\\|(P_{2}^{\prime}-P_{2})\|\psi\rangle\right\\|\leq\sqrt{2}\varepsilon.$ Notice that $\displaystyle\left\\|(P_{2}^{\prime}-P_{2})\|\psi\rangle\right\\|^{2}$ $\displaystyle=\left\\|\sum_{i\in[l^{\prime}]:\theta_{i}\leq\frac{\pi}{4}}\left(\|v_{i,1}\rangle\langle v_{i,1}\|\psi\rangle-\|v_{i,2}\rangle\langle v_{i,2}\|\psi\rangle\right)+\sum_{i\in[l^{\prime}]:\theta_{i}>\frac{\pi}{4}}\left(\|v_{i,1}^{\perp}\rangle\langle v_{i,1}^{\perp}\|\psi\rangle-\|v_{i,2}\rangle\langle v_{i,2}\|\psi\rangle\right)\right\\|^{2}$ (20) $\displaystyle=\sum_{i\in[l^{\prime}]:\theta_{i}\leq\frac{\pi}{4}}\left\\|\|v_{i,1}\rangle\langle v_{i,1}\|\psi\rangle-\|v_{i,2}\rangle\langle v_{i,2}\|\psi\rangle\right\\|^{2}+\sum_{i\in[l^{\prime}]:\theta_{i}>\frac{\pi}{4}}\left\\|\|v_{i,1}^{\perp}\rangle\langle v_{i,1}^{\perp}\|\psi\rangle-\|v_{i,2}\rangle\langle v_{i,2}\|\psi\rangle\right\\|^{2},$ (21) where in the last step we used that $\langle v_{i,b^{\prime}}\|v_{j,b}\rangle=0$ for $i\neq j$. Using that $\|v_{i,2}\rangle=\cos{\theta_{i}}\|v_{i,1}\rangle+\sin{\theta_{i}}\|v_{i,1}^{\perp}\rangle$, we have that if $\theta_{i}\leq\frac{\pi}{4}$, then $\displaystyle\left\\|\langle v_{i,1}\|\psi\rangle\|v_{i,1}\rangle-\langle v_{i,2}\|\psi\rangle\|v_{i,2}\rangle\right\\|^{2}$ $\displaystyle=\sin^{4}\theta_{i}\|\langle v_{i,1}\|\psi\rangle\|^{2}-2\sin^{3}\theta_{i}\cos\theta_{i}\mathfrak{Re}(\langle v_{i,1}^{\perp}\|\psi\rangle\langle v_{i,1}\|\psi\rangle)+\sin^{2}\theta_{i}\cos^{2}\theta_{i}\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}$ $\displaystyle+\sin^{4}\theta_{i}\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}+2\sin^{3}\theta_{i}\cos\theta_{i}\mathfrak{Re}(\langle v_{i,1}\|\psi\rangle\langle v_{i,1}^{\perp}\|\psi\rangle)+\sin^{2}\theta_{i}\cos^{2}\theta_{i}\|\langle v_{i,1}\|\psi\rangle\|^{2}$ $\displaystyle\leq 2\sin^{2}\theta_{i}\cos^{2}\theta_{i}(\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}+\|\langle v_{i,1}\|\psi\rangle\|^{2}),$ (22) where in the inequality we used our assumption that $\theta_{i}\leq\frac{\pi}{4}$ which implies that $\sin\theta_{i}\leq\cos\theta_{i}$. Using similar calculations, we have that if $\theta_{i}\geq\frac{\pi}{4}$ $\displaystyle\left\\|\langle v_{i,1}^{\perp}\|\psi\rangle\|v_{i,1}^{\perp}\rangle-\langle v_{i,2}\|\psi\rangle\|v_{i,2}\rangle\right\\|^{2}\leq 2\sin^{2}\theta_{i}\cos^{2}\theta_{i}(\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}+\|\langle v_{i,1}\|\psi\rangle\|^{2}).$ (23) We will show now that $\sum_{i}\sin^{2}\theta_{i}\cos^{2}\theta_{i}(\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}+\|\langle v_{i,1}\|\psi\rangle\|^{2})=\varepsilon^{2},$ which finishes the proof: $\displaystyle\varepsilon^{2}$ $\displaystyle=\left\\|(P_{2}P_{1}-P_{1}P_{2})\|\psi\rangle\right\\|^{2}$ $\displaystyle=\left\\|\sum_{i\in[l^{\prime}]}\|v_{i,1}\rangle\langle v_{i,1}\|v_{i,2}\rangle\langle v_{i,2}\|\psi\rangle-\|v_{i,2}\rangle\langle v_{i,2}\|v_{i,1}\rangle\langle v_{i,1}\|\psi\rangle.\right\\|^{2}$ $\displaystyle=\left\\|\sum_{i\in[l^{\prime}]}\cos{\theta_{i}}\left(\left(\cos{\theta_{i}}\langle v_{i,1}\|\psi\rangle+\sin{\theta_{i}}\langle v_{i,1}^{\perp}\|\psi\rangle\right)\|v_{i,1}\rangle-\langle v_{i,1}\|\psi\rangle\left(\cos{\theta_{i}}\|v_{i,1}\rangle+\sin{\theta_{i}}\|v_{i,1}^{\perp}\rangle\right)\right)\right\\|^{2}$ $\displaystyle=\left\\|\sum_{i\in[l^{\prime}]}\sin{\theta_{i}}\cos{\theta_{i}}\left(\langle v_{i,1}^{\perp}\|\psi\rangle\|v_{i,1}\rangle-\langle v_{i,1}\|\psi\rangle\|v_{i,1}^{\perp}\rangle\right)\right\\|^{2}$ $\displaystyle=\sum_{i\in[l^{\prime}]}\sin^{2}\theta_{i}\cos^{2}\theta_{i}\left(\|\langle v_{i,1}^{\perp}\|\psi\rangle\|^{2}+\|\langle v_{i,1}\|\psi\rangle\|^{2}\right).$ where in the second equality we again use that $\|v_{i,2}\rangle=\cos{\theta_{i}}\|v_{i,1}\rangle+\sin{\theta_{i}}\|v_{i,1}^{\perp}\rangle$ and in the fourth equality we use the fact that $\langle v_{i,b^{\prime}}\|v_{j,b}\rangle=0$ for $i\neq j$. ∎ Our proof relies solely on Jordan’s Lemma Note that Jordan’s Lemma is sufficient only if we analyze commutation of projectors. Results that show how to make any Hermitian matrices commute [FR96, Has09] are much more complicated to prove and it is not clear how to translate them to the “on-state” case. We stress that our proof only works for two projectors, since Jordan’s lemma does not generalize for three or more projectors. Therefore, we leave as open problem (dis)proving a generalized version of 7 for more projectors. In [Ebr18] the author proves Theorem 7 for $\varepsilon=0$, but with a different proof. They use Halmo’s two rojections theorem instead of Jordan’s lemma. ## References * [BN18] Andreas Bluhm and Ion Nechita. Joint measurability of quantum effects and the matrix diamond. Journal of Mathematical Physics, 59(11):112202, 2018. * [CETU18] Tore Vincent Carstens, Ehsan Ebrahimi, Gelo Noel Tabia, and Dominique Unruh. On quantum indifferentiability. Technical report, Cryptology ePrint Archive, Report 2018/257, 2018. https://eprint. iacr. org/2018/257, 2018. * [CZ83] Gianni Cassinelli and N Zanghi. Conditional probabilities in quantum mechanics. i. – conditioning with respect to a single event. Il Nuovo Cimento B (1971-1996), 73(2):237–245, 1983. * [Cza21] Jan Czajkowski. Github repository “joints-counterexample”, 2021. * [Ebr18] Ehsan Ebrahimi. Post-quantum security in the presence of superposition queries. 2018\. * [Fin73] Arthur Fine. Probability and the interpretation of quantum mechanics. The British Journal for the Philosophy of Science, 24(1):1–37, 1973\. * [Fin82] Arthur Fine. Joint distributions, quantum correlations, and commuting observables. Journal of Mathematical Physics, 23(7):1306–1310, 1982. * [FR96] Peter Friis and Mikael Rørdam. Almost commuting self-adjoint matrices-a short proof of huaxin lin’s theorem. Journal fur die Reine und Angewandte Mathematik, 479:121–132, 1996\. * [GN01] Stan Gudder and Gabriel Nagy. Sequential quantum measurements. Journal of Mathematical Physics, 42(11):5212–5222, 2001. * [GN02] Stan Gudder and Gabriel Nagy. Sequentially independent effects. Proceedings of the American Mathematical Society, 130(4):1125–1130, 2002. * [Has09] Matthew B Hastings. Making almost commuting matrices commute. Communications in Mathematical Physics, 291(2):321–345, 2009. * [HMZ16] Teiko Heinosaari, Takayuki Miyadera, and Mário Ziman. An invitation to quantum incompatibility. Journal of Physics A: Mathematical and Theoretical, 49(12):123001, 2016. * [Jor75] Camille Jordan. Essai sur la géométrie à $n$ dimensions. Bulletin de la Société mathématique de France, 3:103–174, 1875. * [Lin97] Huaxin Lin. Almost commuting selfadjoint matrices and applications. Fields Inst. Commun, 13:193–233, 1997. * [Maa06] Hans Maassen. Quantum Probability and Quantum Information Theory. https://www.math.ru.nl/~maassen/lectures/Trieste.pdf, 2006\. * [Maa10] Hans Maassen. Quantum probability and quantum information theory. In Quantum information, computation and cryptography, pages 65–108. Springer, 2010. * [ME84] WM de Muynck and JPHW van den Eijnde. A derivation of local commutativity from macrocausality using a quantum mechanical theory of measurement. Foundations of physics, 14(2):111–146, 1984. * [NC11] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 10th edition, 2011. * [Nel67] Edward Nelson. Dynamical theories of Brownian motion, volume 3. Princeton university press, 1967. ## Appendix A Numerical values Below we present the state and the projectors that are claimed in the proof of Theorem 5. The script used to generate these values can be found in [Cza21]. Before we write out the state and the projections that we found, let us state our violation of the permutator: $\displaystyle\\|(P_{1}P_{2}P_{3}P_{4}-P_{3}P_{4}P_{1}P_{2})\|\phi\rangle\\|=0.25\pm 3\cdot 10^{-8}.$ (24) All the constraints listed in the proof of Theorem 5 are fulfilled up to the seventh decimal digit of precision, so up to $10^{-7}$. Internal computations of the algorithm are performed with machine precision of $10^{-15}$. The state is $\displaystyle\|\phi\rangle:=\left(\begin{array}[]{c}-0.135381-0.0503468\text{i}\\\ 0.325588\,-0.222403\text{i}\\\ -0.209447-0.0404665\text{i}\\\ -0.418336+0.130098\text{i}\\\ -0.503693-0.299414\text{i}\\\ 0.379842\,+0.205081\text{i}\\\ -0.179291-0.0381456\text{i}\\\ 0.0840381\,-0.125995\text{i}\end{array}\right).$ (33) We define the projectors by their eigenvectors: $\displaystyle P_{1}=\|\pi^{1}\rangle\langle\pi^{1}\|,$ $\displaystyle P_{2}=\|\pi^{2}_{1}\rangle\langle\pi^{2}_{1}\|+\|\pi^{2}_{2}\rangle\langle\pi^{2}_{2}\|,$ (34) $\displaystyle P_{3}=\|\pi^{3}_{1}\rangle\langle\pi^{3}_{1}\|+\|\pi^{3}_{2}\rangle\langle\pi^{3}_{2}\|+\|\pi^{3}_{3}\rangle\langle\pi^{3}_{3}\|,$ $\displaystyle P_{4}=\|\pi^{4}_{1}\rangle\langle\pi^{4}_{1}\|+\|\pi^{4}_{2}\rangle\langle\pi^{4}_{2}\|.$ (35) The eigenvector of $P_{1}$ is: $\displaystyle\|\pi^{1}\rangle:=\left(\begin{array}[]{c}0.440777\,+0.168408\text{i}\\\ 0.208781\,-0.37351\text{i}\\\ 0.247514\,+0.0276065\text{i}\\\ -0.297971+0.0252308\text{i}\\\ 0.118798\,+0.112225\text{i}\\\ -0.293428+0.270889\text{i}\\\ -0.193073+0.218869\text{i}\\\ -0.41405\end{array}\right).$ (44) The eigenvectors of $P_{2}$ are: $\displaystyle\|\pi^{2}_{1}\rangle:=\left(\begin{array}[]{c}-0.497016-0.094035\text{i}\\\ 0.417527\,-0.0737062\text{i}\\\ -0.000125303+0.35123\text{i}\\\ 0.166569\,-0.187245\text{i}\\\ -0.373202+0.205633\text{i}\\\ 0.318452\,-0.251475\text{i}\\\ -0.107473-0.123987i\\\ -0.0711523\end{array}\right),\|\pi^{2}_{2}\rangle:=\left(\begin{array}[]{c}0.365906\,+0.0620997\text{i}\\\ 0.418728\,-0.2059\text{i}\\\ 0.229457\,+0.0557421\text{i}\\\ -0.140393+0.0945029\text{i}\\\ -0.199205-0.188139\text{i}\\\ 0.103617\,+0.279644\text{i}\\\ -0.546498+0.147197\text{i}\\\ 0.275295\end{array}\right).$ (61) The eigenvectors of $P_{3}$ are: $\displaystyle\|\pi^{3}_{1}\rangle:=\left(\begin{array}[]{c}-0.453059+0.181543\text{i}\\\ -0.452841+0.0154095\text{i}\\\ -0.17948-0.222827\text{i}\\\ -0.230355-0.0526756\text{i}\\\ -0.0918752-0.250754\text{i}\\\ 0.242416\,-0.126917\text{i}\\\ 0.300832\,-0.287566\text{i}\\\ 0.315259\end{array}\right),\|\pi^{3}_{2}\rangle:=\left(\begin{array}[]{c}-0.0586669-0.269559\text{i}\\\ -0.280155+0.373271\text{i}\\\ -0.150758-0.158539\text{i}\\\ 0.158793\,-0.0454731\text{i}\\\ 0.165888\,+0.362832\text{i}\\\ -0.110453-0.310755\text{i}\\\ 0.353894\,-0.00811586\text{i}\\\ -0.487537\end{array}\right),$ (78) $\displaystyle\|\pi^{3}_{3}\rangle:=\left(\begin{array}[]{c}-0.182739-0.114718\text{i}\\\ 0.246775\,-0.134678\text{i}\\\ -0.513357-0.193655\text{i}\\\ -0.10451+0.421294\text{i}\\\ 0.111183\,+0.122625\text{i}\\\ -0.200917-0.25897\text{i}\\\ -0.0290851+0.398494\text{i}\\\ 0.30081\end{array}\right).$ (87) The eigenvectors of $P_{4}$ are: $\displaystyle\|\pi^{4}_{1}\rangle:=\left(\begin{array}[]{c}-0.464187+0.213035\text{i}\\\ -0.364421+0.119836\text{i}\\\ -0.324984-0.23097\text{i}\\\ -0.256841+0.0478513\text{i}\\\ -0.0700499-0.192822\text{i}\\\ 0.146148\,-0.225755\text{i}\\\ 0.243944\,-0.284786\text{i}\\\ 0.331272\end{array}\right),\|\pi^{4}_{2}\rangle:=\left(\begin{array}[]{c}0.111757\,+0.151275\text{i}\\\ 0.236223\,-0.323279\text{i}\\\ 0.157312\,-0.115385\text{i}\\\ -0.30864+0.0990552\text{i}\\\ -0.260931-0.236239\text{i}\\\ 0.240497\,+0.13559\text{i}\\\ -0.453404+0.12357\text{i}\\\ 0.490125\end{array}\right).$ (104)
# Extremal Laws for Laplacian Random Matrices ††thanks: Research supported by Conacyt Grant A1-S-976 Santiago Arenas-Velilla Victor Pérez-Abreu CIMAT, Guanajuato, Mexico, <EMAIL_ADDRESS>Politécnico Nacional, Mexico<EMAIL_ADDRESS> ###### Abstract For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non- asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$. An expository review of existing results for the asymptotic spectrum of a Laplacian random matrix is also presented, with the goal of noting the differences from the corresponding classical results for Wigner random matrices. Extensions to Laplacian block random matrices are indicated. ## 1 Introduction The largest eigenvalue of a random Laplacian matrix plays an important role in applications of complex graphs [11], analysis of algorithms for the $\mathbb{Z}_{2}$ Synchronization, and community detection in the Stochastic Block Model [1, 13], among other optimization problems in semidefinite programming [4]. Its limiting behavior both almost surely and in probability has been considered by Bryc, Dembo and Jiang [8] and Ding and Jiang [12]. The present paper considers the limiting law of the largest eigenvalue of a random Laplacian matrix. The famous Tracy–Widom distribution appears in random matrix theory as the limiting law of the largest eigenvalue for a large class of random matrices, including classical Gaussian [25], [26], Wigner [24], Wishart [15], and sparse matrices [17]. However, there are examples of structured random matrices constructed from a number of independent random variables less than the one used to construct Wigner matrices, and for which the limiting law of the largest eigenvalue is Gumbel. This is the case for palindromic circulant random matrices, as shown by Bryc and Sethuraman [9], and for Gaussian Hermitian circulant matrices, as shown by Meckes [19]. The Gumbel distribution is also the limiting law of the spectral radius of Ginibre ensembles [23]. One of the purposes of the present paper is to show that the Gumbel distribution is also the limiting law of the largest eigenvalue of a Laplacian matrix constructed from independent real Gaussian random variables. This kind of random matrix appears in the $\mathbb{Z}_{2}$ Synchronization problem of recovering binary labels with Gaussian noise [4]. More specifically, let $X=(X_{ij})$ be an $n\times n$ symmetric matrix. The Laplacian matrix $L=L_{X}=(L_{ij})$ of $X$ is defined by $L=D_{X}-X$ (1.1) where $D_{X}$ is the diagonal matrix whose entries are given by $(D_{X})_{ii}=\sum_{j=1}^{n}X_{ij}.$ (1.2) Matrices whose row sums are zero are often called Markov matrices, so a Laplacian matrix is an example of a Markov matrix. Let $X=(X_{ij})$ be a Wigner random matrix, that is, an $n\times n$ symmetric random matrix whose off-diagonal entries $\\{X_{ij},1\leq i<j\leq n\\}$ are real-valued, independent and identically distributed random variables with zero mean and unit variance, and the diagonal entries $\\{X_{ii},1\leq i\leq n\\}$ are independent random variables with zero mean and variance two that are independent of the off-diagonal entries. We observe that for a Wigner matrix $X$, the corresponding Laplacian matrix $L$ has $n(n-1)/2$ independent random variables and its diagonal entries are not independent of the non-diagonal entries, while $X$ has $n(n+1)/2$ independent random variables. Let $\lambda_{\max}(X)$ denote the largest eigenvalue of $X$ and $\lambda_{\max}(L)$ denote that of $L$. It is well known that under suitable moment conditions, the limiting spectral distribution of $X/\sqrt{n}$ is the semicircle law in $[-2,2]$, that $\lambda_{\max}(X)/\sqrt{n}$ converges to $2$ almost surely as $n$ goes to infinity, and that the limiting law of $\lambda_{\max}(X)$, after appropriate rescaling, obeys the Tracy–Widom distribution. In contrast, for the Laplacian matrix $L$, under suitable conditions on the entries of $X$, the limiting spectral law of $L/\sqrt{n}$ is a distribution with unbounded support which is the free convolution of the semicircular law with the standard Gaussian distribution [8], [12]. Moreover, $\lambda_{\max}(L)/\sqrt{n\log n}$ converges to $\sqrt{2}$ in probability as $n$ goes to infinity [12]. In Section 2 we provide a brief summary of the above results for Laplacian random matrices, and also indicate some extensions to block matrices. Our main result is the weak convergence of $\lambda_{\max}(L)/\sqrt{n-1}$ with an appropriate rescaling to the Gumbel distribution, as presented in Theorem 4. In order to prove this result, we first show in Section 3 that the limiting law of the largest entry $\max_{i}L_{ii}/\sqrt{n-1}$ of the sequence of the triangular array $\\{L_{ii}=L_{ii}^{(n)}:i=1,\ldots,n\\}$ is a Gumbel distribution for which we use the form of the non-stationary covariance matrix of $\\{L_{ii}\\}$ to produce a useful approximating sequence of independent Gaussian random variables. Then we present in Section 4 suitable non- asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$ for a class of Laplacian matrices, namely, those such that $\max_{i}L_{ii}/\sqrt{n-1}$ grows faster than $\sqrt{\log n}$. Finally, we prove in Section 5 that, after appropriate rescaling, $\max_{i}L_{ii}/\sqrt{n-1}$ and $\lambda_{\max}(L)/\sqrt{n-1}$ have the same limiting law under Gaussian assumptions. ## 2 On the asymptotic spectrum of the Laplacian In this section we present a summary of two kinds of known results for the asymptotic spectrum of a random Laplacian matrix, namely, the weak convergence of the empirical spectral distribution of the Laplacian matrix and the almost sure behavior of its largest eigenvalue. We remark on the differences from the corresponding results for Wigner matrices and also indicate some new useful extensions to block matrices. ### 2.1 Weak convergence of the empirical spectral distribution As is by now well known, the empirical spectral distribution plays an important role in the theory of large dimensional random matrices. For a symmetric $n\times n$ matrix $A_{n}$, let $\lambda_{1}(A_{n})\geq\cdot\cdot\cdot\geq\lambda_{n}(A_{n})$ denote the eigenvalues of the matrix $A_{n}$ and let $\tilde{F}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left\\{\lambda_{i}(A_{n})\leq x\right\\},x\in\mathbb{R}$ be the empirical distribution of its eigenvalues. When $A_{n}$ is a random matrix, $\tilde{F}_{n}$ is a random distribution function in $\mathbb{R}.$ In his pioneering work, Wigner [27] proved that for a Wigner matrix $X$, the empirical spectral distribution of $\frac{1}{\sqrt{n}}X$ converges weakly to the semicircle law $S$ in $[-2,2]$ $S(dx)=\frac{1}{2\pi}\sqrt{4-x^{2}}\mathbb{I}_{\|x\|\leq 2}dx.$ with probability one In the case when $X_{n}$ is a Wigner matrix with Gaussian entries and the sequence $(X_{n})_{n\geq 1}$ is independent of a sequence of diagonal random matrices $(\widetilde{D}_{n})_{n\geq 1}$ also with independent Gaussian entries of zero mean, Pastur and Vasilchuk [21] proved that with probability one the empirical spectral distribution of $\frac{1}{\sqrt{n}}(\widetilde{D}_{n}-X_{n})$ converges weakly to a non-random distribution $\gamma_{M}$, which is the free convolution $S\boxplus\gamma$ of the semicircle law $S$ in $(-2,2)$ with the standard Gaussian distribution $\gamma$. This result is a consequence of a pioneering result by Voiculescu, who established that, under suitable conditions, if $A=(A_{n})_{n\geq 1}$ and $B=(B_{n})_{n\geq 1}$ are independent sequences of $n\times n$ random matrices with empirical spectral distributions $F_{A}$ and $F_{B}$, then $A$ and $B$ become asymptotically free and the asymptotic spectral distribution of the sum $(A_{n}+B_{n})_{n\geq 1}$ is the free convolution $F_{A}\boxplus F_{B}.$ This remarkable fact shows the usefulness of free probability in large dimensional random matrices, since from a knowledge of the eigenvalues of $A_{n}$ and $B_{n}$, it is not easy to find the eigenvalues of the sum $A_{n}+B_{n}$, unless $A_{n}$ and $B_{n}$ commute. We refer to the books [2] and [20] for the definition and properties of the free convolution $\boxplus$ and the asymptotic freeness of random matrices, and to [7] for a study of distributions that are free convolutions with the semicircle distribution. A natural question is whether it is possible to find a corresponding spectral asymptotic result for the Laplacian matrix $L_{n}$ given by (1.1). The challenge is that the elements of the diagonal $D_{n}$ of $L_{n}$ are not independent of the non-diagonal entries of $X_{n}$. However, the result in [21] predicted that the result holds also for $L_{n}$, but the problem is nontrivial because $D_{n}$ strongly depends on $X_{n}$. This was done by Bryc, Dembo and Jiang [8] in the context of Markov random matrices and when the entries $X_{ij}$ are independent and identically distributed random variables. We summarize some of their results in what follows. ###### Theorem 1. Let $L_{n}=L_{X_{n}}$ be the Laplacian matrix of $X_{n}=(X_{ij}^{(n)})$ where $\left\\{X_{ij}^{(n)}:1\leq i<j\leq n\right\\}_{n\geq 1}$ is a collection of independent identically distributed random variables of mean $m$ and finite variance $\sigma^{2}$. Let $\tilde{F}_{n}$ be the empirical spectral distribution of $\frac{1}{\sqrt{n}}L_{n}.$ i) If $m=0$ and $\sigma^{2}=1$, then, with probability one, as $n\rightarrow\infty$, $\tilde{F}_{n}$ converges weakly to the free convolution $\gamma_{M}=S\boxplus\gamma$ of the semicircle law $S$ in $[-2,2]$ with the standard Gaussian distribution $\gamma.$ ii) The measure $\gamma_{M}$ is a nonrandom symmetric probability measure with unbounded support which does not depend on the distribution of $X_{ij}^{(n)}$. iii) If $m$ is not zero and $\sigma^{2}<\infty$, then $\tilde{F}_{n}$ converges weakly to the degenerate distribution $\delta_{-m}$ as $n\rightarrow\infty$. The above result was extended by Ding and Jiang [12] to the case when the random variables $X_{ij}^{(n)}$ are independent but not necessarily identically distributed. More specifically, they make the following assumption. ###### Condition 1. Let $X_{n}=(X_{ij}^{(n)})$ be an $n\times n$ symmetric matrix with $\\{X_{ij}^{(n)};1\leq i<j\leq n,n\geq 2\\}$ random variables defined on the same probability space and $\\{X_{ij}^{(n)};1\leq i<j\leq n\\}$ independent for each $n\geq 2$ (not necessarily identically distributed) with $X_{ij}^{(n)}=X_{ji}^{(n)}$, $\mathbb{E}(X_{ij}^{(n)})=\mu_{n}$, $\text{Var}(X_{ij}^{(n)})=\sigma_{n}^{2}>0$ for all $1\leq i<j\leq n$ $n\geq 2$, and $\sup_{1\leq i<j\leq n,n\geq 2}\mathbb{E}\|(X_{ij}^{(n)}-\mu_{n})/\sigma_{n}\|^{p}<\infty$ for some $p>0$. Assuming this condition for some $p>4,$ it is proved in [12] that if $\tilde{F}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbb{I}\left\\{\frac{\lambda_{i}(L_{n})-n\mu_{n}}{\sqrt{n}\sigma_{n}}\leq x\right\\},\qquad x\in\mathbb{R},$ then, as $n\rightarrow\infty$, with probability one, $\tilde{F}_{n}$ converges weakly to the distribution of the free convolution $\gamma_{M}=S\boxplus\gamma$ of the semicircle law $S$ in $[-2,2]$ with the standard Gaussian distribution $\gamma$. ### 2.2 On the distribution of $\gamma_{M}=S\boxplus\gamma$ Several other properties of the limiting spectral distribution $\gamma_{M}=S\boxplus\gamma$ are known. In addition to being symmetric and with unbounded support, it is also shown in [8] that this distribution has a smooth bounded density $f_{\gamma_{M}}$ and is determined by its moments. Some of these properties are based on [7], where the author consider distributions that are free convolutions with the semicircle distribution. It is also proved in [8] that $\gamma_{M}$ has even moments given by the combinatorial formula $m_{2k}=\sum_{\omega:\left\|\omega\right\|=2k}2^{h(\omega)}$ where $h(\omega)$ is the so-called height function that assigns to a pair partition $\omega$ the number of connected blocks which are of cardinality 2. However, an explicit expression for $\gamma_{M}$ is not known. One can easily simulate this distribution using large dimensional random matrices and predict the form of the empirical spectral distribution. In fact, Figure 1 presents an example of a consistent result of a large simulation study we have done, finding that the density $f_{\gamma_{M}}$ can be statistically approximated by a mixture of a Gaussian distribution and a semicircle distribution, although, in this approximation, the derivative of the density has two singularities. More specifically, if a Laplacian matrix $L$ is constructed from independent identically distributed random variables $\\{X_{ij}^{(n)};1\leq i<j\leq n,n\geq 2\\},$ with zero mean and variance $\sigma^{2},$ a good approximation $\widehat{f}_{\gamma_{M}}$ is $\displaystyle\widehat{f}_{\gamma_{M}}(x)$ $\displaystyle=\frac{\alpha}{2\sigma\sqrt{2}}f_{S}\left(\frac{x}{\sigma\sqrt{2}}\right)+\frac{1-\alpha}{2\sigma\sqrt{2}}f_{\gamma}\left(\frac{x}{\sigma\sqrt{2}}\right)$ (2.1) $\displaystyle=\frac{\alpha}{2\sigma\sqrt{2}}f_{S_{\sigma}}\left(x\right)+\frac{1-\alpha}{2\sigma\sqrt{2}}f_{\gamma_{\sigma}}\left(x\right),\text{ \ \ }x\in\mathbb{R},$ (2.2) where $S_{\sigma}$ is the semicircle distribution on $[-\sqrt{2}\sigma,\sqrt{2}\sigma]$, $\gamma_{\sigma}$ is the Gaussian distribution of zero mean and variance $2\sigma^{2}$, and $\alpha=\sqrt{2}/2$ is the weight of the mixture. Figure 1: Figure 1 ### 2.3 Almost sure behavior of the largest eigenvalue Bai and Yin [3] proved that the largest eigenvalue $\lambda_{\max}(X)$ of a Wigner matrix $X$ satisfies $\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\lambda_{\max}(X)=2\quad\text{a.s.}\quad\text{ as }n\rightarrow\infty.$ (2.3) The next result from Ding and Jiang [12] gives the same almost sure convergence for the largest eigenvalue for symmetric matrices as in the Wigner case, but with unit variance in the diagonal $X$. The second part is the same almost sure convergence of the maximum eigenvalue of symmetric matrices with zero diagonal elements as is the case with adjacency matrices in random graphs. ###### Proposition 1. Let $\mathbf{U}_{n}=(u_{ij}^{(n)})$ be an $n\times n$ symmetric random matrix and assume that for each $n\geq 1$, $\\{u_{ij}^{(n)}:1\leq i\leq j\leq n\\}$ are independent random variables with $\mathbb{E}u_{ij}^{(n)}=0$, $\text{Var}(u_{ij}^{(n)})=1$ for all $1\leq i,j\leq n$, and $\sup_{1\leq i,j,\leq n,n\geq 1}\mathbb{E}\|u_{ij}^{(n)}\|^{6+\delta}<\infty$ for some $\delta>0$. Then: 1. (i) $\lim_{n\to\infty}\frac{\lambda_{\max}(\mathbf{U}_{n})}{\sqrt{n}}=2\quad\text{a.s.}$ 2. (ii) The statement in (i) still holds if $\mathbf{U}_{n}$ is replaced by $\mathbf{U}_{n}-\text{diag}(u_{ij}^{(n)})$. However, for Laplacian matrices, the asymptotic empirical spectral distribution has unbounded support and therefore it is not expected to have the same type of convergence to a finite limit, if at all, with the same normalization $1/\sqrt{n}$. A description of the asymptotic behavior of the largest eigenvalue of Laplacian matrices is also given in [12, Theorem 1]. It turns out that when normalized by $1/\sqrt{n\log n}$, the largest eigenvalue of Laplacian matrices converges to a finite limit in probability. ###### Theorem 2. Let $L_{n}=L_{X_{n}}$ be the Laplacian matrix of an $n\times n$ symmetric matrix $X_{n}$ whose entries satisfy Condition 1 for some $p>6$. If $\mu_{n}=0$ and $\sigma_{n}=1$ for all $n\geq 2$, then 1. (a) $\frac{\lambda_{\max}(L_{n})}{\sqrt{n\log n}}\to\sqrt{2}$ in probability as $n\to\infty$. 1. (b) Furthermore, if $\\{L_{2},L_{3},\ldots\\}$ are independent Laplacian matrices, then $\liminf_{n\rightarrow\infty}\frac{\lambda_{\max}(L_{n})}{\sqrt{n\log n}}=\sqrt{2}\quad\text{a.s.}$ and $\limsup_{n\rightarrow\infty}\frac{\lambda_{\max}(L_{n})}{\sqrt{n\log n}}=2\quad\text{a.s.}$ and the sequence $\\{\lambda_{\max}(L_{n})/\sqrt{n\log n};n\geq 2\\}$ is dense in $[\sqrt{2},2]\quad\text{a.s.}$ As a consequence of Theorem 2, we have the following result (which will be used in Section 4), giving upper and lower bounds for the largest eigenvalue of a Laplacian matrix. We say that an event $\mathcal{C}_{n}$ occurs with high probability ($\mathrm{w.h.p}$) if for every $\eta>0$, there exists an $n_{0}\in\mathbb{N}$ such that $\mathbb{P}(\mathcal{C}_{n})\geq 1-\eta\qquad\text{for }\quad n>n_{0}.$ ###### Corollary 1. Let $L=L_{X}$ be the Laplacian matrix of an $n\times n$ symmetric matrix $X$ whose entries satisfy Condition 1 for some $p>6$. If $\mu_{n}=0$ and $\sigma_{n}=1$ for all $n\geq 2$, then for all $\epsilon>0$ $\lambda_{\max}(L)\leq\sqrt{(2+\epsilon)n\log n}$ (2.4) with high probability, and $\lambda_{\max}(L)\geq(2\sqrt{2}-\sqrt{2+\epsilon})\sqrt{n\log n}.$ (2.5) ###### Proof. This follows from Theorem 2, which ensures that $\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}-\sqrt{2}\right\|>\delta\right)\to 0\qquad\text{as}\quad n\to\infty$ taking $\delta=\sqrt{2+\epsilon}-\sqrt{2}>0$. ∎ Observe that the right side of (2.5) is non-negative if $0<\epsilon<6$. ### 2.4 Block Laplacian matrices We now indicate some extensions of Theorem 2 to certain block diagonal Laplacian matrices $L=\left[\begin{array}[]{cccc}L_{1}&0&\ldots&0\\\ 0&L_{2}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&L_{k}\end{array}\right],$ (2.6) where $\\{L_{1},L_{2},\ldots,L_{k}\\}$ are independent Laplacian random square matrices of the same size. These block matrices and the behavior of the largest eigenvalue of $L$ are important in the optimization problems of Stochastic Block Models: see [4]. As an application of Theorem 2, we find the asymptotic convergence of the largest eigenvalue for block diagonal Laplacian matrices. ###### Proposition 2. Let $L$ be a $k$-block Laplacian diagonal matrix in which each block is of size $n/k$, and the off-diagonal entries in each block are independent. Suppose that Condition 1 holds for some $p>6$. If whenever $L_{ij}$ is a nonzero entry of $L$, $\mathbb{E}[L_{ij}]=0$ and $\text{Var}(L_{ij})=1$ for all $n\geq 2$, then $\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}\to\sqrt{\frac{2}{k}}$ (2.7) in probability as $n\to\infty$. ###### Proof. Let $L_{1},L_{2},\ldots,L_{k}$ be the $k$ blocks of $L$. For $i=1,\ldots,k$ let $\Omega_{i}^{(n)}=\\{\omega\in\Omega:\lambda_{\max}(L_{j})\leq\lambda_{\max}(L_{i})\text{ for all }j\\}.$ Hence $\displaystyle\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\epsilon\right)$ $\displaystyle=\sum_{i=1}^{k}\mathbb{P}\left(\left\\{\left\|\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\epsilon\right\\}\bigcap\Omega_{i}^{(n)}\right)$ $\displaystyle\leq\sum_{i=1}^{k}\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L_{i})}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\epsilon\right)$ $\displaystyle\leq\sum_{i=1}^{k}\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L_{i})}{\sqrt{\frac{n}{k}\log\frac{n}{k}}}\frac{\sqrt{\frac{n}{k}\log\frac{n}{k}}}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\epsilon\right)$ Note that each matrix $L_{i}$ has size $n/k\times n/k$ and satisfies the hypotheses of Theorem 2, so, for $i=1,\ldots,k$, $\frac{\lambda_{\max}(L_{i})}{\sqrt{\frac{n}{k}\log{\frac{n}{k}}}}\to\sqrt{2}$ in probability as $n\to\infty$ and $\frac{\sqrt{\frac{n}{k}\log\frac{n}{k}}}{\sqrt{n\log n}}\to\frac{1}{\sqrt{k}}\qquad\text{as}\qquad n\to\infty.$ Then, by Slutsky’s theorem, we have that for each $i=1,\ldots,k$, $\frac{\lambda_{\max}(L_{i})}{\sqrt{\frac{n}{k}\log{\frac{n}{k}}}}\frac{\sqrt{\frac{n}{k}\log\frac{n}{k}}}{\sqrt{n\log n}}\to\sqrt{\frac{2}{k}}$ in probability as $n\to\infty$. This guarantees that for all $\epsilon>0$, $\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\epsilon\right)\to 0.$ ∎ From Proposition 2 and following similar ideas from Corollary 1, we obtain upper and lower bounds for the largest eigenvalue of a Laplacian block diagonal matrix. ###### Proposition 3. Let $L$ be a $k$-block Laplacian diagonal matrix in which each block is of size $n/k$, where the off-diagonal entries in each block are independent. Suppose that Condition 1 holds for some $p>6$. If $\mathbb{E}[L_{ij}]=0$ and $\text{Var}(L_{ij})=\sigma^{2}$ for all $n\geq 2$, then, for all $\epsilon>0$, $\lambda_{\max}(L)\leq\sigma\sqrt{\left(\frac{2}{k}+\epsilon\right)n\log n}$ (2.8) with high probability, and $\lambda_{\max}(L)\geq\sigma\left(2\sqrt{2}-\sqrt{\frac{2}{k}+\epsilon}\right)\sqrt{n\log n}.$ (2.9) ###### Proof. The proof follows from Proposition 2, which implies that $\mathbb{P}\left(\left\|\frac{\lambda_{\max}(L)}{\sqrt{n\log n}}-\sqrt{\frac{2}{k}}\right\|>\delta\right)\to 0\qquad\text{as}\qquad n\to\infty$ taking $\delta=\sqrt{\frac{2}{k}+\epsilon}-\sqrt{\frac{2}{k}}>0.$ ∎ ## 3 Limiting law for the largest diagonal entry Let $L=\left(L_{ij}\right)$ be an $n\times n$ Laplacian matrix whose off- diagonal entries are independent random variables with zero mean and variance one, and let $D^{(n)}=\frac{1}{\sqrt{n-1}}(L_{11},L_{22},\ldots,L_{nn}).$ The random variables $\\{D_{i}^{(n)}\\}_{i=1,\ldots,n}$ are not independent but rather have the covariance matrix $\Sigma_{D^{(n)}}=(\Sigma_{D^{(n)}})_{ij}=\left\\{\begin{array}[]{ccc}1&\text{ if }&i=j\\\ \frac{1}{n-1}&\text{ if }&i\neq j.\end{array}\right.$ (3.1) To find the limiting law of the largest element of $D^{(n)}$ we first consider a useful representation for $\Sigma_{D^{(n)}}$, which holds in the case of any distribution of $L_{ij}$ with the same covariance matrix. The eigenvalues of $\Sigma_{D^{(n)}}$ are $(n-2)/(n-1)$ with multiplicity $n-1$ and $2$ with multiplicity $1$. Consider the spectral decomposition of $\Sigma_{D^{(n)}}$ $\Sigma_{D^{(n)}}=U\Lambda U^{t}$ where $\Lambda=(\Lambda_{ii})$ is the diagonal matrix with the eigenvalues of $\Sigma_{D^{(n)}}$ and $U$ is an orthogonal matrix whose first $n-1$ columns are an orthonormal basis for the eigenvectors associated to the $n-1$ equal eigenvalues and the last is a normalized eigenvector of the eigenvalue $2$, namely, $(1/\sqrt{n},\ldots,1/\sqrt{n})$. Taking $\tilde{D}^{(n)}=\Sigma_{D^{(n)}}^{-1/2}D^{(n)}$ (3.2) we have that the covariance matrix of $\tilde{D}^{(n)}$ is the identity matrix, so the random variables $\\{\tilde{D}_{i}^{(n)}\\}_{i=1}^{n}$ are uncorrelated. Then $\displaystyle\left(\Sigma_{D^{(n)}}^{1/2}\right)_{ij}$ $\displaystyle=\sum_{l=1}^{n}U_{il}\Lambda_{ll}^{1/2}U_{lj}^{t}$ $\displaystyle=\sum_{l=1}^{n-1}U_{il}\sqrt{\frac{n-2}{n-1}}U_{lj}^{t}+\sqrt{2}U_{in}U_{nj}^{t}$ $\displaystyle=\sqrt{\frac{n-2}{n-1}}\sum_{l=1}^{n}U_{il}U_{lj}^{t}+\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)U_{in}U_{nj}^{t},$ and since $U$ is orthogonal and $U_{in}=1/\sqrt{n}$ for $i=1,\ldots,n$, we obtain $\left(\Sigma_{D^{(n)}}^{1/2}\right)=\left\\{\begin{array}[]{ccc}\left(\sqrt{2}-\sqrt{\frac{n-1}{n-2}}\right)\frac{1}{\sqrt{n}}U_{nj}^{t}&\text{ if }&i\neq j\\\ \sqrt{\frac{n-2}{n-1}}+\left(\sqrt{2}-\sqrt{\frac{n-1}{n-2}}\right)\frac{1}{\sqrt{n}}U_{ni}^{t}&\text{ if }&i=j.\\\ &&\end{array}\right.$ (3.3) Thus $\displaystyle D_{i}^{(n)}$ $\displaystyle=\sum_{\begin{subarray}{c}j=1\\\ j\neq i\end{subarray}}^{n}\left(\sqrt{2}-\sqrt{\frac{n-1}{n-2}}\right)\frac{1}{\sqrt{n}}U_{nj}^{t}\tilde{D}_{j}^{(n)}+\left(\sqrt{\frac{n-2}{n-1}}+\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}U_{nj}^{t}\right)\tilde{D}_{i}^{(n)}$ $\displaystyle=\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}\sum_{j=1}^{n}U_{nj}^{t}\tilde{D}_{j}{(n)}+\sqrt{\frac{n-2}{n-1}}\tilde{D}_{i}^{(n)},$ (3.4) which is a useful explicit representation of the elements of $D^{(n)}$ in terms of the elements of $\tilde{D}^{(n)}$, $\Lambda$, and $U$. We now assume that the entries $L_{ij}$ of $L$ are Gaussian. The following result shows that the Gumbel distribution is the limiting law of the largest diagonal entry of $L/\sqrt{n-1}$. ###### Proposition 4. Let $L=L_{X}$ be an $n\times n$ Laplacian matrix with $X=(X_{ij})$ a symmetric matrix whose off-diagonal entries are independent standard Gaussian random variables. Let $a_{n}=\sqrt{2\log n}\qquad\text{and}\qquad b_{n}=\sqrt{2\log n}-\frac{\log\log n+\log 4\pi}{2\sqrt{2\log n}}.$ (3.5) Then $M_{n}=a_{n}\left(\frac{\max_{i}L_{ii}}{\sqrt{n-1}}-\sqrt{\frac{n-2}{n-1}}b_{n}\right)$ (3.6) converges in distribution when $n\rightarrow\infty$ to a Gumbel random variable. ###### Proof. Since we are assuming Gaussianness, the uncorrelated random variables $\\{D_{i}^{}\\}_{i=1}^{n}$ given by (3.2) are independent standard Gaussian random variables. Then $Z=Z_{n}=\sum_{j=1}^{n}U_{nj}^{t}\tilde{D}_{j}^{(n)}$ has the standard Gaussian distribution for all $n$ and by (3) $D_{i}^{(n)}=\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}Z+\sqrt{\frac{n-2}{n-1}}\tilde{D}_{i}^{(n)}.$ (3.7) We write $\displaystyle a_{n}\left(\max_{1\leq i\leq n}D_{i}^{(n)}-\sqrt{\frac{n-2}{n-1}}b_{n}\right)$ $\displaystyle=a_{n}\left(\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-b_{n}\right)$ $\displaystyle+a_{n}\left(\max_{1\leq i\leq n}D_{i}^{(n)}-\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-\sqrt{\frac{n-2}{n-1}}b_{n}+b_{n}\right).$ (3.8) The first term on the right side of (3) converges in distribution to the Gumbel law by the classical extreme value result for maxima of i.i.d. Gaussian random variables; see [16] or [22]. On the other hand, using (3.7) we obtain $\displaystyle\max_{1\leq i\leq n}D_{i}^{(n)}$ $\displaystyle=\max_{1\leq i\leq n}\left(\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}Z+\sqrt{\frac{n-2}{n-1}}\tilde{D}_{i}^{(n)}\right)$ $\displaystyle=\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}Z+\sqrt{\frac{n-2}{n-1}}\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}.$ Thus $\displaystyle\left\|\max_{1\leq i\leq n}D_{i}^{(n)}-\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-\left(\sqrt{\frac{n-2}{n-1}}-1\right)b_{n}\right\|$ $\displaystyle\leq\left\|\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}Z\right\|+\left\|\sqrt{\frac{n-2}{n-1}}\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-\left(\sqrt{\frac{n-2}{n-1}}-1\right)b_{n}\right\|$ $\displaystyle=\left(\sqrt{2}-\sqrt{\frac{n-2}{n-1}}\right)\frac{1}{\sqrt{n}}\left\|Z\right\|+\left(\sqrt{\frac{n-2}{n-1}}-1\right)\left\|\max_{1\leq i\leq n}\tilde{D}_{i}^{(n)}-b_{n}\right\|.$ Then by Slutsky’s theorem $a_{n}\left\|\max_{1\leq i<n}D_{i}^{(n)}-\max_{1\leq i<n}\tilde{D}_{i}^{(n)}-\left(\sqrt{\frac{n-2}{n-1}}-1\right)b_{n}\right\|\to 0\qquad\text{in probability }$ as $n$ goes to infinity. Hence, the second term on the right side of (3) also goes to zero in probability as $n$ goes to infinity. Then, again using Slutsky’s theorem in (3), we obtain that $M_{n}$ converges in distribution to a Gumbel random variable as $n$ goes to infinity. ∎ ###### Remark. The last result is similar to Theorem 3.1 in [6]. However, in [6], a fixed sequence $\\{X_{n}:n=0,n=\pm 1,\ldots\\}$ of stationary random variables is considered, while Proposition 4 deals with a triangular array $\\{L_{ii}^{(n)}:i=1,\ldots,n\\}$, as is the case in the study of the largest eigenvalue of ensembles of random matrices. ## 4 Non-asymptotic estimates for the largest eigenvalue A consequence of the classical Courant–Fischer min-max Theorem [14] gives that $\max_{i}L_{ii}\leq\lambda_{\max}(L),$ (4.1) without more assumptions than that the matrix $L$ is symmetric. This section establishes a converse inequality, which is valid $\mathrm{w.h.p.}$, for a class of Laplacian matrices of symmetric Wigner matrices. The intuition for this comparison is the following. From Theorem 2(a), $\lambda_{\max}(L)$ grows like $\sqrt{2n\log n}$. On the other hand, the diagonal entries of a Laplacian matrix are sums of independent random variables, so by the Central Limit Theorem they would have an approximately Gaussian distribution for large $n$. It can be proved that for Gaussian random variables $\gamma_{i}$ (not necessarily independent as is the case of $L_{ii}$) $\max_{i}\gamma_{i}$ grows like $\sqrt{n\log n}$. This and (4.1) suggests that $\max_{i}L_{ii}$ and $\lambda_{\max}(L)$ behave similarly when $n$ is large. This comparison of $\max_{i}L_{ii}$ and $\lambda_{\max}(L)$ is known when the Wigner matrix has Gaussian or bounded entries [4, 5]. Our Theorem 3 below makes rigorous the above motivation, and is a general result under the assumption (4.2) and uses Corollary 1, which is a consequence of Theorem 2(a). For the sake of completeness we give a proof that this assumption is satisfied when the Laplacian matrix $L$ is constructed from a Wigner matrix with Gaussian entries, using concentration results for the maximum of Gaussian random variables and Sudakov Minorization which deals with correlated Gaussian random variables. ###### Theorem 3. Let $L=(L_{ij})$ be the Laplacian matrix of an $n\times n$ symmetric Wigner matrix $X=(X_{ij})$ constructed from independent random variables with zero mean, variance $\sigma^{2}$, and finite $p$-th moment for some $p>6$. If there exists a $c>0$ such that $\sigma\sqrt{(n-1)\log n}\leq c\max_{i}L_{ii}\qquad\mathrm{w.h.p}.,$ (4.2) then for all $\epsilon>0$ and $K=c\sqrt{2+\epsilon}>0,$ (4.3) $\lambda_{\max}(L)\leq K\left(1+\frac{1}{\sqrt{n-1}}\right)\max_{i}L_{ii}\qquad\mathrm{w.h.p}.$ (4.4) ###### Proof. Let $\hat{L}$ be the matrix given by $\hat{L}=\frac{1}{\sigma}L.$ We note that for all $1\leq i<j\leq n$, $\mathbb{E}\hat{L}_{ij}=0$ and $\mathbb{E}\hat{L}_{ij}^{2}=1$. From Corollary 1, we have that $\lambda_{\max}(\hat{L})\leq\sqrt{(2+\epsilon)n\log n}\qquad\mathrm{w.h.p}.$ and $\lambda_{\max}(L)\leq\sigma\sqrt{(2+\epsilon)n\log n}\qquad\mathrm{w.h.p}.$ Hence, with high probability $\displaystyle\lambda_{\max}(L)$ $\displaystyle\leq\sigma\sqrt{(2+\epsilon)(n-1)\log n}+\sigma\sqrt{(2+\epsilon)\log n}$ $\displaystyle\leq c\sqrt{2+\epsilon}\max_{i}L_{ii}+\sigma\sqrt{(2+\epsilon)\log n}$ $\displaystyle=\left(c\sqrt{2+\epsilon}+\frac{1}{\max_{i}L_{ii}}\sigma\sqrt{(2+\epsilon)\log n}\right)\max_{i}L_{ii}.$ (4.5) From (4.2) we have that $\frac{1}{\max_{i}L_{ii}}\sigma\sqrt{(2+\epsilon)\log n}\leq\frac{c\sigma\sqrt{(2+\epsilon)\log n}}{\sigma\sqrt{(n-1)\log n}}=\frac{c\sqrt{2+\epsilon}}{\sqrt{n-1}}.$ Now, replacing the last term in (4), we obtain that $\displaystyle\lambda_{\max}(L)$ $\displaystyle\leq\left(c\sqrt{2+\epsilon}+\frac{c\sqrt{2+\epsilon}}{\sqrt{n-1}}\right)\max_{i}L_{ii}$ $\displaystyle=c\sqrt{2+\epsilon}\left(1+\frac{1}{\sqrt{n-1}}\right)\max_{i}L_{ii}.$ Hence for $\epsilon>0$, if we take $K=c\sqrt{2+\epsilon}$, it follows that $\lambda_{\max}(L)\leq K\left(1+\frac{1}{\sqrt{n-1}}\right)\max_{i}L_{ii}\qquad\mathrm{w.h.p}.$ ∎ The inequality (4.2) is satisfied by several distributions. In particular, under Gaussian assumptions, as we now show. ###### Proposition 5. Let $L=(L_{ij})$ be the Laplacian matrix of an $n\times n$ symmetric Wigner matrix $X=(X_{ij})$ constructed from i.i.d. standard Gaussian random variables. Then (4.2) is satisfied. ###### Proof. We note that $\max_{i}L_{ii}=\sqrt{n-1}\max_{i}\frac{L_{ii}}{\sqrt{n-1}}=\sqrt{n-1}\max_{i}\gamma_{i},$ where $\gamma_{i}$ are standard Gaussian random variables with covariance matrix given by (3.1). Now, using concentration results (Theorem 3.12 in [18]) we obtain that for all $\alpha>0$ $\max_{i}\gamma_{i}\geq\mathbb{E}\max_{i}\gamma_{i}-\sqrt{\alpha\log n}\qquad\text{w.h.p.}.$ (4.6) From the structure of the covariance matrix, it follows that for all $i\neq j$ and $n>3$, $\mathbb{E}\left(\gamma_{i}-\gamma_{j}\right)^{2}=2-2\frac{1}{n-1}=2\left(\frac{n-2}{n-1}\right)\geq 1.$ So, Sudakov Minorization (Lemma A.3 in [10]) implies $\mathbb{E}\max_{i}\gamma_{i}\geq C\sqrt{\log n},$ (4.7) where $C$ is a positive constant independent of $n$ and the covariance of the Gaussian random variables $\gamma_{i}$’s. Hence, combining (4.6) and (4.7), with high probability $\displaystyle\max_{i}L_{ii}$ $\displaystyle\geq\sqrt{n-1}\left(C\sqrt{\log n}-\sqrt{\alpha\log n}\right)$ $\displaystyle=(C-\sqrt{\alpha})\sqrt{(n-1)\log n}.$ Then (4.2) is satisfied with $c=(C-\sqrt{\alpha})^{-1}$. ∎ ###### Remark. 1. (a) It is well known (see for example Lemma 2.3 in [18] or Appendix A in [10]) that the expected value of the maximum of $n$ standard Gaussian random variables, even if they are not independent, cannot be much larger that $\sqrt{2\log n}$. For that reason, the universal constant in the Sudakov Minorization is less than or equal to $\sqrt{2}$. So with this, we can take $\alpha$ such that $C\geq 1-\sqrt{\alpha}$, and taking $\epsilon=2((C-\sqrt{\alpha})^{2}-1)\geq 0$ we obtain that in the Gaussian case the constant $K$ can be equal to $\sqrt{2}$. 2. (b) The constant $K$ might not be sharp but it is useful for our weak convergence result in Section 5. For the non-Gaussian case, when the random variables $X_{ij}$ have bounded support, Bandeira proved (proof of Theorem 2.1 in [4]) that condition (4.2) is satisfied. Moreover, the following bound can be established as a consequence of Theorem 2.1 in [4], which considers Laplacian matrices constructed from Wigner matrices with bounded entries but not necessarily with the same distribution. ###### Proposition 6. Let $L=(L_{ij})$ be the Laplacian matrix of an $n\times n$ symmetric Wigner matrix $X=(X_{ij})$ constructed from independent random variables bounded by $M$, with zero mean and variance $\sigma^{2}.$ If there exists a $c>0$ such that $\sigma\sqrt{n-1}\geq cM\sqrt{\log n},$ then there exist positive constants $c_{1},C_{1},\beta$ depending only on $c,$ such that $\lambda_{\max}(L)\leq\left(1+\frac{C_{1}}{\sqrt{\log n}}\right)\max_{i}L_{ii}\qquad$ (4.8) with probability at least $1-c_{1}n^{-\beta}$. ## 5 Limiting law for the largest eigenvalue We are now ready to prove that the Gumbel distribution is the limiting law of the largest eigenvalue of a Laplacian random matrix constructed from Gaussian entries. We write $a_{n}^{\prime}=\frac{a_{n}}{\sqrt{2}}=\sqrt{\log n}\qquad\text{and}\qquad b_{n}^{\prime}=\sqrt{2}b_{n}=2\sqrt{\log n}-\frac{\log\log n+\log 4\pi}{\sqrt{2\log n}}.$ (5.1) ###### Theorem 4. Let $L=L_{X}$ be an $n\times n$ Laplacian matrix with $X=(X_{ij})$ a symmetric matrix whose off-diagonal entries are independent standard Gaussian random variables. Then $R_{n}=a_{n}^{\prime}\left(\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-b_{n}^{\prime}\sqrt{\frac{n-2}{n-1}}\right)$ (5.2) converges in distribution when $n\to\infty$ to a Gumbel random variable with distribution $G(x)=\exp(-e^{-x})$ for all $x\in\mathbb{R}$. We observe that the rescaling sequences $a_{n}^{\prime}$ and $b_{n}^{\prime}$ given by (5.1) have the same order as the appropriate choices for the extreme value theorem for the maxima of a sequence of Gaussian random variables in the i.i.d. and the stationary sequence cases; see [6], [22] respectively and [16]. ###### Proof. We first note, with the notation of Section 2, $\displaystyle a_{n}^{\prime}\left(\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-b_{n}^{\prime}\sqrt{\frac{n-2}{n-1}}\right)$ $\displaystyle=a_{n}\left(\max_{1\leq i\leq n}D_{i}^{(n)}-b_{n}\sqrt{\frac{n-2}{n-1}}\right)$ $\displaystyle+a_{n}^{\prime}\left(\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-\sqrt{2}\max_{1\leq i\leq n}D_{i}^{(n)}\right).$ (5.3) From Proposition 4, the first term on the right side of (5) converges in distribution to a Gumbel random variable. On the other hand, taking $K=\sqrt{2}$ in Theorem 3 and Proposition 5, we have that $\left\|\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-\sqrt{2}\max_{1\leq i\leq n}D_{i}^{(n)}\right\|\leq\left\|\frac{\sqrt{2}}{\sqrt{n-1}}\max_{1\leq i\leq n}D_{i}^{(n)}\right\|$ and $\displaystyle\frac{a_{n}}{\sqrt{2}}\left\|\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-\sqrt{2}\max_{1\leq i\leq n}D_{i}^{(n)}\right\|$ $\displaystyle\leq\frac{a_{n}}{\sqrt{2}}\left\|\frac{\sqrt{2}}{\sqrt{n-1}}\max_{1\leq i\leq n}D_{i}^{(n)}\right\|$ $\displaystyle\leq\frac{a_{n}}{\sqrt{n-1}}\left\|\max_{1\leq i\leq n}D_{i}^{(n)}-b_{n}\sqrt{\frac{n-2}{n-1}}+b_{n}\sqrt{\frac{n-2}{n-1}}\right\|$ $\displaystyle\leq\frac{a_{n}}{\sqrt{n-1}}\left\|\max_{1\leq i\leq n}D_{i}^{(n)}-b_{n}\sqrt{\frac{n-2}{n-1}}\right\|$ $\displaystyle\quad\qquad\qquad+\frac{a_{n}b_{n}}{\sqrt{n-1}}\sqrt{\frac{n-2}{n-1}}.$ (5.4) Now using Proposition 4 and Slutsky’s theorem, the first term in the last inequality goes to zero in probability as $n$ goes to infinity. Hence, using $a_{n}$ and $b_{n}$ in (3.5), we have that the second term in (5) goes to zero as $n$ goes to infinity. Therefore $\frac{a_{n}}{\sqrt{2}}\left\|\frac{\lambda_{\max}(L)}{\sqrt{n-1}}-\sqrt{2}\max_{1\leq i\leq n}D_{i}^{(n)}\right\|\to 0\qquad\text{as}\qquad n\to\infty$ in probability. Thus, the second term of the right side of Equation (5) tends to zero as $n$ goes to infinity, and we obtain that $R_{n}$ in (5.2) converges in distribution to a Gumbel random variable as $n$ goes to infinity. ∎ ###### Remark. It is easy to find the asymptotic distribution of the largest eigenvalue of a $k$-block Laplacian diagonal matrix $L$ given by (2.6) when the blocks are independent and all the entries are Gaussian. 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††institutetext: 1Institut für Experimentalphysik, Universität Hamburg, Germany 2Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 3Berkeley Institute for Data Science, University of California, Berkeley, CA 94720, USA 4NHETC, Department of Physics & Astronomy, Rutgers University, Piscataway, NJ 08854, USA 5Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD 21211, USA 6Google, Mountain View, CA 94043, USA 7Physics Department, Reed College, Portland, OR 97202, USA 8Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia 9Nevis Laboratories, Columbia University, 136 S Broadway, Irvington NY, USA 10Physik Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland 11SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA 12Berkeley Center for Cosmological Physics, University of California, Berkeley 13Departamento de Física da Universidade de Aveiro and CIDMA Campus de Santiago, 3810-183 Aveiro, Portugal 14Institute for Theoretical Physics, University of Heidelberg, Heidelberg, Germany 15Department of Physics & Astronomy, University of Kansas, 1251 Wescoe Hall Dr., Lawrence, KS 66045, USA 16Laboratoire de Physique de Clermont, Université Clermont Auvergne, France 17University of California San Diego, La Jolla, CA 92093, USA 18Laboratory for Nuclear Science, MIT, 77 Massachusetts Ave, Cambridge, MA 02139 19Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia 20Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK 21Center for Theoretical Physics, MIT, 77 Massachusetts Ave, Cambridge, MA 02139 22Physics Department, University of Michigan, Ann Arbor, MI 48109, USA 23Instituto de Física Teórica, IFT-UAM/CSIC, Universidad Autónoma de Madrid, 28049 Madrid, Spain 24Laboratory of Instrumentation and Experimental Particle Physics, Lisbon, Portugal 25European Organization for Nuclear Research (CERN), CH-1211, Geneva 23, Switzerland 26Instituto de Física Corpuscular (IFIC), Universidad de Valencia-CSIC, E-46980, Valencia, Spain 27CSAIL, Massachusetts Institute of Technology, 32 Vassar Street, Cambridge, MA 02139, USA 28International Center for Advanced Studies and CONICET, UNSAM, CP1650, Buenos Aires, Argentina 29Division Office Physics, Math and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA 30Department of Physics, University of Genova, Via Dodecaneso 33, 16146 Genova, Italy # The LHC Olympics 2020 A Community Challenge for Anomaly Detection in High Energy Physics Gregor Kasieczka (ed),1 Benjamin Nachman (ed),2,3 David Shih (ed),4 Oz Amram,5 Anders Andreassen,6 Kees Benkendorfer,2,7 Blaz Bortolato,8 Gustaaf Brooijmans,9 Florencia Canelli,10 Jack H. Collins,11 Biwei Dai,12 Felipe F. De Freitas,13 Barry M. Dillon,8,14 Ioan-Mihail Dinu,5 Zhongtian Dong,15 Julien Donini,16 Javier Duarte,17 D. A. Faroughy10 Julia Gonski,9 Philip Harris,18 Alan Kahn,9 Jernej F. Kamenik,8,19 Charanjit K. Khosa,20,30 Patrick Komiske,21 Luc Le Pottier,2,22 Pablo Martín-Ramiro,2,23 Andrej Matevc,8,19 Eric Metodiev,21 Vinicius Mikuni,10 Inês Ochoa,24 Sang Eon Park,18 Maurizio Pierini,25 Dylan Rankin,18 Veronica Sanz,20,26 Nilai Sarda,27 Uros̆ Seljak,2,3,12 Aleks Smolkovic,8 George Stein,2,12 Cristina Mantilla Suarez,5 Manuel Szewc,28 Jesse Thaler,21 Steven Tsan,17 Silviu-Marian Udrescu,18 Louis Vaslin,16 Jean-Roch Vlimant,29 Daniel Williams,9 Mikaeel Yunus18 <EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract A new paradigm for data-driven, model-agnostic new physics searches at colliders is emerging, and aims to leverage recent breakthroughs in anomaly detection and machine learning. In order to develop and benchmark new anomaly detection methods within this framework, it is essential to have standard datasets. To this end, we have created the LHC Olympics 2020, a community challenge accompanied by a set of simulated collider events. Participants in these Olympics have developed their methods using an R&D dataset and then tested them on black boxes: datasets with an unknown anomaly (or not). This paper will review the LHC Olympics 2020 challenge, including an overview of the competition, a description of methods deployed in the competition, lessons learned from the experience, and implications for data analyses with future datasets as well as future colliders. ## 1 Introduction The field of high energy physics (HEP) has reached an exciting stage in its development. After many decades of searching, the Standard Model (SM) of particle physics was completed in 2012 with the discovery of the Higgs boson Aad:2012tfa ; Chatrchyan:2012ufa . Meanwhile, there are strong motivations for physics beyond the Standard Model (BSM). For example, the nature of dark matter and dark energy, the mass of neutrinos, the minuteness of the neutron dipole moment, and the baryon-anti-baryon asymmetry in the universe are all well-established problems that do not have solutions in the Standard Model. Furthermore, the Higgs boson mass is unstable with respect to quantum corrections, and a consistent theory of quantum gravity remains mysterious. The Large Hadron Collider (LHC) at CERN has the potential to shed light on all of these fundamental challenges. Searching for BSM physics is a major part of the research program at the LHC across experiments atlasexoticstwiki ; atlassusytwiki ; atlashdbspublictwiki ; cmsexoticstwiki ; cmssusytwiki ; cmsb2gtwiki ; lhcbtwiki . The current dominant search paradigm is top-down, meaning searches target specific models. Nearly all of the existing BSM searches at the LHC pick a signal model that addresses one or more of the above experimental or theoretical motivations for BSM physics. Then, high-fidelity synthetic or simulated data are generated using this signal model. These signal events are then often combined with synthetic background events to develop an analysis strategy which is ultimately applied to data. An analysis strategy requires a proposal for selecting signal-like events as well as a method for calibrating the background rate to ensure that the subsequent statistical analysis is unbiased. Many searches provide “model-independent” results, in the form of a limit on cross-section or cross-section times acceptance ungoverned by any theoretical calculation. However, the event selection and background estimation are still strongly model-dependent. These search efforts are constantly improving and are important to continue and expand with new data. However, it is also becoming clear that a complementary search paradigm is critical for fully exploring the complex LHC data. One possible explanation for why we have not discovered new physics yet is that the model dependence of the current search paradigm has created blind spots to unconventional new physics signatures. In fact, despite thousands of BSM searches to date, much of phase space and countless possible signals remain unexplored at present (for many examples just in the realm of 2-body resonances, see Craig:2016rqv ; 1907.06659 ). Model independent searches for new particles have a long history in high energy physics. With a venerable history dating back at least to the discovery of the $\rho$ meson Button:1962bnf , generic searches like the bump hunt111This is a search where signal events present as a localized enhancement on top of a smoothly falling background distribution. assume little about the signal and have been used to discover many new particles, including the Higgs boson Aad:2012tfa ; Chatrchyan:2012ufa . While generic, the bump hunt is not particularly sensitive because it usually does not involve other event properties aside from the resonant feature. More differential signal model independent searches have been performed by D0 sleuth ; Abbott:2000fb ; Abbott:2000gx ; Abbott:2001ke , H1 Aaron:2008aa ; Aktas:2004pz , ALEPH Cranmer:2005zn , CDF Aaltonen:2007dg ; Aaltonen:2007ab ; Aaltonen:2008vt , CMS CMS-PAS-EXO-14-016 ; CMS-PAS-EXO-10-021 ; CMS:2020ohc ; Sirunyan:2020jwk , and ATLAS Aaboud:2018ufy ; ATLAS-CONF-2014-006 ; ATLAS-CONF-2012-107 . The general strategy in these analyses is to directly compare data with simulation in a large number of exclusive final states (bins). Aside from the feature selection, these approaches are truly signal model independent. The cost for signal model independence is sensitivity if there are a large number of bins because of the look elsewhere effect Gross:2010qma . Also, given the extreme reliance on simulation (a form of background model dependence) in these approaches, and they are typically only as sensitive as the simulation is accurate, and characterizing systematic uncertainties across thousands of final states can be challenging. Machine learning offers great potential to enhance and extend model independent searches. In particular, semi-, weak-, or un-supervised training can be used to achieve sensitivity to weak or complex signals with fewer model assumptions than traditional searches. Anomaly detection is an important topic in applied machine learning research, but HEP challenges require dedicated approaches. In particular, single events often contain no useful information — it is only when considering a statistical ensemble that an anomaly becomes apparent. This is a contrast between anomaly detection that is common in industry (“off manifold” or “out-of-sample” anomalies) and that which is the target of searches in high energy physics (“over-densities”). Furthermore, HEP data are systematically different than natural images and other common data types used for anomaly detection in applied machine learning. In order to test the resulting tailored methods, it is essential to have public datasets for developing and benchmarking new approaches. For this purpose, we have developed the LHC Olympics 2020 challenge and corresponding datasets lhco . The name of this community effort is inspired by the first LHC Olympics that took place over a decade ago before the start of the LHC lhco_old1 ; lhco_old2 ; lhco_old3 ; lhco_old4 . In those Olympics, researchers prepared ‘black boxes’ (BBs) of simulated signal events and contestants had to examine these simulations to infer the underlying signal process. These boxes were nearly all signal events and many of the signatures were dramatic (e.g. dilepton mass peaks) and all were findable with simple analysis procedures. While this was an immensely useful exercise, we are now faced with the reality that the new physics is rare or at least hard to find, and characterizing the BSM properties will not be our biggest challenge. The LHC Olympics 2020 challenge is also composed of black boxes. In contrast to the previous Olympics, these contain mostly simulated SM events. The goal of the challenge is to determine if there is new physics in the box and then to identify its properties. As stressed above, calibrating the background prediction is an essential aspect of BSM searches and so we have restricted this search to high energy hadronic final states where sideband methods can be used to estimate the background. We provide lists of the detector- reconstructed final state particles in order to allow contestants to test methods that can process low-level event information. To aid in method development and testing, we also provide a simulated dataset (with no anomalies) and a benchmark signal dataset. These two are combined to form the R&D dataset, and provided in addition to the black box datasets. The goal of this paper is to describe the Winter winterolympics and Summer summerolympics Olympics 2020 competitions. Well over one hundred researchers participated in these events, with over a dozen teams submitting their results for the black boxes. This paper is organized as follows. Section 2 introduces the LHC Olympics competition, including the R&D and black box datasets. A brief description of methods deployed in the competition are provided in Secs. 3, 4, and 5. Each contribution includes an introduction to the method, a concise statement of the results, as well as lessons learned before, during, and/or after the challenge. The results and lessons learned are synthesized in Sec. 6. Implications for data analyses with future datasets and as well as future colliders are discussed in Sec. 7 and the paper concludes in Sec. 8. ## 2 Dataset and Challenge The portal for the LHC Olympics dataset can be found at the challenge website lhco . The datasets described below are all publicly available and downloadable from Zenodo lhc_bb1 . Contestants entered their results in a Google form. On the form, participants were asked to state: * • The black box number (1-3) corresponding to their submission. * • A short abstract describing their method. * • A $p$-value associated with the dataset having no new particles (null hypothesis). * • As complete a description of the new physics as possible. For example: the masses and decay modes of all new particles (and uncertainties on those parameters). * • How many signal events (with the associated uncertainty) are in the dataset (before any selection criteria). Additionally, contestants were encouraged to submit plots or a Jupyter notebook PER-GRA:2007 . The LHC Olympics website includes a basic Jupyter notebook for reading in the data and running basic preprocessing using the pyjet software noel_dawe_2020_4289190 ; Cacciari:2011ma ; Cacciari:2005hq . Further details of the R&D and three black box datasets can be found below. ### 2.1 R&D Dataset The R&D dataset consisted of one million SM events each comprised of two jets produced through the strong interaction, referred to as quantum chromodynamics (QCD) dijet events, and 100,000 $Z^{\prime}\to XY$ events, with $X\to q\bar{q}$ and $Y\to q\bar{q}$, as shown in Fig. 1 for the topology. The masses of the new BSM particles $Z^{\prime}$, $X$, and $Y$ are 3.5 TeV, 500 GeV and 100 GeV, respectively. The events were produced using Pythia 8.219 Sjostrand:2006za ; Sjostrand:2014zea and Delphes 3.4.1 deFavereau:2013fsa ; Mertens:2015kba ; Selvaggi:2014mya , with default settings, and with no pileup or multiparton interactions included. They are selected using a single large- radius ($R=1$) anti-$k_{\mathrm{T}}$ Cacciari:2008gp jet trigger with a $p_{\text{T}}$ threshold of 1.2 TeV. The signal model was discussed in Ref. 1907.06659 and has the feature that existing generic searches for dijet resonances or targeted searches for diboson resonances may not be particularly sensitive. For example, existing searches may register a low significance ($<2\,\sigma$) while automated methods may be able to identify the signal with a high significance. $q$$q$$X$$Y$$q$$q$$q$$q$$Z^{\prime}$ Figure 1: Feynman diagram for signals of R&D dataset and Black Box 1. These events are stored as pandas dataframes mckinney-proc-scipy-2010 saved to compressed HDF5 koranne2011hierarchical format. For each event, all Delphes reconstructed particles in the event are assumed to be massless and are recorded in detector coordinates ($p_{\text{T}}$, $\eta$, $\phi$). More detailed information such as particle charge is not included. Events are zero padded to constant size arrays of 700 particles, with the truth bit appended at the end to dictate whether the event is signal or background. The array format is therefore ($N_{\text{events}}$=1.1 M, 2101). ### 2.2 Black Box 1 Setting \| R&D \| BB1 \| BB3 ---\|---\|---\|--- Tune:pp \| 14 \| 3 \| 10 PDF:pSet \| 13 \| 12 \| 5 TimeShower:alphaSvalue \| 0.1365 \| 0.118 \| 0.16 SpaceShower:alphaSvalue \| 0.1365 \| 0.118 \| 0.16 TimeShower:renormMultFac \| 1 \| 0.5 \| 2 SpaceShower:renormMultFac \| 1 \| 0.5 \| 2 TimeShower:factorMultFac \| 1 \| 1.5 \| 0.5 SpaceShower:factorMultFac \| 1 \| 1.5 \| 0.5 TimeShower:pTmaxMatch \| 1 \| 2 \| 1 SpaceShower:pTmaxMatch \| 0 \| 2 \| 1 Table 1: Pythia settings for the different datasets. For R&D the settings were the Pythia defaults while for BB1 and BB3 they were modified. BB2 is not shown here because it was produced using Herwig++ with default settings. This box contained the same signal topology as the R&D dataset (see Fig. 1) but with masses $m_{Z^{\prime}}=3.823$ TeV, $m_{X}=732$ GeV and $m_{Y}=378$ GeV. A total of 834 signal events were included (out of a total of 1M events in all). This number was chosen so that the approximate local significance inclusively is not significant. In order to emulate reality, the background events in Black Box 1 are different to the ones from the R&D dataset. The background still uses the same generators as for the R&D dataset, but a number of Pythia and Delphes settings were changed from their defaults. For the Pythia settings, see Table222Setting pTmaxMatch = 2 in Pythia invokes a “power shower”, where emissions are allowed to occur all the way to the kinematical limit. With a phase space cut on the hard scattering process, this sculpts a bump-like feature in the multijet background, which was flagged as anomalous by the authors of Section 5.2. Identification of this bump is labeled as “Human NN” in Figure 51. 1. For the Delphes settings, we changed EfficiencyFormula in the ChargedHadronTrackingEfficiency module, ResolutionFormula in the ChargedHadronMomentumSmearing module, and HCalResolutionFormula in the hadronic calorimeter (HCal) module. The tracking variations are approximated using the inner-detector measurements from Ref Aad:2016jkr and the calorimeter energy resolutions are varied by 10% inspired by measurements from Ref. Aaboud:2016hwh . ### 2.3 Black Box 2 This sample of 1M events was background only. The background was produced using Herwig++ Bahr:2008pv instead of Pythia, and used a modified Delphes detector card that is different from Black Box 1 but with similar modifications on top of the R&D dataset card. ### 2.4 Black Box 3 The signal was inspired by Ref. Agashe:2016rle ; Agashe:2016kfr and consisted of a heavy resonance (the KK graviton) with mass $m=4.2$ TeV which had two different decay modes. The first is just to dijets (gluons), while the second is to a lighter neutral resonance $R$ (the IR radion) of mass $m_{R}=2.217$ TeV plus a gluon, with $R\to gg$. 1200 dijet events and 2000 trijet events were included along with QCD backgrounds in Black Box 3. These numbers were chosen so that an analysis that found only one of the two modes would not observe a significant excess. The background events were produced with modified Pythia and Delphes settings (different than the R&D and the other black boxes). For the Pythia settings, see Table 1. $q$$q$$Y$$g$$g$$g$$X$ $q$$q$$q$$q$$X$ Figure 2: Feynman diagrams for signal of Black Box 3. Individual Approaches The following sections describes a variety of approaches to anomaly detection. In addition to an explanation of the method, each section includes a set of results on the LHC Olympics datasets as well as a brief description of lessons learned. We have grouped the various methods into three loose categories: Unsupervised (Sec. 3), Weakly Supervised (Sec. 4), and (Semi)-Supervised (Sec. 5). Supervision refers to the type of label information provided to the machine learning algorithms during training. Unsupervised methods do not provide any label information and learn directly from background-dominated data. Typically, these methods try to look for events with low $p(\text{background})$. (Exceptions exist, see e.g. ANODE in Sec. 3.2 and GIS in Sec. 3.5 which use likelihood ratios.) Weakly supervised methods have noisy labels.333Such a categorisation is not unique, see e.g. zhou2018brief for an alternative way of defining weak supervision. We follow the established usage in applications of machine learning for particle physics. Many of these approaches operate by comparing two datasets with different amounts of a potential signal. The labels are noisy because instead of being pure ‘signal’ and ‘background’, the labels are ‘possibly signal-depleted’ and ‘possibly signal-enriched’. The goal of these methods is to look for events with high $p(\text{possibly signal-depleted})/p(\text{possibly signal-enriched})$. Supervised methods have labels for each event. Semi-supervised methods have labels for some events. Methods that are labeled as (Semi-)Supervised use signal simulations in some way to build signal sensitivity. These three categories are not exact and the boundaries are not rigid. However, this categorization may help to identify similarities and differences between approaches. Within each category, the methods are ordered alphabetically by title. Furthermore, the results on the datasets can be grouped into three types: (i) blinded contributions using the black boxes, (ii) unblinded results or updates on blinded results (and thus, also unblinded) on the black boxes, and (iii) results only using the R&D dataset. All three of these contribution types provide valuable insight, but each serves a different purpose. The first category (i) corresponds to the perspective of a pure challenge that is analogous to a real data analysis. The organizers of the LHCO challenge could not participate in this type of analysis. Section 6.1 provides an overview of the challenge results. The LHC Olympics datasets have utility beyond the initial blinded challenge as well and so contributions of type (ii) and (iii) are also important. Some of the results of these types came from collaborations with the challenge organizers and some came from other groups as well who did not manage (for whatever reason) to deploy their results on the blinded black boxes. A summary of all of the methods and results can be found in Table 2. Note that in some cases, blinded results (of type (i)) were presented at the LHC Olympics workshops, but only a subset (sometimes of type (iii)) appear in the subsequent sections. The table gives precedence to the workshops results, which are also discussed in Sec. 6.1. Section \| Short Name \| Method Type \| Results Type ---\|---\|---\|--- 3.1 \| VRNN \| Unsupervised \| (i) (BB2,3) and (ii) (BB1) 3.2 \| ANODE \| Unsupervised \| (iii) 3.3 \| BuHuLaSpa \| Unsupervised \| (i) (BB2,3) and (ii) (BB1) 3.4 \| GAN-AE \| Unsupervised \| (i) (BB2-3) and (ii) (BB1) 3.5 \| GIS \| Unsupervised \| (i) (BB1) 3.6 \| LDA \| Unsupervised \| (i) (BB1-3) 3.7 \| PGA \| Unsupervised \| (ii) (BB1-2) 3.8 \| Reg. Likelihoods \| Unsupervised \| (iii) 3.9 \| UCluster \| Unsupervised \| (i) (BB2-3) 4.1 \| CWoLa \| Weakly Supervised \| (ii) (BB1-2) 4.2 \| CWoLa AE Compare \| Weakly/Unsupervised \| (iii) 4.3 \| Tag N’ Train \| Weakly Supervised \| (i) (BB1-3) 4.4 \| SALAD \| Weakly Supervised \| (iii) 4.5 \| SA-CWoLa \| Weakly Supervised \| (iii) 5.1 \| Deep Ensemble \| Semisupervised \| (i) (BB1) 5.2 \| Factorized Topics \| Semisupervised \| (iii) 5.3 \| QUAK \| Semisupervised \| (i) (BB2,3) and (ii) (BB1) 5.4 \| LSTM \| Semisupervised \| (i) (BB1-3) Table 2: A categorization in terms of method and result type for all of the results presented in the Sec. 3, Sec. 4, and Sec. 5. ## 3 Unsupervised ### 3.1 Anomalous Jet Identification via Variational Recurrent Neural Network444Authors: Alan Kahn, Julia Gonski, Inês Ochoa, Daniel Williams, and Gustaaf Brooijmans. #### 3.1.1 Method The method described here employs a Variational Recurrent Neural Network (VRNN) to perform jet-level anomaly detection by modeling jets as a sequence of constituents. A VRNN is a sequence-modeling architecture which replaces the standard encoder-decoder architecture of a Recurrent Neural Network with a Variational Autoencoder (VAE) chung2016recurrent . This allows the VRNN to perform both sequence modeling in addition to variational inference, which has been shown to be a very powerful tool for anomaly detection An2015VariationalAB . A sequence-modeling architecture is well-motivated as it is capable of accommodating variable-length inputs, such as lists of constituent four-vectors in a jet, while suppressing the ability of the model to learn correlations with the jet’s constituent multiplicity. By contrast, fixed-length architectures such as VAEs rely on a loss function that is computed between the input layer and the reconstructed output layer. As a result, zero-padded inputs directly affect the value of the loss function, leading to correlations that are difficult to remove when using inputs that are naturally variable in length, but forced to work in a fixed-length framework. Figure 3 shows a diagram of one VRNN cell. The VAE portion of the architecture is displayed on the top row of layers in the diagram, where a constituent’s four-momentum components are input as a vector $x(t)$, which is encoded into a multivariate Gaussian distribution in the latent space $z$, and then decoded to produce a reconstruction of the same input constituent’s components $y(t)$. The variable $t$ refers to the time-step, which advances as the sequence is processed, and can be interpreted as the constituent number currently being processed by the model. Inputs to the VRNN consist of sequences of jet four-vector constituent components $p_{\text{T}}$, $\eta$, and $\phi$, where constituents are assumed to be massless. Jets are reconstructed with FastJet Cacciari:2011ma ; Cacciari:2005hq using the anti-$k_{t}$ algorithm with a radius parameter of 1.0 Cacciari:2008gp . Before training, a pre-processing method is applied which boosts each jet to the same reference mass, energy, and orientation in $\eta-\phi$ space, such that all input jets differ only by their substructure. In addition, our pre-processing method includes a choice of sequence ordering, in which the constituent sequence input into the model is sorted by $k_{t}$-distance instead of by the typical constituent $p_{\text{T}}$. In more detail, the $n^{\text{th}}$ constituent in the list, $c_{n}$, is determined by Eq. 1 to be the constituent with the highest $k_{t}$-distance relative to the previous constituent, with the first constituent in the list being the highest $p_{\text{T}}$ constituent. $c_{n}=\max(p_{Tn}\Delta R_{n,n-1})$ (1) This ordering is chosen such that non-QCD-like substructure, characterized by two or more separate prongs of constituents within the jet, is more easily characterized by the sequence. When compared to $p_{T}$-sorted constituent ordering, the $k_{t}$-sorted sequence consistently travels back and forth between each prong, making their existence readily apparent and easy to model. As a result, a significant boost in performance is observed. The loss function $\mathcal{L}(t)$ for each constituent, defined in Eq. 2, is very similar to that of an ordinary VAE. It consists of a mean-squared-error (MSE) loss between input constituents and generated output constituents as a reconstruction loss, as well as a weighted KL-Divergence from the learned latent space prior to the encoded approximate posterior distribution. Since softer constituents contribute less to the overall classification of jet substructure, each KL-Divergence term, computed constituent-wise, is weighted by the constituent’s $p_{T}$-fraction with respect to the jet’s total $p_{\text{T}}$, averaged over all jets in the dataset to avoid correlations with constituent multiplicity. The weight coefficient of the KL-Divergence term is enforced as a hyperparameter, and has been optimized to a value of 0.1 in dedicated studies. $\mathcal{L}(t)=\text{MSE}+0.1\times\overline{p_{T}}(t)D_{\text{KL}}$ (2) After a jet is fully processed by the VRNN, a total loss function $\mathcal{L}$ is computed as the average of the individual constituent losses over the jet: $\mathcal{L}=\frac{\Sigma\mathcal{L}(t)}{N}$. The architecture is built with 16 dimensional hidden layers, including the hidden state, with a two-dimensional latent space. All hyperparameters used are determined by a hyperparameter optimization scan. The model is trained on the leading and sub-leading jets of each event, where events are taken from the LHC Olympics datasets. After training, each jet in the dataset is assigned an Anomaly Score, defined in Eq. 3, where $D_{\text{KL}}$ is the KL-Divergence from the learned prior distribution to the encoded posterior distribution. $\text{Anomaly Score}=1-e^{-\overline{D_{\text{KL}}}}$ (3) Since the LHC Olympics challenge entails searching for a signal on the event level instead of the jet level, an overall Event Score is determined by choosing the most anomalous score between the leading and sub-leading jets in an event. To ensure consistency between training scenarios, Event Scores are subject to a transformation in which the mean of the resulting distribution is set to a value of 0.5, and Event Scores closer to 1 correspond to more anomalous events. Figure 3: A Variational Recurrent Neural Network cell. The $x(t)$ and $y(t)$ layers represent respectively the input constituent and reconstructed constituents’ four-momentum components $p_{\text{T}}$, $\eta$, and $\phi$. The $\phi_{x}$ and $\phi_{z}$ layers are feature-extracting layers which encode a representation of the features in the input layer $x(t)$ and latent space $z$ respectively. $h(t-1)$ represents the current time-step’s hidden state, which is updated each iteration via a transition function between $h(t-1)$, $\phi_{x}$, and $\phi_{z}$ carried out by a Gated Recurrent Unit (GRU). At each time-step, the prior distribution defined by $\mu_{t}$ and $\sigma_{t}$ is determined from the current hidden state. #### 3.1.2 Results on LHC Olympics The performance of the VRNN was first assessed with the LHC Olympics R&D dataset, which includes a known signal of a beyond-the-Standard-Model $Z^{\prime}$ boson with a mass of 3500 GeV which decays to two hadronically decaying $X$ and $Y$ particles, each reconstructed by a $R=1.0$ jet. This study was used as a validation of the method, with a goal of directly investigating the ability of the Event Score selection to reconstruct the $Z^{\prime}$ mass. Therefore, no selections beyond those described in Section 3.1.1 are applied. The VRNN was trained over a contaminated dataset consisting of 895113 background events and 4498 signal events, corresponding to a signal contamination level of 0.5%. A selection on the Event Score is applied as the sole discriminator, and the invariant mass $m_{JJ}$ of the two jets is then scanned to assess the prominence of the $Z^{\prime}$ mass peak. In this validation analysis, the Event Score is required to exceed a value of 0.65. This value is chosen to significantly enrich the fraction of anomalous jet events over the background, while retaining enough statistics in the background to display its smoothly falling behavior. Figure 4 shows the dijet invariant mass distributions before and after the Event Score selection, along with the local significance of the signal computed in each bin using the BinomialExpZ function from RooStats with a relative background uncertainty of 15% moneta2011roostats . Applying this selection dramatically increases the significance of the excess from $0.18\sigma$ to $2.2\sigma$ without significantly sculpting the shape of the background. Figure 4: Dijet invariant mass distributions before (left) and after (right) a selection on the Event Score, with a two-prong Z’ signal contamination of 0.5%. Once the method was validated in the R&D dataset, it was applied to Black Box 1, with a re-optimized tighter selection on the Event Score of 0.75, as well as a requirement on the pseudorapidity of the leading and sub-leading jets to be less than 0.75, to ensure that central, high momentum transfer events are considered. Figure 5 shows the dijet invariant mass for both the Black Box 1 and Background datasets. The Event Score selection reveals an enhancement in $m_{JJ}$ just below 4000 GeV. This is consistent with the Black Box 1 signal, which is a new $Z^{\prime}$ boson with a mass of 3800 GeV decaying to two new particles, each decaying hadronically. Figure 5: Dijet invariant mass distributions before (left) and after (right) a selection on the Event Score from the Black Box 1 dataset. The signal present is a $Z^{\prime}$ boson with a mass of 3800 GeV. The same method applied to Black Box 2, shown in Fig. 6, results in no significant excesses in the invariant mass distribution. Additionally, the effect of the Event Score selection on the $m_{JJ}$ shapes is similar between the Black Box 2 and Background datasets. Black Box 2 does not contain any beyond-the-Standard-Model events, and therefore these results are consistent with a QCD-only sample. It is important to note that the model was trained independently on each dataset, and the resulting Event Scores are from entirely unique sets of network weights. Figure 6: Dijet invariant mass distributions before (left) and after (right) a selection on the Event Score from the Black Box 2 dataset. No signal is present, and the dataset shown consists entirely of multijet background events. Figure 7 shows results for Black Box 3. The signal in Black Box 3 consists of a new 4200 GeV particle, with varied final states beyond the two-prong large-$R$ jets described earlier. As the model described here is specifically sensitive to substructure within a large-$R$ jet, it is insensitive to the signal present in this Black Box. Figure 7: Dijet invariant mass distributions before (left) and after (right) a selection on the Event Score from the Black Box 3 dataset. The signal present is a new boson with a mass of 4200 GeV. #### 3.1.3 Lessons Learned This challenge presented a highly useful avenue for the development of our model. Results from the R&D and Black Box dataset analyses indicate that the VRNN is capable of identifying anomalies via sequence modeling, as we have shown in the context of searching for anomalous substructure within boosted hadronically decaying objects. We learned that the pre-processing method is hugely influential on the performance of the model, in particular the choice of $k_{t}$-ordered sequencing. We feel that this is a generalizable conclusion from our study which can be applied to the understanding and use of jet substructure in future analyses. Given these lessons, there are a variety of future opportunities with this application of the VRNN architecture to jet-level anomaly detection. Since the VRNN takes constituent information as input and learns jet substructure without explicit reliance on high level variables, it is expected to have less correlation with jet mass than standard substructure variables such as $n$-subjettiness. Further characterization of this point could reveal a key advantage in using such an approach in an analysis context. While we limited our scope in this study to be entirely unsupervised with no signal or background model information, the RNN and VAE elements of the VRNN give potential for accommodating more supervised training scenarios. Furthermore, a number of advancements to the architecture, such as a dedicated adversarial mass de-correlation network, or an additional input layer representing high- level features, are worthwhile avenues of exploration to enhance performance while minimizing unwanted correlations. ### 3.2 Anomaly Detection with Density Estimation555Authors: Benjamin Nachman and David Shih. This section introduces an approach called ANOmaly detection with Density Estimation (ANODE) that is complementary to existing methods and aims to be largely background and signal model agnostic. Density estimation, especially in high dimensions, has traditionally been a difficult problem in unsupervised machine learning. The objective of density estimation is to learn the underlying probability density from which a set of independent and identically distributed examples were drawn. In the past few years, there have been a number of breakthroughs in density estimation using neural networks and the performance of high dimensional density estimation has greatly improved. The idea of ANODE is to make use of these recent breakthroughs in order to directly estimate the probability density of the data. Assuming the signal is localized somewhere, one can attempt to use sideband methods and interpolation to estimate the probability density of the background. Then, one can use this to construct a likelihood ratio generally sensitive to new physics. #### 3.2.1 Method This section will describe the ANODE proposal for an unsupervised method to search for resonant new physics using density estimation. Let $m$ be a feature in which a signal (if it exists) is known to be localized around some $m_{0}$. The value of $m_{0}$ will be scanned for broad sensitivity and the following procedure will be repeated for each window in $m$. It is often the case that the width of the signal in $m$ is fixed by detector properties and is signal model independent. A region $m_{0}\pm\delta$ is called the signal region (SR) and $m\not\in[m_{0}-\delta,m_{0}+\delta]$ is defined as the sideband region (SB). A traditional, unsupervised, model- agnostic search is to perform a bump hunt in $m$, using the SB to interpolate into the SR in order to estimate the background. Let $x\in\mathbb{R}^{d}$ be some additional discriminating features in which the signal density is different than the background density. If we could find the region(s) where the signal differs from the background and then cut on $x$ to select these regions, we could improve the sensitivity of the original bump hunt in $m$. The goal of ANODE is to accomplish this in an unsupervised and model-agnostic way, via density estimation in the feature space $x$. More specifically, ANODE attempts to learn two densities: $p_{\text{data}}(x\|m)$ and $p_{\text{background}}(x\|m)$ for $m\in{\rm SR}$. Then, classification is performed with the likelihood ratio $\displaystyle R(x\|m)=\frac{p_{\text{data}}(x\|m)}{p_{\text{background}}(x\|m)}.$ (4) In the ideal case that $p_{\text{data}}(x\|m)=\alpha\,p_{\text{background}}(x\|m)+(1-\alpha)\,p_{\text{signal}}(x\|m)$ for $0\leq\alpha\leq 1$ and $m\in\text{SR}$, Eq. 4 is the optimal test statistic for identifying the presence of signal. In the absence of signal, $R(x\|m)=1$, so as long as $p_{\text{signal}}(x\|m)\neq p_{\text{background}}(x\|m)$, $R_{\text{data}}(x\|m)$ has a non-zero density away from 1 in a region with no predicted background. In practice, both $p_{\text{data}}(x\|m)$ and $p_{\text{background}}(x\|m)$ are approximations and so $R(x\|m)$ is not unity in the absence of signal. The densities $p(x\|m)$ are estimated using conditional neural density estimation. The function $p_{\text{data}}(x\|m)$ is estimated in the signal region and the function $p_{\text{background}}(x\|m)$ is estimated using the sideband region and then interpolated into the signal region. The interpolation is done automatically by the neural conditional density estimator. Effective density estimation will result in $R(x\|m)$ in the SR that is localized near unity and then one can enhance the presence of signal by applying a threshold $R(x\|m)>R_{\text{cut}}$, for $R_{\text{cut}}>1$. The interpolated $p_{\text{background}}(x\|m)$ can then also be used to estimate the background. The ANODE procedure as described above is completely general with regards to the method of density estimation. In this work we will demonstrate a proof-of- concept using normalizing flow models for density estimation. Since normalizing flows were proposed in Ref. pmlr-v37-rezende15 , they have generated much activity and excitement in the machine learning community, achieving state-of-the-art performance on a variety of benchmark density estimation tasks. #### 3.2.2 Results on LHC Olympics The conditional MAF is optimized666Based on code from https://github.com/ikostrikov/pytorch-flows. using the log likelihood loss function, $\log(p(x\|m))$. All of the neural networks are written in PyTorch NEURIPS2019_9015 . For the hyperparameters, there are 15 MADE blocks (one layer each) with 128 hidden units per block. Networks are optimized with Adam adam using a learning rate $10^{-4}$ and weight decay of $10^{-6}$. The SR and SB density estimators are each trained for 50 epochs. No systematic attempt was made to optimize these hyperparameters and it is likely that better performance could be obtained with further optimization. For the SR density estimator, the last epoch is chosen for simplicity and it was verified that the results are robust against this choice. The SB density estimator significantly varies from epoch to epoch. Averaging the density estimates point-wise over 10 consecutive epochs results in a stable result. Averaging over more epochs does not further improve the stability. All results with ANODE present the SB density estimator with this averaging scheme for the last 10 epochs. Figure 8: Scatter plot of $R(x\|m)$ versus $\log p_{\text{background}}(x\|m)$ across the test set in the SR. Background events are shown (as a two- dimensional histogram) in grayscale and individual signal events are shown in red. Ref. Nachman:2020lpy . Figure 8 shows a scatter plot of $R(x\|m)$ versus $\log p_{\text{background}}(x\|m)$ for the test set in the SR. As desired, the background is mostly concentrated around $R(x\|m)=1$, while there is a long tail for signal events at higher values of $R(x\|m)$ and between $-2<\log p_{\text{background}}(x\|m)<2$. This is exactly what is expected for this signal: it is an over-density ($R>1$) in a region of phase space that is relatively rare for the background ($p_{\text{background}}(x\|m)\ll 1$). The background density in Fig. 8 also shows that the $R(x\|m)$ is narrower around $1$ when $p_{\text{background}}(x\|m)$ is large and more spread out when $p_{\text{background}}(x\|m)\ll 1$. This is evidence that the density estimation is more accurate when the densities are high and worse when the densities are low. This is also to be expected: if there are many data points close to one another, it should be easier to estimate their density than if the data points are very sparse. Figure 9: Receiver Operating Characteristic (ROC) curve (left) and Significance Improvement Characteristic (SIC) curve (right). Figure reproduced from Ref. Nachman:2020lpy . The performance of $R$ as an anomaly detector is further quantified by the Receiver Operating Characteristic (ROC) and Significance Improvement Characteristic (SIC) curves in Fig. 9. These metrics are obtained by scanning $R$ and computing the signal efficiency (true positive rate) and background efficiency (false positive rate) after a threshold requirement on $R$. The Area Under the Curve (AUC) for ANODE is 0.82. For comparison, the CWoLa hunting approach is also shown in the same plots. The CWoLa classifier is trained using sideband regions that are 200 GeV wide on either side of the SR. The sidebands are weighted to have the same number of events as each other and in total, the same as the SR. A single NN with four hidden layers with 64 nodes each is trained using Keras keras and TensorFlow tensorflow . Dropout JMLR:v15:srivastava14a of 10% is used for each intermediate layer. Intermediate layers use rectified linear unit activation functions and the last layer uses a sigmoid. The classifier is optimized using binary cross entropy and is trained for 300 epochs. As with ANODE, 10 epochs are averaged for the reported results777A different regularization procedure was used in Ref. Collins:2018epr ; Collins:2019jip based on the validation loss and $k$-folding. The averaging here is expected to serve a similar purpose.. The performance of ANODE is comparable to CWoLa hunting in Fig. 9, which does slightly better at higher signal efficiencies and much better at lower signal efficiencies. This may be a reflection of the fact that CWoLa makes use of supervised learning and directly approaches the likelihood ratio, while ANODE is unsupervised and attempts to learn both the numerator and denominator of the likelihood ratio. With this dataset, ANODE is able to enhance the signal significance by about a factor of 7 and would therefore be able to achieve a local significance above $5\sigma$ given that the starting value of $S/\sqrt{B}$ is 1.6. #### 3.2.3 Lessons Learned While ANODE appears to be robust to correlations in the data (see Ref. Nachman:2020lpy ), it is challenging to obtain precise estimates of the background density to very values of small $S/B$. Another challenge is extending the density estimation to higher dimensions. While the demonstrations here were based on the innovative MAF density estimation technique, the ANODE method can be used in conjunction with any density estimation algorithm. Indeed, there are numerous other neural density estimation methods from the past few years that claim state-of-the-art performance, including Neural Autoregressive Flows DBLP:journals/corr/abs-1804-00779 and Neural Spline Flows durkan2019neural ; exploring these would be an obvious way to attempt to improve the results in this section. ### 3.3 BuHuLaSpa: Bump Hunting in Latent Space888Authors: Blaz Bortolato, Barry M. Dillon, Andrej Matevc, Jernej F. Kamenik, Aleks Smolkovic. The code used in this project can be found at https://github.com/alekssmolkovic/BuHuLaSpa. #### 3.3.1 Method The BuHuLaSpa method assumes that the LHCO event data was generated through a stochastic process described by an underlying probabilistic generative model with continuous latent variables. We use neural networks as approximators to the likelihood and posterior distributions of the model, and use the variational autoencoder (VAE) architecture as a means of optimising these neural networks. For each event in the dataset we cluster the hadrons, select the resulting two leading $p_{\text{T}}$ jets, and order these by mass, $m_{j_{1}}>m_{j_{2}}$. The data representation we use for the LHCO consists of the following observables for each jet: jet mass $m_{j}$, the $n$-subjettiness observables $\tau_{2}/\tau_{1}$ and $\tau_{3}/\tau_{2}$, and an observable similar to soft drop defined by clustering the jets with the C/A algorithm, then de-clustering them branch by branch, and summing the ratios of parent to daughter subjet masses along the way, stopping at some pre-defined mass scale which we have chosen to be $20$ GeV. We denote these input measurements for the $i^{\text{th}}$ event in the sample by a vector $\vec{x}_{i}$. The probablistic model underlying the VAE architecture can be viewed as a generative process through which the event data is generated from some underlying distributions. The generative process for one event $\vec{x}_{i}$ starts with the sampling of a latent vector $\vec{z}_{i}$ from a prior distribution $p(\vec{z})$. Given this latent vector, the data for a single event is then sampled from the likelihood function $p(\vec{x}_{i}\|\vec{z}_{i})$. The goal is then to approximate the posterior distribution, $p(\vec{z}_{i}\|\vec{x}_{i})$, i.e. perform posterior inference, which maps a single event back to its representation in latent space. The neural networks used as approximators to the posterior and likelihood functions are denoted by, $q_{\phi}(\vec{z}_{i}\|\vec{x}_{i})$ and $p_{\theta}(\vec{x}_{i}\|\vec{z}_{i})$, where $\phi$ and $\theta$ represent the weights and biases (i.e. the free parameters) of the encoder and decoder networks, respectively. The sampling proceure is re-formulated using the re- parameterisation technique which allows the neural networks to be optimised through traditional back-propagation methods. Specifically the encoder network consists of dim$(\vec{x})$ neurons in the input layer, followed by some number of hidden layers, and $2\times$dim$(\vec{z})$ neurons in the output layer. Each element in $\vec{z}_{i}$ corresponds to two neurons in the output layer of the encoder network, one representing the mean and one representing the variance. Elements of the latent vector $\vec{z}_{i}$ are then sampled from Gaussian distributions parameterised by these means and variances. The resulting latent vector $\vec{z_{i}}$ is then fed to the decoder network which consists of dim$(\vec{z})$ neurons in the input layer, some number of hidden layers, and dim$(\vec{x})$ neurons in the output layer. The VAE method is important because it allows us to frame the posterior inference task as an optimisation problem, and the loss function that is optimised is the Stochastic Gradient Variational Bayes (SGVB) estimator: $\mathcal{L}=-D_{\text{KL}}(q_{\phi}(\vec{z}_{i}\|\vec{x}_{i})\|p(\vec{z}_{i}))+\beta_{\text{reco}}\log p_{\theta}(\vec{x}_{i}\|\vec{z}_{i})\,,$ (5) where the first term is the KL divergence between the posterior approximation for event $i$ and the prior, and the second term is the reconstruction loss term. We have added a re-scaling term $\beta_{\text{reco}}$ which alters how much influence the reconstruction loss has over the KL divergence term in the gradient updates. We fix $\beta_{\text{reco}}=5000$ for this work, but our studies indicate that the results are insensitive to within order of magnitude changes to this number. ##### Invariant mass as latent dimension Once we have a fully trained VAE, the goal is then to use the latent representation of the data obtained from the posterior approximation to perform classification on the LHCO events. To search for anomalies we typically look for excesses in the invariant mass distribution of the events. Thus it is important to understand any observed correlations between the latent vectors $\vec{z}_{i}$ and the invariant mass. The latent space dimensions are each some non-linear function of the input observables. In presence of correlations between the input observables and the invariant mass of the events, the latent dimensions are expected to encode some information on the invariant mass of the events. Crucially though, if signal events are localised in the invariant mass distribution and the VAE learns how to accurately encode and reconstruct the signal events, then part of the correlation the VAE networks learn must indeed correspond to the presence of the signal events in the data. We then propose to make the invariant mass dependence of the VAE network explicit by imposing that one of the latent dimensions corresponds exactly to the invariant mass of the events. We do this by modifying the generative process for a single event $\vec{x}_{i}$ with mass $m_{i}$ such that $\vec{z}_{i}$ is sampled from $p(\vec{z}_{i})$, while $\tilde{m}_{i}$ is sampled from a gaussian prior, centered at $m_{i}$ and with a width $\sigma(m_{i})$ reflecting a realistic uncertainty of the invariant mass reconstruction. In the LHCO case we take $\sigma(m_{i})=0.1m_{i}$ for definiteness. Both the latent vector $\vec{z}_{i}$ and the sampled mass variable $\tilde{m}_{i}$ are fed to the decoder which now has dim$(\vec{z})+1$ neurons in the input layer. The encoder remains exactly the same as in the original VAE set-up and in particular can be made completely agnostic to invariant mass by decorrelating the input variables $\vec{x}_{i}$ from $m_{i}$ using standard techniques. Now however the decoder is able to use the invariant mass information for each event to help in the reconstruction of the event data $\vec{x}_{i}$. At the same time the encoder network is no longer incentivized to learn the correlations between $\vec{x}_{i}$ and $m_{i}$ even if these are present in the data. This has a number of potential benefits: 1. 1. The optimisation occurs locally in the invariant mass variable. Events with similar latent representations, i.e. similar $\vec{z}$, but very different invariant masses will now be treated differently by the decoder, therefore the network will no longer be forced to use the latent vector $\vec{z}$ to distinguish between events with different invariant masses. 2. 2. We can visualise the correlations between the latent space and the invariant mass explicitly without relying on data. By scanning over $\vec{z}_{i}$ and $\tilde{m}_{i}$ and feeding the values into the decoder we can visualise the latent space structure in terms of different observables at different invariant masses. This novel way of inferring on what the network has learned could lead to new approaches to bump hunting with machine learning at colliders, or even more broadly to machine learning applications in high- energy physics. ##### Optimization and classification Using the R&D dataset we investigated how best to train the VAE, and then applied what we learned here to the analysis on the black box datasets. After an extensive scan over the hyper-parameters of the model, and monitoring the behaviour of the network throughout the training, we have have come to the following conclusions regarding optimization and classification: * • The Adagrad and Adadelta optimizers consistently outperform momentum-based optimizers like Adam and Nadam, which we expect is due to the smoothing of gradients in the latter which in effect reduce the sensitivity of the gradient updates to outliers in the data. * • The norm of the latent vector $\|\vec{z}_{i}\|$ performs best as a classifier for the signal events. * • Classification performance does not converge throughout training, instead it peaks and then dies off at larger epochs. The epoch at which the peak performance occurs is correlated with a minimum in the reconstruction loss of the signal-only events, indicating that the network begins to ignore outliers in the data in order to reduce the overall reconstruction loss. * • It appears that the reason for this is that at some point during the training the network learns to reconstruct just one or two of the eight observables well, while mostly ignoring the others. What we have found is that this can be avoided if we monitor the variance of the per-observable reconstruction losses through the training, and stop the training at the minima of this variance. This is very strongly correlated with the peak in the classification performance. For the training we used just one latent dimension, SeLU activation functions, two layers of 100 nodes each, the Adadelta optimizer with a learning rate of $0.001$, Mean-Squared-Error reconstruction loss, and batch sizes of $10$k. The correlations used in the early-stopping procedure are more robust and precise when using larger batch sizes. #### 3.3.2 Results on LHC Olympics For the blackbox datasets and the R&D dataset we trained the VAE networks on the whole event sample, without any cuts or binning in invariant mass, and followed the early stopping procedure outlined above. In Fig. 28 we show an example of a ROC curve obtained by training on the R&D data with an S/B of $0.1\%$. In Fig. 29 we show a bump in the invariant mass spectrum in the Black Box 1 data after applying a classifier trained with this method. The bump is at a mass of $\sim 3.8$ TeV and if we study the jet mass (Fig. 30) and $\tau_{2}/\tau_{1}$ distributions of the events that pass the cuts we clearly see that they correspond to events with jet masses $\sim 750$ GeV and $\sim 400$ GeV, with $\tau_{2}/\tau_{1}$ values from the lower end of the spectrum. Our analyses of the Black Box 2 and Black Box 3 data did not result in any clear signals in the data. Figure 10: ROC curve obtained with the VAE classifier on the R&D data. Figure 11: The invariant mass distribution for the blackbox 1 data after applying the VAE classifier. Figure 12: The jet mass distributions for the blackbox 1 data after applying the VAE classifier and restricting to the invariant mass range $[3.6,4.0]$ TeV. #### 3.3.3 Lessons Learned The first interesting lesson learned through this analysis was that the choice of the optimizer can play an important role in different machine-learning tasks. While in standard classification tasks the momentum-based optimizers such as Adam perform very well, we found when using a VAE for anomaly detection this was not the case. Instead, when the VAE is tasked with learning an effective latent representation of the dataset, including a small subset of anomalous signal events, it performs much better when using either the Adagrad or Adadelta optimizers. The reason for this appears to be that the momentum updates in the Adam optimizer tend to smooth out the effects of anomalous events in the gradient updates, in turn ignoring the signal events in the data. This may also be the case for other anomaly detection techniques, but has not been tested here. The second lesson we learned was that after some number of epochs the VAE has a tendancy to ‘over-train’ on just one or two of the eight inputs we used. This results in an overall reduction in the loss function, but interestingly it results in an increase in the loss of signal-only events. This increase in the reconstruction loss of signal-only events is inevitably correlated with a reduction in the peformance of the classifier. We remedied this by introducing an early-stopping procedure in which we stop the training when the variance of the per-observable reconstruction losses reach a minimum. This allowed us to achieve the optimal performance in an entirely unsupervised manner. ### 3.4 GAN-AE and BumpHunter999Authors: Louis Vaslin and Julien Donini. All the scripts used to train and apply the GAN-AE algorithm are given at this link: ”https://github.com/lovaslin/GAN-AE_LHCOlympics”. The implementation of the BumpHunter algorithm used in this work can be found at this link: https://github.com/lovaslin/pyBumpHunter. In near future, it is planed that this implementation of BumpHunter becames a official package to be included in the scikit-HEP toolkit. #### 3.4.1 Method The methods presented in this section combine two independent anomaly detection algorithm. The objective is to have a full analysis workflow that can give a global $p$-value and evaluate the number of signal events in any black-box dataset. ##### GAN-AE The GAN-AE method is an attempt at associating an Auto-Encoder architecture to a discriminant neural network in a GAN-like fashion. The reason for this particular setting is to use information that does not come only from the “reconstruction error” usually used to train AEs. This could be seen as an alternative way to constrain the training of an AE. As discriminant network, a simple feed-forward Multi-Layer Perceptron (MLP) is used. This method been inspired by the GAN algorithm, the two participants (AE and MLP) are trained alternatively with opposite objectives : * • The MLP is trained for a few epochs using the binary crossentropy (BC) loss function on a labeled mixture of original and reconstructed events, the objective being to expose the weaknesses of the AE. * • The AE is trained for a few epochs using a loss function combining the reconstruction error (here, the Mean Euclidean Distance between the input and output, or MED for short) and the BC loss of the MLP. In order to decorrelate as much as possible the reconstruction error and the invariant mass, the distance correlation (DisCo) term is used DiscoFever . The loss is then given by : $\text{loss}_{\text{AE}}=\text{BC}+\varepsilon\times\text{MED}+\alpha\times\text{DisCo}$ With $\varepsilon$ and $\alpha$ two hyperparameters used to balance the weights of each terms. In this case, the BC term is evaluated by giving reconstructed events to the MLP, but this time with the “wrong label”, the objective being to mislead the MLP. * • Then the AE is evaluated on a validation set using a Figure of Merit (FoM) that also combines the reconstruction error and some information from the MLP. The FoM used is given by : $\text{FoM}=\text{MED}+(1-\text{Mean}~{}\text{MLP}~{}\text{output})$ This second term is preferred over the binary crossentropy because it seems to be more stable, which makes it more suitable to set a early stopping condition. As for the reconstruction error, $1-(\text{Mean}~{}\text{MLP}~{}\text{output})$ must be minimized. In fact, the closer to zero is this term, the better the AE is at misleading the MLP. These three steps are repeated in a loop until the FoM fails to improve for five cycles. Once the AE has been trained, the MLP can be discarded since it is not needed anymore. Then, the AE can be used by taking the reconstruction error (Euclidean distance) as discriminative feature. The GAN-AE hyperparameter used for the LHC Olympics are shown in Tab. 3 \| AE \| MLP ---\|---\|--- Neurons per hidden layer \| 30/20/10/20/30 \| 150/100/50 Number of epochs per cycle \| 4 \| 10 Activation function \| ReLU (sigmoid for output) \| LeakyReLU (sigmoid for output) Dropout \| 0.2 (hidden layers only) Early-stopping condition \| 5 cycles without improvment Table 3: Hyperparameters used for the GAN-AE algorithm. ##### BumpHunter The BumpHunter algorithm is a hypertest that compares a data distribution with a reference and evaluates the p-value and significance of any deviation. To do so, BumpHunter will scan the two distributions with a sliding window of variable width. For each position and width of the scan window, the local p-value is calculated. The window corresponding to the most significant deviations is then defined as the one with the smallest local p-value. In order to deal with the look elsewhere effect and evaluate a global p-value, BumpHunter generates pseudo-experiment by sampling from the reference histogram. The scan is then repeated for each pesudo-data histogram by comparing with the original reference. This gives a local p-value distribution that can be compared with the local p-value obtained for the real data. Thus, a global $p$-value and significance is obtained. The BumpHunter hyperparameters used for the LHC Olympics are shown in Tab. 4 min/max window width \| 2/7 bins ---\|--- width step \| 1 bins scan step \| 1 bin number of bins \| 40 number of pseudo-experiments \| 10000 Table 4: Hyperparameters used for the BumpHunter algorithm. ##### Full analysis workflow The objective of this work is to use the Auto-Encoder trained withe the GAN-AE algorithm to reduce the background and then use the BumpHunter algorithm to evaluate the (global) $p$-value of a potential signal. However, the use of this second algorithm requires the use of a ”reference background” to be expected in the data. Unfortunately, such reference is not always available, as it is the case for the LHC Olympics black-box dataset. Thus, in order to use BumpHunter, one must first extract a background model for the data. Another point that has to be taken into account is the fact that, despite the use of the DisCo term, the dijet mass spectrum is not totally independent from the reconstruction error. Thus, simply rescaling the full dataset precut to fit the mass spectrum postcut will not work. One way to do this is to use a small subset of the data to compute a shaping function. The objective of this function is to capture how the mass spectrum behaves when a cut on the reconstruction error is applied. This function is computed bin per bin on the dijet mass histogram by doing the ratio of the bin yields postcut and precut. Of course, the presence of signal in the subset used for this calculation might impact this shaping function. In order to mitigate this effect, the shaping function can be fitted using the tools available in the scikit-learn toolkit. This will minimize the effect of the signal on the shaping function. Once the shaping function is defined, it can be used to reshape the mass spectum precut in order to reproduce the behaviour of the background postcut. With this final step, the full analysis workflow is the following : * • Data preprocessing (anti-$k_{t}$ clusturing, precut on dijet mass) * • Training of GAN-AE on the R&D background * • Application of the trained AE on the black-box dataset * • Use 100k events for the black-box to compute a shaping function * • Use the shaping function to build a reference to use the BumpHunter algorithm #### 3.4.2 Results on LHC Olympics The results shown were obtained with an AE trained with the GAN-AE algorithm on 100k events from the R&D background. Note that before the training and application, cuts were applied on the dijet mass at 2700 GeV and 7000 GeV. ##### R&D dataset Here we discuss the result obtained on the R&D dataset. The trained AE have been tested on 100k background events (not used during the training), as well as on the two signals provided. Fig. 13 shows the Euclidean distance distributions (left) and the corresponding ROC curves (right). This result illustrates the potential of the GAN-AE algorithm to obtain a good discrimination between the background and signals, event though only the background was used during the training. However, if the obtained AUC is good, it also appears that the Euclidean distance is still very correlated with the dijet mass. This might have a negative impact on the bump hunting algorithm performance. Figure 13: Euclidean distance distributions and ROC curves obtained for the R&D dataset. ##### Black Boxe datasets Here we discuss the results obtained for the black box dataset provided for the LHC Olympics challenge. Figure 14 shows the Euclidean distance distribution obtained for each black box. Compared to what was obtained with the R&D background, the distributions seem larger and globally shifted to the right. This is most likely due to the difference between the R&D background and the background generated in the black boxes. This fact shows that the method used is quite sensitive to the modeling of the background. Figure 14: Euclidean distance distributions and ROC curves obtained for the black boxes datasets. Figure 15 shows the shaping function obtained using 100k events from each black box dataset. A preliminary fit was made to each of the distribution. Since the fit is suboptimal this might lead to the appearance of fake bump or fake deficit during the BumpHunter scan. Figure 15: Shaping function obtained for each black box. From left to right, black box 1, 2 and 3. Finally Fig. 16 shows the results obtained with BumpHunter for all black boxes. As foreseen with the poor fit of the shaping functions, the constructed reference backgrounds do not fit well the data after cut on the Euclidean distance. In this condition and at the current stage of this work we can not really evaluate a meaningful $p$-value for a potential signal. If the results were good on the R&D dataset, it seems that the method is more challenging to apply without a good modeling of the background shape. Figure 16: Result of the BumpHunter scan obtained for each black box. From left to right, Black Box 1, 2 and 3. #### 3.4.3 Lessons Learned The LHC Olympics challenge has been a good opportunity to test the potential of the GAN-AE algorithm that we have been developing. This shows the potential of this method with the good results on the R&D dataset, but also its limits. The results obtained revealed the sensibility of GAN-AE to the modeling of the background and to the correlation of the distance distribution with the dijet mass, despite the use of DisCo term. In addition, the fact that no background simulation that fits the black boxes data were available made the use of the BumpHunter algorithm difficult to apply. ### 3.5 Gaussianizing Iterative Slicing (GIS): Unsupervised In-distribution Anomaly Detection through Conditional Density Estimation101010Authors: George Stein, Uros̆ Seljak, Biwei Dai. The Gaussianizing Iterative Slicing (GIS) used in this work was an early form of what is now called Sliced Iterative Generation (SIG). More details on SIG can be found at sig , and code will be made publicly available when ready. The results discussed in this section were also presented in Ref. stein2020unsupervised . We approached the LHC signal detection challenge as an example of in- distribution anomaly detection. Rather than searching for samples near the tails of various data distributions as is typically done in out-of- distribution anomaly detection applications, the strategy we pursue is to look for excess density in a narrow region of a parameter of interest, such as the invariant mass. We term this in-distribution anomaly detection. We perform conditional density estimation with Gaussianizing Iterative Slicing (GIS) sig , and construct a local over-density based in-distribution anomaly score to reveal the signal in a completely blind manner. The results presented here are unchanged from our blind submission to the LHC Olympics in January 2020. Parallel and independent to our development and application of our conditional density estimation method, a similar one was applied in Nachman:2020lpy , to great results on the R&D dataset. The R&D dataset lhc_randd was used for constructing and testing the method, while the first of the ‘black boxes’ lhc_bb1 was the basis of our submission to the winter Olympics challenge. As the up to 700 particles given for each event are likely the result of hadronic decays we expect them to be spatially clustered in a number of jets. By focusing on the jet summary statistics rather than the particle data from an event we are able to vastly reduce the dimensionality of the data space. We note that this form of dimensionality reduction requires a small amount of prior knowledge and understanding of the data, and the assumption that the detected jets contain the anomaly, and other data-agnostic dimensionality reduction methods could instead be used. We used the python interface of FastJet Cacciari:2011ma ; Cacciari:2005hq \- pyjet pyjet \- to perform jet clustering, setting $R=1.0$ as the jet radius and keeping all jets with $\|\eta\|<2.5$. Each jet $J$ is described by a mass $m_{J}$, a linear momentum $p=(p_{\text{T}},\eta,\phi)$, and n-subjettiness ratios $\tau^{J}_{nn-1}$ Thaler:2010tr ; Thaler:2011gf , which describe the number of sub-jets within each jet. A pair of jets has an invariant mass $M_{JJ}$. Additional parameters beyond these few may be necessary in certain scenarios, or at minimum useful, but our lack of familiarity with the field limited our search to use only these standard jet statistics. To construct images of the jets we binned each particles transverse momentum $p_{\text{T}}$ in $(\eta,\phi)$ and oriented using the moment of inertia. For the final black box 1 run we limited events to 2250 GeV $<$ $M_{JJ}$ $<$ 4750 GeV, resulting in 744,217 events remaining after all data cuts. #### 3.5.1 Method Our in-distribution anomaly detection method relies on a framework for conditional density estimation. Current state-of-the-art density estimation methods are those of flow-based models, popularized by realnvp and comprehensively reviewed in normalizing_flows . A conditional normalizing flow (NF) aims to model the conditional distribution $p(x\|x_{c})$ of input data $x$ with conditional parameter $x_{c}$ by introducing a sequence of $N$ differentiable and invertible transformations $f=f_{1}\circ f_{2}\circ\dots\circ f_{N}$ to a random variable $z$ with a simple probability density function $\pi(z)$, generally a unit Gaussian. Through the change of variables formula the probability density of the data can be evaluated as the product of the density of the transformed sample and the associated change in volume introduced by the sequence of transformations: $p(x\|x_{c})=\pi(f_{x_{c}}(x))\left\|\mathrm{det}\left(\frac{\partial f_{x_{c}}(x)}{\partial x}\right)\right\|=\pi(f_{x_{c}}(x))\prod_{i=1}^{i=N}\left\|\mathrm{det}\left(\frac{\partial f_{x_{c},i}(x)}{\partial x}\right)\right\|.$ (6) While various NF implementations make different choices for the form of the transformations $f_{i}$ and their inverse $f_{i}^{-1}$, they are generally chosen such that the determinant of the Jacobian, $\mathrm{det}(\partial f_{x_{c},i}(x)/\partial x)$, is easy to compute. Mainstream NF methods follow the deep learning paradigm: parametrize the transformations using neural networks, train by maximizing the likelihood, and optimize the large number of parameters in each layer through back-propagation. In this work we use an alternative approach to the current deep learning methodology, a new type of normalizing flow - Gaussianizing Iterative Slicing (GIS) sig . GIS works by iteratively matching the 1D marginalized distribution of the data to a Gaussian. At iteration $i$, the transformation of data $X_{i}$, $f_{x_{c},i}$, can be written as $X_{i+1}=X_{i}-W_{i}W_{i}^{T}X_{i}+W_{i}\mathbf{\Psi}_{x_{c},i}(W_{i}^{T}X_{i}),$ (7) where $W_{i}$ is the weight matrix that satisfies $W_{i}^{T}W_{i}=I$, and $\mathbf{\Psi}_{x_{c},i}$ is the 1D marginal Gaussianization of each dimension of $W_{i}^{T}X_{i}$. To improve the efficiency, the directions of the 1D slices $W_{i}$ are chosen to maximize the PDF difference between the data and Gaussian using the Wasserstein distance at each iteration. The conditional dependence on $x_{c}$ is modelled by binning the data in $x_{c}$ and estimating a 1D mapping $\mathbf{\Psi}_{i}$ for each $x_{c}$ bin, then interpolating ($W_{i}$ is the same for different $x_{c}$ bins). GIS can perform an efficient parametrization and calculation of the transformations in Equation 6, with little hyperparameter tuning. We expect that standard conditional normalizing flow methods would also work well for this task, but did not perform any comparisons. With the GIS NF trained to calculate the conditional density, our in- distribution anomaly detection method, illustrated in Fig. 17, works as following: 1. 1. Calculate the conditional density at each data point $p(x\|M_{JJ})$, denoting this $\mathrm{p_{signal}}$, using the jet masses and n-subjettiness ratios as the data $x$ and the invariant mass of a pair of jets $M_{JJ}$ as the conditional parameter. 2. 2. Calculate the density at neighbouring regions along the conditional dimension, $p(x\|M_{JJ}\pm\Delta)$, and interpolate to get a density estimate in the absence of any anomaly. This is denoted $\mathrm{p_{background}}$. Explore various values of $\Delta$ and interpolation/smoothing methods. 3. 3. The local over-density ratio (or anomaly score $\alpha$), $\mathrm{\alpha=p_{signal}/p_{background}}$, will be $\approx 1$ in the presence of a smooth background with no anomaly. A sign of an anomalous event is $\alpha>1$. Individual events can also be selected based on the desired $\alpha$ characteristic. Figure 17: In-distribution anomaly detection through conditional density estimation. Consider samples of a 1D feature $x$ and a conditional parameter of interest $M$ (left panel), drawn from a smooth Gaussian ‘background’ with a small number of anomalous ‘signal’ events added (inside red circle for clarity). The conditional density values at each data point do not allow the anomaly to be distinguished from the background (center left panel), as they only identify the outliers. However, the local over-density anomaly ratio $\mathrm{\alpha}$ peaks at the anomalous data points (center right panel), and implementing a minimum cut on the anomaly ratio reveals the anomalous events (right panel). #### 3.5.2 Results on LHC Olympics We reasoned that if there is an anomalous particle decay in the data, its jet decay products would likely be located in a narrow range of masses corresponding to the mass of the particle itself. For this reason we chose the invariant mass $M_{JJ}$ of two jets as the conditional parameter to conduct the anomaly search along. We iterated on selections of jets $i$ and $k$, and selections of n-subjettiness ratios, and found the most significant anomaly when investigating the lead two jets and the first n-subjettiness ratio, so we used {$M_{JJ}$, $m_{J_{1}}$, $m_{J_{1}}-m_{J_{2}}$, $\tau_{21}^{J_{1}}$, $\tau_{21}^{J_{2}}$} as the 5 parameters describing each event. We also experimented with training a convolutional autoencoder on the jet images, reasoning that rare events (anomalies) would have a higher reconstruction error and different latent space variables than more common ones, as seen in Farina:2018fyg . While we found a larger than average reconstruction error for signal events, and latent space parameters to be noticeably different between background and signal events, on the R&D dataset, these autoencoder-based variables introduced more noise in the density estimation than the physics-based parameters, so they were not used in our final submission. Simple investigations of the dataset showed that it was smoothly distributed, and no anomalies were apparent by eye. We trained the conditional GIS on all events, and evaluated the anomaly score $\alpha$ for each datapoint. On the R&D set we found that point estimates of the conditional densities resulted in a larger noise level than convolving the conditional density with a Gaussian PDF of width $\sigma=\Delta$ (1-PDF convolution for the background), discretely sampled at 10 points, so used the Gaussian-convolved probability estimates. $\mathrm{\sigma=250\ GeV}$ provided the most strongly peaked signal. As seen in Fig. 18, the anomaly score strongly peaks around $\mathrm{M_{JJ}\approx 3750\ GeV}$. If these events are truly from a particle decay we expect that their resulting jet statistics will be clustered around some mean value, unlike if it is simply a result of noise in the model or background. To investigate the anomaly we remove data outside of $\mathrm{3600\ GeV<M_{JJ}<3900\ GeV}$, and look at the events that remain after a series of cuts on the anomaly score $\alpha$. Figure 18: The anomaly score for each event as a function of the invariant mass of the leading two jets. A number of anomalous events are clearly seen near $\mathrm{M_{JJ}\approx 3750GeV}$. Figure 19: Parameter distributions of the events that remain after imposing cuts on the anomaly score $\alpha$, and limiting the mass range to $\mathrm{3600\ GeV<M_{JJ}<3900\ GeV}$. Vertical dashed lines are the true anomalous events that were unveiled after the close of the competition. In Fig. 19 we show the parameter distributions of the events that remain after imposing $\alpha>[1.5,2.5,5.0]$ cuts in the right four panels, and find that the most anomalous events are centered in $M_{J1}$ and $M_{J1}-M_{J2}$, and have small values of n-subjettiness $\tau_{21}$. This strongly indicates that we found a unique over-density of events that do not have similar counterparts at neighbouring $M_{JJ}$ values - i.e. an anomaly. Figure 20: The eight most anomalous events in the black box. Each pair of images visualizes the particles belonging to the lead two jets. Images were constructed by binning the transverse momentum of each particle belonging to the jet in ($\eta$, $\phi$), and oriented along the y axis using using the $p_{\text{T}}$ weighted moment of inertia. Color is log scaled. We visualized the events ranked by decreasing anomaly score in Fig. 20, and found that each of the leading two jets for events with a high anomaly score additionally have very similar visual appearances. Using the events that remain after an $\alpha>2.0$ cut we can summarize the anomalous events as follows: a $\mathrm{3772.9\pm 8.3\ GeV}$ particle decays into 2 particles, one with $\mathrm{M_{1}=727.8\pm 3.8\ GeV}$, and the other with $\mathrm{M_{2}=374.8\pm 3.5\ GeV}$. Each of these decayed into two-pronged jets. Based on the corresponding analysis of the R&D data, by limiting the number of signal events until the results visually resembled Fig. 18, we estimated that there were a total of $1000\pm 200$ of these events included in the black box of a million total events. While this is not a robust technique to estimate the number of events in all cases, as the anomaly characteristics may be much more broad or peaked in a black box than they were in the R&D set, it nevertheless gave an accurate result here. #### 3.5.3 Lessons Learned The availability of a low-noise and robust density estimation method such as GIS was key throughout this work, as the lack of hyperparamater tuning allowed us to focus on the blind search rather than worrying that failing to detect an anomaly may purely stem from some parameters in the method. We also learned plenty of interesting particle physics along the way, and thank the organizers greatly for taking the time to design and implement this challenge. ### 3.6 Latent Dirichlet Allocation111111Authors: B. M. Dillon, D. A. Faroughy, J. F. Kamenik, M. Szewc. The implementation of LDA used here for the unsupervised jet-substructure algorithm is available at http://github.com/barrydillon89/LDA-jet-substructure. Latent Dirichlet allocation (LDA) is a generative probabilistic model for discrete data first introduced to particle physics for unsupervised jet tagging and event classification in Refs. Dillon:2019cqt ; 1797846 . In general, a single collider event can be represented by a set of measurements $(o_{1},o_{2},\ldots)$. For example, the set of all particle four-momenta in the space $(p_{\text{T}},\eta,\phi)$, or any set of substructure observables extracted while declustering jets. The basic assumption of LDA is that individual events can be modelled by mixtures of a finite number of latent distributions, referred to as themes or topics. These themes are multinomial distributions over the binned space of observables where event measurements are generated from. Therefore, sampling a single measurement $o_{i}$ from a theme consists in drawing a bin from a discretized phase space containing the particular measurement. The simplest case is to assume two underlying themes, the two-theme LDA model. In this case the generative process for a single event goes as follows: (i) from a suitable prior distribution draw a random number $\omega$ between zero and one, (ii) select a theme by drawing from the binomial distribution with bias $\omega$, (iii) sample one measurement from the selected theme’s multinomial space of observables. Repeat steps (ii-iii) until all measurements in the event are generated. Repeat the procedure above for each event in the event sample. The above setting can be generalized to more than two themes by replacing the two-theme mixing proportion $\omega$ with a set of mixing proportions $(\omega_{1},\ldots,\omega_{T})$ living in a $(T-1)$-dimensional simplex121212The simplex is the space of all $T$ dimensional vectors satisfying $0\leq\omega_{t}\leq 1$ and $\sum_{t=0}^{T}\omega_{t}=1$. where $T$ is the number of themes. The $\omega_{t}\,$’s reflect the preponderance of each theme within an individual event. The themes are then drawn from the multinomial distributions with biases $\omega_{t}$. In contrast to a mixture model131313In a mixture model all measurements from an individual event are drawn from a single underlying distribution., in a mixed membership model like LDA different measurements within an event can originate from different themes, leading to a more flexible probabilistic model. LDA has a set of hyper-parameters $\alpha$ parametrizing the prior distribution from which the theme mixing proportions $\omega_{t}$ are to be drawn for each event (step (i) of the generative process described above). In particular, the prior is the Dirichlet distribution $\mathcal{D}(\alpha_{0},\ldots,\alpha_{T})$. Different choices of the concentration parameters $\alpha_{t}>0$ yield different shapes over the simplex. For the two-theme model, the Dirichlet reduces to a beta distribution $\mathcal{D}(\alpha_{0},\alpha_{1})$ over the unit interval. Once the Dirichlet hyper-parameter $\alpha$ and the number of themes $T$ is fixed, we can train a LDA model by “reversing” the generative process described above to infer from unlabelled collider data the latent parameters, namely the mixing proportions $\omega_{t}$ of each theme and the multinomial parameters $0\leq\beta_{t,m}\leq 1$ of the theme distributions $p(o\|\beta)$, where $t$ labels the theme and $m$ labels the bins in the space of observables. To learn these parameters in this work we use the standard method of stochastic variational inference (SVI). Once these parameters are learned from the data, we can then use LDA to classify events in an unsupervised fashion. In the case of a two-theme LDA model ($T=2$) we can conveniently use the likelihood ratio of the learned themes of an event $e=(o_{1},\ldots o_{N})$: $L(o_{1},\ldots,o_{N}\|\alpha)=\prod_{i=1}^{N}\frac{p(o_{i}\|\hat{\beta}_{1}(\alpha))}{p(o_{i}\|\hat{\beta}_{2}(\alpha))}\,.$ Here $\hat{\beta}_{t}$ are the estimators of the theme parameters extracted from SVI. Notice that the above expression is dependent on the Dirichlet hyper-parameter $\alpha$ leading to a landscape of classifiers. In principle there are no hard criteria for choosing one set of hyper-parameters over the other. One way to guide the choice is by using the resulting model’s perplexity, see Ref. 1797846 for details. After training LDA models for different points in the landscape, the LDA classifier with the lowest perplexity (corresponding to the LDA model that best fits the data) has been shown in examples to be correlated with truth-level performance measures like the AUC. #### 3.6.1 Method As shown in Refs. Dillon:2019cqt ; 1797846 , the two-theme LDA model can be used for anomaly detection in events with large radius jets. The jets are declustered, and at each splitting a set of substructure observables is extracted and binned. We refer to these binned measurements as $o_{j,i}$, with an added categorical variable that tags the jet to which the splitting belongs to. In the limit of exchangeable splittings, De Finetti’s theorem allows us to derive, with the help of some additional assumptions, the latent substructure of such jets, characteristic of a mixed-membership model. In practice, exchangeability is a reasonable approximation since most of the interesting physical information contained in jet substructure is in the kinematical properties of the splittings, not in their ordering. The choice of data representation and suitable binning are fundamental for LDA performance. Here we refer to data representation as both the kinematical information we use from each splitting as well as the kinematic cuts determining the splittings to be considered. As shown in Ref. 1797846 , the data representation and binning on one hand must allow for discrimination between signal and background, while at the same time produce co-occurrences of measurements within the same event. The former is obvious considering the classification task at hand, while the latter is needed for the SVI procedure to be able to extract the latent distributions. This results in a trade-of of using relatively coarse binning in order to ensure co-occurrence of measurements without sacrificing too much discriminatory power. In a fully unsupervised setting, one does not know a priori which data representation is best for any given possible signal, and any data representation carries some assumptions on how the signal is imprinted in jet substructure. In this work we consider two fairly general bases of jet substructure observables, the so called mass-basis and the Lund-basis. In the mass basis we only include splittings from subjets of mass above $30$ GeV. In the Lund basis we only include splittings from subjets which lie in the primary Lund plane. We emphasise that the resulting two data representations do not only differ in the observables included, but also in the set of splittings kept for each jet due to the different declustering cuts. In our current setting, the number of considered jets in an event is fixed to two (of highest $p_{\text{T}}$).141414When considering a variable number of jets, LDA tends to cluster together events based on jet multiplicity rather then jet substructure. After choosing a suitable data representation and binning, the procedure is as follows: We first split the dataset into overlapping invariant mass bins. In each bin, we perform a hyper-parameter optimization using perplexity to find the best LDA model. Selecting the signal and background themes in the model by looking at the latent distributions of the themes over the vocabulary and the weight distributions of the events, we build a test statistic and define a threshold for data selection. Finally, we perform a bump hunt on the selected data invariant mass distribution. In order to provide a background-only hypothesis, we consider the uncut invariant mass distribution as a background template and fix the total number of background events using the sideband regions. We can then produce a local p-value after also estimating the systematic errors due to possible classifier correlation with the invariant mass using the simulated background sample. #### 3.6.2 Results on LHC Olympics For Black Box 1 we assumed a di-jet resonance and consequently applied the LDA method to the two leading jets in each event using the mass-basis data representation. The invariant mass bin of 2.5-3.5 TeV yields themes shown in Fig. 29. We deem the signal theme to be the one with resonant substructre, uncharacteristic of QCD. Figure 21: Best inferred latent distributions of the two themes (left and right column) for Black Box 1 with the LDA method. Shown is the $m_{0},m_{1}/m_{0}$ plane of the mass-basis for the heavier (top row) and the lighter (bottom row) of the two jets. We perform a bump hunt with this model on Black Box 1 and on the simulated background sample. We show the invariant mass distribution after cutting using this LDA and the resulting BumpHunter excess in Fig. 30. In both cases we also show the background estimation used to compute the p-value. The reported significances are 1.8$\sigma$ and 3.8$\sigma$ for the background sample and the Black Box 1 sample respectively. Figure 22: Invariant mass event distribution of the simulated background (left) and Black Box 1 (right) after performing an LDA-based cut along with the background estimation using the uncut invariant mass distribution. Bottom row displays the corresponding excess found by BumpHunter. When comparing our estimates to the unveiled results, the LDA inferred di-jet resonance mass is not incompatible with the actual value of 3.8 TeV. However, the two decay products of this resonance have masses which are significantly above LDA estimates (732 and 378 GeV). The discrepancy is possibly due to an unfortunate choice of binning, since having bins narrow in $m_{0}$ may have reduced the strength of the co-occurrences, which in turn may have caused the signal features to be washed out by sculpting effects in the jet mass distribution coming from the $p_{\text{T}}$ cut. On the other hand, we could not find compelling new physics candidates in neither Black Box 2, where no signal was present, nor Black Box 3. #### 3.6.3 Lessons Learned The main lesson we take from the LHCO challenges is that a realistic LDA implementation should consider several different data representations and binnings. As we limited ourselves to di-jet jet-substructure observables we missed the characteristics of a rare signal which does not produce a rich jet substructure in the two leading jets. In the future, the search pipeline should allow to consider a larger number of jets but also include data representations which are not focused exclusively on jet substructure, e.g. by considering global jet or event variables. ### 3.7 Particle Graph Autoencoders151515Authors: Steven Tsan, Javier Duarte, Jean-Roch Vlimant, Maurizio Pierini. All code is publicly available at https://github.com/stsan9/AnomalyDetection4Jets. #### 3.7.1 Method We propose particle graph autoencoders (PGAEs) based on graph neural networks 1808887 for unsupervised detection of new physics in multijet final states at the LHC. By embedding particle jet showers as a graph, GNNs are able to exploit particle-particle relationships to efficiently encode and reconstruct particle-level information within jets. We posit that this can improve the capacity of autoencoders to learn a compressed representation of a jet and consequently help identify anomalous beyond-the-standard-model (BSM) multijet signal events from LHC data. In our PGAE model, we represent each input jet as a graph in which each particle of the jet is a node, and each node has an edge connecting it to every other particle in the jet (i.e. a fully-connected particle graph). When encoding and decoding, the graph structure of the data remains the same, but the nodes’ features, initially the particle’s four-momentum $(E,p_{x},p_{y},p_{z})$, have their dimensionality reduced during the encoding phase. We note the model can be expanded to consider additional particle-level information, such as particle type, electromagnetic charge, and pileup probability weight Bertolini:2014bba . For the encoder and decoder, we use the edge convolution layer from Ref. DGCNN , which performs message passing along the edges and aggregation of messages at the nodes of the graphs. A schematic of this is shown in Fig. 23. Figure 23: Schematic of the particle graph autoencoder model proposed. Each input jet is represented as a graph in which each particle of the jet is a node, and each node has an edge connecting it to every other particle in the jet. After an edge convolution layer DGCNN , each particle is encoded in a reduced two-dimensional latent space, before another edge convolution layer reconstructs each particle’s four-momentum $(E,p_{x},p_{y},p_{z})$. The PGAE model is constructed using the PyTorch Geometric library PyTorchGeometric . In this model, the input node features are first processed by a batch normalization layer batchnorm . The encoder is an edge convolution layer DGCNN , built from a fully connected neural network $\phi_{\mathrm{e}}$ with layers of sizes $(8,32,32,2)$ and rectified linear activation unit (ReLU) activation functions relu . The first layer of dimension $8$ represents the input, which is given by $(\bm{p}_{i},\bm{p}_{j}-\bm{p}_{i})$, where $\bm{p}_{i}$ ($\bm{p}_{j}$) is the four-momentum for particle $i$ ($j$) and $i\neq j$. The final layer produces a two-dimensional message vector from each pair of distinct particles. These two-dimensional message vectors are aggregated (using a mean function) for each receiving particle $\bm{h}_{i}=\frac{1}{\|\mathcal{N}(i)\|}\sum_{j\in\mathcal{N}(i)}\phi_{\mathrm{e}}(\bm{p}_{i},\bm{p}_{j}-\bm{p}_{i})\,,$ (8) where $\mathcal{N}(i)$ is the neighborhood of particles connected to the $i$th particle, which corresponds to all other particles in this case. This summed message $\vec{h}_{i}$ is the bottleneck or encoded representation for the $i$th particle. The decoder is also an edge convolution layer, containing a network $\phi_{\mathrm{d}}$ with layers of sizes $(4,32,32,4)$ and ReLU activation functions, except for the final layer, which reconstructs each particle’s momentum. We note that the architecture itself is insensitive to the ordering of the input particles. PyTorch Geometric supports variable-size input graphs so there is no need for zero-padding. The model is trained on the QCD background dataset with two different loss functions. The first is the mean squared error (MSE) between the input and output particles. This choice of loss function violates the permutation invariance of the algorithm because the particles must be reconstructed in the same order as they are input to achieve a small value of the loss function. For this reason, we also investigate a second, alternative loss function, the Chamfer distance loss, whose value does not depend on either the order of the input particles or the reconstructed particles 10.5555/1622943.1622971 ; Fan_2017_CVPR ; Zhang2020FSPool . Given two input sets of particles $\mathcal{M}$ and $\mathcal{N}$, expressed in terms of the momentum vectors $\bm{p}_{i}$ and $\bm{p}_{j}$ (with $i\in\mathcal{M}$ and $j\in\mathcal{N}$), the loss function is defined as $D^{\mathrm{NN}}(\mathcal{M},\mathcal{N})=\frac{1}{\|\mathcal{M}\|}\sum_{i\in\mathcal{M}}\min_{j\in\mathcal{N}}\left(\|\|\bm{p}_{i}-\bm{p}_{j}\|\|\right)^{2}+\frac{1}{\|\mathcal{N}\|}\sum_{j\in\mathcal{N}}\min_{i\in\mathcal{M}}\left(\|\|\bm{p}_{i}-\bm{p}_{j}\|\|\right)^{2}\,,$ (9) where $\|\|\bm{p}_{i}-\bm{p}_{j}\|\|$ is the Euclidean distance. Figure 24: Comparison of input and reconstructed features $E$ (far left), $p_{x}$ (center left), $p_{y}$ (center right), and $p_{z}$ (far right) for the models trained with MSE (top) and Chamfer (bottom) loss functions on the QCD testing dataset. #### 3.7.2 Results on LHC Olympics First, we studied our algorithm on the R&D dataset. As the truth information is provided, we can create a receiver operating characteristic (ROC) curve to determine the effectiveness of the PGAE to identify a signal ($\mathrm{W}^{\prime}\to\mathrm{X}\mathrm{Y}$, $\mathrm{X}\to\mathrm{q}\mathrm{q}$, and $\mathrm{Y}\to\mathrm{q}\mathrm{q}$ with $m_{\mathrm{W}^{\prime}}=3.5$ TeV, $m_{\mathrm{X}}=500$ GeV, and $m_{\mathrm{Y}}=100$ GeV) that it did not observe during training. The ROC curves for both the MSE and Chamfer loss functions are shown in Fig. 25. Although the MSE loss is not permutation invariant, we find it provides better discrimination for a new unseen signal. Figure 25: ROC curves for the PGAE trained with the MSE (left) and Chamfer loss (right). To evaluate our model’s performance for anomaly detection, we perform a resonance search (or “bump hunt”) in the dijet invariant mass $m_{\mathrm{jj}}$, computed from the two jets with highest $p_{\mathrm{T}}$ in the event. We perform this dijet search in black box (BB) 1, which contains a resonant dijet signal at $m_{\mathrm{jj}}\sim 3.8$ TeV, and BB 2, which contains no signal. We require both of the jets to be “outliers,” which we define as jets with a reconstruction loss exceeding a threshold corresponding to the 90% quantile of the loss distribution for the leading two jets in the corresponding evaluation dataset. We note that because our algorithm is jet- focused, it is straightforward to generalize this search to multijet events. For the background prediction in the signal-enriched outlier region, we perform a simplified analysis using the shape of the data in the background- enriched nonoutlier region. Specifically, we fit the ratio of the nonoutlier- to-outlier dijet mass distribution with a fourth-order polynomial to derive a transfer factor (TF). We take nonoutlier data distribution weighted by the TF as an estimate of the expected background in the outlier region. We do not consider systematic uncertainties associated to the TF although these could be taken into account in a more complete analysis in the future. The procedure is illustrated in Fig. 26 for BB 2. Figure 26: Illustration of the simplified background estimation procedure in BB 2 for the GAE trained with MSE loss. A comparison between the nonoutlier and outlier jet mass distribution is shown (upper left). The ratio of the two distributions is fit with a fourth-order polynomial to derive a transfer factor (lower left). The corresponding postfit prediction is also shown (upper right). The postfit ratio is randomly scattered around one as expected for BB 2, which contains no signal. To derive the observed significance with the simplified background prediction, we use the bump hunter (BH) algorithm Choudalakis:2011qn , recently implemented in Python pybumphunter . We choose the variable-width mass binning from the CMS dijet searches Sirunyan:2018xlo in the range from 2659 GeV to 6099 GeV. We look for resonances in windows spanning two to five bins. With the MSE model in BB 1, we identify a possible resonance around $3.9$ TeV with a local significance of $2.1\,\sigma$, which is close to the region of the injected dijet resonance with $m_{\mathrm{Z}}^{\prime}=3823$ GeV. In BB 2 using the same model, the most discernable bump lies around $3.3$ TeV with a small local significance of $0.8\,\sigma$, which agrees with the fact that BB 2 has no injected signal. For the model trained with the Chamfer loss, a $1.5\,\sigma$ excess is seen at $2.8$ TeV in BB 1 and a $-1.4\,\sigma$ excess at $5.1$ TeV in BB 2. Neither is significant. As noted previously, the permutation invariant Chamfer loss performs worse at the unsupervised anomaly detection task. This may be due to the minimization, which will often return a smaller loss value than MSE even for poorly reconstructed, anomalous jets. Fig. 27 shows the BH results for BBs 1 and 2 using the models trained with both losses. Figure 27: BB 1, MSE, $2.1\,\sigma$ at $3.9$ TeV, BB 2, MSE, $0.8\,\sigma$ at $3.3$ TeV, BB 1, Chamfer, $1.5\,\sigma$ at $2.8$ TeV, BB 2, Chamfer, $-1.4\,\sigma$ at $5.1$ TeV. Bump hunt in the dijet invariant mass in BB 1 (left) and 2 (right) using MSE (top) and Chamfer (bottom) as the loss functions. Outlier jets have a reconstruction loss in the top 10% with respect to the corresponding BB. Outlier events are required to have both jets be outliers. BB 1 has an anomalous large-radius dijet signal $\mathrm{Z}^{\prime}\to\mathrm{X}\mathrm{Y}\to(\mathrm{q}\mathrm{q})(\mathrm{q}\mathrm{q})$ injected at $m_{\mathrm{Z}}^{\prime}=3823$ GeV (with $m_{\mathrm{X}}=732$ GeV and $m_{\mathrm{Y}}=378$ GeV), while BB 2 has no injected anomalies. #### 3.7.3 Lessons Learned Graph neural networks, like our proposed particle graph autoencoder, are promising methods for anomaly detection. However, further work is needed to define a permutation-invariant loss function for use with such architectures that is more performant for anomaly detection. In addition, a more generic resonance search procedure, such a multimensional fit in the trijet, dijet, trijet, and single-jet mass distributions possibly using methods like Gaussian process fitting Frate:2017mai , would be appropriate to use in combination with this algorithm. In our experience, the R&D dataset was extremely helpful in preparing our anomaly detection algorithms and gauging whether the algorithm we were developing was on the right track. In the future, more extensive R&D datasets, together with additional black boxes with different signals, may be useful. Finally, it may be productive to host a future competition on a well-known platform, such as Kaggle, to increase engagement with the broader machine learning community. ### 3.8 Regularized Likelihoods161616Authors: Ioan-Mihail Dinu. Most of the machine learning heavy lifting was done with the help of the existing code base from the original $\mathcal{M}$-flow model introduced in Ref. Brehmer:2020vwc by Johann Brehmer and Kyle Cranmer. https://github.com/johannbrehmer/manifold-flow. #### 3.8.1 Method The method presented in this section attempts to use the power of generative models for the downstream task of Anomaly Detection. We have mainly explored the possible applications of flow-based methods, since they have the advantage of providing an explicit likelihood. Normalizing Flows (NF) are one of the best methods available at the moment for density estimation in high-dimensional data (Ref. pmlr-v37-rezende15 ). Those types of models work by learning a bijective mapping between the data distribution and a multivariate gaussian (with the same number of dimensions). Experience shows that, unfortunately, the likelihood that NF models provide is not sufficient as a stand-alone anomaly detection metric. In an attempt to regularize the likelihood obtained with such density estimation techniques we have explored several alternatives to the vanilla NF models. One particularly interesting approach is the $\mathcal{M}$-flow model introduced originally in Ref. Brehmer:2020vwc . ##### $\mathcal{M}$-flows The $\mathcal{M}$-flow model combines the idea of reconstruction error from autoencoders with the tractable density of NF. If there exists a lower- dimensional data manifold embedded in the data space, this method attempts to learn both the shape of this data manifold $\mathcal{M}$ and the density over that manifold. In order to create a $\mathcal{M}$-flow we start with a bijective mapping $\mathrm{f}$ between the latent space $\mathrm{U}\times\mathrm{V}$ to the data space $\mathrm{X}$, as in Eq. 10. The latent space is split in two components: $\mathbf{u}$, which is the latent space representation that maps to the learned manifold, and $\mathbf{v}$, which represents the remaining latent variables that are “off the manifold”. $\begin{split}\mathrm{f}:\mathrm{U}\times\mathrm{V}&\rightarrow\mathrm{X}\\\ u,v&\rightarrow\mathrm{f}(u,v)\end{split}$ (10) The transition from the space $\mathrm{U}\times\mathrm{V}$ space to the space $\mathrm{U}$ is implemented as a projection operation, the $\mathbf{v}$ component being basically discarded. The inverse of this transition is implemented with zero-padding, $\mathbf{u}$ remains unchanged and $\mathbf{v}$ is filled with zeros. We notate the previous operations with the function $\mathrm{g}$, characterizing the transformation of a latent representation $\mathbf{u}$ to a data point $\mathbf{x}$ (shown in Eq. 11). $\begin{split}\mathrm{g}:\mathrm{U}&\rightarrow\mathcal{M}\subset\mathrm{X}\\\ u&\rightarrow\mathrm{g}(u)=\mathrm{f}(u,0)\end{split}$ (11) Finally the density in the space $\mathrm{U}$ is learned using a regular NF model denoted as $\mathrm{h}$. A schematic representation of those operations is presented in Fig. 28. Figure 28: An example representation of dependencies between the data $\mathbf{x}$, latent variables $\mathbf{u}$, $\mathbf{v}$ and the normally distributed variable $\mathbf{z}$. Here the example data has 8 dimensions and the latent space has 5 dimensions. The bijective transformations are learned with Masked Autoregressive Flows (MAFs). The training of this model is split in two phases completed sequentially for every batch. Firstly, the parameters of $\mathrm{f}$ are updated by minimizing reconstruction error from the projection onto the manifold (loss function in Eq. 12). The second phase of training consists in updating the parameters of $\mathrm{h}$ by minimizing the negative log likelihood from Eq. 13. $\mathcal{L}_{manifold}=\lvert\lvert x-g(g^{-1}(x))\lvert\lvert^{2}$ (12) $\mathcal{L}_{density}=\log p_{u}(g^{-1}(x))$ (13) Regarding the preprocessing steps, the LHC Olympics datasets have been clustered and the following features have been selected for each of the two leading jets: $p_{T}$, $\eta$, $E$, $m$, $\tau_{3}/\tau_{2}$, $\tau_{2}/\tau_{1}$, where $\tau_{n}$ is the n-subjettiness. For these 12 features, the best performing manifold size was 8. This model offers the possibility to calculate both the density on the manifold and the reconstruction error from the projection on the manifold. We tried to use both of those metrics in order construct a robust anomaly score as in Eq. 14. This metric performs the anomaly detection task better on the R&D dataset than its components and better than a basic normalizing flow model trained on the same data, judging by the ROC curves in Fig. 29. $\mathcal{R}_{exp}(x)=\frac{\lvert\lvert x-g(g^{-1}(x))\lvert\lvert^{2}}{1+p_{u}(g^{-1}(x))}$ (14) While experimenting with this anomaly score, it became apparent that it generates a bias towards events with high dijet mass ($m_{jj}$). In order to decouple $\mathcal{R}_{exp}$ from $m_{jj}$ we included the marginal likelihood of $m_{jj}$, that was modeled using Kernel Density Estimation (KDE), as a term into the anomaly score. The resulting metric, denoted ${R}_{m_{jj}}$, uses the ratio between the likelihood on the manifold and marginal $m_{jj}$ likelihood as in Eq. 15. $\mathcal{R}_{m_{jj}}(x)=\frac{\lvert\lvert x-g(g^{-1}(x))\lvert\lvert^{2}}{1+\frac{p_{u}(g^{-1}(x))}{p_{KDE}(m^{x}_{jj})}}$ (15) Translating the performance obtained on the R&D data to the black boxes proved to be a big challenge. The small differences in modeling from a black box to another are often enough to introduce significant biases. The only apparent solution seems to be training and applying the method on the same dataset. #### 3.8.2 Results on LHC Olympics The R&D dataset was heavily used for benchmarking different approaches, Fig. 29 shows the anomaly detection performance of different metrics on the R&D dataset. Figure 29: Signal detection ROC curves in the R&D dataset for different anomaly scores In order to evaluate the performance of this method in the absence of pure background training data, a small fraction ($\sim 1\%$) of signal was introduced into a subsample from the R&D dataset. The resulting data sample was used both for training and evaluation of the model. Several cuts have been applied on $\mathcal{R}_{m_{jj}}$ while trying to find any indication of a resonance in the $m_{jj}$ spectrum. Although less apparent, there is still a bias towards identifying higher $m_{jj}$ events as being anomalous. The right plot in Fig. 30 shows the $m_{jj}$ distribution for events above the $50^{th}$ percentile of $\mathcal{R}_{m_{jj}}$ vs events above the $70^{th}$ percentile of $\mathcal{R}_{m_{jj}}$. If we were to take the $50^{th}$ cut as a baseline, it is clear that increasing the threshold has the effect of selecting events with slightly higher $m_{jj}$. Unfortunately there is no sharp peak in the $m_{jj}$ distribution that would indicate a possible resonance, but rather the tail of the distribution seems to get bigger. Figure 30: Overlapping $m_{jj}$ distributions below (left) and above (right) two threshold cuts on $\mathcal{R}_{m_{jj}}$. Distributions for a $50^{th}$ percentile cut are in blue, while distributions for a $70^{th}$ percentile cut are in orange. The $x$ axis is in $GeV/c^{2}$. The results so far suggest that this method can not be used reliably to find the hidden signal within the black-boxes. This behavior is consistent regardless of the choice of $\mathcal{R}_{m_{jj}}$ thresholds. #### 3.8.3 Lessons Learned One of the main lessons learned during this challenge is that: in absence of a good background model, the neural networks by themselves can not achieve good anomaly detection performance. For the winter LHC Olympics, we approached the problem with a simple autoencoder that was trained on the full background black box. Applying that model on Black Box 1 (BB1) introduced a lot of bias that ended up acting like a fake signal. Special precautions should always be taken in order to avoid this scenario. With the experience gained from studying BB1 we were a lot more careful to avoid creating fake signal. The subsequent problem proved to be the lack of a good background model. Since we could not rely on the full background black box, the alternative was to train on data, but this comes with its own issues. All of the attempts so far came short of providing a good background modeling and therefore the current anomaly detection performance leaves a lot to be desired. Those trials taught us that a good machine learning anomaly detection algorithm is not just about the neural network itself, but many other analysis details should be treated with the same amount of attention. ### 3.9 UCluster: Unsupervised Clustering171717Authors: Vinicius Mikuni and Florencia Canelli. UCluster is available at: https://github.com/ViniciusMikuni/UCluster. #### 3.9.1 Method The properties of physics beyond the Standard Model (BSM) are not yet known. However, we can expect anomalous events, from the same physics processes, to carry similar event signatures. In this section, we introduce a method for Unsupervised Clustering (UCluster). The goal of UCluster is to reduce the data dimensionality using a neural network that retains the main properties of the event collision. In this reduced representation, a clustering objective is added to the training to encourage points embedded in this space to be close together when they share similar properties and far apart otherwise. To create meaningful event embeddings, a per-particle jet mass classification is chosen. We first start clustering particles into jets with the Fastjet implementation of the anti-$k_{t}$ algorithm with $R=1.0$ for the jet radius. Each particle associated to a clustered jet receives a label, proportional to the mass of the associated jet. For this task, we require the model to learn the mass of the associated jet the particle belongs to, and which particles should belong to the same jet. This approach is motivated by the fact that the invariant mass of a jet is correlated with jet substructure observables, which often contains useful information for distinguishing different physics processes. The mass labels are then created by defining 20 equidistant intervals from 10 to 1000 GeV. For simplicity, the first 100 particles associated to the two heaviest jets in the event are considered. If a smaller number of particles are found, events are zero-padded up to 100, otherwise truncated. The classification task is achieved by means of a classification loss ($L_{\mathrm{focal}}$), defined by the focal loss DBLP:journals/corr/abs-1708-02002 . The focal loss is usually applied to classification problems with unbalanced labels. This choice was made since a different number of events is expected for different mass intervals. The focal loss expression for a multiclass classification is defined as: $L_{\mathrm{focal}}=-\frac{1}{N}\sum_{j}^{N}\sum_{m}^{M}y_{j,m}(1-p_{\theta,m}(x_{j}))^{\gamma}\log(p_{\theta,m}(x_{j}))$ (16) where $p_{\theta,m}(x_{j})$ is the network’s confidence, for event $x_{j}$ with trainable parameters $\theta$, to be classified as class $m$. The term $y_{j,m}$ is 1 if class $m$ is the correct assignment for event $x_{j}$ and 0 otherwise. In this implementation, the parameter $\gamma=2$ is used. Different values of $\gamma$ were tested resulting in no significant changes in performance. To cluster events with similar properties, a clustering loss ($L_{\mathrm{cluster}}$) is added to the overall loss function. $L_{\mathrm{cluster}}$ was introduced in DBLP:journals/corr/abs-1806-10069 , defined as: $L_{\mathrm{cluster}}=\frac{1}{N}\sum_{k}^{K}\sum_{j}^{N}\left\\|f_{\theta}(x_{j})-\mu_{k}\right\\|^{2}\pi_{jk}.$ (17) The distance between each event $x_{j}$ and cluster centroid $\mu_{k}$ is calculated in the embedding space $f_{\theta}$, created by the classification task. The function $\pi_{jk}$ weighs the importance of each event to the clustering objective of the form: $\pi_{jk}=\frac{e^{-\alpha\left\\|f_{\theta}(x_{j})-\mu_{k}\right\\|}}{\sum_{k^{\prime}}e^{-\alpha\left\\|f_{\theta}(x_{j})-\mu_{k}\right\\|}},$ (18) with hyperparameter $\alpha$. Since $L_{\mathrm{cluster}}$ is differentiable, stochastic gradient descent can be used to optimize jointly the trainable parameters $\theta$ and the centroid positions $\mu_{k}$. The combined loss to be minimized is then: $L=L_{\mathrm{focal}}+\beta L_{\mathrm{cluster}}.$ (19) The hyperparameter $\beta$ controls the relative importance between the two losses. The value of $\beta$=10 is used to give the two components the same relative order of magnitude. As defined in Eq. 17, $L_{\mathrm{cluster}}$ requires an initial value for the cluster centers. While the initial value can be corrected during training, a more stable performance is observed when the model is first pre-trained with only $L_{\mathrm{focal}}$ for 10 epochs. After the pre-training, the centroids are initialized by applying the K-Means algorithm 10.2307/2346830 to the object embeddings. The full training is then carried out with the combined loss defined in 19 for 100 epochs. The $\alpha$ parameter controls the importance of the initial cluster assignment and is set to a starting value of 1, increasing by a factor 2 for every following epoch. UCluster was designed to be independent of the ML architecture. For these studies, ABCNet Mikuni:2020wpr is used as the backbone network. ABCNet is a graph-based implementation where each reconstructed particle is taken as a node in a graph. The importance of each node is then learned by the addition of attention mechanisms described in velikovi2017graph . The 10 nearest neighbors from each particle are used to calculate the GAPLayers 2019arXiv190508705C . The initial distances are calculated in the pseudorapidity-azimuth ($\eta-\phi$) space of the form $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$. The second GAPLayer uses the Euclidean distances in the space created by subsequent fully connected layers. The architecture used and the layer where the embedding space is define are depicted in Fig. 31. No significant changes were observed when varying the number of neighbors and maximum number of training epochs. Additional hyperparameters of ABCNet were kept as is to avoid fine tuning. Figure 31: ABCNet architecture used in UCluster for a batch size N, F input features, and embedding space of size E. Fully connected layers and encoding node sizes are denoted inside “{}”. For each GAPLayer, the number of k-nearest neighbors (k) and heads (H) are given. Full lines represent direct connections while dotted lines denote skip connections. UCluster and ABCNet are implemented in v1.14 of Tensorflow tensorflow . The loss is optimized by Adam adam and back-propagation to compute gradients. The learning rate starts from 0.001 and decreases by a factor 2 every three epochs, until reaching a minimum of 1e-5. The batch size is fixed to 1024. #### 3.9.2 Results on LHC Olympics Results are presented on the R&D data set created for the LHC Olympics 2020. From this data set, 300k events are used for training, 150k for testing and 300k events used to evaluate the performance. The signal fraction in each of these samples is fixed at 1% of the total amount of events. The distributions used as input features for ABCNet are described in Tab. 5. Table 5: Descriptions of each feature used to define a point in the point cloud implementation for multiclass classification. The last two lines are the global information added to parameterize the network. Variable \| Description ---\|--- $\Delta\eta$ \| Pseudorapidity difference between the constituent and the associated jet $\Delta\phi$ \| Azimuthal angle difference between the constituent and the associated jet $\log(p_{\text{T}})$ \| Logarithm of the constituent’s $p_{\text{T}}$ $\log\mathrm{E}$ \| Logarithm of the constituent’s E $\log\frac{p_{\text{T}}}{p_{\text{T}}(\mathrm{jet})}$ \| Logarithm of the ratio between the constituent’s $p_{\text{T}}$ and the associated jet $p_{\text{T}}$ $\log\frac{\mathrm{E}}{\mathrm{E}(\mathrm{jet})}$ \| Logarithm of the ratio between the constituent’s E and the associated jet E $\Delta\mathrm{R}$ \| Distance in the $\eta-\phi$ space between the constituent and the associated jet $\log m_{J\\{1,2\\}}$ \| Logarithm of the masses of the two heaviest jets in the event $\tau_{21}^{\\{1,2\\}}$ \| Ratio of $\tau_{1}$ to $\tau_{2}$ for the two heaviest jets in the event We first evaluate the performance of UCluster by requiring the presence of two clusters in an embedding space of same size. Fig. 32 shows the result of the event embeddings, superimposed for 1000 events of the evaluation sample. Figure 32: Visualisation of the embedding space created for anomaly detection for 1000 events. The true labels are show in the left, while the cluster labels created by UCluster are shown in the right. Figure from Ref. Mikuni:2020qds . A large fraction of BSM events are found in the same cluster, confirming the earlier assumption that anomalous events would end up close together in the embedding space. However, the QCD background contamination in the same cluster only leads to a signal-to-background ratio (S/B) increase from 1% to 2.5%. The S/B can be further enhanced by partitioning the events into more clusters. This assumption is correct if the properties of the anomalous events are different than the QCD signatures. To verify this behavior, the cluster size is varied while keeping all other network parameters fixed. In Fig. 33 (left), the maximum S/B found in a single cluster is shown as a function of the cluster multiplicity. The S/B increases up to around 28% as the number of clusters increases. The effect of the sample size used for training was also checked by varying the amount of training and evaluation examples while keeping the initial S/B fixed. In Fig. 33 (right), the approximate significance (S/$\sqrt{\mathrm{B}}$) is shown as a function of the different sample sizes for UCluster trained with a fixed cluster size of 30. The red markers show the maximum significance found in a single cluster, compared to the initial significance of the sample shown in blue. For initial significances in the range 2-6, we observe enhancements by factors 3-4. The training stability is tested by retraining each model five times. The standard deviation of the independent trainings is shown by the error bars in Fig. 33. When many clusters are used, the clustering stability starts to decrease, as evidenced by larger error bars. Figure 33: Maximum signal-to-background ratio found for different clustering sizes (left) and maximum approximate significance found for UCluster trained and evaluated on different number of events with cluster size fixed to 30 (right). The uncertainty corresponds to the standard deviation of five trainings with different random weight initialization. Figure from Ref. Mikuni:2020qds . #### 3.9.3 Lessons Learned The development of UCluster was carried based on the R&D data set. While the conceptual implementation to cluster events with similar properties was achieved in this data set, an additional step to identify interesting clusters for further inspection was also required. The latter step, while important, was not fully investigated by the time the results were announced, leading to no conclusive results when the method was applied to the black boxes. For future endeavors, an automatic procedure to evaluate the cluster importance will be necessary. The classification task, paired with the clustering objective, is paramount to the ability of UCluster to reduce the data dimensionality while providing meaningful event embeddings. During the development of the method, the substructure observables of each jet in the dijet event carried information to characterize the anomaly. Because of that, a classification task that took advantage of this property was defined. However, for different decay topologies, like the one presented in BB3, this approach would not necessarily be optimal. The reason is that only one of the decay modes presented jets with substructure properties that would differ from the main QCD background. To alleviate this issue, a different classification task could be adapted. However, a more general approach to create the embedding space could be used. In particular, auto-encoders applied to particle physics are suitable candidates for a summary statistic that can encapsulate the event information in a lower dimensional representation. ## 4 Weakly Supervised ### 4.1 CWoLa Hunting181818Authors: Jack H Collins and Benjamin Nachman. The code can be found at https://github.com/Jackadsa/CWoLa-Hunting/tree/tf2/LHCO- code. #### 4.1.1 Method CWoLa (Classification Without Labels) Hunting is a strategy for searching for resonant anomalies, in which the signal is hypothesized to be localized in one chosen resonant variable (e.g. some invariant mass, $m_{\text{res}}$) and the background is known to follow some smooth and simple distribution in that variable. Given a hypothesis resonance mass $m_{\text{hyp}}$ and width, a signal region is constructed by selecting events in a window around the resonance mass hypothesis, and upper and lower sideband regions are constructed by selecting events in windows adjacent to the signal region. Additional features $\\{y\\}$ orthogonal to the resonance mass (e.g. jet substructures) are used to distinguish a potential signal from the background. A binary classifier is trained to distinguish signal region events from sideband events using these additional features. If the features are chosen such that the distribution of background events in the signal region is indistinguishable from those in the sideband, then in the absence of a signal the classifier will be driven by statistical fluctuations between the two event samples and will have poor performance on test data. If, however, the signal region contains an additional population of signal events that is not present or is very rare in the sideband, then the classifier may learn the distribution of the signal events in $\\{y\\}$. Given that the black-box data is simulated with a di-jet invariant mass trigger, we use as our resonant mass variable the invariant mass between the two highest $p_{\text{T}}$ $R=1$ anti-$k_{t}$ jets in the event, and the orthogonal features will be the jet substructure variables $\mathrm{Features:}~{}~{}m_{J,A},\;m_{J,B},\;\tau_{21,A}^{(1)},\;\tau_{21,B}^{(1)},\;\tau_{32,A}^{(1)},\;\tau_{32,B}^{(1)},$ (20) where $A,B$ refer to the two $p_{\text{T}}$-ordered jets. In order to remove some amount of correlation between the jet masses and $m_{JJ}$ in background QCD events, we rescale the jet masses before they are input into the classifiers $m_{J}\rightarrow m_{J}^{\prime}=\frac{m_{J}-30\;\text{GeV}}{m_{JJ}}+\frac{30\;\text{GeV}}{3000\;\text{GeV}}.$ (21) The key part is the rescaling by dividing by $m_{JJ}$, since the $m_{J}$ distributions have a strong scaling with $m_{JJ}$. The additional offset by $30\;\text{GeV}$ is not important, but was judged by eye to result in smaller correlation between $m_{J}$ and $m_{JJ}$. By construction, this strategy is sensitive only to signals that result from the decay of a heavy resonance into two significantly lighter particles which each decay into largely hadronic boosted final states. This still covers a broad range of phenomenological possibilities, as the space of possible jet substructures is large. This has the potential to be sensitive to the the signal in the R&D dataset and BB1, but not to that in BB3. We attempted to apply a modified form to the signal in BB3 without success, as briefly described at the end. The statistical independence of training and test sets is critical, and in order to retain as much statistical power as possible we perform a nested cross-validation procedure to select signal-like events. A detailed explanation follows. There are four training loops including the scan over $m_{\text{hyp}}$, the additional ones running over loops indexed by the labels $k,l,i$. 1. $k$ We split the entire dataset (including events outside the signal and sideband regions) randomly into five $k$-folds, and when searching for a signal in the $k$th fold we train a classifier using the remaining $k-1$ folds. Given a pre- determined threshold efficiency $\epsilon_{\text{sel}}$, that fraction of highest scoring events is chosen from the $k$th fold as judged by the classifier trained on the other folds. The selected events from each fold are then combined into a single histogram in $m_{\text{res}}$. A bump hunt is then performed at $m_{\text{hyp}}$ using a fit of a simple function to data outside the signal region to predict the expected background in the signal region. A simple Poisson hypothesis test is performed on the observed event rate in the signal region compared to the background expectation, with uncertainties in the fit parameters assumed to follow a Gaussian distribution. 2. $l$ The training of the classifier for a single $k$-fold involves another layer of cross-validation. This is due to the difficulty of a single classifier learning a small difference between two distributions that are otherwise identical besides statistical fluctuations, and overfitting to these fluctuations is unavoidable. Multiple classifiers are liable to overfit in different ways, and an ensemble model consisting of an average of multiple individually-trained neural networks tends to be more robust, due to destructive interference of overfitting and constructive interference of a true signal. For each $k$ four classifiers are trained labelled by $1\leq l\leq 5$, $l\neq k$. The $l$th classifier uses the $l$th fold of data as a validation set and the remaining three folds as training data. The ensemble model consists of the mean of the outputs of the individual neural networks. 3. $i$ For each $l$, multiple networks (in this work, three) are trained on the same data and the best performing one is chosen as the corresponding input to the ensemble model. The performance metric (evaluated on validation data) is the selection efficiency on the signal region events of a selection cut on the neural network output above a threshold determined to have a given efficiency $\epsilon_{\text{cut}}$ on sideband events. In the present study $\epsilon_{\text{cut}}$ is chosen to be 0.01, in order to be as small as possible while avoiding being dominated by statistical fluctuations when the number of validation events is small. The neural networks are coded in Keras keras with Tensorflow tensorflow backend. The architecture consists of four hidden layers each with 128 nodes. The activation function of the first hidden layer is Leaky ReLU with inactive gradient of 0.1, while the other hidden layers have elu activation functions. Dropout layers with probability 0.1 are added between each pair of hidden layers. Adam adam is used for optimization with hyperparameters: $\text{lr}=0.001$, $\beta_{1}=0.8$, $\beta_{2}=0.99$, $\text{decay}=5\times 10^{-4}$. The model is trained with batch size of 5000, the large number being chosen to increase the chance of true signal events being in each batch. The metric $\epsilon_{\text{cut}}$ is monitored on validation data; the model is saved at the maximum value and training is halted if 250 epochs pass without improvement. Training and validation events are reweighted so that the lower and upper sidebands each have equal weight (which ensures that one is not favoured over the other in training), and together they have the same total weight as the signal region. No scan or systematic optimization of hyperparameters was performed and many of these choices are likely to be suboptimal. Data is selected in the window $2632\;\text{GeV}\leq m_{JJ}\leq 6000\;\text{GeV}$, and split into 16 equally log-spaced bins. A signal region is defined as three adjacent bins, which corresponds to a width of around 15%. The two bins adjacent above and below the signal region are defined as the upper and lower sidebands. There are therefore ten overlapping signal regions, starting centered at the fourth bin and ending centered at the 13th bin. This strategy was chosen so that a signal cannot hide by being centered at a bin boundary, split equally between signal region and sideband. The signal region background is determined by a fit of the following function to the $m_{JJ}$ distribution in bins outside the signal region $\frac{dN}{dm_{JJ}}=p_{0}\frac{(1-y)^{p_{1}}}{y^{p_{2}+p_{3}\log(y)}},~{}~{}~{}y=\frac{m_{JJ}}{13\;\text{TeV}}$ (22) where $p_{i}$ are four free fit parameters. This function is used in ATLAS and CMS diboson searches Aaboud:2017eta ; Sirunyan:2016cao . #### 4.1.2 Results on LHC Olympics This study was performed on BB1 and BB2 after the signal was unblinded. However, no changes were made in the algorithm compared to the original study Collins:2018epr ; Collins:2019jip that were chosen on the basis of knowledge of the signal. The $p$-values obtained are shown in Fig. 34, for cuts at efficiency 10%, 1%, and 0.2% (the solid black line is the result before any selection). We find no significant excess in BB2, but a large $5\sigma$ excess in BB1 at a resonance mass of $3500\;\mathrm{GeV}$. Fig. 35 shows the distributions in $m_{JJ}$ obtained for the signal region centered around $3500\,\mathrm{GeV}$ for BB2 (left) and BB1 (right) after a series of cuts. Figure 34: $p$-values obtained from the analysis in the resonance mass scan for BB2 (left) and BB1 (right) at selection efficiencies 10%, 1%, 0.2%. The dashed black line is the result with no selection cut. Figure 35: $m_{JJ}$ distributions obtained for BB2 (left) and BB1 (right) for the signal region centered around $3500\,\mathrm{GeV}$ after a series of selection cuts. The top line and data points corresponds to no selection cut. We can study the signal observed in BB1 in more detail by plotting substructure distributions of selected events in the anomalous signal region, Fig. 36. Grey points are the distribution of all events in the signal region sample, while red points are the events in that sample that have been selected by a cut on the classifier output with efficiency 0.5%. In the leftmost plot, we see two clusters of events with jet masses of around $400,750\;\text{GeV}$ and the reverse, indicating that the two fat jets are produced from the decay of boosted particles of these masses. The middle plot indicates that the signal-like events all have small $\tau_{21}$ for both jets, indicating that they have a two-pronged structure. No strong clustering is observed in $\tau_{32}$ (right plot). Figure 36: Substructure distributions in the anomalous BB1 signal region for signal-like (red), and background-like (grey) events. For this figure, signal- like is defined by a selection on the classifier output with efficiency 0.5% #### 4.1.3 Lessons Learned Compared to the original study Collins:2018epr ; Collins:2019jip , we found that rescaling $m_{J}$ by $m_{JJ}$ is effective in sufficiently eliminating the correlation between these variables. In the original study we instead removed events with high jet mass over $500\;\text{GeV}$, since this is where the neural networks focussed on finding these correlations and a cut on high jet masses severely distorts the QCD background shape by rejecting a very high fraction of events at low $m_{JJ}$. The same strategy applied to BB1 would have missed the signal. Of course, the method stricty defined is clearly limited in finding signals that do not look like two fat jets with substructure, and would therefore fail in identifying the signal in BB3. We attempted to apply a modificatied version of the strategy, called ‘Tossed CWoLa SALAD’ (a variation on CWoLa SALAD (Simulation Assisted Likelihood-free Anomaly Detection) 1815227 ). In this attempt, the top ten jets in each event have recorded their 4-momenta which act as the inputs to the classifiers (with zero-padding in the case of fewer jets in an event), and the total invariant mass of the system acts as the resonant variable. The jet-momenta are rescaled by this mass in an attempt to avoid correlations. The classifiers are trained simultaneously on BB3 data and also on QCD simulation from the R&D dataset, but in this second dataset the sideband and signal region labels are reversed (‘tossed’). If the simulated background is similar to the true background, then this training strategy penalizes attempts to learn background correlations. Nonetheless, all attempts to apply this strategy and the original CWoLa SALAD strategy on BB3 led to heavy sculpting of the background $m_{JJ}$ distribution. A more global decorrelation strategy is apparently needed. ### 4.2 CWoLa and Autoencoders: Comparing Weak- and Unsupervised methods for Resonant Anomaly Detection191919Authors: Jack H. Collins, Pablo Martín-Ramiro, Benjamin Nachman, and David Shih. #### 4.2.1 Machine Learning Setup There are two techniques that show great potential at model-independent anomaly detection: Classification Without Labels (CWoLa) Metodiev:2017vrx ; Collins:2018epr ; Collins:2019jip and deep autoencoders Farina:2018fyg ; Heimel:2018mkt ; Cerri:2018anq ; Roy:2019jae ; Blance:2019ibf . These techniques have two important advantages over supervised methods. First, they are model independent and therefore allow to extend the sensitivity of current new physics searches to model-agnostic BSM scenarios. Second, they can learn directly from real data and thus do not rely on simulations that may suffer from potentially large mismodeling effects. In this section, we provide a comparative study between CWoLa and an autoencoder (AE) using a similar signal than the one released in the R&D dataset: two jets with masses $m_{j_{1}}=m_{j_{2}}=500\;\mathrm{GeV}$. We examine the ability of the two methods to identify the new physics signal at different cross sections and to increase the significance of the signal excess. CWoLa is expected to reach an excellent performance for large amounts of signal, while the AE should show a robust performance in the limit of low signal statistics. Therefore, these two approaches may have an intersection in performance at different cross sections that would be of great interest for real experimental searches. The R&D dataset represents a general new physics scenario where a signal is localized in one known dimension of phase space (in this case, the dijet invariant mass $m_{JJ}$) on top of a smooth background. In this scenario, CWoLa and the AE can be trained to exploit the information in the substructure of the two jets to gain discriminating power between the signal and background events. From the full dataset, we select all of the events in the range $m_{JJ}\in[2800,5200]\;\mathrm{GeV}$ and split them uniformly in $\log(m_{JJ})$ in $15$ bins. After selecting this range, $537304$ background events remain in our sample. We consider the following set of input features for each jet: $Y_{i}=\left\\{m_{J},\,\sqrt{\tau_{1}^{(2)}}/\tau_{1}^{(1)},\,\tau_{21},\,\tau_{32},\,\tau_{43},\,n_{\text{trk}}\right\\}\,,$ (23) where $\tau_{ij}^{(\beta)}$ represent fractions of $N$-subjettiness variables (with angular exponent $\beta=1$ unless otherwise specified in the superscript), $n_{\text{trk}}$ denotes the number of tracks of a given jet, and jets are ordered by mass in descending order. For the autoencoder we add two extra input features for each jet: $\left\\{p_{\text{T}_{1}},\,p_{\text{T}_{2}},\eta_{1},\,\eta_{2}\right\\}$, which lead to a significant performance improvement. For CWoLa, using these extra input features produces an undesirable correlation between the jets $p_{\text{T}}$ and $m_{JJ}$, which may help CWoLa learn $m_{JJ}$ and sculpt artifical bumps on this distribution in the absence of signal. ##### Classification Without Labels (CWoLa) The strategy that we follow to implement CWoLa is similar to the approach described in Ref. Collins:2019jip . First, we build a signal region and a sideband region to test for a signal hypothesis with mass $m_{JJ}=m_{\text{peak}}$, where $m_{\text{peak}}$ is the mean mass of the injected signal. The signal region contains all of the events in the three bins centered around $m_{\text{peak}}$, while the sideband region includes all of the events in the two bins below and above the signal region. The width of the signal region is $435\;\mathrm{GeV}$, and the lower and upper sidebands have a width of $262\;\mathrm{GeV}$ and $322\;\mathrm{GeV}$, respectively. Note that in a real search the location of the mass peak of any potential signal would be unknown, and therefore the mass hypothesis must be scanned as described in Ref. Collins:2019jip . After defining these two regions, the CWoLa approach is used to train a fully supervised classifier to distinguish the events of the signal region from the events of the sideband using the set of twelve input features that describe the jet substructure of each event, presented in Eq. (23). If a signal is present in the signal region with anomalous jet substructure, CWoLa should learn the information that is useful to distinguish the signal and sideband regions. This classifier can then be used to select signal-like events, producing a new distribution in the dijet mass that may enhance the significance of the signal excess. Note that the CWoLa performance should be poor when no signal is present in the signal region; in this case, the signal and sideband regions will be statistically identical and thus the classifier should not be able to distinguish between the two regions. The classifier that we use is a dense neural network with four hidden layers. The first layer has $64$ nodes with ReLU activation, and the second through fourth layers have $32$, $16$ and $4$ nodes respectively, with ELU activation. The output layer has a sigmoid activation. The first three hidden layers are followed by dropout layers with a $20\,\%$ dropout rate. We use the binary cross-entropy loss function and the Adam optimizer with learning rate of $0.001$ and learning rate decay of $5\cdot 10^{-4}$, and a batch size of $20480$. The training data is reweighted such that the two sidebands have the same total weight, the signal region has the same total weight as the sum of the sidebands, and the sum of all events weights in the training data is equal to the total number of training events. Although the two sideband regions have different event rates, this reweighting procedure ensures that they contribute equally to the training process and that the classifier output peaks around $0.5$ if no signal is present in data. In order to reduce any potential overfitting, a $5$-fold cross-validation procedure is implemented. After standardizing all the input features, we divide each bin of the full dataset in five parts to build five samples of events of equal size. First, four of these samples are used to perform four rounds of training and validation, using three different subsets for training and one for validation each time, and the other sample is saved for testing. For each cross-validation round, ten neural networks are trained for $200$ epochs on the same training and validation data using different initializations. The performance of each classifier is measured on validation data according to the metric $\epsilon_{\text{val}}$. This metric is defined as the true positive rate for classifying signal region events as such, calculated at a threshold with a false positive rate $z=0.5\,\%$ for incorrectly classifying sideband region events. The best of the ten models is saved at the end of each round, and the four selected models are used to build an ensemble model, which is used to classify the events in the test set. The output of this classifier can then be used to select the $x\,\%$ most signal- like events in the test set. We repeat the same procedure for the five choices of test set and combine the signal-like event subsamples into a final signal- like sample. If a signal is present in data and CWoLa is able to find it, the selected sample of signal-like events will show an enhanced excess in the signal region on the $m_{JJ}$ plane. ##### Autoencoder In order to use all the available information from the events, we build two different autoencoders, Autoencoder I and II, which are trained on Jet 1 and Jet 2, respectively. Both autoencoders are trained and tested on a mixed sample of signal and background events. The reason for this is that the signal contamination ratio in the full sample for the $S/B$ benchmarks that we consider is small enough for the AE to learn the potentially anomalous feature distribution of the signal events. For each jet, we build an autoencoder ensemble that is trained on a randomly selected sample of $50000$ events for only $1$ epoch. We train twenty different models (i.e. the ensemble components) and compute the reconstruction error for each event. The final reconstruction error of an event is obtained by computing the mean over the twenty different ensemble components. The autoencoder ensembles are then used to classify events in the test set, by selecting the $x\,\%$ most signal-like events applying a simultaneous cut in the reconstruction loss of the autoencoders trained on Jet $1$ and Jet $2$. Since the autoencoders are trained mostly on background events, signal events are expected to yield larger reconstruction losses if the signal is sufficiently different to the background. In this work, the two autoencoders that we consider are dense neural networks with seven hidden layers. The autoencoders have an input layer with $8$ nodes. The encoder has three hidden layers of $64$, $32$ and $16$ nodes, and is followed by a bottleneck layer with $1$ node and linear activation. Finally, the decoder has 3 hidden layers of $16$, $32$ and $64$ nodes. All of the hidden layers have ReLU activation. The output layer is made of $8$ nodes and has linear activation. We use the Minimum Squared Error (MSE) loss function, the Adam optimizer with learning rate of $0.001$ and a batch size of $5120$. We standardize all the input features from the training and test sets using training information only. #### 4.2.2 Results on LHC Olympics The goal of this work is to compare the performance of CWoLa and the AE at different cross sections. For this purpose, we define a set of eight benchmarks with the same number of background events and different amounts of injected signal events. In particular, we consider a set of benchmarks distributed over the range $S/B\in[1.2\cdot 10^{-3},6\cdot 10^{-3}]$ in the signal region. To test the consistency of both models in the absence of signal, we consider a final benchmark with no signal events. For each $S/B$ benchmark, we present results averaged over five independent runs using a random subset of signal events each time. The $S/B$ range that we consider is key to observe the complementarity of the two methods for different amounts of signal, and the observed behaviors continue beyond these limits. We analyze the performance of CWoLa and the AE according to two different metrics. First, we measure the performance of the two methods according to the AUC metric. The AUC score is computed using all the available signal events to reduce any potential overfitting. Second, we compare the performance of CWoLa and the AE at increasing the significance of the signal region excess. For this purpose, we use the following $4$-parameter function Aad:2019hjw ; Sirunyan:2018xlo to fit the smooth background distribution: $d\sigma/dm_{JJ}=(p_{0}(1-x)^{p_{1}})/(x^{p_{2}+p_{3}\ln(x)})$. This function is used to estimate the background density outside of the signal region and then the fit result is interpolated into the signal region. The number of expected and observed events in the signal region are compared and p-value is calculated to evaluate the significance of any potential excess. We present results showing the performance of CWoLa and the AE for different $S/B$ ratios according to the two previously defined metrics in Fig. 37. The left plot shows results for the AUC metric, while the right plot shows the models performance at increasing the significance of the signal region excess. First, the AUC metric shows that CWoLa achieves very good discrimination power between signal and background events for large $S/B$ ratios, reaching AUC values above $0.90$ and approaching the $0.98$ score from a fully supervised classifier. As the amount of injected signal in the signal region is decreased, the amount of useful information that allows CWoLa to discriminate between the signal and sideband regions during training is reduced. As a consequence, the classifier struggles to learn the signal features and its performance drops in testing. By contrast, the AE shows a solid performance in the full $S/B$ range. This is caused by the fact that once the AE learns to reconstruct the event sample, its performance remains independent of the amount of signal present in this sample as long as the contamination ratio is sufficiently small. Interestingly, the performance of the AE trained on Jet 2 is superior to the one trained on Jet 1, which suggests that using full event information can be very important. Note that the AUC scores from CWoLa and the AE cross at $S/B\sim 3\cdot 10^{-3}$. The p-values analysis shows two interesting patterns. First, CWoLa is able to enhance the significance of the signal regions excess by $3\sigma-8\sigma$ for $S/B$ ratios above $\sim 3\cdot 10^{-3}$, even when the fit to the full event sample shows no deviation from the background-only hypothesis. Second, the AE shows a superior performance below this range, increasing the significance of the excess by at least $2\sigma-3\sigma$ in the low $S/B$ region where CWoLa is not sensitive to the signal. Crucially, there is again an intersection in the performance of CWoLa and the AE as measured by their ability to enhance the significance of the signal region excess. Therefore, our results show that the two methods are complementary for resonant anomaly detection depending on the amount of signal. Figure 37: Left plot: Performance of CWoLa (blue), the Autoencoder trained on Jet 1 (brown) and Jet 2 (green), and their average (orange), as measured by the AUC metric. The error bars denote the standard deviation on the AUC metric. Right plot: Significance of the signal region excess after applying different cuts for CWoLa (blue) and the Autoencoder (orange). The best cuts for CWoLa and the AE ensemble correspond to the $0.3\%$ and the (Jet 1, Jet 2) = $(80\,\%,2.5\,\%)$ event selections, respectively. The initial significance of the excess ($100\,\%$ selection) is shown in green. Note that the fit to the raw distribution (i.e. no cut applied) is lower than the naive expected significance $S/\sqrt{B}$ due to a downward fluctuation in the number of background events in the signal region. #### 4.2.3 Lessons Learned We have compared weakly-supervised and unsupervised anomaly detection methods in a fully hadronic dijet resonance search in the context of the LHC Olympics $2020$. We used CWoLa and deep autoencoders as representative models of the two classes, and examined their ability to identify the signal and enhance the sensitivity of the signal excess at different cross sections. Our results demonstrate that CWoLa is very effective for sizable amounts of signal, increasing the significance of a negligible excess above the $5\sigma$ discovery limit. The AE showed a solid performance at low signal rates, raising the significance of the excess by up to $3\sigma$ in a region where CWoLa was not sensitive to the signal. Therefore, both techniques are complementary and can be used together for anomaly detection at the LHC and beyond. We feel the LHC Olympics $2020$ has been a very enriching experience that allowed us to deepen our understanding of machine learning methods for LHC physics and learn more about the work that has been done in this field. We really hope to repeat this experience next year. ### 4.3 Tag N’ Train202020Authors: Oz Amram and Cristina Mantilla Suarez. Code to reproduce all of our results can be found on https://github.com/OzAmram/TagNTrain. #### 4.3.1 Method Tag N’ Train Amram:2020ykb is a technique to train classifiers on unlabeled events that is naturally employed in an anomaly search. The Tag N’ Train (TNT) approach is based on the premise that signal events contain two or more anomalous objects (hereafter called Object-1 and Object-2) in them that can be used independently for classification. If this is the case, one can use the Object-1’s in each event to tag examples as signal-like or background-like. These signal-rich and background-rich samples can then be used to train a classifier for Object-2. This training step uses the Classification Without Labels (CWoLa) method Metodiev:2017vrx , in which a classifier is trained by using mixed samples of signal and background rather than fully labeled events. One can then repeat the procedure to train a classifier for Object-1 as well. In order to perform the initial tagging, one must be able to at least weakly classify the anomalous objects to begin with, and so the technique must be seeded by initial classifiers. In a jet-based anomaly search, autoencoders can be used as the initial classifiers because they were previously shown to be effective unsupervised classifiers of anomalous jets Farina:2018fyg ; Heimel:2018mkt . Overall, TNT takes as input a set of unlabeled data events and two initial classifiers, and outputs two new classifiers designed to have improved performance. Because the technique works better if the initial classifier can create a larger separation between signal and background in the mixed samples, multiple iterations of this technique (where the output classifiers are used with a new data sample to train new classifiers) can further improve classification performance until a plateau is reached. The technique is summarized graphically in Fig. 38. Figure 38: An illustration of the Tag N’ Train technique. Here O1 and O2 represent Object-1 and Object-2, the two components of the data one wishes to train classifiers for. The usage of TNT in an anomaly search requires data events to be partitioned into three subsets. The first subset is used to train the autoencoders, the second subset is used to perform the TNT technique that trains improved classifiers, which are then used on the third subset to select anomalous events and search for a signal. A nested cross validation approach, where the different subsets are swapped from being used for training or searching, can be used in order to achieve maximum sensitivity. Our search only targeted dijet resonances, where we took the two highest pT jets as the dijet candidate. In order to apply the Tag N’ Train technique, we treat our Object-1 as the more massive jet and Object-2 as the less massive jet of these two jets. We found that one can incorporate the assumption of a resonant signal by requiring signal-like events fall in particular dijet mass window and scanning this window over the full range during a search (as in Collins:2018epr ; Collins:2019jip ). This requirement helps to better isolate resonant signals and improves the performance of the resulting classifier. Our implementation of the TNT based anomaly search used jet images as the inputs for both the autoencoders and the TNT classifiers and CNN based model architectures trained with the Adam optimizer. We chose a latent size of 6 for the autoencoder based on results of the previous studies in the literature Farina:2018fyg ; Heimel:2018mkt . Based on results on the R&D Dataset we found that the second iteration of the Tag N’ Train technique generally reached the plateau performance and so we used 2 iterations in our search. No optimization of the model architectures and optimizer hyperparameters was attempted. A rough optimization of the selection of signal-like and background-like samples in the TNT technique was performed using the R&D dataset. In the first iteration, we used the 40% of events with the lowest autoencoder reconstruction losses as the background-like sample and the 20% with the highest as signal-like sample during the first iteration. In the second iteration, we once again used the 40% of events with the lowest scores as the background-rich sample, but tightened the signal-like cut to the top 10% of events. On the R&D dataset we found the performance was quite insensitive to the exact background-like cut used (as the resulting sample was always nearly pure background) and moderately sensitive to the signal-like cut used. On the Blackboxes we used 200k events to train the autoencoders, 400k to run Tag N’ Train (200k for each iteration) and searched for a signal in remaining 400k events. Due to limited computational resources, we did not run the full cross validation, but rather switched the 400k events used for training and searching and kept the same autoencoders. Thus only 800k out of the 1M events were actually used to determine the significance of the anomaly. We used the alteration of TNT that assumes a resonance by requiring signal events fall in a dijet mass window and scanned over the dijet mass range of 3000 to 5000 with window sizes of 500 GeV. In searching for a signal, we selected events where both jets scores were in the top 3% most signal-like (for an overall efficiency of roughly 0.1%) and then did a bump hunt. We generally found that cutting as tightly as possible while still having enough statistics for a stable fit maximized our sensitivity. We did not mix events from the two sub- samples but rather fit each simultaneously to produce a single p-value. #### 4.3.2 Results on LHC Olympics On the R&D dataset we compared the performance of the Tag N’ Train classifiers to autoencoders and the CWoLa hunting Collins:2018epr ; Collins:2019jip method for various amounts of signal in the dataset (9%, 1%, 0.3% and 0.1% of the total dataset respectively). We generally found the Tag N’ Train approach to be competitive with these other methods. For the 1% signal test, TNT produced a classifier that is somewhat worse than the one produced with TNT with an additional dijet mass cut (TNT + Mjj), but still had significantly improved performance with respect to the autoencoder. For the 0.3% and 0.1% signal tests, there was too little signal for the TNT classifier to learn from, and TNT performs significantly worse than the autoencoder. The TNT + Mjj classifier performs similarly to the one trained using CWoLa hunting for the 3 tests with larger signal. For the 0.1% test the TNT + Mjj classifier is able to achieve better performance better than that of the CWoLa hunting method, but does not improve with respect to the autoencoders approach. More details along with ROC curves are in the TNT paper Amram:2020ykb . When applying the Tag N’ Train search to Blackbox 1 we found a resonance at around 3800 $\pm$ 50 GeV with a local significance of 4$\sigma$. The bump-hunt plot for one of the subset is shown in Fig. 39. Figure 39: Events in the first data subset after final selection for Blackbox 1. The signal peak can be seen slightly above 3800 GeV. The local p-value for just this subset of the data was around 3$\sigma$. We had difficulty characterizing the nature of the signal as that was not extensively tested on the R&D dataset. We reported that one of the resonance’s daughters had a mass of 270 $\pm$ 40 GeV (this was meant to be the lighter daughter) and did a very rough guess of the total number of signal events present. When doing an initial run over Black boxes 2 and 3 we did not see any significant evidence of a signal and we did not revisit Blackboxes 2 and 3 once the results of Blackbox 1 was revealed. #### 4.3.3 Lessons Learned It is not surprising our technique was not able to find the signal in Blackbox 3 because our implementation of TNT focused only dijet resonances where both jets had anomalous substructure while Blackbox 3 had 3-jet decays and its dijet decays had gluon-jets. We were happy to find the correct resonance on Blackbox 1 but had difficulty characterizing the signal. Because we were using jet images and CNN’s it was not straightforward to interpret what our models had learned was signal-like. For future studies it may be interesting to explore using more sophisticated techniques to attempt to understand what a model like a CNN has learned, or use models with higher level features that are more interpretable. Additionally, we tried plotting the distribution of jet masses of signal-like events, but we knew that our technique distorted the jet mass distribution (selecting higher jet masses as preferentially signal-like). However we had not extensively studied this effect, making it difficult to extract the signal jet masses from the distributions of most signal-like events. We think with more deliberate study of these effects and/or using more interpretable model architectures, characterizing basic aspects of the signal (number of prongs, jet masses, etc.) should be possible. What poses a more significant challenge is trying to estimate the signal cross section (total amount of signal present in the dataset) with an anomaly detection search that features a cut meant to isolate signal events. One can always set a lower bound based on the estimated number of signal events in the final fit, however because these events are selected with quite a low selection efficiency it will usually be a poor lower bound. Without specifying a particular model, one cannot know the signal efficiency of the selection imposed so it is difficult to estimate how far this lower bound is from the true amount of signal. An approach that could be taken would be to try to calibrate the sensitivity of a technique in mock experiments on simulated datasets where the amount of signal is known. However it is likely that such a calibration, a mapping from observed p-value to total amount of signal present, depends on the nature of the signal and will not be universal. Some signals (e.g. those containing more exotic substructures) may be easier to find than others. Thus such a procedure would face difficult to estimate modeling uncertainties even if performed after signal characterization has been attempted. ### 4.4 Simulation Assisted Likelihood-free Anomaly Detection212121Authors: Anders Andreassen, Benjamin Nachman, and David Shih. The code can be found at https://github.com/bnachman/DCTRHunting. While learning directly from data can mitigate model biases, it is also useful to incorporate information from background simulations. These simulations are only an approximation to the Standard Model, but they include a wealth of physics knowledge at all energy scales relevant for collider reactions. This section describes an approach that uses a background simulation in a way that depends as little as possible on the simulations. In particular, a neural network based on the Deep neural networks using Classification for Tuning and Reweighting (Dctr) protocal Andreassen:2019nnm is trained in a region of phase space that is largely devoid of signals. In a resonance search, this region can be isolated using sidebands in the resonant feature. The reweighting function morphs the simulation into the data and is parameterized in the resonant feature(s). The model is then interpolated to the signal region region and the reweighted background simulation can be used for both enhancing signal sensitivity and estimating the background. As deep learning classifiers can naturally probe high dimensional spaces, this reweighting model can in principle exploit the full phase space for both enhancing signal sensitivity and estimating the Standard Model background. #### 4.4.1 Method Let $m$ be a feature (or set of features) that can be used to localize a potential signal in a signal region (SR). Furthermore, let $x$ be another set of features which are useful for isolating a potential signal. For the LHC Olympics, $m$ will be the invariant mass of two jets and $x$ includes information about the substructure of the two jets. The Simulation Assisted Likelihood-free Anomaly Detection (Salad) method then proceeds as follows: 1. 1. Train a classifier $f$ to distinguish data and simulation for $m\not\in\text{SR}$. This classifier is parameterized in $m$ by simply augmenting $x$ with $m$, $f=f(x,m)$ Cranmer:2015bka ; Baldi:2016fzo . If $f$ is trained using the binary cross entropy or the mean squared error loss, then asymptotically, a weight function $w(x\|m)$ is defined by $\displaystyle w(x\|m)\equiv\frac{f(x)}{1-f(x)}=\frac{p(x\|\text{data})}{p(x\|\text{simulation})}\times\frac{p(\text{data})}{p(\text{simulation})},$ (24) where the last factor in Eq. 24 is an overall constant that is the ratio of the total amount of data to the total amount of simulation. This property of neural networks to learn likelihood ratios has been exploited for a variety of full phase space reweighting and parameter estimation proposals in high energy physics (see e.g. Andreassen:2019nnm ; Brehmer:2018hga ; Brehmer:2018eca ; Brehmer:2018kdj ; Cranmer:2015bka ; Andreassen:2019cjw ). 2. 2. Simulated events in the SR are reweighted using $w(x\|m)$. The function $w(x\|m)$ is interpolated automatically by the neural network. A second classifier $g(x)$ is used to distinguish the reweighted simulation from the data. This can be achieved in the usual way with a weighted loss function such as the binary cross-entropy: $\displaystyle\text{loss}(g(x))=-\sum_{m_{i}\in\text{SR}_{\text{data}}}\log g(x_{i})-\sum_{m_{i}\in\text{SR}_{\text{simulation}}}w(x_{i}\|m_{i})\log(1-g(x_{i})).$ (25) Events are then selected with large values of $g(x)$. Asymptotically222222Sufficiently flexible neural network architecture, enough training data, and an effective optimization procedure., $g(x)$ will be monotonically related with the optimal classifier: $\displaystyle\frac{g(x)}{1-g(x)}\propto\frac{p(x\|\text{signal+background})}{p(x\|\text{background})}.$ (26) It is important that the same data are not used for training and testing. The easiest way to achieve this is using different partitions of the data for these two tasks. One can make use of more data with a cross-validation procedure Collins:2018epr ; Collins:2019jip . 3. 3. One could combine the previous step with a standard data-driven background estimation technique like a sideband fit or the ABCD method. However, one can also directly use the weighted simulation to predict the number of events that should pass a threshold requirement on $g(x)$: $\displaystyle N_{\text{predicted}}(c)=\sum_{m_{i}\in\text{SR}_{\text{simulation}}}w(x_{i}\|m_{i})\mathbb{I}[g(x_{i})>c],$ (27) for some threshold value $c$ and where $\mathbb{I}[\cdot]$ is the indicator function that is one when its argument is true and zero otherwise. The advantage of Eq. 27 over other data-based methods is that $g(x)$ could be correlated with $m$; for sideband fits, thresholds requirements on $g$ cannot sculpt local features in the $m$ spectrum. #### 4.4.2 Results on LHC Olympics The R&D dataset was used for the results presented in this section. The first step of the Dctr reweighting procedure is to train a classifier to distinguish the ‘data’ (Pythia) from the ‘simulation’ (Herwig) in a sideband region. The next step for Salad is to interpolate the reweighting function. The neural network is trained conditional on $m_{jj}$ and so it can be evaluated in the SR for values of the invariant mass that were not available during the network training. Note that the signal region must be chosen large enough so that the signal contamination in the sideband does not bias the reweighting function. Figure 40 shows a classifier trained to distinguish ‘data’ and ’simulation’ in the signal region before and after the application of the interpolated Dctr model as well as $\tau_{21}$. As expected, the neural network is a linear function of the likelihood ratio (as seen in the ratio), but this closure is excellent after the interpolated reweighting. Figure 40: A histogram of the classifier output (left) and the subleading $\tau_{21}$ (right) for a neural network trained to distinguish ‘data’ (Pythia) and ‘simulation’ (Herwig) in the signal region. The ratio between the ‘simulation’ (Herwig) or ‘simulation + Dctr’ and ‘data’ (Pythia) is depicted by orange circles (green squares) in the lower panels. Figure from Ref. Andreassen:2020nkr . After reweighting the signal region to match the data, the next step of the search is to train a classifier to distinguish the reweighted simulation from the data in the signal region. If the reweighting works exactly, then this new classifier will asymptotically learn $p(\text{signal}+\text{background})/p(\text{background})$. If the reweighting is suboptimal, then some of the classifier capacity will be diverted to learning the residual difference between the simulation and background data. If the reweighted simulation is nothing like the data, then all of the capacity will go towards this task and it will not be able to identify the signal. There is therefore a tradeoff between how different the (reweighted) simulation is from the data and how different the signal is from the background. If the signal is much more different from the background than the simulation is from the background data, it is possible that a sub-optimally reweighted simulation will still be able to identify the signal. Figure 41 shows the sensitivity of the Salad tagger to signal as a function of the signal-to-background ratio ($S/B$) in the signal region. In all cases, the background is the QCD simulation using Pythia. The Pythia lines correspond to the case where the simulation follows the same statistics as the data ($=$ Pythia). When the $S/B\sim\mathcal{O}(1)$, then the performance in Fig. 41 is similar to a fully supervised classifier. As $S/B\rightarrow 0$, the Pythia curves approach the random classifier, with a max significance improvement of unity. The significance improvement quickly drops to unity for Herwig when $S/B\lesssim 1\%$, indicating the the network is spending more capacity on differentiating Pythia from Herwig than finding signal. Salad significantly improves the performance of the Herwig-only approach. In particular, the Salad tagger is effective to about $S/B\lesssim 0.5\%$, whereas the Herwig-only tagger is only able to provide useful discrimination power down to about $S/B\sim 1\%$. The performance gains can be combined with a sideband background estimation strategy, as long as threshold requirements on the classifier do not sculpt bumps in the $m_{jj}$ spectrum. However, there is also an opportunity to use Salad to directly estimate the background from the interpolated simulation. The right plot of Fig. 41 illustrates the efficacy of the background estimation for a single classifier trained in the absence of signal. Without the Dctr reweighting, the predicted background rate is too low by a factor of two or more below 10% data efficiency. With the interpolated reweighting function, the background prediction is accurate within a few percent down to about 1% data efficiency. Figure 41: Left: the significance improvement at the a fixed 50% signal efficiency as a function of the signal-to-background ratio ($S/B$) in the signal region. The evaluation of these metrics requires signal labels, even though the training of the classifiers themselves do not have signal labels. Error bars correspond to the standard deviation from training five different classifiers. Each classifier is itself the truncated mean over ten random initializations. Right: The predicted efficiency normalized to the true data efficiency in the signal region for various threshold requirements on the NN. The $x$-axis is the data efficiency from the threshold. The error bars are due to statistical uncertainties. Figure from Ref. Andreassen:2020nkr . #### 4.4.3 Lessons Learned In practice, the difficulty in using Salad to directly estimate the background is the estimation of the residual bias. One may be able to use validation regions between the signal region and sideband region, but it will never require as much interpolation as the signal region itself. One can rely on simulation variations and auxiliary measurements to estimate the systematic uncertainty from the direct Salad background estimation, but estimating high- dimensional uncertainties is challenging Nachman:2019dol ; Nachman:2019yfl . With a low-dimensional reweighting or with a proper high-dimensional systematic uncertainty estimate, the parameterized reweighting used in Salad should result in a lower uncertainty than directly estimating the uncertainty from simulation. In particular, any nuisance parameters that affect the sideband region and the signal region in the same way will cancel when reweighting and interpolating. While the numerical Salad results presented here did not fully achieve the performance of a fully supervised classifier trained directly with inside knowledge about the data, there is room for improvement. In particular, a detailed hyperparameter scan could improve the quality of the reweighting. Additionally, calibration techniques could be used to further increase the accuracy Cranmer:2015bka . Future work will investigate the potential of Salad to analyze higher-dimensional feature spaces as well as classifier features that are strongly correlated with the resonant feature. It will also be interesting to compare Salad with other recently proposed model independent methods. When the nominal background simulation is an excellent model of nature, Salad should perform similarly to the methods presented in Ref. DAgnolo:2018cun ; DAgnolo:2019vbw and provide a strong sensitivity to new particles. In other regimes where the background simulation is biased, Salad should continue to provide a physics-informed but still mostly background/signal model-independent approach to extend the search program for new particles at the LHC and beyond. ### 4.5 Simulation-Assisted Decorrelation for Resonant Anomaly Detection232323Authors: Kees Benkendorfer, Luc Le Pottier, Benjamin Nachman. The code can be found at https://github.com/bnachman/DCTRHunting. In this section, two weakly supervised approaches are studied: Classification without Labels (CWoLa) Metodiev:2017vrx ; Collins:2018epr ; Collins:2019jip ; collaboration2020dijet and Simulation Assisted Likelihood-free Anomaly Detection (Salad) Andreassen:2020nkr . CWoLa is a method that does not depend on simulation and achieves signal sensitivity by comparing a signal region with nearby sideband regions in the resonance feature. As a result, CWoLa is particularly sensitive to dependencies between the classification features and the resonant feature. Salad uses a reweighted simulation to achieve signal sensitivity. Since it never directly uses the sideband region, Salad is expected to be more robust than CWoLa to dependencies. In order to recover the performance of CWoLa in the presence of significant dependence between the classification features and the resonant feature, a new method called simulation augmented CWoLa (SA-CWoLa) is introduced. The SA-CWoLa approach augments the CWoLa loss function to penalize the classifier for learning differences between the signal region and the sideband region in simulation, which is signal-free by construction. All of these methods will be investigated using the correlation test proposed in Ref. Nachman:2020lpy . #### 4.5.1 Method For a set of features $(m,x)\in\mathbb{R}^{n+1}$, let $f:\mathbb{R}^{n}\rightarrow[0,1]$ be parameterized by a neural network. The observable $m$ is special, for it is the resonance feature that should be relatively independent from $f(x)$. The signal region (SR) is defined by an interval in $m$ and the sidebands (SB) are neighboring intervals. All neural networks were implemented in Keras keras with the Tensorflow backend tensorflow and optimized with Adam adam . Each network is composed of three hidden layers with 64 nodes each and use the rectified linear unit (ReLU) activation function. The sigmoid function is used after the last layer. Training proceeds for 10 epochs with a batch size of 200. None of these parameters were optimized; it is likely that improved performance could be achieved with an in-situ optimization based on a validation set. ##### Simulation Assisted Likelihood-free Anomaly Detection (SALAD) The Salad network Andreassen:2020nkr is optimized using the following loss: $\displaystyle\mathcal{L}_{\text{SALAD}}[f]$ $\displaystyle=-\sum_{i\in\text{SR,data}}\log(f(x_{i}))-\sum_{i\in\text{SR,sim.}}w(x_{i},m)\log(1-f(x_{i}))\,$ (28) where $w(x_{i},m)=g(x_{i},m)/(1-g(x_{i},m))$ are a set of weights using the Classification for Tuning and Reweighting (Dctr) Andreassen:2019nnm method. The function $g$ is a parameterized classifier Cranmer:2015bka ; Baldi:2016fzo trained to distinguish data and simulation in the sideband: $\displaystyle\mathcal{L}[g]$ $\displaystyle=-\sum_{i\in\text{SB,data}}\log(g(x_{i},m))-\sum_{i\in\text{SB,sim.}}\log(1-g(x_{i},m))\,.$ (29) The above neural networks are optimized with binary cross entropy, but one could use other functions as well, such as the mean-squared error. Intuitively, the idea of Salad is to train a classifier to distinguish data and simulation in the SR. However, there may be significant differences between the background in data and the background simulation, so a reweighting function is learned in the sidebands that makes the simulation look more like the background in data. ##### Simulation Augmented Classification without Labels (SA-CWoLa) The idea of CWoLa Metodiev:2017vrx is to construct two mixed samples of data that are each composed of two classes. Using CWoLa for resonant anomaly detection Collins:2018epr ; Collins:2019jip , one can construct the mixed samples using the SR and SB. In the absence of signal, the SR and SB should be statistically identical and therefore the CWoLa classifier does not learn anything useful. However, if there is a signal, then it can detect the presence of a difference between the SR and SB. In practice, there are small differences between the SR and SB because there are dependencies between $m$ and $x$ and so CWoLa will only be able to find signals that introduce a bigger difference than already present in the background. The CWoLa anomaly detection strategy was recently used in a low-dimensional application by the ATLAS experiment collaboration2020dijet . We propose a modification of the usual CWoLa loss function in order to construct a simulation-augmented (SA) CWoLa classifier: $\displaystyle\mathcal{L}_{\text{SA-CWola}}[f]$ $\displaystyle=-\sum_{i\in\text{SR,data}}\log(f(x_{i}))-\sum_{i\in\text{SB,data}}\log(1-f(x_{i}))$ $\displaystyle\hskip 71.13188pt+\lambda\left(\sum_{i\in\text{SR,sim.}}\log(f(x_{i}))+\sum_{i\in\text{SB,sim.}}\log(1-f(x_{i}))\right)\,,$ (30) where $\lambda>0$ is a hyper-parameter. The limit $\lambda\rightarrow 0$ is the usual CWoLa approach and for $\lambda>0$, the classifier is penalized if it can distinguish the SR from the SB in the (background-only) simulation. In order to help the learning process, the upper and lower sidebands are given the same total weight as each other and together, the same weight as the SR. #### 4.5.2 Results on LHC Olympics The R&D dataset is used for the results presented here. For anomaly detection, the dijet invariant mass $m_{JJ}$ is used as the resonant feature. The classification features used are the invariant mass of the lighter jet, the mass difference between the two leading jets, and the $N$-subjettiness $\tau_{21}$ of the two leading jets. As a benchmark, 1500 signal events corresponding to a fitted significance of about $2\sigma$ are injected into the data for training. For evaluation, the entire signal sample (except for the small number of injected events) is used. In order to demonstrate the breakdown of CWoLa in the presence of dependencies between the classification features and the resonant feature, we strengthen the dependence between the jet masses $m_{J}$ and invariant dijet mass $m_{JJ}$ by setting $m_{J}\mapsto m_{J}+0.1m_{JJ}$, as in Nachman:2020lpy . Figure 42 shows the performance of various configurations. The fully supervised classifier uses high statistics signal and background samples in the SR with full label information. Since the data are not labeled, this is not achievable in practice. A solid red line labeled ‘Optimal CWoLa’ corresponds to a classifier trained using two mixed samples, one composed of pure background in the single region and the other composed of mostly background (independent from the first sample) in the SR with the 1500 signal events. This is optimal in the sense that it removes the effect from phase space differences between the SR and SB for the background. The Optimal CWoLa line is far below the fully supervised classifier because the neural network needs to identify a small difference between the mixed samples over the natural statistical fluctuations in both sets. The actual CWoLa method is shown with a dotted red line. By construction, there is a significant difference between the phase space of the SR and SB and so the classifier is unable to identify the signal. At low efficiency, the CWoLa classifier actually anti-tags because the SR-SB differences are such that the signal is more SB-like then SR-like. Despite this drop in performance, the simulation augmenting modification (solid orange) with $\lambda=0.5$ nearly recovers the full performance of CWoLa. Figure 42: A Receiver Operating Characteristic (ROC) curve (left) and significance improvement curve (right) for various anomaly detection methods described in the text. The significance improvement is defined as the ratio of the signal efficiency to the square root of the background efficiency. A significance improvement of 2 means that the initial significance would be amplified by about a factor of two after employing the anomaly detection strategy. The supervised line is unachievable unless there is no mismodeling and one designed a search for the specific $W^{\prime}$ signal used in this paper. The curve labeled ‘Random’ corresponds to equal efficiency for signal and background. Figure from Ref. 1815227 . For comparison, a classifier trained using simulation directly is also presented in Figure 42. The line labeled ‘Data vs. Sim.’ directly trains a classifier to distinguish the data and simulation in the SR without reweighting. Due to the differences between the background in data and the simulated background, this classifier is not effective. In fact, the signal is more like the background simulation than the data background and so the classifier is worse than random (preferentially removes signal). The performance is significantly improved by adding in the parameterized reweighting, as advocated by Ref. Andreassen:2020nkr . With this reweighting, the Salad classifier is significantly better than random and is comparable to SA-CWoLa. The Optimal CWoLa line also serves as the upper bound in performance for Salad because it corresponds to the case where the background simulation is statistical identical to the background in data. #### 4.5.3 Lessons Learned This section has investigated the impact of dependencies between $m_{jj}$ and classification features for the resonant anomaly detection methods Salad and CWoLa. A new simulation-augmented approach has been proposed to remedy challenges with the CWoLa method. This modification is shown to completely recover the performance of CWoLa from the ideal case where dependences are ignored in the training. In both the Salad and SA-CWoLa methods, background- only simulations provide a critical tool for mitigating the sensitivity of the classifiers on dependences between the resonant feature and the classifier features. Each of the methods examined here have advantages and weaknesses, and it is likely that multiple approaches will be required to achieve broad sensitivity to BSM physics. Therefore, it is critical to study the sensitivity of each technique to dependencies and propose modifications where possible to build robustness. This paper is an important step in the decorrelation program for automated anomaly detection with machine learning. Tools like the ones proposed here may empower higher-dimensional versions of the existing ATLAS search collaboration2020dijet as well as other related searches by other experiments in the near future. ## 5 (Semi)-Supervised ### 5.1 Deep Ensemble Anomaly Detection242424Authors: Felipe F. De Freitas, Charanjit K. Khosa, Veronica Sanz. The codes can be found at the following link https://github.com/FFFreitas/Deep-Ensemble-Anomaly-Detection. #### 5.1.1 Method For the LHC Olympics challenge we opted for a semi-supervised approach. This was partly motivated by lack of time, and partly by the way the challenge itself was set up. Indeed, previous to the releasing of the blackboxes, the organisers had provided warm-up examples including signal and background labels. At the end we focused on a mixture of neural networks, with convolutional layers, and Boosted Decision Trees (BDTs). This hybrid approach was based on previous studies by one of the authors Alves:2019ppy , which proposes to use a two step “pipeline” to assign event-by-event probabilities in categories signal or background. This model uses event input in two forms; raw data as an image as well as high level features (kinematic variables). We train the model for the labelled background and signal data sets (R $\&$ D data set). ResNet is used as a pre-classifier for the $\eta$-$\phi$ 2d images (weighted by pt) of the un-clustered particles of the event. Along with ResNet predictions of signal/background (event-by-event), we used the kinematics of fat jets (zero-padded in case of only one) for the BDTs. This two-step approach provides an AUC increase of about 5$\%$ over the BDT trained only on kinematic observables. ##### Data sets We start by describing the data preparation procedure 1. 1. We first create event images from the data. The images are generated from the uncluttered data display in a $224\times 224$ 2-D grid with the $x$ and $y$ positions given by the $\eta$ and $\phi$ information from the particles in an event. The 2-D grid is further converted into a RGB image with $224\times 224$ pixels, the pixels color values are normalized according to $W_{p_{\text{T}}}=\frac{p_{\text{T}}(i)}{max(p_{\text{T}}(i))}$, where $i$ runs over all the particles found in an event. 2. 2. The tabular data is a comma-separated values (CSV) file, where each row corresponds to an event and the columns are the kinematic and angular observables from the final state particles and jets from the event. In our analysis, we cluster inclusive fat jets with $p_{\text{T}}^{min}>$ 500 GeV and R=1 with the anti-$k_{T}$ algorithm. For this analysis, we consider the first two leading jets. If there were only one jet, then all the kinematics of the second jet were set to zero values. We cluster again the constituents of the fat jets with R=0.4 and $k_{T}$ algorithm with minimum $p_{T}$ condition of 20 GeV. With these jets we construct the following observables : $p_{\text{T}}^{j_{1}}$, $m_{j_{1}}$, $\eta_{j_{1}}$, $\phi_{j_{1}}$, $E_{j_{1}}$, $p_{\text{T}}^{j_{2}}$, $m_{j_{2}}$, $\eta_{j_{2}}$, $\phi_{j_{2}}$, $E_{j_{2}}$, $\delta\eta$, $\delta\phi$, ${m/E}_{j_{1}}$, ${m/E}_{j_{2}}$, $m_{jj}$, $P_{T}^{A}(j_{1}j_{2})$, $\delta R^{j_{1}}_{12}$, $\delta R^{j_{1}}_{13}$, $\delta R^{j_{1}}_{14}$, $\delta R^{j_{1}}_{23}$, $\delta R^{j_{1}}_{24}$, $\delta R^{j_{1}}_{34}$, $\delta R^{j_{2}}_{12}$, $\delta R^{j_{2}}_{13}$, $\delta R^{j_{2}}_{14}$, $\delta R^{j_{2}}_{23}$, $\delta R^{j_{2}}_{24}$, $\delta R^{j_{2}}_{34}$, $n_{subjets1}$,$n_{subjets2}$. Some of the observables are constructed from the fat-jets kinematics and some of them are from their sub-jets. After we have a trained CNN model for the classification of the image dataset, we include in the tabular data the predicted scores from the CNN model for a given event. This additional information helps to improve further the classification power of the BDT model, allowing our framework to predict with fairly good confidence if an event is background or an anomaly. ##### The CNN architecture and training methodology We use a modified pre-trained ResNet-34 as a pre-classifier, the ResNet-34 consists of 34 convolutional (Conv2D) layers. In between each Conv2D layers, one has a series of batch normalizations, average pooling and rectified activations (ReLU). For our task, we replace the last fully connected layers of the ResNet-34, responsible for the classification, with the following sequence of layers: * • An adaptive concatenated pooling layer (AdaptiveConcatPool2d), * • A flatten layer, * • A block with batch normalization, dropout, linear, and ReLU layers, * • A dense linear layer with 2 units as outputs, corresponding to a $signal$ and $background$, and a softmax activation function. The AdaptiveConcatPool2d layer uses adaptive average pooling and adaptive max pooling and concatenates them both. Such procedure provides the model with the information of both methods and improves the overall performance of the CNN. We also make use of the label smoothing methodology DBLP:journals/corr/SzegedyVISW15 as well as the MixUp 2017arXiv171009412Z training method. Due to the high imbalance between the number of signal and background events in the full R & D data set, we generate a smaller sample with 93390 images, with the same proportion of images for the signal and background events. We further separate these images into a training set containing 74712 (80%) images and 18678 (20%) for the validation and test sets. We train the ResNet-34 for 25 epochs using fit one-cycle method 2018arXiv180309820S . Our modified ResNet-34 achieves an accuracy of 92%. We then use this CNN to get the predictions for each image and append these values to the tabular data, so we have both the kinematic information for a given event and also the score from the CNN of the same event to belong to $signal$ or $background$. ##### The BDT model After we gather the predictions from our modified ResNet-34 and the kinematic information, described in Sec. 5.3.1, with the appended tabular dataset we make use of scikit-learn scikit-learn and DEAP DEAP_JMLR2012 to build an evolutionary search algorithm in order to find the best hyper-parameters of the BDT which maximize the accuracy. After we perform a evolutionary search over the space of hyper-parameters, we found a BDT with 95.38% accuracy were achieved by the following configuration: * • Estimators : 700 * • Learning rate : 0.1 * • Algorithm : SAMME.R * • Max depth : 3 * • Criterion : entropy * • Splitter : random The BDT model gives us the final classification of events where we estimate all metrics presented in the section 5.4.2. In Fig. 43 and 44 we show the BDT score and ROC curves for the test data sets. #### 5.1.2 Results on LHC Olympics Figure 43: Left: BDT scores using the kinematic observables and the scores from ResNet-34. Right: BDT scores using the kinematic observables only. Figure 44: Left: ROC curve for a BDT using the kinematic observables and the scores from ResNet-34. Right: ROC curve for a BDT using the kinematic observables only. Figure 45: Data Features for the Blackbox data 1. The dark blue line (background) refers to the labeled dataset, whereas the other three lines are distributions from the blackbox. Figure 46: Data Features for the Blackbox data 1. Using the training from the previous section and applying it to the blackbox, we found that the new physics signal is likely to be a heavy resonance of mass 3.5 GeV which further decays to two particles of mass 100 GeV and 500 GeV, as in the labeled data. We show the feature distributions in Fig. 45 and 46. Note that, due to lack of time, we have not analysed the whole blackbox, just a subsample of 200K events. #### 5.1.3 Lessons Learned Although the method outlined above is able to identify the new physics events, it is not robust enough to predict the correct range of mass of the heavy resonance. It is specific to the type of new physics it was trained on. This exercise and the issue of lack of generalization to new blackboxes, led to two of the authors to develop a semi-supervised algorithm, called Anomaly Awareness Khosa:2020qrz , with the focus on robustness respect to the types of anomalies. Initially we spent a significant amount of time on preliminary investigations to get the feeling of what could be the best approach. Maybe a better strategy would have been to try different methods in parallel. Different approaches have different systematic errors and sensitivities, and we would have liked to develop further the other proposals we thought of for the challenge. Alas, we had to settle for the quickest analysis to reach the deadline. Another point we would like to mention is the following: in the list of observables we included correlated variables such as $m_{j_{1}}$ and $m_{j_{2}}$. A more appropriate option would have been for example $m_{j_{1}}$ and $m_{j_{1}}-m_{j_{2}}$. ### 5.2 Factorized Topic Modeling252525Authors: Patrick Komiske, Eric Metodiev, Nilai Sarda, and Jesse Thaler. Code will be made available at the following link: https://github.com/nilais/factorized-topic-modeling In this contribution, we propose and evaluate a new technique to statistically model and discriminate between dijet configurations, called “factorized topic modeling”. The cross section for dijet production satisfies a leading-order factorization theorem, which implies that both jets in a dijet event are statistically independent, conditioned on their joint types. This is an extremely powerful statement, motivated from first principles, which we convert into a statistical constraint on a generative model. Starting from the framework of “jet topics” Metodiev:2018ftz , we leverage a factorization theorem for jet substructure to build a generative model, and demonstrate a procedure for optimizing it to recover both the relative fractions of different types of jets within a mixed sample, as well as the component distribution for a given observable. We use factorized topic modeling in the context of anomaly detection to identify exotic jet types in the LHC Olympics R&D dataset. #### 5.2.1 Method ##### Review of factorization Factorization is the statement that the cross-section for dijet production can be decomposed into the product of independent probability functions. Each component of the cross-section corresponds to a different physical process contributing to the observed jet pair. Concretely, to leading order in a power expansion, the cross-section for dijet production in proton-proton collisions can be written as Ellis:1991qcd : $d\sigma=\sum_{ab\to cd}f_{a}(\xi_{a})\otimes f_{b}(\xi_{b})\otimes\mathcal{H}_{ab\to cd}\otimes\mathcal{J}_{c}(z_{c})\otimes\mathcal{J}_{d}(z_{d}),$ (31) where $f_{a}$ is the parton distribution function for parton $a$ inside the proton, $\xi_{a}$ is that parton’s momentum fraction, $\mathcal{H}$ is the partonic cross section for the short-range hard scattering process $(ab\to cd)$, and $\mathcal{J}$ are the jet branching functions. In this work, as our goal is jet tagging, we focus on the part of this equation that governs the substructure of the two jets: $d\sigma\propto\sum_{cd}\mathcal{H}_{cd}\otimes\mathcal{J}_{c}(z_{c})\otimes\mathcal{J}_{d}(z_{d}).$ (32) Our goal is to translate this physical, leading-order factorization theorem into a statistical constraint on the probability distribution over jet observables. For dijets, we consider each observation to be a pair $(\mathbf{x}_{1},\mathbf{x_{2}})$, corresponding to the value of a given observable for the hardest and second-hardest jet in the event, respectively. Using eq. (32) as a starting point, we will write down a generative model for dijet production – more specifically, a topic model. ##### Review of topic modeling Topic modeling was first applied to jet physics in Metodiev:2018ftz and has since been studied in both phenomenological and experimental contexts Komiske:2018vkc ; Aad:2019onw ; Sirunyan:2019jud ; Aad:2020cws . This body of work leverages the statistical connection between themes in text corpora and jet flavors in event samples to propose a new data-driven method for defining classes of jets. We first consider an unfactorized topic model in a single observable $\mathbf{x}$. For a mixed sample $\mathcal{M}$, this corresponds to a generative process with the following structure: $p_{\mathcal{M}}(\mathbf{x})=\sum_{k}f_{\mathcal{M}}(k)\cdot p_{k}(\mathbf{x}),\qquad\text{s.t.}\quad\int_{\mathcal{X}}d\mathbf{x}\,p_{k}(\mathbf{x})=1\quad\forall k,\quad\sum_{k}f_{\mathcal{M}}(k)=1.$ (33) Each component $k$ corresponds to a jet class (i.e. an anomalous jet or an ordinary quark/gluon jet from QCD). The mixture components $\\{p_{k}\\}$ correspond to the distributions of any given jet observable $\mathbf{x}$, while the fractions $f(k)$ represent the fraction of the total sample which belongs to each component. The goal of a topic model is to simultaneously learn the components $\\{p_{k}\\}$ and fractions $f(k)$ from a set of samples $\\{\mathcal{M}_{i}\\}$. ##### Factorized topic models Unlike the univariate topic model described in eq. (33), factorized topic modeling operates on pairs of observables $\mathbf{x}_{1},\mathbf{x}_{2}$, corresponding to the leading and subleading jets in an event. The multivariate formula for the sample distribution is then given by: $p_{\mathcal{M}}(\mathbf{x}_{1},\mathbf{x}_{2})=\sum_{k}f_{\mathcal{M}}(k)p_{k}(\mathbf{x}_{1},\mathbf{x}_{2}).$ (34) To specify the form for $p(\mathbf{x}_{1},\mathbf{x}_{2})$, we must explicitly state our constraints in a statistical language: 1. 1. _Sample independence_ : The model assumes that, to leading order, the jet observable $\mathbf{x}$ depends only on the initiating parton. In reality, there is some dependence on the process in addition to the parton flavor. However, experimental studies have shown a high degree of empirical independence, and we suggest that these differences can be considered negligible for our model Komiske:2018vkc . Defining $p^{(1)},p^{(2)}$ as the distribution functions for the hardest and second-hardest jet, respectively, sample independence implies: $p^{(1)}_{k}(\mathbf{x})=p^{(2)}_{k}(\mathbf{x}).$ (35) 2. 2. _Factorization:_ The leading-order factorization theorem tells us that the two jets in an event are statistically independent, conditioned on convolution through the matrix element describing the short-range scattering. From a statistical perspective, the factorization theorem given above is mathematically equivalent to stating that our topic model for dijets must be an _mixture of products_. In other words, $(\mathbf{x}_{1}\|k_{1},k_{2})\text{ and }(\mathbf{x}_{2}\|k_{1},k_{2})\text{ are conditionally independent.}$ (36) Note that by simply replacing the structure of the sample-level probability distribution given above with these constraints, the mapping between the factorization theorem and statistical language can directly give us a topic model. The model can be expressed as follows. $p_{\mathcal{M}}(\mathbf{x}_{1},\mathbf{x}_{2})=\sum_{k_{1},k_{2}}f_{\mathcal{M}}(k_{1},k_{2})\cdot p_{k_{1}}(\mathbf{x}_{1})\cdot p_{k_{2}}(\mathbf{x}_{2}).$ (37) ##### Algorithm to solve the model Figure 47: The paired observables from a dijet sample can be represented as a histogram, shown as the matrix $\mathbf{D}$. The generative process we describe can be visualized as the matrix product $\mathbf{PFP}^{\mathsf{\scriptscriptstyle T}}$, shown as a decomposition on the right. This example is for separating dijet events into quark and gluon categories, where the observable is jet constituent multiplicity. Our goal is to find the parameters of the topic model that give the best fit to the true distribution for the mixed sample $p_{\mathcal{M}}$. First, we discretize the sample distribution into binned histograms, which allows us to reformulate eq. (37) as a matrix decomposition. Define the matrix $\mathbf{D}$ to be the 2-dimensional histogram generated by jointly binning the sampled data across $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$. Similarly, let $\mathbf{P}$ be the matrix whose columns are $n$-bin histograms representing each component $\mathbf{p}_{k}$. By rewriting the model in terms of histograms and bins, we arrive at the following non-convex program: ###### Problem 5.1 $\displaystyle\min_{\begin{subarray}{c}\mathbf{F}\in\mathbbm{R}^{k\times k}\\\ \mathbf{P}\in\mathbbm{R}^{n\times k}\end{subarray}}\left\\|\mathbf{D}-\mathbf{PFP}^{\mathsf{\scriptscriptstyle T}}\right\\|^{2}_{F},\qquad\text{s.t.}\quad\mathbf{P}^{\mathsf{\scriptscriptstyle T}}\mathbbm{1}_{n}=\mathbbm{1}_{k},\quad\mathbbm{1}_{k}^{\mathsf{\scriptscriptstyle T}}\mathbf{F}\mathbbm{1}_{k}=1,\quad\mathbf{P,F}\geq 0,$ where $\mathbbm{1}_{n}$ is the $n$-dimensional vector of all ones, and we have taken the Frobenius norm $\\|\mathbf{A}-\mathbf{B}\\|_{F}=\sqrt{\sum_{ij}(\mathbf{A}_{ij}-\mathbf{B}_{ij})^{2}}$ as our loss function. A pictorial representation of this discretization is given in figure 47. While this problem is non-convex, and thus finding global optima is not guaranteed, we employed a scheme based on alternating minimization to recover locally optimal $\mathbf{P},\mathbf{F}$. #### 5.2.2 Results on the LHC Olympics Figure 48: Anomalous components at 10% signal, Background component at 10% signal, Anomalous components at 1% signal, Background component at 1% signal The components retrieved from factorized topic modeling of the LHC Olympics R&D dataset, using jet mass as our observable. Our method shows good agreement between the learned topics and the ground truth on the jet mass observable. We are able to recover both of the new physics resonant masses (at 100 GeV and 500 GeV) with signal fraction of 10% (top row) and 1% (bottom row). The dips in the background model at the resonance masses arise because the topic finding procedure attempts to identify the most orthogonal components. In figure 48, we demonstrate the performance of our algorithm in recovering the jet mass distributions for dijet events from the R&D dataset, using jet mass as our observable. We learn a model with 3 topics, corresponding to $p_{X},p_{Y},p_{\text{QCD}}$, respectively. To generate these figures, we consider a signal fraction of 10%, and 1% respectively, solve the topic model, and then re-weight the component distributions by subtracting the overall background distribution and renormalizing. As the algorithms used to optimize the model returns extremal points in the polytope of all feasible solutions, the solution forces the components to be as orthogonal as possible, which is why we see characteristic dips in the background components near the $m_{X}$ and $m_{Y}$ resonance masses. For signal fractions of both 10% and 1%, we are able to recover the known exotic jet masses of $m_{X}=100~{}\text{GeV}$ and $m_{Y}=500~{}\text{GeV}$. As expected, the noise in the recovered distributions is noticeably larger at the lower signal fractions. (The behavior of this model in the low-signal regime with $<0.1\%$ signal is still under investigation – as we currently formulate it, performance degrades considerably, mostly likely due to our choice of histogram bins.) Even in the presence of this noise, our model has significant discriminative power. In particular, the model can infer which process any event was generated from using the likelihood ratio: $\displaystyle\mathcal{L}(\mathbf{x}_{1},\mathbf{x}_{2})$ $\displaystyle=\frac{f(\text{signal})\cdot p_{\text{signal}}(\mathbf{x}_{1},\mathbf{x}_{2})}{f(\text{background})\cdot p_{\text{background}}(\mathbf{x}_{1},\mathbf{x}_{2})}$ (38) $\displaystyle=\frac{f(X,Y)\,p_{X}(\mathbf{x}_{1})\,p_{Y}(\mathbf{x}_{2})+f(Y,X)\,p_{Y}(\mathbf{x}_{1})\,p_{X}(\mathbf{x}_{2})}{f(\text{QCD, QCD})\,p_{\text{QCD}}(\mathbf{x}_{1})\,p_{\text{QCD}}(\mathbf{x}_{2})}.$ (39) Using this likelihood ratio as a discriminant, we can test the ability of our model to classify events relative to the ground truth in the dataset. In both cases, the model performs well, with AUCs of 0.88 and 0.81, respectively. #### 5.2.3 Lessons Learned Leveraging the leading-order factorization theorem in eq. (32), we designed a statistically constrained, non-parametric, generative model to disentangle anomalous jet components from the LHC Olympics R&D dataset. For large enough signal fractions, our minimization algorithm finds a robust solution to Problem 5.1, though performance degrades at lower signal fractions. Since the input to our model is simply a 2-dimensional histogram, an interesting direction for future research could be to use this as a drop-in replacement for density estimation steps in other anomaly detection methods. More crucially, we see this model as a proof-of-concept for the idea of encoding physical constraints on scattering processes into a statistical language. ### 5.3 QUAK: Quasi-Anomalous Knowledge for Anomaly Detection262626Authors: Sang Eon Park, Dylan Rankin, Silviu-Marian Udrescu, Mikaeel Yunus, Philip Harris. Further details can be found in Ref. Park:2020pak . Deep-learning-based anomaly detection within physics has largely focused on searching for anomalous signatures in the complete absence of a signal prior. In this scenario, two fundamental approaches have been considered: * • Isolate two distinct datasets, one which may contain a signal, and one which does not; try to find a deviation between the two. * • Embed our knowledge of known physics processes into a simulation or a deep learning algorithm and look for events with a low likelihood of being a known physics process. In the first approach, colloquially referred to as classification without labels (CWoLa), conventional discrimination algorithms are used to separate the two datasets Metodiev:2017vrx ; Collins:2018epr ; Collins:2019jip ; Nachman:2020lpy . Care must be taken to ensure that selection biases are mitigated so that the only discernible difference within the discrimination algorithm is the presence of an unknown physics signal. The second approach attempts to embed a complete knowledge of physics processes within a selected region into a likelihood discriminant. An excess of events with a low likelihood of being from the selected region constitutes a new physics signature. Complete knowledge of all expected physical processes within a large, high dimensional dataset can become quite complicated and can lead to reduced sensitivity Heimel:2018mkt ; Farina:2018fyg ; Cerri:2018anq ; Kuusela_2012 . This method broadly comprises models that utilize deep learning autoencoders. When comparing the two approaches, the CWoLa approach is often more sensitive, provided a signal region is present. This increase in sensitivity results from the fact that an implicit signal assumption is placed on this type of anomaly search: a signal is localized to be within a specific kinematic region. This constitutes a signal prior to the model and enhances discrimination power. For many new physics models, there are fundamental assumptions that broadly apply to all anomalies. For example, if a massive particle decays, its decay products fall within a cone determined by the particle’s energy and Lorentz invariance. Neural net algorithms, on the other hand, have to learn about Lorentz invariance Butter:2019cae . By relying on one anomaly metric that measures the deviation from the background, we miss the chance to apply fundamental physical laws about how new physics may appear at the LHC, thus wasting our prior knowledge about existing physics. However, if we can incorporate this prior knowledge into the search, it should be possible to either improve the sensitivity of our search or restrict the size of the network, since additional constraints help to limit the scope of the exploration needed to construct the model. In this section, we extend the concept of placing signal priors on anomaly searches by developing a mechanism to add signal priors without degrading the sensitivity of the pre-existing model-independent search. Through our approach, signal priors, which may or may not be accurate signal descriptions, can be embedded within an anomaly search. By inserting additional signal priors, we enhance sensitivity to signal models with characteristics similar to the embedded signal priors. Since priors are systematically added to construct information, we refer to this technique as Quasi-Anamalous Knowledge, or simply QUAK. #### 5.3.1 Method Figure 49: The QUAK approach In natural language processing, new models have emerged that utilize semi- supervised learning to embed these constraints on models in a natural way chen2020big ; ouali2020overview . Semi-supervision works by training on both labeled and unlabeled data. With the unlabeled data, an unsupervised network is used. With the labeled data, a supervised classification is applied using the intermediate latent space of the unsupervised network. The unsupervised network constructs a latent space of self-organized patterns; the labeled data identifies the most useful characteristics of this space. The use of a self- supervised space has been found to be robust against variations in the data, and classifiers, in some cases, exceed that of supervised training hendrycks2019using . Semi-supervision has been found to be effective for anomaly detection ruff2020deep ; hendrycks2019deep , even, very recently, within physics Cheng:2020dal . This paper differs from this previous approach in the construction of the network architecture and the abstraction of the latent space. To construct QUAK, we follow a multi-step procedure whereby we * • choose a background and set of approximate signal priors that capture features of a new physics signature, * • train $N$ separate unsupervised probabilistic models for each signal prior or background prior, * • construct an $N$-dimensional “QUAK” space consisting of the loss on each unsupervised probabilistic model, and * • exploit QUAK space to search for anomalies. The construction is semi-supervised in that we use the signal priors as labels for the construction of QUAK space. Figure 49 illustrates the concept of QUAK. By constructing an N-dimensional space, we allow for the placement of certain physics processes within specific regions of the space. A background event will have low background loss and high signal loss. An anomalous event similar to the chosen signal prior will have a low signal loss and large background loss. An anomalous event that is different from both will have large signal and background loss. By searching in the region where other proxy signals are present, we aim to isolate broadly similar anomalies, but not necessarily the same. In the following sections, we present this idea in various setups, including the LHC Olympics dataset and the MNIST dataset. #### 5.3.2 Results on LHC Olympics To perform the anomaly search, we first construct high-level jet features and then feed these inputs into the network for training. The high-level features consist of n-subjettiness ratios ranging from 1-prong to 4-prong and the raw jet mass of the individual jets Thaler:2010tr ; Datta:2017rhs . Training and testing are performed with 12 variables for each event, 6 variables for each jet (4 n-subjettiness ratios, a total number of tracks, and the jet mass). In the construction of the unsupervised probabilistic model, an optimized scan for the studies with the LHC Olympics dataset converged on variational autoencoders (VAEs) kingma2014autoencoding with normalizing flows rezende2015variational for improved posterior approximation. Among a wide variety of normalizing flow architectures, we find Masked Autoregressive Flow papamakarios2017masked yields optimized results. For the training, we apply a loss metric of mean squared error reconstruction on each of the 12 variables with a KL-divergence term to regularize the sampling parameters. We choose a value of KL-divergence scale $\beta=10$ Higgins2017betaVAELB . Additionally, we choose a latent dimension $z_{dim}=4$, with a fully connected neural network with 3 hidden layers on either end of the VAE layer. In computing the loss within QUAK space, we remove the KL-divergence term. With QUAK applied to the BSM search, we train multiple separate autoencoders on the QCD(background) and individual signal priors. We first utilize the single QCD autoencoder loss (1D QUAK). We progressively add additional approximate priors to the search with 2D QUAK, including one approximate signal prior, and 3D QUAK, including two approximate signal priors. To construct the ROC curve, we systematically scan the 2D space integrating from the region of minimum signal loss and maximum QCD loss. Alternative, more sophisticated approaches, such as a fit within the n-dimensional space, are not investigated here. The performance comparison of adding additional priors to the search is shown in Fig 50. By comparing dotted, dashed, and solid lines, we see that we can increase the sensitivity of the search by adding more approximate priors in training. The addition of the approximate priors approaches, and, in some places, exceeds, a fully supervised discriminator computed by training the same inputs on the known signal. Interestingly, much of the gain in discrimination of the 3-prong signal arises by adding a 2-prong signal prior. Therefore, we observe that the addition of signal priors preserves the model- independent sensitivity of the search. Even if the signal priors are not accurate, we gain sizable performance improvement. We interpret this to mean that the added information present in the signal loss helps isolate “signal”-like anomalies from other anomalous features present within the background. Through the construction of QUAK space, we also demonstrate that incorrect signal priors, whether they result from inaccurate simulation or different signal model choice, can still be a powerful discriminant when searching for new physics. Figure 50: (Left) Receiver Operator Characteristic (ROC) for signal versus background selection for different test priors. Performance comparison of the 1D (QCD prior only), 2D (QCD prior and two prong $(m_{jj},m_{j1},m_{j2})=(4500,500,150)$), 3D (QCD prior, two prong $(m_{jj},m_{j1},m_{j2})=(4500,500,150)$ prior, and three prong $(m_{jj},m_{j1},m_{j2})=(5000,500,500)$) with fully supervised training on the correct signal prior (red). Jet masses $(m_{j1},m_{j2})$ are excluded in the training of the supervised classifier to mitigate model dependence and to allow for the potential of signal extraction through mass fits. (Right) ROC for signal versus background selection for 2D QUAK (solid) and a fixed supervised network (dashed). For both QUAK and the supervised network a signal prior of $(m_{jj},m_{j1},m_{j2})=(4500,500,150)$ is used in the training. To contrast this with conventional new physics searches, we consider constructing a supervised classifier where we choose a signal prior and apply it to a broad range of different signal models, due to uncertainties in signal simulation and detector modeling. Nearly every signal model is inconsistent with data to a certain degree. Figure 50 compares two-dimensional QUAK, trained with QCD events and a 2-prong prior, with a supervised classifier trained with the same raw inputs and signal prior. A fully-connected network is used for both the learnable mapping from data to the latent space for the VAEs, and the supervised classifier (4 hidden layers with batch normalization batchnorm and dropout hinton2012improving ). With the supervised training, we observe a general trend where the supervised classifier performs gradually worse as the test data deviates further from the 2-prong prior used to train the supervised classifier. With the 3-prong signal, we find abysmal performance with the supervised model. With QUAK, we observe relatively stable discrimination against background as the test signal further deviates from the signal prior. We conclude that QUAK incorporates signal priors in a more efficient way than supervised classifiers, and by using QUAK, we can do a more efficient scan of the whole possible BSM space. For searches where the signal prior is partially known (to within uncertainties), QUAK has the potential to mitigate loss in sensitivity. #### 5.3.3 Lessons Learned In summary, we propose the exploration of a new algorithm, QUAK, to perform model independent searches. We demonstrate this work in the context of new physics search at the LHC. We observe that the addition of approximate priors to anomaly loss allows for enhanced identification of anomalies by providing generic “signal-like” or “background-like” features to help in identification. With QUAK, we have presented an approach that effectively adds these priors without degrading the sensitivity of a prior-free model. QUAK is broadly applicable to a number of different problems and can help to improve both anomaly searches, and searches where large uncertainties are present on the signal modeling. ### 5.4 Simple Supervised learning with LSTM layers272727Authors: Zhongtian Dong. #### 5.4.1 Method Recurrent neural networks have had some success in jet-tagging tasks Guest:2018yhq . The goal of this section is to examine how a simple model with a small number of parameters can perform on the complicated anomaly detection task. As a first step, the raw hadron data where clustered into jets using the anti-$k_{t}$ algorithm with pyjet. The radius $R$ is varied during the training phase to obtain optimal test performance and is set around $R=0.7$ as a result. The input into the network is the sequence of four-momentum of jets ($p_{\text{T}}$, $\eta$, $\phi$, mass). The jets are ordered by their $p_{\text{T}}$, from largest to smallest. The length of the sequence $N$ is varied for the best performance for each data set; events that had fewer jets were be zero-padded to the same number. Typically, $N$ is chosen between $6$ and $10$. The neural network model has four hidden layers: two LSTM layers with $64$ and $128$ units, followed by two dense layers with $256$ units before a single output node. The intermediate layers have ReLU activation, and the output has a sigmoid activation. All training was done using Tensorflow through keras with the adam optimizer. $10\%$ of the R&D data set is used as the test data set, the rest is used for training and validation. The training is done with about $30$ epochs, when the model is able to successfully identify $95\%$ of the signals in the test data. #### 5.4.2 Results on LHC Olympics The model performs well on the R&D data set, which is unsurprising for a supervised learning model, but it has relatively poor performances on the black boxes. In Black Box 1, it identifies some events as signals but with relatively low confidence, i.e. the output scores given by the model are not as high as given by the R&D data. Compared to the actual signals presented in the data sets, the number of events identified as signals by the model is relatively large, with a higher average mass. It is possible that the model does not actually capture any real signal and incorrectly labels events as signals or backgrounds. In Black Box 2 and 3, the output results are similar to the results when running the model over the pure background data set. In retrospect, this happens perhaps for a good reason. The signal type in Black Box 3 is quite different from what is presented in the training set, which is probably the reason the neural network cannot identify them correctly. #### 5.4.3 Lessons Learned Supervised models tend to perform well on identifying specific signals comparing to a consistent background, and is probably inappropriate for anomaly detection with varying backgrounds. In general, it seems such a model is incapable of handling data that is different from the ones presented in the training set. Maybe we can still use the network structure to learn important features of the events but not doing classification tasks as it is. Rather than continuing on this type of model, I would like to study in a different direction next. Suppose we can obtain $n$ independent features for each event, and divide each feature into $2$ regions. We would have $2^{n}$ regions, some would be populated by ”background” generic QCD events, and some other region would be not. We can focus on these regions that are rarely populated by QCD events, if many events occupy one such region in particular data sets, we may consider this an anomaly. Perhaps it is even wrong to divide into only two regions for each feature, more regions for each feature would result in a high dimensional grid, generic QCD events would rarely appear in some grid, and they can be considered as our anomaly grid. We can use machine learning to find novel features as mentioned in some previous works Datta:2017lxt . We may also use methods such as distance correlation to make sure features we find are independent of each other DiscoFever . The majority of what has been done so far are studied with dense layer networks, it would be interesting to see if we can find exotic features with more complicated network structures. Of course, there will be a lot of potential problems with this approach, one being how to make sure that data we use for training covers all of the regions that suppose to be the backgrounds, and does not falsely label signal region as background. ## 6 Discussion The important thing in the Olympic Games is not to win, but to take part; the important thing in Life is not triumph, but the struggle; the essential thing is not to have conquered but to have fought well.282828Pierre de Coubertin, founder of the International Olympic Committee, as quoted in The Olympian (1984) by Peter L. Dixon, p. 210 The results of the LHCO are to be understood in a similar way. The goal is not to declare one superior method, but to foster the development of novel tools for the unsupervised detection of anomalies. With this in mind, we now turn to a discussion and comparison of the various algorithms’ performance on the LHCO2020 Black Box datasets. Knowing which algorithms achieved accurate results in blinded and unblinded tests is important information, as it will provide crucial feedback for the further refinement and improvement of each approach. Also, it is important to keep in mind that an approach which did not perform well in this challenge may have its strengths elsewhere and may turn out to be better suited for a different part of phase space. We discuss the results in Sec. 6.1 and review the lessons learned — both in terms of anomaly detection as well as in future directions for the LHCO — in Sec. 6.2. ### 6.1 Overall Results In the following we will review the results submitted during the two LHC Olympics sessions as well as additional contributions received for this paper. As approaches were allowed to change and improve between the sessions and in preparation of this document, we chronologically walk through results at these three stages. As discussed in Sec. 2.2, the signal in Black Box 1 consists of 834 anomalous events with the same topology as the R&D dataset and masses $m_{W^{\prime}}=3.823$ TeV, $m_{X}=732$ GeV and $m_{Y}=378$ GeV and were unblinded during the LHCO session at the 2020 ML4Jets workshop winterolympics . Nine blind approaches were submitted and are summarised in Fig. 51: ResNet + BDT (Sec. 5.1), PCA (Principal component analysis used as an outlier detector), LSTM (Sec. 5.4.1), High-level features AE (encoding kinematic and substructure variables using an autoencoder, selecting signal as events with high reconstruction MSE loss), Tag N Train (Sec. 4.3), Density Estimation (Sec. 3.5), VRNN (Sec. 3.1), Latent Dirichlet Allocation (Sec. 3.6), and Human NN (manual search). Of these submissions, four approaches identified the correct resonance mass either within the claimed error (PCA) or within a window of $\pm 200$ GeV (LSTM, Tag N Train, Density Estimation). Accurate predictions for the other observables were achieved only by the Density Estimation method. Figure 51: Results of unblinding the first black box. Shown are the predicted resonance mass (top left), the number of signal events (top right), the mass of the first daughter particle (bottom left), and the mass of the second daughter particle (bottom right). Horizontal bars indicate the uncertainty (only if provided by the submitting groups). In a smaller panel the pull (answer-true)/uncertainty is given. Descriptions of the tested models are provided in the text. Next, Black Boxes 2 and 3 were unblinded in Summer 2020 summerolympics . For Black Box 2, a resonance at 4.8 TeV (PCA), at 4.2 TeV (VRNN, Sec. 3.1), 4.6 TeV (embedding clustering, Sec. 3.9), and 5 TeV (QUAK, Sec. 5.3) were predicted. For LDA (Sec. 3.6), the absence of signal as di-jet resonance was reported. As Black Box 2 did not contain any injected signal, these results highlight a possible vulnerability of anomaly detection methods in the tail of statistical distributions. For Black Box 3 a resonance decaying to hadrons and invisible particles (PCA), a resonance with a mass between 5.4 and 6.4 TeV (LDA), at 3.1 TeV (embedding clustering), and between 5 and 5.5 TeV (QUAK) was reported. No signal was observed by one approach (VRNN). The true injected resonance with a mass of 4.2 TeV and two competing decay modes was not detected by any approach. After unveiling the black boxes, further submissions and improvements to the anomaly detectors were made. The VRNN and BuHuLaSpa (Sec. 3.3) approaches now report an enhancement at an invariant mass below 4 TeV for black box 1, while no signal is observed for the other two black boxes. With deep ensemble anomaly detection (Sec. 5.1) a resonance at 3.5 TeV is seen for the first black box and for Latent Dirichlet Allocation a resonance not incompatible with 3.8 TeV is observed. Another new submission was Particle Graph Autoencoders (Sec 3.7) which detected a resonance at 3.9 TeV for the first black box. Finally, a resonance at 3.5 TeV was seen using CWoLa hunting (Sec. 4.1). For Black Box two and three, no additional observations of a signal were reported after unblinding. ### 6.2 Overall Lessons Learned This large and diverse number of submissions on the blinded and unblinded datasets is very encouraging. Even better, the resonance in the first black box was successfully detected multiple times even before unblinding. Of the three methods finding a mass resonance mass closest to the true value, two were based on building a signal-to-background likelihood ratio (Tag N Train, Density Estimation) while one used a signal likelihood (LSTM), and likely benefitted from the same topology between the provided development signal and the first black box. However, there still is substantial room for improvement for anomaly detection in the realm of particle physics. First, no confident statement of the absence of signal for the second black box could be made, with a number of false positives at high values of the invariant mass. Second, the resonance in Black Box 3 was not detected. The structure of this signal was different from the shared topology of the development data and Black Box 1 which was likely to cause issues for models too finely tuned on these signals. Furthermore Black Box 3 featured two different decay modes which need to be combined to achieve a significant observation. Finally, substructure offered a less useful handle here as one decay mode involved the direct production of a pair of gluon jets. Despite all this, the signal in Black Box 3 still decayed as hadronic resonance with a well-defined mass-peak and visible particles in the final state. Future developments therefore will need to both improve the sensitivity as well as the statistical interpretation of anomaly detectors. Beyond the reported results on the black box datasets, we also observe the use of the initial dataset for the development of new approaches. Overall, the volume of work and results shows the value of targeted community studies. For anomaly detection, a new frontier would lie in the inclusion of more complex detector effects and observables such as track and vertex information, although first a credible detection or rejection of anomalies similar to Black Box 3 would could be desireable. While toy studies will play an important role in developing new methods, we keenly await experimental results with these tools. ## 7 Outlook: Anomaly Detection in Run 3, the HL-LHC and for Future Colliders ### 7.1 Prospects for the (HL)-LHC While there are already many search results from the LHC collaborations using the full Run 2 dataset, many more will be published in the coming years. Notably, almost none of the methods described in this paper have been applied yet to collider data. The ATLAS Collaboration has produced a first version of the CWoLa hunting analysis using low-dimensional features collaboration2020dijet , which is likely the start of a growing set of searches. At this juncture, it is useful to consider what is possible with the full LHC dataset and what is the best way of organizing these efforts going forward. Figure 52: The organization of physics analysis groups in ATLAS and CMS. The large circles on the left represent analysis groups that are primarily focused on measuring properties of the Standard Model. The group called SM is focused on the electroweak and QCD aspects of the SM that are not covered by the other groups. The large circles on the right represent the analysis groups primarily focused on searches for new particles. Selected supporting organizations that are connected to both measurement and search groups are depicted in smaller circles in the middle. The ATLAS CWoLa hunting search was performed in the HDBS analysis group in ATLAS (as a ‘model agnostic extension of the diboson resonance search’) and the ATLAS and CMS data-versus-simulation analyses are performed in the Exotics/Exotics groups. First of all, it is clear that there is no one approach which is strictly better than every other approach. Therefore, we envision a group of searches using complementary methodologies and targeting a variety of final states. Currently, analyses in ATLAS and CMS are organized by physics models: there is a group focusing on supersymmetry (SUSY) models, one focused on Higgs-like particles (HDBS in ATLAS) and 3rd generation particles (B2G in CMS), and one focused on exotic particles (Exotics in ATLAS and Exotica in CMS). It is not obvious that a model agnostic search program fits within the scope of this existing model-dependent group structure. At the same time, the commonalities across model agnostic methods would benefit from a coherent strategy. Therefore, a new analysis group or at least a new analysis subgroup may be required. This is illustrated in Fig. 52. There are clearly strong connections with supporting groups as well, including the statistics and machine learning fora. The analysis group home of these searches is not just a sociological question — the technical development and physics review is primarily carried out by the analysis groups so this choice can have important implications for the success of this program. The LHC Olympics focused on resonant new physics because there is a natural scheme for estimating backgrounds. However, there is still a non-trivial relationship between classification and background estimation. In particular, if the classifier is dependent on the resonant feature (e.g. the invariant mass of pairs of jets), then an artificial bump could be sculpted in the absence of any signal. This is related to the challenge of exploring higher dimensional feature spaces, which is required to achieve the broadest sensitivity. In some cases, automated decorrelation techniques for model- dependent searches can be adapted; in other cases, these methods would mask potential signals and so new approaches are required. None of the methods deployed for the LHC Olympics were able to find anomalies using the full list of hadron four-vectors directly — successful approaches all used some model- inspired dimensionality reduction. Scaling up these approaches to high dimensional feature spaces is a key challenge for the next years and will require both methodological and computational innovation. Exploring anomaly detection in the non-resonant case is more challenging because there is no general approach for estimating the background. Some of the methods deployed for the LHC Olympics can achieve signal sensitivity for non-resonant signals, but significant research is required in order to combine these and possibly new approaches with background estimation strategies. Strategies that directly compare data and background simulation are promising for final states where the background model is accurate and when the uncertainty is well-known. A key challenge for these approaches is scaling up to high-dimensional features where the full systematic uncertainty covariance matrix may not be known. This is a general challenge that is also faced by model-dependent approaches, where signal model uncertainties in many dimensions may not be well constrained. Another independent dimension to consider is when in the data processing pipeline the anomaly detection happens. The LHC Olympics is designed as an offline analysis, where standard trigger algorithms are used to collect the data. There is significant unexplored phase space from existing triggers, but there is also a vast phase space that is deleted in real time before it can be explored. The development of online anomaly detection will be a significant innovation to complement offline analysis. Recent innovations have shown that machine learning inference can be fast enough to fit within the strict trigger latency requirements (see e.g. Duarte:2018ite ; CERN-LHCC-2020-004 ). However, the same algorithms applied offline may not be applicable online. For example, offline methods can make multiple passes through the dataset in order to identify anomalous regions of the phase space. In contrast, the trigger only sees collision once before a decision to save the event or not must be made. Even if a method could identify anomalous events within the required bandwidth, this is only a partial solution because strange collisions are only useful if we can quantify their level of strangeness. This is one key difference between anomaly detection in high energy physics and the typical anomaly detection protocols developed in industry; we are almost never able to declare a discovery with a single collision. Our expectation is that new physics will manifest as an ‘over-density’ in phase space rather than being ‘off-manifold’. By analogy, we are not looking for flying elephants, but instead a few extra elephants than usual at the local watering hole. The only way to know that the number of elephants is anomalous is to have a precise understanding of the usual rate of elephants. In addition to the rich research and development program required to fully exploit the potential of these searches, there are a variety of complications involved in the interpretation of results. The most pressing question is what to do in the case of a positive signal detection. No fundamental particle that was not already precisely predicated by an existing theory has been discovered in decades. Would the high energy physics community believe a significant anomaly? It is important to start a conversation about the community plan in the case of a significant anomaly detected by one of these approaches. If an anomaly is found before the full high-luminosity LHC (HL-LHC) dataset is recorded, then a targeted search could be conducted using an independent dataset. What if the anomaly is only identified using the full HL-LHC dataset? What post-hoc analysis can and should be done? It is also important to ensure sensitivity to complex signals, where there may be multiple possible final states (as exemplified by Black Box 3). Figure 53: An illustration of the nested loops required for signal model- dependent interpretation of a model-agnostic search. The parenthetical remark for the signal cross section refers to the fact that if the number of predicted signal events is small, one may need to repeat the injection many times due to the large statistical fluctuations in the phase space. This is not a problem for model-dependent search where one can use all simulated signal events and scale by the predicted cross section. Unsupervised approaches may be able to avoid certain steps if they do not change in response to modifications in the data. In the absence of a signal detection, there is a significant challenge to quantify the sensitivity of a search. For a model-dependent search, quantifying what was not seen is relatively straightforward for a given model, one can provide an upper limit on the cross section. However, model agnostic methods are sensitive to many models all at once and it is challenging to define the sensitive volume. This is particularly challenging in many dimensions. One way to map out the sensitivity is to pick a small set of representative signal models. Signal model dependent limits can be significantly more difficult to compute for these searches than for standard searches. In particular, any time the anomaly classifier depends directly on the data in the signal-sensitive region, the entire analysis procedure must be repeated for every variation of the signal hypothesis. This is represented schematically in Fig. 53. Since the analysis selection depends on the data, the classifier must be retrained every time a different signal model cross section is injected into the data. For example, the final exclusion plots in Ref collaboration2020dijet required training tens of thousands of neural networks. Computing may become a bottleneck in the future when there is more data and higher dimensional features. Heterogeneous High Performance Computing Centers with significant GPU resources may provide a solution to this challenge. The dependence of the event selection on the data also complicates the usability of these data for reanalysis and reinterpretation. One cannot simply recast published results because if a new signal was in the data, then the event selection would have been different. If the network training and statistical analysis can be automated, then a system like RECAST Cranmer:2010hk may be possible whereby signal models could be submitted to collaborations for inference. Note that this is one more level of automation beyond typical analysis preservation: in addition to preserving the analysis selection, we also need to preserve the analysis optimization procedure which itself needs to be reproducible. ### 7.2 Prospects for Future Colliders and Beyond All of the challenges described in the previous section also apply to future colliders beyond the HL-LHC. However, a new machine opens up the possibility to imagine the co-design of accelerator/detector and analysis procedure. What operational conditions and detector configurations are most interesting for anomaly detection? The methods developed for colliders may also be more broadly applicable. Anomaly detection at other fundamental physics experiments shares many features with collider physics. In fact, a presentation at the Summer Olympics described an anomaly detection method developed using the LHC Olympics that is now being studied for astrophysical data streams . ### 7.3 The Role of Theory and Theorists While this paper is about making a search program that is model agnostic, this does not mean we should proceed without theory and without theorists. The most powerful methods will likely employ physics-informed machine learning techniques, whereby symmetries and other physical principles are part of the learning. These tools may allow us to find rarer signals and design procedures that are interpretable and robust. Furthermore, there is a continuum of model independence. Building theoretically motivated, but still relatively broad priors may achieve more powerful sensitivity to a wide class of models. Machine learning is in general a unifying subject, where there have been many rich collaborations between experimentalists and theorists as well as between high energy physicists and machine learning researchers and practitioners. About half of the participants in the LHC Olympics are ‘experimentalists’ and half are ‘theorists’. It is critical for the success of the anomaly detection program that model agnostic method development be a respected form of theory work and that machine learning method development and implementation be an appreciated from of experimental work. Furthermore, barriers between theory, experiment, and computation/statistics should be as low as possible so we can make the best use of our data. Public challenges like the LHC Olympics are an important step in this direction, but this is only one part of a bigger program of engagement. ### 7.4 The Future of the LHC Olympics This round of the LHC Olympics was driven by a need from the community to develop and test a growing number of machine learning anomaly detection methods. With a diverse set of submissions, we believe that this exercise has succeeded and has added value to the community. However, there is always room for improvement. In no particular order: * • Unlike other challenges in high energy physics such as the top tagging competition Kasieczka:2019dbj and the challenges on the Kaggle platform like the HiggsML Challenge pmlr-v42-cowa14 , the Flavours of Physics Challenge flavorofphyiscs , and the TrackML Challenge Amrouche:2019wmx , there was no single metric for determine a winner and therefore it was not possible to directly compare methods. (See Rousseau:2020rnz for a recent overview of these competitions.) This is similar to the correlation aspect of the Flavours of Physics Challenge and the efficiency-versus-fake-rate aspect of the TrackML challenge, but it even more acute for the LHC Olympics in part because the estimation of the false positive rate is non-trivial. * • Without a platform like Kaggle that offers broad exposure and a monetary prize, few ML experts outside of HEP participated in the LHC Olympics. Additionally, accessibility to non-experts could be improved. Code to read in the data and cluster jets were provided to participants, but given that nearly every group performed additional dimensionality reduction first suggests that additional information could have been useful. * • One of the biggest difficulties with selecting the Black Boxes was that the anomalies should be easy enough to find that the challenge is doable, but not too easy that one could find them without new methods. Some checks were performed before releasing the Black Boxes, but with significant work, this could have been more robust and streamlined. There are many possibilities for variations on the LHC Olympics 2020. Additional signal models could be considered as black boxes and more signal topologies could be studied including final state leptons, heavy flavor quarks, and long-lived particles. We look forward to the deployment and impact of new methods developed from the LHC Olympics 2020 as well as future iterations. ## 8 Conclusions Given the current lack of convincing evidence for new fundamental particles or new forces of nature from HEP experiments, it is critical that the program of dedicated searches be complemented with more model agnostic methods. While there has been a long history of signal model agnostic methods based on binned data-simulation comparisons, there has been a recent explosion of new ideas for less model dependent approaches. Many of the new proposals make use of machine learning to aid in the less-than-supervised exploration of collider data. The methods presented in this paper represent a snapshot292929See Ref. livingreview for a more updated list of papers in this area. of the rapidly developing area of machine learning for anomaly detection in HEP. To address this challenge, we introduced the LHC Olympics, a community effort to develop and test anomaly detection methods in a relatively realistic setting. A set of datasets were produced to emulate the typical setting where data are unlabeled but there is a corresponding labeled dataset for research and development. In the LHC Olympics, three black boxes were the analog of collider data, each with a different SM background simulation and a different potential anomaly. Many teams developed and implemented a variety of techniques on these datasets covering at least 18 different methods (some submissions compared multiple distinct methods). In addition to results with the R&D dataset, many teams deployed their techniques on the black boxes. At the Winter and Summer Olympics workshops, teams presented their results on these boxes before even knowing the nature of the signal in the datasets analyzed. While some strategies were closer to the correct answer than others, every team followed the scientific method and gained valuable insight and experience. In several cases, strategies were refined between the two workshops using feedback from the unveiling of the first black box. Many of these strategies continue to be refined as they are prepared for the application to collider data in the near future. These methods use a variety of unsupervised, semisupervised, and fully supervised machine learning methods based on neural networks and other approaches. While there are unique advantages and disadvantages to each method, there are also common challenges across techniques, such as scaling to higher dimensions. The ultimate performance is likely to include a combination of approaches, and new method development will be required to reach the full physics potential of the data. A data-driven revolution has started with machine learning as its catalyst. We are well-equipped to explore the complex LHC data in new ways with immense potential for discovery. The Run 2 data collection is over, but our exploration of these precious collisions in their natural high dimensionality is only beginning. This LHC Olympics has been a starting point for a new chapter in collider physics that will produce exciting physics results from the current datasets as well from the datasets of the future at the LHC and beyond. ## Acknowledgments We thank the organizers and participants in the ML4Jets2020 workshop hosted at New York University and at the anomaly detection workshop hosted (virtually) by the University of Hamburg for many interesting discussions at the Winter and Summer Olympics, respectively. B. Nachman and G. Kasieczka are grateful to the NHETC Visitor Program at Rutgers University for the generous support and hospitality during the spring of 2019 where the idea for the LHC Olympics 2020 was conceived. A. Kahn, J. Gonski, D. Williams, and G. Brooijmans are supported by the National Science Foundation (NSF) under Grant No. PHY-2013070. I. Ochoa is supported by the fellowship LCF/BQ/PI20/11760025 from “la Caixa” Foundation (ID 100010434) and by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłlodowska-Curie grant agreement No 847648. S. E. Park, S. Udrescu, M. Yunus, P. Harris are supported by the NSF Grants #1934700 and #1931469. Cloud credits for training were supported by the Internet2/NSF Grant #190444. V. Mikuni and F. Canelli are supported in part by the Swiss National Science Foundation (SNF) under contract No. 200020-182037. F. F. Freitas is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020 and the project PTDC/FIS- PAR/31000/2017. C. K. Khosa is supported by the Royal Society, UK under the Newton International Fellowship programme (NF171488). K. Benkendorfer was supported in part by NSF PHY REU Grant 1949923. B. Bortolato, B. Dillon, A. Matevc, J. Kamenik, A. Smolkovic acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0035 and J1-8137). D. A. Faroughy is supported by SNF under contract 200021-159720. M. Szewc would like to thank the Jozef Stefan Institute for its enormous hospitality. P. Komiske, E. Metodiev, N. Sarda, and J. Thaler are supported by the Office of Nuclear Physics of the U.S. Department of Energy (DOE) under grant DE- SC-0011090 and by the DOE Office of High Energy Physics under grant DE- SC0012567. N. Sarda was additionally supported by the QCRI-CSAIL Computer Research Program. P. Komiske, E. Metodiev, N. Sarda, and J. Thaler are grateful to Benjamin Nachman and Justin Solomon for helpful conversations. B. Nachman and J. Collins were supported by the DOE under contracts DE- AC02-05CH11231 and DE-AC02-76SF00515, respectively. P. Martín-Ramiro acknowledges Berkeley LBNL, where part of this work has been developed. P. Martín-Ramiro further acknowledges support from the Spanish Research Agency (Agencia Estatal de Investigación) through the contract FPA2016-78022-P and IFT Centro de Excelencia Severo Ochoa under grant SEV-2016-0597. P. Martín- Ramiro also received funding/support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłlodowska-Curie grant agreement No 690575 (RISE InvisiblesPlus). S. Tsan, J. Duarte, J.-R. Vilmant, and M. Pierini thank the University of California San Diego Triton Research and Experiential Learning Scholars (TRELS) program for supporting this research, CENIC for the 100 Gpbs networks, and Joosep Pata for helpful discussions. They are additionally supported in part by NSF awards CNS-1730158, ACI-1540112, ACI-1541349, OAC-1826967, the University of California Office of the President, and the University of California San Diego’s California Institute for Telecommunications and Information Technology/Qualcomm Institute. J. Duarte is supported by the DOE, Office of Science, Office of High Energy Physics Early Career Research program under Award No. DE-SC0021187. M. Pierini is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369). J-R. Vilmant is partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 772369) and by the DOE, Office of Science, Office of High Energy Physics under Award No. DE-SC0011925, DE-SC0019227, and DE-AC02-07CH11359. D. Shih is supported by DOE grant DOE- SC0010008. GK acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. ## References * (1) ATLAS collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1–29, [1207.7214]. * (2) CMS collaboration, S. 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# Observations and predictions from past lightcones Martin Lesourd Black Hole Initiative, Harvard University <EMAIL_ADDRESS> ###### Abstract. In a general Lorentzian manifold $M$, the past lightcone of a point is a proper subset of $M$ that does not carry enough information to determine the rest of $M$. That said, if $M$ is a globally hyperbolic Cauchy development of vacuum initial data on a Cauchy surface $S$ and there is a point whose past lightcone contains $S$, then the contents of such a lightcone determines all of $M$ (up to isometry). We show some results that describe what properties of $M$ guarantee that past lightcones do indeed determine all or at least significant portions of $M$. Null lines and observer horizons, which are well known features of the de-Sitter spacetime, play a prominent role. ## 1\. Introduction In Lorentzian geometry, an observer at a given time in a spacetime $(M,g)$ is represented by a timelike curve with future endpoint $p\in M$, and the past lightcone $J^{-}(p)\subset M$ of $p$ represents all signals in $M$ that can reach the observer at $p$. One can then ask the following. 1. (A) Can an observer at $p\in M$ know the global structure of $M$ on the basis of $J^{-}(p)$? 2. (B) Can an observer at $p\in M$ make predictions about $M\backslash J^{-}(p)$ on the basis of $J^{-}(p)$? In this short note, we describe some of what is known about (A) and (B), we prove various further results, and we list some natural further questions. Part of the appeal in (A) and (B) is that they are subject to somewhat surprising examples. As described below, two (inextendible and globally hyperbolic) spacetimes $(M^{\prime},g^{\prime})$ and $(M,g)$ can be non- isometric, in spite of the fact that each member of the countable collection of past lightcones $\\{I^{-}(p_{i})\\}$ that covers $M$ can be isometrically embedded into $(M^{\prime},g^{\prime})$ and likewise with $M$ and $M^{\prime}$ interchanged. Here we will show when this cannot happen. We now recall some basic definitions of causal theory, cf. [1], [14], [16] for some classic references and [15] for a more recent authorative survey. A spacetime $(M,g)$ is a connected $C^{\infty}$ Hausdorff manifold $M$ of dimension two or greater with a Lorentzian metric $g$ of signature $(-,+,+,...)$, and we will assume a time and space orientation. Since regularity is not the issue here, for simplicity of expression we take $g$ to be smooth, but many of the arguments can be extended to lower regularity. The lightcone structure inherited on the tangent space $T_{p}M$ at each $p$ leads to the notion of a causal curve, which in turn leads to defining $J^{-}(p)$ (or $I^{-}(p)$) as the collection of all points $q\in M$ from which there exists a causal (or timelike) curve with future endpoint $p$ and past endpoint $q$. A causal curve between $p$ and $q\in J^{+}(p)$ is achronal iff $q\notin I^{+}(p)$. A null line is an inextendible achronal causal curve. A set is achronal (acausal) iff no two members of it can be connected by a timelike (causal) curve. The domain of dependence $D(S)$ of a set $S\subset M$ is given by $D(S)=D^{+}(S)\cup D^{-}(S)$ and $D^{+}(S)$ is defined as the collection of all points $q$ in $M$ such that any inextendible ($C^{1}$) past directed causal curve passing through $q$ intersects $S$. $\tilde{D}(S)$ is defined identically except that curves are timelike, rather than causal. From the perspective of the Cauchy problem of general relativity, $D^{+}(S)$ represents the maximal portion of $M$ that could be determined by initial data on $S$. If $S$ is closed as a subset of $M$ then $\overline{D^{+}(S)}=\tilde{D}^{+}(S)$. The future Cauchy horizon is defined as $H^{+}(S)\equiv\overline{D^{+}(S)}\backslash I^{-}(D^{+}(S))$ and $H(S)=H^{+}(S)\cup H^{-}(S)$. $S$ is a partial Cauchy hypersurface if it an edgeless acausal set, cf. Definition 14.27 of [1] for the definition of $\text{edge}(S)$ for an achronal set $S$. A spacetime $(M,g)$ is globally hyperbolic iff there exists a partial Cauchy hypersurface $S$ such that $M=D(S)$. By [2], a smooth globally hyperbolic spacetime $(M,g)$ is isometric to $(\mathbb{R}\times S,-f(t)dt^{2}+h(t))$ where $f(t)$ is smooth and $h(t)$ a Riemannian metric on $S$. From the perspective of causal structure, global hyperbolicity is equivalent to the spacetime being causal111No closed causal curves. and $J^{+}(p)\cap J^{-}(q)$ being compact for all $p,q\in M$. If $S$ is acausal and $S\cap\text{edge}(S)=\emptyset$ (eg., $S$ is a partial Cauchy hypersurface), then $D(S)$ is non-empty, open, and $D(S)\cap H(S)=\emptyset$. Note that the openness of $D(S)$ may be ruined if we take $S$ to be achronal rather than acausal. A spacetime $(M,g)$ is causally simple iff it is causal and $J^{+(-)}(p)$ is closed for all $p\in M$. Global hyperbolicity is strictly stronger than causal simplicity. An isometric embedding of $(M,g)$ into $(M^{\prime},g^{\prime})$ is an injective (but not necessarily surjective) map $\phi:M\hookrightarrow M^{\prime}$ such that $\phi$ is a diffeomorphism onto its image $\phi(M)$ and $\phi^{}(g^{\prime})=g$. If $\phi$ maps $M$ surjectively onto $M^{\prime}$ then we write $\phi:M\to M^{\prime}$ and we say that $(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric. A spacetime $(M,g)$ is inextendible when there exists no isometric embedding $\phi$ of $M$ into $M^{\prime}$ such that $M^{\prime}\backslash\phi(M)\neq\emptyset$. A spacetime is future holed if there is a partial Cauchy hypersurface $S^{\prime}\subset M^{\prime}$ and an isometric embedding $\phi:\tilde{D}(S^{\prime})\hookrightarrow M$ such that $\phi(S^{\prime})$ is acausal in $(M,g)$ where $(M,g)$ is any spacetime, and $\phi(H^{+}(S^{\prime}))\cap D^{+}(\phi(S^{\prime}))\neq\emptyset$. A spacetime is hole-free if it lacks future and past holes. Minguzzi [12] shows that causally simple, inextendible spacetimes are hole free. Acknowledgements. We thank the Gordon Betty Moore Foundation and John Templeton Foundation for their support of Harvard’s Black Hole Initiative. We thank Professors Erik Curiel, JB Manchak, Ettore Minguzzi, Chris Timpson, and James Weatherall for valuable comments that improved the paper. ## 2\. Previous Work (A). A natural definition coming from [8] (whose terminology is slightly different) is the following. ###### Definition 2.1. A spacetime $(M,g)$ is weakly observationally indistinguishable from $(M^{\prime},g^{\prime})$ just in case there is an isometric embedding $\phi:I^{-}(p)\hookrightarrow M^{\prime}$ for every point $p\in M$. If this is also true with $M$ and $M^{\prime}$ interchanged, then the spacetimes are strongly observationally indistinguishable. One could replace $I^{-}(p)$ with $J^{-}(p)$, and although this makes no real difference to any of the results in [8] and [9] or indeed to what follows, that would capture a somewhat more honest sense of distinguishability because observers can certainly receive signals from $J^{-}(p)\backslash I^{-}(p)$. One can also take observers to be inextendible timelike curves $\sigma$ (as is also done in [8]), but in that case $J^{-}(\sigma)\subset M$ would represent “all possible observations that could be made supposing that the observer lasts forever with respect to $M$”, which, though interesting, is stronger than what happens in practice. Malament pg. 65-6 [8] gives examples of spacetimes that are strongly observationally indistinguishable but non-isometric. His examples are globally hyperbolic, inextendible, and exploit the presence of observer horizons in the de-Sitter spacetime. Based on an explicit cut and paste argument, Manchak [9] shows the following. ###### Proposition 2.2 (Manchak [9]). Given any non-causally bizarre222A spacetime is causally bizarre if there is a point $p\in M$ such that $M\subset I^{-}(p)$. spacetime $(M,g)$, there exists a spacetime $(M^{\prime},g^{\prime})$ that is weakly observationally indistinguishable from $(M,g)$ but not isometric to $(M,g)$. Although Manchak’s construction of $(M^{\prime},g^{\prime})$ works for any (non-causally bizarre) spacetime $(M,g)$, it relies on introducing a countably infinite collection of holes in $(M^{\prime},g^{\prime})$. It is unknown to us whether Proposition 2.2 holds for strong observational indistinguishability. Viewed together, [8] and [9] lead one to the following. Question. Find conditions $\\{A,B,...\\}$ satisfied by $(M,g)$ and $(M^{\prime},g^{\prime})$ such that: ‘Weakly (or strongly) observationally indistinguishable $+A+B+...$’ $\Leftrightarrow$ ‘$(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric’ Proposition 3.6 and Corollary 3.11 below are in this direction. (B). Geroch [7] defines prediction in general relativity as follows. ###### Definition 2.3. $p\in M$ is predictable from $q$, written as $p\in P(q)$, iff $p\in D^{+}(S)$ for some closed, achronal set $S\subset J^{-}(q)$. $p\in M$ is verifiable iff $p\in P(q)$ and $p\in I^{+}(q)\backslash J^{-}(q)$. Manchak observes the following. ###### Proposition 2.4 (Manchak [10]). If $P(q)\cap(I^{+}(q)\backslash J^{-}(q))\neq\emptyset$, then $(M,g)$ admits an edgless compact achronal set. A slightly different notion of prediction considered in [10] is as follows. ###### Definition 2.5. $p\in M$ is genuinely predictable from $q$, written $\mathcal{P}(q)$, iff $p\in P(q)$ and for all inextendible spacetimes $(M^{\prime},g^{\prime})$, if there is an isometric embedding $\phi:J^{-}(q)\hookrightarrow M^{\prime}$, then there is an isometric embedding $\phi^{\prime}:J^{-}(q)\cup J^{-}(p)\hookrightarrow M^{\prime}$ such that $\phi=\phi^{\prime}_{\mid J^{-}(q)}$. The idea here is that genuine predictions guarantee that observers with the same past make the same predictions. By a short cut-and-paste argument, Manchak observes the following. ###### Proposition 2.6 (Manchak [10]). Let $(M,g)$ be any spacetime and $q$ a point in $M$. Then $\mathcal{P}(q)\subseteq\partial J^{-}(q)$. Thus the domain of genuine predictions from $q$, if non-empty, is on the verge of being a retrodiction. ## 3\. Some Observations We assume that spacetimes satisfy the field equations (3.1) $G_{g}\equiv\text{Ric}_{g}-\frac{1}{2}g\>\text{Scal}_{g}=T_{\phi,...}-\Lambda\>g$ where $\Lambda\in\mathbb{R}$ is a constant, and with $T_{\phi,...}$ the stress-energy tensor associated with possible matter fields $\\{\phi,...\\}$ in $M$.333Here we think of the Cauchy problem from the perspective of ‘initial data’, as opposed to ‘initial and boundary data’, though the latter is more natural for $\Lambda<0$. The system (3.1) leads to the formulation of a Cauchy problem on a spacelike initial data set $S$. In the vacuum setting $T=0$, $\Lambda=0$, the Cauchy problem was shown [3] to be well posed in the sense that there exists a unique, up to isometry, maximal globally hyperbolic development of $S$ obtained by Cauchy evolution of the initial data on $S$ according to (3.1) with $T=0$, $\Lambda=0$. This well posedness has been extended to more general settings and we take it as a pre-condition for the spacetimes we consider. ###### Definition 3.1. Fix the constant $\Lambda$ and fix an expression for $T_{\phi,...}$. Given a spacetime $(M,g)$ and an acausal edgeless connected set $S\subset M$, we say that $(\tilde{D}(S),g\|_{\tilde{D}(S)})$ is a faithful development if it is uniquely determined, up to isometry, by Cauchy evolution of the initial data on $S$ according to (3.1). If $(M,g)$ admits a connected acausal edgeless set, we say that it is locally Cauchy if, for any connected acausal edgeless set $S$, $(\tilde{D}(S),g\|_{\tilde{D}(S)})$ is a faithful development. ###### Remark 3.2. Note that this is slightly unorthodox in the sense that it is usually $D(S)$, rather than $\tilde{D}(S)$, which we think of as being determined by $S$. Given that $S$ is closed for the definition of locally Cauchy, we have $\overline{D(S)}=\tilde{D}(S)$, and so being locally Cauchy is only slightly stronger than asking for $D(S)$ to be determined up to isometry. ###### Remark 3.3. Since we want to guarantee isometric embeddings, we want to rule out examples of regions that are globally hyperbolic but not determined by Cauchy evolution. To see a trivial example444Examples like this suggest the following problem. Given a Lorentzian manifold $(M,g)$ with an arbitrary geodesically complete Lorentzian metric $g$, what conditions on $M$ and $g$ make it possible to solve for a function $\Omega:M\to\mathbb{R}$ such that $(M,\Omega^{2}g)$ is complete and vacuum? smooth functions transformations $\Omega$ , start with Minkowski spacetime $(\mathbb{R}^{1,n},\eta)$, identify some open set $O\subset\mathbb{R}^{1,n}$ lying above $t=0$, and modify it by a conformal transformation $\eta\to\Omega^{2}\eta$. In that case, in spite of $M\backslash O$ being vacuum, $O$ will in general have a non-vanishing Einstein tensor $G_{g}$. But since there are Cauchy hypersurfaces (in $M\backslash O$) for $(\mathbb{R}^{1,n},\Omega^{2}\eta)$ which are exactly flat555In the language of the constraints, initial data sets of the form ($\mathbb{R}^{n},g_{E},0$)., $(\mathbb{R}^{1,n},\Omega^{2}\eta)$ is not locally Cauchy if it is not isometric to $(\mathbb{R}^{1,n},\eta)$. We also make use of the following. ###### Definition 3.4. Given a partial Cauchy hypersurface $S$ in a spacetime $M$ we say that an open subset of $M$ is an $\epsilon$-development of $S$, denoted $D_{\epsilon}(S)(\supset S)$ if there is exists an $\epsilon>0\>(\in\mathbb{R})$ such that $D_{\epsilon}(S)$ admits a Cauchy surface $S_{\epsilon}$ every point of which lies at distance $\geq\epsilon>0$ in the future of $S$, as measured by a normalized timelike vector field orthogonal to $S$. ###### Remark 3.5. The non-empty interior of a causal diamond $J^{+}(p)\cap J^{-}(q)\neq\emptyset$ with $p,q\in(\mathbb{R}^{1,n},\eta)$ is not an $\epsilon$-development because its Cauchy surfaces are anchored at $\partial J^{+}(p)\cap\partial J^{+}(q)$. We now observe the following. ###### Proposition 3.6. Let $(M,g)$ and $(M^{\prime},g^{\prime})$ be inextendible and locally Cauchy. Suppose that 1. (i) $(M,g)$ has a compact Cauchy surface and no null lines, 2. (ii) $(M^{\prime},g^{\prime})$ is causal and hole-free. Then $(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric iff they are weakly observationally indistinguishable. Note that by the examples in [8], Proposition 3.6 is false without the assumption that $(M,g)$ lacks null lines. In either case of weakly o.i. or strongly o.i., it would be interesting to settle whether the compactness in (i) is necessary, cf. Proposition 3.13 below. ###### Proof. The proof of Proposition 3.6 starts by strengthening Theorem 1 of [6].666In [6], the authors assume that $(M,g)$ is null geodesically complete, satisfies the null energy condition, and the null generic condition. These assumptions implies the absence of null lines. It was then observed by Galloway that one can prove the theorem by instead assuming that $(M,g)$ lacks null lines (cf. footnote for Theorem 1 of [6]), but since those details never appeared we include them for completeness. ###### Lemma 3.7. Let $(M,g)$ be a spacetime without null lines. Then given any compact region $K$, there exists another compact $K^{\prime}\supset K$ such that if $p,q\notin K^{\prime}$ and $q\in J^{+}(p)-I^{+}(p)$ , then any causal curve $\gamma$ connecting $p$ to $q$ cannot intersect $K$. ###### Proof. Suppose otherwise for some $K_{0}$ and $K_{1}\supset K_{0}$. Then consider a sequence of ever bigger compact sets $K_{i+1}\subset K^{\prime}_{i}$. By assumption, each $K_{i}$ will have horismos777The future horismos of $p$ is defined as $J^{+}\backslash I^{+}(p)$. related outer points $p_{i},q_{i}\notin K_{i}$, $q_{i}\in J^{-}(p_{i})-I^{-}(p_{i})$ that are connected by a causal curve, necessarily achronal, that intersects $K_{0}$. In considering larger compact sets, we make these causal curves longer in the sense of an auxiliary Riemannian metric. All of these curves intersect $K_{0}$. Taking the limit, by compactness of $K_{0}$, there is a limit point for the sequence of points lying in $K_{0}$ for each causal curve linking $p_{i},q_{i}$, which moreover is in $K_{0}$. By a standard limit curve arguments, cf. Proposition 3.1 of [1], there passes an inextendible limit curve through this limit point. The limit curve is also straightforwardly seen to be achronal.888See [11] for significantly stronger limit curve statements. Thus we have a null line. ∎ With the same proof as in Corollary 1 of [6], Lemma 3.7 implies the following.999Lemma 3.8 strengthens the main result of [6]. ###### Lemma 3.8. Let $(M,g)$ be a spacetime with compact Cauchy surface that does not admit any null lines. Then $(M,g)$ admits a point $p\in M$ such that $S\subset I^{-}(p)$ for some Cauchy surface $S$. ###### Proof. We include this for completeness, but this argument is exactly as in [6] except that Lemma 3.7 plays the role of Theorem 1 of [6]. Since $(M,g)$ is globally hyperbolic, there exists a continuous global time function $t:M\to\mathbb{R}$, such that each surface of constant $t$ is a Cauchy surface. Let $K=\Sigma$ and $K^{\prime}$ be as in Lemma 3.7. Let $t_{1}$ and $t_{2}$ denote, respectively, the minmum and maximum values of $t$ on $K^{\prime}$. Let $\Sigma_{1}$ be any Cauchy surface with $t<t_{1}$ and let $\Sigma_{2}$ denote the Cauchy surface $t=t_{2}$. Let $q\in I^{+}(\Sigma_{2})$, $p\in\Sigma_{1}$ and suppose that $p\in\partial I^{-}(q)$. Since $(M,g)$ is globally hyperbolic, $J^{-}(q)$ is closed and so $p\in J^{-}(q)\backslash I^{-}(q)$ and thus there is a causal curve connecting $p$ and $q$. It follows from Lemma 3.7 that this causal curve does not intersect $\Sigma$. However, this contradicts the fact that $\Sigma$ is a Cauchy surface. Consequently, there cannot exist a $p\in\Sigma_{1}$ and such that $p\in\partial I^{-}(q)$, i.e., $\partial I^{-}(q)\cap\Sigma_{1}=\emptyset$. But $I^{-}(q)$ is open and since $\partial I^{-}(q)\cap\Sigma_{1}=\emptyset$, the complement of $I^{-}(q)$ in $\Sigma_{1}$ is also open. Since we have $I^{-}(q)\cap\Sigma_{1}$ and $\Sigma_{1}$ is connected, this implies $\Sigma_{1}\subset I^{-}(q)$. ∎ We now finish the proof of Proposition 3.6. By Lemma 3.8 there is a point $p\in M$ with $S\subset I^{-}(p)$ where $S$ is a Cauchy surface of $M$. $\phi(S)$ is compact. By weak observational indistinguishability, there is an isometric embedding $\phi:I^{-}(p)\hookrightarrow M^{\prime}$. Because $\phi$ is a diffeomorphism onto its image, the map $\phi\|_{S}$, $\phi$ restricted to $S$, is a diffeomorphism of $S$ onto its image and thus $\phi(S)$ is compact in $M^{\prime}$. $\phi(S)$ is achronal. Suppose otherwise that $\gamma^{\prime}$ is a past directed timelike curve from $x^{\prime}$ to $y^{\prime}$ with $x,y\in\phi(S)$. Extend $\gamma^{\prime}$ to $\sigma^{\prime}$ so that $\sigma^{\prime}$ is a past directed timelike curve from $q^{\prime}$ to $x^{\prime}$ to $y^{\prime}$. Note that the isometric embedding forces $\phi(S)$ to be locally achronal; that is, there is an open neighborhood $O$ around around $S$ with two connected boundary components $\partial_{+}O(\subset I^{+}(S))$, $\partial_{-}O$ (allocated using the orientation in $M$), such that no two distinct points in $O$ can be joined by a timelike curve in $O$. Since $O$ isometrically embeds into $M^{\prime}$, a locally achronal neighborhood $O^{\prime}$ exists around $\phi(S)$ in $M^{\prime}$. As such, the curve $\sigma^{\prime}$ must leave $\partial_{-}O^{\prime}$ and re-enter $O^{\prime}$ via either $\partial_{-}O^{\prime}$ or $\partial_{+}O^{\prime}$. In the former case, we can build a closed piecewise smooth timelike curve from $q^{\prime}$ and back, which violates causality of $M^{\prime}$. In the latter case, we will obtain a contradiction with the inextendibility of $M$. Since $I^{-}(S)\subset D^{-}(S)\subset I^{-}(p)$, we must have that $\sigma^{\prime}$ leaves $\phi(D^{-}(S))$, say at some point $r^{\prime}\in\partial\phi(D^{-}(S))$, and re-enter $\phi(I^{-}(p))\cap I^{+}(\phi(S))$. Since the global hyperbolicity of $M$ implies that every future directed timelike curve in $\phi(I^{-}(p))$ must eventually leave $\phi(I^{-}(p))$ when sufficiently extended in the future direction, we can take the re-entry point to lie on the ‘future’ boundary of $\phi(I^{-}(p))$; that is, which is the endpoint of a timelike curve whose $\phi^{-1}$ pre-image has endpoint on $\partial I^{-}(p)$. Now consider an open neighborhood $Z^{\prime}$ of $r^{\prime}$ in $M^{\prime}\cap\partial\phi(D^{-}(S))$. Consider the open subset $\phi(I^{-}(p))\cup Z^{\prime}$ of $M^{\prime}$. Now define a new spacetime $M^{\prime\prime}$ by $\phi(I^{-}(p))\cup Z^{\prime}\cup J^{+}(\partial I^{-}(p))$. We know this can be done because the global hyperbolicity of $M$ and the locally Cauchy property of $M$ and $M^{\prime}$ imply that the regions $\overline{I^{-}(p)}$ and $\overline{\phi(I^{-}(p))}\backslash\partial\phi(D^{-}(S))$ are isometric. We now have a spacetime $M^{\prime\prime}$ into which $M$ can be isometrically embedded as a proper subset (in virtue of the extra $Z^{\prime}$ beyond $\phi(D^{-}(S))$), contradicting the inextendibility of $M$. $\phi(S)$ is edgeless. Compactness of $S$ and achronality of $\phi(S)$ straightforwardly implies that $\phi(S)$ is edgeless. ###### Remark 3.9. At this point if $(M^{\prime},g^{\prime})$ is assumed globally hyperbolic, $\phi(S)$ being an edgeless compact connected achronal set means that $\phi(S)$ can be taken to be a Cauchy surface of $M^{\prime}$. In that case, both $(M,g)$ and $(M^{\prime},g^{\prime})$ are representatives of the unique, up to isometry, maximal globally hyperbolic development of $S$, and are thus isometric. We will instead show that $(M^{\prime},g^{\prime})$ is globally hyperbolic. In $(M,g)$ we can consider an $\epsilon$-development $D_{\epsilon}(S)\subset M$. Within $D_{\epsilon}(S)$, we can then find an acausal hypersurface $S_{\epsilon}$ which is still Cauchy for $M$. We know that $D_{\epsilon}(S)\subset M$ isometrically embeds in $M^{\prime}$ as a small neighborhood around $\phi(S)$. The image $\phi(S_{\epsilon})$ of $S_{\epsilon}$, now denoted $S^{\prime}_{\epsilon}$, is acausal in $M^{\prime}$ (by a causal version of the argument for the achronality of $\phi(S)$) and edgeless. We now have partial Cauchy surfaces $S_{\epsilon}$ and $S^{\prime}_{\epsilon}$ in $M$ and $M^{\prime}$ respectively. We have the inclusion $D^{-}(S^{\prime}_{\epsilon})\supseteq M^{\prime}\backslash I^{+}(S^{\prime}_{\epsilon})$. Since $(M,g)$ is globally hyperbolic and inextendible, we have $D^{-}(S_{\epsilon})\supseteq J^{-}(S_{\epsilon})$. Since $J^{-}(S_{\epsilon})$ isometrically embeds into $(M^{\prime},g^{\prime})$, if $D^{-}(S^{\prime}_{\epsilon})$ fails to cover $M^{\prime}\backslash I^{+}(S^{\prime}_{\epsilon})$, then $S^{\prime}_{\epsilon}$ must have a past Cauchy horizon $H^{-}(S^{\prime}_{\epsilon})\neq\emptyset$ in $M^{\prime}$. In that case, by the locally Cauchy property we can isometrically embed $S^{\prime}_{\epsilon}$ and $\tilde{D}^{-}(S^{\prime}_{\epsilon})$ back into $M$ using $\phi^{-1}$, and by the global hyperbolicity and inextendibility of $M$, we have that $D^{-}(\phi^{-1}(S^{\prime}_{\epsilon}))\supset\phi^{-1}(H^{-}(S^{\prime}_{\epsilon}))$, which contradicts the past hole-freeness of $(M^{\prime},g^{\prime})$. We have the inclusion $D^{+}(S^{\prime}_{\epsilon})\supseteq M^{\prime}\backslash I^{-}(S^{\prime}_{\epsilon})$. This follows by the same argument. Since we now have $D(S_{\epsilon})=M$ and $D(S^{\prime}_{\epsilon})=M^{\prime}$, the conclusion follows from locally Cauchy. ∎ In view of the role of null lines, we note the rigidity theorem of Galloway- Solis [5]. ###### Theorem 3.10 (Galloway-Solis [5]). Assume that the $4$-dimensional spacetime $(M^{4},g)$ 1. (i) satisfies (3.1) with $T=0$101010Generalizations to Einstein-Maxwell are possible. and $\Lambda>0$, 2. (ii) is asymptotically de-Sitter111111cf. [5] for definitions of asymptotically de- Sitter and the associated hypersurfaces $\mathcal{J}^{+(-)}$ in that context., 3. (iii) is globally hyperbolic, 4. (iv) there is a null line with endpoints on $\mathcal{J}^{+}$ and $\mathcal{J}^{-}$. Then $(M^{4},g)$ isometrically embeds as an open subset of the de-Sitter spacetime containing a Cauchy surface. Together, Proposition 3.6 and Theorem 3.10 imply the following. ###### Corollary 3.11. Given two $4$-dimensional spacetimes $(M,g)$ and $(M^{\prime},g^{\prime})$ assume that 1. (i) $(M,g)$ and $(M^{\prime},g^{\prime})$ satisfy (3.1) with $T=0$ and $\Lambda>0$, 2. (ii) $(M,g)$ is inextendible and has a compact Cauchy surface, 3. (iii) $(M,g)$ is asymptotically de-Sitter but not isometric to de-Sitter, 4. (iv) $(M^{\prime},g^{\prime})$ is inextendible, causal and hole-free. Then $(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric iff they are weakly observationally indistinguishable. Note that the assumptions of Corollary 3.11 just falls short of astrophysical relevance on account of the assumption that $(M,g)$ be past asymptotically de- Sitter, which is not supported by current data. A more desirable statement would be welcome.121212[13] contains results precluding the existence of null lines based on astrophysically interesting assumptions. (B). We say that a spacetime $(M,g)$ is Cauchy friendly if it is weakly locally Cauchy131313Weakly locally Cauchy replaces $\tilde{D}(S)$ with $D(S)$. and there are no points $p\in M$ such that $J^{-}(p)\supseteq M$. We now show the following, which guarantees that genuine predictions extend a little beyond what is suggested in Proposition 2.6. ###### Proposition 3.12. Given two Cauchy friendly spacetimes $(M,g)$ and $(M^{\prime},g^{\prime})$, assume that 1. (i) there is a partial Cauchy surface $S\subset J^{-}(q)\subseteq D(S)$ for some point $q\in M$, 2. (ii) there is an isometry $\phi:J^{-}(q)\to J^{-}(q^{\prime})$. Then there is an isometric embedding $\psi:A\hookrightarrow M^{\prime}$ for some $A\supsetneq J^{-}(q)$ such that • $\psi_{\mid J^{-}(q)}=\phi$, * • $A$ and $\psi(A)$ contain points in the domain of verifiable prediction of $q$ and $q^{\prime}$. ###### Proof. First we make some basic observations, and throughout we denote $S^{\prime}\equiv\phi(S)$. We have $J^{-}(q)\subsetneq D(S)$. Since $J^{-}(q)$ lies in a globally hyperbolic set $D(S)$, $J^{-}(q)$ is closed. Moreover, since $S$ is acausal and edgeless, $D(S)$ must be open. From this it follows that the inclusion of $J^{-}(q)\subseteq D(S)$ is strict. Suppose otherwise that $J^{-}(q)=D(S)$. In that case $D(S)$ is both open and closed in $M$, and since $M$ is connected, that implies $D(S)=M$. But then $M=J^{-}(q)$, in contradiction with Cauchy friendly. Thus $J^{-}(q)$ is a closed proper subset of the open set $D(S)$. $S^{\prime}$ is a partial Cauchy hypersurface. Since $\phi$ is a diffeomorphism onto its image, we know that $S^{\prime}$ is compact. Unlike the arguments given in Proposition 3.6, $\phi(S)$ is acausal and edgeless by the fact that $\phi$ is an isometry (as opposed to merely an embedding). Since $S^{\prime}$ belongs to $J^{-}(q^{\prime})$, if $S^{\prime}$ is acausal in $M^{\prime}$, then $S$ is also acausal in $M$, which contradicts (ii). Thus $S^{\prime}$ is a partial Cauchy surface in $M^{\prime}$. We have $J^{-}(x^{\prime})\cap J^{+}(S^{\prime})\subseteq D^{+}(S^{\prime})$ for any $x^{\prime}\in\phi(J^{-}(q))$. We seek to show that any past inextendible causal curve in $J^{-}(x^{\prime})$ with with future endpoint $x^{\prime}\in\phi(J^{-}(q))$ intersects $S^{\prime}$. By assumption, we have an isometric embedding $\phi:J^{-}(q)\cap D^{+}(S)\hookrightarrow M^{\prime}$ where $S$ is a closed achronal set. Consider any point $x\in J^{-}(q)\cap J^{+}(S)$. By the isometry $\phi$, we have $\phi(J^{-}(x))=J^{-}(x^{\prime})$. Note first that by definition, there is a neighborhood of $0\in T_{x}M$ such that the exponential map of the past non-spacelike vectors in that neighborhood is contained in $D^{+}(S)$. By the isometry $\phi$, the same is true for $x^{\prime}$ with respect to $\phi(D^{+}(S))$, in particular there is a neighborhood $U_{x^{\prime}}$ of $x^{\prime}$ such that $U_{x^{\prime}}\cap J^{-}(x^{\prime})\subset\phi(D^{+}(S))$. Seeking a contradiction, suppose there is a past inextendible causal curve $\gamma^{\prime}$ with future endpoint $x^{\prime}$ that does not intersect $S^{\prime}$. By the property aforementioned, there is at least a segment of $\gamma^{\prime}$ contained in $\phi(D^{+}(S))$. The curve defined by $\gamma\equiv\phi^{-1}(\gamma^{\prime})$ is causal and ends at $x$, and is thus entirely contained in $D^{+}(S)$. But then $\gamma$ does not intersect $S$, which is a contradiction. Similarly, we have $J^{-}(S)\subseteq D^{-}(S^{\prime})$. This follows from (ii), the isometry $\phi$, and the argument just above. We have $J^{-}(q^{\prime})\subsetneq D(S^{\prime})$. This proceeds as above, which now leads to a contradiction with the Cauchy friendliness of $M^{\prime}$. We have two closed sets $J^{-}(q^{\prime})$ and $J^{-}(q)$ each strictly contained in the open sets $D(S^{\prime})$ and $D(S)$. We now seek to show the existence of $\psi$. Although there may be no isometric embedding of $D(S)$ into $M^{\prime}$, we need only show that it is possible to non-trivially extend the pre-image of $\phi$ beyond $J^{-}(q)$, which will thus enter $D(S)\backslash J^{-}(q)$. Consider now the unique (up to isometry) maximal globally hyperbolic development $X(S)$ of $S$, where $X(S)$ denotes one representative among all isometric developments. By the isometry $\phi$, we know that $S$ and and $S^{\prime}$ are isometric as initial data sets, and thus that $X(S)$ is the unique (up to isometry) maximal globally hyperbolic development of both $S$ and $S^{\prime}$. By locally Cauchy, it follows that both $D(S)$ and $D(S^{\prime})$ can be isometrically embedded into $X(S)$.141414Note here that we could use a weaker version of locally Cauchy here that involves $D(S)$ rather than $\tilde{D}(S)$. Denote these isometries by $\rho:D(S)\hookrightarrow X(S)$ and $\rho^{\prime}:D(S^{\prime})\hookrightarrow X(S)$. It is obvious that $\rho(D(S))\cap\rho^{\prime}(D(S^{\prime}))\neq\emptyset$ and since $\rho$ and $\rho^{\prime}$ are local diffeomorphisms, both $\rho(D(S))$ and $\rho(D(S^{\prime}))$ are open in $X(S)$, and moreover both $\rho(J^{-}(q))$ and $\rho^{\prime}(J^{-}(q^{\prime})$ are closed in $X(S)$. We now define the following set in $X(S)$ $[\rho(D(S))-\rho(J^{-}(q))]\cap[\rho^{\prime}(D(S^{\prime}))-\rho(J^{-}(q^{\prime}))]\equiv I$ The openness of $\rho(D(S)),\rho^{\prime}(D(S^{\prime}))$, and the fact that we may choose $\rho,\rho^{\prime}$ such that $\rho(J^{-}(q))=\rho^{\prime}(J^{-}(q^{\prime}))$ means that the strict inclusions $D(S)\backslash J^{-}(q),D(S^{\prime})\backslash J^{-}(q^{\prime})$ and $\neq\emptyset$ extend to $\rho$ and $\rho^{\prime}$, i.e., $\rho(D(S))\backslash\rho(J^{-}(q))\neq\emptyset$ and $\rho(D(S^{\prime}))\backslash\rho(J^{-}(q^{\prime}))\neq\emptyset$. It follows that $I\neq\emptyset$. We can now identify the set $A\equiv\rho^{-1}[\rho(J^{-}(q))\cup I]$ as having the desired properties, i.e. there exists an isometric embedding of $\psi:A\hookrightarrow M^{\prime}$ such that $\psi\|_{J^{-}(q)}=\phi$. ∎ We can also consider what happens after lifting the compactness assumption on $S$. ###### Proposition 3.13. Let $S$ be a partial Cauchy hypersurface in $M$ and $D_{\epsilon}(S)$ an $\epsilon$-development of $S$. Let $\phi:D_{\epsilon}(S)\hookrightarrow M^{\prime}$ be an isometric embedding into a hole-free spacetime $M^{\prime}$. Then either * • $\phi(S)$ is causal in $M^{\prime}$, * • or $\phi(S)$ is a partial Cauchy hypersurface in $M^{\prime}$. In the latter case, if $M,M^{\prime}$ are locally Cauchy and $M^{\prime}$ is inextendible, then there is an isometric embedding $\psi:D(S)\hookrightarrow M^{\prime}$ with $\psi\|_{D_{\epsilon}(S)}=\phi$. Thus, after basic assumptions like hole-freeness, locally Cauchy and inextendibility, the only obstruction concerns the acausality of $\phi(S)$. It may be that $\phi(S)$ is always acausal if $M^{\prime}$ satisfies some causality assumption, eg. causally simple, causally continuous151515cf. pg. 59 of [1], etc. Note also that $\phi(S)$ need not be a partial Cauchy hypersurface if we replace $D_{\epsilon}(S)$ by ‘an open globally hyperbolic subset of $D(S)$’ (delete a half-space at $t=0$ from $(\mathbb{R}^{1,n},\eta)$). ###### Proof. First we show the second statement. If $\phi(S)$ is a partial Cauchy hypersurface, we know that $D(S)$ and $D(\phi(S))$ are both open subsets of $M$ and $M^{\prime}$ respectively, and since $M,M^{\prime}$ are locally Cauchy, we know that $D(S)$ and $D(\phi(S))$ share a common (isometric) subset extending beyond $D_{\epsilon}(S)$. Let $\mathcal{D}(S)$ denote the maximal open globally hyperbolic subset of $D(S)$ for which there is an isometric embedding $\psi^{\prime}:\mathcal{D}(S)\hookrightarrow M^{\prime}$. Now suppose that $D(S)$ does not isometrically embed into $M^{\prime}$, i.e. $\mathcal{D}(S)\subsetneq D(S)$. If $H(\psi^{\prime}(S))\neq\emptyset$ then by locally Cauchy we can use $\psi^{\prime-1}$ to embed $\tilde{D}(\psi^{\prime}(S))$ into $M$ and contradict the hole-freeness of $M^{\prime}$. If $H(\psi^{\prime}(S))=\emptyset$, then $M^{\prime}=D(\psi^{\prime}(S))$. But in that case, by locally Cauchy, $M^{\prime}$ isometrically embeds into $M$ as a proper subset of $M$, contradicting the inextendibility of $M^{\prime}$. Now we show the first part: if $\phi(S)$ is acausal, then it is a partial Cauchy hypersurface in $M^{\prime}$. Supposing that $\text{edge}(\phi(S))\neq\emptyset$, we will show that $\text{edge}({\phi(S)})\cap\phi(S)=\emptyset$, and we then show that this implies $(M^{\prime},g^{\prime})$ is holed. Let $q^{\prime}$ be a point in $\text{edge}(\phi(S))\cap\phi(S)$. In that case denote $q=\phi^{-1}(q^{\prime})\in S$ and take a future directed timelike curve $\sigma$ from $q$ to $q_{\epsilon}\in I^{+}(S)\cap D_{\epsilon}(S)$, and set $\sigma$ to be past inextendible in $M$. Then there is a timelike curve $\sigma^{\prime}=\phi(\sigma)\subset M^{\prime}$ passing through $q^{\prime}$. Take $U_{i}(q^{\prime})$ to be a system of increasingly small neighborhoods $U_{i}(q^{\prime})\supsetneq U_{i+1}(q^{\prime})$, each containing points in $I^{+}(q^{\prime})$ and $I^{-}(q^{\prime})$, such that $\\{U_{i}(q^{\prime})\\}$ has accumulation point $q^{\prime}$. Define a collection of curves $\\{\gamma_{i}^{\prime}\\}$ by taking $\sigma^{\prime}$, removing from $\sigma^{\prime}$ the portion $\sigma^{\prime}\cap U_{i}(q^{\prime})$, and replacing that portion with timelike segments with endpoints in $I^{+}(q^{\prime})$ and $I^{-}(q^{\prime})$ which miss $\phi(S)$. Although this might produce only piecewise smooth timelike curves, the curves $\\{\gamma_{i}^{\prime}\\}$ can be approximated by $C^{1}$ causal curves (still missing $\phi(S)$), which we relabel as $\\{\gamma_{i}^{\prime}\\}$. Now consider $\\{\phi^{-1}(\gamma_{i}^{\prime})\\}$. This defines a collection of $C^{1}$ causal curves in $D_{\epsilon}(S)$ that approach $\sigma$ but which do not intersect $S$. We now recall a well known fact. In any spacetime $L$, there exists a sufficiently small neighborhood $N(x)\subset L$ around some point $x$ such that $J^{+}(y)\cap J^{-}(z)$ is compact for all $y$, $z$ $\in N(x)$. Since a sufficiently small $N(x)$ is causal, every point in a spacetime lives in a small globally hyperbolic neighborhood. Consider such a globally hyperbolic neighborhood $N(q)\subset M$ centered at $q$. Without loss of generality, we can take $N(q)$ to have Cauchy surface $S\cap N(q)$. For some sufficiently large $n\in\mathbb{N}$, the causal curves $\\{\phi^{-1}(\gamma_{i\geq n}^{\prime})\\}$ lie in $N(q)$ and are inextendible therein. But since these do not intersect $S$, we contradict the global hyperbolicity of $N(q)$. It now follows that $\text{edge}(\phi(S))$, if not empty, lies outside of $\phi(S)$. By standard results in causal theory, $H(\phi(S))$ is ruled by null geodesics intersecting $\text{edge}(\phi(S))$. By the definition of $D_{\epsilon}(S)$, we can use $\phi$ to pull back $H(\phi(S))\cap\phi(D_{\epsilon}(S))$ into $D_{\epsilon}(S)\cap M$. By the definition of $D_{\epsilon}(S)$, it is then clear that $\phi^{-1}\left[H(\phi(S))\cap\phi(D_{\epsilon}(S)\right]\cap D(S)\neq\emptyset$, and thus $(M^{\prime},g^{\prime})$ is holed. ∎ ## References * [1] Beem, J., Ehrlich, P., Easley, K., Global Lorentzian geometry, second ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker Inc., New York, 1996. * [2] Bernal, A. N., Sánchez, M., On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, 2003, Commun.Math.Phys. 243:461-470. * [3] Choquet-Bruhat, Y., Geroch, R., Global aspects of the Cauchy problem in general relativity, 1969, Commun.Math.Phys. 14:329–335. * [4] Earman, J., Glymour, C., Stachel, J. (eds.), Foundations of Space-Time Theories. Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977. * [5] Galloway, G. J. G., Solis, D., Uniqueness of de Sitter space, 2007, Class.Quant.Grav. 24:3125-3138. * [6] Gao, S., Wald, R. M., Theorems on gravitational time delay and related issues, 2000, Class. Quant.Grav. 17:4999-5008. * [7] Geroch, R., Prediction in General Relativity, in Earman, J., Glymour, C., Stachel, J. (eds.) Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977. * [8] Malament, D., Observationally Indistinguishable Space-Times, in Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977. * [9] Manchak, JB., Can We Know the Global Structure of Spacetime ?, 2009, Studies in History and Philosophy of Modern Physics 40:53–56. * [10] Manchak, JB., Is prediction possible in general relativity ?, 2008, Foundations of Physics 38(4):317-321. * [11] Minguzzi, E., Limit curve theorems in Lorentzian geometry, 2008, J.Math.Phys. 49:092501. * [12] Minguzzi, E., Causally simple inextendible spacetimes are hole free, 2012, J.Math.Phys. 53:062501. * [13] Minguzzi, E., Chronological spacetimes without lightlike lines are stably causal , 2009, Commun. Math. 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# “This Whole Thing Smacks of Gender”: Algorithmic Exclusion in Bioimpedance- based Body Composition Analysis Kendra Albert<EMAIL_ADDRESS>Harvard Law SchoolCambridge, MA and Maggie Delano<EMAIL_ADDRESS>Swarthmore CollegeSwarthmore, PA (2021) ###### Abstract. Smart weight scales offer bioimpedance-based body composition analysis as a supplement to pure body weight measurement. Companies such as Withings and Fitbit tout composition analysis as providing self-knowledge and the ability to make more informed decisions. However, these aspirational statements elide the reality that these numbers are a product of proprietary regression equations that require a binary sex/gender as their input. Our paper combines transgender studies-influenced personal narrative with an analysis of the scientific basis of bioimpedance technology used as part of the Withings smart scale. Attempting to include nonbinary people reveals that bioelectrical impedance analysis has always rested on physiologically shaky ground. White nonbinary people are merely the tip of the iceberg of those who may find that their smart scale is not so intelligent when it comes to their bodies. Using body composition analysis as an example, we explore how the problem of trans and nonbinary inclusion in personal health tech goes beyond the issues of adding a third “gender” box or slapping a rainbow flag on the packaging. We also provide recommendations as to how to approach creating more inclusive technologies even while still relying on exclusionary data. data collection and curation, sex/gender, bioelectrical impedance analysis, body composition, critical data/algorithm studies, science and technology studies, critical HCI and the design of algorithmic systems ††copyright: rightsretained††journalyear: 2021††conference: Conference on Fairness, Accountability, and Transparency; March 3–10, 2021; Virtual Event, Canada††booktitle: Conference on Fairness, Accountability, and Transparency (FAccT ’21), March 3–10, 2021, Virtual Event, Canada††doi: 10.1145/3442188.3445898††isbn: 978-1-4503-8309-7/21/03††ccs: Human-centered computing Ubiquitous and mobile computing††ccs: Social and professional topics Gender††ccs: Social and professional topics Race and ethnicity††ccs: Social and professional topics People with disabilities††ccs: Social and professional topics Geographic characteristics††ccs: Social and professional topics Cultural characteristics††ccs: Social and professional topics Remote medicine††ccs: Applied computing Consumer health ## 1\. Introduction Kendra: As a nonbinary person who was assigned female at birth, I’ve always had an uncomfortable relationship with my body weight. To be honest, before I stepped on the Withings scale, I hadn’t weighed myself in some time. But after undergoing some medical transition steps, I found myself much more curious about what my body fat percentage was. Especially since I have put on a fair amount of muscle, I was interested to see how the numbers matched or didn’t match my self perception. Because of this discomfort with weight numbers, I asked Maggie to make a profile for me when I was getting started with the scale. She started going through the registration flow for a new user, and quickly encountered a roadblock - the system required a gender. Wait, that’s not right. It didn’t require a gender, it required you to pick one of two images labeled gender - a person wearing pants or a person wearing a skirt (see Figure 1). When Maggie asked me which one I preferred, I told her to pick one and not tell me. When I did finally step on the scale (with my shiny new profile), I looked at the numbers and felt…well, some sort of way. To be honest, I’m happier with my body now than I remember ever being before, but the numbers seemed very off. Figure 1. The Withings profile view in their Health Mate mobile application. A view of the Withings profile view. There is an option for Gender with a person wearing pants and a person wearing a skirt, an option for height (5 ft 3) and an option for Athlete mode, which is set. The next morning, we were talking about the scale at breakfast, and that was when Maggie first told me that the fat percentage was an estimation based on a technique called bioelectrical impedance analysis (BIA). There was an equation behind that number - it didn’t actually measure my body fat percentage directly. I was shocked, and asked how that related to my gender. We decided to do some testing. When Maggie changed my “gender” from skirt-wearing to pants-wearing, my body fat percentage dropped 10% points. 10! The huge difference seemed to confirm everything I’d thought about the numbers feeling wonky. But which one was right? Or failing that, which one closer? Could we look at how the algorithm calculated the percentage in order to see which one was likely to be more accurate for me? Maggie: Kendra’s reaction to the scale was an interesting experience because I realized that the knowledge I had about how BIA worked was non-obvious to anyone not familiar with the tech. I had already been thinking about how sex is often coded in the underlying BIA equations as 1 or 0 because I had given a guest lecture about it recently, but I hadn’t fully thought through the implications for trans and nonbinary people. Seeing the numbers on the scale jump by so much was jarring. We talked a lot about how to interpret the results and what the exact percentages really mean. There is a little progress bar on the scale that indicated that the numbers (regardless of gender) were above the “normal range.” But that normal range itself appears to be a “normal range” for an athlete because we were both well above that line. In terms of making the scale work for Kendra, the answer I was proposing was to pick skirt-wearing or pants-wearing and then track that number over time. Kendra’s interest in using the Withings smart scale, and figuring out what “gender option” was right drove us to reexamine how BIA works as a technology, and how assumptions about gender and sex are built into the fundamental equations that drive the algorithmic models used for estimating body fat percentage. This paper, the result of that analysis, combines transgender studies- influenced personal narratives with an analysis of the scientific basis behind the bioimpedance technology used as part of the Withings smart scale. With these different forms of knowledge, we explore how the problem of trans and nonbinary inclusion in personal health tech goes beyond adding a third “gender” option or slapping a rainbow flag on some packaging. Rather, nonbinary exclusion is the tip of an iceberg of flawed assumptions and exclusionary clinical testing, resulting in algorithms that are advertised for self-knowledge but prove to allow anything but. ## 2\. Background This paper draws on previous work related to self-tracking, transgender studies, human-computer interaction, and the study of sex and gender in biomedical research. In this section, we provide a brief summary of related work in these disciplines to situate our findings. While the Withings weight scale is not the first commercially available scale to estimate body fat percentage, the device was one of the original “smart” (i.e. connected) weight scales (Bod, 2020). The device was first sold in the early 2010s at the beginning of a surge of interest in self-tracking and the advent of the “quantified self” movement (Lupton, 2016; Neff and Nafus, 2016; Nafus, 2016). The quantified self movement included a variety of stakeholders including individual self-trackers, organizations such as Quantified Self, companies and toolmakers, academic researchers, and physicians (with considerable overlap between these categories) (Boesel, 2013). Participants are broadly interested in the capabilities of self-tracking to provide unique, actionable, and personalized insights (Lupton, 2020; Wolf and De Groot, 2020). Self-trackers engage deeply with their data through a process sociologist Deborah Lupton refers to as “data sense-making” (Lupton, 2018b). Many self- trackers believe that data can “speak for itself” and should be involved in medical care (Fiore-Gartland and Neff, 2015; Omer, 2016). However, the use of data collected from commercial “wellness” devices such as the Withings scale or activity trackers like Fitbits is controversial as these devices don’t always perform well in third-party validations, and often involve proprietary algorithms and metrics (e.g. steps, sleep scores). Previous research investigating self-tracking devices and wearable monitors has shown that these devices, like devices in other categories, are designed primarily for an unmarked user (Cifor and Garcia, 2020; Costanza-Chock, 2020). That is, the user is assumed to be a White, cisgender, non-disabled man. Cifor and Garcia, for example, use duoethnography to evaluate gendered assumptions in the design and app of the Jawbone UP3 fitness tracker. They illustrate that while the device itself appeared “genderless,” the design of the device and the app reinforced masculinist values such as competition (Cifor and Garcia, 2020). Such issues are also present in the design of algorithms - for example, Fitbit devices count steps less reliably at slower gait speeds and with softer steps, which decreases step count accuracy for older people or people with mobility related disabilities (Feehan et al., 2018; Javorina, 2020). Early implementations of the hardware and algorithms used to estimate heart rate on wearables were less accurate for users with darker skin tones (Shcherbina et al., 2017; Hailu, 2019), though recent evidence suggests these disparities may have been addressed by improvements to the device’s algorithms (Bent et al., 2020). The development of algorithms without a diverse set of users creates algorithmic exclusion. Populations are excluded from the use of algorithmic tools because they were not included as part of the original data used in development, or because information was not gathered in such a way as to make their needs visible. This algorithmic exclusion means that the performance of these algorithms for individuals not in the original dataset are unknown; the practical implication is that these algorithms likely work less well for those not included in the original dataset. Algorithmic exclusion can have real world impacts as individuals rely more and more on these data, especially when these data are used by physicians. For example, pulse oximeter devices that measure blood oxygenation (using a more involved technique similar to that used by wearables manufacturers for measuring heart rate) overestimate blood oxygenation in individuals with darker skin tones (Bickler et al., 2005; Feiner et al., 2007). Renewed interest in these disparities during the COVID-19 epidemic led to a study that showed that Black patients had nearly three times the frequency of occult hypoxemia (oxygen depravation) that was not detected by pulse oximetry than White patients (Moran-Thomas, 2020; Sjoding et al., 2020), potentially leading to higher mortality rates for Black patients when the seriousness of their COVID-19 cases were underestimated. These issues have not escaped notice within communities that build technological tools. There has been increasing discussion in different design communities about how to create technology that is more inclusive and/or addresses some of the disparities discussed above. In human-computer interaction (HCI) and artificial intelligence (AI) research, for example, there have been efforts including analytical reviews, software analysis of datasets, and guidelines about increasing “gender sensitivity” (Burtscher and Spiel, 2020; Scheuerman et al., 2019; Scheuerman et al., 2020; Hamidi et al., 2018; Keyes, 2018) and more intersectional approaches to addressing disparities, such as the intersection of gender and race (Buolamwini and Gebru, 2018; Scheuerman et al., 2018). There have been multiple guides and recommendations for including transgender people in user interface design (Morgan Klaus Scheuerman et al., 2020; Burtscher and Spiel, 2020) and in surveys (Spiel et al., 2019). These recommendations include allowing all individuals to self-identify their gender, not assuming binary gender, using the language users use, and protecting the privacy of participants. In the case of dealing with medical research and “embodiment,” the guidelines recommend measuring physiological parameters such as hormone levels directly, rather than assuming them based on gender. However, embodiment is a tricky line to draw. When one considers the terms “sex” and “gender,” the common assumption is that sex is biological and gender is social. If there is any relationship between the two, it is assumed that sex influences gender, and transgender and intersex people are seen as outliers whose needs vary from “normal” populations. However, Springer et al. argue that it is _sex_ that cannot be purely decoupled from social factors (i.e. gender) (Springer et al., 2012). A “biosocial turn” is now beginning in the study of sex and gender (Shattuck-Heidorn and Richardson, 2019). Many mechanisms that were previously thought to be due to biological “sex” differences, are in fact mechanisms that involve differences based on socialization that manifest in biological differences. Springer et al. recommend using “gender” to refer to social relations and gender roles and the term “sex/gender” to refer to those biological and biosocial factors associated with individual physiology. In this paper, we will use the terms sex/gender and gender, unless we are referring to how these terms are used in a specific work. ## 3\. Approach Our work draws heavily from transgender studies as an approach, while having some similarity to Black feminist methods, specifically Jennifer Nash’s love politics in the form of witnessing (Nash, 2019). We include conversations between the two of us throughout throughout the paper. Personal narrative, especially dialogue, can help uncover “common pain points and overlooked opportunities” (Cifor and Garcia, 2020). Where duoethnography, used by previous studies, is a research method that employs personal narrative to “simultaneously generate, interpret, and articulate data” about a shared experience (Norris, 2008), we include personal narratives throughout this paper to combine, in the words of Susan Stryker, “the embodied experience of the speaking subject” (i.e. our experiences using the weight scale) with “the specialized domain of scholarship”, (i.e. the specifics of the theory and practice of BIA for at home body composition monitoring) (Stryker, 2013; Spade, 2003). Personal narrative allows for a starting point to a broader conversation about smart weight scales and the implications the system and algorithm design have for technology and biomedical research more broadly. We are approaching this topic as a White nonbinary person (Kendra), and as a White cisgender woman (Maggie). Neither of us are disabled in ways that are likely to affect our readings from or interactions with the Withings scale. Both of us have considerable background in technology and gender. Kendra is a lawyer teaching at a technology clinic who also teaches in transgender studies. They have, at times, engaged in self-tracking, although not previously around weight. Maggie is an assistant professor at a small liberal arts school where she teaches digital/embedded systems and inclusive engineering design. Her research involves using bioimpedance to help patients manage fluid overload. She is also a self-tracker and has presented her work at several Quantified Self conferences. ## 4\. Bioelectrical Impedance Analysis Critical analysis of the sort that we deploy in this paper requires the knowledge of how the measurement technology inside smart weight scales works. In this section, we present a background on Bioelectrical Impedance Analysis (BIA). We should note, however, that because the specific testing and equations used by the Withings scale are not publicly available, this background will leverage knowledge from public and peer-reviewed sources and may or may not reflect the specific approaches that the Withings or other consumer-facing scales employ. At the most basic level, the body can be divided into two main “compartments:” fat mass (FM) and fat free mass (FFM) (Kyle et al., 2004; Lukaski, 2013; Khalil et al., 2014). FM includes all the fat in the human body, including what we think of as body fat and also visceral fat around vital organs. FFM is the rest of the tissue; it is about 73% water, about 7% bone, and the rest is proteins. BIA leverages the fact that the water in FFM is conductive; by driving a small, painless current through the body via a pair of electrodes (in weight scales these are two of the electrodes on the scale), the resulting voltage can be measured by another pair of electrodes (also on the scale) and related to the electrical properties of the underlying tissue. If one assumes the body is a homogeneous cylinder with cross-sectional area A, the measured resistance $R$ (defined as the real part of the measured voltage divided by the current) is equal to: (1) $R=\frac{\rho L}{A}$ where $\rho$ is the conductivity of the cylinder, $L$ is the length of the cylinder, and $A$ is the cross-sectional area of the cylinder. Most BIA equations assume that $L$ is proportional to the body height $H$. Multiplying both sides of the equation by $L/L$, the resistance can be related to the inverse of the volume, assuming that $V=L\times A$: (2) $R=\frac{\rho L}{A}\cdot\frac{L}{L}=\frac{\rho L^{2}}{V}$ If one moves the volume to the other side, there is then: (3) $V=\frac{\rho L^{2}}{R}$ This volume $V$ corresponds to what is called the “total body water” or the volume of all water in the body, which is assumed to be about 73% of the volume of the FFM. If one multiplies this volume by the presumed density of the FFM to obtain the FFM, the FM and body fat percentage (BF%) can then be calculated as: (4) $\displaystyle FM$ $\displaystyle=Weight-FFM$ (5) $\displaystyle BF\%$ $\displaystyle=\frac{FM}{FM+FFM}\cdot 100$ ### 4.1. Assumptions of BIA The methods described above require a number of assumptions related to the body. In order for these assumptions to be valid, the resistivity $\rho$ of the measured volume must be homogeneous, and the cross-sectional area must be constant throughout the body such that $V=L\times A$. The assumption that the FFM is 73% water must also hold, along with the assumed density of the FFM. Finally, it must be assumed that the current penetrates through the whole body in a uniform manner such that the estimated volume is truly reflective of the total body water, and not just a fraction of it. Of course, these assumptions are not realistic; the body is not a single homogeneous cylinder with precisely known body composition. Instead, BIA leverages different variables that correlate with “gold standard” estimations of the FFM to estimate the FFM based on the BIA itself. An example BIA equation for estimating FFM might look like the following: (Kyle et al., 2001a): (6) $FFM=-4.104+(0.518\ \times\ H^{2}/R)+(0.231\ \times\ weight)+(0.130\ \times\ X)\\\ +(4.229\ \times\ sex:men=1,women=0)$ This equation involves a number of key terms: the $H^{2}/R$ term, the weight term, the $X$ term (reactance or imaginary part of measured bioimpedance), and sex. Each of these terms is associated with a coefficient in front (along with a constant at the beginning of the equation) that are calculated based on the best fit of the regression equation that minimizes the error between the estimation via BIA and the estimation via the gold standard for the population under test (in this case, the gold standard used was a technology called dual x-ray absorptiometry or DXA). Precisely which parameters are included in the regression equations and their corresponding coefficients depends on the population used to calculate the equations and researcher preference. Other researchers also include factors such as the age of the participants and whether or not participants are athletes (Khalil et al., 2014). In some cases these parameters are all incorporated into a single equation (such as the one above that has “options” for participant sex), or multiple equations are generated, such as one for “lean” participants, one for “average” participants, and one for “obese” participants (Segal et al., 1988). Parameters included in FFM estimation equations often do a lot of “work” and their role is not always clearly understood. These parameters and their coefficients “stand in” for things such as body density, which can vary depending on both factors included in the equations and those typically excluded from the equations such as ethnicity. For example, age is sometimes included as a parameter because there tends to be a decrease in FFM and an increase in FM with age, and sex is included because males on average have lower FM than females. We unpack these assumptions and coefficient parameters in more depth in Section 6. ## 5\. How BIA Is Marketed Kendra: Of course, I didn’t know how BIA worked before using the scale. Nor would looking at the Withings website have revealed any of the fraughtness of BIA to me - when I look at their ads now, they call the technology “body composition.” It’s not obvious from their advertising that it’s estimating body fat percentage based on a set of assumptions and an algorithm, rather than providing an individual-level ground truth. If you don’t know how the technology works, it’s actually quite easy to draw the conclusion that the scale just magically knows your actual body fat percentage. Even if I review the “technical specifications,” the information contained requires quite a bit of context to determine that what is produced isn’t an individualized number. The bullet points say “Bioelectrical Impedance Analysis / Athlete and non-athlete mode / Unit: body fat %, total body water %, muscle mass kg or lb, bone mass kg or lb.” There’s nothing there that tells me, as an end-user without a lot of expertise in BIA, that it’s engaged in an estimation based on plugging particular values into an equation. That brings me to the question, Maggie, what were you thinking when you bought the Withings scale? How did the body composition stuff play into it? Maggie: I’ve had the scale for a long time - since 2012. That was also the time when the number of people talking about self-tracking was growing, and organizations such as Quantified Self were facilitating some of the first meetups and conferences in this area. Quantified Self emphasized self- knowledge through numbers, often using the frame “self-research” or “personal science” (Wolf and De Groot, 2020). Over the next few years, the idea of self- tracking would become very hyped, and an entire commercial ecosystem Whitney Erin Boesel dubs “quantified self” (i.e. little q, little s, vs big Q, big S) was formed (Boesel, 2013). Looking back at my data, my first weigh in was March 23rd, 2012. I wanted to learn more about the tech that was out there and see what I could do to make sense of things, and was also inspired by a Quantified Self “Show & Tell” talk by Amelia Greenhall about how she weighed herself everyday and sustained her weight loss long term (Greenhall, 2013). I was excited about the possibility of self-tracking and consistent habits improving my fitness and my life. I wanted to learn more and then translate that knowledge to help others. Kendra: This idea of self-knowledge is really exciting. That’s what I was hoping for when I stepped on the scale as well - some numbers to help me quantify what I was feeling about my body. But of course, that’s not what I got. As a White nonbinary person, what I learned is that this tech isn’t build for me - in part because of the choices that technology companies make, and in part because of the failure to meaningfully account for transgender experience as part of the underlying clinical testing. And it’s worse for non-White nonbinary or intersex folks, who are both not represented in the studies in terms of sex/gender or race/ethnicity. So much for smart scales. ## 6\. Critiques of BIA Many limitations of BIA have been well established in the medical literature (Khalil et al., 2014), though some researchers argue that the techniques are still appropriate to use under the correct circumstances (Ward, 2019). Researchers suggest caution when using BIA, especially when working with populations that have altered body composition, such as patients with fluid overload. In these cases, some researchers have developed equations specifically for a particular patient population, or have used alternative methods of body composition assessment that don’t rely on regressions (see e.g. (Keane et al., 2017)). A major challenge with body composition using BIA is that the two compartment model of “fat mass” (FM) vs “fat free mass” (FFM) inherently requires assumptions about the composition of the two different compartments (in addition to other assumptions such as homogeneous composition as discussed previously in Section 4.1). Uniformity of the FM is a fairly reasonable assumption across individuals (Martin et al., 1994), but assumptions about the composition of the FFM are not (Côté and Adams, 1993). Additionally, when using the regression equations, the assumptions about the composition of the FFM are obscured into the coefficients associated with the included variables such as age, weight, and sex. Because the linear regressions are optimized at a population level and most studies do not examine the accuracy of the estimations at an individual level, there is no guarantee that a specific equation is accurate for any given individual. Additionally, once one begins to consider any individual that is not perfectly matched for the population that was used to create the equations, the role of these variables becomes increasingly murky but also increasingly important in order to design equations that work for populations historically not included in the populations used to generate the equations. This includes non-White people, trans and nonbinary people, intersex people, people with chronic illnesses, and those at the intersections of these categories. ### 6.1. Unpacking “Sex” We begin with the assumptions and role of “sex” in the equations.111We use the term “sex” here given that this is the term used by the researchers. However, sex and gender are entangled as described in Section 2. Sex in BIA is either coded as 0 for female and 1 for male (effectively changing the offset of the equations, as in Equation 6) or there are separate equations created for male participants and female participants, as in Sun et al. (Sun et al., 2003). What the literature means by “male” or “female” is unclear, and these terms are often confounded with gender identities of “man” and “woman” as in Equation 6. As we discussed in Section 2, “sex” as a concept is just as fraught and contingent as gender (Fausto-Sterling, 2000; Springer et al., 2012; Davis and Preves, 2017). This is not a problem unique to (or caused by) the existence of trans (or intersex) people. Although the methods by which “sex” was evaluated in the BIA literature is unclear, it is common for reported sex to be a participant’s sex assigned at birth. And “sex assigned at birth” is generally only a proxy for someone’s external genitalia at birth, which is only one of the many characteristics that are often encompassed under the word sex (Davis and Preves, 2017; Fausto- Sterling, 2000). Others include hormone balance, chromosomal makeup, and internal reproductive organs. We could not find an example of a study that produces BIA estimates that discusses what sexual characteristics they round up to a determination of sex, and it is generally not clear how the identification of sex was made (i.e. whether self-identified or identified by the researchers). This lack of specificity is one of the first and most significant barriers to creating a more inclusive algorithm for transgender people. Given how large the sample sizes were for some of the populations used to create BIA equations (upwards of 5,000 in (Kyle et al., 2001b)), it is statistically unlikely that no transgender people were involved. But we don’t know how they were accounted for or counted in the calculations. The use of “sex” itself is also a “stand in” for other parameters. What the researchers presumably intend is for “sex” to stand in for the roughly dimorphic variation in body composition between those assigned male at birth and those assigned female at birth (Davis and Preves, 2017). However, it is not the fact that a person is assigned male at birth (i.e., they have genital erectile tissue of a certain size) that makes those assigned male at birth have lower FM on average compared with those assigned female at birth. In fact, FM and the distribution of fat may actually be biosocial phenomena: they may depend on both sex AND gender (sex/gender). For example, one effect of larger quantities of testosterone is changes in fat distribution and increases in muscle production, resulting in lower body fat percentages (Spanos et al., 2020). However, most research into sex differences in body composition do not explore other explanations, such as the differences in diet between men and women in some cultural contexts (Prättälä et al., 2007; Holm and Møhl, 2000). The BIA literature also does not disaggregate sociocultural context that might be caught up in the word “sex”, nor account for the myriad ways in which someone assigned male at birth might not match the presumed physiological composition associated with those assigned male at birth on average. With only two sex or gender options, some intersex and most nonbinary users experience what D.E. Wittkower has termed a dysaffordance – they have to pretend to be something they are not in order to access the service (Wittkower, 2016; Costanza-Chock, 2020). The lack of transparency in the BIA equations makes it difficult to tell exactly what adjustments might need to be made to make equations more inclusive, or to at the least advise individuals as to which “sex” they should choose when using consumer technologies that incorporate BIA. On the other hand, the current setup of the Withings scale that asks for user gender may actually produce more accurate results for trans women and trans men who have undergone gender-affirming hormone therapy, as their body composition may resemble that of a cis woman or cis man, rather than their sex assigned at birth (Klaver and T’Sjoen, 2018; Spanos et al., 2020). ### 6.2. Other Issues With Published Studies White nonbinary or intersex people, despite experiencing a dysaffordance, may be better off than their siblings of color because the composition of the study populations used in BIA equations in the Withings scale are unclear, and there is no way to enter racial or ethnic information.222We use both the terms race and ethnicity because the literature is mixed on which are relevant factors, likely because of the entanglement of biological and social factors similar to sex/gender as discussed in Sections 2 and 6. The majority of measurement studies involving BIA include primarily Caucasian subjects and assume that Caucasian subjects are to serve as a “reference” for other ethnicities.333The use of White people as a default mirrors Kimberle Crenshaw’s critiques of US law, as well as Ruha Benjamin’s critique of photographic film (Benjamin, 2019; Crenshaw, 1989). This is an issue because body composition can vary among ethnic and racial groups due to the environment, diet, cultural factors, and anthropomorphic measurements such as limb length and body size (Deurenberg et al., 1991). Researchers or device manufacturers interested in using BIA equations validated in a Caucasian population therefore need to cross-validate the equations in the population of study. As with sex, race or ethnicity is likely standing in as a proxy for some other variables, such as environmental racism, socioeconomic status, or some actual relationship between FFM composition and certain genetic factors. Some studies suggest that ethnicity specific compensations are needed to use body composition equations in different ethnic groups (Cohn et al., 1977; Schutte et al., 1984), whereas other studies have shown that stratification based on adiposity (i.e. body fat percentage) is more important than race (Ainsworth et al., 1997). Regardless, cross-validation studies are necessary to ensure that the assumptions of equations produce accurate results for all populations. Another major issue with BIA is that the regression equations themselves are validated against “gold standards” that in turn have their own assumptions. For example, BIA equations are often compared against air displacement plethysmography or hydrostatic weighing. But both of these techniques require assumptions about the density of the FFM. The density of the FFM depends on numerous factors such as age, sex, and ethnicity (Ellis, 2000). Therefore any “gold standards” that likewise depend on the two-compartment model (i.e. FM and FFM) are themselves are only valid in the same population, and most of those are also primarily tested on White (presumably cis and binary gender) subjects. Even techniques that do not depend on the two-compartment model such as dual x-ray absorptiometry (DXA) are found to significantly underestimate fat mass when compared with CT scans (Kullberg et al., 2009). Ultimately, the only way to validate a body composition device is using cadaver studies and chemical analysis, which have some of the same issues - the results would then only be validated for those similar to the cadavers (Shepherd et al., 2017). Although measurement problems of this type are common in all areas of science, the lack of physiological basic science about how and why FFM varies with respect to both social and biological factors in an intersectional way means that it is difficult to determine which assumptions will hold across populations. Because of this, the lack of population-specific testing in each individual “gold standard” testing regime complicates the possibility of meaningful validation for folks who do not fit the default. ### 6.3. Why (anti-)Black Boxes? The Lack of Regulatory Oversight The gap between promises and reality in the smart medical device space is in part due to the limited scope of review by the United States’ Food and Drug Administration (FDA). The vast majority of medical devices, including “smart” and “connected” devices, are approved through a program at the FDA known as the 510k approval process. This process requires no clinical trials and very little oversight - device manufacturers merely need to prove that their product is “substantially equivalent” to that of an already approved medical device. The Withings scale discussed in this paper received 510(k) clearance (Wit, 2012); similar scales on the market such as the Fitbit Aria scale received approval even after the devices were already on the market (Fit, 2014). In 2014, the FDA announced their intentions to cease requiring even 510(k) approval for devices such as smart weight scales. This means that there is very little regulation of these devices, and certainly no required clinical validation of the algorithms used to calculate body composition, despite the label of “clinically tested” on the Withings website. This lack of regulatory oversight results in most consumer-focused deployments of technologies like BIA being “black box” algorithms. Popularized in the context of technology studies by scholar Bruno Latour, an algorithm is considered a black box “when a statement is simply presented as raw fact without any reference to its genesis or even its author.” (Harman, 2010). When using the Withings scale, only the final body fat percentage is made available, and there is no explicit reference to an algorithm in the app or in most of the marketing materials. Additionally, Withings does not release the equations its scales use to calculate FFM or the populations that it used to calculate those equations. The power of the black box is that we cannot thoroughly investigate a subject about which nothing is known (Pasquale, 2015). We are left with the assumption that, unless proven otherwise, Withings’ internal algorithm production mirrors the biases of the research at large, but again, it is impossible to tell. All that we know are the inputs that Withings asks users for (binary gender, height, age and athlete status). We can draw some conclusions even with just publicly available information. The binary approach to the gender question without any explanatory text both erases nonbinary people, and does not ask the right question about the embodiment of the user. The Withings scale does not prompt the user to enter information related to ethnicity, so it is not possible that the scale is using an equation that adjusts variables to compensate for different FFM factors in racial or ethnic groups. Because of the marketing of the technology, non-White users may not even know that it might be relevant. The Withings scale’s algorithm is, in the words of Ruha Benjamin, an anti-Black box (Benjamin, 2019). Furthermore, what evidence we do have points to BIA being unreliable even on the populations that it is theoretically well positioned for. We reviewed the list of studies that Withings shared on their page for researchers and found only one study that specifically evaluated the body composition aspects of the scales (Collier et al., 2020). The results of the study compared the performance of the newer Body Cardio scales with an air displacement plethysmography device called the Bod Pod (a “gold standard” discussed in Section 6.2). The mean absolute percentage error of the body fat estimation of the Body Cardio weight scale compared with the Bod Pod was greater than 25%, well above a previously established 1.5% threshold deemed as an acceptable error (Collier et al., 2020).444It is important to note, however, that while air displacement plethysmography is often used as a comparison device/gold standard, it relies on assumptions that can compromise its effectiveness as we discuss in Section 4.1. Withings also argues that these scales should be used to indicate trends rather than for absolute assessment (Worley and Messer, 2016). This suggests that any results from the Withings scales should be interpreted with extreme caution, even on the target population who is most well represented in the studies likely used to create equations. ## 7\. Denying Self-Knowledge Kendra: The distance between Withings’ promise of self-knowledge and the reality of regression equations is upsetting. They advertise all of these benefits to self-quantification, but it’s actually limited by the technology (Felber, 2012). As with many algorithmic technologies, the creation of regression equations based on a limited sample cannot and will not create accurate self-knowledge amongst those who do not fit within those samples. If the scale was actually individually calculating a ground truth number, as opposed to using a regression based on height and skirt-wearing vs. pants- wearing, being nonbinary wouldn’t matter! To be honest, 90% of the time when I’m asked my gender or sex, it doesn’t matter. It’s an arbitrary box checking exercise. So why would the Withings scale be different? There’s this sleight of hand involved in not revealing to people how the technology works that creates the situation in which a nonbinary person could go in expecting self-knowledge and getting a prediction totally disconnected from what they themselves could tell you about their body. I could have told the scale more about me to make the answer more accurate, but that wasn’t an option. I don’t necessarily mind sharing information about my hormone status, my sex assigned at birth, or other facts about my body if they’re actually useful. And although my views on volunteering race are shaped by my membership in a privileged racial group, I suspect many users would prefer to share race or ethnicity information if it meant that they would get more accurate results. Withings asks for my gender, but it doesn’t want it, even aside from the app’s confusion between gender and sex. I know things about myself that are relevant to its guessing, but there’s no way to translate this knowledge within the limited frame of reference produced by the clinical trials. There’s no way to come out with a more accurate picture or contest the bounded understanding of the system. That feels erasing, even more than the mere existence of the binary prompt. It makes me wonder about all of the other random gender/sex requests that I’ve encountered when using technologies around health. Does Fitbit want my gender/sex for calorie burning calculation? Does Apple? What about the Ring Fit? How deep does the rabbit hole go? ## 8\. Paths Forward and Recommendations Trans and nonbinary people of all races deserve to have access to inclusive self-tracking technologies that do not collapse their identities or force them to ignore relevant self-knowledge. What can be done to improve these technologies? We evaluate three options for how to handle sex/gender under the hood of a BIA-calculating device such as the Withings scale, and then provide overall recommendations as to how to handle the use of regression equations based on limited medical testing. It would be inappropriate to continue without noting that BIA as a technology deployed in smart scales may be fundamentally inseparable from the fatphobic medical and cultural context that has created concerns about body fat in the first place (Lupton, 2018a). Withings may claim that “there is more to you than just weight” (see Figure 2), but the subtext of its advertising indicates that you should want to weigh less. That is not fixable through recommendations around the handling of sex (or gender, or hormone status). It might be reasonably asked - given all its flaws, should BIA be used in consumer devices at all? We don’t seek to answer that question in this paper. Our aims are more modest. Nonbinary folk and transgender folks deserve access to technologies of self-knowledge, even as those technologies may be used both by companies and individuals to suggest harmful body standards. We provide a series of options for making weight scales based on BIA more inclusive, with recommendations for users, and both manufacturers and researchers. Most of our recommendations specifically focus on sex/gender, but it is worth noting that overall, BIA also has a long way to go when it comes to race/ethnicity, which we leave for future work to explore in more depth. Although our recommendations are designed based on the specific context of BIA smart scales, they could potentially apply to many areas where binary sex/gender is used as part of an algorithmic system. Figure 2. A portion of the front page of the Withings website advertising a version of their smart scale. Image of the front page of the Withings website advertising a square weight scale. The text reads “There is more to you than just weight.” ### 8.1. Option 1: Eliminate Sex/Gender as A Variable One option for making systems more inclusive of nonbinary people would be the elimination of sex/gender as a variable, using one equation for all users. Practically speaking, this is not difficult. Manufacturers would not have to acquire new data, only re-run the regression to find the best fit without sex/gender as a variable in the regression equation(s). However, a drawback of this option is that elimination of sex / gender as a variable for all users would result in readings that were less accurate in aggregate, as the inclusion of sex does reduce the mean error of the regression equations (Khalil et al., 2014). Given the lack of accuracy of the technology as a whole, this may or may not be hugely significant - however, users who find a sex/gender binary appropriate for their bodies might be upset to lose accuracy in order to make the technology more inclusive. ### 8.2. Option 2: Add a Third Option The next option is to add a third option to the menu where one currently chooses sex/gender. One method of implementing a nonbinary option would be the optional elimination of sex/gender as a variable as listed above for people who select a third option. There would then be three options: “male,” “female,” and “sex/gender neutral.” This option could be helpful for some intersex people, nonbinary people who have not medically transitioned but who would prefer potentially less accurate results to having to volunteer a binary marker, nonbinary people who have taken some medical transition steps, and anyone else for whom the binary sex options are unlikely to produce accurate results. A cautionary note: having a third option in the menu does not a nonbinary inclusive app make. As Anna Lauren Hoffman explains in her generative work on data violence and inclusion, without taking meaningful steps to change the power dynamics present in the product, inclusion is at best, lip service (Hoffmann, 2020). For example, when Facebook allowed for self-identification with a wider variety of gender identities on their platform, they did not fundamentally change the binary advertising logic of male or female, making their claims of nonbinary inclusion questionable (Bivens, 2017). Thus, adding a third option is only appropriate if there is an underlying algorithmic change. ### 8.3. Option 3: Stop Using Sex/Gender as a Proxy Ultimately, the ideal outcome of this work would be for the field to take a step back and consider the role that sex/gender are playing as a “stand in” for things like body fat distribution and anthropomorphic information. This is exactly the kind of work that HCI researchers have recommended when considering trans embodiment (Morgan Klaus Scheuerman et al., 2020; Burtscher and Spiel, 2020). But it is more difficult in this case than many others because of how pervasive assumptions about sex and gender are in clinical research (Tannenbaum et al., 2019; Springer et al., 2012; Moseson et al., 2020). The full, complicated role that sex and gender play in BIA equations and beyond are not well understood. Significant fundamental research is necessary to begin to understand which additional factors to measure and how to measure them in cost-effective and reliable ways. A deeper understanding of sex/gender and body composition will require “slow science” (Stengers, 2018). With more information about the role that these factors are playing, additional information could be provided by end users - everything from body shape (i.e., “apple”, “pear”) to hormone status. This information could even be made optional to not place an additional burden on those unfamiliar with the specifics or who want to do the basics. Fundamentally, this approach is the most well aligned with the promises that companies such as Withings make about their technologies, but would also require the most fundamental research. Table 1. Recommended best practices for trans, nonbinary and intersex inclusion in regression based technologies such as BIA. Table design inspired by (Moseson et al., 2020). Context \| Marginalizing Practices \| Inclusive Practices ---\|---\|--- For manufacturers and researchers \| Assume sex is purely biological and gender is purely social. \| Review literature on sex/gender and select the appropriate measures for the specific project. If none are available, consider conducting more basic research, and consider and articulate the limitations of the state of the art. \| \| Assume a biosocial explanation for physiological differences unless evidence clearly suggests otherwise. \| Ignore the existence of trans, non-binary, and intersex people. \| Acknowledge the existence of trans, non-binary, and intersex people and how their physiology or experiences might be different from other users. \| \| Acknowledge, if needed, the limitations of the current results and how trans and nonbinary people can still obtain the best results for them. \| Analyze results only at the population level. \| Analyze results for sub-groups and at the individual level. \| \| Evaluate results across racial and ethnic groups, implementing race/ethnicity selection or inclusion of non-proxy variable as appropriate. For manufacturers \| Elide the measurement precision and assumptions. \| Explicitly state the precision of the measurement system, along with assumptions and constants used. \| Represent sex and/or gender with pictograms. \| Use clear terminology based on the underlying research or known physiology to select a term. \| \| Be explicit why sex and/or gender are being used so that trans, non-binary and/or intersex users can choose the best option for them. For researchers \| Assume “gold standard” has no built-in assumptions. \| Discuss the assumptions embedded in the gold standard methods themselves, and how those assumptions influence the results. \| Assume the collection of sex and/or gender information is obvious or straightforward, and therefore does not need to be discussed in depth. \| Include whether reported sex and/or gender was based on self-report, and if so, what options were available for participants to choose between. Were there only two options? Was there a free response option? \| \| Explicitly state how sex and gender are being used and what they stand in for. Example: “sex is a stand-in for the dimorphic distribution of body fat in the human population.” ### 8.4. Recommendations First, some advice to transgender, nonbinary, and intersex people who wish to use technology that incorporates BIA but presents binary sex options. Based on studies looking at changes in body composition, hormone status is a very significant variable for body composition of the bodily characteristics, perhaps more significant than other variables that are encompassed by the word sex (Spanos et al., 2020; Elbers et al., 1997). So we would recommend that if folks must pick a binary category, they pick the one most closely aligned with their hormonal balance - male if in a normal male range for testosterone, female if not. In any case, because of the study populations used to produce equations and the black box nature of these algorithms, the actual value produced is unlikely to be accurate and should be used primarily to track change over time rather than for its absolute value. (This recommendation also holds true for any person of any gender using the scale.) In Table 1, we lay out additional recommendations for researchers and manufacturers who wish to build more inclusive regression-based technologies. Elaboration on some of these recommendations and additional recommendations for different contexts can be found in the following references: (Tannenbaum et al., 2019; Moseson et al., 2020; Morgan Klaus Scheuerman et al., 2020; Springer et al., 2012; Spiel et al., 2019; Burtscher and Spiel, 2020; Gebru, 2019). In general our recommendations can be summarized as a) acknowledge the existence of non-gender normative people, b) make fewer assumptions, and c) explain in more detail the limitations of technology. Of course, it may be difficult for companies to fully be honest about measurement accuracy and precision. But if being explicit and honest with customers about these errors and assumptions would make them think twice about purchasing a product, perhaps the best next step is for companies to reconsider their business model. Of course, including trans people after the fact is not ideal. Participatory methods that incorporate transgender people in problem definition around medical devices - to design, in the words of Haimson et al., trans technologies - would be preferable to all of these stopgap measures (Haimson et al., 2020; Costanza-Chock, 2020; Shamas and Afsaneh Rigot, 2019; Gebru, 2019). However, as practitioners who work with participatory methods, we understand that such practices are unlikely to arise overnight. Until it’s widely accepted that designing for those on the margins can create better medical devices, participatory design may never fully adopted by those who commercialize them. ## 9\. Conclusion It can be easy to assume that the use of “sex” in quasi-medical applications is neutral, just another fact about one’s body that allows for a more accurate complete picture. But, in the immortal words of dril, “this whole thing smacks of gender” (dril, 2012). When the lived realities of nonbinary folks cause us to scratch below the surface, the lack of careful thought around assumptions that go into technologies like smart scales becomes clear. Cultural beliefs about gender are driving the bus when it comes to engagement with “sex differences.” And because of that, sex, even in clinically tested BIA equations, is holding space for too many variables, supported by too little basic research. When inadequately validated, (anti)-Black box algorithms built on these shaky foundations deceive their users. They harm people who do not line up with the assumed user base, promising knowledge of self but instead merely reproducing the violence of erasing clinical systems (Hoffmann, 2020). It doesn’t have to be this way. Even without additional clinical testing or regulation, there are clear steps that manufacturers can take to mitigate some of the harms caused by these systems. First, they can educate users as to the population-level accuracy of metrics like BIA, rather than advertising body composition analysis as if it was accurate on an individual basis. Second, as discussed above, they can make clear how the technology does or does not work on transgender or nonbinary people, while also identifying other factors (such as race and/or ethnicity) that make results more or less accurate. Finally, manufacturers could release as much information about their equations as possible, including validation studies, in order to facilitate cross- validation by independent researchers. Admittedly, these solutions may reify technical expertise and serve to legitimize the ideas of these types of body measurement, as Greene, Stark, and Hoffman, point out in their work on technological ethics (Greene et al., 2019). Ultimately, BIA is just one example of how regression-based body measurements, whether implemented in technologies like smart scales or described in scientific papers, harm those who are not presumed to be the ideal. And that whole thing is worth overturning. ###### Acknowledgements. 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# The H$\alpha$ Dots Survey. IV. A Fourth List of Faint Emission–Line Objects Joseph D. Watkins Department of Astronomy, Indiana University, 727 East Third Street, Bloomington, IN 47405, USA John J. Salzer Department of Astronomy, Indiana University, 727 East Third Street, Bloomington, IN 47405, USA Angela Van Sistine Department of Astronomy, Indiana University, 727 East Third Street, Bloomington, IN 47405, USA Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin-Milwaukee, 3135 N Maryland Ave Milwaukee, Wisconsin 53211, USA Ana Hayslip Lowell Observatory, 1400 W. Mars Hill Rd. Flagstaff. Arizona. 86001. USA. Eric Hoar School of Materials Science & Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Rayna Rampalli Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA ###### Abstract We present the fourth catalog of serendipitously discovered compact extragalactic emission-line sources – H$\alpha$ Dots. A total of 454 newly discovered objects are included in the current survey list. These objects have been detected in searches of moderately deep narrow-band images acquired for the ALFALFA H$\alpha$ project (Van Sistine et al., 2016). The catalog of H$\alpha$ Dots presented in the current paper was derived from searches carried out using ALFALFA H$\alpha$ images obtained with the KPNO 2.1 m telescope. This results in a substantially deeper sample of Dots compared to our previous lists, which were all discovered in images taken with the WIYN 0.9 m telescope. The median R-band magnitude of the current catalog is 21.59, more than 1.6 magnitudes fainter than the median for the 0.9 m sample (factor of 4.4$\times$ fainter). Likewise, the median emission-line flux of the detected sources is a factor of 4.3$\times$ fainter. The line-flux completeness limit of the current sample is approximately 3 $\times$ 10-16 erg s-1 cm-2. We present accurate coordinates, apparent magnitudes and narrow-band line fluxes for each object in the sample. Unlike our previous lists of H$\alpha$ Dots, the current sample does not include follow-up spectroscopy. ††journal: Astrophysical Journal Supplement Series††facilities: KPNO:2.1m††software: IRAF ## 1 Introduction We present the latest installment of the H$\alpha$ Dots survey. H$\alpha$ Dots are compact emission-line galaxies (ELGs) discovered in narrow-band images (e.g., Salzer et al., 2020). The nature of our selection method results in catalogs of strong-lined objects that identify very specific types of star- forming and active galaxies detected via a number of different emission lines. In particular, the H$\alpha$ Dots detected via the H$\alpha$ line are all dwarf star-forming galaxies (including many blue compact dwarf (BCD) galaxies) or outlying H II regions in nearby spiral galaxies, while those detected via the [O III]$\lambda$5007 line are typically Green Pea galaxies (e.g., Cardamone et al., 2009; Brunker et al., 2020) or Seyfert 2 galaxies (Salzer et al., 2020). The H$\alpha$ Dots survey also detects high redshift QSOs via one of several UV emission lines (e.g., Mg II $\lambda$2798, C IV $\lambda$1549 or Ly$\alpha$). The H$\alpha$ Dots survey is carried out using images acquired for the ALFALFA H$\alpha$ project (Van Sistine et al., 2016, hereafter AHA). The goal of the AHA project was to measure accurately the star-formation rate density in the local universe by obtaining H$\alpha$ images of a volume-limited sample of H I-selected galaxies from the ALFALFA survey (Giovanelli et al., 2005; Haynes et al., 2011, 2018). The H$\alpha$ Dots project originated with the serendipitous discovery of point-like emission-line sources located in the narrow-band AHA images (Kellar et al., 2012, hereafter K12). The initial discovery was made by an undergraduate student who was processing early AHA data in collaboration with the senior author. All subsequent searches for H$\alpha$ Dots in the AHA images have been carried out exclusively by undergraduate students. Previous H$\alpha$ Dots survey lists were derived from AHA images taken with the WIYN 0.9 m telescope (K12, Salzer et al., 2020). A third list of H$\alpha$ Dots detected using WIYN 0.9 m images is in preparation. The current survey catalog has been created by analyzing narrow-band images obtained with the Kitt Peak National Observatory (KPNO)111The Kitt Peak National Observatory is part of the National Optical-Infrared Research Laboratory (NOIRLab) . NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. 2.1 m telescope. As we show in the subsequent sections of this paper, the use of a larger telescope naturally results in a sample of ELGs that reaches to substantially fainter flux levels compared to the previous survey lists. Our paper is organized as follows: Section 2 describes the observational data and the preliminary data processing steps, while Section 3 details our selection methodology. The latter follows closely the methods adopted in the previous H$\alpha$ Dots papers. Section 4 presents our new list of H$\alpha$ Dots and provides an overview of the properties of the samples using the available survey data. Section 5 presents the limited spectroscopic information available in the literature for the newly cataloged H$\alpha$ Dots and discusses potential applications of this deep sample of ELGs once follow- up spectra are obtained. Section 6 summarizes the main results of this study. ## 2 Observational Data The current study analyzes all ten observing runs of AHA data obtained with the KPNO 2.1 m telescope. Images were taken using three different CCD detectors during the course of the AHA project: T2KB, with a FOV of 10 by 10 arcmin, and STA2 and STA3, which both had FOVs of 10 by 6.6 arcmin. All three CCDs had a pixel scale of 0.30 arcsec/pixel. Both continuum $R$-band images and narrow-band images were acquired, the latter using the KPNO narrow-band filter set (see Figure 1). Three of the four narrow-band filters were used in the observational data set analyzed in this paper (KP1564, KP1565 and KP1566), although the majority of the AHA targets observed with the KPNO 2.1 m telescope are located in the two higher redshift filters (KP1565 and KP1566). For each target AHA galaxy, three images were obtained. First, a 15 minute narrow-band image was taken at the beginning of the observing sequence, then a 3 minute $R$-band image, followed by another 15 minute narrow-band image. Figure 1: The filter transmission curves for the narrow-band filter set used for the current survey. The filters are designated in the survey catalog (Table 2) as filters 1 through 4: filter 1 = KP1563, filter 2 = KP1564, filter 3 = KP1565, filter 4 = KP1566. The vast majority of the AHA targets observed with the 2.1 m telescope were observed in filters 3 and 4. The AHA project required accurate flux calibration for the narrow-band data. Multiple observations of spectrophotometric standard stars were acquired each night of the program. If the photometric conditions were at all dubious on any night where data were obtained, fields taken on those nights were flagged and re-observed on a later night using short “post-calibration” observations. All narrow-band fluxes measured from AHA images have the zero-point of the flux scale measured to better than 2% accuracy (and usually $\sim$1%). Full details are included in Van Sistine et al. (2016). Because of the careful approach to flux calibration employed by the AHA program, the narrow-band fluxes measured for the H$\alpha$ Dots should be quite accuarate. Standard corrections for instrumental signatures were performed on each image. This included overscan correction, mean bias subtraction, flat field correction, and bad pixel cleaning (see K12 for details). All preliminary image processing utilized the Image Reduction and Analysis Facility (IRAF). Cosmic rays were removed from the images using the L.A. Cosmic script (van Dokkum, 2001). A software pipeline was developed to process the AHA images; details of this pipeline are given in AHA. First, the code aligns the $R$-band and two narrow- band images to a common center and applies an astrometric solution to each image. They are then Gaussian filtered to a common stellar FWHM, after which all images have their fluxes scaled to the flux level of the first narrow-band image using bright, unsaturated stars in the field. The scaled $R$-band image is subtracted from each narrow-band image, producing the continuum-subtracted images. Finally, the two continuum-subtracted images are added together to get the combined continuum-subtracted image. Figure 2: H$\alpha$ Dot 1010 is located in the upper-left portion of the field of AGC 330247. The upper image is the $R$-band continuum image, and the lower image is the continuum-subtracted narrow-band image. The isolated point source of residual narrow-band flux is circled in red. It has an R magnitude of 20.67. The target AHA galaxy, AGC 330247 = CGCG 476-011, is located in the bottom-left of the field (below the two bright stars), and is seen to possess strong H$\alpha$ emission. The spiral galaxy in the lower right is UGC 12514; it exhibits H$\alpha$ emission from a number of disk H II regions. The total field-of-view of these images is 10.0 $\times$ 6.3 arcminutes. Examples of a continuum $R$-band image and a continuum-subtracted narrow-band image are shown in Figure 2. The images show a cut-out of the field for AHA target AGC 330247, taken with the T2KB detector. Circled in the upper-left corner of both images is an isolated, point-like source of residual narrow- band flux. It is unresolved in our images and is located far from either of the two larger galaxies in this field. This object is what we call an “H$\alpha$ Dot.” When viewed in SDSS color images it exhibits a greenish tint, strongly suggesting that it is an [O III]-detected galaxy located at z $\sim$ 0.34. The target AHA galaxy is located in the lower-left of this image; it is a strong H$\alpha$ emitter. The spiral galaxy located in the lower right is UGC 12514. It is also an ALFALFA H I source, and exhibits emission from many H II regions in its spiral disk. Table 1: Observing Run Summary for KPNO 2.1 m H$\alpha$ Dot Catalog Observing Run \| Detector \| Number Nights \| Number Fields \| H$\alpha$ Dots ---\|---\|---\|---\|--- September 2010 \| T2KB \| 4 \| 25 \| 24 November 2010 \| T2KB \| 5 \| 45 \| 26 March 2011 \| T2KB \| 7 \| 72 \| 78 May 2011 \| T2KB \| 7 \| 48 \| 53 October 2011 \| T2KB \| 8 \| 88 \| 54 March 2012 \| T2KB \| 9 \| 87 \| 89 September 2012 \| STA2 \| 8 \| 80 \| 45 March 2013 \| T2KB \| 5 \| 45 \| 28 April 2013 \| T2KB \| 8 \| 68 \| 34 March 2014 \| STA3 \| 5 \| 53 \| 23 Totals \| \| 66 \| 611 \| 454 A total of 611 ALFALFA H$\alpha$ fields were observed over the course of 10 observing runs (Table 1). These 10 runs can be broken up into Fall sample and Spring samples. The Fall sample is approximately contained within a region between R.A. of $22^{h}$ to $3^{h}$ and Dec. of $+24^{\circ}$ to $+29^{\circ}$, and the Spring sample is approximately contained between R.A. of $7^{h}30^{m}$ to $16^{h}40^{m}$ and Dec. of $+3^{\circ}$ to $+17^{\circ}$. The images collected during the course of these ten observing runs were searched for H$\alpha$ Dots. Our methodology for detecting H$\alpha$ Dots in the processed AHA narrow-band images is described in the next section. ## 3 2.1 m H$\alpha$ Dot Survey: Identifying H$\alpha$ Dots For an object in the field to be considered to be an H$\alpha$ Dot, it must satisfy two primary criteria. First, it must have a statistically significant excess of flux in the narrow-band filter relative to the R-band flux. Second, it must be morphologically compact. This usually means the object is either unresolved or barely resolved in our CCD images. The first criterion is readily quantified (see below), while the second is admittedly somewhat subjective. In particular, it will vary from field-to-field depending on the size of the point-spread function (a.k.a. “seeing”) associated with each image. All H$\alpha$ Dot candidates are evaluated by at least two members of the project team in order to try to invoke a uniform assessment of compactness. A software package was developed in order to automatically and systematically search for H$\alpha$ Dots in the AHA images. The software employs routines designed to identify every object in an image, compare their fluxes in the continuum and narrow-band filters, and then calculate a magnitude difference and its uncertainty for each source. Potential candidates are reviewed by members of the H$\alpha$ Dot team before the list of ELG candidates is finalized. For more details about the software package, see K12. The software takes as input the $R$-band and composite narrow-band images and identifies every object present in the field using a modified version of DAOFIND (Stetson, 1987). Next the software performs photometry with a constant aperture size on each object in the scaled $R$-band and the unsubtracted narrow-band images to construct a magnitude difference, calculated as $\Delta m=m_{NB}-m_{R}.$ (1) Here the magnitudes used are simple instrumental magnitudes. Because the images being used have all been scaled to a common flux level, objects with no emission lines (e.g., most stars) will have $\Delta$m = 0.0. Large negative values of $\Delta$m indicate an object with a significant excess of flux in the narrow-band image. The software also computes the ratio of the absolute value of the magnitude difference to the error in the magnitude difference, as $ratio=\frac{\|\Delta m\|}{\sigma_{\Delta m}},$ (2) where $\sigma_{\Delta m}$ is generated by taking the errors associated with the $R$-band and narrow-band magnitudes and summing them in quadrature: $\sigma_{\Delta m}={\sqrt{\sigma_{NB}^{2}+\sigma_{R}^{2}}}.$ (3) The $ratio$ parameter serves as a pseudo signal-to-noise (SNR) ratio. Small values of $ratio$ represent either objects with little or no emission-line flux (small $\Delta$m) or noisy sources (large $\sigma_{\Delta m}$). For each field analyzed, the software generates a diagnostic plot (see Figure 3 for an example). Each “$\mathsf{X}$” in the plot indicates a single object in the images. The left-hand graph plots $\Delta m$ against the instrumental R-band magnitude. The bright stars in the field are clumped at $\Delta m$ = 0 on the left side of the plot; the locus of stars remains centered on zero but spreads out to larger values of $\Delta m$ for the fainter stars with large photometric errors. Objects with a negative magnitude difference indicate more flux in the narrow-band image than in the $R$-band image. The right-hand graph plots $\Delta m$ against the $ratio$ parameter. The vertical and horizontal lines drawn on the diagnostic plot represent the threshold values for $\Delta m$ and $ratio$ that are used to select emission-line candidates. We set the values for inclusion in the H$\alpha$ Dot sample at $\Delta m$ $<$ $-$0.4 and $ratio$ $>$ 4.5. These values were found by K12 to optimize the detection of faint objects and minimize the number of false detections. Objects located in the lower-right quadrant of the right-hand plot of Figure 3 represent candidate H$\alpha$ Dots. Figure 3: The diagnostic plot produced after using the dot-finding software on the field of AGC 330247 (see Figure 2). The left panel plots $\Delta m$ (eq. 1) vs. instrumental R-band magnitude. Brighter stars are located to the left and lie along a line around $\Delta m=0$ since the narrow-band and broad-band flux levels are normalized to a common value by our software. Objects with a negative $\Delta m$ indicate residual narrow-band flux. The right panel plots $\Delta m$ vs. $ratio$ (eq. 2). Objects of interest have a large negative $\Delta m$ and large $ratio$ values, corresponding to the bottom-right quadrant of the plot. The solid red lines indicate the limiting values for $\Delta m$ and $ratio$ for inclusion in the sample (see text). The large number of putative detections in this field is caused by the many H II regions located in AGC 330247 and UGC 12514 (Figures 2 and 5). The H II regions (n = 52) are marked in both panels with red squares, while the single H$\alpha$ Dot candidate (H$\alpha$ Dot 1010) is shown as a green circle. The remaining objects in the lower right quadrant, indicated by the “$\mathsf{X}$” symbol, are all image artifacts that have been rejected. We also point out the many bright (RInst $<$ $-$8) sources with large postive $\Delta m$ in the upper left portion of the left panel. These are all image artifacts caused by the long saturation “bleed” trail from the bright star visible in Figure 2. Once the software has selected all possible candidates, these objects must be visually examined to ascertain their nature. This verification step is essential, since the automated software and our high-quality data combine to yield numerous sources that are not true emission-line objects. Our survey images typically include numerous sources that can lead to false detections, including uncleaned cosmic rays, saturated stars, satellite or meteor trails, and noise spikes. It is also common that star–forming regions in the AHA target are also selected by passing the $\Delta m$ and $ratio$ criteria listed above. For example, many of the emission-line candidates in the diagnostic plot shown in Figure 3 are H II regions in UGC 12514 (see Figures 2 and 5). The object review process is necessary to separate these three types of detections: real H$\alpha$ Dots, H II regions, or image artifacts. Figure 4: Example of image cutouts used to evaluate each H$\alpha$ Dot candidate. The field is centered on H$\alpha$ Dot 1447. From left to right, these images are the $R$-band continuum image, the combined narrow-band image, and the continuum-subtracted narrow-band image. The compact appearance and significant residual flux present in the continuum-subtracted image is characteristic of an H$\alpha$ Dot. Each sub-image is 200 $\times$ 200 pixels, or 60 arcseconds on a side. Figure 5: This set of image cutouts shows an example where an H II region in a large spiral galaxy has been detected by the dot-finding software (indicated by the red circle). The software often detects dozens of H II regions in the central AHA galaxy because they surpass the $\Delta m$ and $ratio$ thresholds. In the case of this particular galaxy (UGC 12514) a total of 37 H II regions were selected by our software. The review of all objects located in the lower right quadrant of the diagnostic diagram (e.g., Figure 3) allows the user to categorize this detection properly as an H II region and not an H$\alpha$ Dot. Our software produces image cut-outs for each object found in the bottom-right quadrant of the diagnostic plot. These are three sub-images (200 $\times$ 200 pixels in size) centered on the object in question and displayed horizontally. The cut-outs contain the object as seen in the $R$-band image, the combined narrow-band image, and the continuum-subtracted narrow-band image (see Figures 4 and 5). Using the image cut-outs, the user visually examines each object located in the bottom–right quadrant of the right plot in Figure 3 and categorizes them into one of the three object types specified above. Objects that are classified as image artifacts are discarded. Objects flagged as H$\alpha$ Dots or HII regions are cataloged into separate lists for subsequent analysis. An object must be compact in appearance, spatially separate from the central AHA galaxy, and contain significant emission in the narrow-band image in order to be selected as an H$\alpha$ Dot candidate. These criteria are discussed in detail in the previous H$\alpha$ Dots survey papers (Kellar et al., 2012; Salzer et al., 2020). The compactness criteria was instituted to avoid cataloging large, extended galaxies that were already known. In particular, we did not wish to include the target AHA galaxy in our survey. The compactness and separation requirements are somewhat subjective, although the visual checking by at least two independent team members helps to ensure a fairly uniform approach. We note that the separation criterion does not prevent the survey from detecting isolated / outlying H II regions that are associated with the AHA target galaxy. Several examples are included in Salzer et al. (2020), who emphasize that the identification of such objects requires follow- up spectroscopy. Examples of objects detected in our emission-line searches are shown in Figures 4 and 5. The object in Figure 4 is H$\alpha$ Dot 1447, which has a measured R-band magnitude of 19.87 $\pm$ 0.03. While very compact in nature, is is seen to be resolved in the broad-band image. This implies that the emission line detected in our narrow-band filter is most likely H$\alpha$. Since it was observed in filter KP1566 (i.e., filter 4), its redshift is most likely to fall between 5300 and 7800 km/s ((Van Sistine et al., 2016). This would make H$\alpha$ Dot 1447 a dwarf star-forming galaxy, which is consistent with its blue appearance in the Sloan Digital Sky Survey (SDSS) color images (York et al., 2000; Abolfathi et al., 2018). Figure 5 shows a number of H II regions located in the nearly face-on spiral UGC 12514 (see Figure 2). Large spiral galaxies like this are commonly detected multiple times by our software. In the example shown the detected H II region is the faint object located at the center of the cut-outs. As mentioned above, these H II regions are cataloged separately from the H$\alpha$ Dots. They are not discussed further as part of the current study. After the review process is completed, objects flagged as H$\alpha$ Dots are remeasured more carefully in order to obtain accurate $R$-band magnitudes and narrow-band line fluxes for each source. The photometric calibrations derived for the AHA project are utilized to place the measurements on an accurate flux scale. Once the list of H$\alpha$ Dots discovered in the AHA 2.1 m images was finalized, the entire catalog was crossed-matched with objects in the SDSS Data Release 14 (Abolfathi et al., 2018). Using the coordinates for the H$\alpha$ Dots obtained from our astrometric solution (see Section 2), SDSS positions were retrieved for all H$\alpha$ Dots that were within the SDSS footprint. We then visually compared the location given by the coordinates for each H$\alpha$ Dot to verify that the query had returned the correct object. If it returned the wrong object, the correct object was located using the SDSS navigate window, and new SDSS coordinates were obtained by centering the cursor on the object in the navigate window and reading off the corresponding Right Ascension and Declination. Fully 83.7% of the 2.1 m H$\alpha$ Dots (380 of 454) were included in the SDSS photometric catalog. The H$\alpha$ Dots not in SDSS were either too faint (65 of 454, or 14.3%) or are located outside of the SDSS footprint (9 or 454, or 2.0%). For the objects in common between the two surveys we queried SDSS DR14 again to obtain the full set of SDSS ugriz magnitudes and errors. This information was then merged into the H$\alpha$ Dots database. ## 4 2.1 m H$\alpha$ Dot Survey: Results The final catalog of H$\alpha$ Dots detected using the KPNO 2.1 m AHA images contains 454 newly discovered ELGs. This list of objects was arrived at after searching 611 AHA fields. The total sky coverage of these images represents 15.494 square degrees, resulting in a surface density of new H$\alpha$ Dots of 29.30 deg-2. This number, which is a key figure of merit for such surveys, is substantially higher than the surface densities of objects detected in the previous H$\alpha$ Dots survey lists (K12, Salzer et al., 2020). The latter had surface densities of 5.22 and 5.24 ELGs/deg2, respectively, more than a factor of 5.5$\times$ lower. Table 2: Fourth List of H$\alpha$ Dots H$\alpha$ Dot # \| RA(2000) \| DEC(2000) \| Obs. Run \| Filter \| $\Delta$m \| Ratio \| mR \| NB Line Flux \| SDSS r ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| degrees \| degrees \| \| \| \| \| mag \| x10-14 erg/s/cm2 \| mag (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| (10) 1001 \| 330.44398 \| 24.18998 \| Sep2010 \| 4 \| -1.21 \| 23.91 \| 21.84 $\pm$ 0.06 \| 0.062 $\pm$ 0.003 \| 22.05 $\pm$ 0.10 1002 \| 330.44587 \| 24.20362 \| Sep2010 \| 4 \| -0.92 \| 72.05 \| 19.27 $\pm$ 0.02 \| 0.329 $\pm$ 0.004 \| 19.45 $\pm$ 0.02 1003 \| 330.45553 \| 24.21585 \| Sep2010 \| 4 \| -0.60 \| 34.08 \| 19.69 $\pm$ 0.02 \| 0.179 $\pm$ 0.004 \| 1004 \| 334.14431 \| 27.94493 \| Sep2010 \| 4 \| -1.20 \| 17.60 \| 21.39 $\pm$ 0.04 \| 0.058 $\pm$ 0.004 \| 1005 \| 340.46256 \| 27.76333 \| Sep2010 \| 4 \| -0.49 \| 10.80 \| 21.13 $\pm$ 0.05 \| 0.043 $\pm$ 0.004 \| 1006 \| 340.55283 \| 27.69124 \| Sep2010 \| 4 \| -1.25 \| 18.51 \| 22.13 $\pm$ 0.10 \| 0.059 $\pm$ 0.004 \| 21.87 $\pm$ 0.11 1007 \| 340.59996 \| 27.81721 \| Sep2010 \| 4 \| -1.89 \| 12.94 \| 22.94 $\pm$ 0.18 \| 0.057 $\pm$ 0.004 \| 1008 \| 344.40155 \| 26.40989 \| Sep2010 \| 4 \| -0.72 \| 30.60 \| 19.98 $\pm$ 0.03 \| 0.135 $\pm$ 0.005 \| 20.19 $\pm$ 0.03 1009 \| 349.23530 \| 27.93332 \| Sep2010 \| 4 \| -1.10 \| 7.39 \| 22.97 $\pm$ 0.28 \| 0.033 $\pm$ 0.006 \| 22.87 $\pm$ 0.19 1010 \| 350.08029 \| 26.09913 \| Sep2010 \| 4 \| -1.64 \| 52.78 \| 20.67 $\pm$ 0.05 \| 0.348 $\pm$ 0.007 \| 20.66 $\pm$ 0.04 1011 \| 352.82919 \| 25.12090 \| Sep2010 \| 4 \| -0.54 \| 17.84 \| 20.48 $\pm$ 0.03 \| 0.092 $\pm$ 0.006 \| 20.82 $\pm$ 0.05 1012 \| 355.59838 \| 27.96453 \| Sep2010 \| 4 \| -0.42 \| 7.10 \| 21.37 $\pm$ 0.05 \| 0.026 $\pm$ 0.004 \| 21.81 $\pm$ 0.16 1013 \| 355.70529 \| 27.97838 \| Sep2010 \| 4 \| -1.63 \| 106.59 \| 19.44 $\pm$ 0.02 \| 0.762 $\pm$ 0.008 \| 19.78 $\pm$ 0.04 1014 \| 355.70809 \| 28.07243 \| Sep2010 \| 4 \| -1.87 \| 7.21 \| 23.75 $\pm$ 0.58 \| 0.017 $\pm$ 0.004 \| 1015 \| 355.99726 \| 27.15461 \| Sep2010 \| 4 \| -2.17 \| 8.94 \| 22.51 $\pm$ 0.12 \| 0.053 $\pm$ 0.006 \| 1016 \| 356.00110 \| 27.15570 \| Sep2010 \| 4 \| -1.63 \| 7.17 \| 22.02 $\pm$ 0.11 \| 0.057 $\pm$ 0.006 \| 1017 \| 356.00053 \| 27.12553 \| Sep2010 \| 4 \| -0.50 \| 11.08 \| 21.16 $\pm$ 0.04 \| 0.054 $\pm$ 0.006 \| 21.07 $\pm$ 0.09 1018 \| 356.05400 \| 27.19430 \| Sep2010 \| 4 \| -0.72 \| 7.08 \| 22.38 $\pm$ 0.10 \| 0.033 $\pm$ 0.004 \| 22.00 $\pm$ 0.15 1019 \| 357.03115 \| 24.32929 \| Sep2010 \| 4 \| -1.48 \| 6.31 \| 24.87 $\pm$ 2.26 \| 0.041 $\pm$ 0.006 \| 1020 \| 357.18135 \| 27.98432 \| Sep2010 \| 4 \| -0.41 \| 4.92 \| 21.38 $\pm$ 0.09 \| 0.017 $\pm$ 0.003 \| 21.43 $\pm$ 0.09 1021 \| 357.18551 \| 27.94486 \| Sep2010 \| 4 \| -1.28 \| 17.00 \| 21.27 $\pm$ 0.11 \| 0.152 $\pm$ 0.007 \| 21.13 $\pm$ 0.07 1022 \| 357.38800 \| 27.91175 \| Sep2010 \| 4 \| -1.46 \| 9.06 \| 22.53 $\pm$ 0.24 \| 0.055 $\pm$ 0.006 \| 1023 \| 17.12467 \| 24.68193 \| Sep2010 \| 4 \| -0.55 \| 13.64 \| 20.81 $\pm$ 0.04 \| 0.053 $\pm$ 0.006 \| 20.77 $\pm$ 0.06 1024 \| 17.17512 \| 24.68174 \| Sep2010 \| 4 \| -0.59 \| 9.91 \| 21.75 $\pm$ 0.07 \| 0.034 $\pm$ 0.004 \| 21.91 $\pm$ 0.10 1025 \| 331.97791 \| 27.02233 \| Nov2010 \| 4 \| -0.42 \| 6.76 \| 20.30 $\pm$ 0.05 \| 0.032 $\pm$ 0.007 \| 1026 \| 332.20645 \| 24.72709 \| Nov2010 \| 4 \| -1.17 \| 17.20 \| 21.23 $\pm$ 0.05 \| 0.071 $\pm$ 0.004 \| 21.87 $\pm$ 0.18 1027 \| 332.27611 \| 24.61840 \| Nov2010 \| 4 \| -0.46 \| 7.05 \| 21.30 $\pm$ 0.07 \| 0.028 $\pm$ 0.005 \| 21.94 $\pm$ 0.08 1028 \| 332.53514 \| 25.44871 \| Nov2010 \| 4 \| -1.04 \| 10.36 \| 21.55 $\pm$ 0.12 \| 0.062 $\pm$ 0.006 \| 22.15 $\pm$ 0.11 1029 \| 334.34985 \| 27.68723 \| Nov2010 \| 4 \| -0.45 \| 5.71 \| 21.97 $\pm$ 0.07 \| 0.019 $\pm$ 0.005 \| 22.12 $\pm$ 0.14 1030 \| 334.51783 \| 27.55964 \| Nov2010 \| 4 \| -1.33 \| 11.42 \| 23.14 $\pm$ 0.22 \| 0.033 $\pm$ 0.004 \| 1031 \| 350.37149 \| 24.25764 \| Nov2010 \| 4 \| -0.91 \| 21.42 \| 21.68 $\pm$ 0.05 \| 0.045 $\pm$ 0.003 \| 21.90 $\pm$ 0.09 1032 \| 351.29051 \| 25.85756 \| Nov2010 \| 4 \| -0.69 \| 8.87 \| 20.90 $\pm$ 0.10 \| 0.050 $\pm$ 0.006 \| 21.07 $\pm$ 0.12 1033 \| 351.68935 \| 25.68903 \| Nov2010 \| 4 \| -1.21 \| 20.38 \| 22.72 $\pm$ 0.12 \| 0.045 $\pm$ 0.004 \| 22.25 $\pm$ 0.19 1034 \| 352.94941 \| 25.83481 \| Nov2010 \| 4 \| -0.74 \| 47.50 \| 19.11 $\pm$ 0.02 \| 0.274 $\pm$ 0.006 \| 19.24 $\pm$ 0.03 1035 \| 354.34425 \| 25.64395 \| Nov2010 \| 4 \| -1.98 \| 8.47 \| 23.51 $\pm$ 0.43 \| 0.046 $\pm$ 0.004 \| 1036 \| 354.49840 \| 25.69983 \| Nov2010 \| 4 \| -1.93 \| 9.46 \| 22.71 $\pm$ 0.27 \| 0.051 $\pm$ 0.004 \| 1037 \| 357.03530 \| 27.60221 \| Nov2010 \| 4 \| -0.65 \| 15.31 \| 20.87 $\pm$ 0.06 \| 0.057 $\pm$ 0.006 \| 20.76 $\pm$ 0.07 1038 \| 357.31563 \| 25.53299 \| Nov2010 \| 4 \| -1.51 \| 16.02 \| 21.68 $\pm$ 0.14 \| 0.164 $\pm$ 0.008 \| 1039 \| 358.17888 \| 27.30178 \| Nov2010 \| 4 \| -1.08 \| 5.74 \| 22.26 $\pm$ 0.17 \| 0.016 $\pm$ 0.004 \| 1040 \| 358.24607 \| 27.29628 \| Nov2010 \| 4 \| -0.89 \| 10.03 \| 22.06 $\pm$ 0.14 \| 0.042 $\pm$ 0.006 \| 22.18 $\pm$ 0.14 Note. — Table 2 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content. After the analysis is completed on the data from each of the AHA observing runs, every H$\alpha$ Dot candidate in the final list is assigned a unique H$\alpha$ Dot number. Following the convention established in K12, the H$\alpha$ Dots are ordered by increasing right ascension within each observing run. The observing runs are ordered chronologically, with the H$\alpha$ Dot numbering sequence proceeding continuously from one run to the next. To avoid any confusion with the numbering sequence established for the 0.9 m H$\alpha$ Dot catalogs, the 2.1 m H$\alpha$ Dot numbers start with 1001. Hence, the first list of 24 H$\alpha$ Dot candidates from the September 2010 observing run are numbered 1001 through 1024, the H$\alpha$ Dot numbers for the 26 ELGs discovered in the November 2010 data run from 1025 to 1050, and so on. Our fourth catalog of H$\alpha$ Dots is presented in Table 2. The table includes, for each H$\alpha$ Dot, its identifying H$\alpha$ Dot number (column 1), SDSS Right Ascension and Declination (epoch J2000) (columns 2 and 3), the observing run from which the AHA data originates (column 4; see also Table 1), and the narrow-band filter used for the observation of that field (column 5, see Figure 1). Columns 6 and 7 give the magnitude difference $\Delta m$ and $ratio$ as defined in Section 3. These represent two of the primary selection parameters used in creating the H$\alpha$ Dots survey. The measured R-band magnitude and its associated error is given in column 8, while column 9 gives the measured narrow-band line flux and its error. These measured quantities are derived directly from our survey images. Finally, column 10 lists the SDSS r-band magnitude and its error for objects where this has been measured. We make a few notes about the data presented in Table 2. The decision to use SDSS coordinates as opposed to our own astrometry was based on a direct comparison of the two sets of positional data. The accuracy of our astrometry is, in general, typically quite good. The median positional offset between our astrometry and SDSS positions is 1.02 arcsec for the full sample. However, we found that while the positional agreement is good at the centers of our images, it tends to be less reliable near the edges of our frames. Given the wide-field accuracy of the SDSS astrometry, we have opted to adopt it whenever possible. In cases where the H$\alpha$ Dot is located outside the footprint of the SDSS (nine galaxies), the coordinates are those derived from our analysis. We include the SDSS r-band magnitudes for those H$\alpha$ Dots that match with an object in the SDSS photometric catalog (380 objects). In general, there is good agreement between the SDSS and H$\alpha$ Dots photometry for objects with R $<$ 22 (with some notable exceptions). The median SDSS r magnitudes are systematically 0.10-0.15 magnitudes fainter than our R-band measurements due to well known offsets in the two photometric systems (e.g., Jordi et al., 2006). For R $>$ 22.5, the photometric errors in both measurements get quite large (typically larger than 0.2 magnitudes). As expected, the R-band magnitudes for the 65 H$\alpha$ Dots that were not cataloged by SDSS are substantially fainter than those that are: $\langle$R$\rangle_{In\ SDSS}$ = 21.34, compared with $\langle$R$\rangle_{Not\ in\ SDSS}$ = 22.32. Figure 6 presents a histogram of the R-band magnitudes for the full sample of 454 H$\alpha$ Dots listed in Table 2 (red histogram). The brightest object in the catalog is H$\alpha$ Dot 1144, with R = 17.07, while the faintest object, H$\alpha$ Dot 1117, has R = 25.35. For comparison, the corresponding histogram for the entire catalog of H$\alpha$ Dots discovered using the 0.9 m telescope (K12, Salzer et al., 2020, plus additional objects from the upcoming third catalog list) is shown in black. The median R magnitude in the 2.1 m catalog is 21.59, 1.62 magnitudes fainter than the median R magnitude found in the 0.9 m catalog. This corresponds to a factor of 4.45 in brightness. The two sample medians are indicated with arrows in the figure. The relative photometric depths of the two samples of ELGs is very similar to the ratio of the surface density of objects discussed above, and is consistent with the expectations based on the ratio of the light- collecting areas of the two telescopes of (84 inches/37 inches)2 = 5.15. Figure 6: Histogram of the R-band magnitudes for all 454 2.1 m H$\alpha$ Dots (red histogram). For comparison we plot the corresponding histogram for the full sample of 0.9 m detected H$\alpha$ Dots (black histogram). The median values for the two distributions are given in the legend and are indicated by the two arrows. The median apparent magnitude for the 2.1 m H$\alpha$ Dots is seen to be 1.62 magnitudes fainter (factor of 4.45$\times$ fainter). Figure 7: Histogram of the emission-line flux measured in our narrow-band images for the full set of 454 2.1 m H$\alpha$ Dots (red histogram). For comparison we plot the corresponding histogram for the entire sample of 358 0.9 m detected H$\alpha$ Dots (black histogram). The median values for the two distributions are given in the legend and are indicated by the two arrows. The median line flux for the 2.1 m H$\alpha$ Dots is seen to be 0.63 dex fainter (factor of 4.27$\times$ fainter). While the distribution of apparent magnitudes is a useful indicator of the depth of the H$\alpha$ Dots survey, this is a sample of galaxies detected based on the strength of their emission lines located within the bandpasses of our narrow-band filters. Hence, the proper way to evaluate the depth of the survey is by examination of the distribution of emission-line fluxes. This distribution is shown in Figure 7. As with Figure 6, we plot line-flux histograms for both the 2.1 m H$\alpha$ Dots from the current study (red histogram) as well as the sample of H$\alpha$ Dots detected in the 0.9 m data (black histogram) in Figure 7. The median values of the two distributions are indicated in the legend, and denoted by arrows in the plot. Once again, the 2.1 m H$\alpha$ Dots sample is seen to be substantially deeper than that found with the 0.9 m telescope. The median line flux found for the 2.1 m data is 4.57 $\times$ 10-16 erg s-1 cm-2, which is a factor of 4.27 times fainter than the median for the 0.9 m H$\alpha$ Dots. The flux distribution of the current sample peaks at log(flux) = $-$15.5 (3.2 $\times$ 10-16 erg s-1 cm-2), beyond which the distribution falls off rapidly. We adopt this value as the approximate line-flux completeness limit for the 2.1 m H$\alpha$ Dots sample. ## 5 Discussion ### 5.1 Spectroscopic Follow-up of the New H$\alpha$ Dots Previous H$\alpha$ Dot survey lists (K12, Salzer et al., 2020) included information from follow-up spectra. These spectra provide confirmation of the emission-line detection, as well as identifying the line present in the narrow-band images. Typically these follow-up spectra yield accurate redshifts and emission-line ratios that allow for the determination of the activity type of each ELG (e.g., star forming vs. AGN). A similar spectroscopic campaign for objects in the current H$\alpha$ Dot catalog was not possible, in part because of the larger number of objects, and in part because the objects are, on average, significantly fainter than those in the previous catalogs. Hence, we are presenting our latest and deepest list of H$\alpha$ Dots without benefit of spectroscopic confirmations. During our cross-matching with the SDSS, we noted a small number of H$\alpha$ Dots with SDSS spectra. A search of Data Release 16 (Ahumada et al., 2020) reveals that 17 H$\alpha$ Dots from the current survey list (3.7% of the total) have existing spectra in SDSS. These objects are listed in Table 3. There are four H$\alpha$ Dots with low redshifts where the emission line in our narrow-band filter was H$\alpha$ (based on our inspection of the SDSS spectra). These four objects are all brighter than R = 18.0, and are among the brightest of the H$\alpha$ Dots in the current sample (e.g., Figure 6). One of these is H$\alpha$ Dot 1144, the brightest object in our catalog. All four of these were observed as part of the legacy SDSS redshift survey (e.g., Strauss et al., 2002). The remaining H$\alpha$ Dots with SDSS spectra were observed as part of the BOSS (Dawson et al., 2013) or eBOSS (Dawson et al., 2016) projects. One of these, H$\alpha$ Dot 1021, was detected via strong [O III]$\lambda$5007 being located in our survey filter. This object has a spectrum similar to the Green Pea galaxies (Cardamone et al., 2009; Brunker et al., 2020). All of the remaining 12 H$\alpha$ Dots with SDSS spectra are QSOs. Of these, nine are detected due to redshifted Mg II emission, one via C III], and two via C IV. Table 3 lists the H$\alpha$ Dot number, the emission line detected in our narrow-band filter, the spectroscopic redshift, and the activity type (either QSO or star-forming galaxy (SFG)) for each object. Table 3: H$\alpha$ Dots with SDSS Spectra H$\alpha$ Dot # \| Detected Line \| z \| ELG Type ---\|---\|---\|--- (1) \| (2) \| (3) \| (4) 1011 \| Mg II $\lambda$2798 \| 1.3955 \| QSO 1021 \| [O III] $\lambda$5007 \| 0.3406 \| SFG 1110 \| H$\alpha$ \| 0.0165 \| SFG 1142 \| H$\alpha$ \| 0.0241 \| SFG 1144 \| H$\alpha$ \| 0.0239 \| SFG 1192 \| Mg II $\lambda$2798 \| 1.4085 \| QSO 1193 \| Mg II $\lambda$2798 \| 1.3925 \| QSO 1214 \| Mg II $\lambda$2798 \| 1.3858 \| QSO 1216 \| C IV $\lambda$1549 \| 3.2969 \| QSO 1217 \| C IV $\lambda$1549 \| 3.3161 \| QSO 1220 \| Mg II $\lambda$2798 \| 1.3930 \| QSO 1260 \| C III] $\lambda$1909 \| 2.4814 \| QSO 1329 \| Mg II $\lambda$2798 \| 1.3896 \| QSO 1341 \| Mg II $\lambda$2798 \| 1.3803 \| QSO 1356 \| Mg II $\lambda$2798 \| 1.3837 \| QSO 1358 \| Mg II $\lambda$2798 \| 1.3717 \| QSO 1392 \| H$\alpha$ \| 0.0214 \| SFG The distribution of redshifts for the 17 H$\alpha$ Dots from the current catalog with SDSS spectra is dramatically different from the one for the 0.9 m Dots (e.g., Salzer et al., 2020). The latter sample is dominated by objects detected via their H$\alpha$ or [O III] lines. Specifically, 92% of the H$\alpha$ Dots from the first two catalogs were detected either by H$\alpha$ (55%) or [O III]$\lambda$5007 (37%). The remaining galaxies were discovered either by the [O II]$\lambda$3727 doublet (2%) or one of the UV lines common to QSO spectra (6%). The fact that the “emission-line detection function” for the H$\alpha$ Dots found with the KPNO 2.1 m telescope is so different from the one observed for the previous H$\alpha$ Dot survey lists (WIYN 0.9 m component) should be no surprise. Many of the objects with SDSS spectra were pre-selected for the BOSS and eBOSS surveys as having broad-band colors consistent with QSOs (e.g., Dawson et al., 2013, 2016). Hence, the large percentage of QSOs is no accident. However, even before seeing the SDSS spectra, we were anticipating that the new catalog of H$\alpha$ Dots would be different. The increased depth of the 2.1 m sample compared to the 0.9 m Dots, coupled with the fixed redshift range accessible for each emission line, implies that the fainter H$\alpha$ Dots should preferentially be objects selected at higher redshifts rather than being lower luminosity versions of the the objects found with the 0.9 m telescope. We expect a higher portion of [O III]-detected Dots in the current catalog than were present on the first two survey lists, as well as substantially higher numbers of [O II]-detected galaxies and high redshift QSOs. As time and telescope resources allow, we will hopefully get the opportunity to test these hypotheses as we obtain follow-up spectra for these new H$\alpha$ Dots. ### 5.2 Applications of the H$\alpha$ Dots As highlighted in the previous survey lists based on sources detected in WIYN 0.9 m images (K12, Salzer et al., 2020), the H$\alpha$ Dots have a number of interesting science applications. These include studying large samples of dwarf star-forming galaxies, including Blue Compact Dwarfs (BCDs), and the detection of strong [O III] emitters like Green Peas and Seyfert 2 galaxies. The lack of existing follow-up spectroscopy for the current sample of H$\alpha$ Dots naturally limits its immediate impact on addressing relevant science questions. Nonetheless, we draw attention to the high-impact scientific applications these objects can be used to address. #### 5.2.1 Low Luminosity Star-forming Galaxies As outlined above, we anticipate that a lower percentage of the H$\alpha$ Dots in the current catalog will be low redshift H$\alpha$ detections. Still, we expect that a significant fraction will have been detected via the H$\alpha$ line. Given the wavelength coverage of the narrow-band filters employed, the resulting redshift range of the H$\alpha$-selected galaxies will be 0.005 – 0.026 (velocity range 1460 – 7810 km/s). This redshift range, coupled with the apparent magnitude distribution of the current sample (e.g., Figure 6), implies that the H$\alpha$-detected H$\alpha$ Dots will all be dwarf star- forming systems (e.g., see Figure 8 in Salzer et al., 2020). Since this catalog of galaxies represents a statistically complete, line-flux-limited sample, it could be used for accurately establishing the volume density of low-luminosity star-forming galaxies in the local universe. Since the H$\alpha$ Dots are pre-selected as possessing strong emission lines, this sample of dwarf galaxies will also be ideal for measuring metal abundances for a large, statistically-complete sample of objects. A similar study is currently underway utilizing the brighter H$\alpha$ Dots from K12 and Salzer et al. (2020) (A. Hirschauer, in preparation). An important application of the current catalog of objects is that they extend the detection of strong- lined ELGs to substantially fainter magnitudes. Hence, we expect that the 2.1 m survey of H$\alpha$ Dots will include lower-luminosity dwarfs than those found in the previous survey lists. This in turn should result in a larger yield of extremely low abundance systems, similar to HADot 303 = AGC 198691 (a.k.a. the Leoncino Dwarf; Hirschauer et al., 2016; McQuinn et al., 2020). #### 5.2.2 [O III]-detected Star-forming Galaxies The H$\alpha$ Dots discovered with the WIYN 0.9 m telescope included a large number of [O III]-detected galaxies (37% of the objects in the first two survey lists). The expectation is that the 2.1 m Dots will include a comparable or somewhat larger fraction of systems detected by the [O III]$\lambda$5007 line. In fact, the current catalog might well be dominated by such objects. The relevant redshift range for detection by [O III]$\lambda$5007 is z=0.317-0.345. The [O III]-detected systems found in the previous H$\alpha$ Dots catalogs included a mix of Green Pea-like galaxies (Cardamone et al., 2009; Brunker et al., 2020) and Seyfert 2 galaxies. We expect that the current catalog will detect many additional Green Pea candidates. Additionally, given the increased depth of the current list of H$\alpha$ Dots, we fully expect that many less extreme [O III]-selected star-forming galaxies will also come within reach of detection. It is well known that the strength of the [O III]$\lambda$5007 line peaks at metal abundances of $\sim$10% solar (log(O/H)+12 $\sim$ 7.7). In the local universe (z $<$ 0.1), actively star-forming galaxies with B-band absolute magnitudes in the range $-$16 $\leq$ MB $\leq$ $-$18 are often found in [O III] line-selected samples such as the UM survey (e.g., MacAlpine et al., 1977; MacAlpine & Williams, 1981; Salzer et al., 1989), the Case survey (e.g., Sanduleak & Pesch, 1982; Pesch & Sanduleak, 1983; Salzer et al., 1995), and the KISSB survey (Salzer et al., 2002). The depth of the current sample should allow for the detection of this population of objects at the higher redshifts probed by the [O III]$\lambda$5007 line with our filters. A key attribute of both the Green Pea galaxies and the intermediate luminosity [O III]-detected SFGs is that they very often exhibit spectra with such strong emission lines that the temperature-sensitive [O III]$\lambda$4363 auroral line is present in their follow-up spectra. Hence, we expect that the [O III]-detected H$\alpha$ Dots in the current list will include dozens of sources from which accurate direct abundances will be measurable. #### 5.2.3 [O III]-detected Seyfert 2 Galaxies The additional depth of the current H$\alpha$ Dot catalog will likely result in a deeper and presumably more comprehensive sample of [O III]-selected Seyfert 2 galaxies in the redshift window z=0.317-0.345. While the previous H$\alpha$ Dots list include a significant number of Seyfert 2s (11% of the sample overall, and 30% of the [O III]-detected Dots), they tend to be objects with extreme spectral characteristics. The Seyfert 2s included in the previous H$\alpha$ Dots catalogs tend to exhibit very high [O III]/H$\beta$ ratios; they are nearly all very high-excitation objects (see Figure 7 in Salzer et al. (2020)). Because the great depth of the current survey, the Seyfert 2s cataloged in the current paper should include both the high-excitation objects as well as many with lower [O III]/H$\beta$ ratios. Overall we expect an even higher percentage of Seyfert 2s compared to the previous lists. We also expect that this new sample of Seyferts will be more representative, rather than being biased toward the more extreme examples. Once again, the line-flux limited nature of the H$\alpha$ Dots survey method will allow us to generate an accurate volume density for Seyfert 2 population in the redshift window covered by our filters. The strong-lined nature of the Seyfert 2 sample will also allow for the study of the metallicity of the AGN at these redshifts (which represents a lookback time of $\sim$3.7 Gyr). A preliminary analysis of the [O III]-detected Seyfert 2s from the previous survey catalogs has indicated the possibility of a modest drop in the average metal abundance compared to low redshift counterparts (D. Carr, in preparation). #### 5.2.4 [O II]-detected SFGs and AGN Another expectation of the current list of H$\alpha$ Dots is that it will include a higher proportion of [O II]-detected SFGs and AGN at z = 0.770-0.807. This population of ELGs is just barely detectable with the 0.9 m telescope portion of the H$\alpha$ Dots survey. Only three [O II]-detected galaxies are included in the first two survey lists, and these are all found in the fainter portion of the sample (average line flux of 1.3 $\times$ 10-15 ergs/s/cm2). The increased depth of the 2.1 m survey list should result in a substantial increase in the number of [O II]-selected ELGs. This will allow the survey to probe the star-forming and AGN galaxy populations at these cosmologically interesting distances (lookback times of $\sim$6.8 Gyr, or 50% the age of the universe). #### 5.2.5 QSOs Finally, we mention the rather obvious presence of numerous QSOs within the current H$\alpha$ Dots catalog. While our survey is not capable of producing a comprehensive QSO survey, it does detect substantial numbers of quasars in specific, narrow redshift windows: z = 1.357-1.408 for the Mg II $\lambda$2798 line, z = 2.454-2.528 for C III] $\lambda$1909, z = 3.257-3.347 for C IV $\lambda$1549, and z = 4.428-4.543 for Ly$\alpha$ $\lambda$1215\. If detected in sufficient numbers, the H$\alpha$ Dots QSOs could provide accurate “hard points” for their volume densities in these redshift windows. While it is clear that the large fraction of QSOs among the H$\alpha$ Dots with available SDSS spectra is a selection effect, these spectra provide a glimpse of the potential science applications of the line-flux limited H$\alpha$ Dots survey for probing the QSO population at a range of redshifts. ## 6 Summary & Conclusions We present the latest list of H$\alpha$ Dots, based on images obtained for the ALFALFA H$\alpha$ project (Van Sistine et al., 2016). H$\alpha$ Dots are compact emission-line sources detected serendipitously in narrow-band images. Previous survey catalogs have presented lists of H$\alpha$ Dots detected in images obtained with the WIYN 0.9 m telescope (K12, Salzer et al., 2020). Our new list of H$\alpha$ Dots has been created by analyzing 611 ALFALFA H$\alpha$ fields observed with the KPNO 2.1 m telescope. The current H$\alpha$ Dot catalog contains 454 unique H$\alpha$ Dots. All the new H$\alpha$ Dots were identified in 2.1 m images using the same software packages developed for the previous H$\alpha$ Dots catalogs. Hence, the only significant difference with the previous survey lists is in the depth of the sample. The 2.1 m H$\alpha$ Dots survey is sensitive to fainter objects, detecting sources with a median apparent R magnitude of 21.59 and median line fluxes of 4.57 $\times$ 10-16 erg s-1 cm-2. In both metrics, the current survey list probes a factor of $\sim$4.4 times fainter than the 0.9 m H$\alpha$ Dots catalogs. The approximate emission-line flux completeness limit of the current sample is 3 $\times$ 10-16 erg s-1 cm-2. While the previous H$\alpha$ Dots catalogs included information from follow-up spectroscopy, we do not have corresponding spectral data for the current list of ELGs. We speculate that the additional depth of the H$\alpha$ Dots list generated using 2.1 m telescope images will result in a significantly different mix of objects being discovered, relative to the previous catalogs. While we expect that the current catalog includes numerous low-luminosity star-forming dwarf galaxies detected via their H$\alpha$ lines, we expect that this population will account for a much smaller fraction of the overall ELG catalog when compared to the lists generated from the 0.9 m data (where the H$\alpha$-detected fraction was 55%). We anticipate large fractions of the current catalog will be found to have been detected via their [O III] or [O II] lines. Plans for carrying out follow-up spectroscopy of the 2.1 m H$\alpha$ Dots are being formulated. We would like to thank the anonymous referee who made a number of suggestions that have improved the paper. We gratefully acknowledge the financial support of the College of Arts and Sciences and the Department of Astronomy at Indiana University, which helped make this ongoing undergraduate research project possible. The H$\alpha$ Dots survey project is based on data obtained for the ALFALFA H$\alpha$ project, which was carried out with the support of a National Science Foundation grant to JJS (NSF-AST-0823801). We would also like to acknowledge the Maria Mitchell Association, which provided a summer research internship to RR as part of their NSF-funded REU program (with JJS serving as an associate mentor). This project made use of Sloan Digital Sky Survey data. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. 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# Density-based clustering of social networks Giovanna Menardi Department of Statistical Sciences, University of Padova, Italy<EMAIL_ADDRESS>Domenico De Stefano Department of Political and Social Sciences, University of Trieste, Italy<EMAIL_ADDRESS> ###### Abstract The idea underlying the modal formulation of density-based clustering is to associate groups with the regions around the modes of the probability density function underlying the data. This correspondence between clusters and dense regions in the sample space is here exploited to discuss an extension of this approach to the analysis of social networks. Such extension seems particularly appealing: conceptually, the notion of high-density cluster fits well the one of community in a network, regarded to as a collection of individuals with dense local ties in its neighbourhood. The lack of a probabilistic notion of density in networks is turned into a major strength of the proposed method, where node-wise measures that quantify the role and position of actors may be used to derive different community configurations. The approach allows for the identification of a hierarchical structure of clusters, which may catch different degrees of resolution of the clustering structure. This feature well fits the nature of social networks, disentangling a different involvement of individuals in social aggregations. ## 1 Introduction ### 1.1 Background and motivation Within large social communities, individuals sparsely interact with each others and usually set a tight relationship with a limited number of subjects. Interactions favour individuals to aggregate into groups, where the relationships are stronger and the information flow is more intense than outside. The generating mechanism of these groups, albeit pervasive, is complex and often difficult to be disclosed. On one hand, different kinds of relationship may be established, from friendship to professional collaboration, each of them possibly with different levels of intensity. On the other hand, aggregation may be driven by diverse, sometimes unobserved, social mechanisms – homophily, popularity, ranking or influence. Depending on the context, cohesive communities may be formed, where even relationships connect each actor with most of other actors. This configuration characterizes, for instance, individual interactions, communication system, sport and team relationships (Carron and Brawley, 2000). A different dynamic arises when one or few influential actors drive the aggregation and shape the whole organization of the community (Ghalmane et al., 2019). Examples of this latter behaviour are opinion or news spreading in online communities where followers are attached to influencers (e.g. Wang et al., 2017b); epidemic diffusion where few prominent actors govern the outbreak (Medo et al., 2009), scientific collaborations and citations where communities develop around the so-called star scientists (De Stefano et al., 2013). Here, the nature of leadership may be associated to various roles which actors carve out within the groups, acting for instance as hubs or brokers. In this context, Social network analysis (SNA) exploits the framework offered by graph theory to translate these ideas into operational tools: any community is suitably described by a graph where nodes represent the actors and the links between them their interactions, possibly of different strength. A wide range of methods, among which centrality or equivalence measures are just simple examples, have been spawned to express notions of social role and position. A standard accounts is Wasserman and Faust (1994). While the underlying scope to find groups in network may follow different routes, these are usually defined as locally densely connected set of nodes. The correspondence between groups of subjects and their inner connection density, as well as the possible role of influential individuals within communities, suggest us to extend the ideas underlying the density-based approach for clustering nonrelational data to the network framework. The _modal_ formulation of this approach associates clusters with the domains of attraction of the modes of the density function underlying the observed data, namely clusters correspond to dense regions of the sample space. While network data unarguably prevent the definition of a probabilistic notion of a density function defined on the nodes, the two notions of group are in agreement conceptually. Operationally, modal clustering often resorts to graph theory to detect clusters, which further favours the extension of this formulation to network data. As a fortunate side effect the modal approach allows for the identification of a hierarchical structure of clusters, which may catch different degrees of resolution of the clustering structure. Based on these ideas, the aim of this work is to discuss a method to find clusters of nodes within a network structure, while accounting for relationships of different strength. Consistently with the cluster notion shared by the nonrelational density-based approach, we focus on aggregation mechanisms driven by the attraction exerted by influential actors, on the basis of different “leadership” roles as detected by means of alternative node-wise measures. Note that this perspective is largely neglected by the inherent literature, most focusing on the concept of mutual cohesiveness within communities. The paper is organised as follows. After a brief review of clustering approaches for networks, we overview the modal clustering formulation in metric spaces. Then, we discuss its extension to network data, in both, the unweighted and weighted network framework. The procedure is illustrated on some simple archetypal networks characterized by different community configurations, on a number of benchmark examples with a known community structure, and on a comprehensively complex original dataset to identify groups of researchers within the community of the Italian academic statisticians. A discussion concludes the paper. ### 1.2 Overview of the related literature _Community detection_ refers to the general aim of partitioning networks in subsets of nodes, which share some common properties and play similar roles in the relational structure. Similarly to the nonrelational framework, this task is, in fact, far from being accurately defined. Thus, while the general purpose usually translates into the task of identifying assortative groups with dense inner connections, a different perspective would also include the search of disassortative structures with weaker interactions within, rather than between communities. The lack of a precise definition of cluster, along with the unsupervised nature of the problem, have led on one hand to the proliferation of a voluminous amount of literature on this topic and, on the other hand, to confusing taxonomies of methods designed for the scope. A lack of a consistent terminology has determined expressions as _network_ or _graph clustering_ , _module_ , _block_ or _community detection_ to be either used interchangeably, or carry slightly different, yet ambiguous, connotations. In this confounding panorama, methods are easier classified on the basis of their technical details and algorithmic implementations (e.g., Fortunato, 2010; Azaouzi et al., 2019), which yet disguises the more relevant notion of cluster underlying them. Reviewing all these methods is then an awkward task which we cannot engage without crossing over the scope of the paper. For our purpose, we limit to set some boundaries by providing a coarse overview of the main different goals and motivations for finding groups in networks, and refer back to the insightful review of Rosvall et al. (2019), where the reader will find further details and references. At the same time, we use the terms cluster, community, groups and so on exchangeably in the rest of the paper. The first, perhaps most widespread approach to find clusters in networks aims at identifying densely interconnected nodes compared to the other nodes. Due to the generality of this principle, methods differ in the way it is translated into operational implementations. Several formulations rely on detection of actors or edges with high centrality, as for instance, the very popular method of Girvan-Newman (GN, Newman and Girvan, 2004), a divisive algorithm for undirected and unweighted graphs based on edge-betweenness, afterwards generalized by Chen and Yuan (2006). Further methods relying on a similar ground build on the optimisation of the cluster modularity (Danon et al., 2005), so that each community will include a larger number of inner edges than expected by chance. The Louvain method is unarguably one of the most popular representative of this category (Blondel et al., 2008). The aforementioned methods result in cohesive communities where transitivity is high and each actor is highly connected to each other inside the group. Notwithstanding, the idea of high density within a group may be also intended as the one arising in star-shaped clusters, where density is concentrated in the figure of some hubs attracting less prominent actors. Evidence of such a theoretical mechanism of aggregation has been explained by Goyal et al. (2006) as a combination of small-world behavior guided by the presence of interlinked stars. In fact, this principle has been largely neglected by SNA, with the works of Kloster and Gleich (2014), based on the local optimization of the so- called conductance and, to some extent, Falkowski et al. (2007) representing an exception. This is also the route we follow. A further facet of the clustering problem in networks, known as _cut-based_ perspective, aims at partitioning networks in a fixed number of balanced groups with a small number of edges between them, and no guarantees about a possible denser structure of inner connection. In this context, networks are often of a mesh- or grid-like form. Methods in this class refer back to the seminal work of Kernighan and Lin (1970) and often build on the spectrum of of the data. Examples are Pothen et al. (1990); Wang et al. (2017a). The block-modeling approach follows a completely different purpose, relying on the fundamental concept of node equivalence, of which _structural equivalence_ is the most used. Disregarding the similarity of nodes, groups are here based on more general patterns that include disassortative communities and include nodes that serve, within the network, a similar structural role in terms of their connectivity profile. A first formalization in terms of non-stochastic blocks can be found in Lorrain and White (1971), while Holland et al. (1983) gave rise to the stochastic counterpart, later generalized to the weighted framework (Aicher et al., 2015) and largely applied in various contexts See Lee and Wilkinson (2019) for a recent review. ## 2 Clusters as dense sets ### 2.1 Modal clustering of non-relational data Modal clustering delineates a class of methods for grouping non-relational data defined on a metric, continuous space, and building on the concept of clusters as “regions of high density separated from other such regions by regions of low density” (Hartigan, 1975, p. 205) Formally, the observed data $(x_{1},\ldots,x_{n})^{\prime}$, $x_{i}\in\mathbb{R}^{d}$, $i=1,\ldots,n$, are supposed to be a sample from a random vector with (unknown) probability density function $f$. The modes of $f$ are regarded to as the archetypes of the clusters, which are in turn represented by their domain of attraction. The practical identification of the modal regions may occur according to different directions. One of them associates the clusters to disconnected density level sets of the sample space, without attempting explicitly the difficult task of mode detection. The key idea is that, when there is no clustering structure, $f$ is unimodal, and any section of $f$, at a given level $\lambda$, singles out a connected (upper) level set: $L(\lambda)=\\{x\in\mathbb{R}^{d}:f(x)\geq\lambda\\}$. Conversely, when $f$ is multimodal, $L(\lambda)$ may be either connected or disconnected, depending on $\lambda$. In the latter case, it is formed by a number of connected components, each of them associated with a region of the sample space including at least one mode of $f$. Since a single section of $f$ could not reveal all the modes of $f$, $\lambda$ is moved along its feasible range, giving rise to a hierarchical structure, known as the _cluster tree_ , which provides the number of connected components for each $\lambda$. Each leaf of the tree describes a _cluster core_ , defined as the largest connected component of the density level sets which includes one mode. Figure 1 illustrates a simple example of these ideas: cluster cores associated with the two highest modes are identified by the smallest $\lambda$ larger than $\lambda_{3}$, while the smallest $\lambda$ larger than $\lambda_{1}$ identifies two connected components whose one is the cluster core associated to the lowest mode. Note that while the cluster tree resembles a dendrogram, the whole procedure cannot be included in the class of hierarchical techniques. These explore, within the same run, all the partitions with a number of clusters ranging from one to $n,$ by subsequent splits (divisive algorithms) or aggregations (agglomerative algorithms). Conversely, in the cluster tree, the leaves are themsevels veritable clusters, instead of single observations, and their number is then an estimate of the number of clusters. Hence, with respect to a dendrogram, the cluster tree enjoys a different, more insightful interpretation. The height of the leaves corresponds to the density level at which the associated mode appears, thus providing an indication of the cluster prominence. Finally, the hierarchical structure of the tree allows for catching possible different degrees of resolution of the clustering. In the example illustrated in Figure 1 the number of modes is three, but the two highest ones pertain to the same macro-group, at a lower level of resolution, hence the leaves associated to the two groups collapse to a single branch accordingly. Figure 1: A sample from three subpopulations and the associated contour set at a level $\lambda_{0}$ (left). The threshold $\lambda_{0}$ defines a section of the trimodal underlying density function (center) and identifies two connected regions. On the right, the cluster tree indicates the number of connected components for varying $\lambda$ and the total number of clusters, corresponding to the leaves. As the union of the cluster cores is not a partition of the sample space, unallocated points are assigned to the cluster cores according to a supervised scheme of classification, generally accounting for their density. Operationally, clustering involves two main choices: first, a density estimator is required and this is typically selected among the nonparametric methods. Second, for each examined threshold $\lambda$ it is to establish whether the associated level set is connected and what are its components. Since there is no obvious method to identify connected sets in a multidimensional space, graph theory comes to this aid. A graph is built on the sample points and the connected components of the subgraphs induced by the level sets are then easily detected. The reader is referred to Menardi (2016) for further details about modal clustering. ### 2.2 Modal clustering of social networks #### 2.2.1 Defining density on networks For the current formulation, we regard to social networks as undirected graphs $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$ consisting of a set $\mathcal{V}=\\{v_{1},\dots,v_{n}\\}$ of nodes – the actors of the network– and a set $\mathcal{E}=\\{e_{ij}\\}$ of $m$ links or edges, $i\neq j=1,\ldots,n,$ representing relations between pairs of nodes. Depending on the nature of the observed relationships, the elements of $\mathcal{E}$ assume different forms: in binary networks the $e_{ij}$ will take values in $\\{0,1\\}$, denoting the absence and the presence of a link, respectively, while real nonnegative values of $e_{ij}$ will account for different strengths of the relationship in weighted networks. In order to represent a given network $\mathcal{G}$ it is possible to define a $n\times n$ adjacency matrix $\mathbf{A}$ whose elements $a_{ij}=e_{ij}$. The notion of high-density regions highlighted in the previous section suggests a natural counterpart in network analysis, where clusters are often referred to as sets of actors with dense relationship patterns (see, among others, Moody, 2001). However, network objects are subject to an inherent limitation, as their properties can be established in geodesic terms only. In particular, a probabilistic notion of density cannot be defined and shall be intended in a less formal way, reflecting some intuitive meaning of cohesiveness. We are naturally tempted to borrow the concept of density and akin notions from graph theory. The density of a subgraph $\mathcal{H}\subseteq\mathcal{G}$ is defined as the proportion of all possible edges of $\mathcal{H}$ which are actually observed. In fact, density definition as a node-wise measure is arbitrary as a subgraph $\mathcal{H}_{v}$ is required to be associated to each node $v$. For instance, one could set $\mathcal{H}_{v}=\\{\mathcal{V}_{v},\mathcal{E}_{v}\\}$ as the subgraph having the nearest neighbours of $v$ as nodes, or focus on the single node $\mathcal{V}_{v}=v$ and its incident edges $\mathcal{E}_{v}$ thus recasting to the notion of (possible weighted) degree. In fact, consistently with the previous one, a wider set of candidates to quantify local density is represented by measures of connectivity or measures of centrality, which evaluate, somehow, the role as well as the prominence of each actor in a network. It is worthwhile to observe that the choice of a node-wise density measure is not inconsequential with regard to the subsequent interpretation of clusters, and different choices would entail a different concept of cluster. For example, the notion of degree accounts for the rate of the activity of individual nodes in the network, so that high-degree actors act as “hubs” and play a central role in the overall connectivity. Alternatively, by measuring the proportion of times a node works as a broker connecting nodes otherwise disconnected in the network, betweenness evaluates the influence of the actors with respect to the information flow in the network. Despite in the following we adopt well-known node centrality measures only, any function defined on the node set $\mathcal{V}$ or alternative node-wise measures that allow to quantify the role and/or position of each node in the network can be used. This allows our procedure to be more flexible than other methods based on optimisation of a given node (or edge) function. While, in general, the above mentioned measures do not sum up to one, as it would be required by a density function, they can be easily normalised to this purpose, but for the subsequent developments this is not strictly necessary. #### 2.2.2 Clustering of unweighted networks Consider a binary network $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, where $\mathcal{E}=\\{e_{ij}\\}$ and $e_{ij}\in\\{0,1\\}.$ To perform clustering, we select a node-wise measure of density $\delta:\mathcal{V}\mapsto\mathrm{R}^{+}\cup\\{0\\}$ as discussed in the previous section. Afterwards, we may proceed to cluster the nodes according to the modal formulation illustrated in Section 2.1, i.e. actors are clustered together when they have density above the examined threshold and they are connected. With respect to the nonrelational framework above, we further benefit of the fact that the connected components of the high-density level sets may be identified as the connected components of the induced subgraphs, namely the maximal set of nodes such that each pair of nodes is connected by a path. An operational route is a represented by the following scheme: 1. 1. Compute the density of the relationships of each actor: $\delta(v_{1}),\dots,\delta(v_{i}),$ $\dots,\delta(v_{n})$. Clusters will be formed around the modal actors, namely actors with the densest relationship patterns. 2. 2. For $0<\lambda<\max_{i}\delta(v_{i}):$ * • Determine the upper level set $\mathcal{V}_{\lambda}=\\{v_{i}\in\mathcal{V}:\delta(v_{i})\geq\lambda\\},$ * • Build the subgraph $\mathcal{G}_{\lambda}=(\mathcal{V}_{\lambda},\mathcal{E}_{\lambda})\subset\mathcal{G}$ where $\mathcal{E}_{\lambda}=\\{e_{ij}(\lambda)\\}$ and $e_{ij}(\lambda)=\left\\{\begin{array}[]{ll}e_{ij}&\mbox{if }(v_{i},v_{j}\in\mathcal{V}_{\lambda})\\\ 0&\mbox{otherwise}\end{array}\right.$ * • Find the connected components of $\mathcal{G}_{\lambda}$. 3. 3. Build the cluster tree by associating each level $\lambda$ to the number of connected components of $\mathcal{G}_{\lambda}$. 4. 4. Identify all the lowest $\lambda$ for which the branches of the tree represent the leaves, and form the cluster cores as the connected components of the different associated $\mathcal{G}_{\lambda}$. Essentially, at each threshold $\lambda$ we evaluate the connected components of $\mathcal{G}_{\lambda}$, the subgraph formed by the nodes with density above $\lambda$ and the only connections between them. The scheme usually leaves unallocated a number of actors with low-density patterns, when they do not univocally pertain to a modal actor. Depending on the aim of clustering and on subjects-matter considerations, part, or all of them may be either left unallocated or assigned to the cluster for which they present the highest density $\delta(\cdot)$. The described way of proceeding entails the early identification of clusters as formed by actors with the highest density, i.e. the leaders of the community, and the subsequent aggregation to the formed clusters of actors with less prominent role. In this sense, and consistently with the non- relational version of modal clustering, the final clusters are then described by the domains of attraction of the community leaders. #### 2.2.3 Clustering of weighted networks Let us now consider a weighted network $\mathcal{G}=\\{\mathcal{V},\mathcal{E}\\}$, where $\mathcal{E}=\\{e_{ij}\\}$ and $e_{ij}\in\mathbb{R}^{+}\cup\\{0\\}$, _i.e._ the link weight is proportional to the strength of the relationship between the two incident nodes and it is set to zero when the two nodes are not linked. As a first natural ploy to account for real-valued edges, we consider density measures for weighted networks. Indeed, the generalisation of these measures to weighted networks has been historically a somewhat controversial matter which cannot be tackled without considering the nature of the data, the goal of the analysis, and subject-matter knowledge. However, for most of the mentioned candidate measures $\delta$, there exist a reasonable weighted counterpart. The degree, for instance, is easily extended to measure centrality in weighted networks by summing up the weights incident with each node. This allows considering prominent an actor not only when he has many connections, but also when the strength of these connection is large. We refer the reader to the existing literature for a discussion about the specification of descriptive measures for weighted networks (Opsahl et al., 2010). In the presence of relationships of different strengths, we need to further adjust the presented procedure. Indeed, a possible weak connection between two high density actors does not appear as a sufficient condition for them to be clustered. Thus, we account for the weights on the basis of the following simple idea: two actors are clustered together when they have density above the examined threshold and they are _strongly_ connected. Actors presenting a weak relationship with their neighbours are merged into the same cluster at a lower level of density. Here, the strength of the connection is intended as relative to the set of connections of each node. While this is consistent with the natural idea that prominent actors exercise more influence over their strong connections and less influence over their weak connections, its implementation may take various forms. The following scheme provides two options of possible operational routes: 1. 1. Compute the density of each actor, $\delta(v_{1}),\dots,\delta(v_{i}),$ $\dots,\delta(v_{n})$, with $\delta$ an appropriate measure of node-wise density accounting for the weights of the edges; 2. 2. For each node $v_{i},i=1,\ldots,n$, identify the incident edge with maximum weight $e_{im}=\underset{j:\mbox{ }e_{ij}\in\mathcal{E}}{\max}{e_{ij}};$ 3. 3. For $0<\lambda<\max_{i}\delta(v_{i}):$ * • Determine the upper level set $\mathcal{V}(\lambda)=\\{v_{i}\in\mathcal{V}:\delta(v_{i})\geq\lambda\\}$ * • Build the subgraph $\mathcal{G}_{\lambda}=\\{\mathcal{V}_{\lambda},\mathcal{E}_{\lambda}\\}$, where $\mathcal{E}_{\lambda}=\\{e_{ij}(\lambda)\\}$ and $e_{ij}(\lambda)$ can be defined according to the two alternative options, denoted by ‘AND’ and ‘OR’ respectively. option AND $e_{ij}(\lambda)=\left\\{\begin{array}[]{ll}e_{ij}&\mbox{if }(v_{i},v_{j}\in\mathcal{V}_{\lambda})\cap((e_{im}=e_{ij})\cap(e_{jm}=e_{ij}))\\\ 0&\mbox{otherwise}\end{array}\right.$ option OR $e_{ij}(\lambda)=\left\\{\begin{array}[]{ll}e_{ij}&\mbox{if }(v_{i},v_{j}\in\mathcal{V}_{\lambda})\cap((e_{im}=e_{ij})\cup(e_{jm}=e_{ij}))\\\ 0&\mbox{otherwise}\end{array}\right.$ * • find the connected components of $\mathcal{G}_{\lambda}$ * • update $e_{im}=\underset{j:\mbox{ }e_{ij}\in\mathcal{E}\setminus\mathcal{E}_{\lambda}}{\max}{e_{i,j}};$ 4. 4. Find the connected components of $\mathcal{G}_{\lambda}$. 5. 5. Build the cluster tree by associating each level $\lambda$ to the number of connected components of $\mathcal{G}_{\lambda}$. 6. 6. Identify all the lowest $\lambda$ for which the branches of the tree represent the leaves, and form the cluster cores as connected components of the different associated $\mathcal{G}_{\lambda}$. Essentially, at each $\lambda$, we identify the connected components of $\mathcal{G}_{\lambda}$ which are formed by the nodes with density above $\lambda$. According to “option AND” the additional condition for aggregation is that these nodes represent their reciprocal strongest connection among those not examined yet; conversely, according to “option OR” the condition is loosen by requiring that such connection is the strongest for just one of the actors. The two options, albeit not exhaustive, correspond to different ways of disentangling network complexity and defining the underlying network group structure. With the tight AND option, aggregation is harder to occur, hence leading to a large number of highly homogeneous clusters. The resulting partition is mostly driven by the importance of the relations among nodes rather than by their relative importance within the whole network. According to the “OR option”, where the aggregation condition is more frequently satisfied, more parsimonious partitions are created, with clusters mostly driven by the attraction hold by the high density nodes, namely the leaders, on the lower-density ones. Note that this way of proceeding does not guarantee that all the weights are scanned while scanning the density values, i.e. at the lowest considered $\lambda$, the weakest connections between some pairs of actors might not be accounted for. Since in practice these connections are negligible as, by construction, the weakest ones, we simply circumvent this problem by identifying, at the end of the density scanning, the connected components of the network disregarding the weights of the connections. The clustering procedure eventually entails the formation of singleton clusters: suppose that three connected nodes $u,v,$ and $z$ have all density above a given $\lambda$, but while the strongest relationship of $u$ is with $v$, the strongest relationship of $v$ is with $z$ and _viceversa_. Then, with the AND option, $v$ and $z$ will fall in the same cluster while $u$ will be a singleton cluster which will be aggregated to the other at a lower $\lambda$. Unallocated actors are finally classified to the cluster core at which they present highest density, like in the unweighted setting. ## 3 Empirical analysis ### 3.1 Aims and implementation details The current section aims to illustrate the aggregation mechanism at the basis of the proposed method for different community configurations, also with respect to the selected node-wise measure. We consider as density measures three alternative indexes of centrality designed to catch different roles and community configurations within a network: degree centrality evaluates the actor importance in terms of number of relationships with other members of the community; betwenness centrality, on the other hand, by counting the number of times actor work as bridges to connect other members, evaluates their strategic role in terms of brokerage; finally, local density, by shifting the focus from single actors to their nearest neighbourhood (i.e., nodes at geodesic distance equal to one from the focal actor), relaxes the focus on centralized groups and identifies shared leaderships. The considered measures have been consistently adjusted for their use in weighted networks. With the aim of comparison, the considered examples are run also by considering a few competing community detection methods, mostly selected because of their wide popularity: the Girvan Newman (GN) method and its extension to weighted networks, the Louvain method and Stochastic Block Models (SBM). All the analyses are run in the R computing environment R Core Team (2020) with the aid of libraries igraph (Csardi and Nepusz, 2006), sna(Butts, 2020), sbm(Chiquet et al., 2020). The proposed method has been implemented within the DeCoDe package (Density-based Community Detection), available on the author webpage111https://homes.stat.unipd.it/giovannamenardi/content/software. ### 3.2 A simple illustrative example For the sake of illustration, we consider as a first example of our empirical analysis some unweighted archetypal networks where the community structure is determined by the presence of high-density nodes. The simple network displayed in the top row of Figure 2 highlights 4 hubs standing out among 28 actors, labelled as $5,8,15$, and $22$. Each of the four hubs drives the information flow from and towards six actors having a less prominent role. Density-based clustering built on degree centrality reflects the hub dominance by identifying 4 clusters headed by the leaders (Figure 2 a1). In fact, if the leaders were connected - middle panel of the Figure, where a tie links actors $1$ and $8$ \- the clustering configuration would change accordingly, and a single group would be formed by all the followers of the leader dyad (Figure 2 b1). In the lack of hubs - bottom row of Figure 2, where actors $5,8,15,22$ have been removed from the network - density-based clustering built on the degree fails to identify groups (Figure 2 c1), which are better identified by alternative node-wise measures accounting for a decentralized leadership. By considering, for example, the local density based on the nearest neighborhood, modal clustering detects four clusters in all the three versions of network (second column of Figure 2). Conversely, if the the analysis focuses on the strategic role of the actors, the leadership is rather drawn by actors $12$ and $25$, acting as brokers which connect nodes otherwise disconnected in the network. With this changed aim in mind, a different structure characterizes the network, as the whole community is compact around the leaders. Consistently, density-based clustering built on betweenness detects just one cluster in all the three versions of network, leaded by the connected brokers (third column of Figure 2). The cluster trees provide further information by identifying the hierarchy of the communities. Thus, in the four clusters configurations the more central communities aggregate first, whereas in the three clusters configuration the first merge occurs between the largest cluster and the closest one (bottom panel of Figure 2). Compared to density-based clustering, GN and the Louvain method, mostly driven by the idea of modularity within a community, identify 4 clusters in all variants of the network, thus behaving like modal clustering with local density (Figure 2, fourth and fifth columns). SBM identifies an optimal partition in two clusters in the presence of hubs, and one cluster only in the absence of hubs (Figure 2 a6, b6, and c6 respectively). Figure 2: At each row a slightly changed version of the same toy network: in the first one four hubs not linked directly; in the second row two of them are connected by a link; in the third row the hubs have been removed. At each column the clustering produced by the density-based procedure built on different measures, GN, Louvain, and SBM. Clusters are marked with different colors. In the first three columns actor size is proportional to their density. In the bottom panel, the cluster trees associated with the density- based partitions into 4 and 3 clusters. ### 3.3 Benchmark examples As a second step of the empirical analysis, we explore the behaviour of our method in some popular real datasets where a ground truth community membership is assigned. The choice of evaluating results in term of a true labeling, rather common in clustering, is motivated by our will of not being biased towards specific community configurations. On the other hand, it is worth highlighting that the possible identification of community structures diverse from the defined true labels would not necessarily imply a failure of the applied clustering method. Such possible result would just reflect that the true clusters have a configuration different from the one that each method is designed to detect. The agreement between the true and the detected membership has been measured in terms of normalised mutual information (NMI, Danon et al., 2005) which increases for improved quality and associates the maximum value 1 to a perfect agreement. ##### Zachary Karate Club network The well-known Karate Club data (Zachary, 1977) describes the network of friendships between 34 members of a karate club at a US university in the 1970s. The network is in principle weighted, with the strength of connections given by the number of common activities of the club members. In fact, we run the empirical analysis on both the weighted network and on its binary version, built by neglecting the strength of connections. Due to a dispute between the administrator ‘John A’ and the instructor ‘Mr Hi’, the club split into two factions, here representing the benchmark membership. The two factions are then built around the leadership of John A and Mr Hi, which play a special role in terms of both direct influence on the club members, and influence on the information flow to and from the actors. binary network --- G-N \| Louvain \| SBM \| Density-based clustering \| \| \| degree \| loc. density \| betw. \| \| \| 0.58 \| 0.59 \| 0.01 \| 1 \| 0.36 \| 1 \| \| \| weighted network G-N \| Louvain \| SBM \| Density-based clustering \| \| \| OR option \| AND option \| \| \| degree \| loc. density \| betw. \| degree \| loc. density \| betw. 0.56 \| 0.69 \| 0.01 \| 1 \| 0.44 \| 0.85 \| 0.61 \| 0.36 \| 0.42 Figure 3: Zachary Karate Club network with true communities marked with different colours. Below, NMI results of different community detection methods. In agreement with these considerations, in the binary setting, an essentially perfect agreement is found between the two factions and the density-based partition detected with both degree and betweenness as node-wise measures. Conversely, since local density accounts for the maximum number of ties each actor can set in its neighbourhood, it results in depowering the leaders of star-based community, thus proving not to be adequate as a node-wise measure to recover the true factions. A slightly better performance arises from the application of both Louvain and GN methods, while SBM cannot reconstruct the benchmark factions. Note, however, that none of the competitors is designed to detect hub headed communities. See Figure 3. When considering the weighted network with the OR option, the two factions are again perfectly recovered with the degree used as a node-wise measure. Betwenness overall makes a remarkable job as well, althought it identifies three community instead of two. Accounting for the link weights, in fact, allows to distinguish a new leader beyond Mr Hi and John A, namely actor 32, with a high prominent bridging role. The inadequacy of local density to find clusters arising from a leadership is confirmed also in the weighted setting. The AND option gives rise, by construction, to a larger number of homogeneous clusters, with the highest density ones still led by John A and Mr Hi. For this reason the NMI stands at decreased values. Disregarding the employed density, in general, the presence of more peripheral actors is enhanced, as with the AND option individual connections are accounted for in clustering formation, rather than the leader influence. In fact, despite the true cluster membership, driven by a forced choice of each actor to line up with one of the leaders, data show that the relationships among the peripheral actors are generally stronger than the ones they have with the leaders. The Louvain method produces improved results with respect to the binary case, while SBM and GN stand at about the same level than in the binary counterpart. ##### Les Misérables character network binary network --- G-N \| Louvain \| SBM \| Density-based clustering \| \| \| degree \| loc. density \| betw. \| \| \| 0.76 \| 0.63 \| 0.54 \| 0 \| 0.76 \| 0 \| \| \| weighted network G-N \| Louvain \| SBM \| Density-based clustering \| \| \| OR option \| AND option \| \| \| degree \| loc. density \| betw. \| degree \| loc. density \| betw. 0.35 \| 0.63 \| 0.59 \| 0.48 \| 0.43 \| 0.61 \| 0.76 \| 0.78 \| 0.78 Figure 4: Les Misérables character network. Cf. Figure 3. This popular network describes the interactions between 77 characters of the Victor Hugo’s novel Les Misérables (Knuth, 1993). The network is in principle weighted, with edge strength set to the number of co-appearance of characters in one or more scenes of the novel. Like in the previous example, we also analyse its binary version. With the aim of an objective evaluation, we pursue the assignment of a ground truth membership by associating each character to the book of his/her early appearance. This eventually results in 20 small communities having an assorted attachment mechanism, with some communities formed around more relevant characters and other more cohesive communities (Figure 4). The partition provides an overall fair summary of the novel plot, yet we shall account with some limitations. Beyond three ambiguous references to unnamed actors, the cluster membership of a few main characters should rather have overlapping nature. Hence, results evaluation requires some further insights beyond the mere observation of the NMI values. Neglecting the strenght of relationships, has little impact on the minor characters, generally claiming a small number of weak interactions. Thus, in the binary version of the network, cohesive communities are anyway easily detected, whereby GN, Louvain and modal clustering based on local density stand out from the other methods at high values of accuracy. Conversely, the loss of information on the weights affects the classification of the main characters, all being connected to each other, yet with a different extent which pinpoints their role. Hence, modal clustering with degree and betweenness identifies in the binary network just one community, built around the protagonist Jean Valjean. When the weight strength is accounted for, modal clustering works remarkably with the AND option, which tends to inflate the segmentation and highlights minor groups. The OR option underperforms the AND version compared to the true labels, yet results are anyway highly interpretable. The use of both degree and betwenness gives rise to 6 clusters. In the former case most communities are built around one main character, whereas when betwenness centrality kicks in, being its values larger for those characters having a protagonist role in multiple books, modal clustering is able to isolate all the main characters in just one group (that is the “main plot” cluster) together with other 5 smaller sized groups of actors whose story is standalone within the whole plot. ##### US politics books co-purchasing network G-N \| Louvain \| SBM \| Density-based clustering ---\|---\|---\|--- \| \| \| degree \| loc. density \| betw. 0.56 \| 0.51 \| 0.45 \| 0.60 \| 0.31 \| 0.07 Figure 5: US politics books co-purchasing network. Cf. Figure 3. As a further example of star-shaped communities in networks, the US politics books co-purchasing data222http://www.orgnet.com/ include 105 books about US politics published around the presidential election in 2004 and sold online at Amazon.com. The 441 ties between them represent co-purchasing of books by the same buyers. Community membership is given by the book political alignment: liberal, neutral, or conservative. Within communities there exists a slighlty centralized organization of links, especially among liberal and conservative thinkings, with bestsellers representing high-density nodes, often bought in bundle with a variety of less popular other books. Results (Figure 5) reflect such behaviour, as the density-based partition built on degree centrality outperforms both the other centrality measures and the competitors. While the latter tend to oversegment the network yet achieving acceptable results, the former result not appropriate to describe the community configuration. Without exception, methods are not able to identify the least characterized neutral books. ##### Email-EU-core network G-N \| Louvain \| SBM \| Density-based clustering ---\|---\|---\|--- \| \| \| degree \| loc. density \| betw. 0.56 \| 0.53 \| - \| 0.26 \| 0.58 \| 0.26 Figure 6: Email-EU-core network network. Cf. Figure 3. The Email-EU-core network (Leskovec et al., 2007; Yin et al., 2017) describes the email exchange between the members of 42 departments of an European research institution. The network is regarded to as undirected by setting an edge whenever there has been at least one either outgoing or incoming email between two members. True clusters are the Departments of affiliation. Distribution of actors among Departments is rather unbalanced, ranging from 1 to 107 individuals. Since the network includes a few isolated nodes, we focus on the giant component only, consisting of 986 individuals (98% of the total) connected by 25552 ties (99.9%). The network is far more complex than the ones examined above. While of difficult inspection, Figure 6 shows that the community configuration is hardly caught by the link description. Research collaborations, indeed, are possibly conducted also by email, and likely not to be limited to the members of a Department. Additionally, there is little evidence of some attachment mechanism guided by the presence of prominent individuals in terms of their degree; also due to the unavailability of weights, conversely, it is likely to expect quite a homogeneous distribution of links within each Department and possible clusters not built around some leaders. Results confirm the expectations, as local density is the only centrality measure able to catch the gross community structure via density-based clustering. GN and Lovain method stand on about the same level of accuracy of classification. The application of SBM is computationally unfeasible on this network, due to an inner limitation of the R routines included in package sbm, which requires the joint estimation of models for any number of communities and the subsequent selection of the best models. Hence, networks with a large number of clusters, like in this case, run into a memory error. ##### American college football network The American college football network, described by Girvan and Newman (2002), represents the schedule of Division I American football games for the 2000 season. Nodes represent teams and ties between two teams represent regular- season games they dispute. The 115 teams are divided into 12 conferences, representing the benchmark community memberships. In most conferences, inner games are more frequent than games with external teams, with an average of about seven intraconference games and four interconference games in the reference season (Girvan and Newman, 2002, p.7824). The example is here explored to show the inadequacy of density-based community detection in the lack of leadership. Games configuration, indeed, leads to a grid-like organization of links within communities. In this situation the competing methods are able to recover accurately the community structure, while our proposal fails by setting either degree or betwenness as node-wise measure. Local density in this case, allows just for a slight improvement. See Figure 7 for details. GN \| Louvain \| SBM \| Density-based clustering ---\|---\|---\|--- \| \| \| degree \| loc.dens. \| betw. 0.88 \| 0.89 \| 0.89 \| 0.33 \| 0.55 \| 0.13 Figure 7: American college football network. Cf. Figure 3. ### 3.4 Finding clusters within the community of Italian academic statisticians The aim of the case study here considered is to characterise the scientific community of the Italian academic scholars in Statistics and related fields, via the identification of the clusters formed on the basis of the relationships between them, possibly of different nature and strength, and of the leading aggregation mechanism. This can be useful, for instance, for the creation of new projects and synergies, or more generally, to understand who are, within the community, the leading actors with respect to specific topics. The main hypothesis underlying the data collection is that, to characterise a researcher within the community, we broadly answer to the questions: _Where does he/she work? What is his/her macro-area of research? Who does he/she work with? What does he/she work on?_ As a consequence, we have built a weighted network having in principle a multiplex structure, divided in four layers associated with the questions above: (1) affiliation adjacency matrix (AFF): two actors are connected when they share the same university department affiliation; (2) macro-sector adjacency matrix (MS): two actors are connected when they belong to the same macro-sector, within the area of Statistics and related fields, and as defined by the Italian Ministry of Education, Universities, and Research MIUR (statistics, economic statistics, demography and social statistics, mathematical methods for economy, actuarial and financial sciences) (3) co-authorship network (PUBS): two actors are connected with a link weighted as the number of publications they co-authored; (4) common keywords adjacency matrix (KW): two actors are connected with a link weighted as the number of common keywords in their publications. Data have been collected in November 2019 and refer to 1160 among professors and researchers of the academic community of statisticians, as recorded by the MIUR database333http://cercauniversita.cineca.it. where information about the university affiliation and the scientific macro-sector have been drawn. Information about the publications and the keywords have been extracted from the ISI-WoS database444https://apps.webofknowledge.com . Handling the latter one has been troublesome, due to an awkward operation of author matching, especially in the case of homonymy or when a researcher has changed his affiliation at some time and the WoS database does not recognise it. In fact, we shall live with the likely, hopefully not relevant, distortion in the assessment of both the publications and the inherent keywords. A summarising description of the single layers is provided in Table 1. All networks at the individual layers are composed of the 1160 nodes representing the members of the scientific community under study. Given the exclusivity of the affiliation, the associated network is composed by as many components as the number of observed University departments (namely, 194), within which every actor is connected with all the other actors. The number of researchers within departments is pretty heterogeneous, ranging from 1 to 54. A similar behaviour is observed in the network associated to the macro-sector, where each actor is connected with all other researchers in the same macro-sector. The number of connected components in this layer is equal to the number of considered macro-sectors and these components have diverse sizes (both the statistics and mathematical methods for economy, actuarial and financial sciences areas count more that 400 researchers, while each of the two further sectors count about 150 researchers). The co-authorship layer represents an updated, enriched version of one of the databases employed by De Stefano et al. (2013). We observe 255 isolated researchers, either because they have not published on ISI journals, or because their publications have never been co- authored by any other Italian academic statistician currently on the MIUR list. Their publications have been in any case considered to extract the keywords for the fourth layer of the network, where the number of isolated researchers reduces to 109. Table 1: Italian academic statisticians network: descriptive statistics for the individual layer networks (AFF - Department affiliation, MS - Macro-sector, CA - Coauthorship network, KW - common keywords) and overall weighted networks. \| AFF \| MS \| PUBS \| KW \| Overall ---\|---\|---\|---\|---\|--- # of isolated nodes \| 67 \| 0 \| 255 \| 109 \| 0 # of components \| 194 \| 4 \| 292 \| 110 \| 1 Network Density \| 0.014 \| 0.308 \| 0.002 \| 0.523 \| 0.734 Global transitivity \| 1 \| 1 \| 0.306 \| 0.820 \| 0.833 Degree centralization \| 0.032 \| 0.082 \| 0.016 \| 0.346 \| 0.240 In order to aggregate the four layers into a single, weighted network, we have first normalised the edge weights, measured on different scales, depending on the represented relationship. In principle, there are many procedures to choose among for the purpose.We opt for the simple idea of dividing each weight by the sum of weights within the layer. Then, stemming from the four normalised networks, we have built the associated _overlapping_ network (Battiston et al., 2014) by simply summing up the edge weights associated to the same actor across different layers. The overall network is relatively dense and cohesive with no isolates since all nodes are comprised in the unique component (see Table 1). The strength of the links in the overall network is largely governed by the sparsest co-authorship layer because of the used weighting system. Among the community detection methods, we have been able to run the Louvain method only, whereas the application of both GN and SBM has turned out computationally unfeasible. It is worth reporting, however, that we run SBM up to the maximum number of communities allowed by our computational resources, i.e. 64. The detected partition is unarguably suboptimal, with one community gathering the $2/3$ of the actors but the remaining 63 clusters do not differ much from the ones obtained with our procedure. The Louvain algorithm identifies $23$ communities, of size ranging from 13 to 102 researchers. Cluster homogeneity with respect to the scientific macro-sector and the affiliation has been evaluated via the complement to one of the Gini index. As for the publications, for each researcher, the proportion of works coauthored by members of the same cluster has been evaluated and the cluster average used as a summarising measure of cluster homogeneity. The same index has been computed for the keywords. To look at the assortative mixing within the detected communities and find if actors within clusters tend to exhibit dense connections among them rather than with actors in different clusters, modularity of the clusters has been also evaluated. Results are summarized in Figure 8. Due to the large size of the detected clusters, clusters are somewhat homogeneous for the scientific sector and the affiliation only, while communities are scarcely associated to co-authorship and research topics. While, by construction, modularity of the Louvain-based partition is maximised, it does not show a remarkably high value. To this respect, it is worth noting that even if the detected partition corresponds to the global maximum of the modularity, in scientific applications this solution is not guaranteed to be more meaningful than the ones obtained by local maxima (Good et al., 2010). Furthermore, results are affected by the so-called resolution limit for which small, plausible communities cannot be identified if the network is large and heterogeneous clusters tend to be formed (Fortunato and Barthélemy, 2007). Modal clustering has been run building on the degree of the actors, as it appears the most sensible and easiest to interpret choice in such a complex application. Both the options “OR” and “AND” have been run. Summarising results in terms of size of clusters, modularity, and homogeneity with respect to the considered relationships are reported in Figure 8. The OR option identifies 139 groups of size ranging from 4 to 49 scholars. Some heterogeneity with respect to the considered relationships is unavoidable, but clusters are far more homogeneous than those identified by the Louvain method. In fact, while actors working either together or on similar topics tend to be aggregated into the same cluster, the same membership is often shared by other researchers. In this case, cluster aggregation is mostly driven by the attraction hold by a few leaders towards minor actors which often exhibit pretty diverse characteristics. Conversely, option AND gives rise to a very sensible partition, counting 499 clusters, overall a realistic value in the overview of the statistical community where, excluded applied interdisciplinary scholars, researchers tend to work within very small-sizes teams, and publications are generally co-authored by 2/3 researchers at most. The majority of clusters is fully homogeneous with respect to the scientific macro-sector and affiliation, and gathers researchers who are known to belong to the same research group. Note that this result is largely acknowledged to occur in social contexts, where social groups are limited in size even if social actors are embedded in relatively large networks (Dunbar, 1992). Louvain \| Density-based clustering ---\|--- \| \| Option OR \| Option AND size \| 45.69 (22.90) \| size \| 7.56 (7.97) \| size \| 2.11 (1.42) modularity \| 0.42 \| modularity \| 0.24 \| modularity \| 0.18 \| \| Figure 8: Italian community of statisticians. Top panel: average size of clusters (and standard deviation) found via the Louvain method and both the options OR/AND of the density-based method and modularity. The boxplots display the homogeneity of actors across clusters with respect to the considered relationships. Figure 9: Cluster tree of the Italian academic statisticians (rotated for better readability, with high density levels at the right side). An insight of the highlighted area is provided in Figure 10. However, the clustering is not to be interpreted solely in term of final group membership \- a not that different partition could be trivially obtained by aggregating pairs of maximally connected actors. In fact, the generation process of collaboration among researchers is pretty peculiar: there may be solitary researchers, sparsely collaborating with other subjects; also, there are researchers who mostly focus on a specific research topic, but also collaborate with different groups of people on a variety of different areas. Both keeping these researchers separated or merging them into the same group may be a stretch. To this aim, a relevant interpretation derives from the exploration of the cluster tree, where clusters are subsequently aggregated at lower levels of the hierarchy, to form larger clusters with a lower resolution (Figure 9). For the sake of interpretation, one of its branches, including 17 clusters and a total of 30 researchers, is detailed in Figure 10, along with the associated subnetwork highlighting cluster aggregations at the different levels of the cluster tree. The forming leaves of the branch mostly include either researchers affiliated to the Department of Statistical Science at University of Padova, or scholars who have spent at that Department part of their academic career. Actor aggregation in clusters mostly relies on the strength of connections, hence lead by co-authorship which weights most on the overlapping network. At a lower level of the tree, cluster merging is driven by research topics, with the largest branch on the left associated to likelihood theory, and the other branches including scholars working on its more applied declinations. A link between branches derives from the eclecticism of some of the researchers, working on different research topics. The size of the tree prevents an overall interpretation, but similar traits of homogeneity can be easily identified by picking any branch of the tree. Of course, the lower the level of aggregation of the branches, the lower the homogeneity of the branch. Figure 10: Detailed visualization of the subtree highlighted in Figure 9 and associated subnetwork with clusters marked in different colors and superimposed the cluster aggregations at the different levels of the cluster tree. Actor size is proportional to the their density and different shapes are associated to different macro-sectors. Actor colour is associated to the affiliation. Edge width is proportional to the number of common keywords and coauthored publications. ## 4 Discussion Due to the unsupervised nature of the problem, and to the further lack of a ground truth against which to evaluate the quality of a partition, clustering is an ill-posed task, which cannot be performed fully automatically, i.e. without some amount of human intervention and disregarding subject-matter considerations. The methodology here presented makes no exception in the clustering panorama, as it both has required during its planning and still requires the user to make a few thorny choices. A first choice concerns the density measure. The lack of a probabilistic notion of density at a node-wise level implies the loss, for network data, of the probabilistic framework of the original approach defined for non-relational data. Hence, the proposed procedure cannot enjoy the mathematical rigour of other well known stochastic procedures. On the other hand, it follows the opportunity of selecting the measure of density among a wide set of candidates which quantify connectivity or centrality roles of the actors. Different group structures arise according to the chosen density measure and those structures account for different aspects of subnetworks cohesiveness. In fact, we believe that leaving unspecified this measure represents a strength of the procedure. Depending on subject-matter considerations, this provides the procedure with the flexibility of adapting to different notions of clusters, each of them associated with a specific selection of the density and consistently with the intrinsic ill-posedness of the clustering problem. A second choice concerns the way to handle relationships of different strength. Unlike the unweighted framework, there is no obvious way to extend modal clustering in the presence of weighted links. Our strategy aggregates strongly connected individuals at a higher density level than individuals which are weakly connected. While this choice is consistent with the considered aggregation mechanism, based on the most prominent actors exerting influence over their neighbours, the actual implementation of this idea may take various forms. The AND option aggregates two actors with density above a threshold, when they represent their reciprocal strongest connection among those not examined yet. Alternatively, the condition may be loosen via the OR option, by requiring that such connection is the strongest for just one of the actors. A further alternative route would consist in proceeding in a block- sequential manner, aggregating several actors with density above the threshold at a time, as long as their relationship has, at least, a given strenght. The possibility of choosing among these options in cluster formation allows for looking at a given network structure from a different granularity of the representation. As showed in the proposed applications, the OR mechanism tends to minimize the network partition in a smaller number of internal densely connected clusters with loose connections with other clusters. On the other hand, the AND mechanism maximizes the internal homogeneity of clusters detecting a larger number of smaller groups. Here again the choice of the mechanism to handle weights depends on the purpose of the analysis. For instance in the Karate network, the choice of the OR option would reflect an interest in the big picture after collapsing the relations in the community. Conversely, in the Italian statisticians network, the choice of the AND option would entail small scale groups of actors and reflect the purpose of looking for cohesive research clusters. Although featuring these different options of analysis, the proposed density- based procedure does not suffer from the arbitrariness matters which are typical of standard clustering procedures. While the number of clusters is determined within the procedure, the partitioning accounts for different levels of cluster resolution, via the group hierarchy provided by the cluster tree. In this sense, the cluster tree represents a somewhat formal instrument to emulate the human cognitive system and allows for getting over the resolution limit of modularity-based methods. 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Proof of a Conjecture on the Wiener Index of Eulerian Graphs Peter Dankelmann111Financial support by the South African National Research Foundation, grant 118521, is gratefully acknowledged. (University of Johannesburg) ###### Abstract The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. A connected graph is Eulerian if its vertex degrees are all even. In [Gutman, Cruz, Rada, Wiener index of Eulerian Graphs, Discrete Applied Mathematics 132 (2014), 247-250] the authors proved that the cycle is the unique graph maximising the Wiener index among all Eulerian graphs of given order. They also conjectured that for Eulerian graphs of order $n\geq 26$ the graph consisting of a cycle on $n-2$ vertices and a triangle that share a vertex is the unique Eulerian graph with second largest Wiener index. The conjecture is known to hold for all $n\leq 25$ with exception of six values. In this paper we prove the conjecture. Keywords: Wiener index; average distance; mean distance; total distance; Eulerian graph; degree MSC-class: 05C12 (primary) 92E10 (secondary) ## 1 Introduction Let $G=(V,E)$ be a finite, connected graph. The Wiener index of $G$ is defined by $W(G)=\sum_{\\{u,v\\}\subseteq V}d_{G}(u,v),$ where $d_{G}(u,v)$ denotes the usual distance between vertices $u$ and $v$ of $G$, i.e., the minimum number of edges on a $(u,v)$\- path in $G$. The Wiener index, originally conceived by the chemist Wiener [25], has been investigated extensively in the mathematical and chemical literature, often under different names, such as transmission, distance, total distance or gross status. Several of these results were originally obtained for the closely related average distance, also called mean distance, which is defined as $\binom{n}{2}^{-1}W(G)$, where $n$ is the order of the graph $G$. For some of its chemical applications see, for example, [21]. One of the most basic results on the Wiener index states that $W(G)\leq\binom{n+1}{3}$ for every connected graph on $n$ vertices, and equality holds if and only if $G$ is a path. A path has only vertices of degree one and two, so it is reasonable to expect that better bounds can be obtained if restrictions are placed on the values of the degrees. Upper bounds on the Wiener index that take into account not only the order, but also the minimum degree were given, for example, in [2, 7, 17], and it was shown in [1] that stronger bounds hold in the presence of a vertex of large degree. The Wiener index in relation to the inverse degree, i.e., the sum of the inverses of all vertex degrees, was considered by Erdös, Pach, and Spencer [11]. Bounds on the Wiener index of trees in terms of vertex degree have also been considered extensively. Every tree has minimum degree $1$, so it is natural to ask how large or small the Wiener index can be in trees of given maximum degree. Answering this question for the maximum value of the Wiener index is fairly straightforward (see [20] and [23]), however the determination of the minimum Wiener index by Fischermann, Hoffmann, Rautenbach, Székely and Volkmann [12] required much more effort. For the more general problem of determining the extremal values of the Wiener index of a tree with given degree sequence see, for example, [6], [22] and [24]. A good survey of results on the Wiener index of trees before 2000 was given in [10]. Not the actual value, but the parity of the degrees has been used to bound the Wiener index. Trees in which all vertices have odd degree were considered by Lin [18], who determined their smallest and largest possible Wiener index of trees. This result was extended in [14] with the determination of all such trees of order $n$ with the largest $\lfloor\frac{n}{4}\rfloor+1$ values of the Wiener index, see also [13]. The smallest and largest Wiener index of a tree whose order and number of vertices of even degree are given, was determined in [19]. The Wiener index of connected graphs in which all vertices have even degrees, that is, Eulerian graphs, was considered by Gutman, Cruz and Rada [15], who obtained the following theorem. ###### Theorem 1 (Gutman, Cruz and Rada [15]). Let $G$ be an Eulerian graph of order $n$. Then $W(G)\leq W(C_{n}),$ where $C_{n}$ is the cycle on $n$ vertices. Equality holds if and only if $G=C_{n}$. The authors of Theorem 1 gave a direct proof of their result. However, since every Eulerian graph is $2$-edge-connected, Theorem 1 can also be obtained as a consequence of Theoremm 3(a) below, which states that the cycle is the unique graph maximising the Wiener index among all $2$-edge-connected graphs of given order. Gutman, Cruz and Rada [15] also presented a conjecture on the question which Eulerian graph of given order has the second largest Wiener index. For $n\geq 5$ let $C_{n,3}$ be the graph of order $n$ obtained from the disjoint union of two cycles on $n-2$ vertices and $3$ vertices, respectively, by identifying two vertices, one from each cycle. Their conjecture states that $C_{n,3}$ is the unique graph that has the second largest Wiener index among all Eulerian graphs of order $n$ for $n\geq 26$. It is the aim of this paper to give a proof of this conjecture. It was verified in [15] that the conjecture holds for all values of $n$ up to $25$ except $n\in\\{7,9\\}$, for which there are other extremal graphs of larger Wiener index than $C_{n,3}$, and $n\in\\{8,10,11,13\\}$, for which $C_{n,3}$ has second largest Wiener index, but there exists another graph of the same Wiener index. All Eulerian graphs of order $7,8,9,10,11,13$ that have the second largest Wiener index are shown in Figure 1. The main result of this paper reads as follows. ###### Theorem 2. Let $G$ be an Eulerian graph of order $n$ with $n\geq 26$ that is not a cycle. Then $W(G)\leq W(C_{n,3})$ Equality holds if and only if $G=C_{n,3}$. Figure 1: The Eulerian graphs of second largest Wiener index for $n=7,8,9,10,11,13$. The determination of the unique Eulerian graph with second largest Wiener index is reminiscent of the corresponding problem for $2$-connected graphs. A cutvertex of a connected graph is a vertex whose removal disconnects the graph. A connected graph with no cutvertex is said to be $2$-connected, and a block of a graph is a maximal subgraph that is $2$-connected. Plesník [20] proved that among all $2$-connected graphs of given order, the cycle is the unique graph maximising the Wiener index. While the proof of this result is relatively straightforward, determining the $2$-connected graph of given order with the second largest Wiener index requires significantly more effort, see the paper by Bessy, Dross, Knor and S̆krekovski [3]. The proof of Theorem 2 given in the present paper suggest that the situation is no different for Eulerian graphs. We note that Plesník’s result on $2$-connected graphs was, asymptotically, extended to $k$-connected graphs in [8]. We note also a certain analogy between our result, and the determination of the largest Wiener index of a connected graph with given order and number of cutvertices in [4] and [5]. If the number of cutvertices is sufficiently small relative to the order, then the extremal graph consists of a path, whose one end is attached to a cycle. So among all graphs with exactly one cutvertex, the extremal graph has two blocks whose order is as unequal as possible. Similarly, the extremal graph $C_{n,3}$ has two blocks, which are as unequal as possible, given the restriction that every block of an Eulerian graph has at least three vertices. The notation we use is as follows. If $G$ is a graph, then $V(G)$ and $E(G)$ denote the vertex set and the edge set, respectively, of $G$. The order $n(G)$ and the size $m(G)$ are the number of vertices and edges, respectively, of $G$. If $G$ and $H$ are graphs with $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$, then we say that $H$ is a subgraph of $G$ and write $H\leq G$. If $A\subseteq V(G)$, then $G[A]$ denotes the subgraph of $G$ induced by $A$, i.e., the graph whose vertex set is $A$, and whose edges are exactly the edges of $G$ joining two vertices of $A$. If $v$ is a vertex of $G$, then $N_{G}(v)$ denotes the neighbourhood of $v$, i.e., the set of all vertices of $G$ adjacent to $v$. For $i\in\mathbb{N}$ we define the $i$-th neighbourhood of $v$, $N_{i}(v)$, to be the set of vertices at distance exactly $i$ from $v$, and we let $n_{i}(v)=\|N_{i}(v)\|$. The degree of $v$ in $G$, i.e., the value $n_{1}(v)$, is denoted by ${\rm deg}_{G}(v)$. A cutset of $G$ is a set $S\subseteq V$ such that $G-S$, the graph obtained from deleting all vertices in $S$ and all edges incident with vertices in $S$ from $G$, is disconnected. An edge-cut of $G$ is a set $E_{1}\subseteq E(G)$ such that $G-E_{1}$, the graph obtained from $G$ by deleting all edges in $E_{1}$, is disconnected. Let $k\in\mathbb{N}$. We say that $G$ is $k$-connected ($k$-edge-connected) if $G$ contains no cutset (no edge-cut) with fewer than $k$ elements. A cutvertex is a vertex $v$ with the property that $\\{v\\}$ is a cutset. An endblock a graph $G$ is a block of $G$ that contains only one cutvertex. It is known that every connected graph that is not $2$-connected has at least two endblocks. If $S$ is a cutset of $G$ and $H$ a component of $G-S$, then we say that $G[V(H)\cup S]$ is a branch of $G$ at $S$. If $S=\\{v\\}$, then we say that $H$ is a branch at $v$. The total distance of a vertex $v$, $\sigma_{G}(v)$, is defined as the sum $\sum_{y\in V(G)}d_{G}(v,y)$. By $\sigma_{G}(A)$ we mean $\sum_{y\in V(G)-A}d_{G}(y,A)$, where $d_{G}(y,A)$ is defined as $\min_{a\in A}d_{G}(y,a)$. The eccentricity $e(v)$ of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$ in $G$. ## 2 Preliminary Results In this section we present definitions and results that will be used in the proof of Theorem 2. We begin with some bounds on the Wiener index and on the total distance of vertices in $2$-connected and $2$-edge-connected graphs. ###### Theorem 3 (Plesník [20]). (a) Let $G$ be a $2$-edge-connected graph of order $n$. Then $W(G)\leq\left\\{\begin{array}[]{cc}\frac{n^{3}}{8}&\textrm{if $n$ is even,}\\\ \frac{n^{3}-n}{8}&\textrm{if $n$ is odd.}\end{array}\right.$ Equality holds if and only if $G$ is a cycle. (b) Let $G$ be a $2$-connected graph of order $n$ and $v$ a vertex of $G$. Then $\sigma_{G}(v)\leq\left\\{\begin{array}[]{cc}\frac{n^{2}}{4}&\textrm{if $n$ is even,}\\\ \frac{n^{2}-1}{4}&\textrm{if $n$ is odd.}\end{array}\right.$ Equality holds if $G$ is a cycle. (c) Let $G$ be a $2$-edge-connected graph of order $n$ and $v$ a vertex of $G$. Then $\sigma_{G}(v)\leq\frac{n(n-1)}{3}.$ ###### Corollary 1. Let $G$ be a $2$-connected graph of order $n$ and $u,w$ two vertices of $G$. Let $u_{1},u_{2}$ be two adjacent vertices of the cycle $C_{n}$. Then $\sigma_{G}(\\{u,w\\})\leq\sigma_{C_{n}}(\\{u_{1},u_{2}\\}).$ Proof: Let $G^{\prime}$ be the $2$-connected graph obtained from $G$ by adding a new vertex $z$ and joining it to $u$ and $w$. Then $\sigma(z,G^{\prime})=\sum_{x\in V(G)}\big{(}1+d_{G}(x,\\{u,w\\})\big{)}=n+\sigma_{G}(\\{u,w\\}).$ Let $C_{n}^{\prime}$ be the graph obtained from $C_{n}$ by adding adding a new vertex $y$ and joining it to two adjacent vertices $u_{1}$ and $u_{2}$ of $C_{n}$. As above, $\sigma(y,C_{n}^{\prime})=\sum_{x\in V(C_{n})}\big{(}1+d_{C_{n}}(x,\\{u_{1},u_{2}\\})\big{)}=n+\sigma_{C_{n}}(\\{u_{1},u_{2}\\}).$ Clearly, removing the edge $u_{1}u_{2}$ from $C_{n}^{\prime}$ does not change $\sigma(y)$. But $C_{n}^{\prime}-u_{1}u_{2}$ is $C_{n+1}$, so by Theorem 3(b), we have $\sigma(z,G^{\prime})\leq\sigma(y,C_{n}^{\prime})$, which implies the statement of the lemma. $\Box$ $C_{8,3}$ $F_{7,4}$ Figure 2: Graphs defined in Definitions 1. ###### Definition 1. (a) Let $n,a\in\mathbb{N}$ with $3\leq a\leq n-2$. Then $C_{n,a}$ denotes the graph of order $n$ obtained from two disjoint cycles $C_{a}$ and $C_{n+1-a}$ by identifying a vertex of $C_{a}$ with a vertex of $C_{n+1-a}$. (b) Let $n,a\in\mathbb{N}$ with $3\leq a\leq n-1$. Then $F_{n,a}$ denotes the graph of order $n$ obtained from two disjoint cycles $C_{a}$ and $C_{n+2-a}$ by choosing two adjacent vertices $u,v$ of $C_{a}$ and two adjacent vertices $u^{\prime},v^{\prime}$ of $C_{n+2-a}$ and identifying $u$ with $u^{\prime}$ and $v$ with $v^{\prime}$. The Wiener index of the graph $C_{n,a}$ was evaluated in [15]. Specifically for $C_{n,3}$ we have $W(C_{n,3})=\left\\{\begin{array}[]{cc}\frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{3}{2}n-2&\textrm{if $n$ is even,}\\\ \frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{11}{8}n-\frac{9}{4}&\textrm{if $n$ is odd.}\end{array}\right.$ (1) In our proofs below we make use of the fact that $\frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{11}{8}n-\frac{9}{4}\leq W(C_{n,3})\leq\frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{3}{2}n-2$ for all $n\in\mathbb{N}$ with $n\geq 5$, irrespective of the parity of $n$. ###### Lemma 1. (Gutman, Cruz, Rada [15] If $n\in\mathbb{N}$ is even, $n\geq 6$, then $W(C_{n,3})>W(C_{n,4})>\ldots>W(C_{n,n/2-1})>W(C_{n,n/2}).$ If $n\in\mathbb{N}$ is odd, $n\geq 11$ and $n=4k+3$ for some $k\in\mathbb{N}$, then $W(C_{n,3})>W(C_{n,4})>\ldots>W(C_{n,2k})>W(C_{n,2k+2})>W(C_{n,2k+1}).$ If $n\in\mathbb{N}$ is odd, $n\geq 11$ and $n=4k+1$ for some $k\in\mathbb{N}$, then $W(C_{n,3})>W(C_{n,4})>\ldots>W(C_{n,2k-2})>W(C_{n,2k})>W(C_{n,2k-1})>W(C_{n,2k+1}).$ For $n=7,9$ we have $W(C_{7,4})>W(C_{7,3})$ and $W(C_{9,4})>W(C_{9,3})>W(C_{9,5})$. ###### Lemma 2. Let $n\geq 26$ and $4\leq a\leq n-2$. Then $W(F_{n,a})\leq W(C_{n,3}),$ with equality only if $a=4$ or $a=n-2$. Proof: A tedious but straightforward calculation yields that $W(F_{n,a})=\left\\{\begin{array}[]{cc}\frac{1}{8}\big{[}a(n-2)(a-n-2)+n(n^{2}+2n-4)\big{]}&\textrm{if $n$ even, $a$ even,}\\\ \frac{1}{8}\big{[}a(n-2)(a-n-2)+n(n^{2}+2n-4)-3n+6\big{]}&\textrm{if $n$ even, $a$ odd,}\\\ \frac{1}{8}\big{[}a(n-2)(a-n-2)+n(n^{2}+2n-4)-n-a+2\big{]}&\textrm{if $n$ odd, $a$ even,}\\\ \frac{1}{8}\big{[}a(n-2)(a-n-2)+n(n^{2}+2n-4)+a-2n\big{]}&\textrm{if $n$ odd, $a$ odd.}\end{array}\right.$ Since $F_{n,a}=F_{n,n+2-a}$, we may assume that $a\leq\lfloor\frac{n+2}{2}\rfloor$. The derivative with respect to $a$ of the four terms on the right hand side above equals $\frac{1}{8}((n-2)(2a-n-2))$ if $n$ is even, $\frac{1}{8}((n-2)(2a-n-2)-1)$ if $n$ is odd and $a$ is even, and $\frac{1}{8}((n-2)(2a-n-2)+1)$ if $n$ is even and $a$ is odd. Hence each of these four terms is strictly decreasing in $a$. It thus follows that $W(F_{n,a})\leq W(F_{n,4})$ if $a$ is even, with equality only if $a=4$, and $W(F_{n,a})\leq W(F_{n,5})$ if $a$ is odd, with equality only if $a=5$. Now an easy calculation shows that $W(F_{n,5})<W(F_{n,4})=W(C_{n,3})$. Hence the lemma follows. $\Box$ ###### Corollary 2. Let $G$ be a graph of order $n\geq 26$ obtained from a cycle $C_{n}$ by adding three edges between vertices of $C_{n}$ that are not in $E(C_{n})$ and that form a triangle. Then $W(G)<W(C_{n,3})$. Proof: Let the three edges added to $C_{n}$ be $uv,vw,wu$. Then for at least one of these three edges, $uv$ say, we have $C_{n}+uv=F_{n,a}$ for some $a$ with $4\leq a\leq\frac{n+1}{2}$. Applying Lemma 1 yields that $W(G)<W(F_{n,a})\leq W(C_{n,3})$. $\Box$ ## 3 Excluding $2$-Connected Counterexamples The goal of this section is to prove that an Eulerian graph of given order that is not a cycle, and which has maximum Wiener index among such graphs, cannot be $2$-connected. Hence it will suffice to prove Theorem 2 for graphs that have a cutvertex. We begin by showing that Theorem 2 holds for graphs that are obtained from two $2$-connected graphs by gluing them together at two vertices, provided one of them contains a spanning cycle. ###### Lemma 3. (a) Let $G$ be a $2$-connected graph of order $n$. If $G$ contains a cutset $\\{u,w\\}$ with the property that the union of some, but not all, branches of $G$ at $\\{u,w\\}$ has exactly $a$ vertices and contains a spanning cycle, and the union of the remaining branches is $2$-connected, then $W(G)\leq W(F_{n,a}).$ Equality implies that $G=F_{n,a}$. (b) If, in addition, $G$ is Eulerian, $n\geq 26$ and $4\leq a\leq n-2$, then $W(G)<W(C_{n,3})$. Proof: Let $A$ be the vertex set of the union of the branches at $\\{u,w\\}$ that contains a spanning cycle, and let $B$ be the vertex set of the union of the remaining branches. Let $a=\|A\|$ and $b=\|B\|$. Then $A\cap B=\\{u,w\\}$. Let $H=G[B]$ and let $C$ be a spanning cycle of $G[A]$. We denote the set of vertices $x$ of $A-\\{u,w\\}$ for which $d_{C}(u,x)<d_{C}(w,x)$ ($d_{C}(u,x)>d_{C}(w,x)$, $d_{C}(u,x)=d_{C}(w,x)$) by $U$ ($W$, $S$). Then $\displaystyle W(G)$ $\displaystyle=$ $\displaystyle W_{G}(A)+W_{G}(B)-d_{G}(u,w)+\sum_{x\in U\cup W\cup S,y\in B-\\{u,w\\}}d_{G}(x,y)$ (2) $\displaystyle\leq$ $\displaystyle W(C)+W(H)-d_{G}(u,w)+\sum_{x\in U,y\in V(H)-\\{u,w\\}}\big{(}d_{C}(x,u)+d_{H}(u,y)\big{)}$ $\displaystyle+\sum_{x\in W,y\in V(H)-\\{u,w\\}}\big{(}d_{C}(x,w)+d_{H}(w,y)\big{)}+\sum_{x\in S,y\in V(H)-\\{u,w\\}}\big{(}d_{C}(x,\\{u,w\\})+d_{H}(\\{u,w\\},y)\big{)}$ $\displaystyle=$ $\displaystyle W(C)+W(H)-d_{G}(u,w)+(b-2)\sigma_{C}(u,U)+\|U\|(\sigma_{H}(u)-d_{H}(u,w))$ $\displaystyle+(b-2)\sigma_{C}(w,W)+\|W\|(\sigma_{H}(w)-d_{H}(u,w))+(b-2)\sigma_{C}(\\{u,w\\},S)$ $\displaystyle+\|S\|\sigma_{H}(\\{u,w\\}).$ Let $C_{a}$ and $C_{b}$ be the cycles of the graph $F_{a,b}$ defined above, and let $u^{\prime}$ and $w^{\prime}$ be the two adjacent vertices of $F_{a,b}$ shared by $C_{a}$ and $C_{b}$. Let $U^{\prime}$ ($W^{\prime}$, $S^{\prime}$) be the set of vertices $x$ of $C_{a}-\\{u,w\\}$ with $d(u^{\prime},x)<d(w^{\prime},x)$ ($d(u^{\prime},x)>d(w^{\prime},x)$, $d(u^{\prime},x)=d(w^{\prime},x)$). As above, we have $\displaystyle W(F_{a,b})$ $\displaystyle=$ $\displaystyle W(C_{a})+W(C_{b})-d_{F_{a,b}}(u^{\prime},w^{\prime})+(b-2)\sigma_{C_{a}}(u^{\prime},U^{\prime})+\|U^{\prime}\|(\sigma_{C_{b}}(u^{\prime})-d_{F_{a,b}}(u^{\prime},w^{\prime}))$ (3) $\displaystyle+(b-2)\sigma_{C_{a}}(w^{\prime},W^{\prime})+\|W^{\prime}\|(\sigma_{C_{b}}(w^{\prime})-d_{F_{a,b}}(u^{\prime},w^{\prime}))+(b-2)\sigma_{C_{a}}(\\{u^{\prime},w^{\prime}\\},S^{\prime})$ $\displaystyle+\|S^{\prime}\|\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\}).$ Since $C$ and $C_{a}$ are cycles, we have $\|U\|=\|W\|$ and $\|U^{\prime}\|=\|W^{\prime}\|$. Subtracting (3) from (2) yields thus $\displaystyle W(F_{a,b})-W(G)$ $\displaystyle\geq$ $\displaystyle\big{(}W(C_{a})-W(C)\big{)}+\big{(}W(C_{b})-W(H)\big{)}+\big{(}d_{G}(u,w)-d_{F_{a,b}}(u^{\prime},w^{\prime})\big{)}$ $\displaystyle+(b-2)\big{[}\sigma_{C_{a}}(u^{\prime},U^{\prime})+\sigma_{C_{a}}(w^{\prime},W^{\prime})+\sigma_{C_{a}}(\\{u^{\prime},w^{\prime}\\},S^{\prime})-\sigma_{C}(u,U)$ $\displaystyle-\sigma_{C}(w,W)-\sigma_{C}(\\{u,w\\},S)\big{]}+\|U^{\prime}\|\big{[}\sigma_{C_{b}}(u^{\prime})+\sigma_{C_{b}}(w^{\prime})-2d_{F_{a,b}}(u^{\prime},w^{\prime})\big{]}$ $\displaystyle+\|S^{\prime}\|\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\})-\|U\|\big{[}\sigma_{H}(u)+\sigma_{H}(w)-2d_{H}(u,w)\big{]}-\|S\|\sigma_{H}(\\{u,w\\})$ We now argue that the right hand side of the last inequality is nonnegative. Clearly, $C$ and $C_{a}$ are isomorphic, so $W(C)-W(C_{a})=0$. Since $H$ is $2$-connected, and thus $2$-edge-connected, we have $W(C_{b})-W(H)\geq 0$ by Theorem 3(a). Since $d_{F_{a,b}}(u^{\prime},w^{\prime})=1$ we have $d_{G}(u,w)-d_{F_{a,b}}(u^{\prime},w^{\prime})\geq 0$. Also $\sigma_{C_{a}}(u^{\prime},U^{\prime})+\sigma_{C_{a}}(w^{\prime},W^{\prime})+\sigma_{C_{a}}(\\{u^{\prime},w^{\prime}\\},S^{\prime})-\sigma_{C}(u,U)-\sigma_{C}(w,W)-\sigma_{C}(\\{u,w\\},S)=\sigma_{C_{a}}(\\{u^{\prime}w^{\prime}\\})-\sigma_{C}(\\{u,w\\})$, but $\sigma_{C_{a}}(\\{u^{\prime}w^{\prime}\\})-\sigma_{C}(\\{u,w\\})\geq 0$ by Lemma 1. We now bound the remaining expression, $\|U^{\prime}\|\big{(}\sigma_{C_{b}}(u^{\prime})+\sigma_{C_{b}}(w^{\prime})-2d_{C_{b}}(u^{\prime},w^{\prime})\big{)}+\|S^{\prime}\|\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\})-\|U\|\big{(}\sigma_{H}(u)+\sigma_{H}(w)-2d_{H}(u,w)\big{)}-\|S\|\sigma_{H}(\\{u,w\\})$, which we denote by $f$. In order to complete the proof of the lemma it remains to show that $f\geq 0$. We have $a=2\|U^{\prime}\|+\|S^{\prime}\|=2\|U\|+\|S\|$. In $C_{a}$, vertices $u^{\prime}$ and $w^{\prime}$ are adjacent, so there is exactly one vertex equidistant from $u^{\prime}$ and $w^{\prime}$ if $a$ is odd, and there is no vertex equidistant from $u^{\prime}$ and $w^{\prime}$ if $a$ is even. Hence $\|S^{\prime}\|=1$ if $a$ is odd, and $\|S^{\prime}\|=0$ if $a$ is even. In $C$ the vertices $u$ and $w$ are not necessarily adjacent, so we have $\|S\|=1$ if $a$ is odd, and $\|S\|\in\\{0,2\\}$ if $a$ is even. We conclude that if $a$ is odd, then $\|U\|=\|U^{\prime}\|$ and $\|S\|=\|S^{\prime}\|$, and if $a$ is even then either $\|U\|=\|U^{\prime}\|$ and $\|S\|=\|S^{\prime}\|$, or $\|U^{\prime}\|=\|U\|+1$, $\|S^{\prime}\|=0$, and $\|S\|=2$. If $\|U^{\prime}\|=\|U\|$ and $\|S\|=\|S\|$, then $f=\|U\|\big{(}\sigma_{C_{b}}(u^{\prime})-\sigma_{H}(u)+\sigma_{C_{b}}(w^{\prime})-\sigma_{H}(w)+2d_{H}(u,w)-2d_{C_{b}}(u^{\prime},w^{\prime})\big{)}+\|S\|\big{(}\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\}-\sigma_{H}(\\{u,w\\})\big{)}$. Each of the terms $\|U\|$, $\|S\|$, $\sigma_{C_{b}}(u^{\prime})-\sigma_{H}(u)$, $\sigma_{C_{b}}(w^{\prime})-\sigma_{H}(w)$, $2d_{H}(u,w)-2d_{C_{b}}(u^{\prime},w^{\prime})$, and $\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\}-\sigma_{H}(\\{u,w\\})$ is nonnegative, hence $f\geq 0$ in this case. If $\|U^{\prime}\|=\|U\|+1$ and $\|S\|=2$, $\|S^{\prime}\|=0$, then $f=\|U\|\big{(}\sigma_{C_{b}}(u^{\prime})-\sigma_{H}(u)+\sigma_{C_{b}}(w^{\prime})-\sigma_{H}(w)+2d_{H}(u,w)-2d_{C_{b}}(u^{\prime},w^{\prime})\big{)}+\sigma_{C_{b}}(u^{\prime})+\sigma_{C_{b}}(w^{\prime})-2d_{C_{b}}(u^{\prime},w^{\prime})-2\sigma_{H}(\\{u,w\\})$. As above, each of the terms $\|U\|$, $\sigma_{C_{b}}(u^{\prime})-\sigma_{H}(u)$, $\sigma_{C_{b}}(w^{\prime})-\sigma_{H}(w)$, $2d_{H}(u,w)-2d_{C_{b}}(u^{\prime},w^{\prime})$, is nonnegative. We also have $\sigma_{C_{b}}(u^{\prime})+\sigma_{C_{b}}(w^{\prime})-2d_{C_{b}}(u^{\prime},w^{\prime})-2\sigma_{H}(\\{u,w\\})\geq 0$ since $\sigma_{C_{b}}(u^{\prime})+\sigma_{C_{b}}(w^{\prime})-2d_{C_{b}}(u^{\prime},w^{\prime})=\sum_{x\in V(C_{b})-\\{u^{\prime},w^{\prime}\\}}\big{(}d_{C_{b}}(u^{\prime},x)+d_{C_{b}}(w^{\prime},x)\big{)}\geq\sum_{x\in V(C_{b})-\\{u^{\prime},w^{\prime}\\}}2\min\\{d_{C_{b}}(u^{\prime},x),d_{C_{b}}(w^{\prime},x)\\}=2\sigma_{C_{b}}(\\{u^{\prime},w^{\prime}\\})$. Hence $f\geq 0$ also in this case. This proves the desired bound on $W(G)$. Now assume that $W(G)=W(F_{n,a})$. Then we have equality between the corresponding terms in (2) and (3), in particular $W(G[A])=W(C_{a})$ and $W(H)=W(C_{b})$. This implies by Theorem 3(a) that $G[A]$ and $H$ are cycles of length $a$ and $b$, respectively. We also have $d_{C}(u,w)=1$. It follows that $G=F_{n,a}$. (b) If $G$ is Eulerian, then $G\neq F_{n,a}$ and so $W(G)<W(F_{n,a})$. By Lemma 2 we have $W(F_{n,a})\leq W(C_{n,3})$, and (b) follows. $\Box$ ###### Lemma 4. Let $n\in\mathbb{N}$ with $n\geq 26$. Among all Eulerian graphs of order $n$ that are not cycles, let $G$ be one that has maximum Wiener index. Then $G$ has a cutvertex. Proof: Suppose to the contrary that $G$ is $2$-connected. We first prove that every triangle of $G$ contains a vertex of degree $2$. (4) Suppose to the contrary that $G$ contains a triangle $u_{1}u_{2}u_{3}$ with ${\rm deg}(u_{i})>2$ for $i=1,2,3$. Let $E^{\prime}$ be the edge set of this triangle. Then $G-E^{\prime}$ is connected since otherwise, if $G-E^{\prime}$ is disconnected, the vertices $u_{1},u_{2}$ and $u_{3}$ are not all in the same component of $G-E^{\prime}$, so there exists a component of $G-E^{\prime}$ containing only one vertex, $u_{1}$ say, of the triangle. This implies that $u_{1}$ is a cutvertex of $G$, a contradiction to $G$ being $2$-connected. Hence $G-E^{\prime}$ is connected. Clearly, $G-E^{\prime}$ is also Eulerian, and $W(G-E^{\prime})>W(G)$. By our choice of $G$, the graph $G-E^{\prime}$ is a cycle. But then $G$ is obtained from a cycle by adding the edges of a triangle, and so $W(G)<W(C_{n,3})$ by Corollary 2. This contradicts the choice of $G$ as having maximum Wiener index, and so (4) follows. $\overline{v}$$v$$w$$u_{1}$$u_{2}$ $v$$\overline{v}$$w$$u_{1}$$u_{2}$$u_{3}$ $v$$\overline{v}$$w$$u_{1}$$u_{2}$$u_{3}$ $v$$\overline{v}$$w$$u_{1}$$u_{2}$$u_{3}$ Figure 3: Cases 1, 2A, 2B, and 2C in the proof of Lemma 4. Since $G$ is not a cycle, it has a vertex of degree greater than $2$. For $v\in V(G)$ we define $\overline{v}$ to be a nearest vertex of degree greater than $2$ (with ties broken arbitrarily) and let $f(v)=d(v,\overline{v})$. Note that $v=\overline{v}$ if and only if ${\rm deg}(v)>2$. Since $G$ is Eulerian, we have ${\rm deg}(\overline{v})\geq 4$ for every $v\in V$. For $v\in V$, we have $\overline{v}\in N_{f(v)}(v)$, and thus $N(\overline{v})\subseteq N_{f(v)-1}(v)\cup N_{f(v)}(v)\cup N_{f(v)+1}(v)$. We claim that $\textrm{If ${\rm deg}(v)=2$, then}\ \|N(\overline{v})\cap N_{f(v)-1}(v)\|=1.$ (5) Indeed, if ${\rm deg}(v)=2$ then the neighbour, $w$ say, of $\overline{v}$ on a shortest $(v,\overline{v})$-path is in $N_{f(v)-1}(v)$. If there was a second neighbour $w^{\prime}$ of $\overline{v}$ in $N_{f(v)-1}(v)$, then the vertices in $\bigcup_{i=0}^{f(v)-1}N_{i}(v)\cup\\{\overline{v}\\}$ would induce a cycle as a subgraph whose only vertex of degree greater than $2$ in $G$ is $\overline{v}$, implying that $\overline{v}$ is a cutvertex of $G$, contradicting the $2$-connectedness of $G$. This proves (5). Let $v$ be a vertex of degree $2$. By the definition of $f(v)$, all vertices in $\bigcup_{i=0}^{f(v)-1}N_{i}(v)$ have degree $2$. It is easy to see that, since $G$ is $2$-connected, this implies $n_{1}(v)=n_{2}(v)=\cdots=n_{f(v)}(v)=2.$ (6) Let $N_{f(v)}(v)=\\{\overline{v},w\\}$. It follows from (5) that $n_{f(v)}(v)+n_{f(v)+1}(v)\geq{\rm deg}(\overline{v})\geq 4$, so $n_{f(v)+1}(v)\geq 2$. We consider three cases, depending on the value $n_{f(v)+1}(v)$. Case 1: There exists $v\in V(G)$ with $n_{f(v)+1}(v)=2$. Let $N_{f(v)+1}(v)=\\{u_{1},u_{2}\\}$. Since ${\rm deg}_{G}(\overline{v})\geq 4$ it follows that $\overline{v}$ is adjacent to $u_{1},u_{2}$, $w$ and a vertex in $N_{f(v)-1}(v)$, we have ${\rm deg}_{G}(\overline{v})=4$. Since $w$ is adjacent to a vertex in $N_{f(v)+1}(v)$, otherwise $\overline{v}$ would be a cutvertex, to a vertex in $N_{f(v)-1}(v)$, and also to $\overline{v}$, it follows that ${\rm deg}_{G}(w)>2$, and thus ${\rm deg}_{G}(w)=4$, so $w$ is also adjacent to $u_{1}$ and $u_{2}$. Now $\overline{v}$, $w$ and $u_{i}$ form a triangle for $i=1,2$. Since ${\rm deg}_{G}(\overline{v})={\rm deg}_{G}(w)=4$, it follows by (4) that ${\rm deg}_{G}(u_{1})={\rm deg}_{G}(u_{2})=2$. So the vertices in $N_{f(v)+1}(v)$ have only neighbours in $N_{f(v)+1}(v)\cup N_{f(v)}(v)$. This implies that $e_{G}(v)=f(v)+1$, and so $V(G)=\bigcup_{i=0}^{f(v)+1}N_{i}(v)$. It follows that $G$ consists of the cycle induced by $\bigcup_{i=0}^{f(v)}N_{i}(v)$ and the two additional vertices $u_{1}$ and $u_{2}$ of degree two, both adjacent to $\overline{v}$ and $w$. Hence $G$ is the first graph depicted in Figure 3. Applying Lemma 3 to the cutset $\\{\overline{v},w\\}$ now yields that $W(G)<W(C_{n,3})$. This contradiction to the maximality of $W(G)$ proves the lemma in Case 1. Case 2: There exists $v\in V(G)$ with $n_{f(v)+1}=3$. Let $N_{f(v)+1}(v)=\\{u_{1},u_{2},u_{3}\\}$. We consider subcases as follows. Case 2a: $\overline{v}w\in E(G)$. The set $\\{\overline{v},w\\}$ is a cutset. Its branch containing $v$ is a cycle of length $2f(v)+1$, and the union of the other branches is $2$-connected since $\overline{v}$ and $w$ are adjacent. Hence we have $W(G)\leq W(F_{n,2f(v)+1})$ by Lemma 3(a). If $f(v)\geq 2$, then $4\leq 2f(v)+1\leq n-2$ and so $W(F_{n,2f(v)+1})<W(C_{n,3})$ by Lemma 3(b), a contradiction to the maximality of $W(G)$. Hence $f(v)=1$ and $\overline{v}\in N_{1}(v)$. Then ${\rm deg}(w)>2$, since otherwise $w$ would not have neighbours in $N_{2}(v)$, and $\overline{v}$ would be a cutvertex, a contradiction. Hence ${\rm deg}(w)\geq 4$. From (5) and $n_{1}(v)+n_{2}(v)=5$ we conclude that both, $\overline{v}$ and $w$, have degree $4$, and both are adjacent to exactly two vertices in $N_{2}(v)$. We may assume that $\overline{v}$ is adjacent to $u_{1}$ and $u_{2}$, while $w$ is adjacent to $u_{2}$ and $u_{3}$. The situation is depicted in the second graph of Figure 3. Since $\overline{v}$, $w$ and $u_{2}$ form a triangle, we have ${\rm deg}(u_{2})=2$ by (4). This implies that $\\{\overline{v},w\\}$ is a cutset of adjacent vertices with at least three branches, and the branches containing $v$ and $u_{2}$ are $3$-cycles, so their union contains a spanning cycle, whose length is $4$. Since $\overline{v}$ and $w$ are adjacent, the union of the remaining branches is $2$-connected. Hence it follows by Lemma 3 that $W(G)<W(C_{n,3})$. This contradiction to the maximality of $W(G)$ proves the lemma in this case. Case 2b: $\overline{v}w\notin E(G)$ and ${\rm deg}(w)=2$. Then $w$ has a unique neighbour, $u_{3}$ say, in $N_{f(v)+1}(v)$. The set $\bigcup_{i=0}^{f(v)}N_{i}(v)\cup\\{u_{3}\\}$ induces a cycle in $G$ in which only $\overline{w}$ and possibly $u_{3}$ have degree greater than $2$. The situation is depicted in the third graph of Figure 3. The set $\\{\overline{v},u_{3}\\}$ is a cutset. The branch containing $v$ and $w$ induces a cycle of length $2f(v)+2$, and the union of the remaining branches is $2$-connected since $\overline{v}$ and $u_{3}$ are adjacent. Since $4\leq 2f(v)+2\leq n-2$, we have $W(G)<W(C_{n,3})$ by Lemma 3, a contradiction to the maximality of $W(G)$. Case 2c: $\overline{v}w\notin E(G)$ and ${\rm deg}(w)>2$. By (5) and $n_{f(v)+1}(v)=3$ if follows that $\overline{v}$ and $w$ are both adjacent to $u_{1}$, $u_{2}$ and $u_{3}$. If at least one vertex in $\\{u_{1},u_{2},u_{3}\\}$, $u_{1}$ say, has degree $2$, then the union of the two branches at the cutset $\\{\overline{v},w\\}$ containing $v$ and $u_{1}$ has at least four vertices and a spanning cycle, while the union of the remaining branches is $2$-connected. Hence it follows from Lemma 3 that $W(G)<W(C_{n,3})$, contradicting the maximality of $W(G)$. So we may assume that $u_{1}$, $u_{2}$ and $u_{3}$ all have degree greater than $2$. The situation is depicted in the fourth graph of Figure 3. Let $E^{\prime}$ be the edge set of the $4$-cycle $\overline{v},u_{1},w,u_{2},\overline{v}$. Then $G-E^{\prime}$ is connected since otherwise, similarly to the proof of (4), one of the vertices $u_{1}$, $u_{2}$ or $u_{3}$ would be a cutvertex of $G$. Since all vertices in $G-E^{\prime}$ have even degree, it follows that $G-E^{\prime}$ is Eulerian. Since at least one vertex of $G-E^{\prime}$ has degree greater than $2$, viz $u_{3}$, we conclude that $G-E^{\prime}$ is not a cycle. But $W(G-E^{\prime})>W(G)$, a contradiction to the maximality of $W(G)$. Case 3: $n_{f(v)+1}(v)\geq 4$ for all $v\in V$. Let $v\in V(G)$ be fixed. We first show that $\sigma_{G}(v)\leq\left\\{\begin{array}[]{cc}\frac{1}{4}n^{2}-n+\frac{11}{4}+2f(v)&\textrm{if $n$ is odd,}\\\ \frac{1}{4}n^{2}-n+3+2f(v)&\textrm{if $n$ is even}\end{array}\right.$ (7) We note that $n_{0}(v)=1$, $n_{1}(v)=n_{2}(v)=\ldots=n_{f(v)}(v)=2$, and $n_{f(v)+1}(v)\geq 4$ imply that $n\geq 5+2f(v)$, so $f(v)\leq\lfloor\frac{n-5}{2}\rfloor$. Let $k=e(v)$. Then $\sigma_{G}(v)=\sum_{i=0}^{k}in_{i}(v)$. The values $n_{i}(v)$ satisfy the following conditions: (i) $n_{0}(v)=1$ and (ii) $\sum_{i=0}^{k}n_{i}(v)=n$. Since $G$ is $2$-connected, we have (iii) $n_{i}(v)\geq 2$ for $i=1,2,\ldots,k-1$, and (iv) $n_{f(v)+1}(v)\geq 4$ by the defining condition of Case 3. In order to bound $\sum_{i=1}^{k}in_{i}(v)$ from above, assume that $n$ and $f(v)$ are fixed, and that integers $k,n_{0},n_{1},\ldots,n_{k}$ are chosen to maximise $\sum_{i=1}^{k}in_{i}$ subject to conditions (i)-(iv). Then $n_{0}=1$, and $n_{i}=2$ for all $i\in\\{1,2,\ldots,k-1\\}-\\{f(v)+1\\}$, since otherwise, if $n_{i}>2$, we can modify the sequence $n_{0},\ldots,n_{k}$ by decreasing $n_{i}$ by $1$ and increasing $n_{i+1}$ by $1$ to obtain a new sequence $n_{0}^{\prime},\ldots,n_{k}^{\prime}$ which satisfies (i)-(iv), but for which $\sum_{i=0}^{k}in_{i}^{\prime}>\sum_{i=0}^{k}in_{i}$, a contradiction. The same argument yields that $n_{f(v)+1}=4$, and also that $n_{k}\in\\{1,2\\}$ if $k>f(v)+1$. Therefore, if $n$ is odd we have $k=\frac{n-3}{2}$ and $\sum_{i=0}^{k}in_{i}=\frac{1}{4}n^{2}-n+\frac{11}{4}+2f(v)$, and if $n$ is even we have $k=\frac{n-2}{2}$, $n_{k}=1$ and $\sum_{i=0}^{k}in_{i}=\frac{1}{4}n^{2}-n+30+2f(v)$, which is (7). Summation of (7) over all $v\in V(G)$ yields $2W(G)=\sum_{v\in V(G)}\sigma_{G}(v)\leq\left\\{\begin{array}[]{cc}\frac{1}{4}n^{3}-n^{2}+\frac{11}{4}n+2\sum_{v\in V(G)}f(v)&\textrm{if $n$ is odd,}\\\ \frac{1}{4}n^{3}-n^{2}+3n+2\sum_{v\in V(G)}f(v)&\textrm{if $n$ is even.}\end{array}\right.$ (8) We now bound $\sum_{v\in V(G)}f(v)$. Since $G$ is an Eulerian graph but not a cycle, $G$ contains a vertex $w$ of degree at least $4$. Since for $i\in\\{0,1,\ldots,e(w)\\}$ every vertex $v\in N_{i}(w)$ satisfies $f(v)\leq d(v,w)$, we have $\sum_{v\in V(G)}f(v)\leq\sum_{v\in V(G)}d(v,w)=\sigma_{G}(w).$ Now $G$ has more than one vertex of degree greater than two, since otherwise such a vertex would be a cutvertex, contradicting the $2$-connectedness of $G$. That implies that the strict inequality $\sum_{v\in V(G)}f(v)<\sigma_{G}(w)$ holds. Noting that $f(w)=0$, we obtain by (7) that $\sum_{v\in V(G)}f(v)<\sigma_{G}(w)\leq\left\\{\begin{array}[]{cc}\frac{1}{4}n^{2}-n+\frac{11}{4}&\textrm{if $n$ is odd,}\\\ \frac{1}{4}n^{2}-n+3&\textrm{if $n$ is even}.\end{array}\right.$ (9) From (8) and (9) we get $W(G)<\left\\{\begin{array}[]{cc}\frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{3}{8}n+\frac{11}{4}&\textrm{if $n$ is odd,}\\\ \frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{1}{2}n+3&\textrm{if $n$ is even.}\end{array}\right.$ But the right hand side of the last inequality equals $W(C_{n,3})$. This contradiction to the maximality of $W(G)$ completes the proof. $\Box$ ## 4 Completing the proof of Theorem 2 Proof of Theorem 2: Suppose to the contrary that the theorem is false, and let $n$ be the smallest value with $n\geq 26$ for which the theorem fails. Let $G$ be an Eulerian gaph of order $n$ that is not a cycle, and that has maximum Wiener index among all such graphs. By Lemma 4, $G$ has a cutvertex, so $G$ is not $2$-connected. Then $G$ hat at least two endblocks. Let $H$ be a smallest endblock of $G$, let $v$ be the cutvertex of $G$ contained in $H$, and let $K$ be the union of the branches at $v$ distinct from $H$. Let $A$ and $B$ be the vertex set of $H$ and $K$, respectively, and let $a=\|A\|$ and $b=\|B\|$. Then $b=n-a+1$, and since $H$ is a smallest endblock we have $a\leq\frac{n+1}{2}$. We have $\displaystyle W(G)$ $\displaystyle=$ $\displaystyle\sum_{\\{x,y\\}\subseteq A}d_{H}(x,y)+\sum_{\\{x,y\\}\subseteq B}d_{K}(x,y)+\sum_{x\in A-\\{v\\}}\sum_{y\in B-\\{v\\}}\big{(}d_{H}(x,v)+d_{K}(v,y)\big{)}$ (10) $\displaystyle=$ $\displaystyle W(H)+W(K)+(a-1)\sigma_{K}(v)+(b-1)\sigma_{H}(v).$ Since $H$ is an endblock, $H$ is $2$-connected, but $K$ may or may not be $2$-connected. Case 1: $K$ is $2$-connected. Similarly to (10) we obtain for the graph $C_{n,a}$ and its two blocks $C_{a}$ and $C_{b}$ that $W(C_{n,a})=W(C_{a})+W(C_{b})+(a-1)\sigma_{C_{b}}(w)+(b-1)\sigma_{C_{a}}(w),$ where $w$ is the cutvertex of $C_{n,a}$. Since $H$ and $K$ are $2$-connected, we have by Theorem 3 that $W(H)\leq W(C_{a})$, $W(K)\leq W(C_{b})$, $\sigma_{K}(v)\leq\sigma_{C_{b}}(w)$ and $\sigma_{H}(v)\leq\sigma_{C_{a}}(w)$. Hence we have $W(G)\leq W(C_{n,a})$. By Lemma 1 we have $W(C_{n,a})\leq W(C_{n,3})$, and so we have $W(G)\leq W(C_{n,3})$, as desired. Assume that $W(G)=W(C_{n,3})$. Then $W(K)=W(C_{b})$, and so $K=C_{b}$, and similarly $H=C_{a}$ by Theorem 3. Now Lemma 1 implies that $a=3$. It follows that $G=C_{n,3}$, and so the theorem holds in Case 1. Case 2: $K$ is not $2$-connected. We now bound each term on the right hand side of (10) separately. Clearly, $K$ is an Eulerian graph of order $n-a+1$ but not a cycle. Since $G$ is a smallest counterexample to Theorem 2, the bound in Theorem 2 holds for $K$ unless $b<5$ or $b\in\\{7,9\\}$. However, since $b\geq\frac{n+1}{2}$ and $n\geq 26$, $b$ is not one of these exceptional values and Theorem 2 holds for $K$. Therefore, $W(K)\leq W(C_{n-a+1,3})\leq\frac{1}{8}(n-a+1)^{3}-\frac{1}{4}(n-a+1)^{2}+\frac{3}{2}(n-a+1)-2.$ (11) It follows from Theorem 3(c) that $\sigma_{K}(v)\leq\frac{1}{3}(n-a+1)(n-a).$ (12) As in Case 1, Theorem 3 yields the following bounds for $W(H)$ and $\sigma_{H}(v)$ $W(H)\leq W(C_{a})\leq\frac{1}{3}a^{3}\ \ \textrm{and}\ \ \sigma_{H}(v)\leq\frac{1}{4}a^{2}.$ (13) Substituting (13), (11) and (12) into (10) yields that $\displaystyle W(G)$ $\displaystyle\leq$ $\displaystyle\frac{1}{8}a^{3}+\frac{1}{8}(n-a+1)^{3}-\frac{1}{4}(n-a+1)^{2}+\frac{3}{2}(n-a+1)-2$ $\displaystyle+\frac{1}{3}(a-1)(n-a+1)(n-a)+\frac{1}{4}(n-a)a^{2}.$ From equation (1) and the remark following it, we have $W(C_{n,3})\geq\frac{1}{8}n^{3}-\frac{1}{4}n^{2}+\frac{11}{8}n-\frac{9}{4}.$ Subtracting these two bounds we obtain, after simplification, $W(C_{n,3})-W(G)\geq\frac{1}{24}\Big{\\{}(a-1)n^{2}+(a^{2}-18a+8)n-2a^{3}+13a^{2}+25a-39\Big{\\}}.$ Denote the right hand side of the above inequality by $f(n,a)$. To complete the proof of the Lemma it suffices to show that $f(n,a)>0$ for $n\geq 26$ and $3\leq a\leq\frac{n+1}{2}$. Now $\frac{\partial f}{\partial a}=\frac{1}{24}\big{\\{}n^{2}+(2a-18)n-6a^{2}+26a+25\big{\\}}$. For constant $n$, this is a quadratic function of $a$ which is concave down and thus it attains its minimum for $a\in[3,\frac{n+1}{2}]$ at $a=3$ or $a=\frac{n+1}{2}$. Since for $a=3$ we have $\frac{\partial f}{\partial a}=\frac{1}{24}(n^{2}-12n+49)>0$, and for $a=\frac{n+1}{2}$ we have $\frac{\partial f}{\partial a}=\frac{1}{48}(n^{2}-14n+73)>0$, the derivative $\frac{\partial f}{\partial a}$ is positive for $3\leq a\leq\frac{n+1}{2}$. It follows that the function $f$ is increasing in $a$ for constant $n$, and thus $W(C_{n,3})-W(G)\geq f(3)=\frac{1}{24}\big{(}2n^{2}-37n+99\big{)},$ which is greater than $0$ for $n\geq 26$. This completes the proof of Theorem 2 $\Box$ ## 5 Eulerian Graphs with Small Wiener Index A natural question that arises in the context of of the Wiener index of Eulerian graphs is how small the Wiener index of an Eulerian graph can be. For Eulerian graphs of given order, this was answered in [15]. ###### Proposition 1 (Gutman, Cruz and Rada [15]). Let $G$ be an Eulerian graph of order $n$, where $n\geq 3$. Then $W(G)\geq\left\\{\begin{array}[]{cc}\binom{n}{2}&\textrm{if $n$ is odd,}\\\ \binom{n}{2}+\frac{n}{2}&\textrm{if $n$ is even.}\end{array}\right.$ Equality holds if and only if $G$ is complete (for odd $n$), or $G$ is obtained from the complete graph by removing the edges of a perfect matching (for even $n$). Finding the minimum value of the Wiener index of Eulerian graphs becomes more challenging if not only the order, but also the size of the graph is considered. We have the following lower bound on the Wiener index due to Plesník [20]. ###### Proposition 2. Let $G$ be a connected graph with $n$ vertices and $m$ edges. Then $W(G)\geq 2\binom{n}{2}-m.$ Equality holds if and only if the diameter of $G$ is at most $2$. Proposition 2 yields a lower bound on the Wiener index of Eulerian graphs of given order and size. However, if $m$ is so small relative to $n$ that there is no Eulerian graph of diameter two of order $n$ and size $m$, then this bound is not sharp. The following result determines the minimum size of an Eulerian graph of order $n$ and diameter $2$. In the proof we use the fact that the minimum size of a graph of order $n$ and diameter $2$ not containing a vertex of degree $n-1$ is $2n-5$, which was proved by Erdős and Rényi [9], see also [16]. ###### Proposition 3. Let $G$ be an Eulerian graph of order $n$ and diameter two. Then $m(G)\geq\left\\{\begin{array}[]{cc}\frac{3}{2}(n-1)&\textrm{if $n$ is odd,}\\\ 2n-5&\textrm{if $n$ is even.}\end{array}\right.$ This bound is sharp for $n\geq 9$. Proof: First let $n$ be even. Since $G$ contains only vertices of even degree, $G$ has no vertex of degree $n-1$. The above-mentioned result by Erdős and Rényi [9] now proves that $m(G)\geq 2n-5$, as desired. To see that the bound is sharp consider the graph obtained from a triangle with vertices $a$, $b$ and $c$ and a star $K_{1,n-4}$ by joining two of the leaves of the star to $a$, joining two other leaves to $b$, and joining the remaining $n-8$ leaves to $c$. (We note that this graph was already described in [16].) Now let $n$ be odd. If $G$ contains no vertex of degree $n-1$, then we have $m\geq 2n-5$ as above, and the result follows. If $G$ contains a vertex of degree $n-1$, then all other vertices have degree at least $2$, and so the degree sum of $G$ is at least $n-1+2(n-1)$, and so $m\geq\frac{3}{2}(n-1)$, as desired. 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# Structured Time-Delay Models for Dynamical Systems with Connections to Frenet-Serret Frame Seth M. Hirsh Department of Physics, University of Washington, Seattle, WA ([email protected]). Sara M. Ichinaga Applied and Computational Mathematical Sciences Program, University of Washington, Seattle, WA ([email protected]). Steven L. Brunton Department of Mechanical Engineering, University of Washington, Seattle, WA<EMAIL_ADDRESS>J. Nathan Kutz Department of Applied Mathematics, University of Washington, Seattle, WA<EMAIL_ADDRESS>Bingni W. Brunton Department of Biology, University of Washington, Seattle, WA ([email protected]). ###### Abstract Time-delay embeddings and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition (DMD) on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal with a zero diagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modeling procedure on data from several nonlinear synthetic and real-world examples. Keywords: Dynamic mode decomposition, Time-delay coordinates, Frenet-Serret, Koopman operator, Hankel matrix. ## 1 Introduction Discovering meaningful models of complex, nonlinear systems from measurement data has the potential to improve characterization, prediction, and control. Focus has increasingly turned from first-principles modeling towards data- driven techniques to discover governing equations that are as simple as possible while accurately describing the data [1, 2, 3, 4]. However, available measurements may not be in the right coordinates for which the system admits a simple representation. Thus, considerable effort has gone into learning effective coordinate transformations of the measurement data [5, 6, 7], especially those that allow nonlinear dynamics to be approximated by a linear system. These coordinates are related to eigenfunctions of the Koopman operator [8, 9, 10, 11, 12, 13], with dynamic mode decomposition (DMD) [14] being the leading computational algorithm for high-dimensional spatiotemporal data [11, 15, 13]. For low-dimensional data, time-delay embedding [16] has been shown to provide accurate linear models of nonlinear systems [5, 17, 18]. Linear time-delay models have a rich history [19, 20], and recently, DMD on delay coordinates [15, 21] has been rigorously connected to these linearizing coordinate systems in the Hankel alternative view of Koopman (HAVOK) approach [5, 17, 7]. In this work, we establish a new connection between HAVOK and the Frenet-Serret frame from differential geometry, which inspires an extension to the algorithm that improves the stability of these models. Time-delay embedding is a widely used technique to characterize dynamical systems from limited measurements. In delay embedding, incomplete measurements are used to reconstruct a representation of the latent high-dimensional system by augmenting the present measurement with a time-history of previous measurements. Takens showed that under certain conditions, time-delay embedding produces an attractor that is diffeomorphic to the attractor of the latent system [16]. Time-delay embeddings have also been extensively used for signal processing and modeling [20, 19, 22, 23, 24, 25, 26, 27], for example, in singular spectrum analysis (SSA) [19, 22] and the eigensystem realization algorithm (ERA) [20]. In both cases, a time history of augmented delay vectors are arranged as columns of a Hankel matrix, and the singular value decomposition (SVD) is used to extract _eigen_ -time-delay coordinates in a dimensionality reduction stage. More recently, these historical approaches have been connected to the modern DMD algorithm [15], and it has become commonplace to compute DMD models on time delay coordinates [15, 21]. The HAVOK approach established a rigorous connection between DMD on delay coordinates and eigenfunctions of the Koopman operator [5]; HAVOK [5] is also referred to as Hankel DMD [17] or delay DMD [15]. HAVOK produces linear models where the matrix representation of the dynamics has a peculiar and particular structure. These matrices tend to be skew- symmetric and dominantly tridiagonal, with zero diagonal (see Fig. 2 for an example). In the original HAVOK paper, this structure was observed in some systems, but not others, with the structure being more pronounced in noise- free examples with an abundance of data. It has been unclear how to interpret this structure and whether or not it is a universal feature of HAVOK models. Moreover, the eigen-time-delay modes closely resemble Legendre polynomials; these polynomials were explored further in Kamb et al. [28]. The present work directly resolves this mysterious structure by establishing a connection to the Frenet-Serret frame from differential geometry. The structure of HAVOK models may be understood by introducing intrinsic coordinates from differential geometry [29]. One popular set of intrinsic coordinates is the Frenet-Serret frame, which is formed by applying the Gram- Schmidt procedure to the derivatives of the trajectory $\dot{\bm{x}}(t),\ddot{\bm{x}}(t),\dddot{\bm{x}}(t),\ldots$ [30, 31, 32]. Alvarez-Vizoso et al. [33] showed that the SVD of trajectory data converges locally to the Frenet-Serret frame in the limit of an infinitesimal time step. The Frenet-Serret frame results in an orthogonal basis of polynomials, which we will connect to the observed Legendre basis of HAVOK [5, 28]. Moreover, we show that the dynamics, when represented in these coordinates, have the same tridiagonal structure as the HAVOK models. Importantly, the terms along the sub- and super-diagonals have a specific physical interpretation as intrinsic curvatures. By enforcing this structure, HAVOK models are more robust to noisy and limited data. In this work, we present a new theoretical connection between time-delay embedding models and the Frenet-Serret frame from differential geometry. Our unifying perspective sheds light on the antisymmetric, tridiagonal structure of the HAVOK model. We use this understanding to develop _structured_ HAVOK models that are more accurate for noisy and limited data. Section 2 provides a review of dimensionality reduction methods, time delay embeddings, and the Frenet-Serret frame. This section also discusses current connections between these fields. In Section 3, we establish the main result of this work, connecting linear time-delay models with the Frenet-Serret frame, explaining the tridiagonal, antisymmetric structure seen in Figure 2. We then illustrate this theory on a synthetic example. In Section 4, we explore the limitations and requirements of the theory, giving recommendations for achieving this structure in practice. In Section 5, based on this theory, we develop a modified HAVOK method, called _structured_ HAVOK (sHAVOK), which promotes tridiagonal, antisymmetric models. We demonstrate this approach on three nonlinear synthetic examples and two real-world datasets, namely measurements of a double pendulum experiment and measles outbreak data, and show that sHAVOK yields more stable and accurate models from significantly less data. Figure 1: In this work, we unify key results from dimensionality reduction, time-delay embedding and the Frenet-Serret frame to show that a dynamical system may be decomposed into a sparse linear model plus a forcing term. Further, this linear model has a particular structure: it is an antisymmetric tridiagonal matrix with nonzero elements only along the super- and sub- diagonals. These nonzero elements are interpretable as they are intrinsic curvatures of the system in the Frenet-Serret frame. ## 2 Related Work Our work relates and extends results from three fields: dimensionality reduction, time-delay embedding, and the Frenet-Serret coordinate frame from differential geometry. There is an extensive literature on each of these fields, and here we give a brief introduction of the related work to establish a common notation on which we build a unifying framework in Section 3. ### 2.1 Dimensionality Reduction Recent advancements in sensor and measurement technologies have led to a significant increase in the collection of time-series data from complex, spatio-temporal systems. Although such data is typically high dimensional, in many cases it can be well approximated with a low dimensional representation. One central goal is to learn the underlying structure of this data. Although there are many data-driven dimensionality reduction methods, here we focus on linear techniques because of their effectiveness and analytic tractability. In particular, given a data matrix $\bm{X}\in\mathbb{R}^{m\times n}$, the goal of these techniques is to decompose $\bm{X}$ into the matrix product $\bm{X}=\bm{U}\bm{V}^{\intercal},$ (1) where $\bm{U}\in\mathbb{R}^{m\times k}$ and $\bm{V}\in\mathbb{R}^{n\times k}$ are low rank ($k<\min(m,n)$). The task of solving for $\bm{U}$ and $\bm{V}$ is highly underdetermined, and different solutions may be obtained when different assumptions are made. Here we review two popular linear dimensionality reduction techniques: singular value decomposition (SVD) [34, 35] and dynamic mode decomposition (DMD) [36, 15, 13]. Both of these methods are key components of the HAVOK algorithm and play a key role in determining the underlying tridiagonal antisymmetric structure in Figure 2. #### 2.1.1 Singular Value Decomposition (SVD) The SVD is one of the most popular dimensionality reduction methods, and it has been applied in a wide range of applications, including genomics [37], physics [38], and image processing [39]. SVD is the underlying algorithm for principal component analysis (PCA). Given the data matrix $\bm{X}\in\mathbb{R}^{m\times n}$, the SVD decomposes $\bm{X}$ into the product of three matrices, $\bm{X}=\bm{U}\bm{\Sigma}\bm{V}^{\intercal},$ where $\bm{U}\in\mathbb{R}^{m\times m}$ and $\bm{V}\in\mathbb{R}^{n\times n}$ are unitary matrices, and $\bm{\Sigma}\in\mathbb{R}^{m\times n}$ is a diagonal matrix with nonnegative entries [34, 35]. We denote the $i$th columns of $\bm{U}$ and $\bm{V}$ as $\bm{u}_{i}$ and $\bm{v}_{i}$, respectively. The diagonal elements of $\bm{\Sigma}$, $\sigma_{i}$, are known as the singular values of $\bm{X}$, and they are written in descending order. The rank of the data is defined to be $R$, which equals the number of nonzero singular values. Consider the low rank matrix approximation $\bm{X}_{r}=\sum_{j=1}^{r}\bm{u}_{j}\sigma_{j}\bm{v}_{j}^{T},$ with $r\leq R$. An important property of $\bm{X}_{r}$ is that it is the best rank $r$ approximation to $\bm{X}$ in the least squares sense. In other words, $\bm{X}_{r}=\operatorname{argmin}_{\bm{Y}}\left\lVert\bm{X}-\bm{Y}\right\rVert\quad\text{such that}\text{ rank}(\bm{Y})=r,$ with respect to both the $l_{2}$ and Frobenius norms. Further, the relative error in this rank-$r$ approximation using the $l_{2}$ norm is $\frac{\left\lVert\bm{X}-\bm{X}_{r}\right\rVert_{l_{2}}}{\left\lVert\bm{X}\right\rVert_{l_{2}}}=\frac{\sigma_{r+1}}{\sigma_{1}}.$ (2) From (2), we immediately see that if the singular values decay rapidly, ($\sigma_{j+1}\ll\sigma_{j}$), then $\bm{X}_{r}$ is a good low-rank approximation to $\bm{X}$. This property makes the SVD a popular tool for compressing data. #### 2.1.2 Dynamic Mode Decomposition (DMD) DMD [14, 15, 13] is another linear dimensionality reduction technique that incorporates an assumption that the measurements are time series data generated by a linear dynamical system in time. DMD has become a popular tool for modeling dynamical systems in such diverse fields, including fluid mechanics [11, 14], neuroscience [21], disease modeling [40], robotics [41], plasma modeling [42], resolvent analysis [43], and computer vision [44, 45]. Like the SVD, for DMD we begin with a data matrix $\bm{X}\in\mathbb{R}^{m\times n}$. Here we assume that our data is generated by an unknown dynamical system so that the columns of $\bm{X}$, $\bm{x}(t_{k})$, are time snapshots related by the map $\bm{x}(t_{k+1})=\bm{F}(\bm{x}(t_{k}))$. While $\bm{F}$ may be nonlinear, the goal of DMD is to determine the best-fit linear operator $\bm{A}:\mathbb{R}^{m}\to\mathbb{R}^{m}$ such that $\bm{x}(t_{k+1})\approx\bm{A}\bm{x}(t_{k}).$ If we define the two time-shifted data matrices, $\bm{X}_{1}^{n-1}=\begin{bmatrix}\|&\|&\cdots&\|\\\ \bm{x}(t_{1})&\bm{x}_{2}(t_{2})&\cdots&x(t_{n-1})\\\ \|&\|&\cdots&\|\\\ \end{bmatrix}\text{, and }\bm{X}_{2}^{n}=\begin{bmatrix}\|&\|&\cdots&\|\\\ x(t_{2})&x(t_{3})&\cdots&x(t_{n})\\\ \|&\|&\cdots&\|\\\ \end{bmatrix},$ then we can equivalently define $\bm{A}\in\mathbb{R}^{m\times m}$ to be the operator such that $\bm{X}_{2}^{n}\approx\bm{A}\bm{X}_{1}^{n-1}.$ It follows that $\bm{A}$ is the solution to the minimization problem $\bm{A}=\min_{\bm{A^{\prime}}}\left\lVert\bm{X}_{2}^{n}-\bm{A^{\prime}}\bm{X}_{1}^{n-1}\right\rVert_{F},$ where $\left\lVert\cdot\right\rVert_{F}$ denotes the Frobenius norm. A unique solution to this problem can be obtained using the exact DMD method and the Moore-Penrose pseudo-inverse $\hat{\bm{A}}=\bm{X}_{2}^{n}\left(\bm{X}_{1}^{n-1}\right)^{\dagger}$ [15, 13]. Alternative algorithms have been shown to perform better for noisy measurement data, including optimized DMD [46], forward-backward DMD [47], and total-least squares DMD [48]. One key benefit of DMD is that it builds an explicit temporal model and supports short-term future state prediction. Defining $\left\\{\lambda_{j}\right\\}$ and $\left\\{\bm{v}_{j}\right\\}$ to be the eigenvalues and eigenvectors of $\bm{A}$, respectively, then we can write $\bm{x}(t_{k})=\sum_{j=1}^{r}\bm{v}_{j}e^{\omega_{j}t_{k}},$ (3) where $\omega_{j}=\ln(\lambda_{j})/\Delta t$ are eigenvalues normalized by the sampling interval $\Delta t$, and the eigenvectors are normalized such that $\sum_{j=1}^{r}\bm{v}_{j}=\bm{x}(t_{1})$. Thus, to compute the state at an arbitrary time $t$, we can simply evaluate (3) at that time. Further, letting $\bm{v}_{j}$ be the columns of $\bm{U}$ and $\\{\exp(\omega_{j}t_{k})\text{ for }k=1,\ldots r\\}$ be the columns of $\bm{V}$, then we can express data in the form of (1). ### 2.2 Time Delay Embedding Suppose we are interested in a dynamical system $\frac{d\bm{\xi}}{dt}=\bm{F}(\bm{\xi}),$ where $\bm{\xi}(t)\in\mathbb{R}^{l}$ are states whose dynamics are governed by some unknown nonlinear differential equation. Typically, we measure some possibly nonlinear projection of $\bm{\xi}$, $\bm{x}(\bm{\xi})\in\mathbb{R}^{d}$ at discrete time points $t=0,\Delta t,\ldots,q\Delta t$. In general, the dimensionality of the underlying dynamics is unknown, and the choice of measurements are limited by practical constraints. Consequently, it is difficult to know whether the measurements $\bm{x}$ are sufficient for modeling the system. For example, $d$ may be smaller than $m$. In this work we are primarily interested in the case of $d=1$; in other words, we have only a single one-dimensional time series measurement for the system. We can construct an embedding of our system using successive time delays of the measurement $x$, at $x(t-\tau)$. Given a single measurement of our dynamical system $x(t)\in\mathbb{R}$, for $t=0,\Delta t,\ldots q\Delta t$, we can form the Hankel matrix $\bm{H}\in\mathbb{R}^{m\times n}$ by stacking time shifted snapshots of $x$ [49], $\bm{H}=\begin{bmatrix}x_{1}&x_{2}&x_{3}&x_{4}&\cdots&x_{n}\\\ x_{2}&x_{3}&x_{4}&x_{5}&\cdots&x_{n+1}\\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ x_{m}&x_{m+1}&x_{m+2}&x_{m+3}&\cdots&x_{q+1}\end{bmatrix}.$ (4) Each column may be thought of as an augmented state space that includes a short, $m$-dimensional trajectory in time. Our data matrix $\bm{H}$ is then this $m$-dimensional trajectory measured over $n$ snapshots in time. There are several key benefits of using time delay embeddings. Most notably, given a chaotic attractor, Taken’s embedding theorem states that a sufficiently high dimensional time delay embedding of the system is diffeomorphic to the original attractor [16], as illustrated in Figure 1. In addition, recent results have shown that time delay matrices are guaranteed to have strongly decaying singular value spectra. In particular, Beckerman et al. [50] prove the following theorem: ###### Theorem 1. Let $\bm{H}_{n}\in\mathbb{R}^{n\times n}$ be a positive definite Hankel matrix, with singular values $\sigma_{1},\ldots,\sigma_{n}$. Then $\sigma_{j}\leq C\rho^{-j/\log{n}}\sigma_{1}$ for constants $C$ and $\rho$ and for $j=1,\ldots,n$. Equivalently, $\bm{H}_{n}$ can be approximated up to an accuracy of $\epsilon\left\lVert\bm{H}_{n}\right\rVert_{2}$ by a rank $\mathcal{O}(\log{n}\log{1/\epsilon})$ matrix. From this, we see that $\bm{H}_{n}$ can be well-approximated by a low-rank matrix. Many methods have been developed to take advantage of this structure of the Hankel matrix, including the eigensystem realization algorithm (ERA) [20], singular spectrum analysis (SSA) [19], and nonlinear Laplacian spectrum analysis [22]. DMD may also be computed on delay coordinates from the Hankel matrix [15, 51, 21], and it has been shown that this approach may provide a Koopman invariant subspace [52, 5]. In addition, this structure has also been incorporated into neural network architectures [53]. ### 2.3 HAVOK: Dimensionality Reduction and Time Delay Embeddings Leveraging dimensionality reduction and time delay embeddings, the Hankel alternative view of Koopman (HAVOK) algorithm constructs low dimensional models of dynamical systems [5]. Specifically, HAVOK learns effective measurement coordinates of the system and estimate its intrinsic dimensionality. Remarkably, HAVOK models are simple, consisting of a linear model and a forcing term that can be used for short term forecasting. Figure 2: Outline of steps in HAVOK method. First, given a dynamical system a single variable $x(t)$ is measured. Time-shifted copies of $x(t)$ are stacked to form a Hankel matrix $\bm{H}$. The singular value decomposition (SVD) is applied to $\bm{H}$, producing a low dimensional representation $\bm{V}$. The dynamic mode decomposition (DMD) is then applied to $\bm{V}$ to form a linear dynamical model and a forcing term. We illustrate this method in Figure 2 for the Lorenz system (see section 5.2 for details about this system). To do so, we begin with a one dimensional time series $x(t)$ for $t=0,\Delta t,\ldots,q\Delta t$. We construct a higher dimensional representation using time delay embeddings, producing a Hankel matrix $\bm{H}\in\mathbb{R}^{m\times n}$ as in (4) and computes its SVD, $\bm{H}=\bm{U}\bm{\Sigma}\bm{V}^{\intercal}.$ If $\bm{H}$ is sufficiently low rank (with rank $r$), then we need only consider the reduced SVD, $\bm{H}_{r}=\bm{U}_{r}\bm{\Sigma}_{r}\bm{V}_{r}^{\intercal},$ where $\bm{U}_{r}\in\mathbb{R}^{m\times r}$ and $\bm{V}_{r}\in\mathbb{R}^{n\times r}$ are orthogonal matrices and $\bm{\Sigma}_{r}\in\mathbb{R}^{r\times r}$ is diagonal. Rearranging the terms, $\bm{V}_{r}^{\intercal}=\bm{\Sigma}_{r}^{-1}\bm{U}_{r}^{\intercal}\bm{H}_{r}$ and we can think of $\bm{V}_{r}^{\intercal}=\begin{bmatrix}\bm{v}_{1}&\bm{v}_{2}&\cdots&\bm{v}_{n}\end{bmatrix}$ (5) as a lower dimensional representation of our high dimensional trajectory. For quasi-periodic systems, the SVD decomposition of the Hankel matrix results in principal component trajectories (PCT) [54], which reconstruct dynamical trajectories in terms of periodic orbits. To discover the linear dynamics, we apply DMD. In particular, we construct the time shifted matrices, $\bm{V}_{1}=\begin{bmatrix}\bm{v}_{1}&\bm{v}_{2}&\cdots&\bm{v}_{n-1}\end{bmatrix}\mbox{ and }\bm{V}_{2}=\begin{bmatrix}\bm{v}_{2}&\bm{v}_{3}&\cdots&\bm{v}_{n}\end{bmatrix}.$ (6) We then compute the linear approximation $\hat{\bm{A}}$ such that $\bm{V}_{2}=\hat{\bm{A}}\bm{V}_{1}$, where $\hat{\bm{A}}=\bm{V}_{2}\bm{V}_{1}^{\dagger}$. This yields a model $\bm{v}_{i+1}=\hat{\bm{A}}\bm{v}_{i}$. In the continuous case, $\dot{\bm{v}}(t)=\bm{A}\bm{v}(t)$ (7) which is related to first order in $\Delta t$ to the discrete case by $\bm{A}\approx\left(\hat{\bm{A}}-\bm{I}\right)/\Delta t.$ For a general nonlinear dynamical system, this linear model yields a poor reconstruction. Instead, [5] proposed a linear model plus a nonlinear forcing term in the last component of $\bm{v}$ (Figure 2): $\dot{\bm{v}}(t)=\bm{A}\bm{v}(t)+\bm{B}v_{r}(t),$ (8) where $\bm{v}(t)\in\mathbb{R}^{r-1}$, $\bm{A}\in\mathbb{R}^{r-1\times r-1}$, and $\bm{B}\in\mathbb{R}^{r-1}$. In this case, $\bm{V}_{2}$ is defined as columns $2$ to $n$ of the SVD singular vectors with an $r-1$ rank truncation $\bm{V}_{r-1}^{\intercal}$. $\hat{\bm{A}}\in\mathbb{R}^{r-1\times r-1}$ and $\hat{\bm{B}}\in\mathbb{R}^{r-1\times 1}$ are computed as $\left[\hat{\bm{A}},\hat{\bm{B}}\right]=\bm{V}_{2}\bm{V}_{1}^{\dagger}$. The continuous analog of $\hat{\bm{B}}$, $\bm{B}$, is computed by $\bm{B}\approx(\hat{\bm{B}}-\bm{I})/\Delta t$. HAVOK was shown to be a successful model for a variety of systems, including a double pendulum and switchings of Earth’s magnetic field. In addition, the linear portion of the HAVOK model has been observed to adopt a very particular structure: the dynamics matrix was antisymmetric, with nonzero elements only on the superdiagonal and subdiagonal (Figure 2). Much work has been done to study the properties of HAVOK. Arbabi et al. [17] showed that, in the limit of an infinite number of time delays ($m\to\infty$), $\bm{A}$ converges to the Koopman operator for ergodic systems. Bozzo et al. [55] showed that in a similar limit, for periodic data, HAVOK converges to the temporal discrete Fourier transform. Kamb et al. [28] connects HAVOK to the use of convolutional coordinates. The primary goal of this current work is to connect HAVOK to the concept of curvature in differential geometry, and with these new insights, improve the HAVOK algorithm to take advantage of this structure in the dynamics matrix. In contrast with much of the previous work, we focus on the limit where only small amounts of noisy data are available. ### 2.4 The Frenet-Serret Coordinate Frame Suppose we have a smooth curve $\bm{\gamma}(t)\in\mathbb{R}^{m}$ measured over some time interval $t\in[a,b]$. As before, we would like to determine an effective set of coordinates in which to represent our data. When using SVD or DMD, the basis discovered corresponds to the spatial modes of the data and is constant in time. However, for many systems, it is sometimes natural to express both the coordinates and basis as functions of time [56, 57]. One popular method for developing this noninertial frame is the Frenet-Serret coordinate system, which has been applied in a wide range of fields, including robotics [58, 59], aerodynamics [60], and general relativity [61, 62]. Let us assume that $\bm{\gamma}(t)$ has $r$ nonzero continuous derivatives, $\bm{\gamma}^{\prime},(t),\bm{\gamma}^{\prime\prime}(t),\ldots\bm{\gamma}^{(r)}(t)$. We further assume that these derivatives are linearly independent and $\left\lVert\bm{\gamma}^{\prime}(t)\right\rVert\neq\bm{0}$ for all $t$. Using the Gram-Schmidt process, we can form the orthonormal basis, $\bm{e}_{1},\bm{e}_{2},\ldots,\bm{e}_{r}$, $\displaystyle\begin{split}\bm{e}_{1}(t)&=\frac{\bm{\gamma}^{\prime}(t)}{\left\lVert\bm{\gamma}^{\prime}(t)\right\rVert},\\\ \bm{e}_{2}(t)&=\frac{\bm{\gamma}^{\prime\prime}(t)-\langle\bm{\gamma}^{\prime\prime}(t),\bm{e}_{1}(t)\rangle\bm{e}_{1}(t)}{\left\lVert\bm{\gamma}^{\prime\prime}(t)-\langle\bm{\gamma}^{\prime\prime}(t),\bm{e}_{1}(t)\rangle\bm{e}_{1}(t)\right\rVert},\\\ &\leavevmode\nobreak\ \kern 1.66672pt\vdots\\\ \bm{e}_{r}(t)&=\frac{\bm{\gamma}^{(r)}(t)-\sum_{k=1}^{r-1}\langle\bm{\gamma}^{(r)}(t),\bm{e}_{k}(t)\rangle\bm{e}_{k}(t)}{\left\lVert\bm{\gamma}^{(r)}(t)-\sum_{k=1}^{r-1}\langle\bm{\gamma}^{(r)}(t),\bm{e}_{k}(t)\rangle\bm{e}_{k}(t)\right\rVert}.\end{split}$ (9) Here $\langle\cdot,\cdot\rangle$ denotes an inner product, and we choose $r\leq m$ so that these vectors are linearly independent and hence form an orthonormal basis basis. This set of basis vectors define the Frenet-Serret frame. To derive the evolution of this basis, let us define the matrix formed by stacking these vectors $\bm{Q}(t)=[\bm{e}_{1}(t),\bm{e}_{2}(t),\ldots,\bm{e}_{r}(t)]^{\intercal}\in\mathbb{R}^{r\times m}$, so that $\bm{Q}(t)$ satisfies the following time-varying linear dynamics, $\frac{d\bm{Q}}{dt}=\left\lVert\bm{\gamma}^{\prime}(t)\right\rVert\bm{K}(t)\bm{Q},$ (10) where $\bm{K}(t)\in\mathbb{R}^{r\times r}$. By factoring out the term $\left\lVert\bm{\gamma}^{\prime}(t)\right\rVert$ from $\bm{K}(t)$, it is guaranteed that $\bm{K}(t)$ does not depend on the parametrization of the curve (i.e. the speed of the trajectory), but only on its geometry. The matrix $\bm{K}(t)$ is highly structured and sparse; the nonzero elements of $\kappa_{i}(t)$ are defined to be the curvatures of the trajectory. The curvatures $\kappa_{i}(t)$ combined with the basis vectors $\bm{e}_{i}(t)$ define the Frenet-Serret apparatus, which fully characterizes the trajectory up to translation [33]. To understand the structure of $\bm{K}(t)$ we derive two key properties: 1. 1. $\bm{K}_{i,j}(t)=-\bm{K}_{j,i}(t)$ (antisymmetry): ###### Proof. Since $r\leq m$, then by construction $\bm{Q}(t)$ is a unitary matrix with $\bm{QQ}^{\intercal}=\bm{I}$. Taking the derivative with respect to $t$, $\frac{d\bm{Q}}{dt}\bm{Q}^{T}+\bm{Q}\frac{d\bm{Q}^{\intercal}}{dt}=0$, or equivalently $\frac{d\bm{Q}}{dt}\bm{Q}^{\intercal}=-\left(\frac{d\bm{Q}}{dt}\bm{Q}^{\intercal}\right)^{\intercal}.$ Since $\bm{Q}$ is unitary, then $\bm{Q}^{-1}=\bm{Q}^{\intercal}$, and hence $\bm{K}(t)=\frac{1}{\left\lVert\bm{\gamma}^{\prime}(t)\right\rVert}\frac{d\bm{Q}}{dt}\bm{Q}^{\intercal},$ from which we immediately see that $\bm{K}(t)=-\bm{K}(t)^{\intercal}$. ∎ 2. 2. $\bm{K}_{i,j}(t)=0$ for $j\geq i+2$: We first note that since $\bm{e}_{i}(t)\in\text{span}\\{\bm{\gamma}^{\prime}(t),\ldots,\bm{\gamma}^{i}(t)\\}$, its derivative must satisfy $\bm{e}_{i}^{\prime}(t)\in\text{span}\\{\bm{\gamma}^{\prime}(t),\ldots,\bm{\gamma}^{(i+1)}(t)\\}$. Now by construction, using the Gram-Schmidt method, $\bm{e}_{j}$ is orthogonal to $\text{span}\\{\bm{\gamma}^{\prime}(t),\ldots,\bm{\gamma}^{(i+1)}(t)\\}$ for $j\geq i+2$. Since $\bm{e}_{i}^{\prime}(t)$ is in the span of this set, then $\bm{e}_{j}$ must be orthogonal to $\bm{e}^{\prime}_{i}$ for $j\geq i+2$. Thus, $\bm{K}_{i,j}(t)=\langle\bm{e}_{i}^{\prime}(t),\bm{e}_{j}\rangle=0$ for $j\geq i+2$. With these two constraints, $\bm{K}(t)$ takes the form, $\bm{K}(t)=\begin{bmatrix}0&\kappa_{1}(t)&&0\\\ -\kappa_{1}(t)&\ddots&\ddots&&\\\ &\ddots&0&\kappa_{r-1}(t)\\\ 0&&-\kappa_{r-1}(t)&0\end{bmatrix}.$ (11) Thus $\bm{K}(t)$ is antisymmetric with nonzero elements only along the superdiagonal and subdiagonal, and the values $\kappa_{1}(t),\ldots,\kappa_{r-1}(t)$ are defined to be the curvatures of the trajectory. From a geometric perspective, $\bm{e}_{1}(t),\ldots,\bm{e}_{r}(t)$ form an instantaneous (local) coordinate frame, which moves with the trajectory. The curvatures define how quickly this frame changes with time. If the trajectory is a straight line the curvatures are all zero. If $\kappa_{1}$ is constant and nonzero, while all other curvatures are zero, then the trajectory lies on a circle. If $\kappa_{1}$ and $\kappa_{2}$ are constant and nonzero with all other curvatures zero, then the trajectory lies on a helix. Comparing the structure of (11) to Figure 2 we immediately see a similarity. Over the following sections we will shed light on this connection. ### 2.5 SVD and Curvature Given time series data, the SVD constructs an orthonormal basis that is fixed in time, whereas the Frenet-Serret frame constructs an orthonormal basis that moves with the trajectory. In recent work, Alvarez-Vizoso et al. [33] showed how these frames are related. In particular, the Frenet-Serret frame converges to the SVD frame in the limit as the time interval of the trajectory goes to zero. To understand this further, consider a trajectory $\bm{\gamma}(t)\in\mathbb{R}^{m}$ as described in Section 2.4. If we assume that our measurements are from a small neighborhood $t\in(-\epsilon,\epsilon)$ (where $\epsilon\ll 1$), then $\bm{\gamma}(t)$ is well-approximated by its Taylor expansion, $\bm{\gamma}(t)-\bm{\gamma}(0)=\bm{\gamma}^{\prime}(0)t+\frac{\bm{\gamma}^{\prime\prime}(0)}{2}t^{2}+\frac{\bm{\gamma}^{\prime\prime\prime}(0)}{6}t^{3}+\cdots$ Writing this in matrix form, we have that $\bm{\gamma}(t)-\bm{\gamma}(0)=\underbrace{\begin{bmatrix}\|&\|&\|&\|\\\ \bm{\gamma}^{\prime}(0)&\bm{\gamma}^{\prime\prime}(0)&\bm{\gamma}^{\prime\prime\prime}(0)&\cdots\\\ \|&\|&\|&\|\end{bmatrix}}_{\bm{\Gamma}}\underbrace{\begin{bmatrix}1&&&\\\ &\frac{1}{2}&&\\\ &&\frac{1}{6}&\\\ &&&\ddots\end{bmatrix}}_{\bm{\Sigma}}\underbrace{\begin{bmatrix}-&t&-\\\ -&t^{2}&-\\\ -&t^{3}&-\\\ -&\vdots&-\end{bmatrix}}_{\bm{T}^{\intercal}}.$ (12) Recall one key property of the SVD is that the $r$th rank truncation in the expansion is the best rank-$r$ approximation to the data in the least squares sense. Since $\epsilon\ll 1$, then each subsequent term in this expansion is much smaller than the previous term, $\left\lVert\bm{\gamma}^{\prime}(0)t\right\rVert_{2}\ll\left\lVert\frac{\bm{\gamma}^{\prime\prime}(0)}{2}t^{2}\right\rVert_{2}\ll\left\lVert\frac{\bm{\gamma}^{\prime\prime\prime}(0)}{6}t^{3}\right\rVert_{2}\ll\ldots.$ (13) From this, we see that the expansion in (12) is strongly related to the SVD. However, in the SVD we have the constraint that the $\bm{U}$ and $\bm{V}$ matrices are orthogonal, while for the Taylor expansion $\bm{\Gamma}$ and $\bm{T}$ have no such constraint. Alvarez et al. [33] show that in the limit as $\epsilon\to 0$, then $\bm{U}$ is the result of applying the Gram-Schmidt process to the columns of $\bm{\Gamma}$, and $\bm{V}$ is the result of applying the Gram-Schmidt process to the columns of $\bm{T}$. Comparing this to above, we see that $\bm{U}=\begin{bmatrix}\|&\|&\|&\|\\\ \bm{e}_{1}(0)&\bm{e}_{2}(0)&\bm{e}_{3}(0)&\cdots\\\ \|&\|&\|&\|\end{bmatrix}\mbox{ and }\bm{V}=\begin{bmatrix}\|&\|&\|&\|\\\ p_{1}(t)&p_{2}(t)&p_{3}(t)&\cdots\\\ \|&\|&\|&\|\end{bmatrix},$ where $\bm{e}_{1}(t),\bm{e}_{2}(t),\ldots,\bm{e}_{r}(t)$ is the basis for the Frenet-Serret frame defined in (9) and $p_{i}(t)=\frac{t^{i}-\sum_{j=1}^{i-1}\left\langle t^{i},p_{j}(t)\right\rangle p_{j}(t)}{\left\lVert t^{i}-\sum_{j=1}^{i-1}\left\langle t^{i},p_{j}(t)\right\rangle p_{j}(t)\right\rVert}\text{ for }i=1,2,3,\ldots$ (14) We note that the $p_{i}(t)$’s form a set of orthogonal polynomials independent of the dataset. In this limit, the curvatures depend solely on the singular values, $\kappa_{i}(t)=\sqrt{a_{i}}\frac{\sigma_{i+1}}{\sigma_{1}(t)\sigma_{i}(t)}\text{, where }a_{i-1}=\left(\frac{i}{i+(-1)^{i}}\right)^{2}\frac{4i^{2}-1}{3}.$ ## 3 Unifying Singular Value Decomposition, Time Delay Embeddings, and the Frenet-Serret Frame \begin{overpic}[width=433.62pt]{Figures/Fig3.pdf} \end{overpic} Figure 3: An illustration of how a highly structured, antisymmetric linear model arises from time delay data. Starting with a one dimensional time-series, we construct a $m\times n$ Hankel matrix using time-shifted copies of the data. Assume that $n\gg m$, in which case $\bm{H}$ can be thought of as an $m$ dimensional trajectory over a long period ($n$ snapshots in time). Similarly, the transpose of $\bm{H}$ may be thought of as a high dimensional ($n$ dimensional) trajectory over a short period ($m$ snapshots) in time. With this interpretation, by the results of [33], the singular vectors of $\bm{H}$ after applying centering yield the Frenet-Serret frame. Regression on the dynamics in the Frenet-Serret frame yields the tridiagonal antisymmetric linear model with an additional forcing term, which is nonzero only in the last component. In this section, we show that time series data from a dynamical system may be decomposed into a sparse linear dynamical model with nonlinear forcing, and the nonzero elements along the sub- and super-diagonals of the linear part of this model have a clear geometric meaning: they are curvatures of the system. In Section 3.1, we combine key results about the Frenet-Serret frame, time delays, and SVD to explain this structure. Following this theory, Section 3.2 illustrates this approach with a simple synthetic example. The decomposition yields a set of orthogonal polynomials that form a coordinate basis for the time-delay embedding. In Section 3.3, we explicitly describe these polynomials and compare their properties to the Legendre polynomials. ### 3.1 Connecting SVD, Time Delay Embeddings, and Frenet-Serret Frame Here we connect the properties of the SVD, time delay embeddings, and the Frenet-Serret to decompose a dynamical model into a linear dynamical model with nonlinear forcing, where the linear model is both antisymmetric and tridiagonal. To do this, we follow the steps of the HAVOK method with slight modifications and show how they give rise to these structured dynamics. This process is illustrated in Figure 3. Following the notation introduced in Section 2.3, let’s begin with the time series $x(t)$ for $t=0,\Delta t,\ldots,q\Delta t$. We construct a time delay embedding $\bm{H}\in\mathbb{R}^{m\times n}$, where we assume $m\ll n$. Next we compute the SVD of $\bm{H}$ and show that the singular vectors correspond to the Frenet-Serret frame at a fixed point in time. In particular, to compute the SVD of this matrix, we consider the transpose $\bm{H}^{\intercal}\in\mathbb{R}^{n\times m}$, which is also be a Hankel matrix. Thus, the columns of $\bm{h}(t)$ can be thought of as a trajectory $\bm{h}(t)\in\mathbb{R}^{n}$ for $t=0,\Delta t,\ldots,(m-1)\Delta t$. For simplicity, we shift the origin of time so that $\bm{h}(t)$ spans $t=-(m-1)\Delta t/2,\ldots,0,\ldots(m-1)\Delta t/2$, and we denote $\bm{h}(i\Delta t)$ as $\bm{h}_{i}$. In this form, $\bm{H}^{\intercal}=\begin{bmatrix}\|&\cdots&\|&\cdots&\|\\\ \bm{h}_{(-m+1)/2}&\cdots&\bm{h}_{0}&\cdots&\bm{h}_{(m-1)/2}\\\ \|&\cdots&\|&\cdots&\|\\\ \end{bmatrix}.$ Subtracting the central column $\bm{h}_{0}$ from $\bm{H}^{\intercal}$ (or equivalently, the central row of $\bm{H}$) yields the centered matrix $\bar{\bm{H}}^{\intercal}=\bm{H}^{\intercal}-\bm{h}_{0}\bm{1}^{\intercal}.$ (15) We can then express $\bm{h}_{i}$ as a Taylor expansion about $\bm{h}_{0}$, $\bm{h}_{i}-\bm{h}_{0}=\bm{h}_{0}^{{}^{\prime}}i\Delta t+\frac{1}{2}\bm{h}_{0}^{\prime\prime}(i\Delta t)^{2}+\frac{1}{3!}\bm{h}_{0}^{\prime\prime\prime}(i\Delta t)^{3}+\cdots.$ With this in mind, applying the results of [33] described in Section 2.5 yields the SVD111We define the left singular matrix as $\bm{V}$ and the right singular matrix as $\bm{U}$. This definition can be thought of as taking the SVD of the transpose of the matrix $\bm{H}-\bm{1}\bm{h}_{0}^{\intercal}$. This keeps the definitions of the matrices more inline with the notation used in HAVOK., $\bar{\bm{H}}^{\intercal}=\underbrace{\begin{bmatrix}\|&\|&\|&\\\ \bm{e}_{0}^{1}&\bm{e}_{0}^{2}&\bm{e}_{0}^{3}&\cdots\\\ \|&\|&\|&\end{bmatrix}}_{\bm{V}}\underbrace{\begin{bmatrix}\sigma_{1}&&&\\\ &\sigma_{2}&&\\\ &&\sigma_{3}&\\\ &&&\ddots\end{bmatrix}}_{\bm{\Sigma}}\underbrace{\begin{bmatrix}-&\bm{p}_{1}&-\\\ -&\bm{p}_{3}&-\\\ -&\bm{p}_{3}&-\\\ &\vdots&\end{bmatrix}}_{\bm{U}^{\intercal}}.$ (16) The singular vectors in $\bm{V}$ correspond to the Frenet-Serret frame (the Gram-Schmidt method applied to the vectors, $\bm{h}^{\prime}_{0},\bm{h}^{\prime\prime}_{0},\bm{h}^{\prime\prime\prime}_{0}$), $\displaystyle\bm{e}_{0}$ $\displaystyle=\frac{\bm{h}_{0}^{\prime}}{\left\lVert\bm{h}_{0}^{\prime}\right\rVert}$ $\displaystyle\bm{e}_{0}^{i}$ $\displaystyle=\frac{\bm{h}_{0}^{(i)}-\sum_{j=1}^{i-1}\langle\bm{h}_{0}^{(i)},\bm{e}_{0}^{j}\rangle\bm{e}_{0}^{j}}{\left\lVert\bm{h}_{0}^{(i)}-\sum_{j=1}^{i-1}\langle\bm{h}_{0}^{(i)},\bm{e}_{0}^{j}\rangle\bm{e}_{0}^{j}\right\rVert}.$ The matrix $\bm{U}$ is similarly defined by the discrete orthogonal polynomials $\displaystyle\bm{p}_{1}$ $\displaystyle=\frac{1}{c}_{1}\bm{p}$ $\displaystyle\bm{p}_{i}$ $\displaystyle=\frac{1}{c_{i}}\left(\bm{p}^{i}-\sum_{j=1}^{i-1}\langle\bm{p}^{i},\bm{p}_{j}\rangle\bm{p}_{j}\right),$ where $\bm{p}$ is the vector $\bm{p}=\begin{bmatrix}(-m+1)/2&(-m+2)/2&\cdots&0&\cdots&(m-2)/2&(m-1)/2\end{bmatrix},$ (17) and where $c_{i}$ is a normalization constant so that $\langle\bm{p}_{i},\bm{p}_{i}\rangle=1$. Note that $\bm{p}^{i}$ here means raise $\bm{p}$ to the power $i$ element-wise. These polynomials are similar to the discrete orthogonal polynomials defined in [63], except $\bm{p}$ is the normalized ones vector $\frac{1}{c_{1}}\left[1\cdots 1\right]$. These polynomials will be discussed further in Section 3.3. Next, we build a regression model of the dynamics. We first consider the case where the system is closed (i.e. $\bar{\bm{H}}$ has rank $r$). Thinking of $\bm{V}$ as the Frenet-Serret frame at a fixed point in time, then following the Frenet-Serret equations (10), $\frac{\bm{d\bm{V}}}{dt}^{\intercal}=\bm{A}\bm{V}^{\intercal},$ (18) where $\bm{A}=\left\lVert\bm{h}^{\prime}_{0}\right\rVert\bm{K}$. Here $\bm{K}$ is a constant tridiagonal and antisymmetric matrix, which corresponds to the curvatures at $t=0$. From the dual perspective, we can think about this set of vectors $\bm{e}^{0}$ as an $r$-dimensional time series over $n$ snapshots in time, $\bm{V}^{\intercal}=\begin{bmatrix}-&v_{1}(t)&-\\\ -&v_{2}(t)&-\\\ &\vdots&\\\ -&v_{r}(t)&-\\\ \end{bmatrix}=\begin{bmatrix}-&\bm{e}_{0}^{1}&-\\\ -&\bm{e}_{0}^{2}&-\\\ &\vdots&\\\ -&\bm{e}_{0}^{r}&-\\\ \end{bmatrix}\in\mathbb{R}^{r\times n}.$ (19) Here $\bm{v}(t)=[v_{1}(t),v_{2}(t),\cdots v_{r}(t)]^{\intercal}\in\mathbb{R}^{r}$ denotes the $r$-dimensional trajectory, which corresponds to the $r$-dimensional coordinates considered in (5) for HAVOK. From (18), these dynamics must therefore satisfy $\dot{\bm{v}}(t)=\bm{A}\bm{v}(t),$ where $\bm{A}$ is a skew-symmetric tridiagonal matrix. If the system is not closed, the dynamics take the form $\begin{bmatrix}\dot{v}_{1}\\\ \dot{v}_{2}\\\ \vdots\\\ \dot{v}_{r}\\\ \dot{v}_{r+1}\\\ \vdots\end{bmatrix}=\left\lVert\bm{h}^{\prime}_{0}\right\rVert\begin{bmatrix}0&\kappa_{1}&&&&\\\ -\kappa_{1}&\ddots&\ddots&&&&\\\ &\ddots&0&\ddots&\\\ &&-\kappa_{r-1}&0&\kappa_{r}&\\\ &&&-\kappa_{r}&0&\ddots\\\ &&&&\ddots&\ddots&\end{bmatrix}\begin{bmatrix}v_{1}\\\ v_{2}\\\ \vdots\\\ v_{r}\\\ v_{r+1}\\\ \vdots\end{bmatrix}.$ We note that, due to the tridiagonal structure of $\bm{K}$, the governing dynamics of the first $r-1$ coordinates $v_{1}(t),\ldots v_{r-1}(t)$ are the same as in the unforced case. The dynamics of the last coordinate includes an additional term $\dot{v}_{r}=-\kappa_{r-1}v_{r-1}+\kappa_{r+1}v_{r+1}$. The dynamics therefore take the form, $\frac{d\bm{v}}{dt}=\bm{A}\bm{v}(t)+\bm{B}v_{r+1}(t),$ where $\bm{B}$ is a vector that is nonzero only its last coordinate. Thus, we recover a model as in (8), but with the desired tridiagonal skewsymmetric structure. The matrix of curvatures is simply given by $\bm{K}=\bm{A}/\left\lVert\bm{h}_{0}^{\prime}\right\rVert$. To compute $\bm{A}$, similar to (6), we define two time shifted matrices $\bm{V}_{1}=\begin{bmatrix}\bm{v}(t_{1})&\bm{v}(t_{2})&\cdots&\bm{v}(t_{m-1})\end{bmatrix}\quad\bm{V}_{2}=\begin{bmatrix}\bm{v}(t_{2})&\bm{v}(t_{3})&\cdots&\bm{v}(t_{m})\end{bmatrix}.$ (20) The matrix $\bm{A}$ may then be approximated as $\bm{A}=\frac{d\bm{V}}{dt}^{\intercal}\bm{V}^{\intercal^{\dagger}}\approx\left(\frac{\bm{V}_{2}-\bm{V}_{1}}{\Delta t}\right)\bm{V}_{1}^{\dagger}.$ (21) In summary, we have shown here that the trajectories of singular vectors $\bm{v}(t)$ from a time-delay embedding are governed by approximately tridiagonal antisymmetric dynamics, with a forcing term nonzero only in the last component. Comparing these steps to those described in Section 2.3, we see that the estimation of $\bm{K}$ is nearly identical to the steps in HAVOK. In particular, $\left\lVert\bm{h}_{0}\right\rVert\bm{K}$ is the linear dynamics matrix $\bm{A}$ in HAVOK. The only difference is the centering step in (15), which is further discussed in Section 3.3. ### 3.2 HAVOK Computes Approximate Curvatures in a Synthetic Example To illustrate the correspondence between nonzero elements of the HAVOK dynamics matrix and curvatures, we start by considering an analytically tractable synthetic example. We start by applying the steps of HAVOK as described in [5] with an additional centering step. The resultant modes and terms on the sub- and superdiagonals of the dynamics matrix are then compared to curvatures computed with an analytic expression, and we show that they are approximately the same, scaled by a factor of $\left\lVert\bm{h}^{\prime}_{0}\right\rVert$. We consider data from the one dimensional system governed by $x(t)=\sin(t)+\sin(2t),$ for $t\in[0,10]$ and sampled at $\Delta t=0.001$. Following HAVOK, we form the time delay matrix $\bm{H}\in\mathbb{R}^{41\times 9961}$ then center the data, subtracting the middle row $\bm{h}_{0}$ from all other rows, which forms $\bar{\bm{H}}$. We next apply the SVD to $\bar{\bm{H}}^{\intercal}=\bm{V}\bm{\Sigma}\bm{U}^{\intercal}$. Figure 4 shows the columns of $\bm{U}\in\mathbb{R}^{41\times 4}$ and the columns of $\bm{V}\in\mathbb{R}^{9961\times 4}$. The columns of $\bm{U}$ correspond to the orthogonal polynomials described in Section 3.3 and the columns of $\bm{V}$ are the instantaneous basis vectors $\bm{e}_{i}$ for the $9961$ dimensional Frenet-Serret frame. To compute the derivative of the state we now treat $\bm{V}$ as a 4 dimensional trajectory with $9961$ snapshots. Applying DMD to $\bm{V}$ yields the $\bm{A}$ matrix, $\bm{A}=\begin{bmatrix}-1.245\times 10^{-3}&\hbox{\pagecolor{Orange!70}$\displaystyle 1.205\times 10^{-2}$}&4.033\times 10^{-6}&1.444\times 10^{-7}\\\ \hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-1.224\times 10^{-2}$}&3.529\times 10^{-4}&\hbox{\pagecolor{Orange!70}$\displaystyle 4.458\times 10^{-3}$}&2.283\times 10^{-6}\\\ -9.390\times 10^{-4}&\hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-3.467\times 10^{-3}$}&5.758\times 10^{-4}&\hbox{\pagecolor{Orange!70}$\displaystyle 6.617\times 10^{-3}$}\\\ 3.970\times 10^{-4}&-6.568\times 10^{-4}&\hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-7.451\times 10^{-3}$}&2.835\times 10^{-4}\\\ \end{bmatrix}.$ (22) This matrix is approximately antisymmetric and tridiagonal as we expect. Next, we compute the Frenet-Serret frame for the time delay embedding using analytic expressions and show that HAVOK indeed extracts the curvatures of the system multiplied by $\left\lVert\bm{h}^{\prime}_{0}\right\rVert$. Forming the time delay matrix, we can easily compute $\bm{h}_{0}=[x_{0.02},x_{0.02+\Delta t}\ldots,x_{9.98}]$. $\bm{h}_{0}=\begin{bmatrix}\sin(t)+\sin(2t)\text{ for }t\in[0.02,0.021,\ldots,9.98]\end{bmatrix}$ and the corresponding derivatives, $\displaystyle\dot{\bm{h}}_{0}=\begin{bmatrix}\cos(t)+2\cos(2t)\text{ for }t\in[0.02,0.021,\ldots,9.98]\end{bmatrix}$ $\displaystyle\ddot{\bm{h}}_{0}=\begin{bmatrix}-\sin(t)-4\sin(2t)\text{ for }t\in[0.02,0.021,\ldots,9.98]\end{bmatrix}$ $\displaystyle\dddot{\bm{h}}_{0}=\begin{bmatrix}-\cos(t)-8\cos(2t)\text{ for }t\in[0.02,0.021,\ldots,9.98]\end{bmatrix}$ $\displaystyle\bm{h}^{(4)}_{0}=\begin{bmatrix}\sin(t)+16\sin(2t)\text{ for }t\in[0.02,0.021,\ldots,9.98]\end{bmatrix}.$ The $5$th derivative $\bm{h}^{(5)}$ is given by $\cos(t)+32\cos(2t)$ and can be expressed as a linear combination of the previous derivatives, namely, $\bm{h}_{0}^{(5)}=-5\dddot{\bm{h}}_{0}-4\dot{\bm{h}}_{0}$. This can also be shown using the fact that $x(t)$ satisfies the $4$th order ordinary differential equation $x^{(4)}+5\ddot{x}+4x=0$. Since only the first four derivatives are linearly independent, only the first three curvatures are nonzero. Further, exact values of the first three curvatures can be computed analytically using the following formulas from [64], $\kappa_{1}=\frac{\sqrt{\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix})}}{\left\lVert\dot{\bm{h}}_{0}\right\rVert^{3/2}}\mbox{, \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ }\kappa_{2}=\frac{\sqrt{\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}\end{bmatrix})}}{\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix})},$ $\kappa_{3}=\frac{\sqrt{\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}&\bm{h}^{(4)}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}&\bm{h}^{(4)}_{0}\end{bmatrix})\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}\end{bmatrix})}}{\det(\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}\end{bmatrix}^{\intercal}\begin{bmatrix}\dot{\bm{h}}_{0}&\ddot{\bm{h}}_{0}&\dddot{\bm{h}}_{0}\end{bmatrix})\left\lVert\bm{h}_{0}\right\rVert}.$ These formulas yields the values $\kappa_{1}=1.205\times 10^{-2}$, $\kappa_{2}=4.46\times 10^{-3}$, and $\kappa_{3}=6.62\times 10^{-3}$. As expected, these curvature values are very close to those computed with HAVOK, highlighted in (22). In particular, the superdiagonal entries of the matrix appear to be a very good approximations to the curvatures. The reasons why the superdiagonal, but not the subdiagonal, is so close in value to the true curvatures is not yet well understood. Further, in Section 5, we use the theoretical insights from Section 3.1 to propose a modification to the HAVOK algorithm that yields an even better approximation to curvatures in the Frenet-Serret frame. \begin{overpic}[width=433.62pt]{Figures/Fig4.pdf} \end{overpic} Figure 4: Frenet-Serret frame (left) and corresponding orthogonal polynomials (right) for HAVOK applied to time-series generated by $x(t)=\sin(t)+\sin(2t)$. The orthogonal polynomials and the Frenet-Serret frame are the right singular vectors $\bm{U}$ and left singular vectors $\bm{V}$ of $\bar{\bm{H}}$, respectively. ### 3.3 Orthogonal Polynomials and Centering In the decomposition in (16), we define a set of orthonormal polynomials. Here we discuss the properties of these polynomials, comparing them to the Legendre polynomials and providing explicit expressions for the first several terms in this series. In Section 3.1, we apply the SVD to the centered matrix $\bar{\bm{H}}$, as in (16). The columns of $\bm{U}$ in this decomposition yield a set of orthonormal polynomials, which are defined by (14). In the continuous case, the inner product in (14) is $\langle a(t),b(t)\rangle=\int_{-p}^{p}a(t)b(t)dt$, while in the discrete case $\langle a,b\rangle=\sum_{j=-p}^{p}a_{j}b_{j}$. The first five polynomials in the discrete case may be found in Appendix A. The first five of these polynomials $p_{i}(x)$ in the continuous case are: $\displaystyle p_{1}(x)=\frac{x}{c_{1}(p)}\text{, where }c_{1}(p)=\frac{\sqrt{6}\,\sqrt{p^{3}}}{3}$ $\displaystyle p_{2}(x)=\frac{x^{2}}{c_{2}(p)}\text{, where }c_{2}(p)=\frac{\sqrt{10}\,\sqrt{p^{5}}}{5}$ $\displaystyle p_{3}(x)=\frac{1}{c_{3}(p)}\left(x^{3}-\frac{3}{5}p^{2}x\right)\text{, where }c_{3}(p)=\frac{2\,\sqrt{14}\,\sqrt{p^{7}}}{35}$ $\displaystyle p_{4}(x)=\frac{1}{c_{4}(p)}\left(x^{4}-\frac{5}{7}p^{2}x^{2}\right)\text{, where }c_{4}(p)=\frac{2\,\sqrt{2}\,\sqrt{p^{9}}}{21}$ $\displaystyle p_{5}(x)=\frac{1}{c_{5}(p)}\left(x^{5}+\frac{5}{21}p^{4}x-\frac{10}{9}p^{2}x^{3}\right)\text{, where }c_{5}(p)=\frac{8\,\sqrt{22}\,\sqrt{p^{11}}}{693}.$ By construction, $p_{i}(t)$ form a set of orthonormal polynomials, where $p_{i}(t)$ has degree $i$. Interestingly, these orthogonal polynomials are similar to the Legendre polynomials $\bm{l}_{i}$ [65, 66], which are defined by the recursive relation $\displaystyle\bm{l}_{1}=\frac{1}{c}_{1}\begin{bmatrix}1&1&\cdots&1\end{bmatrix}$ $\displaystyle\bm{l}_{i}=\frac{1}{p_{i}}\left(\bm{p}^{i}-\sum_{k=1}^{i-1}\langle\bm{p}^{i},\bm{l}_{k}\rangle\right),$ where $\bm{p}$ is as defined in (17). For the corresponding Legendre polynomials normalized over $[-p,p]$, we refer the reader to [63]. The key difference between these two sets of polynomials is that the first polynomial $\bm{p}_{1}$ is linear, while the first Legendre polynomial is constant (i.e., corresponding in the discrete case to the normalized ones vector). In particular, if $\bm{H}$ is not centered before decomposition by SVD, the resulting columns of $\bm{U}$ will be the Legendre polynomials. However, without centering, the resulting $\bm{V}$ will no longer be the Frenet-Serret frame. Instead, the resulting frame corresponds to applying the Gram-Schmidt method to the set $\left\\{\bm{\gamma}(t),,\bm{\gamma}^{\prime}(t),\bm{\gamma}^{\prime\prime}(t),...\right\\}$ instead of $\left\\{\bm{\gamma}^{\prime}(t),\bm{\gamma}^{\prime\prime}(t),\bm{\gamma}^{\prime\prime\prime}(t),...\right\\}$. Recently it has been shown that using centering as a preprocessing step is beneficial for the dynamic mode decomposition [67]. That being said, since the derivation of the tridiagonal and antisymmetric structure seen in the Frenet- Serret frame is based on the properties of the derivatives and orthogonality, this same structure can be computed without the centering step. ## 4 Limits and Requirements Section 3.1 has shown how HAVOK yields a good approximation to the Frenet- Serret frame in the limit that the time interval spanned by each row $\bm{H}$ goes to zero. To be more precise, HAVOK yields the Frenet-Serret frame if (13) is satisfied. However, this property can be difficult to check in practice. Here we establish several rules for choosing and structuring the data so that the HAVOK dynamics matrix adopts the structure we expect from theory. Figure 5: Increasing sampling frequency and number of columns yields more structured HAVOK models for the Lorenz system. Given the Hankel matrix $\bm{H}$, the linear dynamical model is plotted for values of sampling period $\Delta t$ equal to $0.01,0.005,0.001,0.0005$ for a fixed number of rows and fixed time span of measurement (top). Similarly, the model is plotted for values of number of columns $n$ equal to $1001,2001,5001,$ and $10001$ for fixed sampling frequency and time span of measurement $q\Delta t$(bottom). As we increase the sampling frequency and the number of columns of the data, $\bm{A}$ becomes more antisymmetric with nonzero elements only on the super- and sub-diagonals. These trends illustrate the results in Section 4. Choose $\Delta t$ to be small. The specific constraint we have from (13) is $\left\lVert\bm{h}^{\prime}_{0}t_{i}\right\rVert\gg\left\lVert\frac{\bm{h}^{\prime\prime}_{0}}{2}t_{i}^{2}\right\rVert\gg\left\lVert\frac{\bm{h}_{0}^{\prime\prime\prime}}{6}t_{i}^{3}\right\rVert\gg\cdots\gg\left\lVert\frac{\bm{h}_{0}^{(k)}}{k!}t_{i}^{k}\right\rVert,$ for $-m\Delta t/2\leq t_{i}\leq m\Delta t/2$ or more simply $\lvert t_{i}\rvert\leq m\Delta t$, where $\Delta t$ is the sampling period of the data and $m$ is the number of delays in the Hankel matrix $\bm{H}$. If we assume that $m\Delta t<1$, then rearranging, $m\Delta t\ll\frac{2\left\lVert\bm{h}_{0}^{\prime}\right\rVert}{\left\lVert\bm{h}_{0}^{\prime\prime}\right\rVert},\frac{3\left\lVert\bm{h}_{0}^{\prime\prime}\right\rVert}{\left\lVert\bm{h}_{0}^{\prime\prime\prime}\right\rVert},\ldots,\frac{k\left\lVert\bm{h}_{0}^{(k-1)}\right\rVert}{\left\lVert\bm{h}_{0}^{(k)}\right\rVert}.$ (23) In practice, since the series of ratios of derivatives defined in (23) grows, it is only necessary to check the first inequality. By choosing the sampling period of the data to be small, we can constrain the data to satisfy this inequality. To illustrate the effect of decreasing $\Delta t$, Figure 5 (top) shows the dynamics matrices $\bm{A}$ computed by the HAVOK algorithm for the Lorenz system for a fixed number of rows of data and fixed time span of the simulation. As $\Delta t$ becomes smaller, $\bm{A}$ becomes more structured in that it is antisymmetric and tridiagonal. Choose the number of columns $n$ to be large. The number of columns comes into the Taylor expansion through the derivatives $\left\lVert\bm{h}_{0}^{(k)}\right\rVert$, since $\bm{h}_{0}^{(k)}\in\mathbb{R}^{n}$. For the synthetic example $x(t)=\sin(t)+2\sin(t)$, we can show that the ratio $2\left\lVert\bm{h}^{\prime}_{0}\right\rVert/\left\lVert\bm{h}^{\prime\prime}_{0}\right\rVert$ saturates to a fixed value in the limit as $n$ goes to infinity (see Appendix B). However, for short time series (small values of $n$), this ratio can be arbitrarily small, and hence (23) will be difficult to satisfy. We illustrate this in Figure 5 using data from the Lorenz system. We compute and plot the HAVOK linear dynamics matrix for a varying number of columns $n$, while fixing the sampling frequency and time span of measurements $q\Delta t$. We see that as we increase the number of columns, the dynamics becomes more skew symmetric and tridiagonal. In general, due to practical constraints and restrictions, it may be difficult to guarantee that given data satisfies these two requirements. In Sections 4.1 and 5, we propose methods to tackle this challenge. ### 4.1 Interpolation From the first requirement, we see that the sampling frequency $\Delta t$ needs to be sufficiently small to recover the antisymmetric structure in $\bm{A}$. However, in practice, it is not always possible to satisfy this sampling criterion. One solution to remedy this is to use data interpolation. To be precise, we can increase the sampling rate by spline interpolation, then construct $\bm{H}$ from the interpolated data that satisfies $\eqref{eq:series}$. The ratio of the derivatives $\left\lVert\bm{h}^{\prime}_{0}\right\rVert/\left\lVert\bm{h}^{\prime\prime}_{0}\right\rVert,\left\lVert\bm{h}^{\prime\prime}_{0}\right\rVert/\left\lVert\bm{h}^{\prime\prime\prime}_{0}\right\rVert,\ldots$ may also contain some dependence on $\Delta t$, but we observe that this dependence is not significantly affected in practice. Figure 6: In the case where a dynamical system is sparsely sampled, interpolation can be used to recover a more tridiagonal and antisymmetric matrix for the linear model in HAVOK. First, we simulate the Lorenz system, measuring $x(t)$ with a sampling period of $\Delta t=0.1$. The resulting dynamics model $\bm{A}$ and corresponding singular vectors of $\bm{U}$ are plotted. Due to the low sampling frequency these values do not satisfy the requirements in (23). Consequently the dynamics matrix is not antisymmetric and the singular vectors do not correspond to the orthogonal polynomials in Section 3.3. Next, the data is interpolated using cubic splines and subsequently sampled using a sampling period of $\Delta t=0.001$. In this case the data satisfies the assumptions in (23), which yields the tridiagonal antisymmetric structure for $\bm{A}$ and orthogonal polynomials for $\bm{U}$ as predicted. As an example, we consider a set of time series measurements generated from the Lorenz system (see Section 5 for more details about this system). We start with a sampling period of $\Delta t=0.1$ (Figure 6, top row). Note that here we have simulated the Lorenz system at high temporal resolution then subsampled to produce this timeseries data. Applying HAVOK with centering and $m=201$, we see that $\bm{A}$ is not antisymmetric and the columns of $\bm{U}$ are not the orthogonal polynomials like in the synthetic example shown in Figure 4. Next, we apply cubic spline interpolation to this data, evaluating at a sampling rate of $\Delta t=0.001$ (Figure 6, bottom row). We note that, especially for real-world data with measurement noise, this interpolation procedure also serves to smooth the data, making the computation of its derivatives more tractable [68]. Applying HAVOK to this interpolated data yields a new antisymmetric $\bm{A}$ matrix and the $\bm{U}$ corresponds to the orthogonal polynomials described in Section 3.3. ## 5 Promoting structure in the HAVOK decomposition HAVOK yields a linear model of a dynamical system explained by the Frenet- Serret frame, and by leveraging these theoretical connections, here we propose a modification of the HAVOK algorithm to promote this antisymmetric structure. We refer to this algorithm as structured HAVOK (sHAVOK) and describe it in Section 5.1. Compared to HAVOK, sHAVOK yields structured dynamics matrices that better approximate the Frenet-Serret frame and more closely estimate the curvatures. Importantly, sHAVOK also produces better models of the system using significantly less data. We demonstrate its application to three nonlinear synthetic example systems in Section 5.2 and two real-world datasets in Section 5.3. ### 5.1 The Structured HAVOK (sHAVOK) Algorithm We propose a modification to the HAVOK algorithm that more closely induces the antisymmetric structure in the dynamics matrix, especially for shorter data with a smaller number of delays $n$. The key innovation in sHAVOK is the application of two SVD’s applied separately to time-shifted Hankel matrices (compare Figure 2 and Figure 7). This simple modification enforces that the singular vector bases on which the dynamics matrix is computed are orthogonal, and thus more closely approximate the Frenet-Serret frame. Figure 7: Outline of steps in structured HAVOK (sHAVOK). First, given a dynamical system a single variable $x(t)$ is measured. Time-shifted copies of $x(t)$ are stacked to form a Hankel matrix $\bm{H}$. $\bm{H}$ is split into two time-shifted matrices, $\bm{H}_{1}$ and $\bm{H}_{2}$. The singular value decomposition (SVD) is applied to these two matrices individually. This results in reduced order representations, $\bm{V}_{1}$ and $\bm{V}_{2}$, of $\bm{H}_{1}$ and $\bm{H}_{2}$, respectively. The matrices, $\bm{V}_{1}$ and $\bm{V}_{2}$ are then used to construct an approximation to this low dimensional state and its derivative. Finally, linear regression is performed on these two matrices to form a linear dynamical model with an additional forcing term in the last component. Building on the HAVOK algorithm as summarized in Section 2.3, we focus on the step where the singular vectors $\bm{V}$ are split into $\bm{V}_{1}$ and $\bm{V}_{2}$. In the Frenet-Serret framework, we are interested in the evolution of the orthonormal frame $\bm{e}_{1}(t),\bm{e}_{2}(t),\ldots,\bm{e}_{r}(t)$. In HAVOK, $\bm{V}_{1}$ and $\bm{V}_{2}$ correspond to instances of this orthonormal frame. Although $\bm{V}$ is a unitary matrix, $\bm{V}_{1}$ and $\bm{V}_{2}$—which each consist of removing a column from $\bm{V}$—are not. To enforce this orthogonality, we propose to split $\bar{\bm{H}}$ into two time-shifted matrices $\bar{\bm{H}}_{1}$ and $\bar{\bm{H}}_{2}$ (Figure 7) and then compute two SVDs with rank truncation $r$, $\bar{\bm{H}}_{1}=\bm{U}_{1}\bm{\Sigma}_{1}\bm{V}_{1}^{\intercal}\text{ and }\bar{\bm{H}}_{2}=\bm{U}_{2}\bm{\Sigma}_{2}\bm{V}_{2}^{\intercal}.$ By construction, $\bm{V}_{1}$ and $\bm{V}_{2}$ are now orthogonal matrices. Like in HAVOK, our goal is to estimate the dynamics matrix $\bm{A}$ such that $\dot{\bm{v}}(t)=\bm{A}\bm{v}(t).$ To do so, we use the matrices $\bm{V}_{1}$ and $\bm{V}_{2}$ to construct the state and its derivative, $\displaystyle\bm{V}=\bm{V}_{1}$ $\displaystyle\frac{d\bm{V}}{dt}=\frac{\bm{V}_{2}-\bm{V}_{1}}{\Delta t}.$ $\bm{A}$ then satisfies $\bm{A}=d\bm{V}/dt\bm{V}^{\intercal^{\dagger}}\approx\left(\bm{V}_{2}^{\intercal}-\bm{V}_{1}\right)/\Delta t\bm{V}_{1}^{\intercal^{\dagger}}$ (24) If this system is not closed (nonzero forcing term), then $\bm{V}_{2}$ is defined as columns $2$ to $n-1$ of the SVD singular vectors with an $r-1$ rank truncation $\bm{V}_{r-1}^{\intercal}$, and $\hat{\bm{A}}\in\mathbb{R}^{r-1\times r-1}$ and $\hat{\bm{B}}\in\mathbb{R}^{r-1\times 1}$ are computed as $\left[\hat{\bm{A}},\hat{\bm{B}}\right]=\bm{V}_{2}^{\intercal}\bm{V}_{1}$. The corresponding pseudocode is elaborated in Appendix C. As a simple analytic example, we apply sHAVOK to the same system described in Section 3.2 generated by $x(t)=\sin(t)+\sin(2t)$. The resulting dynamics matrix is $\bm{A}=\begin{bmatrix}-1.116\times 10^{-5}&\hbox{\pagecolor{Orange!70}$\displaystyle 1.204\times 10^{-2}$}&-1.227\times 10^{-5}&8.728\times 10^{-8}\\\ \hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-1.204\times 10^{-2}$}&-1.269\times 10^{-5}&\hbox{\pagecolor{Orange!70}$\displaystyle 4.458\times 10^{-3}$}&4.650\times 10^{-6}\\\ 2.053\times 10^{-5}&\hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-4.458\times 10^{-3}$}&-4.897\times 10^{-6}&\hbox{\pagecolor{Orange!70}$\displaystyle 6.617\times 10^{-3}$}\\\ -9.956\times 10^{-8}&-1.118\times 10^{-7}&\hbox{\pagecolor{rgb:red!60,0.1216;green!60,0.466666;blue!60,0.705882}$\displaystyle-6.617\times 10^{-3}$}&-3.368\times 10^{-6}\\\ \end{bmatrix}.$ We see immediately that, with this small modification, $\bm{A}$ has become much more structured compared to (22). Specifically, the estimates of the curvatures both below and above the diagonal are now equal, and the rest of the elements in the matrix, which should be zero, are almost all smaller by an order of magnitude. In addition, the curvatures are equal to the true analytic values up to three decimal places. ### 5.2 Comparison of HAVOK and sHAVOK for Three Synthetic Examples The results of HAVOK and sHAVOK converge in the limit of infinite data, and the models they produce are most different in cases of shorter time series data, where we may not have measurements over long periods of time. Using synthetic data from three nonlinear example systems, we compute models using both methods and compare the corresponding dynamics matrices $\bm{A}$ (Figure 8). In every case, the $\bm{A}$ matrix computed using the sHAVOK algorithm is more antisymmetric and has a stronger tridiagonal structure than the corresponding matrix computed using HAVOK. In addition to the dynamics matrices, we also show in Figure 8 the eigenvalues of $\bm{A}$, $\omega_{k}\in\mathbb{C}$ for $k=1,\ldots r$ for HAVOK (teal) and sHAVOK (maroon). We additionally plot the eigenvalues (black crosses) corresponding to those computed from the data measured in the large data limit, but at the same sampling frequency. In this large data limit, both sHAVOK and HAVOK yield the same antisymmetric tridiagonal dynamics matrix and corresponding eigenvalues. Comparing the eigenvalues, we immediately see that eigenvalues from sHAVOK more closely match those computed in the large data limit. Thus, even with a short trajectory, we can still recover models and key features of the underlying dynamics. Below, we describe each of the systems and their configurations. Figure 8: Structured HAVOK (sHAVOK) yields more structured models from short trajectories than HAVOK on three examples. For each system, we simulated a trajectory extracting a single coordinate in time (gray). We then apply HAVOK and sHAVOK to data $x(t)$ from a short subset of this trajectory, shown in black. The middle columns show the resulting dynamics matrices $\bm{A}$ from the models. Compared to HAVOK, the resulting model for sHAVOK consistently shows stronger structure in that they are antisymmetric with nonzero elements only along the sub- and super-diagonals. The corresponding eigenvalue spectra of $\bm{A}$ for HAVOK and sHAVOK are plotted in teal and maroon, respectively, in addition to eigenvalues from HAVOK for the full (gray) trajectory. In all cases, the sHAVOK eigenvalues are much closer in value to those from the long trajectory limit than HAVOK. Lorenz Attractor: We first illustrate these two methods on the Lorenz system. Originally developed in the fluids community, the Lorenz (1963) system is governed by three first order differential equations [69]: $\displaystyle\dot{x}=\sigma(y-x)$ $\displaystyle\dot{y}=x(\rho-z)-y$ $\displaystyle\dot{z}=xy-\beta z.$ The Lorenz system has since been used to model systems in a wide variety of fields, including chemistry [70], optics [71], and circuits [72]. We simulate $3,000$ samples with initial condition $[-8,8,27]$ and a stepsize of $\Delta t=0.001$, measuring the variable $x(t)$. We use the common parameters $\sigma=10,\rho=28$, and $\beta=8/3$. This trajectory is shown in Figure 8 and corresponds to a few oscillations about a fixed point. We compare the spectra to that of a longer trajectory containing $300,000$ samples, which we take to be an approximation of the true spectrum of the system. Rössler Attractor: The Rössler attractor is given by the following nonlinear differential equations [73, 74]: $\displaystyle\dot{x}=-y-z$ $\displaystyle\dot{y}=x+ay$ $\displaystyle\dot{z}=b+z(x-c).$ We choose to measure the variable $x(t)$. This attractor is a canonical example of chaos, like the Lorenz attractor. Here we perform a simulation with $70,000$ samples and a stepsize of $\Delta t=0.001$. We choose the following common values of $a=0.1$, $b=0.1$ and $c=14$ and the initial condition $x_{0}=y_{0}=z_{0}=1$. We similarly plot the trajectory and dynamics matrices. We compare the spectra in this case to a longer trajectory using a simulation for $300,000$ samples. Double Pendulum: The double pendulum is a similar nonlinear differential equation, which models the motion of a pendulum which is connected at the end to another pendulum [75]. This system is typically represented by its Lagrangian, $\mathcal{L}=\frac{1}{6}ml^{2}\left(\dot{\theta}_{2}^{2}+4\dot{\theta}_{1}^{2}+3\dot{\theta}_{1}\dot{\theta}_{2}\cos{\left(\theta_{1}-\theta_{2}\right)}\right)+\frac{1}{2}mgl\left(3\cos{\theta_{1}}+\cos{\theta_{2}}\right),$ (25) where $\theta_{1}$ and $\theta_{2}$ are the angles between the top and bottom pendula and the vertical axis, respectively. $m$ is the mass at the end of each pendulum, $l$ is the length of each pendulum and $g$ is the acceleration constant due to gravity. Using the Euler-Lagrange equations, $\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{i}}-\frac{\partial\mathcal{L}}{\partial\theta_{i}}=0\text{ for }i=1,2,$ we can construct two second order differential equations of motion. The trajectory is computed using a variational integrator to approximate $\delta\int_{a}^{b}\mathcal{L}(\theta_{1},\theta_{2},\dot{\theta}_{1},\dot{\theta}_{2})dt=0.$ We simulate this system with a stepsize of $\Delta t=0.001$ and for $1200$ samples. We choose $m_{1}=m_{2}=l_{1}=l_{2}=1$ and $g=10$, and use initial conditions $\theta_{1}=\theta_{2}=\pi/2$, $\dot{\theta}_{1}=-0.01$ and $\dot{\theta}_{2}=-0.005$. As our measurement for HAVOK and sHAVOK we use $x(t)=\sin(\theta_{1}(t))$ and compare our data to a long trajectory containing $100,000$ samples. ### 5.3 sHAVOK Applied to Real-world Datasets Here we apply sHAVOK to two real world time series datasets, the trajectory of a double pendulum and measles outbreak data. Similar to the synthetic examples, we find that the the dynamics matrix from sHAVOK is much more antisymmetric and tridiagonal compared to the dynamics matrix for HAVOK. In both cases, some of the HAVOK eigenvalues contain positive real components; in other words, these models have unstable dynamics. However, the sHAVOK spectra do not contain positive real components, resulting in much more accurate and stable models (Figure 9). Figure 9: Comparison of HAVOK and structured HAVOK (sHAVOK) for two real world systems: a double pendulum and measles outbreak data. For each system, we measure a trajectory extracting a single coordinate (gray). We then apply HAVOK and sHAVOK to a subset of this trajectory, shown in black. The $\bm{A}$ matrices for the resulting linear dynamical models are shown. sHAVOK yields models with an antisymmetric structure, with nonzero elements only along the subdiagonal and superdiagonal. The corresponding eigenvalue spectra for HAVOK and sHAVOK are additionally plotted in teal and maroon, respectively, along with eigenvalues from HAVOK for a long trajectory. In both cases, the eigenvalues of sHAVOK are much closer in value to those in the long trajectory limit than HAVOK. Some of the eigenvalues of HAVOK are unstable and have positive real components. The corresponding reconstructions of the first singular vector of the corresponding Hankel matrices are shown along with the real data. Note that the HAVOK models are unstable, growing exponentially due to the unstable eigenvalues, while the sHAVOK models do not. Credit for images on left : (double pendulum) [76] and (measles) CDC/ Cynthia S. Goldsmith; William Bellini, Ph.D. Double Pendulum: We first look at measurements of a double pendulum [76]. A picture of the setup can be found in Figure 9. The Lagrangian in this case is very similar to that in (25). One key difference in the synthetic case is that all of the mass is contained at the joints, while in this experiment, the mass is spread over each arm. To accommodate this, the Lagrangian can be slightly modified, $\mathcal{L}=\frac{1}{2}\left(m_{1}(\dot{x}_{1}^{2}+\dot{y}_{1}^{2})+m_{2}(\dot{x}_{2}^{2}+\dot{y}_{2}^{2})\right)+\frac{1}{2}\left(I_{1}\dot{\theta}_{1}^{2}+I_{2}\dot{\theta}_{2}^{2}\right)-\left(m_{1}y_{1}+m_{2}y_{2}\right)g,$ where $x_{1}=a_{1}\sin(\theta_{1})$, $x_{2}=l_{1}\sin(\theta_{1})+a_{2}\sin(\theta_{2})$, $y_{1}=a_{1}\cos(\theta_{1})$, and $y_{2}=l_{1}\cos(\theta_{1})+a_{2}\cos(\theta_{2})$. $m_{1}$, and $m_{2}$ are the masses of the pendula, $l_{1}$ and $l_{2}$ are the lengths of the pendula, $a_{1}$ and $a_{2}$ are the distances from the joints to the center of masses of each arm, and $I_{1}$ and $I_{2}$ are the moments of inertia for each arm. When $m_{1}=m_{2}=m$, $a_{1}=a_{2}=l_{1}=l_{2}$, and $I_{1}=I_{2}=ml^{2}$ we recover (25). We sample the data at $\Delta t=0.001$s and plot $\sin(\theta_{2}(t))$ over a 15s time interval. The data over this interval appears approximately periodic. Measles Outbreaks: As a second example we apply measles outbreak data from New York City between 1928 to 1964 [77]. The case history of measles over time has been shown to exhibit chaotic behavior [78, 79], and [5] applied HAVOK to measles data and successfully showed that the method could extract transient behavior. For both systems, we apply sHAVOK to a subset of the data corresponding to the black trajectories $x(t)$ shown in Figure 9. We then compare that to HAVOK applied over the same interval. We use $m=101$ delays with a $r=5$ rank truncation for the double pendulum, and $m=51$ delays and a $r=6$ rank truncation for the measles data. For the measles data, prior to applying sHAVOK and HAVOK the data is first interpolated and sampled at a rate of $\Delta t=0.0018$ years. Like in previous examples, the resulting sHAVOK dynamics is tridiagonal and antisymmetric while the HAVOK dynamics matrix is not. Next, we plot the corresponding spectra for these two methods, in addition to the eigenvalues applied to HAVOK over the entire time series. Most noticeably, the eigenvalues from sHAVOK are closer to the long data limit values. In addition, two of the HAVOK eigenvalues lie to the right of the real axis, and thus have positive real components. All of the sHAVOK eigenvalues, on the other hand, have negative real components. This difference is most prominent in the reconstructions of the first singular vector. In particular, since two of the eigenvalues from HAVOK are positive, the reconstructed time series grows exponentially. In contrast, for sHAVOK the corresponding time- series remains bounded providing a much better model of the true data. ## 6 Discussion In this paper, we describe a new theoretical connection between models constructed from time-delay embeddings, specifically using the HAVOK approach, and the Frenet-Serret frame from differential geometry. This unifying perspective explains the peculiar antisymmetric, tridiagonal structure of HAVOK models: namely, the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet-Serret frame. Inspired by this theoretical insight, we develop an extension we call _structured_ HAVOK that effectively yields models with this structure. Importantly, we demonstrate that this modified algorithm improves the stability and accuracy of time-delay embedding models, especially when data is noisy and limited in length. All code is available at https://github.com/sethhirsh/sHAVOK. Establishing theoretical connections between time-delay embedding, dimensionality reduction, and differential geometry opens the door for a wide variety of applications and future work. By understanding this new perspective, we now better understand the requirements and limitations of HAVOK and have proposed simple modifications to the method which improve its performance on data. However, the full implications of this theory remain unknown. Differential geometry, dimensionality reduction and time delay embeddings are all well-established fields, and by understanding these connections we can develop more robust and interpretable methods for modeling time series. For instance, by connecting HAVOK to the Frenet-Serret frame, we recognize the importance of enforcing orthogonality for $\bm{V}_{1}$ and $\bm{V}_{2}$ and inspired development of sHAVOK. With this theory, we can incorporate further improvements on the method. For example, sHAVOK can be thought of as a first order forward difference method, approximating the derivative and state by $\left(\bm{V}_{2}-\bm{V}_{1}\right)/\Delta t$ and $\bm{V}_{1}$, respectively. By employing a central difference scheme, such as approximating the state by $\bm{V}$, we have observed this to further enforce the antisymmetry in the dynamics matrix and move the corresponding eigenvalues towards the imaginary axis. Throughout this analysis, we have focused purely on linear methods. In recent years, nonlinear methods for dimensionality reduction, such as autoencoders and diffusion maps, have gained popularity [80, 81, 7]. Nonlinear models similarly benefit from promoting sparsity and interpretability. By understanding the structures of linear models, we hope to generalize these methods to create more accurate and robust methods that can accurately model a greater class of functions. ## Acknowledgments We are grateful for discussions with S. H. Singh, and K. D. Harris; and to K. Kaheman for providing the double pendulum dataset. We thank to thank A. G. Nair for providing valuable insights and feedback in designing the analysis. This work was funded by the Army Research Office (W911NF-17-1-0306 to SLB); Air Force Office of Scientific Research (FA9550-17-1-0329 to JNK); the Air Force Research Lab (FA8651-16-1-0003 to BWB); the National Science Foundation (award 1514556 to BWB); the Alfred P. 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Knapp, “Sines and cosines of angles in arithmetic progression,” _Mathematics magazine_ , vol. 82, no. 5, p. 371, 2009. ## Appendix A Discrete Orthogonal Polynomials In Section 3.3 we introduced a set of orthogonal polynomials that appear in HAVOK, and listed these polynomials in the continuous case. The first five polynomials in the discrete case are listed below. $\displaystyle p_{1}(n)=\frac{n}{c_{1}}$ $\displaystyle p_{2}(n)=\frac{n^{2}}{c_{2}}$ $\displaystyle p_{3}(n)=\frac{1}{c_{3}}\left(n^{3}-\frac{n\,\left(3\,p^{2}+3\,p-1\right)}{5}\right)$ $\displaystyle p_{4}(n)=\frac{1}{c_{4}}\left(n^{4}-\frac{5\,n^{2}\,\left(3\,p^{4}+6\,p^{3}-3\,p+1\right)}{7\,\left(3\,p^{2}+3\,p-1\right)}\right)$ $\displaystyle p_{5}(n)=\frac{1}{c_{5}}\left(\frac{5\,\left(\frac{n\,\left(3\,p^{2}+3\,p-1\right)}{5}-n^{3}\right)\,\left(2\,p^{2}+2\,p-3\right)}{9}-\frac{n\,\left(3\,p^{4}+6\,p^{3}-3\,p+1\right)}{7}+n^{5}\right)$ $\displaystyle c_{1}=\sqrt{\frac{p\,\left(2\,p+1\right)\,\left(p+1\right)}{3}}$ $\displaystyle c_{2}=\sqrt{\frac{p\,\left(2\,p+1\right)\,\left(p+1\right)\,\left(3\,p^{2}+3\,p-1\right)}{15}}$ $\displaystyle c_{3}=\sqrt{\frac{p\,\left(2\,p-1\right)\,\left(2\,p+1\right)\,\left(2\,p+3\right)\,\left(p-1\right)\,\left(p+1\right)\,\left(p+2\right)}{175}}$ $\displaystyle c_{4}=\sqrt{\frac{p\,\left(2\,p-1\right)\,\left(2\,p+1\right)\,\left(2\,p+3\right)\,\left(p-1\right)\,\left(p+1\right)\,\left(p+2\right)\,\left(15\,p^{4}+30\,p^{3}-35\,p^{2}-50\,p+12\right)}{2205\,\left(3\,p^{2}+3\,p-1\right)}}$ $\displaystyle c_{5}=\sqrt{\frac{4\,p\,\left(2\,p-1\right)\,\left(2\,p+1\right)\,\left(2\,p-3\right)\,\left(2\,p+3\right)\,\left(2\,p+5\right)\,\left(p-1\right)\,\left(p+1\right)\,\left(p-2\right)\,\left(p+2\right)\,\left(p+3\right)}{43659}}$ ## Appendix B Column Rule for Synthetic Example In Section 4, we state that when applying HAVOK to the synthetic example in 3.2 in the limit as the number of columns $n$ in the Hankel matrix $\bm{H}$ goes to infinity, the derivatives in (23) converge to fixed values. Here we prove that the first ratio in the series $\frac{2\left\lVert\bm{h}^{\prime}_{0}\right\rVert}{\left\lVert\bm{h^{\prime\prime}_{0}}\right\rVert}$ approaches a constant as $n\to\infty$. Further terms in the sequence, can be shown to have the same behavior using a similar proof. We start with the system $x(t)=\sin(t)+\sin(2t)$. The central row of the matrix $\bm{h}_{0}$ will be of the form $x(t)$ for some $a,b\in\mathbb{Z}$ such that $t=\begin{bmatrix}a\Delta t&(a+1)\Delta t&(a+2)\Delta t&\dots&b\Delta t\end{bmatrix}.$ In particular $b=n+a$. Thus, showing that the limit as $n\to\infty$ is equivalent to the limit as $b\to\infty$. $\begin{split}\frac{\|\|\bm{h}_{0}^{\prime\prime}\|\|}{\|\|\bm{h}^{\prime}_{0}\|\|}&=\frac{\|\|-\sin(t)-4\sin(2t)\|\|}{\|\|\cos(t)+2\cos(2t)\|\|}\\\ &=\frac{\|\|\begin{bmatrix}-\sin(a\Delta t)-4\sin(2a\Delta t)&\dots&-\sin(b\Delta t)-4\sin(2b\Delta t)\end{bmatrix}\|\|}{\|\|\begin{bmatrix}\cos(a\Delta t)+2\cos(2a\Delta t)&\dots&\cos(b\Delta t)+2\cos(2b\Delta t)\end{bmatrix}\|\|}\\\ &=\sqrt{\frac{\sum_{k=a}^{b}(\sin(k\Delta t)+4\sin(2k\Delta t))^{2}}{\sum_{k=a}^{b}(\cos(k\Delta t)+2\cos(2k\Delta t))^{2}}}\\\ &=\sqrt{\frac{\sum_{k=a}^{b}(\sin^{2}(k\Delta t)+8\sin(k\Delta t)\sin(2k\Delta t)+16\sin^{2}(2k\Delta t))}{\sum_{k=a}^{b}(\cos^{2}(k\Delta t)+4\cos(k\Delta t)\cos(2k\Delta t)+4\cos^{2}(2k\Delta t))}}\\\ &=\sqrt{\frac{\sum_{k=a}^{b}(\frac{17}{2}+4\cos(k\Delta t)-\frac{1}{2}\cos(2k\Delta t)-4\cos(3k\Delta t)-8\cos(4k\Delta t))}{\sum_{k=a}^{b}(\frac{5}{2}+2\cos(k\Delta t)+\frac{1}{2}\cos(2k\Delta t)+2\cos(3k\Delta t)+2\cos(4k\Delta t))}}.\end{split}$ In the last step we have used the trigonometric identities $\sin^{2}(a)=\frac{1}{2}(1-\cos(2a))$, and $\cos^{2}(a)=\frac{1}{2}(1+\cos(2a))$. Using [82], we have the identity $\sum_{k=0}^{q}\cos(Bk)=\frac{\sin\big{[}(\frac{q+1}{2})B\big{]}\cos\big{[}(\frac{q}{2})B\big{]}}{\sin(\frac{B}{2})},\quad B,q\in\mathbb{R}.$ $\sum_{k=a}^{b}\cos(Bk)=\frac{\sin\big{[}(\frac{b+1}{2})B\big{]}\cos\big{[}(\frac{b}{2})B\big{]}-\sin\big{[}(\frac{a}{2})B\big{]}\cos\big{[}(\frac{a-1}{2})B\big{]}}{\sin(\frac{B}{2})},\quad B,a,b\in\mathbb{R}.$ Defining $g(b)$ and $h(b)$ as the numerator and denominator under the radical, $\displaystyle g(b)$ $\displaystyle=\sum_{k=a}^{b}(4\cos(k\Delta t)-\frac{1}{2}\cos(2k\Delta t)-4\cos(3k\Delta t)-8\cos(4k\Delta t))$ $\displaystyle=\frac{4(\sin[(b+1)(\frac{\Delta t}{2})]\cos[b(\frac{\Delta t}{2})]-\sin[a(\frac{\Delta t}{2})]\cos[(a-1)(\frac{\Delta t}{2})])}{\sin[\frac{\Delta t}{2}]}$ $\displaystyle-\frac{\sin[(b+1)\Delta t]\cos[b\Delta t]-\sin[a\Delta t]\cos[(a-1)\Delta t]}{2\sin[\Delta t]}$ $\displaystyle-\frac{4(\sin[(b+1)(\frac{3\Delta t}{2})]\cos[b(\frac{3\Delta t}{2})]-\sin[a(\frac{3\Delta t}{2})]\cos[(a-1)(\frac{3\Delta t}{2})])}{\sin[\frac{3\Delta t}{2}]}$ $\displaystyle-\frac{8(\sin[(b+1)(2\Delta t)]\cos[b(2\Delta t)]-\sin[a(2\Delta t)]\cos[(a-1)(2\Delta t)])}{\sin[2\Delta t]}$ $\displaystyle h(b)$ $\displaystyle=\sum_{k=a}^{b}(2\cos(k\Delta t)+\frac{1}{2}\cos(2k\Delta t)+2\cos(3k\Delta t)+2\cos(4k\Delta t))$ $\displaystyle=\frac{2(\sin[(b+1)(\frac{\Delta t}{2})]\cos[b(\frac{\Delta t}{2})]-\sin[a(\frac{\Delta t}{2})]\cos[(a-1)(\frac{\Delta t}{2})])}{\sin[\frac{\Delta t}{2}]}$ $\displaystyle+\frac{\sin[(b+1)\Delta t]\cos[b\Delta t]-\sin[a\Delta t]\cos[(a-1)\Delta t]}{2\sin[\Delta t]}$ $\displaystyle+\frac{2(\sin[(b+1)(\frac{3\Delta t}{2})]\cos[b(\frac{3\Delta t}{2})]-\sin[a(\frac{3\Delta t}{2})]\cos[(a-1)(\frac{3\Delta t}{2})])}{\sin[\frac{3\Delta t}{2}]}$ $\displaystyle+\frac{2(\sin[(b+1)(2\Delta t)]\cos[b(2\Delta t)]-\sin[a(2\Delta t)]\cos[(a-1)(2\Delta t)])}{\sin[2\Delta t]}.$ Note that we have the following: $\lim_{b\to\infty}\frac{g(b)}{b}=0\quad\text{and}\quad\lim_{b\to\infty}\frac{h(b)}{b}=0.$ Using this fact, then $\begin{split}\lim_{b\to\infty}\frac{2\|\|\bm{h}_{0}^{\prime\prime}\|\|}{\|\|\bm{h}^{\prime}_{0}\|\|}&=2\lim_{b\to\infty}\sqrt{\frac{\sum_{k=a}^{b}\frac{17}{2}+\sum_{k=a}^{b}(4\cos(k\Delta t)-\frac{1}{2}\cos(2k\Delta t)-4\cos(3k\Delta t)-8\cos(4k\Delta t))}{\sum_{k=a}^{b}\frac{5}{2}+\sum_{k=a}^{b}(2\cos(k\Delta t)+\frac{1}{2}\cos(2k\Delta t)+2\cos(3k\Delta t)+2\cos(4k\Delta t))}}\\\ &=2\lim_{b\to\infty}\sqrt{\frac{\frac{17}{2}(b-a+1)+g(b)}{\frac{5}{2}(b-a+1)+h(b)}}\\\ &=2\lim_{b\to\infty}\sqrt{\frac{\frac{17}{2}-\frac{17a}{2b}+\frac{17}{2b}+\frac{g(b)}{b}}{\frac{5}{2}-\frac{5a}{2b}+\frac{5}{2b}+\frac{h(b)}{b}}}\\\ &=2\sqrt{\frac{17}{5}}.\end{split}$ ## Appendix C Structured HAVOK (sHAVOK) algorithm Here we present pseudocode for the sHAVOK algorithms with and without forcing terms. Algorithm 1 Structured HAVOK (sHAVOK) without forcing Input: Measured signal $x(t)$, number of delays $m$, and rank of Hankel Matrix $\bm{r}$. Output: Dynamics matrix $\hat{\bm{A}}\in\mathbb{R}^{r\times r}$. $\bm{H}:=\text{Hankel}(x(t),m)$ $\bm{H}_{1}:=\bm{H}[:,1:n-1]$ $\bm{H}_{2}:=\bm{H}[:,2:n]$ $\bm{U}_{1}\bm{\Sigma}_{1}\bm{V}_{1}^{\intercal}:=\text{SVD}(\bm{H}_{1},r)$ $\bm{U}_{2}\bm{\Sigma}_{2}\bm{V}_{2}^{\intercal}:=\text{SVD}(\bm{H}_{2},r)$ $\hat{\bm{A}}:=\bm{V}_{2}^{\intercal}\bm{V}_{1}$ Algorithm 2 Structured HAVOK (sHAVOK) with forcing Input: Measured signal $x(t)$, number of delays $m$, and rank of Hankel Matrix $\bm{r}$. Output: Dynamics matrix $\hat{\bm{A}}\in\mathbb{R}^{r-1\times r-1}$ and forcing term $\hat{\bm{B}}\in\mathbb{R}^{r-1}$. $\bm{H}:=\text{Hankel}(x(t),m)$ $\bm{H}_{1}:=\bm{H}[:,1:n-1]$ $\bm{H}_{2}:=\bm{H}[:,2:n]$ $\bm{U}_{1}\bm{\Sigma}_{1}\bm{V}_{1}^{\intercal}:=\text{SVD}(\bm{H}_{1},r)$ $\bm{U}_{2}\bm{\Sigma}_{2}\bm{V}_{2}^{\intercal}:=\text{SVD}(\bm{H}_{2},r-1)$ $[\hat{\bm{A}},\hat{\bm{B}}]:=\bm{V}_{2}^{\intercal}\bm{V}_{1}$
G. Menardi Giovanna Menardi, via C. Battisti 241, 35121 Padova, Italy # Nonparametric clustering for image segmentation Giovanna Menardi Department of Statistical Sciences, University of Padova, Padova, Italy<EMAIL_ADDRESS>Menardi G (26 April 2016; 6 June 2016; 6 June 2016) ###### Abstract [Summary]Image segmentation aims at identifying regions of interest within an image, by grouping pixels according to their properties. This task resembles the statistical one of clustering, yet many standard clustering methods fail to meet the basic requirements of image segmentation: segment shapes are often biased toward predetermined shapes and their number is rarely determined automatically. Nonparametric clustering is, in principle, free from these limitations and turns out to be particularly suitable for the task of image segmentation. This is also witnessed by several operational analogies, as, for instance, the resort to topological data analysis and spatial tessellation in both the frameworks. We discuss the application of nonparametric clustering to image segmentation and provide an algorithm specific for this task. Pixel similarity is evaluated in terms of density of the color representation and the adjacency structure of the pixels is exploited to introduce a simple, yet effective method to identify image segments as disconnected high-density regions. The proposed method works both to segment an image and to detect its boundaries and can be seen as a generalization to color images of the class of thresholding methods. A revised version of this manuscript is the following: , (2020), Nonparametric clustering for image segmentation, Statistical Analysis and DataMining: The ASA Data Science Journal, 13(1), p. 83-97. ###### keywords: Image Segmentation, Kernel smoothing, Mode, Nonparametric Density Estimation ††articletype: Research Article ## 1 Introduction and motivation In the recent years, the need of analyzing large amounts of image information has become relevant in several contexts and applications. Daily examples include medical diagnosis based on X-ray or magnetic resonance images, video surveillance and geographic information system applications, and image tagging. A possible goal of image analysis is the one of _segmentation_ , the automatic process of identifying salient regions and single objects in an image, with the purpose of content retrieval, object detection or recognition, occlusion boundary, image compression or editing. Digital images are created by a variety of input devices, such as cameras or scanners, and they have usually a fixed resolution, _i.e._ they are represented by a fixed number of digital values, known as _pixels_. Pixels are the smallest individual element in an image, holding quantized values that represent the brightness of a given color at any specific location of the image. When an image is segmented, a label is assigned to each pixel, so that pixels with the same label share similar characteristics in terms of color, intensity, or texture. This task recalls closely the aim of cluster analysis, and thereby clustering methods have been featured as a standard tool to segment images. Within this framework, an approach which naturally lends itself to the task of image segmentation is known as _nonparametric_ or _modal_ clustering. With respect to most of clustering methods, relying on heuristic ideas of similarity between objects, the nonparametric formulation claims a sounder theoretical ground, since the definition of a precise notion of clusters allows for a “ground truth” to aim at when evaluating a clustering configuration or comparing alternatives. Specifically, a probability density function is assumed to underlie the data and clusters are defined as the domains of attraction of the modes of the density function, estimated nonparametrically. The correspondence between clusters and regions around the modes of the data distribution entails some reasons of attractiveness. First, the number of clusters is an intrinsic property of the data generator mechanism, thereby conceptually well defined, and its determination is itself an integral part of the clustering procedure. Additionally, modal regions comply with the geometric intuition about the notion of clusters, also because they are not bound to any particular shape. These reasons make nonparametric clustering particularly suitable for the segmentation of digital images, as segments shall be allowed to assume arbitrary shapes and an automatic determination of the number of segments would be desirable. In this work the use of nonparametric clustering for image segmentation is discussed. Pixel similarity is evaluated in terms of density of the color representation and the adjacency structure of the pixels is exploited to introduce a simple method to assess the connectedness of the modal density regions. In the following, an overview about nonparametric clustering is provided along with its connection with methods for image segmentation. Within this framework, a novel method specifically conceived for image segmentation is proposed and discussed, and several applications illustrated. ## 2 Background ### 2.1 Overview of nonparametric clustering \| \| ---\|---\|--- Figure 1: A section of a density function at a level $\lambda_{0}$ (left), the identified level set (middle panel), formed by two disconnected regions and the associated cluster tree, with leaves corresponding to the modes. The horizontal line is at the level $\lambda_{0}$ (right). Nonparametric, or modal, clustering hinges on the assumption that the observed data $(x_{1},\ldots,x_{n})^{\prime}$ are a sample from a probability density function $f:\mathbb{R}^{d}\mapsto\mathbb{R}^{+}$. The modes of $f$ are regarded as the archetypes of the clusters, which are in turn represented by the surrounding regions. The identification of the modal regions is performed according to two alternative directions. One strand of methods looks for an explicit representation of the modes of the density and associates each cluster to the set of points along the steepest ascent path towards a mode. Optimization method are applied to find the local maxima of the density, such as the _mean-shift_ algorithm 13, and a number of its variants 9, 30, 5. We consider, alternatively, a second strand, which associates the clusters to disconnected density level sets of the sample space, without attempting the explicit task of mode detection. Specifically, a section of $f$ at a given level $\lambda$ singles out the (upper) level set $L(\lambda)=\\{x\in\mathbb{R}^{d}:f(x)\geq\lambda\\},\quad 0\leq\lambda\leq\max f$ which may be connected or disconnected. In the latter case, it consists of a number of connected components, each of them associated with a cluster at the level $\lambda$. While there may not exist a single $\lambda$ which catches all the modal regions, any connected component of $L(\lambda)$ includes at least one mode of the density and, on the other hand, for each mode there exists some $\lambda$ for which one of the connected components of the associated $L(\lambda)$ includes this mode at most. Hence, not only it is not necessary to define a specific level $\lambda$ to identify the groups, which would be difficult and often not effective in providing the overall number of modes, but conversely, all the modal regions may be detected by identifying the connected components of $L(\lambda)$ for different $\lambda$s. Varying $\lambda$ along its range gives rise to a hierarchical structure of the high-density sets, known as the _cluster tree_. For each $\lambda$, it provides the number of connected components of $L(\lambda),$ and each of its leaves corresponds to a _cluster core_ , _i.e._ the largest connected component of $L(\lambda)$ including one mode only. Figure 1 illustrates a simple example of this idea: cluster cores associated with the highest modes $2$ and $3$ are identified by the smallest $\lambda$ larger than $\lambda_{3}$, while the smallest $\lambda$ larger than $\lambda_{1}$ identifies the cluster core associated to mode $1$. In some sense, the cluster tree provides a tool to inspect data with different degree of sharpness: clusters 2 and 3 are distinct, but they merge together to create a lower resolution cluster. Instead of indexing the tree with $\lambda,$ it is equivalent to consider the probability associated to $L(\lambda)$, which varies inversely with $\lambda$. For this reason the tree is usually plotted upside down. This will be also the convention considered for the rest of the paper. From the operational point of view, two choices are required to implement the ideas underlying nonparametric clustering. Since $f$ is unknown, a nonparametric estimator $\hat{f}$ is employed to obtain its representation. A common choice for multidimensional data is the product kernel estimator see, e.g., 22: $\hat{f}(x;h)=\sum_{i=1}^{n}\frac{1}{nh_{1}\cdot\ldots\cdot h_{d}}\prod_{j=1}^{d}K\left(\frac{x^{(j)}-x_{i}^{(j)}}{h_{j}}\right)$ (1) where $x^{(j)}$ is the $j-th$ component of $x$, the univariate kernel $K$ is usually taken to be a unimodal, nonnegative function centered at zero and integrating to one, and a different smoothing parameter $h_{j}$ is chosen for each component. In fact, for the development of the method, it does not really matter which specific estimator is adopted, provided that $\hat{f}$ is positive and finite at the observed points. A second choice derives from the lack, in multidimensional sample spaces, of an obvious method to identify the connected components of a level set. For these reasons the inherent literature has mainly focused on developing efficient methods for this task _e.g._ 25, 18. Note that the union of the cluster cores does not produce a partition of the sample space, as regions at the tails or at the valleys of $f$, where the attraction of each mode is low, are initially left unallocated. However, the availability of a density measure allows for providing each unallocated observation with a degree of confidence of belonging to the cluster cores. Depending on the application, the evaluation of such confidence may be exploited to force the assignment or may result in the opportunity of fuzzy clustering schemes. ### 2.2 Related works on image segmentation Also due to the extensiveness of its applicability, several different methods have been proposed to pursue the task of image segmentation. These are broadly ascribable to two alternative routes see, for a review, 11: _noncontextual_ techniques ignore the relationships among the features in the image and assign the same label to pixels sharing some global attribute, such as the gray level or the color brightness. _Thresholding_ , for instance, compares the intensity of each pixel with a suitable threshold and associates higher values to the foreground of the image, of main interest, and lower values to the background. Recent contributions within this framework are 1, 12. _Contextual_ techniques, conversely, also account for pixel location or color gradient. Within this class, _region-based methods_ mainly rely on the assumption that the neighboring pixels within one region have similar value see, e.g. 8, 16, 19. Boundary-based methods as _edge detection_ and _active contours_ build on finding pixel differences rather than similarities, to determine a closed boundary between the foreground and the background of the image. Examples of recent procedures following these approaches are 15, 10 and, respectively, 29, 31. _Watershed_ segmentation builds a distance map of a gray-scale image or of its gradient and considers it as topographic relief, to be flooded from its minima. When two lakes merge, a dam is built, representing the boundary between two segments e.g. 23. Within the framework of clustering methods, $K-$means clustering is diffusely used for image segmentation 26, perhaps due to its simplicity, but a few severe limitations prevent its effectiveness. First, $K-$means clustering is known to produce sub-optimal solutions as it highly depends on the initialization of the centroids. Additionally, it requires a prior specification of the number of clusters. In image segmentation this operation is undoubtedly easier than in other clustering applications. On the other hand, the need of human intervention vanishes the effort to automate the segmentation procedure. Finally, $K-$means is known to be biased toward the identification of spherical clusters, which can be restrictive in image data where segments may assume arbitrarily odd shapes. While nonparametric clustering is rarely mentioned as the underlying approach to perform image analysis, it features some connection with a number of segmentation algorithms. By exploiting some notions from Morse theory, in 7 an elegant formalization of the notion of modal (nonparametric) cluster, which closely recalls the ideas of watershed segmentation is provided: intuitively, if the density underlying the data is figured as a mountainous landscape, and modes are its peaks, clusters are the “regions that would be flooded by a fountain emanating from a peak of the mountain range”. Furthermore, segmentation _thresholding_ is often performed by looking for the minimum of the histogram of the grey intensities, i.e. the antimode of a histogram-based density estimator of the grey intensities. Since any antimode lies between two modes, the approach can be interpreted as a naive, single-$\lambda$, implementation of the density level set formulation above mentioned, where gray intensities are employed as a measure of density. While without a specific reference to nonparametric clustering, gradient ascent algorithms in the guise of mean-shift, have been also sometimes applied for image segmentation 27, 17, 32. Indeed, by climbing the modes of a kernel density estimate, the mean-shift is rightly ascribable to the class of non parametric clustering methods. Similar instruments are also at the basis of active contours models, where a suitable measure of energy is iteratively minimized by a gradient descent algorithm to identify the segment contours. As a further link, even when applied to different goals, image analysis and nonparametric clustering share several tools: an example is provided by spatial tessellation as the Voronoi or Delaunay diagrams, which have been used in nonparametric clustering to identify density level sets connected components 3 and are frequently employed in image analysis for thinning and skeletonization. ## 3 A nonparametric method for image segmentation ### 3.1 Description of the procedure Let $\mathcal{I}=\\{p_{1},\ldots,p_{n}\\}$ be a digital image, where the ordered set of pixels $p_{i}=((x_{i},y_{i}),z_{i}),i=1,\ldots,n,$ is described by the pair $(x_{i},y_{i})$ denoting the coordinates of the pixels location, and by the vector $z_{i}$ denoting the adopted color model, _e.g._ $z_{i}=(z_{i}^{(r)},z_{i}^{(g)},z_{i}^{(b)})$ in the RGB color representation 24. In grayscale images, $z_{i}$ is a scalar quantifying the gray intensity. The particularization of nonparametric clustering in the framework of image analysis requires a density function to be defined at the pixels. A sensible choice builds $\hat{f}$ based on the color coordinates $z_{i}$. The specification of the (1) is then: $\hat{f}(z)=\frac{1}{nh_{r}h_{g}h_{b}}\sum_{i=1}^{n}K\left(\frac{z^{(r)}-z_{i}^{(r)}}{h_{r}}\right)K\left(\frac{z^{(g)}-z_{i}^{(g)}}{h_{g}}\right)K\left(\frac{z^{(b)}-z_{i}^{(b)}}{h_{b}}\right).$ (2) For example, if the Uniform kernel $K(u)=\tfrac{1}{2}\mathbf{1}_{\\{\|u\|<1\\}}$ is selected and $h_{j}\to 0$, each pixel is provided with a density proportional to the frequency of its color in the image. Similar interpretations hold with different kernel functions. Consider, as an example, the image in the top panel of Figure 2. For the sake of illustration, the image has been built by using colors entirely characterized by the red and blue RGB channels (_i.e._ each pixel has the green channel set to 0). The image is clearly composed by 4 segments, each of them featured by a main color pattern appearing with different shades. The bottom panel displays the color intensities of each pixel, disregarding their location within the image. Pixels sharing a similar color have similar red and blue intensities, and cluster together in different areas of the plane. Hence, the color density estimate exhibts 4 modes, associated to the 4 color patterns. As it will be discussed in Section 3.2, an alternative to (2) would also account for the spatial coordinates of the pixels. Once that the color density has been estimated, the associated upper level sets $\hat{L}(\lambda)=\\{p_{i}\in\mathcal{I}:\hat{f}(z_{i})\geq\lambda\\}\qquad 0\leq\lambda<\max\hat{f}$ are easily determined for a grid of $\lambda$ values. Next step is the identification of the connected components of the $\hat{L}(\lambda)^{\prime}$s. Unlike the above mentioned case of clustering data on $\mathbb{R}^{d}$, where the identification of connected regions is ambiguous, the notion of connectedness is (almost) unequivocally determined in the image framework, due to the spatial structure of the pixels. This justifies the procedure here proposed, which builds on the level set formulation of nonparametric clustering but naturally exploits the adjacency structure of the pixels to identify the connected components of the modal regions. For a given $\lambda$, the connected components of $\hat{L}(\lambda)$ are approximated as follows: (i) for each pixel of $\hat{L}(\lambda)$, identify the adjacent pixels forming its _4-neighbourhood_ , _i.e._ a central pixel has four connected neighbors - top, bottom, right and left. * (ii) approximate the connected components of $\hat{L}(\lambda)$ by the union of adjacent pixels in $\hat{L}(\lambda)$. Hence, in order to identify a segment, not only the pixels need to share a similar color, but a constraint of spatial connectedness is also imposed by the procedure. For example, if a black spot was embedded in the violet segment in the toy image of Figure 2, the associated pixels would contribute to form (and hence raise) the mode of the density at the bottom corner of the lowermost panel of the Figure. However, at a level $\lambda$ for which the black colored pixels would have higher density, $\hat{L}(\lambda)$ would be disconnected, because formed by both the pixels of the black square already in the image, and the pixels of the embedded, not adjacent, black spot. Algorithm 1 Nonparametric density-based segmentation: main steps of the procedure 1:$h=(h_{r},h_{g},h_{b})$; $K(\cdot)$; $\epsilon$; Class $\in$ {TRUE, FALSE} (set Class:= FALSE to not allocate pixels not belonging to the segment cores;) 2:Identify the set $N_{4}(p_{i})$ of pixels in the 4-neighborhood of $p_{i},i=1,\ldots,n$ 3:Compute $\hat{f}(z)=\sum_{i=1}^{n}\frac{1}{nh_{r}h_{g}h_{b}}\prod_{j=\\{r,g,b\\}}K\left(\frac{z^{(j)}-z_{i}^{(j)}}{h_{j}}\right)$, $\quad\forall z\in\\{z_{i}\\}_{i=1,\ldots,n}$ 4:Set $\lambda:=0$ 5:while $0\leq\lambda\leq\max\hat{f}$ do 6: identify $\hat{L}(\lambda)=\\{p_{i}:\hat{f}(z_{i})\geq\lambda\\}$; 7: find the connected components of $\hat{L}(\lambda)$ (as the union of adjacent pixels in $\hat{L}(\lambda)$) 8: $\lambda:=\lambda+\epsilon$ 9:end while 10:Build the hierarchy of the connected components of $\hat{L}{(\lambda)}^{\prime}s$ and obtain the cluster tree 11:Denote core pixels as $p_{c}$ and unallocated pixels as $p_{u}$ 12:Assign the label $\ell_{c}\in\\{1,\ldots,M\\}$ to each core pixel $p_{c}$ 13:Set Isolated:= $\emptyset$ 14:if Class = TRUE then 15: while $\\{p_{u}\\}\neq\emptyset\,\,\vee$ {Isolated} $=\\{p_{u}\\}$ do 16: Set Isolated:= $\emptyset$ 17: for all $p_{u}$ do 18: $\hat{f}_{m}(z_{u})=\sum_{c:\ell_{c}=m}\frac{1}{nh_{r}h_{g}h_{b}}\prod_{j=\\{r,g,b\\}}K\left(\frac{z^{(j)}-z_{c}^{(j)}}{h_{j}}\right)$, $m=1\ldots,M$ 19: set $m_{0}:=\mbox{argmax}_{m}\hat{f}_{m}(z_{u})$ 20: if $\exists$ $p_{c}$ such that ($p_{c}$ $\in$ $N_{4}(p_{u})\wedge\ell_{c}=m_{0}$) then 21: assign the label $\ell_{c}:=m_{0}$ to $p_{u}$ 22: else 23: Isolated: = {Isolated $\cup\,p_{u}$} 24: end if 25: end for 26: update $\\{p_{u}\\}$ and $\\{p_{c}\\}$ 27: end while 28:else 29: set $\ell_{u}:=0$ 30:end if 31:RETURN: $\ell_{1},\ldots,\ell_{n}$ For varying $\lambda$, the procedure described so far creates $M$ groups of pixels $\ell_{m}$ ($m=1,\ldots,M$), which we call, in analogy with the clustering problem, (segment) _cores_ , and it leaves a number of pixels unlabeled. Depending on the application at hand, we can either decide to force their assignment to the existing segment cores or to leave these pixels unallocated. In fact, a peculiar aspect is that the unlabeled points are not positioned randomly in the image, but are inevitably on the outskirts of the existing segment cores. As will be illustrated in the Section 4, unallocated pixels include (or correspond to) the contours of the segments. The possible allocation of the unlabeled pixels to the existing groups is essentially a classification problem that may be faced according to a wide range of choices. To remain within the same kind of approach pursued so far, and consistently with the purpose of identifying segments as connected sets of pixels, we propose the following idea to classify an unallocated pixel $p_{u}$: compute the $M$ estimated densities $\hat{f}_{m}(z_{u})$, each based on the pixels already assigned to the $m^{th}$ core only ($m=1,\ldots,M)$; then, set $m_{0}=\mbox{argmax}_{m}\hat{f}_{m}(z_{u})$ (3) and assign $p_{u}$ to the group with label $m_{0}$ provided that at least one of the pixels already assigned to the $m_{0}^{th}$ segment core is adjacent to $p_{u}$. The operational implementation of this idea is here performed in a sequential manner, as detailed in the reported pseudo-code, along with the main steps of the whole segmentation procedure. It may be the case that none of the pixels already assigned to the segment presenting maximum density (3) is adjacent to $p_{u}$, (_i.e._ the color of $p_{u}$ is not similar to the color of any other adjacent segment). These pixels usually lie at the borders of the segments and may either be left unallocated, or the assignment can be forces to the highest density segment, disregarding spatial adjacency, or to a novel further segment. Figure 2: A 4-segments toy image entirely characterized by red and blue channels (top); red and blue intensities of the pixels and, superimposed, their kernel density estimate, highlighting 4 clear modes corresponding to the 4 segments (bottom). Figure 3: A simple gray-scale image (left), and the estimate of color density superimposed to the spatial coordinates to the pixels, represented both as the level set density (middle panel) and as a perspective plot (right panel). ### 3.2 Discussion Since the procedure illustrated so far accounts for both the colors and the connectivity of the image patterns, it emulates, in some sense, the behavior of the human eye, which instinctively, perceives different segments in a picture as either disconnected set of pixels or image patterns with a diverse color intensity. A simple illustration of this latter aspect is witnessed by the grayscale image in Figure 3. Even if the gray intensities of the foreground (the inner square) and the background are similar, the density estimator (2) perfectly distinguishes the two density levels and the isoline identifies the contours of the foreground segment. Conversely, with respect to the former aspect, a major limitation of nonparametric clustering, in principle inherited by the proposed segmentation procedure, derives from the definition of mode itself, which requires a separating “gap” between dense patterns. In Figure 3, the density shows itself like a squared hole, and there is no lower density area between the background and the foreground. This prevents $\hat{L}(\lambda)$ to be a disconnected set for any $\lambda$, which would guarantee the identification of two segments. This behavior is somewhat paradoxical, as the neater the image, the less ideal the setting for the procedure to work effectively: within an image, dripped contours of a segment, indeed, manifest themselves as small changes of colors at the borders with respect to the interior. Since the perimeter of a shape is always smaller than its area, and the density of a pixel is positively associated with the frequency of its color, dripped contours would guarantee that the color density along the contour of a segment is lower than its inner density, and hence a valley would arise between a segment and its background. In fact, the considered example has been built ad hoc by setting the gray intensity for each pixel. In practice, many images have segment contours not defined with such neatness, no matter what the image resolution is. This is especially true with segments having either curve or sloped contours, since the shades of colors along the border of the segments allow to prevent a sawtoothed rendering. See Figure 4 for an illustration. Figure 4: A simple gray-scale image (left), and a zoommed detail (middle), showing that when a segment has either curve or sloped contours, the color is ripped along the border, to prevent a sawtoothed rendering. Due to this feature, the perspective plot of the density estimate, based on the (2) highlights a valley at the border of the foreground (right). When the image does not features itself with dripped contours, it is possible to overcome the issue of lacking valleys in the density by introducing some blurring of the image. To this aim, given that the identification of pixel neighbors is required anyway for the identification of disconnected regions, a simple strategy is to replace each pixel value with an average of the values in its neighborhood. In fact, a quite common practice prior to thresholding segmentation is to smooth an image using averaging or median mask. See 11 §10.2.1. A further, somewhat related, issue concerns the choice of the density measure. While we choose to build $\hat{f}$ based on the color intensities only, an alternative route would consist in exploiting the whole available information in terms of both the color coordinates and the spatial coordinates, _i.e._ : $\hat{f}(p)=\frac{1}{n\prod_{j}h_{j}}\sum_{i=1}^{n}K\left(\frac{z^{(r)}-z_{i}^{(r)}}{h_{r}}\right)K\left(\frac{z^{(g)}-z_{i}^{(g)}}{h_{g}}\right)K\left(\frac{z^{(b)}-z_{i}^{(b)}}{h_{b}}\right)K\left(\frac{x-x_{i}}{h_{x}}\right)K\left(\frac{y-y_{i}}{h_{y}}\right).$ (4) Note that, depending on the choice of both the kernel and the bandwidth, the (4) can be a function of the distance between $(x,y)$ and $(x_{i},y_{i})$. Let, for instance, $h_{x}=h_{y}=1.$ Then, the last two factors are proportional to $e^{d_{2}((x,y),(x_{i},y_{i}))^{2}}$ or to $e^{d_{1}((x,y),(x_{i},y_{i}))}$ when a Gaussian or, respectively, a Laplace kernel is selected, with $d_{p}(\cdot,\cdot)$ the $L_{p}$ distance. Estimating the density as in (4) would also overcome the above mentioned problem of lacking valleys at the borders of the segments: in (2), the largest contribution to the density of a generic pixel is provided by all the pixels having similar colors; conversely, if also the spatial coordinates are involved in $\hat{f}$, the density of a generic pixel depends on pixels with similar colors _and_ close spatially. Hence, at the borders of a segment, where part of the adjacent pixels have a different color, the density turns out to be lower than the interior pixels (see Figure 5 for an illustration). While this behavior is desirable for the purpose of segmentation, the interpretation of $\hat{f}$ in terms of color frequency fails and, indeed, a higher computational effort is required. Additionally, some empirical work not included in the manuscript has proven that estimating $f$ via the (4) results in oversegmenting the image, hence using the only color coordinates to estimate density is overall preferable. In fact, two further aspects concerning the estimation of $\hat{f}$ need to be accounted for, concerning the selection of the kernel function $K$ and the smoothing vector $h$. With respect to the former choice, it has been well- established that it does not have a strong impact on the density estimate see, e.g. 28, 6. However, the use of a bounded-support kernel, such as the uniform one (which would be overall more easily interpretable in this context) is in general not advisable for image segmentation, as it entails that in the evaluation of the density of a pixel, other pixels have weight either constant and positive, or null, depending on $h$ and on the difference between color intensities. In this sense, different hues of the same colors could be evaluated as either the same colors or as completely different colors. For this reason, and especially when colors are not homogeneous within the image, a longer-tail kernel is more secure. Concerning the choice of $h,$ on the other hand, in clustering it is not as critical as it is in density estimation, since a rough indication about the high density location may suffice. To this aim, one may resort to one of the numerous bandwidth selectors proposed in literature about kernel density estimation, such as the ones based on cross-validation or plug-in strategies. The reader may refer to 6 for a recent review. In any case, both choices are certainly issues to be tackled, and will represent object of empirical investigation in the next section. As a final note of this section, observe that, since the proposed procedure relies on the definition of a segment as a cluster of connected pixels sharing the same color intensity, it cannot identify segments which configure themselves as isolated pixels within some differently colored background. Depending on the application, and on the image resolution, this may represent a limitation, or not. For example, the stars in the image of a starry sky might appear as single isolated yellow pixels if the image has low resolution (_i.e._ it is small), thus likely required to be identified as small segments. On the other hand, single differently-colored pixels might be just spots or imperfections, which would be preferably not segmented. Figure 5: Density estimate superimposed to the grayscale image in Figure 3 when $\hat{f}$ is based both on the colors and the spatial coordinates: level set representation (top panel) and perspective plot (bottom panel). ## 4 Empirical work ### 4.1 Simulation study As a first step of the empirical analysis, we present the results of a compendious simulation study aimed at understanding and systematizing the strengths and the limits of the proposed segmentation method, also with respect to the sensitivity of the density function to different kernels and smoothing amounts. To provide the images with the randomness required to run simulations, yet guaranteeing a given segmentation structure, images are generated as follows: each pixel is associated a priori with a nominal color intensity, which defines the segment membership; then, the actual intensity is randomly drawn from a normal distribution having the nominal intensity as a mean. Four main simulation settings are considered. Stemming from a reference image, with two convex shaped segments and no clefts (setting A1), one characteristic is changed at a time for each simulation setting, in order to control for its effect on the resulting segmentation. A larger number of smaller segments is considered (setting B1), non-convex shapes (setting C1), as well as the presence of clefts in the image, where the color intensity might mix up with the color intensity of neighbour segments (setting D1). Within each of these settings, different degrees of color homogeneity are simulated by increasing the variance of the normal distribution from which the color intensity of each pixel is drawn (setting A2, B2, C2, D2). Additionally, shaded segment contours are considered, determined by setting the border color intensity at the mean of the nominal intensities of the adjacent segments (setting A3, B3, C3, D3). One example of image generated from each setting and subsetting is displayed in the top of Tables 1, 2, 3, 4. Table 1: Simulation results for setting A: each entry displays the Monte Carlo average of the Adjusted Rand Index (ARI) and of the number of detected segments (the true number of segments is 2). Standard deviations are reported in brackets. \| \| A1: benchmark \| A2: shaded contours \| A3: heterogeneous colors ---\|---\|---\|---\|--- \| ARI \| 0.99 \| 0.98 \| 0.91 Normal K \| \| (0.08) \| (0.03) \| (0.14) $h=h_{N}$ \| # segments \| 3.10 \| 2.92 \| 2.71 \| \| (1.54) \| (1.52) \| (1.22) \| ARI \| 1.00 \| 0.99 \| 0.60 Normal K \| \| (0.00) \| (0.02) \| (0.20) $h=0.75h_{N}$ \| # segments \| 2.00 \| 2.00 \| 5.60 \| \| (0.00) \| (0.09) \| (2.22) \| ARI \| 1.00 \| 0.96 \| 0.77 Normal K \| \| (0.09) \| (0.06) \| (0.22) $h=1.25h_{N}$ \| # segments \| 2.00 \| 2.27 \| 4.07 \| \| (0.00) \| (0.83) \| (2.22) \| ARI \| 1.00 \| 0.54 \| 0.57 Uniform K \| \| (0.00) \| (0.45) \| (0.21) $h=h_{N}$ \| # segments \| 2.00 \| 1.86 \| 5.68 \| \| (0.00) \| (1.08) \| (2.06) \| ARI \| 1.00 \| 0.95 \| 0.66 Uniform K \| \| (0.00) \| (0.11) \| (0.23) $h=0.75h_{N}$ \| # segments \| 2.00 \| 2.18 \| 4.92 \| \| (0.00) \| (0.73) \| (2.33) \| ARI \| 0.55 \| 0.48 \| 0.48 Uniform K \| \| (0.16) \| (0.38) \| (0.16) $h=1.25h_{N}$ \| # segments \| 6.72 \| 2.82 \| 6.55 \| \| (1.86) \| (1.84) \| (1.78) Table 2: Simulation results for setting B: each entry displays the Monte Carlo average of the Adjusted Rand Index (ARI) and of the number of detected segments (the true number of segments is 9). Standard deviations are reported in brackets. \| \| B1: benchmark \| B2: shaded contours \| B3: heterogeneous colors ---\|---\|---\|---\|--- \| ARI \| 0.99 \| 0.83 \| 0.87 Normal K \| \| (0.02) \| (0.04) \| (0.10) $h=h_{N}$ \| # segments \| 9.51 \| 9.31 \| 8.75 \| \| (0.70) \| (0.79) \| (1.36) \| ARI \| 0.99 \| 0.84 \| 0.93 Normal K \| \| (0.02) \| (0.02) \| (0.06) $h=0.75h_{N}$ \| # segments \| 9.16 \| 9.19 \| 9.95 \| \| (0.39) \| (0.44) \| (1.18) \| ARI \| 0.84 \| 0.78 \| 0.82 Normal K \| \| (0.13) \| (0.08) \| (0.10) $h=1.25h_{N}$ \| # segments \| 6.99 \| 9.00 \| 8.66 \| \| (1.13) \| (1.34) \| (1.53) \| ARI \| 1.00 \| 0.78 \| 0.91 Uniform K \| \| (0.02) \| (0.06) \| (0.07) $h=h_{N}$ \| # segments \| 8.98 \| 9.30 \| 9.68 \| \| (0.20) \| (0.98) \| (1.23) \| ARI \| 0.65 \| 0.78 \| 0.96 Uniform K \| \| (0.00) \| (0.07) \| (0.04) $h=0.75h_{N}$ \| # segments \| 5.00 \| 8.97 \| 9.34 \| \| (0.00) \| (0.83) \| (0.64) \| ARI \| 0.94 \| 0.65 \| 0.85 Uniform K \| \| (0.11) \| (0.12) \| (0.09) $h=1.25h_{N}$ \| # segments \| 8.48 \| 7.99 \| 8.96 \| \| (1.42) \| (1.54) \| (1.50) For the analyses, density is estimated via (2), with both Gaussian and Uniform kernels. The bandwidth $h$ is selected as the asymptotically optimal solution for data following a Normal distribution (in the following, $h=h_{N}$). This selection criterion is, by construction, sub-optimal. Indeed, any assumption of Normality of the data color distribution cannot hold since multi-segment images have, in principle, a multimodal density. Nevertheless, this rule of thumb has resulted quite effective in clustering applications and has been then used in this analysis due to its simplicity and low computational burden. In order to understand the sensitivity of the procedure to different amounts of smoothing, the bandwidth has been both shrinked and enlarged slightly to the values $h=0.75h_{N}$ and $h=1.25h_{N}$ to evaluate the effect on the segmentation. The agreement between the true and the detected segment membership has been measured in terms of number of detected segments and Adjusted Rand Index 14, which takes increasing values for improved quality of the segmentation (the maximum value $1$ is associated to a perfect segmentation). In each simulation setting 500 grayscale images of $320$ pixels are generated, as this allows for keeping feasible the computational burden. All the empirical work has been performed in the `R` environment 21. Images have been handled via the packages `EBIimage` 20 and `BiOps` 4, while the segmentation routines have been built as adjustments of the clustering routines available in the package `pdfCluster` 2. Simulation results are summarized in Tables 1, 2, 3, 4. The procedure works comprehensively in a satisfactory way. With a Normal kernel, it provides almost perfect segmentations in the simplest setting A1, A2, A3, where results are robust to different amounts of smoothing. In the more challenging settings featured by shaded contours and heterogeneous color segments the performance of the segmentation worsens, yet remaining remarkably good in most of the considered frameworks. Neither non-convex segment shapes or the presence of many small segments do compromise the segmentation. In the latter case, segmentation may worsen for large smoothing, due to the risk of cluster merging when the density is oversmoothed. The greatest challenge is represented by the setting D (Table 4), since within interstices of the segments (like the thin frame of the main image) the color intensity mixes up with the intensity of the neighbours segments and prevents the edge detection. Increasing the smoothing amount may help to reduce the entailed oversegmentation. In fact, this behaviour is exacerbated by the small size of the segmented images, as thin clefts in high-resolution images can be anyway associated to relatively large spatial areas, and the color mixing is not expected to affect the whole areas. It shall be noticed, however, that oversegmentation appears as a general tendency which seems to feature the proposed method, since the number of detected segments is, on average, higher than the true one. Both choices of the kernel appear appropriate at a first sight and such choice is overall not relevant in the easiest settings. However, the use of a bounded Uniform kernel is riskier as expected, and results show higher variability and high sensitivity to a bad selection of the smoothing amount. Hence, the general preference for unbounded-support kernels is confirmed. Table 3: Simulation results for setting C: each entry displays the Monte Carlo average of the Adjusted Rand Index (ARI) and of the number of detected segments (the true number of segments is 2). Standard deviations are reported in brackets. \| \| C1: benchmark \| C2: shaded contours \| C3: heterogeneous colors ---\|---\|---\|---\|--- \| ARI \| 1.00 \| 1.00 \| 0.80 Normal K \| \| (0.00) \| (0.01) \| (0.20) $h=h_{N}$ \| # segments \| 2.00 \| 2.00 \| 3.87 \| \| (0.00) \| (0.00) \| (2.14) \| ARI \| 1.00 \| 0.94 \| 0.57 Normal K \| \| (0.00) \| (0.18) \| (0.17) $h=0.75h_{N}$ \| # segments \| 2.00 \| 2.19 \| 6.21 \| \| (0.00) \| (0.78) \| (2.02) \| ARI \| 1.00 \| 1.00 \| 0.67 Normal K \| \| (0.00) \| (0.01) \| (0.20) $h=1.25h_{N}$ \| # segments \| 2.00 \| 2.00 \| 5.15 \| \| (0.00) \| (0.04) \| (2.18) \| ARI \| 1.00 \| 0.98 \| 0.94 Uniform K \| \| (0.00) \| (0.07) \| (0.11) $h=h_{N}$ \| # segments \| 2.00 \| 2.13 \| 2.54 \| \| (0.00) \| (0.58) \| (1.10) \| ARI \| 1.00 \| 0.97 \| 0.83 Uniform K \| \| (0.00) \| (0.15) \| (0.17) $h=0.75h_{N}$ \| # segments \| 2.00 \| 2.01 \| 3.55 \| \| (0.00) \| (0.29) \| (1.67) \| ARI \| 0.29 \| 0.59 \| 0.71 Uniform K \| \| (0.32) \| (0.46) \| (0.37) $h=1.25h_{N}$ \| # segments \| 2.56 \| 2.02 \| 2.82 \| \| (1.55) \| (1.23) \| (1.88) Table 4: Simulation results for setting D: each entry display the Monte Carlo average of the Adjusted Rand Index (ARI) and of the number of detected segments (the true number of segments is 3). Standard deviations are reported in brackets. \| \| D1: benchmark \| D2: shaded contours \| D3: heterogeneous colors ---\|---\|---\|---\|--- \| ARI \| 0.49 \| 0.24 \| 0.38 Normal K \| \| (0.09) \| (0.25) \| (0.07) $h=h_{N}$ \| # segments \| 7.16 \| 2.77 \| 9.28 \| \| (1.47) \| (2.01) \| (1.85) \| ARI \| 0.42 \| 0.12 \| 0.36 Normal K \| \| (0.09) \| (0.19) \| (0.06) $h=0.75h_{N}$ \| # segments \| 8.94 \| 2.55 \| 10.59 \| \| (1.69) \| (2.28) \| (1.72) \| ARI \| 0.61 \| 0.13 \| 0.39 Normal K \| \| (0.34) \| (0.20) \| (0.12) $h=1.25h_{N}$ \| # segments \| 3.12 \| 2.93 \| 7.97 \| \| (1.97) \| (2.24) \| (2.01) \| ARI \| 1.00 \| 0.28 \| 0.36 Uniform K \| \| (0.00) \| (0.24) \| (0.07) $h=h_{N}$ \| # segments \| 3.00 \| 3.60 \| 9.87 \| \| (0.00) \| (2.53) \| (1.90) \| ARI \| 1.00 \| 0.42 \| 0.42 Uniform K \| \| (0.00) \| (0.17) \| (0.13) $h=0.75h_{N}$ \| # segments \| 3.00 \| 3.70 \| 7.53 \| \| (0.00) \| (2.22) \| (1.89) \| ARI \| 0.81 \| 0.39 \| 0.36 Uniform K \| \| (0.26) \| (0.30) \| (0.06) $h=1.25h_{N}$ \| # segments \| 3.63 \| 3.72 \| 10.37 \| \| (1.88) \| (2.31) \| (1.85) ### 4.2 Real images illustration In this section, some examples are presented to illustrate the proposed procedure in action, and to evaluate its effectiveness in identifying the salient regions for varying characteristics of more challenging images. Further examples are presented in the online Supplementary Material. Grayscale and multichannel images are considered, and selected for featuring either shaded and neat colors, and possible highlighted contours; both convex and nonconvex shaped segments have been analyzed. The procedure is here compared with the performance of some competitors: as a benchmark methods, $K-$means clustering is considered, thresholding based on the Otsu algorithm, and an edge-detection algorithm, based on the Sobel filter see, for details, 11 §10.2.1 and, respectively, §4.1. The first method has been given a head start by setting the number of clusters to its true number, as it is intuitively perceived by the author. The choice is not always obvious, especially for photos or, in general, shaded-color images. The second algorithm assumes that the image contains two classes of pixels -black/white- grossly corresponding to two modal regions of the histogram built on the gray intensities. It calculates the optimum threshold separating the two classes based on the minimization of the intra-class variance. Although it is designed for grayscale images only, thus requiring a prior preprocessing of multi-channel images, it has been considered for comparison as it represents a rough, yet effective, binary variant of the proposed procedure. The third method works in a slightly different logic, as it is aimed at edge rather than segment detection. Then, segments are assumed to be the sets of pixels within the linked edges. A further aim of the empirical work is to investigate the use of the density function as a tool to identify the main features of an image: one question of interest is its ability to detect edges of the segments. Also, in agreement with the previous section, its sensitivity to different smoothing amounts is evaluated, by testing $h\in\\{h_{N},0.75,h_{N},1.25h_{N}\\}$. Due to the considerations arisen in the simulation study, a Gaussian kernel only has been considered here. Additionally, the ability of the cluster tree to identify a hierarchy of cluster merging that is meaningful with respect to the image has been tested. Finally, the allocation of low-density pixels, not belonging to the segment cores has been observed and commented. The top left panels of Figures 6, 7 display two examples of grayscale images, selected for the analysis due to their different characteristics and different degrees of difficulty in segmentation. Multichannel images are displayed in the top left panels of Figures 8 and 9 . original image \| $K-$means segmentation, $K=5$ \| thresholding segmentation \| Sobel segmentation \| nonparametric segmentation, $h=h_{N}$ ---\|---\|---\|---\|--- \| \| \| \| segment cores, $h=h_{N}$ \| cluster tree, $h=h_{N}$ \| density contours, $h=h_{N}$ \| nonparametric segmentation, $h=0.75h_{N}$ \| nonparametric segmentation, $h=1.25h_{N}$ \| \| \| \| Figure 6: Segmentation results. Segments have been assigned arbitrary colors, except for the thresholding segmentation, where segments are either black or white by construction. The benchmark procedures work satisfactorily, in grayscale and multichannel images. Thresholding is intrinsically limited, as it identifies two segments only by construction. It is, nevertheless, able to reconstruct the broad features of the image, even when it is particularly challenging. See, _e.g._ the famous Einstein’s grimace, in Figure 7, not easy at all to be segmented, yet very well recognizable even in the binary reconstruction via thresholding. A self-evident limitation occurs when similar color shades characterize adjacent segments, since the dichotomous choice between black or white segments unavoidably determine a corrupt reconstruction of the image. See Figures 6 as an example of this behaviour. Despite its simplicity, $K-$means behave very well, being able to reconstruct accurately most of the images where the segment distinction is unarguable (Figures 6, 8). On the other hand, the procedure requires the number of segments to be known in advance, and a remarkable head start has been then granted by providing the true number of segments as an input. In some applications such number might be known a priori; consider, for example, X-ray or magnetic resonance medical images, where the number of segments can be set based on anatomy knowledge. However, if the purpose of image segmentation would be to attempt a first automatization of the diagnostic process, setting the number of segments to its ‘normal’ value, would prevent detection of fractures or anomalies, thus going in the opposite direction. A further feature of $K-$means is its complete noncontextuality; since distances from the cluster centroids are computed on the basis of the color only, disconnected segments might be assigned to the same cluster just for sharing a similar color, disregarding the adjacency structure. Depending on applications, this characteristic may be sensible or not. For instance, in Figure 8, assigning head and body to the same segment is particularly consistent. On the other hand, a complete noncontextuality may lead to the identification of meaningless clusters, as it happens for Figure 7 whose segmentation by $K-$means results in a pixelated image. The Sobel algorithm recognizes the edges between different segments generally in a very detailed way. This is particularly evident not only in simple examples with neat contours, like Bart Simpson (Figure 8) but also in challenging images like the fish (Figure 9). On the other hand, detection of the edges can be even too much detailed (Figure 7), with a risk of not identifying the salient regions of the image. Similarly to thresholding, it fails when there is small change in the color shades between adjacent segments, as it occurs with Figure 6. Furthermore, the edges do not form closed boundaries, thus possibly requiring the possible application of some further algorithm to complete the scope and define the segments. original image \| $K-$means segmentation, $K=14$ \| thresholding segmentation \| Sobel segmentation \| nonparametric segmentation, $h=h_{N}$ ---\|---\|---\|---\|--- \| \| \| \| segment cores, $h=h_{N}$ \| cluster tree, $h=h_{N}$ \| density contours, $h=h_{N}$ \| nonparametric segmentation, $h=0.75h_{N}$ \| nonparametric segmentation, $h=1.25h_{N}$ \| \| \| \| Figure 7: Segmentation results. Segments have been assigned arbitrary colors, except for the thresholding segmentation, where segments are either black or white by construction. original image \| $K-$means segmentation, $K=5$ \| thresholding segmentation \| Sobel segmentation \| nonparametric segmentation, $h=h_{N}$ ---\|---\|---\|---\|--- \| \| \| \| segment cores, $h=h_{N}$ \| cluster tree, $h=h_{N}$ \| density contours, $h=h_{N}$ \| nonparametric segmentation, $h=0.75h_{N}$ \| nonparametric segmentation, $h=1.25h_{N}$ \| \| \| \| Figure 8: Segmentation results. Segments have been assigned arbitrary colors, except for the thresholding segmentation, where segments are either black or white by construction. original image \| $K-$means segmentation, $K=5$ \| thresholding segmentation \| Sobel segmentation \| nonparametric segmentation, $h=h_{N}$ ---\|---\|---\|---\|--- \| \| \| \| segment cores, $h=h_{N}$ \| cluster tree, $h=h_{N}$ \| density contours, $h=h_{N}$ \| nonparametric segmentation, $h=0.75h_{N}$ \| nonparametric segmentation, $h=1.25h_{N}$ \| \| \| \| Figure 9: Segmentation results. Segments have been assigned arbitrary colors, except for the thresholding segmentation, where segments are either black or white by construction. The performance of the proposed nonparametric procedure are generally satisfactory when applied to both multichannel and grayscale images. The procedure does not result challenged by the need of distinguishing contours of assorted segments, both for size and nonconvex shapes, as especially evidenced by Figures 7 and 8. Consistently with the results of the simulation study, it is especially able to identify segments as connected regions characterized by uniformity of color, but performs well also when applied to image featured by shaded colors. On the con’s side, the procedure is somewhat sensitive to perceive color differences even when they are not distinguishable with the unaided eye at once, thus resulting in oversegmenting the image. As the method mainly hinges on the density, which is built on the image colors, it is in principle framed within the class of the noncontextual algorithms. However, it takes in some information about the spatial relationship between the pixels since each segment is, by construction, a (high density) connected set which is disconnected from the other segments. This characteristic prevents pixeled segmentations like $K-$means and, on the other hand does not allow the identification of unique segments as disconnected regions sharing the same color (see Figure 8, where Bart’s body and head are classified as different segments). Note that most of the segmented images include a number of black-colored regions, corresponding to unallocated pixels, and typically located at the borders of the segments. These are associated to low-density areas evaluated in the second step of the segmentation procedure. As discussed at the end of Section 3.1, in those cases, none of the pixels already assigned to the segment presenting maximum density (3) is adjacent to the ones under evaluation, _i.e._ , their color is not similar to the color of any other adjacent segment. In the analysis, these pixels are left unallocated, to enlighten the low degree of confidence in their classification. Related to this aspect, and depending on subject-matter considerations, the procedure allows the opportunity of not allocating pixels that are not belonging to the cluster cores. The values of $\hat{f}_{m}$ in (3), suitably normalized, provide a degree of confidence in the allocation, in the guise of fuzzy clustering schemes. This is especially useful in all the images where colors are homogeneous and segments well separated, as unallocated pixels mostly identify the boundaries of the segments (first bottom panel of Figures 6 to 9). With regard to the cluster tree (second bottom panel of Figures 6 to 9), it works effectively with multichannel images in establishing a hierarchy of cluster-merging which can be associated to different levels of resolution. In general, when scanning the density for varying $\lambda$, two segments appear disconnected because they are separated by some pixels having lower density color. At a lower level of $\lambda$, also these latter pixels have density above the threshold and merging can occur due to the spatial connectedness of all the involved pixels. In the Bart image, for example, clusters that are kept separated due to small color differences (see the face and the neck) are the first to be aggregated through the cluster tree. Similarly, in the fish image (Figure 9), at some high density level the different segments composing the sea are aggregated. In grayscale images similar conclusions may be drawn. In the Einstein image, for instance, highest-density aggregations of the tree branches entail merging the segments of the background. Yet, establishing a meaningful hierarchy of the segments is less easy with grayscale images, where also the human eye is challenged to aggregate clusters without resorting to subject-matter considerations and on the basis of the color only. In general, the density function results in an effective tools to identify the main features of the images, and density contours work well as edge detectors of the segments when $h=h_{N}$. See the third bottom panel of Figures 6 to 9. Additionally, the segmentation is quite robust to variation of $h$. This is especially true for simple images with neat contours as the Greyscale raws and Bart Simpson (IV and V bottom panels of Figures 6 and 8), where comparable segmentations are produced over the range of considered values of $h$. As expected, more challenging images tend to be oversegmented for small $h$ as seen, for instance, in the background of the Einstein and fish images (IV bottom panel of Figures 7 and 9). A large value of $h$, on the other hand, smoothes the density and results in segment aggregation, as seen in the last bottom panel of Figures 7 and 9. This can be seen as a way to reduce the risk of over-segmentation. ## 5 Concluding remarks Image segmentation is a complicated task whose implementation cannot, in general, leave aside subject-matter considerations. All this considered, the proposed procedure is framed halfway between contextual and noncontextual segmentation algorithms, and may be then applied to a variety of situations. It can be either applied fully automatically, or be richly customized, depending on the goals of the segmentation. 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Generalized Tilings with Height Functions Olivier Bodini 111LIP, École Normale Supérieure de Lyon, 46, Allée d’Italie, 69364 Lyon Cedex 05, France<EMAIL_ADDRESS>and Matthieu Latapy 222LIAFA, Université Paris 7, 2, place Jussieu, 75005 Paris, France. <EMAIL_ADDRESS> “Height functions are cool!” Cris Moore, DM-CCG’01 talk [LMN01]. Abstract: In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected flip-accessibility graph can be determined, and the directed flip- accessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce. Keywords: Tilings, Height Functions, Tilability, Distributive Lattices, Random Sampling, Potentials, Flows. ## 1 Preliminaries Given a finite set of elementary shapes, called _tiles_ , a _tiling_ of a given region is a set of translated tiles such that the union of the tiles covers exactly the region, and such that there is no overlapping between any tiles. See for example Figure 1 for a tiling of a polyomino (set of squares on a two-dimensional grid) with dominoes ($1\times 2$ and $2\times 1$ rectangles). Tilings are widely used in physics to modelize natural objects and phenomena. For example, quasicrystals are modelized by Penrose tilings [CF96] and dimers on a lattice are modelized by domino tilings [Ken00]. Tilings appeared in computer science with the famous undecidability of the question of whether the plane is tilable or not using a given finite set of tiles [Ber66]. Since then, many studies appeared concerning these objects, which are also strongly related to many important combinatorial problems [Lat00]. Figure 1: From left to right: the two possible tiles (called _dominoes_), a polyomino (_i.e._ a set of squares) to tile, and a possible tiling of the polyomino with dominoes. A local transformation is often defined over tilings. This transformation, called _flip_ , is a local rearrangement of some tiles which makes possible to obtain a new tiling from a given one. One then defines the (undirected) _flip- accessibility graph_ of the tilings of a region $R$, denoted by $\overline{A_{R}}$, as follows: the vertices of $\overline{A_{R}}$ are all the tilings of $R$, and $\\{t,t^{\prime}\\}$ is an (undirected) edge of $\overline{A_{R}}$ if and only if there is a flip between $t$ and $t^{\prime}$. See Figure 2 for an example. The flip notion is a key element for the enumeration of the tilings of a given region, and for many algorithmical questions. For example, we will see in the following that the structure of $\overline{A_{R}}$ may give a way to sample randomly a tiling of $R$, which is crucial for physicists. This notion is also a key element to study the entropy of the physical object [LR93], and to examine some of its properties like frozen areas, weaknesses, and others [JPS01]. Figure 2: From left to right: the flip operation over dominoes, and two examples of tilings which can be obtained from the one shown in Figure 1 by one flip. In these tilings, we shaded the tiles which moved during the flip. On some classes of tilings which can be drawn on a regular grid, it is possible to define a _height function_ which associates an integer to any node of the grid (it is called the _height_ of the point). For example, one can define such a function over domino tilings as follows. As already noticed, a domino tiling can be drawn on a two dimensional square grid. We can draw the squares of the grid in black and white like on a chessboard. Let us consider a polyomino $P$ and a domino tiling $T$ of $P$, and let us distinguish a particular point $p$ on the boundary of $P$, say the one with smaller coordinates. We say that $p$ is of height $0$, and that the height of any other point $p^{\prime}$ of $P$ is computed as follows: initialize a counter to zero, and go from $p$ to $p^{\prime}$ using a path composed only of edges of dominoes in $T$, increasing the counter when the square on the right is black and decreasing it when the square is white. The height of $p^{\prime}$ is the value of the counter when one reaches $p^{\prime}$. One can prove that this definition is consistent and can be used as the height function for domino tilings [Thu90]. See Figure 3 for an example. Figure 3: The directed flip-accessibility graph of the tilings of a polyomino by dominoes. The height of each point of the polyomino is shown for each tiling. The set of all the tilings of this polyomino is ordered by the flip relation directed with respect to the height functions. These height functions make it possible to define $A_{R}$, the _directed_ flip-accessibility graph of the tilings of $R$: the vertices of $A_{R}$ are the tilings of $R$ and there is a directed edge $(t,t^{\prime})$ if and only if $t$ can be transformed into $t^{\prime}$ by a flip which decreases the sum of the heights of all the points. See Figure 3 for an example with domino tilings. The generalized tilings we introduce in this paper are based on these height functions, and most of our results are induced by them. These notions of height functions are close to classical notions of flows theory in graphs. Let $G=(V,E)$ be a directed graph. A flow on $G$ is a map from $E$ into $\mathbb{C}$ (actually, we will only use flows with values in $\mathbb{Z}$). Given two vertices $v$ and $v^{\prime}$ of $G$, a travel from $s$ to $s^{\prime}$ is a set of edges of $G$ such that, if one forgets their orientations, then one obtains a path from $s$ to $s^{\prime}$. Given a flow $C$, the flux of $C$ on the travel $T$ is $F_{T}(C)\ =\ \sum_{e\in T^{+}}C(e)\ -\ \sum_{e\in T^{-}}C(e)$ where $T^{+}$ is the set of vertices of $T$ which are traveled in the right direction when one goes from $s$ to $s^{\prime}$, and $T^{-}$ is the set of vertices traveled in the reverse direction. One can easily notice that the flux is additive by concatenation of travels: if $T_{1}$ and $T_{2}$ are two travels such that the ending point of $T_{1}$ is equal to the starting point of $T_{2}$, then $F_{T_{1}\cdot T_{2}}(C)\ =\ F_{T_{1}}(C)+F_{T_{2}}(C)$. See [Ahu93] for more details about flows theory in graphs. Since there is no circuit in the graph $A_{R}$ (there exists no nonempty sequence of flips which transforms a tiling into itself), it induces an order relation over all the tilings of $R$: $t\leq t^{\prime}$ if and only if $t^{\prime}$ can be obtained from $t$ by a sequence of (directed) flips. In Section 3, we will study $A_{R}$ under the order theory point of view, and we will meet some special classes of orders, which we introduce now. A lattice is an order $L$ such that any two elements $x$ and $y$ of $L$ have a greatest lower bound, called the _infimum_ of $x$ and $y$ and denoted by $x\wedge y$, and a lowest greater bound, called the _supremum_ of $x$ and $y$ and denoted by $x\vee y$. The infimum of $x$ and $y$ is nothing but the greatest element among the ones which are lower than both $x$ and $y$. The supremum is defined dually. A lattice $L$ is _distributive_ if for all $x$, $y$ and $z$ in $L$, $x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z)$ and $x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)$. For example, it is known that the flip-accessibility graph of the domino tilings of a polyomino without holes is always a distributive lattice [Rem99b]. Therefore, this is the case of the flip-accessibility graph shown in Figure 3 (notice that the maximal element of the order is at the bottom, and the minimal one at the top of the diagram since we used the discrete dynamical models convention: the flips go from top to bottom). Lattices (and especially distributive lattices) are strongly structured sets. Their study is an important part of order theory, and many results about them exist. In particular, various codings and algorithms are known about lattices and distributive lattices. For example, there exists a generic algorithm to sample randomly an element of any distributive lattice [Pro98]. For more details about orders and lattices, we refer to [DP90]. Finally, let us introduce a useful notation about graphs. Given a directed graph $G=(V,E)$, the undirected graph $\overline{G}=(\overline{V},\overline{E})$ is the graph obtained from $G$ by removing the orientations of the edges. In other words, $\overline{V}=V$, and $\overline{E}$ is the set of undirected edges $\\{v,v^{\prime}\\}$ such that $(v,v^{\prime})\in E$. We will also call $\overline{G}$ the _undirected version_ of $G$. Notice that this is consistent with our definitions of $A_{R}$ and $\overline{A_{R}}$. In this paper, we introduce a generalization of tilings on which a height function can be defined, and show how some known results may be understood in this more general context. All along this paper, like we did in the present section, we will use the tilings with dominoes as a reference to illustrate our definitions and results. We used this unique example because it is very famous and simple, and permits to give clear figures. We emphasize however on the fact that our definitions and results are much more general, as explained in the last section of the paper. ## 2 Generalized tilings. In this section, we give all the definitions of the generalized notions we introduce, starting from the objects we tile to the notions of tilings, height functions, and flips. The first definitions are very general, therefore we will only consider some classes of the obtained objects, in order to make the more specific notions (mainly height functions and flips) relevant in this context. However, the general objects introduced may be useful in other cases. Let $G$ be a simple ($G$ has no multiple edges, no loops, and if $(v,v^{\prime})$ is an edge then $(v^{\prime},v)$ can not be an edge) directed graph. We consider a set $\Theta$ of elementary circuits of $G$, which we will call cells. Then, a polycell is any set of cells in $\Theta$. Given a polycell $P$, we call the edges of cells in $P$ the _edges of $P$_, and their vertices the _vertices of $P$_. A polycell $P$ is $k$-regular if and only if there exists an integer $k$ such that each cell of $P$ is a circuit of length $k$. Given a polycell $P$, the _boundary_ of $P$, denoted by $\partial P$, is a (arbitrarily) distinguished set of edges of $P$. We say that a vertex of $P$ is on the boundary of $P$ if it is the extremity of an edge in $\partial P$. A polycell $P$ is full if the undirected boundary $\overline{\partial P}$ is connected. Given an edge $e$ of $P$ which is not in $\partial P$, we call the set of all the cells in $P$ which have $e$ in common a tile. We will always suppose that, given any set of cells of $P$, they have at most one edge in common. A _tiling_ $Q$ of a polycell $P$ is then a partition of the set of cells of $P$ into tiles. A polycell $P$ which admits at least a tiling $Q$ is tilable. Notice that, if one considers a tiling $Q$, then one has a natural bijection $\pi$ between the tiles of $Q$ and a set of edges of $G$: if $t$ is a tile in $Q$, then $\pi(t)$ is nothing but the edge which is in each of the cells which define $t$ (recall that we have made the assumption that this edge is unique). The edges in $\pi(Q)=\\{\pi(t),\ t\in Q\\}$ are called the tiling edges of $Q$. See Figure 4 and Figure 5 for some examples. Notice that if we distinguish exactly one edge of each cell of a polycell $P$, then the distinguished edges can be viewed as the tiling edges of $P$. Indeed, each edge induces a tile (the set of cells which have this edge in common), and each cell is in exactly in one tile. Figure 4: From left to right: a polycell $P$ (the boundary $\partial P$ is composed of all the edges except the dotted ones), the three tiles of $P$, and a tiling of $P$ represented by its tiling edges (the dotted edges). Figure 5: Left: a $4$-regular polycell $P$, the boundary of which is composed of those edges which belong to only one cell. Right: a tiling of $P$ represented by its tiling edges (the dotted edges). Notice that this figure is very close to Figure 1. Let $P$ be a $k$-regular tilable polycell and $Q$ be a tiling of $P$. We associate to $Q$ a flow $C_{Q}$ on $\Theta$ (seen as a graph): $C_{Q}(e)=\left\\{\begin{array}[]{ll}1-k&\mbox{if the edge $e$ is a tiling edge of $Q$}\\\ 1&\mbox{otherwise.}\end{array}\right.$ For each cell $c$, we define $T_{c}$ as the travel which contains exactly the edges in $c$ (in other words, it consists in turning around $c$). Notice that the flux of $C_{Q}$ on the travel $T_{c}$ is always null: $F_{T_{c}}(C_{Q})=0$ since each cell contains exactly a tiling edge, valued $1-k$, and $k-1$ other edges, valued $1$. Moreover, for each edge $e\in\partial P$, we have $C_{Q}(e)=1$. Let us consider a polycell $P$ and a flow $C$ on the edges of $P$ such that $C(e)=1$ for all edge $e$ in $\partial P$. If for all closed travel $T$ (_i.e._ a cycle when one forgets the orientation of each edge) on the boundary of $P$ we have $F_{T}(C)=0$, then we say that $P$ has a balanced boundary. More specifically, if for all closed travel $T$ in $P$ (not only on the boundary) we have $F_{T}(C)=0$, then the flow $C$ is called a tension. Finally, a polycell $P$ is contractible if it satisfies the two following properties: * • $P$ has a balanced boundary. * • $C$ is a tension if and only if for all cell $c$, $F_{T_{c}}(C)=0$. Notice that if $P$ is a contractible $k$-regular polycell and $Q$ is a tiling of $P$, then the flow $C_{Q}$ is a tension. Now, if we (arbitrarily) distinguish a vertex $\nu$ on the boundary of $P$, we can associate to the tension $C_{Q}$ a potential $\varphi_{Q}$, defined over the vertices of $P$: * • $\varphi_{Q}(\nu)=0$. * • for all vertices $x$ and $y$ of $P$, $\varphi_{Q}(y)-\varphi_{Q}(x)=F_{T_{x,y}}(C_{Q})$ where $T_{x,y}$ is a travel between $x$ and $y$. The distinguished vertex is needed else $\varphi_{Q}$ would only be defined at almost a constant, but one can choose any vertex on the boundary. Notice that this potential can be viewed as a _height function_ associated to $Q$, and we will see that it indeed plays this role in the following. Therefore, we will call the potential $\varphi_{Q}$ the _height function_ of $Q$. See Figure 6 for an example. Figure 6: From left to right: a tiling $Q$ of a polycell (represented by its tiling edges, the dotted ones), the tension $C_{Q}$ and the height function (or potential) $\varphi_{Q}$ it induces. Again, this figure may be compared to Figure 3 (topmost tiling). We now have all the main notions we need about tilings of polycells, including height functions, except the notion of flips. In order to introduce it, we need to prove the following: ###### Theorem 2.1. Let $P$ be a $k$-regular contractible polycell. There is a bijection between the tilings of $P$ and the tensions $C$ on $P$ which verify: * • for all edge $e$ in $\partial P$, $C(e)=1$, * • and for all edge $e$ of $P$, $C(e)\in\\{1-k,1\\}$. ###### Proof. For all tiling $Q$ of $P$, we have defined above a flow $C_{Q}$ which verifies the property in the claim, and such that for all cell $c$, $F_{T_{c}}(C_{Q})=0$. Since $P$ is contractible, this last point implies that $C_{Q}$ is a tension. Conversely, let us consider a tension $C$ which satisfies the hypotheses. Since each cell is of length $k$, and since $C(e)\in\\{1-k,1\\}$, the fact that $F_{T_{c}}(C)=0$ implies that each cell has exactly one negative edge. These negative edges can be considered as the tiling edges of a tiling of $P$, which ends the proof. ∎ Given a $k$-regular contractible polycell $P$ defined over a graph $G$, this theorem allows us to make no distinction between a tiling $Q$ and the associated tension $C_{Q}$. This makes it possible to define the notion of flip as follows. Suppose there is a vertex $x$ in $P$ which is not on the boundary and such that its height, with respect to the height function of $Q$, is greater than the height of each of its neighbors in $\overline{G}$. We will call such a vertex a _maximal_ vertex. The neighbors of $x$ in $\overline{G}$ have a smaller height than $x$, therefore the outgoing edges of $x$ in $G$ are tiling edges of $Q$ and the incoming edges of $x$ in $G$ are not. Let us consider function $C_{Q^{\prime}}$ defined as follows: $C_{Q^{\prime}}(e)=\left\\{\begin{array}[]{ll}1-k&\mbox{if $e$ is an outgoing edge of $x$}\\\ 1&\mbox{if $e$ is an incoming edge of $x$}\\\ C_{Q}(e)&\mbox{else.}\end{array}\right.$ Each cell $c$ which contains $x$ contains exactly one outgoing edge of $x$ and one incoming edge of $x$, therefore we still have $F_{T_{c}}(C_{Q^{\prime}})=0$. Therefore, $C_{Q^{\prime}}$ is a tension, and so it induces from Theorem 2.1 a tiling $Q^{\prime}$. We say that $Q^{\prime}$ is obtained from $Q$ by a _flip around $x$_, or simply by a _flip_. Notice that $Q^{\prime}$ can also be defined as the tiling associated to the height function obtained from the one of $Q$ by decreasing the height of $x$ by $k$, and without changing anything else. This corresponds to what happens with classical tilings (see for example [Rem99b]). See Figure 7 for an example. Figure 7: A flip which transforms a tiling $Q$ of a polycell $P$ into another tiling $Q^{\prime}$ of $P$. From left to right, the flip is represented between the tilings, then between the associated tensions, and finally between the associated height functions. We now have all the material needed to define and study $A_{P}$, the (directed) flip-accessibility graph of the tilings of $P$: $A_{P}=(V_{P},E_{P})$ is the directed graph where $V_{P}$ is the set of all the tilings of $P$ and $(Q,Q^{\prime})$ is an edge in $E_{P}$ if $Q$ can be transformed into $Q^{\prime}$ by a flip. We will also study the undirected flip-accessibility graph $\overline{A_{P}}$. The properties of these graphs are crucial for many questions about tilings, like enumeration, generation and sampling. ## 3 Structure of the flip-accessibility graph. Let us consider a $k$-regular contractible polycell $P$ and a tiling $Q$ of $P$. Let $h$ be the maximal value among the heights of all the points with respect to the height function of $Q$. If $Q$ is such that all the vertices of height $h$ are on the boundary of $P$, then it is said to be a _maximal tiling_. For a given $P$, we denote by $\mbox{Tmax}_{P}$ the set of the maximal tilings of $P$. We will see that these tilings play a particular role in the graph $A_{P}$. In particular, we will give an explicit relation between them and the number of connected components of $\overline{A_{P}}$. Recall that we defined the maximal vertices of $Q$ as the vertices which have a height greater than the height of each of their neighbors, with respect to the height function of $Q$ (they are _local_ maximals). ###### Lemma 3.1. Let $P$ be a $k$-regular tilable contractible polycell ($P$ is not necessarily full). There exists a maximal tiling $Q$ of $P$. ###### Proof. Let $V$ be the set of vertices of $P$, and let $Q$ be a tiling of $P$ such that for all tiling $Q^{\prime}$ of $P$, we have: $\sum_{x\in V}\varphi_{Q}(x)\leq\sum_{x\in V}\varphi_{Q^{\prime}}(x).$ We will prove that $Q$ is a maximal tiling. Suppose there is a maximal vertex $x_{m}$ which is not on the boundary. Therefore, one can transform $Q$ into $Q^{\prime}$ by a flip around $x_{m}$. Then $\sum_{x\in V}\varphi_{Q^{\prime}}(x)\ =\ \sum_{x\in V}\varphi_{Q}(x)-k$, which is in contradiction with the hypothesis. ∎ ###### Lemma 3.2. For all tiling $Q$ of a $k$-regular contractible polycell $P$, there exists a unique tiling in $\mbox{Tmax}_{P}$ reachable from $Q$ by a sequence of flips. ###### Proof. It is clear that at least one tiling in $\mbox{Tmax}_{P}$ can be reached from $Q$ by a sequence of flips, since the flip operation decreases the sum of the heights, and since we know from the proof of Lemma 3.1 that a tiling such that this sum is minimal is always in $\mbox{Tmax}_{P}$. We now have to prove that the tiling in $\mbox{Tmax}_{P}$ we obtain does not depend on the order in which we flip around the successive maximal vertices. Since making a flip around a maximal point $x$ is nothing but decreasing its height by $k$ and keeping the other values, if we have two maximal vertices $x$ and $x^{\prime}$ then it is equivalent to make first the flip around $x$ and after the flip around $x^{\prime}$ or the converse. ∎ ###### Lemma 3.3. Let $P$ be a $k$-regular contractible and tilable polycell. A tiling $Q$ in $\mbox{Tmax}_{P}$ is totally determined by the values of $\varphi_{Q}$ on $\partial P$. ###### Proof. The proof is by induction over the number of cells in $P$. Let $x$ be a maximal vertex for $\varphi_{Q}$ in $\partial P$. For all outgoing edges $e$ of $x$, $C_{Q}(e)=1-k$ (otherwise $\varphi(x)$ would not be maximal). Therefore, these edges can be considered as tiling edges, and determine some tiles of a tiling $Q$ of $P$. Iterating this process, one finally obtains $Q$. See Figure 8 for an example. ∎ ###### Theorem 3.4. Let $P$ be a $k$-regular contractible and tilable polycell. The number of connected components in $\overline{A_{P}}$ is equal to the cardinal of $\mbox{Tmax}_{P}$. ###### Proof. Immediate from Lemma 3.2. ∎ This theorem is very general and can explain many results which appeared in previous papers. We obtain for example the following corollary, which generalizes the one saying that any domino tiling of a polyomino can be transformed into any other one by a sequence of flips [BNRR95]. ###### Corollary 3.5. Let $P$ be a full $k$-regular contractible and tilable polycell. There is a unique element in $\mbox{Tmax}_{P}$, which implies that $\overline{A_{P}}$ is connected. ###### Proof. Since $\overline{\partial P}$ is connected, the heights of the points in $\partial P$ are totally determined by the orientation of the edges of $\partial P$ and do not depend on any tiling $Q$. Therefore, from Lemma 3.3, there is a unique tiling in $\mbox{Tmax}_{P}$. ∎ As a consequence, if $P$ is a full tilable and contractible polycell, the height of a vertex $x$ on the boundary of $P$ is independent of the considered tiling. In the case of full polyominoes, this restriction of $\varphi_{Q}$ to the boundary of $P$ is called height on the boundary [Fou97] and has been introduced in [Thu90]. Notice that this height on the boundary can be defined in the more general case where $P$ has a balanced boundary. Notice also that the proof of Lemma 3.3 gives an algorithm to build the unique maximal tiling of any $k$-regular contractible and tilable full polycell $P$, since the height function on the boundary of $P$ can be computed without knowing any tiling of $P$. See Algorithm 1 and Figure 8. This algorithm gives in polynomial time a tiling of $P$ if it is tilable. It can also be used to decide whether $P$ is tilable or not. Therefore, it generalizes the result of Thurston [Thu90] saying that it can be decided in polynomial time if a given polyomino is tilable with dominoes. Input: A full $k$-regular contractible polycell $P$, its boundary $\partial P$ and a distinguished vertex $\nu$ on this boundary. Output: An array _tension_ on integers indexed by the edges of $P$ and another one _height_ indexed by the vertices of $P$. The first gives the tension associated to the maximal tiling, and the second gives its height function. begin $P^{\prime}\leftarrow P$; $\mbox{height}[\nu]\leftarrow 0$; for each _edge $e=(v,v^{\prime})$ in $\partial P^{\prime}$_ do $\mbox{tension}[e]\leftarrow 1$; for each _vertex $v$ on the boundary of $P^{\prime}$_ do Compute $\mbox{height}[v]$ using the values in tension; repeat for each _vertex $v$ on the boundary of $P^{\prime}$ which has the minimal height among the heights of all the vertices on the boundary_ do for each _incoming edge $e$ of $v$_ do $\mbox{tension}[e]\leftarrow 1-k$; for each _edge $e^{\prime}$ in a cell containing $e$_ do $\mbox{tension}[e^{\prime}]\leftarrow 1$; for each _edge $e=(v,v^{\prime})$ such that $\mbox{tension}[e]$ has newly be computed_ do Compute $\mbox{height}[v]$ and $\mbox{height}[v^{\prime}]$ using the values in tension; Remove in $P^{\prime}$ the cells which contain a negative edge; Compute the boundary of $P^{\prime}$: it is composed of all the vertices of $P^{\prime}$ which have a computed height; until _$P^{\prime}$ is empty_; end 1 Computation of the maximal tiling of a full $k$-regular contractible polycell. Figure 8: An example of execution of Algorithm 1. From left to right, we give the polycell, the result of the computation of the height on the boundary, and then the results of each iteration of the addition of tiles and removing of tiles process. In this example, the first iteration of the algorithm gives one vertical tile, and the second (and last) iteration gives four horizontal tiles. With these results, we obtained much information concerning a central question of tilings: the connectivity of the undirected flip-accessibility graph. We did not only give a condition under which this graph is connected, but we also gave a relation between the number of its connected components and some special tilings. We will now deepen the study of the structure induced by the flip relation by studying the directed flip-accessibility graph, and in particular the partial order it induces over the tilings: $t\leq t^{\prime}$ if and only if $t^{\prime}$ can be obtained from $t$ by a sequence of (directed) flips. ###### Lemma 3.6. Let $Q$ and $Q^{\prime}$ be two tilings in the same connected component of $A_{P}$ for a given $k$-regular contractible polycell $P$. Let us consider $x_{m}$ such that $\left\|\varphi_{Q}(x_{m})-\varphi_{Q^{\prime}}(x_{m})\right\|$ is maximal in $\\{\left\|\varphi_{Q}(x)-\varphi_{Q^{\prime}}(x)\right\|,\ x\mbox{ is a vertex of $P$}\\}$. Then, one can make a flip around $x_{m}$ from $Q$ or $Q^{\prime}$. ###### Proof. We can suppose that $\varphi_{Q^{\prime}}(x_{m})<\varphi_{Q}(x_{m})$ (otherwise we exchange $Q$ and $Q^{\prime}$). We will show that the height function $\varphi$ defined by $\varphi(x_{m})=\varphi_{Q}(x_{m})-k$ and $\varphi(x)=\varphi_{Q}(x)$ for all vertex $x\not=x_{m}$ defines a tiling of $P$ (which is therefore obtained from $Q$ by a flip around $x_{m}$). Let us consider any circuit which contains $x_{m}$. Therefore, it contains an incoming edge $(x_{p},x_{m})$ and an outgoing edge $(x_{m},x_{s})$ of $x_{m}$. We will prove that $\varphi_{Q}(x_{p})=\varphi_{Q}(x_{m})-1$ and $\varphi_{Q}(x_{s})=\varphi_{Q}(x_{m})-k+1$, which will prove the claim since it proves that $x_{m}$ is a maximal vertex. The couple $(\varphi_{Q}(x_{p}),\varphi_{Q}(x_{s}))$ can have three values: $(\varphi_{Q}(x_{m})-1,\varphi_{Q}(x_{m})+1)$, $(\varphi_{Q}(x_{m})-1,\varphi_{Q}(x_{m})-k+1)$, or $(\varphi_{Q}(x_{m})+k-1,\varphi_{Q}(x_{m})+1)$. But, if $\varphi_{Q}(x_{s})=\varphi_{Q}(x_{m})+1$ then $\varphi_{Q^{\prime}}(x_{s})=\varphi(x_{m})+1$, and so $\varphi_{Q^{\prime}}(x_{m})=\varphi(x_{m})+k$, which is a contradiction. If $\varphi_{Q}(x_{p})=\varphi_{Q}(x_{m})+k-1$ then $\varphi_{Q^{\prime}}(x_{p})=\varphi_{Q}(x_{m})+k-1$, and so $\varphi_{Q^{\prime}}(x_{m})>\varphi_{Q}(x_{m})$, which is a contradiction again. Therefore, $(\varphi_{Q}(x_{p}),\varphi_{Q}(x_{s}))$ must be equal to $(\varphi_{Q}(x_{m})-1,\varphi_{Q}(x_{m})-k+1)$ for all circuit which contain $x_{m}$, which is what we needed to prove. ∎ Let us now consider two tilings $Q$ and $Q^{\prime}$ of a $k$-regular contractible polycell $P$. Let us define $\max(\varphi_{Q},\varphi_{Q^{\prime}})$ as the height function such that its value at each point is the maximal between the values of $\varphi_{Q}$ and $\varphi_{Q^{\prime}}$ at this point. Let us define $\min(\varphi_{Q},\varphi_{Q^{\prime}})$ dually. Then, we have the following result: ###### Lemma 3.7. Given two tilings $Q$ and $Q^{\prime}$ of a $k$-regular contractible polycell $P$, $\max(\varphi_{Q},\varphi_{Q^{\prime}})$ and $\min(\varphi_{Q},\varphi_{Q^{\prime}})$ are the height functions of tilings of $P$. ###### Proof. We can see that $\max(\varphi_{Q},\varphi_{Q^{\prime}})$ is the height function of a tiling of $P$ by iterating Lemma 3.6: $\sum\limits_{x}\left\|\varphi_{Q}(x)-\varphi_{Q^{\prime}}(x)\right\|$ can be decreased until $Q=Q^{\prime}$. The proof for $\min(\varphi_{Q},\varphi_{Q^{\prime}})$ is symmetric. ∎ ###### Theorem 3.8. If $P$ is a $k$-regular contractible polycell, then $A_{P}$ induces a distributive lattice structure over the tilings of $P$. ###### Proof. Given two tilings $Q$ and $Q^{\prime}$ in $A_{P}$, let us define the following binary operations: $\varphi_{Q}\wedge\varphi_{Q^{\prime}}=\min(\varphi_{Q},\varphi_{Q^{\prime}})$ and $\varphi_{Q}\vee\varphi_{Q^{\prime}}=\max(\varphi_{Q},\varphi_{Q^{\prime}})$. It is clear from the previous results that this defines the infimum and supremum of $Q$ and $Q^{\prime}$. To show that the obtained lattice is _distributive_ , it suffices now to verify that these relations are distributive together. ∎ As already discussed, this last theorem gives much information on the structure of the flip-accessibility graphs of tilings of polycells. It also gives the possibility to use in the context of tilings the numerous results known about distributive lattices, in particular the generic random sampling algorithm described in [Pro98]. To finish this section, we give another proof of Theorem 3.8 using only discrete dynamical models notions. This proof is very simple and has the advantage of putting two combinatorial object in a relation which may help understanding them. However, the reader not interested in discrete dynamical models may skip the end of this section. An Edge Firing Game (EFG) is defined by a connected undirected graph $G$ with a distinguished vertex $\nu$, and an orientation $O$ of $G$. In other words, $\overline{O}=G$. We then consider the set of obtainable orientations when we iterate the following rule: if a vertex $v\not=\nu$ only has incoming edges (it is a _sink_) then one can reverse all these edges. This set of orientations is ordered by the reflexive and transitive closure of the evolution rule, and it is proved in [Pro93] that it is a distributive lattice. We will show that the set of tilings of any $k$-regular contractible polycell $P$ Theorem 3.8. Let us consider a $k$-regular contractible polycell $P$ defined over a graph $G$, and $G^{\prime}$ the sub-graph of $G$ which contains exactly the vertices and edges in $P$. Let us now consider the height function $\varphi_{Q}$ of a tiling $Q$ of $P$, and let us define the orientation $\pi(Q)$ of $\overline{G^{\prime}}$ as follows: each undirected edge $\\{v,v^{\prime}\\}$ in $\overline{G^{\prime}}$ is directed from $v$ to $v^{\prime}$ in $\pi(Q)$ if $\varphi_{Q}(v^{\prime})>\varphi_{Q}(v)$. Then, the maximal vertices of $Q$ are exactly the ones which have only incoming edges in $\pi(Q)$, and applying the EFG rule to a vertex of $\pi(Q)$ is clearly equivalent to making a flip around this vertex in $Q$. Therefore, the configuration space of the EFG is isomorphic to the flip-accessibility graph $A_{P}$, which proves Theorem 3.8. ## 4 Some applications. In this section, we study some examples which appear in the literature with the help of our generalized framework. We show how these classes of tiling problems can be seen as special cases of $k$-regular contractible polycells tilings. We therefore obtain as corollaries some known results about these problems, as well as some new results. ### 4.1 Polycell drawn on the plane. Let us consider a set of vertices $V$ and a set $\Theta$ of elementary (undirected) cycles of length $k$, with vertices in $V$, such that any couple of cycles in $\Theta$ have at most one edge in common. Not let us consider the undirected graph $G=(V,E)$ such that $e$ is an edge of $G$ if and only if it is an edge of a cycle in $\Theta$. Moreover, let us restrict ourselves to the case where $G$ is a planar graph which can be drawn in such a way that no cycle of $\Theta$ is drawn inside another one. $G$ is 2-dual-colorable if one can color in black and white each bounded face in such a way that two faces which have an edge in common have different colors. See for example Figure 9. Figure 9: Two examples of graphs which satisfy all the properties given in the text. The leftmost is composed of cycles of length $3$ and has a hole. The rightmost one is composed of cycles of length $4$. Figure 10: A tiling of each of the objects shown in Figure 9, obtained using the polycells formailsm. The fact that $G$ has the properties above, including being 2-dual-colorable, makes it possible to encode tilings with bifaces (the tiles are two adjacent faces) as tilings of polycells. This includes tilings with dominoes, and tilings with calissons. Following Thurston [Thu90], let us define an oriented version of G as follows: the edges which constitute the white cycles boundaries are directed to travel the cycle in the clockwise orientation, and the edges which constitute the black cycles boundaries are directed counterclockwise. One can then verify that a balanced boundary polycell defined this way is always contractible. Therefore, our results can be applied, which generalizes the results of Chaboud [Cha96] and Thurston [Thu90]. ### 4.2 Rhombus tiling in higher dimension. Let us consider the canonical basis $\\{e_{1},\dots,e_{d}\\}$ of the $d$-dimensional affine space ${\mathbb{R}}^{d}$, and let us define $e_{d+1}=\sum_{i=1}^{d}e_{i}$. For all $\alpha$ between $1$ and $d+1$, let us define the zonotope $Z_{d,d}^{\alpha}$ as the following set of points: $Z_{d,d}^{\alpha}\ =\ \\{x\in{\mathbb{R}}^{d}\mbox{ such that }x=\sum_{i=1,i\not=\alpha}^{d+1}\lambda_{i}e_{i},\mbox{ with }-1\leq\lambda_{i}\leq 1\\}.$ In other words, the $Z_{d,d}^{\alpha}$ is the zonotope defined by all the vectors $\varepsilon_{i}$ except the $\alpha$-th. We are interested in the tilability of a given solid $S$ when the set of allowed tiles is $\\{Z_{d,d}^{\alpha},\ 1\leq\alpha\leq d+1\\}$. These tilings are called _codimension one rhombus tilings_ , and they are very important as a physical model of quasicristals [DMB97]. If $d=2$, they are nothing but the tilings of regions of the plane with three parallelograms which tile an hexagon, which have been widely studied. See Figure 11 for an example in dimension $2$, and Figure 13 for an example in dimension $3$. In order to encode this problem by a problem over polycells, let us consider the directed graph $G$ with vertices in ${\mathbb{Z}}^{d}$ and such that $e=(x,y)$ is an edge if and only if $y=x+\varepsilon_{j}$ for an integer $j$ between $1$ and $d$ or $y=x-\varepsilon_{d+1}$. We will call diagonal edges the edges which correspond to the second case. This graph can be viewed as a $d$-dimensional directed grid (the direction are given by the order on the coordinates), to which we add a diagonal edge in the reverse direction, in each element of the grid. An example in dimension $3$, is given in Figure 12. Figure 11: If one forgets the orientations and removes the dotted edges, then the rightmost object is a classical codimencion one rhombus tiling of a part of the plane ($d=2$). From the polycells point of view, the leftmost object represents the underlying graph $G$, the middle object represents a polucell $P$ (the boundary of which is the set of the edges which belong to only one cell), and the rightmost object represents a tiling of $P$ (the dottes edges are the tiling edges). Figure 12: The $3$-dimensional grid is obtained by a concatenation of cubes like this one. Figure 13: A codimension one rhombus tiling with $d=3$ (first line, rightmost object). It is composed of four different three dimensional tiles, and the first line shows how it can be constructing by adding successive tiles. The second line shows the position of each tile with respect to the cube. Each edge is clearly in a one-to-one correspondence with a copy of a $Z_{d,d}^{\alpha}$ translated by an integer vector: this is the copy on the $d$-dimensional grid of which it is a diagonal. The set $\Theta$ of the cells we will consider is the set of all the circuits of length $d+1$ which contain exactly one diagonal edge. Therefore, each edge belongs to a $d!$ cells, and so the tiles will be themselves composed of $d!$ cells. Given a polycell $P$ defined over $\Theta$, we define $\partial P$ as the set of the edges of $P$ which do not belong to $d!$ circuits of $P$. First notice that a full polycell defined over $G$ is always contractible. Therefore, our previous results can be applied, which generalizes some results presented in [DMB97] and [LM99, LMN01]. We also generalize some results about the 2-dimensional case, which has been widely studied. ## 5 Conclusion and Perspectives. In conclusion, we gave in this paper a generalized framework to study the tiling problems over which a height function can be defined. This includes the famous tilings of polyominoes with dominoes, as well as various other classes, like codimension one rhombus tilings, tilings on torus, on spheres, three- dimensional tilings, and others we did not detail here. We gave some results on our generalized tilings which made it possible to obtain a large set of known results as corollaries, as well as to obtain new results on tiling problems which appear in the scientific literature. Many other problems may exist which can be modelized in the general framework we have introduced, and we hope that this paper will help understanding them. Many tiling problems, however, do not lead to the definition of any height function. The key element to make such a function exist is the presence of a strong underlying structure (the $k$-regularity of the polycell, for example). Some important tiling problems (for example tilings of zonotopes) do not have this property, and so we can not apply our results in this context. Some of these problems do not have the strong properties we obtained on the tilings of $k$-regular contractible polycells, but may be included in our framework, since our basic definitions of polycells and tilings being very general. This would lead to general results on more complex polycells, for example polycells which are not $k$-regular. Acknowledgments: The authors thank Frédéric Chavanon for useful comments on preliminary versions, which deeply improved the manuscript quality.
Further author information: (Send correspondence to P.B.) P.B.: E-mail<EMAIL_ADDRESS> S.L.: E-mail<EMAIL_ADDRESS> # Physical Reservoir Computing with Origami and its Application to Robotic Crawling Priyanka Bhovad Department of Mechanical Engineering, Clemson University, Clemson, SC, US Suyi Li Department of Mechanical Engineering, Clemson University, Clemson, SC, US ###### Abstract A new paradigm called physical reservoir computing has recently emerged, where the nonlinear dynamics of high-dimensional and fixed physical systems are harnessed as a computational resource to achieve complex tasks. Via extensive simulations based on a dynamic truss-frame model, this study shows that an origami structure can perform as a dynamic reservoir with sufficient computing power to emulate high-order nonlinear systems, generate stable limit cycles, and modulate outputs according to dynamic inputs. This study also uncovers the linkages between the origami reservoir’s physical designs and its computing power, offering a guideline to optimize the computing performance. Comprehensive parametric studies show that selecting optimal feedback crease distribution and fine-tuning the underlying origami folding designs are the most effective approach to improve computing performance. Furthermore, this study shows how origami’s physical reservoir computing power can apply to soft robotic control problems by a case study of earthworm-like peristaltic crawling without traditional controllers. These results can pave the way for origami-based robots with embodied mechanical intelligence. ###### keywords: Physical reservoir computing, Origami, Morphological computation, Soft robotics, Peristaltic locomotion ## 1 INTRODUCTION The animal kingdom is an endless source of inspiration for soft robotics [1, 2]. Researchers have constructed compliant robots that can mimic all kinds of animal motions, like octopus locomotion [3], elephant trunk grasping [4], insect flying [5], jellyfish and fish swimming [6, 7, 8], as well as snake and insects crawling [9, 10, 11]. These robots share many similarities with animals regarding their shape and motion kinematics; however, their underlying sensing, actuation, and control architectures could be fundamentally different. Our engineered soft robots typically rely on a centralized controller (aka. an “electronic brain”) that takes up all computing work to process sensor information, generate control commands, and make decisions. This approach often struggles to achieve high actuation speed and control effectiveness as soft robots exhibit virtually infinite degrees of freedom and complicated dynamic characteristics. On the other hand, animals have highly interconnected networks of nerves and muscles that can share the workload with the brain [12, 13, 14]. The animal body’s morphology is an integral part of its actuation, control, and ultimately its “brain’s” decision-making process, leading to far superior efficiency than our engineered soft robots. Motivated by this disparity, an increasing number of researchers have embraced soft bodies’ nonlinear dynamics as a computational resource to create an embodied intelligence and control [15, 16, 17, 18, 19, 20, 21]. As a result, a new computational paradigm called morphological computation has emerged in which the physical body of the robot itself takes part in performing low-level control tasks, such as locomotion coordination and modulation, to simplify the overall control architecture significantly [15, 16, 18, 17, 22]. The contributions of body morphology to cognition and control involve three major categories [20]: (1) Morphology facilitating control: wherein the physical design enables certain behaviors such as motion sequencing (e.g., passive dynamic walker [23]). (2) Morphology facilitating perception: wherein the physical design enables sensing (e.g., the nonuniform distribution of cells in the compound eyes of fly [24]). (3) Morphological computation, such as the _physical reservoir computing_ (PRC), wherein a physical body performs genuine computations. Among these, physical reservoir computing shows promising potentials because of its balanced simplicity and versatility to perform applicable computation with encoding and decoding [20]. Reservoir computing is a computational framework based on artificial recurrent neural networks (RNNs), which have been used extensively for problems involving time-series prediction like the stock market and weather forecasting, robotic motion planning and control, text and speech recognition [25, 26, 27, 28, 29, 30, 21, 31]. In RNNs, the output of the current time step depends on the results from the previous time step in addition to the current input. Since RNNs involve both forward and back-propagation of input data, training them became a challenging task. To address this difficulty, Jaeger introduced the concept of a _fixed_ recurrent neural network as Echo State Networks (ESNs) [25], and Maass introduced Liquid State Machines (LSMs) [26]. Later, these two concepts merged under the umbrella of reservoir computing (RC). In RC, the neural network (aka. the “reservoir”) has fixed interconnections and input weights, and only the linear output readout weights are trained by simple techniques like linear or ridge regression. These reservoirs’ dynamics transform the input data stream into a high-dimensional state space, capturing its non-linearities and time-dependent information for computation tasks. More importantly, the reservoir’s fixed nature opens up the possibility of using physical bodies — such as a random network of nonlinear spring and mass oscillators [18, 32, 33], tensegrity structures [15, 16, 34, 17], and soft robotic arms [19, 35, 36] — to conduct computation, hence the paradigm of Physical Reservoir Computing. These physical systems have shown to possess sufficient computational power to achieve complex computing tasks like emulating other non-linear dynamic systems, pattern generation [34, 17, 18, 32, 19, 21], speech recognition [37], and machine learning [36, 21, 31, 33]. More importantly, robotic bodies with sufficient nonlinear dynamics can also perform like a physical reservoir and directly generate locomotion gait without using the traditional controllers [17, 21, 38, 39, 40]. In this study, we investigate the use of origami as a physical reservoir and harness its computing power for robotic locomotion generation. Origami is an ancient art of folding paper into sophisticated and three-dimensional shapes. Over the past decades, it has evolved into an engineering framework for constructing deployable structures [41, 42, 43], advanced materials [44, 45, 46, 47, 48, 49], and robotics [50, 51, 52, 53, 54, 55, 56]. Origami has many appealing advantages for use in robotics. It is compact, easy to fabricate, and scale-independent (aka. Origami robots can be fabricated at different scales but still follow similar folding principles [57, 50, 58, 59]). Moreover, the nonlinear mechanics and dynamics induced by folding could enhance robotic performance [60, 61]. We show that origami’s nonlinear folding dynamics also possess significant computing power. A mechanical system must exhibit several essential properties to perform as a reservoir [21]. The first one is high-dimensionality, which allows the reservoir to gather as much information possible from the input data stream, separating its spatio-temporal dependencies and projecting it onto a high-dimensional state-space. The second one is non-linearity so that the reservoir acts as a nonlinear filter to map the information from the input stream. All the computation complexity is associated with this nonlinear mapping, thus training the linear static readout becomes a straightforward task. The third one is fading memory (or short-term memory), ensuring that only the recent input history influences the current output. The fourth one is separation property to classify and segregate different response signals correctly, even with small disturbances or fluctuations. Moreover, if two input time series differed in the past, the reservoir should produce different states at subsequent time points [62]. Our physics-informed numerical simulations prove that origami inherently satisfies these four requirements and can complete computation tasks like emulation, pattern generation, and output modulation. Moreover, we conduct extensive numerical simulations to uncover the linkage between origami design and its computing power, providing the guideline to optimize computing performance. Finally, we demonstrate how to directly embed reservoir computing in an origami robotic body to generate earthworm-like peristalsis crawling without using any traditional controllers. This study’s results could foster a new family of origami-based soft robots that operate with simple mechatronics, interact with the environment through distributed sensor and actuator networks, and respond to external disturbances by modulating their activities. In what follows: Section (2) details the construction of an origami reservoir, including the lattice framework used to simulate its nonlinear dynamics. Section 3 elucidates the origami reservoir’s computing power through various numerical experiments. Section 4 discusses the parametric analysis that uncovers the linkages between computing performance and physical design. Section 5 applies the reservoir computing to an origami robot’s crawling problem. Finally, Section 6 concludes this paper with a summary and discussion. ## 2 Constructing The Origami Reservoir In this study, we construct a physical reservoir using the classical Miura-ori sheets. It is essentially a periodic tessellation of unit cells, each consisting of four identical quadrilateral _facets_ with _crease_ lengths $a$ $b$ and an internal sector angle $\gamma$ (Figure 1 (a)) [63, 44]. The folded geometry of Miura-ori can be fully defined with a dihedral _folding angle_ $\theta$ ($\in{[-\pi/2,\pi/2]}$) between the $x$-$y$ reference plane and its facets. The reservoir size is defined as $n\times m$, where $n$ and $m$ are the number of origami _nodes_ (aka. vertices where crease lines meet) in $x$ and $y$-directions, respectively. $N$ is the total number of creases in the origami reservoir. ### 2.1 Dynamics Modeling of the Origami To investigate this origami reservoir’s computing capacity, one must first obtain its time responses under dynamic excitation. To this end, we adopt and expand the lattice framework approach to simulate its nonlinear dynamics [63, 64, 65]. In this approach, origami creases are represented by pin-jointed stretchable truss elements with prescribed spring coefficient $K_{s}$. Folding (or bending) along the crease line is simulated by assigning torsional spring coefficient $K_{b}$ (Figure 1 (b)). We further triangulate the quadrilateral facets with additional truss elements to estimate the facet bending with additional torsional stiffness (typically, $K_{b}$ across the facets is larger than those along the creases). Therefore, this approach discretizes the continuous origami sheet into a network of pin-jointed truss elements connected at the nodes. A typical reservoir consists of an interconnected network of units governed by nonlinear dynamics, and the origami reservoir, in this case, consists of a network of nodes with their interconnections defined by the underlying crease pattern. The corresponding governing equations of motion, in terms of node #p’s displacement ($\mathbf{x}_{p}$) as an example, are: $m_{p}\ddot{\mathbf{x}}_{p}^{(j)}=\mathbf{F}_{p}^{(j)},$ (1) where the superscript “$(j)$” represents the $j^{\text{th}}$ time step in numerical simulation, and $m_{p}$ is the equivalent nodal mass. Unless noted otherwise, the mass of the origami sheet is assumed to be equally distributed to all its nodes. $\mathbf{F}_{p}^{(j)}$ is the summation of internal and external forces acting on this node in that $\mathbf{F}_{p}^{(j)}=\sum\mathbf{F}_{s,p}^{(j)}+\sum\mathbf{F}_{b,p}^{(j)}+\mathbf{F}_{d,p}^{j}+\mathbf{F}_{a,p}^{(j)}+m_{p}\textbf{g},$ (2) where the five terms on the right hand side are the forces from truss stretching, crease/facet bending, equivalent damping, external actuation, and gravity, respectively. The formulation of these forces are detailed below. Figure 1: The nonlinear Truss-frame approach for simulating the origami dynamics. (a) The crease pattern of the classical Miura-ori, with a unit cell highlighted. (b) The rigid-folding kinematics of the Miura-ori. (c) The truss- frame approach discretizes the Miura-ori unit cell, showing the distribution of truss elements along the creases and across the facets, as well as the nodal masses. (d) Detailed kinematics and mechanics set up to analyze the bending and stretching along the truss #$pq$. Notice that $\mathbf{m}^{(j)}$ and $\mathbf{n}^{(j)}$ are the current surface normal vectors defined by triangles #$pqr$ and #$pqv$, respectively. (e) The bending of the Miura-ori sheet under its weight. This simulation serves to validate appropriate material property assignments. Truss stretching forces: The truss elements are essentially elastic springs with axial stretching stiffness ($K_{s}^{(j)}=EA/l^{(j)}$). Here, $EA$ is the material constant, and $l^{(j)}$ is the truss element’s length at the current $j^{\text{th}}$ time step. Thus, the axial stiffness is updated at each time- step, accommodating the truss element’s increase in stiffness as it is compressed and vice-a-versa. The stretching forces from a truss connecting node #p and one of its neighbouring nodes #$q$ is, $\mathbf{F}_{s,p}^{(j)}=-K_{s}^{(j)}\left(l_{pq}^{(j)}-l_{pq}^{(0)}\right)\frac{\mathbf{r}_{p}^{(j)}-\mathbf{r}_{q}^{(j)}}{\|\mathbf{r}_{p}^{(j)}-\mathbf{r}_{q}^{(j)}\|}$ (3) where $l_{pq}^{(0)}$ is the truss length at its initial resting state. $\mathbf{r}_{p}^{(j)}$ and $\mathbf{r}_{q}^{(j)}$ are the current position vectors of these two nodes, respectively. To calculate the total truss stretching forces acting on node #$p$, similar equations apply to all of its neighbour nodes through trusses (e.g., node $q$, $r$, $s$, $t$, $u$, and $v$ in Figure 1(c)). Crease/facet bending forces: The crease folding and facet bending are simulated with torsional spring coefficient ($K_{b}^{(j)}=k_{b}l^{(j)}$), where $k_{b}$ is torsional stiffness _per unit length_. Here, we adopt the formulation developed by Liu and Paulino [64]. For example, the force acting on nodes #$p$ due to the crease folding along the truss between #$p$ and #$q$ is: $\mathbf{F}_{b,p}^{(j)}=-K_{b}^{(j)}(\varphi_{pq}^{(j)}-\varphi_{pq}^{(0)})\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{p}^{(j)}}$ (4) where $\varphi_{pq}^{(j)}$ is the current dihedral angle along truss $pq$ (aka. the dihedral angle between the triangles #$pqr$ and #$pqv$ in 1(d)), and $\varphi_{pq}^{(0)}$ is the corresponding initial value. $\varphi_{pq}^{(j)}$ can be calculated as $\displaystyle\varphi_{pq}^{(j)}$ $\displaystyle=\eta\arccos\left(\frac{\mathbf{m}^{(j)}\cdot\mathbf{n}^{(j)}}{\|\mathbf{m}^{(j)}\|\|\mathbf{n}^{(j)}\|}\right)\text{ modulo }2\pi$ (5) $\displaystyle\eta$ $\displaystyle=\begin{cases}\text{sign}\left(\mathbf{m}^{(j)}\cdot\mathbf{r}_{pv}^{(j)}\right),&\mathbf{m}^{(j)}\cdot\mathbf{r}_{pv}^{(j)}\neq 0\\\ 1.&\mathbf{m}^{(j)}\cdot\mathbf{r}_{pv}^{(j)}=0\end{cases}$ (6) Here, $\mathbf{m}^{(j)}$ and $\mathbf{n}^{(j)}$ are current surface normal vector of the triangles #$pqr$ and #$pqv$, respectively, in that $\mathbf{m}^{(j)}=\mathbf{r}_{rq}^{(j)}\times\mathbf{r}_{pq}^{(j)}$ and $\mathbf{n}^{(j)}=\mathbf{r}_{pq}^{(j)}\times\mathbf{r}_{pv}^{(j)}$. In addition, $\mathbf{r}_{pq}^{(j)}=\mathbf{r}_{p}^{(j)}-\mathbf{r}_{q}^{(j)}$, $\mathbf{r}_{rq}^{(j)}=\mathbf{r}_{r}^{(j)}-\mathbf{r}_{q}^{(j)}$, and $\mathbf{r}_{pv}^{(j)}=\mathbf{r}_{p}^{(j)}-\mathbf{r}_{v}^{(j)}$. This definition of $\varphi_{pq}^{(j)}$ ensures that the folding angle for valley crease lies in $(0,\pi]$ and the folding angle for mountain crease lies in $(\pi,2\pi]$. The derivative between folding angle $\varphi_{pq}^{(j)}$ and the nodal #$p$’s current position vector is $\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{p}^{(j)}}=\left(\frac{\mathbf{r}_{pv}^{(j)}\cdot\mathbf{r}_{pq}^{(j)}}{\|\mathbf{r}_{pq}^{(j)}\|^{2}}-1\right)\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{v}^{(j)}}-\left(\frac{\mathbf{r}_{rq}^{(j)}\cdot\mathbf{r}_{pq}^{(j)}}{\|\mathbf{r}_{pq}^{(j)}\|^{2}}\right)\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{r}^{(j)}}$ (7) where $\displaystyle\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{r}^{(j)}}$ $\displaystyle=\frac{\|\mathbf{r}_{pq}^{(j)}\|}{\|\mathbf{m}^{(j)}\|^{2}}\mathbf{m}^{(j)},$ (8) $\displaystyle\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{v}^{(j)}}$ $\displaystyle=-\frac{\|\mathbf{r}_{pq}^{(j)}\|}{\|\mathbf{n}^{(j)}\|^{2}}\mathbf{n}^{(j)}.$ (9) Again, to calculate the total crease folding and facet bending forces acting on node #$q$, similar equations apply to trusses connected to this node (e.g., truss $pq$, $pr$, $ps$, $pt$, $pu$, and $pv$ in Figure 1(b)). Damping forces: Estimating damping ratio and damping force is essential to achieve realistic dynamic responses and reduce numerical simulation error accumulation. In this study, we follow the formulation developed in [66, 65]. This formulation first calculates the average velocity of a node with respect to its neighboring nodes ($\mathbf{v}_{\text{avg}}^{(j)}$) to effectively remove the rigid body motion components from the relative velocities and ensure that these components are not damped. Then damping force $\mathbf{F}_{d,p}^{(j)}$ applied on node #$p$ is given by $\displaystyle\mathbf{F}_{d,p}^{(j)}$ $\displaystyle=-c_{d}^{(j)}(\mathbf{v}_{p}^{(j)}-\mathbf{v}_{\text{avg}}^{(j)})$ (10) $\displaystyle c_{d}^{(j)}$ $\displaystyle=2\zeta\sqrt{K_{s}^{(j)}m_{p}}$ (11) where $c_{d}^{(j)}$ is the equivalent damping coefficient, and $\zeta$ is the damping ratio. Actuation force: In the origami reservoir, two types of creases receive actuation. The first type is “input creases,” and they receive input signal $u(t)$ required for emulation and output modulation tasks. The second type is “feedback creases,” and they receive reference or current output signal $z(t)$ required by all computing tasks in this study except for the emulation task (more on the applications of input and feedback creases in Section 2.2). In the case of multiple outputs, different groups of feedback creases are present. Here, the selection of input and feedback creases are random. There are many methods to implement actuation to deliver input $u(t)$ and reference/feedback signal $z(t)$ to the reservoir. For example, the actuation can take the form of nodal forces on a mass-spring-damper network [18, 32], motor generated base rotation on octopus-inspired soft arm [19], or spring resting length changes in a tensegrity structure [34]. In origami, the actuation can take the form of moments that can fold or unfold the selected creases. We assume that the resting angle $\varphi^{(0)}$ of the input and feedback creases will change — in response to the actuation at every time step — to a new equilibrium $\varphi_{a,0}^{(j)}$ in that [67, 34] $\displaystyle\varphi_{a,0}^{(j)}$ $\displaystyle=W_{\text{in}}\tanh(u^{(j)})+\varphi^{(0)}\quad\text{for input creases;}$ (12) $\displaystyle\varphi_{a,0}^{(j)}$ $\displaystyle=W_{\text{fb}}\tanh(z^{(j)})+\varphi^{(0)}\quad\text{for feedback creases.}$ (13) where $W_{\text{in}}$ and $W_{\text{fb}}$ are the input and feedback weight associated with these actuated creases. They are assigned before the training and remain unchanged after that. $u^{(j)}$ and $z^{(j)}$ are the input and feedback signal at the $j^{\text{th}}$ time step. The magnitude of $W_{\text{in}}$ and $W_{\text{fb}}$ are selected such that $\varphi_{a,0}^{(j)}\in[0,2\pi]$ and consistent with the folding angle assignment. This approach of assigning new equilibrium folding angles is similar to traditional neural network studies that use $\tanh$ as a nonlinear activation function to transform function $z(t)$ into a new one with magnitudes between $[-1,1]$. Additionally, it prevents actuator saturation due to spurious extreme values of $z(t)$. Denote the torsional stiffness of actuated creases by $K_{b,a}^{(j)}$, and we can update Equation (4) for the actuated creases (using node #$p$ as an example) $\displaystyle\mathbf{F}_{a,p}^{(j)}=-K_{b,a}^{(j)}\left(\varphi_{pq}^{(j)}-\varphi_{a,0,pq}^{(j)}\right)\frac{\partial\varphi_{pq}^{(j)}}{\partial\mathbf{r}_{p}^{(j)}},$ (14) The calculation of other terms in this equation are the same as those in the force from crease folding and facet bending. Once the governing equations of motion are formulated, they are solved using MATLAB’s ode45 solver with $10^{-3}$ second time-steps. Although the governing equation of motions use nodal displacement $\mathbf{x}^{(j)}$ as the independent variables, we use the dihedral crease angles $\varphi^{(j)}$ as the _reservoir state_ variables to characterize the origami’s time responses. This is because measuring crease angles is easier to implement by embedded sensors, and $\varphi^{(j)}$ can be directly calculated from $\mathbf{x}^{(j)}$ via the Equations 5 and 6. ### 2.2 Setting Up Reservoir Computing Similar to the actuated creases (aka. input creases and feedback creases), we designate “sensor creases” for measuring the reservoir states. We denote $N_{a}$ as the number of actuated creases, and $N_{s}$ for sensor creases. It is worth noting that, the actuated creases are typically small subset of all origami creases (i.e., $N_{a}<N$). The sensor creases, on the other hand, can be all of the origami creases ($N_{s}=N$) or a small subset as well ($N_{s}<N$). Once the selections of input, feedback, and sensor creases are completed, one can proceed to the computing. Physical reservoir computing for tasks that require feedback (e.g., pattern generations in Section 3.2, and output modulation in 3.3) consists of two phases: The “training phase” and “closed- loop phase.” While the emulation tasks require the training phase only (Section 3.1). Training phase: In this phase, we use the teacher forcing to obtain the readout weights $W_{i}$ corresponding to every reservoir state (aka. the dihedral angles of the sensor creases). Suppose one wants to train the reservoir to generate a nonlinear time series $z(t)$ (aka. the reference output). The feedback creases receive the reference output and it dynamically excites the origami reservoir under an open-loop condition without feedback (Figure 2(a)). The reservoir states $\varphi^{(j)}$ at every time step are measured and then compiled into a matrix $\mathbf{\Phi}$. Once the numerical simulation is over, we segregate the reservoir state matrix $\mathbf{\Phi}$ into the washout step, training step, and testing step. The washout step data is discarded to eliminate the initial transient responses. We then calculate the output readout weights $W_{i}$ using the training step data via simple linear regression: $\displaystyle\mathbf{W}_{\text{out}}=[\mathbf{1}\;\mathbf{\Phi}]^{+}\mathbf{Z}=\mathbf{\bar{\Phi}}^{+}\mathbf{Z}$ (15) where, $[.]^{+}$ refers to the Moore-Penrose pseudo-inverse to accommodate non-square matrix. $\mathbf{1}$ is a column of ones for calculating the bias term $W_{\text{out},0}$ to shift the fitted function when necessary. $\mathbf{Z}$ contains the reference signals at each time step, and it is a matrix if more than one references present. Lastly, we use testing step data to verify reservoir performance. It is worth noting that white noise of amplitude $10^{-3}$ is superimposed on the reservoir state matrix during training to ensure the robustness of the readout result against numerical imperfections, external perturbations [32], and instrument noise in “real- world” applications. Closed-loop phase: Once the training phase is over and readout weights are obtained, we run the reservoir in the closed-loop condition. That is, instead of using the reference output $z(t)$, the current output $z^{}(t)$ is sent to the feedback creases (Figure 2(b)), and $\displaystyle z^{}(t)=W_{\text{out},0}+\sum_{i=1}^{N_{s}}W_{\text{out},i}\varphi_{i}(t)=\mathbf{W}_{out}^{T}\bar{\mathbf{\Phi}}$ (16) where, $N_{s}$ is the number of sensor creases, and $\bar{\mathbf{\Phi}}=[\mathbf{1}\;\mathbf{\Phi}]$. Thus, the reservoir runs autonomously in the closed-loop phase without any external interventions. Figure 2: The setup of physical reservoir computing with origami. (a) The training phase. The feedback creases receive the reference (or targeted) output $z(t)$; while white noise is added to the reservoir state vector $\mathbf{\Phi}(t)$ before calculating output weights $\mathbf{W}_{\text{out}}$; (b) The closed-loop phase. The output weights obtained in the training phase are used to calculate the current output, which is fed back to the feedback creases. We study the closed loop performance of reservoir by calculating the Mean Squared Error (MSE) using M time-steps as follows: $\text{MSE}=\frac{1}{M}\sum_{j=1}^{M}\left(z(j)-z^{}(j)\right)^{2}$ (17) To estimate performance when multiple reference outputs are present, we combine the MSEs by taking a norm over the individual MSEs. ## 3 Computation Tasks By the Origami Reservoir In this section, we use the origami reservoir to emulate multiple non-linear filters simultaneously, perform pattern generation, and modulate outputs. The baseline variables for the origami geometric design, material properties, and reservoir parameters are given in Table 1. Figure 3: Emulation tasks with the origami reservoir. (a) The Miura-ori reservoir used for this task with input creases highlighted. Appropriate boundary conditions are also necessary. (b) Examples of trajectories generated in the emulation task including (from top to bottom) input signal $u(t)$, 2nd order, 10th order system, and Volterra series. Dashed curves are the targeted trajectories, and solid curves are the result of the reservoir. (c) Error analysis of the emulation tasks. Circles are the standard deviation of MSE, and horizontal bars are the corresponding extreme values. Table 1: Design of a baseline origami reservoir in this study Reservoir size and material properties --- Parameter \| Value Size \| 9$\times$9 Nodal Mass \| 7 g $k_{s}$ \| 100 N/m $k_{c}^{a}$ \| 1 N/(m-rad) $k_{c}$ \| 0.2525 N/(m-rad) $K_{f}$ \| 10 N/(m-rad) $\zeta$ \| 0.2 Geometric design of Miura-ori Parameter \| Value $a$ \| 16 mm $b$ \| 10 mm $\gamma$ \| 48$\degree$ $\theta$ \| 60$\degree$ Actuator and sensor creases Parameter \| Value No. of sensors ($N_{s}$) \| $N$ No. of actuators ($N_{a}$) \| 0.45$N$ No. of Feedback creases \| 0.3$N$ No. of Input creases \| 0.15$N$ Table 2: Emulation task functions Type \| Functions in discretized form (at $j^{th}$ time step) ---\|--- Input \| $u(j)=0.2\sin(2\pi f_{1}j\Delta t)\sin(2\pi f_{2}j\Delta t)\sin(2\pi f_{3}j\Delta t)$ \| $f_{1}=2.11$ Hz, $f_{2}=3.73$ Hz, $f_{3}=4.33$ Hz $2^{\text{nd}}$ order system \| $z_{1}(j+1)=0.4z_{1}(j)+0.4z_{1}(j)z_{1}(j-1)+0.6(u(j\Delta t))^{3}+0.1$ $10^{\text{th}}$-order system \| $\displaystyle z_{2}(j+1)=0.3z_{2}(j-1)+0.05z_{2}(j-1)\sum_{i=1}^{10}z_{2}(j-i)$ \| $+1.5u((j-10)\Delta t)u((j-1)\Delta t)+0.1$ Discrete Volterra series \| $\displaystyle z_{3}(j+1)=100\sum_{\tau_{1}=0}^{T}\sum_{\tau_{2}=0}^{T}h(\tau_{1},\tau_{2})u(j-\tau_{1})u(nj-\tau_{2})$ \| $\displaystyle h(\tau_{1},\tau_{2})=\exp\left(\frac{(\tau_{1}\Delta t-\mu_{1})^{2}}{2\sigma_{1}^{2}}+\frac{(\tau_{2}\Delta t-\mu_{2})^{2}}{2\sigma_{2}^{2}}\right)$ \| $\mu_{1}=\mu_{2}=0.1,\sigma_{1}=\sigma_{2}=0.05,\Delta t=10^{-3}$ ### 3.1 Emulation Task This sub-section shows that the origami reservoir can emulate multiple nonlinear filters simultaneously using a single input. Such emulation is a benchmark task for evaluating the performance in RNN training [68] and prove the multi-tasking capability of physical reservoirs [18, 19]. Note that the emulation task involves only the training phase, so there are no feedback creases in this case. Consequently, we excite the reservoir by sending the input function $u(t)$ to the input creases and train it to find three sets of readout weights in parallel via linear regression. Here, $u(t)$ is a product of three sinusoidal functions with different frequencies, and the three target non-linear filters are $2^{\text{nd}}$-order non-linear dynamic system $z_{1}(t)$, a $10^{\text{th}}$-order non-linear dynamic system $z_{2}(t)$, and discrete Volterra series $z_{3}(n)$ (detailed in Table 2). We use a $9\times 9$ Miura-ori reservoir in this task, exciting the reservoir from complete rest and training it for 100 seconds. We discard the first 50 seconds of data as the washout step, use the data from the next 45 seconds to calculate the optimum static readout weights, and then use the last 5 seconds of data to calculate the MSE for performance assessments. Results in Figure 3 show that the origami reservoir can emulate these three nonlinear filters. As the nonlinearity and complexity of the nonlinear filter increases, MSE also increases (Figure 3(b)). Moreover, we compare the emulation performance when all $N$ creases are used as sensor creases versus when only actuated creases are used as sensors ($N_{s}=N_{a}=pN$). The increase in MSE is marginal in the latter case. Therefore, the origami satisfies the previously mentioned nonlinearity and fading memory requirements to be a physical reservoir, and one only needs to use the input creases angles as the reservoir states to simplify the reservoir setup. ### 3.2 Pattern Generation Task Pattern generation tasks are essential for achieving periodic activities such as robotic locomotion gait generation and manipulator control where persistent memory is required. That is, by embedding these patterns (or limit cycles) in the origami reservoir, one can generate periodic trajectories in the closed- loop. We again use a $9\times 9$ Miura-ori reservoir and randomly select $30\%$ of its creases as the feedback creases (this task does not require input creases). These feedback creases are divided into two groups for the two components of 2D trajectories. We run the training phase for 100 seconds for each pattern, discard the initial 15 seconds of data as the washout step and use the next 51 seconds’ data to calculate the optimum output readout weights. Generating non-linear Limit cycles: In the following results, the origami reservoir demonstrates its computation capability via generating quadratic limit cycles (LC), Van der Pol limit cycles, and the Lissajous curve in closed-loop. The quadratic limit cycle is defined by two differential equations: $\displaystyle\dot{x}_{1}$ $\displaystyle=x_{1}+x_{2}-\epsilon(t)x_{1}\left(x_{1}^{2}+x_{2}^{2}\right),$ (18) $\displaystyle\dot{x}_{2}$ $\displaystyle=-2x_{1}+x_{2}-x_{2}\left(x_{1}^{2}+x_{2}^{2}\right),$ (19) where the parameter $\epsilon(t)$ determines the shape of the limit cycle ($\epsilon(t)=1$ in this case). The Van der Pol limit cycle is defined by: $\displaystyle\dot{x}_{1}$ $\displaystyle=x_{2},$ (20) $\displaystyle\dot{x}_{2}$ $\displaystyle=-x_{1}+\left(1-x_{1}^{2}\right)x_{2}.$ (21) The Lissajous curve is a graph of two sinusoidal signals parameterized by their frequency ratio ($f_{1}/f_{2}=0.5$) and phase difference ($\delta=\pi/2$): $\displaystyle x_{1}$ $\displaystyle=\sin\left(f_{1}t+\delta\right)$ (22) $\displaystyle x_{2}$ $\displaystyle=\sin\left(f_{2}t\right)$ (23) Figure 4: Stable pattern generation under closed-loop using the Miura-ori reservoir. (a) This task’s origami reservoir includes two groups of feedback creases required to generate 2D limit cycles. (b-d) The closed-loop trajectories of quadratic limit cycle, Van der Pol oscillator, and the Lissajous curve, respectively. In these plots, the first row of time responses shows the closed-loop output after 100s of training. The third row of time responses shows how the trained reservoir can recover the targeted limit cycles from an initial resting condition. The corresponding phase portraits are as shown in the second row. Here, the dashed curves are targeted trajectories, and the solid curves are the reservoir’s outputs. (e) Van der Pol limit cycle recovery after the temporary failure of sensor and actuator creases. The two simulations are the same except for the number of sensor creases ($N_{s}=N$ for the first test, $N_{s}=0.3N$ for the second). The insert figures show the corresponding phase-portraits. As shown in Figure 4(b), the origami reservoir can generate all three periodic trajectories just by changing the output readout weights. The MSE for Quadratic LC, Van der Pol LC, and Lissajous curves, calculated using the data for first 10 seconds’ closed-loop run (M = 10000), are $3.28\times 10^{-7}$, $2.03\times 10^{-5}$, and $5.5\times 10^{-4}$, respectively. As expected, MSE increases as the complexity of the curve increases. Stability and robustness of the pattern generation: After finding the readout weights, we test the stability of these three limit cycles by starting the origami reservoir from total rest in the close-loop and running it for more than 1000 seconds. The limit cycle is stable if and only it can recover the pattern from zero initial conditions and stays on target for at least 1000 seconds of simulation [32, 19]. The results in Figure 4(c) indicates that the torsional moments generated from the feedback signals on the feedback creases are sufficient to recover and maintain the three limit cycles from total rest. Small phase differences occur between generated trajectories and the targets because the reservoir takes a slightly different path than the target, and the Lissajous curve takes more than 15 seconds to recover fully. Nonetheless, the origami reservoir successfully passes this test. To further analyze the robustness of reservoir-generated limit cycles, we simulate actuator and sensor failures. As the origami reservoir generates the Van der Pol limit cycles in these tests, all feedback and sensor creases stop working (aka. their signals set to zero) for 10 seconds. We conduct these tests when all creases are used as sensor creases ($N_{s}=N$) and when only feedback creases are sensor creases ($N_{s}=N_{a}=0.3N$). The simulation results in Figure 4(e) show that, although the reservoir diverges to a trajectory far away from the target during the actuator and sensor failure, it can immediately recover the Van der Pol limit cycles after the end of these failures. Figure 5: Results of modulation task under closed-loop using The Miura-ori reservoir. (a) This task’s origami reservoir includes two groups of feedback creases and input creases. (b) Quadratic limit cycle trajectories under closed-loop and the corresponding input signal $\epsilon(t)$. The results are obtained after 500 seconds of training. (c) Closed-loop trajectory recovery from the initial resting conditions. (d) The corresponding phase-portraits, where the targeted trajectories are overlaid on top of the reservoir output. ### 3.3 Output Modulation Task Output modulation capability allows the reservoir to adjust its output according to a randomly varying input signal without changing the readout weights. This ability is also essential for soft robotic control applications because it allows the robot to switch behaviors according to external stimuli or environmental changes. In this task, we randomly select input creases, which account for $15\%$ of the total creases, in addition to the feedback creases (Figure 5(a)). Moreover, all creases are used as sensor creases ($N_{s}=N$). The simulation results in Figure 5(b, c) shows the generated quadratic limit cycles with modulated input (Equation (18, 19)). The origami reservoir can react to the input and modulate the magnitude of the quadratic limit cycles. The MSE is $3.8\times 10^{-4}$, which is remarkably small, considering this task’s complexity. ## 4 Correlating Physical Design and Computing Performance In this section, we use the mean squared error (MSE) as the metric to examine the connections between the origami reservoir’s design and computing performance. In particular, This analysis aims to investigate the sensitivity of MSE to different parameter changes and identify the optimal origami designs. To this end, in-depth parametric analyses are conducted to examine the effect of (1) reservoir size and material properties, (2) crease pattern geometry, and (3) feedback and sensor crease distribution. We use both Van der Pol and quadratic limit cycle generation tasks to ensure the broad applicability of parametric study results. Table 3: Variables for reservoir size and material properties parametric study Parameter \| Base value \| Distribution ---\|---\|--- Nodal mass (g) \| 7 \| [1,50] Geometric \| Standard \| $\sigma=\chi\exp(\frac{-\|\|(N_{i}-N_{j})\|\|}{l})$ imperfections \| Miura-ori \| $\mu=0$, $\chi=0.4a$, $l=4a$ Truss torsional \| $K_{b}^{a}=1$, \| $K_{b}^{a}=1$, stiffness N/(m-rad) \| $K_{b}=0.2525$ \| $K_{b}\in[0.005,0.5]$ ### 4.1 Reservoir Size, Material Properties, and Vertices Perturbation We observe that feedback crease distribution affects reservoir computing performance quite significantly. In particular, poorly distributed feedback creases might result in failed pattern generating tasks. Therefore, we first conduct numerical simulations by randomly changing the feedback crease distributions (72 unique designs in total) and identifying the best performing one (with the least MSE). We refer to this best performing feedback crease distribution as the _base design_ (Figure 6(a, c)) for the following parametric studies. Then, we conduct another parametric study regarding the nodal mass, crease stiffness, and vertices perturbation. We vary these three parameters, one at a time, for 72 randomly selected designs (six batches of jobs in parallel on a computer with 12 cores). The baseline values and range of the parameters are in Table 3. The origami reservoir performance turns out to be highly sensitive to the nodal mass variation. As opposed to the uniform nodal mass in base design, a randomly distributed nodal mass can significantly increase or decrease the MSE for both pattern generation tasks. However, randomly distributing mass in an origami sheet is quite challenging in practical applications. So the use of varying mass distribution should be judicially done based on the particular application at hand. On the other hand, the origami performance is much less sensitive to the crease torsional stiffness. By randomly changing the stiffness, one can achieve performance at par with the base design. Moreover, we investigate the effects of random geometric imperfection in the base designs of origami reservoir. To this end, we adopt the formulation introduced by Liu et al. [69], which introduce small perturbations to the nodal positions in folded origami. Such imperfections are inevitable in practice due to various manufacturing defects. It is found that these small imperfections do not worsen the MSE significantly and in fact could reduce the MSE by a moderate degree (Figure 6(a),(b)). It is also worth noting that the larger $9\times 9$ Miura origami reservoir performs better than the smaller one because larger origami contains more folding angles to constitute the reservoir state matrix. Therefore, the high- dimensionality of a reservoir is desirable to produce smaller MSE. Figure 6: Effect of reservoir size and material properties on the reservoir computing performance. (a) The distribution of MSE from the Quadratic limit cycle simulations using random feedback crease distributions and different design parameter distributions. Here “FB” stands for feedback crease distribution, “M” stands for nodal mass distribution, “V” stands for origami vertices geometry perturbation, and “$K_{f}$” stands for crease torsional stiffness distribution. It is worth emphasizing that the “FB” results come from one parametric study of 72 unique designs, and the “M,” “V,” and “$K_{f}$” are results of the subsequent simulation. The bar charts depict the average value, standard deviation (circles), and extreme values (horizontal bars) of MSE. (b) A similar result from the Van der Pol limit cycle generation task. (c) The feedback crease distributions of the four different baseline designs used in this parametric study. ### 4.2 Origami Design Figure 7: Effect of Miura-ori geometric design on the reservoir performance. (a-c) The Miura-ori geometry and the corresponding landscape of MSE distribution when $\theta=50\degree$, $60\degree$, and $70\degree$, respectively. The lighter and darker regions correspond to larger and smaller errors, respectively, while the white regions depict origami designs that failed the computing task. (d) The unit cell geometry of four representative designs with the same crease $a$ length but different sector angles $\gamma$ and crease length ratios $a/b$. A unique advantage of origami based structures and materials is their considerable freedom to tailor the geometric design. To this end, we start from the Base Design I of $9\times 9$ Miura-ori reservoir, vary its crease length ratio $(a/b)$ and internal sector angle $(\gamma)$, and then run the quadratic limit cycle task with 100 crease length and sector angle combinations at three folding angles $(\theta=50\degree,60\degree,70\degree)$. The results of the parametric analysis are shown in Figure 7. We observe that, at lower folding angles (flatter origami), the origami reservoir has a higher possibility to fail the pattern generation tasks. The computing performance improves significantly with a reduced MSE as the origami folds more (or as $\theta$ increases). This trend is probably because highly folded origami offers an increased range of folding motion. Moreover, there are two design sets with the lowest MES: $a/b\approx 1.5$, $\gamma\approx 45\degree$, and $a/b\approx 2.5$, $\gamma\approx 60\degree$. Generally speaking, a moderate to high crease-length ratio and small sector angles can create “skewed” origami patterns that appear to give better computing performance across all values folding angles. The best designs here have MSEs at the order of $10^{-7}$, which is of the same magnitude as we found previously by tailoring the nodal mass and crease stiffness. Figure 8: Effect of varying the number of actuator and sensor creases. ### 4.3 Actuator and Sensors Distribution Finally, it is important, for practical applications, to find the minimum amount of input/feedback and sensor creases required for achieving acceptable computing performance. To this end, we start with the $9\times 9$ Miura-ori reservoir and conduct two tests. In the first test, we vary the percentage of feedback creases ($N_{a}=0.2N,0.3N,0.4N,0.5N$, each with 24 randomly generated crease distributions) while using all crease dihedral angles to constitute the reservoir state matrix (i.e., $N_{s}=N$). In the second test, we use the same feedback crease design and only use these feedback creases’ dihedral angles to formulate the reservoir state matrix (i.e., $N_{s}=N_{a}$). We find that if only $20\%$ of crease are used for feedback, the origami reservoir might fail the quadratic limit cycle task. On the other hand, the MSE reduces only marginally as we increase the percentage of feedback creases beyond $30\%$ (Figure 8). Therefore, we can conclude that using only $30\%-40\%$ of total creases as the feedback and sensors crease will provide us an adequate computing performance. This result is significant because it shows that, even though a large size (high-dimensionality) of the reservoir is essential for computing performance, one does not need to measure (readout) every reservoir state. In this way, the practical implementation of the origami reservoir can be significantly simplified. In conclusion, the parametric analyses lay out the strategy to optimize the origami reservoir performance by tailoring the underlying physical and computational design. A larger origami with a higher-dimension can ensure low computational error, but one only needs to use $30\%$ $40\%$ of its creases as the feedback and sensor creases to tap into the origami’s computing capacity. Meanwhile, the distribution of these feedback and sensor creases must be carefully chosen with extensive simulations. To further improve computing performance, one can tailor the origami’s mass distribution, crease stiffness, and geometric design. Among these options, optimizing the folding geometry should be the most effective because it is easy to implement in practical applications. ## 5 Application to soft robotic crawling This section demonstrates the application of origami reservoir computing to generate an earthworm-inspired peristaltic crawling gait in a robotic system. The earthworm uses peristalsis to navigate uneven terrain, burrow through soil, and move in confined spaces. The lack of complex external appendages (aka., legs or wings) makes earthworm-inspired robots ideal for field exploration, disaster relief, or tunnel drilling [70, 71, 72]. The body of an earthworm consists of segments (metamerism) grouped into multiple “driving modules” [73, 60]. Each driving module includes contracting, anchoring, and extending segments actuated by antagonistic muscles (Figure 9(a)). During peristaltic locomotion, these segments alternately contract, anchor (to the environment with the help of setae), and extend to create a propagating peristalsis wave, thus moving the body forward. We design an earthworm-inspired origami robot consisting of two $3\times 9$ Miura-ori reservoir connected via a stiff central bridge (9(b)). The left and right half of the robots are symmetric in design, and the central bridge design allows differential motion between the two halves to facilitate turning in response to the external input. In each origami reservoir, we embed two groups of feedback creases (Figure 9(b)) with feedback weights assigned such that their values for the front and back-half are equal but opposite to each other. This arrangement reduces the number of reference outputs needed to generate a crawling gait. To create a peristalsis locomotion gait, we train the origami reservoirs to generate multiple harmonic signals with a phase difference of $\pi/2$ among them (aka. a pattern generation task shown Figure 9(b)). We train the robot for 100 seconds and discard the first 15 seconds of data as the washout step. Figure 9: Reservoir computing powered crawling origami robot. (a) The kinematics of a peristaltic locomotion cycle in an earthworm. For clarity, the earthworm body is simplified and consists of six identical segments organized into two driving modules. The earthworm body moves forward while the peristaltic wave of anchoring segments (or driving modules) propagates backward. (b) The design of an earthworm inspired origami crawling robot that features two stripes of Miura-ori connected by a zig-zag shaped “ridge.” This robot has four groups of feedback creases. (c) The closed-loop trajectory generated by the feedback creases after training. (d) Peristaltic locomotion cycle in the origami robot as a result of the generated trajectory. Also, we apply ideal anchors to the bottom origami creases that are in contact with the surface below. These anchors are assumed to be kinematically attached to the ground when the associated origami crease folds and relaxed as the crease unfolds (or flattens). Such anchor design is feasible by leveraging the origami facets’ folding motion, as shown in the author’s previous study [60]. Figure 9(d) illustrates the robotic locomotion generated by reservoir computing, while Figure 9(c) depicts the closed-loop response and the limit cycle recovery from total rest (MSE is $3.9\times 10^{-04})$. As the origami reservoir generates the multiple harmonic signals with a phase difference, its folding motion naturally “synchronizes” to these signals, generating a peristaltic wave of folding and unfolding. As a result, the robot crawls forward like an earthworm, without using any traditional controllers. ## 6 Summary and Conclusion We demonstrate the physical reservoir computing capability of origami via extensive benchmark simulations and parametric studies. First, we develop a simulation environment to study the nonlinear origami dynamics and detail the origami reservoir setup. This reservoir successfully achieves many computing tasks such as emulation, pattern generation, and modulation, all of which are relevant to robotic applications. We also conduct comprehensive parametric analysis to uncover the linkage between origami reservoir design and its computing performance. This new knowledge base offers us a guideline to optimize computing performance. To the authors’ best knowledge, this is the first study to rigorously examine the performance of physical reservoir computer from the lens of the physical design. Finally, we demonstrate how to embed reservoir computing into an origami robot for control without traditional controllers through the example of peristaltic crawling. We list four requirements for a mechanical system to be a reservoir in the introduction, and origami satisfies all these requirements. The tessellated origami structures are inherently high-dimensional. For example, a $7\times 7$ Miura-ori with 49 nodes contains $N=60$ crease dihedral angles, all or a small portion of them can serve as the reservoir states. The nonlinearity of origami partly originates from the nonlinear kinematic relationships between these crease angles and external geometry. Also, since origami patterns are highly structured (ordered), small perturbations in the material properties, imperfections of crease geometry, and the introduction of local actuation are sufficient to destroy the regularity and create disorder. These properties make origami highly nonlinear dynamic reservoirs. The origami reservoir’s performance in the emulation task proves that it can act as a nonlinear filter and satisfies fading memory property. Nonlinear patterns can be embedded into the origami reservoir, and the resulting pattern generation is robust against external disturbances and recoverable under different initial conditions, proving separation property. Finally, adding the feedback can create persistent memory, which is conducive to learning new tasks. For future robots to work autonomously in unstructured and dynamic environments, the robot body and brain have to work together by continuously exchanging information about the current condition, processing this information, and taking appropriate actions. The physical reservoir computing embodied robots shown in this study presents a step toward this vision. The reservoir embedded in the robot body directly gathers information from the distributed sensor-actuator network to perform low-level control tasks like locomotion generation. The resulting soft robot can generate the global target behavior autonomously without controlling every element individually. Moreover, the generated trajectories could be robust against external disturbances and modulated according to changing working conditions. A challenge in implementing physical reservoir computing is the many sensors and actuators required, even though these sensors and actuators can be simple individually. Our results contribute in this regard by showing that only a small portion of origami creases are required to be equipped with sensors and actuators to tap into the reservoir computing power. In summary, origami reservoir computing provides an attractive pathway for facilitating synergistic collaboration between the soft robot’s body and the brain. 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Mechanistic determination of tear film thinning via fitting simplified models to tear breakup Fitting simplified models to tear breakup 1Department of Mathematical Sciences, University of Delaware, Newark, USA 2School of Optometry, Indiana University, Bloomington, USA 1Rayanne A.Luke 1Richard J.Braun 2Carolyn G.Begley Department of Mathematical Sciences, University of Delaware, 222 S. Chapel St, Newark, USA<EMAIL_ADDRESS>dry eye fluorescent imaging optimization tear breakup tear film To determine whether evaporation, tangential flow, or a combination of the two cause tear film breakup in a variety of instances; to estimate related breakup parameters that cannot be measured in breakup during subject trials; and to validate our procedure against previous work. Five ordinary differential equation models for tear film thinning were designed that model evaporation, osmosis, and various types of flow. Eight tear film breakup instances of five healthy subjects that were identified in fluorescence images in previous work were fit with these five models. The fitting procedure used a nonlinear least squares optimization that minimized the difference of the computed theoretical fluorescent intensity from the models and the experimental fluorescent intensity from the images. The optimization was conducted over the evaporation rate and up to three flow rate parameters. The smallest norm of the difference was determined to correspond to the model that best explained the tear film dynamics. All of the breakup instances were best fit by models with time-dependent flow. Our optimal parameter values and thinning rate and fluid flow profiles compare well with previous partial differential equation model results in most instances. Our fitting procedure suggests that the combination of the Marangoni effect and evaporation cause most of the breakup instances. Comparison with results from previous work suggests that the simplified models can capture the essential tear film dynamics in most cases, thereby validating this procedure as one that could be used on many other instances. ## 1 Introduction Tear film breakup (TBU) occurs when a thinned region forms in the tear film (TF). Clinically, this is defined as the first dark area that is observed in the fluorescent (FL) TF following instillation of fluoresecein dyenorn1969. Various mechanisms are thought to cause different types of TBU: evaporationlemp2007, willcox2017tf, king2018 causes relatively slow thinningking2010, and rapid thinning may be explained by Marangoni-driven tangential flowzhong2019 or, plausibly, dewetting in cases of circular TBUyokoi2013, yokoi2019. The Marangoni effect drives outward flow at the aqueous/lipid interface induced by shear stress as a result of a lipid concentration gradientberger74, craster2009. Dewetting from a defective corneal surface region has been hypothesized to drive outward tangential flow from pressure gradients due to van der Waals type forces sharma85, sharma99, zhang03. A related term is full-thickness tear film breakup (FT-TBU), which is when the aqueous layer has thinned to the point where the lipid layer and glycocalyx touchbegley13, king2018. The effects of evaporation and the Marangoni effect on TBU have been extensively studied and modeled separatelyking2013, peng2014, braun2018, zhong2018; only recently have they been explored in combination to explain breakup occurring on an intermediate time scalezhong2019. Zhong et al.zhong2019 developed a partial differential equation (PDE) model with one spatial variable that incorporated both mechanisms. Luke et al.luke2021 fit FT-TBU data from fluorescent (FL) images from healthy subjects with a rescaled version of the Zhong et al.zhong2019 model. The optimization was conducted via nonlinear least squares minimization of the theoretical and experimental FL intensity, and TBU parameters were estimated for the model. The PDE fitting process is time-consuming and limited to spots or streaks; more complicated shapes could be fit using two spatial dimensions. Ordinary differential equation (ODE) models have been designed to capture TF thinning without spatial variation; exact solutions exist for some cases. Braun et al.braun2019 extended an ODE model without flow from previous workbraun2014 to include time-independent extensional fluid flow; a time- dependent version has recently been developed and is presented here. The flow is divergent from the origin and can be considered with evaporative loss and osmotic supply. Winter _et al._ winter2010 (PDE model) and Braunbraun2012 (ODE model) included van der Waals forces to stop thinning with a zero permeability condition at the tear/cornea interface; such terms and forces are omitted from the models in this work. Luke et al.luke2020 fit TBU data with ODE models with evaporation, with or without osmosis, but without tangential flow. Neither the PDE nor ODE model gave the best fit for all of the TBU instances. The instances best fit by the ODE models had rates of FL intensity decrease most closely approximated by a constant. Both TF instability and hyperosmolarity are important to study because they are proposed as etiological causes of dry eye syndrome gilbard1978, craig2017defn, willcox2017tf. Osmolarity is the osmotically-active salt ion concentration in the aqueous layertomlinson09, stahl2012. A concentration difference between the corneal epithelium and aqueous layer induces osmotic flow from the cornea to the TFpeng2014, braun2015. TF osmolarity may be measured in the inferior meniscus clinicallylemp2011; the healthy range is 296-302 Osm/(m${}^{3})$lemp2011, tomlinson2006, versura2010. Dry eye measurements in the same location can reach 316-360 mOsm/(m${}^{3})$gilbard1978, tomlinson2006, sullivan2010, dartt2013 but estimates for the TF over the cornea reach 900 mOsm/m3 or higherliu09, braun2015, peng2014, luke2020. High levels of TF osmolarity are associated with pain, inflammation and cellular changespflugfelder2011, belmonte2015, liu09. In support of these potentially high levels of TF osmolarity over the cornea, mathematical models without spatial variation have estimated peak osmolarities up to ten times the isotonic concentrationbraun2012, braun2015. The modeling work of Peng et al.peng2014 found that evaporation elevates osmolarity in breakup regions. TF thinning rates have been measured experimentally or estimated in many studies. A few experimental methods include spectral interferometry nichols2005, kimball2010, king2010, an open chamberhamano1981, an evaporimeterpeng2014b, and averaging pure water and functioning lipid layer rates over the cornea obtained by heat transfer analysis and thermal imagingdursch2018. In Braun et al.braun2018, both peak and background evaporation rates in TBU instances, as well as the width of the evaporation distribution, were studied parametrically. Subsequently, parameter estimation schemes were developed in Luke et al.luke2020, luke2021 for fitting PDE models to experimental FL intensity distributions. They found evaporation rates ranging from -36.9 to 4.91 $\mu$m/min (the upper bound indicating thickening) and overall TF thinning rates ranging from -23.5 to -1.85 $\mu$m/min. These thinning rates were comparable to, or a little faster than, previous experimental rates measured there were not specifically seeking TBU instances nichols2005. In this paper, we fit a hierarchy of ODE models to the same dataset as in Luke et al.luke2021. The authors fit TBU instances with PDE models that incorporated evaporation and the Marangoni effectluke2021. We use these PDE results as a guide when determining whether our results have captured what we believe to be the correct dynamics. In some cases, the ODE models are better able to follow the experimental data than the PDEs, suggesting different conclusions may be drawn for those particular instances. ## 2 Methods ### 2.1 FL images The data was taken from twenty-five normal, healthy subjects in a study conducted at Indiana University awisigyau2020 as discussed in several papersluke2020, luke2021. Approval was received from the Biomedical Institutional Review Board of Indiana University, Declaration of Helsinki principles were followed during data collection, and informed consent was obtained from all subjects. Subjects were seated behind a slit lamp biomicroscope and 2% sodium fluorescein solution was instilled in the patient’s eye. A light with a cobalt blue excitation filter illuminated the eye so that the aqueous layer of the TF fluoresced greencarlson2004. A trial is the sequence of images of the subject’s eye following a few quick blinks. The trial records the fluorescence of the aqueous part of the TF. The trials typically start with an FL concentration close to 0.2%, which is the so-called critical concentration where peak fluorescence occurs for thin TFswebber86. The critical FL concentration can be expressed in molar as 0.0053 M; see Appendix A.2. (a) (b) (c) (d) (e) (f) (g) (h) Figure 1: (a)-(g): the last image in each trial. The bright rectangle, called the Purkinje image, is due to the reflection from the light source. The images have been brightened for better visibility. (h): Surface plot of the FL intensity over time for subject S27v2t2 5:00 shown in (g). For our purposes, FT-TBU is thinning to what is evidently a very small aqueous thickness, as determined by the aid of a computer. We fit the central data of the same extracted data from spot- or streak-shaped FT-TBU instances in Luke et al.luke2021. All instances reported in this paper are shown in Figure 1. The time resolution of our dynamic data is restricted by the frame rate, which is 4 or 5 $s^{-1}$ depending on the trial. ### 2.2 Models There is a hierarchy of ODE models we explore; the most complicated is derived in Appendix A.3. Each variation of the model is determined by which mechanisms are assumed to affect the TF. The options are a combination of evaporation, osmosis, and flow. If flow is present, there are several choices we use. These simple models are designed to capture the key ingredients in thinning in order to distinguish which is the dominant mechanism causing thinning: evaporation, outward tangential flow, or a combination of the two. Evaporation-dominated thinning is characterized by inward tangential flow, if anyluke2020, while Marangoni flow is characterized by strong outward tangential flow that decreases in strength as the trial progressesluke2021. In all models that follow, $h(t)$ denotes the nondimensional TF thickness. #### 2.2.1 Initial conditions In all cases of the model, we have uniform nondimensional initial conditions $c(0)=1,\ h(0)=1,\mbox{ and }f(0)=f_{0}.$ (1) After eliminating $c$ and $f$, the only initial condition needed is the one for $h$. #### 2.2.2 Case E model The simplest variations of the model assumes constant evaporation is the only mechanism affecting the TF thickness. The nondimensional evaporation rate is $v$. The differential equation for $h$ , which is conservation of water, is given by $\dot{h}=-v.$ (2) A dot indicates a time derivative. #### 2.2.3 Case O model We assume constant evaporation and osmosis affect the TF thickness. Figure 2 shows a sketch of the model (on left). Osmolarity is quantified in the nondimensional variable $c$. Osmosis is written as a concentration difference between the cornea and the TF; a greater TF osmolarity will drive osmotic flow from the cornea into the TF. The differential equation is given by $\dot{h}=-v+P_{c}(c-1),$ (3) where $P_{c}$ is nondimensional permeability constant. Mass conservation of solute (osmolarity), namely, $ch=1$, allows us to eliminate $c$braun2014 to obtain a single ODE for $h$: $\dot{h}=-v+P_{c}\left[\frac{1}{h}-1\right].$ (4) Cases E and O model situations where evaporation is the dominant mechanism affecting TF thinning and where flow is not important. #### 2.2.4 Case F model We assume that evaporation, osmosis, and flow affect the thickness of the TF. Figure 2 shows a sketch of this model (on right). We introduce nondimensional velocity along the TF, $u(x,t)$. In this first case, $u(x,t)=u(x)=ax,$ (5) where $\partial_{x}u=a$ is the strain rate. This flow may be thought of as stretching taffy if $a>0$. A single curve of Figure 3 illustrates this flow profile. Mass conservation becomes $ch=e^{-at}$, and the differential equation for the TF thickness is $\dot{h}=-v+P_{c}\left[\frac{\exp(-at)}{h}-1\right]-ah.$ (6) For this and all following models, solute conservation relates the derivative of the solute mass ($hc$) to the strain rate. For the case F model, this is given by $\dot{h}=-(\partial_{x}u)h=-ah.$ (7) The sign of the strain rate $a$ suggests the kind of flow present. If $a<0$, the flow is inward, mimicking healing capillary flow. This characterizes evaporation-dominated thinning. If $a>0$, the flow is outward, mimicking Marangoni flow, driven by interfacial shear stress. This characterizes Marangoni effect-dominated thinning. Figure 2: Schematic for the Case O and F models. #### 2.2.5 Case D model This model is designed to mimic the time-dependent flow seen in TBU instances where flow is dominated by the Marangoni effect. We assume evaporation, osmosis, and decaying extensional flow affect the TF thickness. Here, $u(x,t)=b_{1}e^{-b_{2}t}x,$ (8) where $b_{1}$ and $b_{2}$ are flow rate and decay rate parameters, respectively. In this case, the strain rate is $\partial_{x}u=b_{1}e^{-b_{2}t}$. The differential equation for TF thickness $h$ is $\dot{h}=-v+P_{c}\left\\{\frac{1}{h}\exp\left[\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right]-1\right\\}-b_{1}e^{-b_{2}t}h,$ (9) where the first term inside the brackets is the result of using mass conservation to eliminate $c$. #### 2.2.6 Case M model Our most complicated the model is an extension of case D to allow the flow to decay to a constant, nonzero value. We assume that evaporation, osmosis, and a combination of constant and decaying extensional flow affect the TF thickness. This model allows for the flow to change direction: for example, it may start outward and strong, but then decay to a weakly inward, constant value. The nondimensional horizontal fluid profile is given by $u(x,t)=(a+b_{1}e^{-b_{2}t})x.$ (10) The exponential term will greatly diminish that part of the flow after $1/b_{2}$ units of time. The strain rate is $\partial_{x}u=a+b_{1}e^{-b_{2}t}$. An example of this fluid profile is shown in Figure 3. Figure 3: An example of the fluid flow profile $u$ from Equation 10. This simulation models strong outward tangential flow that dies off and then inward capillary flow persists. The nondimensional parameters used are $a=-0.25,b_{1}=1,$ and $b_{2}=2$. A final nondimensional time of 2 was used. Arrows indicate increasing time. The nondimensional film thickness, $h(t)$, is governed by $\dot{h}=-v+P_{c}\left\\{\frac{1}{h}\exp\left[-at+\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right]-1\right\\}-(a+b_{1}e^{-b_{2}t})h.$ (11) Appendix A.3 shows how $c$ can be eliminated in 11 via solute conservation. ### 2.3 FL intensity Nondimensional FL intensity $I$ from the TF is given by $I=I_{0}\frac{1-\exp(-\phi hf)}{(1+f^{2})}.$ (12) Similarly as for the equation for $h$, $f$ can be eliminated so that the FL intensity $I$ for the Case M model is given by $I=I_{0}\frac{1-\exp\left\\{-\phi f_{0}\exp\left[-at+\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right]\right\\}}{1+\left\\{f_{0}\exp\left[-at+\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right]/h\right\\}^{2}}.$ (13) The expressions for $I$ for the other cases can be obtained by setting $b_{1}=0$ if there is no time-dependence in the flow, by setting $a=0$ if there is only time-dependent flow, and by setting $a=b_{1}=0$ if there is no flow at all. ### 2.4 Estimating initial physical quantities We estimate the initial FL concentration following Wu et al. wu2015. This value is assumed to be uniform throughout the TF. By inverting the dimensional version of Equation 12 (Equation 32 in Appendix A.3) for $h^{\prime}$, we obtain an initial TF thickness estimate: $h_{0}^{\prime}=-\frac{1}{\epsilon_{f}f_{0}^{\prime}}\log\left\\{1-\frac{I}{I_{0}}\left[1+\left(\frac{f_{0}^{\prime}}{f_{\text{cr}}}\right)^{2}\right]\right\\}.$ (14) Model eye calculationswu2015 determine $I_{0}$ through a least squares problem. The relative initial FL intensity $I$ at the center of the TBU instance, where the minimum FL intensity in the corneal region of interest has been subtracted off, is used. More details about the procedure can be found in Luke et al.luke2020. ### 2.5 Optimization We follow the process described in Luke et al.luke2020; a summary is given below. #### 2.5.1 Data preparation We use custom MATLAB codes to convert the images in each trial from RGB color to grayscale, smooth the images with a Gaussian filter, and stabilize the images using the Purkinje imageawisigyau2020, a bright artefact reflecting the light source. We use the same roughly linear or circular FT-TBU instances that were chosen and fit by a PDE model in Luke et al.luke2021 to compare with the PDE results. We fit our theoretical FL intensity to the central data of a subset of about 6-10 time levels of experimental FL intensity data from the trial. The starting frame is the last frame before the FL intensity data starts decreasing. The first few frames of a trial are taken at a lower light setting to obtain an initial FL concentration estimate, and in some trials there is evidence that thinning has begun during this interval. As a result, the first bright image already exhibits significant decrease in FL intensity in the center of breakup. Luke et al.luke2021 remedied this issue by introducing “ghost” time levels, allowing the model solution to start with a uniform time level that is not compared to the experimental FL data. This is a product of the low time resolution of our data. In this work, we also use ghost times as appropriate. The last frame is the final frame before the FL intensity data stop decreasing. #### 2.5.2 Optimization problem We discuss the optimization for the case M model. Expressed in continuous variables, we seek to minimize $\|\|I_{\text{th}}(t)-I_{\text{ex}}(t)\|\|_{2}^{2}$ over the parameters $v^{\prime}$, the evaporation rate, $a^{\prime}$, the constant extensional flow rate, $b_{1}^{\prime}$, the decaying extensional flow rate, and $b_{2}^{\prime}$, the decay rate. Here, $t$ corresponds to the time after the light source brightness has been increased to the high setting. This variable has been nondimensionalized with the scalings given in Appendix A.2. The norm is over all $t\in[0,T]$ excluding any “ghost” time levels from the theoretical FL intensity, where $T$ corresponds to the total length of the trial. The optimization problem may be written $\operatorname*{argmin}_{v^{\prime},a^{\prime},b_{1}^{\prime},b_{2}^{\prime}}\|\|I_{\text{th}}(t;v^{\prime},a^{\prime},b_{1}^{\prime},b_{2}^{\prime})-I_{\text{ex}}(t)\|\|_{2}^{2}.$ (15) Theoretical intensity, $I_{\text{th}}$, is computed after solving the differential equation for film thickness, $h$. Similar optimizations are conducted for each of the other models. #### 2.5.3 Numerical method and stopping criterion The five ODEs for $h$ are solved using ode15s in MATLAB (MathWorks, Natick, MA, USA). For the optimization, we use a nonlinear least squares minimization implemented by lsqnonlin in MATLAB with the trust-region reflective algorithmnocedal2006 and we add a second order finite difference approximation of the Jacobianleveque2007 to improve performance. To generate initial guesses for optimization, forward computations were conducted until the theoretical dynamics were close to the experimental. For each instance, the solver stopped because the change in residual was less than the specified tolerance. Optimization tolerances of roughly the square root of the solver tolerances were used. ## 3 Results ### 3.1 Exact solutions Exact solutions exist for the case E model and for case F with $P_{c}=0$. For the case E model, using our initial condition, the nondimensional exact solution is $h(t)=1-vt.$ (16) This solution ignores the physical reality that evaporation ceases once the TF reaches zero thickness; thus, the solution is only relevant for $t\in[0,1/v].$ If we assume that time-independent flow is the only mechanism affecting the TF thickness in case F, then $v=P_{c}=0$ in (6) applies: $\dot{h}=-ah.$ (17) Using our initial condition, we find that $h(t)=e^{-at}.$ (18) To model TF thinning, we assume $a>0$ in this instance. Thus, as expected, the TF thins to zero as time increaseshowell1996. ### 3.2 Numerical solutions In Figure 4 we plot nondimensional theoretical solutions for all of the models. For comparison purposes, we have used the same parameter values for each of the five models. In particular, both flow parameters are positive, indicating outward flow. The nondimensional parameters that result from our scalings are $a=0.45,b_{1}=0.9,b_{2}=2.4,v=0.5,$ and $P_{c}=0.0653$. We see that the case O solution thins slightly less than the case E solution due to osmosis, which adds fluid to the TF. For the three models involving flow, since we have selected both $a,b_{1}>0$, the case M model shows the most thinning since the outward flow is strongest of all the models. As expected, the case F and D models are similar early on, but become increasingly similar once the flow is shut off in the D model. Osmolarity and normalized FL concentration solutions are identical in the absence of spatial variation. Both quantities are inversely related to TF thickness and increase at the origin in the presence of outward flow; the profiles reach the highest peak value for the case M model, which exhibits the greatest decrease in TF thickness and the strongest fluid flow. (a) (b) (c) Figure 4: Nondimensional theoretical solutions for the five cases of the model with $v^{\prime}=30\ \mu$m/min, $a^{\prime}=0.15$ /s, $b_{1}^{\prime}=0.3$ /s, $b_{2}^{\prime}=0.8$ /s, $f_{0}^{\prime}=0.3$%, $d=3\ \mu$m, and $t_{s}=3$ s. ### 3.3 Fitting results We begin by presenting our aggregate results of fitting the same instances that were fit with the mixed-mechanism PDE model of Luke et al.luke2021. The best fit results as determined by the smallest norm are shown in Table 1. Each FT-TBU instance is labeled by subject, visit, and trial number, the location of the breakup as a clock reading, and the type of breakup (streak or spot). Images showing the FT-TBU instances can be found in Section 2.1. A combination of the evaporation rate, constant flow rate, decaying flow rate, and decay rate are adjusted to accomplish the fit. The optimal parameters are given for the case of the model with the smallest norm. Section 3.4 shows examples of the experimental data, fits, and resulting theoretical solutions using the optimal parameters found by nonlinear least squares minimization. The S18v2t4 7:30 spot was originally fit with a single ghost time level in Luke et al.luke2021 but alternatively fit with two in the supplementary material; we choose to fit with two here as well. Five of the FT-TBU instances are best fit by the case M model and the other three are best fit by the case D model. Notably, the versions of the model without flow produce theoretical solutions with worse fits in all eight instances; we take this as strong evidence that flow plays a crucial role in causing the TF thinning. This is expected as each FT-TBU was previously fit with a model that combined evaporation and strong Marangoni flowluke2021. It is worth mentioning a few other fits: the S27v2t2 5:00 streak is also fit well with the case D model, with the case F model not far behind; the case M model fits the S18v2t4 7:30 spot data well; and the S13v2t10 6:30 spot data also matches reasonably well with the case F model. Table 1 shows a wide range of evaporation rates. Notably, the S9v2t5 4:30 spot instance, which was categorized as evaporation-dominated in Luke et al.luke2021, has the highest optimal evaporation rate. In contrast, the S10v1t6 12:30 spot and S27v2t2 5:00 streak are faster instances that were categorized as Marangoni effect-dominated in the aforementioned paper; these cases exhibit the two smallest evaporation rates seen in Table 1. The S18v2t4 7:30 and S10v1t6 12:30 spots have the strongest outward flow of all instances, with initial strain rates close to or over 2 $s^{-1}$. Further, all five Marangoni effect-dominated FT-TBUs from the previous paper exhibit outward flow, the characteristic direction of Marangoni flow. The S9v2t1 3:00 streak and S9v2t5 4:00 and 4:30 spots are the three instances designated as evaporation-dominated or transitional thinning in the PDE paper; these all show some amount of inward flow, which is characteristic of evaporation- dominated thinning. Trial \| FT-TBU ID \| $h_{0}^{\prime}$ ($\mu$m) \| \| $f_{0}^{\prime}$ --- (%) $v^{\prime}$ ($\frac{\mu\text{m}}{\text{min}}$) \| $a^{\prime}$ ($s^{-1}$) \| $b_{1}^{\prime}$ ($s^{-1}$) \| $b_{2}^{\prime}$ ($s^{-1}$) \| Norm \| Model S9v1t4+ \| 4:00 — \| 3.32 \| .324 \| 24.1 \| .0316 \| .418 \| 5.75 \| .203 \| M S9v2t1 \| 3:00 — \| 5.01 \| .292 \| 27.3 \| .461 \| -.490 \| .0715 \| .110 \| M S9v2t5 \| 4:00 $\circ$ \| 2.1 \| .299 \| 22.4 \| .217 \| -.417 \| .882 \| .118 \| M S9v2t5 \| 4:30 $\circ$ \| 2.33 \| .299 \| 50.9 \| .360 \| -.564 \| .367 \| .192 \| M S10v1t6++ \| 12:30 $\circ$ \| 3.08 \| .293 \| 1.27 \| \| 1.95 \| .277 \| .0280 \| D S13v2t10+ \| 6:30 — \| 3.59 \| .259 \| 26.4 \| \| .138 \| .102 \| .121 \| D S18v2t4++ \| 7:30 $\circ$ \| 2.48 \| .363 \| 25.2 \| \| 2.41 \| 8.85 \| .111 \| D S27v2t2+ \| 5:00 — \| 1.91 \| .4 \| 9.32 \| .714 \| -.368 \| .540 \| .0271 \| M Table 1: Results from fitting various ODE models (up to four parameters). The subject (S) number, visit (v) number and (t) trial number are listed. A $+$ denotes using a “ghost” first time level in the PDE fit and “ghost” time in the ODE fit. The FT-TBU location is a clock reading taken from the center of the pupil. FT-TBU type is denoted by — for a streak, and $\circ$ for a spot. The initial TF thickness and FL concentration estimates are given. The optimal parameters are given for the case of the model with the smallest norm. The evaporative thinning rates are given by $v^{\prime}$, constant extensional flow rate by $a^{\prime}$ and decaying extensional flow and decay rates by $b_{1}^{\prime}$ and $b_{2}^{\prime}$. ### 3.4 Fitting examples The S10v1t6 12:30 spot is shown as an example of our fitting procedure in Figure 5. Figure 5c shows the line of data extracted for the PDE fit recorded in Luke et al.luke2021; we fit the breakup data at the midpoint of the line with our ODE models. The results for each of the six ODE models are recorded in Table 2. In order to determine the model selected to report in Table 1, we compare the 2-norms of the difference between the theoretical and experimental FL intensities and select the case of the model corresponding to the smallest value. The first six seconds of the trial are obscured by eyelashes and the upper eyelid. The spot has already started to form and darkens quickly after the breakup region is revealed around six seconds into the trial (see Figure 5b). In order to fit the data with our model, we use “ghost” time levels for 0.5 seconds. Figure 5d shows that the experimental FL intensity drops to less than 10% of its initial value. (a) (b) (c) (d) Figure 5: Extracted data and best fit results for the S10v1t6 12:30 spot. In (c) the image has been brightened and contrast-enhanced. Case (c) evaporation (see Luke et al.luke2021) was used in the PDE fit. Model \| \| $v^{\prime}$ --- ($\frac{\mu\text{m}}{\text{min}}$) \| $a^{\prime}$ --- ($s^{-1}$) \| $b_{1}^{\prime}$ --- ($s^{-1}$) \| $b_{2}^{\prime}$ --- ($s^{-1}$) Residual \| Norm Evap only (E) \| 120 \| \| \| \| 3.88 $\times 10^{-2}$ \| 1.97 $\times 10^{-1}$ Evap + osm (O) \| 122 \| \| \| \| 3.47 $\times 10^{-2}$ \| 1.86 $\times 10^{-1}$ Evap, osm, flow (F) \| 0.00 \| 1.74 \| \| \| 2.0 $\times 10^{-3}$ \| 4.50 $\times 10^{-2}$ Evap, osm, dec. flow (D) \| 1.27 \| \| 1.95 \| 0.277 \| 7.86 $\times 10^{-4}$ \| 2.80 $\times 10^{-2}$ Evap, osm, mixed flow (M) \| 4.91 \| 0.656 \| 1.19 \| 0.423 \| 6.10 $\times 10^{-4}$ \| 4.04 $\times 10^{-2}$ Mixed-mech PDE center \| 5.92 \| \| \| \| \| 5.58 $\times 10^{-2}$ Table 2: S10v1t6 12:30 center of spot data fit with the five cases of models. The central data of the best PDE fit is shown for comparison. In Table 2, the two ODE models without flow select unrealistic evaporation rates in an attempt to match the rapid thinning of the S10v1t6 12:30 spot. On average, the evaporation rates chosen by the ODE models with flow are among the smallest optimal values for all mixed-mechanism fit instances. This is likely due to the relatively large flow rate parameters–unlike any other trial, the initial flow value $b_{1}^{\prime}$ or $a^{\prime}$ is above 1 $s^{-1}$ for each of the three models that involve flow. We take this as strong evidence that the Marangoni effect is the dominant mechanism causing the thinning. Further evidence of this statement is the fact that the case F model selected zero evaporation. This instance was fit well with a Marangoni effect-only PDE model which ignored evaporation, which is consistent with our small or zero optimal evaporation rates. The case D model produces the smallest residual. This FT-TBU instance exhibits the largest drop in FL intensity of all eight; the decaying model gives the best fit because the TF has likely thinned to almost zero thickness, allowing little flow, if any. The S27v2t2 5:00 streak data and fits are shown in Figure 6. As in Luke et al.luke2021, we use a quarter second of “ghost” time at the start of the fit. This instance is of particular interest because the center of the mixed- mechanism PDE theoretical FL intensity does not capture the dynamics of the experimental data well. In Luke et al.luke2021, the S27v2t2 5:00 streak was categorized as Marangoni-effect dominated due to the large Marangoni number and outward flow of the best fit. The best fit case M model selects outward flow; however, the close second-best (D) and third-best (F) cases select inward flow. These latter two models also select a significantly larger evaporation rate than the others. The S27v2t2 5:00 streak was also fit with an evaporation-only modelbraun2018 in Luke et al.luke2021. That fit (E PDE) is shown along with the mixed-mechanism fit (MM PDE) and the ODE results in Figure 6b and outperforms the mixed-mechanism PDE fit. Further, the optimal peak evaporation rate for the evaporation-only PDE fit is 35.3 $\mu$m/min, which is a large but plausible evaporation rate. This suggests that evaporation may play a larger role in this instance than previously thought. (a) (b) Figure 6: Extracted data and best fit results for the S27v2t2 5:00 streak. Uniform evaporation was used in the mixed-mechanism PDE fit. In Figures 7a-c we show the S9v2t5 4:00 spot data and in Figure 7b we show the fits. We have plotted the central data from both the best-fit mixed-mechanism (MM PDE) and evaporation-only (E PDE) models for comparison because this instance is also fit well with the latter model and was categorized as evaporation-dominated in Luke et al.luke2021. All three ODE models with flow select some amount of inward flow, which aligns well with the PDE model, whose flow profile changes sign at the origin as time progresses (see Table 4). In all cases, the flow is of a significantly smaller magnitude than the Marangoni effect-dominated or transitional thinning instances. The case M model gives the smallest residual. Notably, the evaporation-only fits for this instance give closer residuals to the best fit model than other instances; this suggests that evaporation is the dominant mechanism causing the thinning. (a) (b) Figure 7: Extracted data and best fit results for the S9v2t5 4:00 spot data. Case (c) evaporation (see Luke et al.luke2021) was used in the mixed-mechanism PDE fit. ## 4 Discussion The quantities recorded in Table 1 show more variation than the PDE results in some cases, but the qualitative similarities in the solutions are an important takeaway. For each TBU instance, the best fit ODE model includes time- dependent flow. This is strong evidence that evaporation alone cannot explain this thinning and that the Marangoni effect played a role, since it is characterized by non-constant thinning. In Figure 8 we show the various time derivatives $\dot{h}^{\prime}$ computed from the optimal values of the ODE models as well as the optimal $\partial_{t^{\prime}}h^{\prime}$ measured at the origin for the three examples of mixed-mechanism fitting shown in Section 3.4. The average starting two seconds in or the value at the final time point is recorded in Table 3. This delay in averaging matches the approach in Luke et al.luke2021 and mimics experimental proceduresnichols2005. These values are shown along with the optimal evaporation rates for comparison. The S10v1t6 12:30 spot, which showed the most rapid thinning when fit with the mixed-mechanism PDE model, shows dynamic rates of thinning for a large portion of the trial for many of the ODE fits in Figure 8a. The case (D) and (M) ODE model $\dot{h}^{\prime}$ values are very close, which we expect since they gave similar residuals when fit to the data. The S9v2t5 4:00 spot shows non-constant dynamics near the end of the trial in Figure 8c and we see further qualitative and quantitative agreement between the mixed-mechanism and evaporation-only PDE results. The case (D) and (M) models for this instance also show $\dot{h}^{\prime}>0$ in the first quarter second; the theoretical TF thickness solution is in fact slightly positive early on. This is likely an attempt by the optimization to fit the concave down portion of the data in the first second or so. In general, the PDE models produce $\partial_{t^{\prime}}h^{\prime}$ values in the first quarter second that are much larger than the corresponding ODE numbers. (a) (b) (c) Figure 8: $\dot{h}^{\prime}$ from the five ODE models are plotted alongside $\partial_{t^{\prime}}h^{\prime}$ from the PDE model. The time point at which averaging will begin is shown as a dashed vertical line. In Table 3 we record the optimal evaporation rate (the top number) and an average thinning rate (the bottom number) for the PDE fit and each of the five ODE fits. The average thinning rate is either taken starting two seconds into the trial, or if the trial is two seconds or less, the value at the final time is recorded. The values corresponding to the best fit by an ODE model as determined by the smallest residual are shaded. In all instances, the overall thinning rate of the best ODE fit is larger than that of the PDE. This may reflect the tendency of the theoretical PDE FL intensity to lag the experimental FL intensity in later times. In most instances, the best fit overall thinning rate is larger than the evaporation rate, indicative of outward thinning that supports the notion that the Marangoni effect contributed to the thinning. Notably, for the S9v2t5 4:30 spot, which was categorized as evaporation-dominated in Luke et al.luke2021, the evaporation rate is larger than the thinning rate, suggesting inward capillary flow combats the thinning. Some short trials exhibit rapid dynamics which occur early on; the recorded thinning rate may not represent the entirety of the trial. Trial \| \| FT-TBU --- ID \| $v^{\prime}$ --- $\partial_{t^{\prime}}h^{\prime}$ \| $v^{\prime}_{E}$ --- $\dot{h^{\prime}}$ \| $v^{\prime}_{O}$ --- $\dot{h^{\prime}}$ \| $v^{\prime}_{F}$ --- $\dot{h^{\prime}}$ \| $v^{\prime}_{D}$ --- $\dot{h^{\prime}}$ \| $v^{\prime}_{M}$ --- $\dot{h^{\prime}}$ S9v1t4 \| 4:00 — \| \| -6.26 --- -15.8 \| -28.7 --- -28.7 \| -30.2 --- -26.4 \| -13.5 --- -26.3 \| -16.4 --- -26.0 \| -24.1 --- -24.0 S9v2t1 \| 3:00 — \| \| -30.3 --- -21.2 \| -30.4 --- -30.4 \| -31.9 --- -28.5 \| -29.1 --- -29.2 \| -38.6 --- -20.6 \| -27.3 --- -38.1 S9v2t5 \| 4:00 $\circ$ \| \| -26.2 --- -18.2 \| -22.2 --- -22.2 \| -23.4 --- -20.1 \| -27.7 --- -20.2 \| -37.5 --- -25.1 \| -22.4 --- -30.0 S9v2t5 \| 4:30 $\circ$ \| \| -36.9 --- -23.4 \| -42.8 --- -42.8 \| -44.6 --- -35.7 \| -50.0 --- -36.5 \| -48.7 --- -38.2 \| -50.9 --- -44.9 S10v1t6 \| 12:30 $\circ$ \| \| -5.92 --- -7.60 \| -120 --- -120 \| -122 --- -0.0131 \| 0 --- -9.92 \| -1.27 --- -10.4 \| -4.91 --- -11.5 S13v2t10 \| 6:30 — \| \| -13.6 --- -21.8 \| -37.1 --- -37.1 \| -38.9 --- -33.6 \| -21.5 --- -31.7 \| -26.4 --- -31.1 \| -20.4 --- -30.8 S18v2t4 \| 7:30 $\circ$ \| \| -13.1 --- -16.3 \| -37.1 --- -37.1 \| -39.0 --- -32.4 \| -0.0011 --- -18.9 \| -25.2 --- 21.0 \| -20.3 --- -20.5 S27v2t2 \| 5:00 — \| \| -6.11 --- -16.0 \| -31.1 --- -31.1 \| -32.0 --- -28.6 \| -43.6 --- -23.9 \| -47.8 --- -38.8 \| -9.32 --- -30.9 Table 3: The optimal evaporation rates are recorded along with estimates of average $\dot{h^{\prime}}$ for the mixed-mechanism model fits (starting two seconds into the trial). All rates are measured in $\mu$m/min. PDE values are given by $v^{\prime}$ and $\partial_{t^{\prime}}h^{\prime}$; the rest are from the various ODE models. The value at the last time was used for trials less than two seconds in length. The five cases of ODE models are in order and denoted by subscripts on the evaporation value: $v^{\prime}_{E}$, $v_{O}^{\prime}$, $v_{F}^{\prime}$, $v_{D}^{\prime}$ and $v_{M}^{\prime}$. The shaded entries correspond to the model giving the best fit as determined by the smallest norm. We plot $\partial_{r^{\prime}}\bar{u}^{\prime}$ or $\partial_{x^{\prime}}\bar{u}^{\prime}$ for the three examples from Section 3.4 in Figure 9 along with $\partial_{x^{\prime}}u^{\prime}$ from the relevant ODE models. In Figures 9b,c we also plot the evaporation-only PDE profiles. At least one ODE model does a decent job approximating the qualitative behavior of the PDE flow profile after the first quarter second or so. A notable exception is the constant extensional flow option for the S10v1t6 12:30 spot; this suggests the flow profile is highly time-dependent. An average value for each model and instance is taken over the whole trial and recorded in Table 4. (a) (b) (c) Figure 9: The mixed-mechanism $\partial_{r^{\prime}}\bar{u}^{\prime}$ or $\partial_{x^{\prime}}\bar{u}^{\prime}$ are plotted against the strain rate $\partial_{x^{\prime}}u^{\prime}$ for each ODE model with flow. In Table 4 we record the flow profile data for each instance. The shaded entries correspond to the best fit. From left to right, the parameters correspond to the coefficients of the flow terms from the case F, D, and M models, respectively. For most cases, the optimal flow directions match the PDE results, an indicator that they were correctly classified in Luke et al.luke2021. The three instances classified as evaporation-dominated or transitional thinning in Luke et al.luke2021 show some amount of inward flow in both the PDE and ODE best fits, which is consistent since evaporation- dominated thinning is characterized by inward flow. In the cases of the S9v1t4 4:00 streak, S10v1t6 12:30 spot, S13v2t10 6:30 streak, and S18v2t4 7:30 spot, the ODE strain rates are always positive, which match the signs of the average strain rates of their corresponding PDE fits. This is evidence that outward tangential flow is important for explaining these instances, and so the thinning is likely influenced by the Marangoni effect. In the case of the S27v2t2 5:00 streak, the final strain rate signs of the best fit ODE and mixed-mechanism PDE models do not match. In this instance, the evaporation- only and mixed-mechanism PDEs show qualitative differences, and the evaporation-only PDE gives a better fit to the central data than the mixed- mechanism version. Perhaps the breakup dynamics of the streak in fact includes inward flow. For the S27v2t2 5:00 streak, this theory is supported by the text in Section 3.4. Trial \| FT-TBU ID \| $\dot{\gamma}^{\prime}$ \| $a^{\prime}_{F}$ \| $b_{1D}^{\prime}$ \| \| $b_{2D}^{\prime}$ --- $a_{M}^{\prime}$ \| \| $b_{1M}^{\prime}$ --- \| $b_{2M}^{\prime}$ --- S9v1t4 \| 4:00 — \| .150 \| .189 \| .164 \| .0421 \| .0316 \| .418 \| 5.75 S9v2t1 \| 3:00 — \| -.0218 \| .0199 \| -.0894 \| .385 \| .461 \| -.490 \| .0715 S9v2t5 \| 4:00 $\circ$ \| -.0427 \| -.0631 \| -.312 \| .517 \| .217 \| -.417 \| .882 S9v2t5 \| 4:30 $\circ$ \| -.0786 \| -.0757 \| -.963 \| .708 \| .360 \| -.564 \| .367 S10v1t6 \| 12:30 $\circ$ \| .572 \| 1.74 \| 1.95 \| .277 \| .656 \| 1.19 \| .423 S13v2t10 \| 6:30 — \| .147 \| .172 \| .138 \| .102 \| .173 \| .0257 \| .812 S18v2t4 \| 7:30 $\circ$ \| .172 \| .674 \| 2.41 \| 8.85 \| .0733 \| 3.69 \| 12.8 S27v2t2 \| 5:00 — \| .343 \| -.201 \| -.282 \| .0761 \| .714 \| -.368 \| .540 Table 4: Estimates of the extensional rate $\dot{\gamma}^{\prime}$, which is either $\partial_{r^{\prime}}\bar{u}^{\prime}$ for spots or $\partial_{x^{\prime}}\bar{u}^{\prime}$ for streaks, at the origin taken over the entire trial length in ($s^{-1}$) for the mixed-mechanism model fits. These are compared with the optimal values from the three ODE models with flow. The shaded entries correspond to the model giving the best fit as determined by the smallest norm. In general, the ODE models do a good job of capturing the essence of the dynamics of the PDE model fits. Each instance that we expect to have some outward flow has at least one positive flow parameter, and vice versa for the inward flow instances (see Table 4). The ODE models are able to be a little more fine-tuned, as the PDE fit is done over space as well; as such, we might expect the central PDE data that is shown for comparison not to match the data quite as closely. The PDE fits often struggle to keep up with the experimental data in later times; this is reflected in the slower (in general) average thinning rate $\partial_{t^{\prime}}h^{\prime}$ as compared to the ODE values $\dot{h}^{\prime}$ (see Table 3). However, the ODE data and fit is a simplification of the overall breakup dynamics, and viewing temporospatial data has value on its own. (a) (b) Figure 10: Histograms of rates of change plotted against experimental point measurements from Nichols et al.nichols2005; note that the experiment cannot distiguish between $v^{\prime}$ and $\dot{h}^{\prime}$. The best fit ODE model data is shown as determined by the smallest norm. Figure 10 compares the PDE and ODE best fit evaporation and thinning rate results to experimental point measurements reported in Nichols et al.nichols2005. For both histograms, a bin size of 5 $\mu$m/min was used. While the ODE rates show a wider range than the PDE results, there is significant overlap. The overall thinning rate is more comparable to the Nichols et al.nichols2005 data since that study could not separate evaporation from the other mechanisms affecting TF thickness. We expect our thinning rates to be larger than the point measurements since the Nichols et al.nichols2005 study did not target breakup. While some of the ODE data lies outside the experimental distribution, many ODE thinning rate values are comparable, suggesting these simplified models return physically relevant quantities that cannot be otherwise estimated. (a) (b) Figure 11: Histograms of maximum osmolarity and minimum TF thickness (final times of fit). In Figure 11a we compare the maximum osmolarity values of the PDE and best fit ODE models. A bin size of 50 mOsmol/L was used. Both PDE and ODE peak osmolarity estimates are reasonable compared to other experimental and theoretical valuesliu09, peng2014. The ODE results show greater variation and exhibit larger maximal values on average. Peng et al.peng2014b and Braun et al.braun2018 showed that diffusion reduces the peak osmolarity in TBU, which is only relevant for a model with spatial variation. As such, the lower PDE maximum osmolarity values are expected. Further, Braun et al.braun2015 reported theoretical osmolarity values up to ten times the isotonic level, and so our largest osmolarity value, which is just over five times isotonic, is not unreasonable. Figure 11b records the minimum thicknesses of the PDE and best ODE fits. A bin size of 0.2 $\mu$m was used. There is significant overlap of the PDE and ODE model results, suggesting that the simplified version can capture the end behavior of TF dynamics with a sufficient level of accuracy. The minimum TF thickness values are larger on average for the PDE models; this may be explained by the lag of the theoretical FL intensity behind the experimental data at later times in many of the PDE fitsluke2020, luke2021. Overall, there is more variability in the ODE results than the PDE results. We may overfit the subtleties of the dynamics with four parameters in the case (M) model, especially when the few data points of the central dynamics are essentially linear. Further, in some instances, the dynamics of the case (D) and (M) models are nearly indistinguishable, suggesting the additional parameter in the (M) model that mimics steady outward flow may not be necessary. The PDE data, which combines spatial and temporal information, is only fit with three parameters, reducing the likelihood of overfitting. Slight differences in the experimental data that are likely due to noise can affect the optimal parameters. Better time resolution would help get rid of the influence of outlier time levels on the optimization. In order to compare with the PDE results, we scale with a characteristic length and horizontal velocity whose meanings are less clear in the context of the ODE model. The time scale we use is more of a characteristic time to bend the curve rather than the time to an overall decrease in FL intensity. We show the PDE results only for comparison; this ODE fitting process can be used on many other instances. ## 5 Conclusions and future perspectives We fit the same data as in Luke et al.luke2021 with simplified models to validate our ODE fitting procedure and find good qualitative agreement of PDE and ODE results in most instances. The ODE fitting procedure provides a relatively quick process that returns important information about the TBU instance, including parameters that cannot currently be measured directly in vivo. We are working on a machine learning approach to automatically identify breakup instances and fit the central data with our ODE models. This strategy could be applied on a large scale to obtain statistical information about a wide range of TBU shapes. ## Acknowledgements This work was supported by National Science Foundation grant DMS 1909846. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding sources. ## Appendix A Appendix ### A.1 Governing dimensional equations We derive the case (M) model. We use Cartesian coordinates $(x^{\prime},z^{\prime})$ to denote the position and $\vec{u}^{\prime}=(u^{\prime},v^{\prime})$ to denote the fluid velocity. Primes denote dimensional quantities. The TF is modeled as an incompressible Newtonian fluid on $0<x^{\prime}<X_{0}$ and $0<z^{\prime}<h^{\prime}(x^{\prime},t^{\prime})$, where $h^{\prime}(x^{\prime},t^{\prime})$ denotes the thickness of the film. Conservation of mass of TF fluid is given by $\nabla^{\prime}\cdot\vec{u}^{\prime}=0.$ (19) At the film/cornea interface $z^{\prime}=0$, we require osmosis across a perfect semipermeable membrane: $u^{\prime}=0,\quad v^{\prime}=P_{o}V_{w}(c^{\prime}-c_{0}),$ (20) where $c^{\prime}$ is the osmolarity. The membrane permeability is given by $P_{o}$, the molar volume of water is $V_{w}$, and $c_{0}$ is the isotonic osmolarity. The kinematic condition at the fluid/air interface is given by $\partial_{t^{\prime}}h^{\prime}=v^{\prime}\|_{z^{\prime}=h^{\prime}}-u^{\prime}\|_{z^{\prime}=h^{\prime}}\partial_{x^{\prime}}h^{\prime}-J^{\prime},$ (21) where $J^{\prime}$ is evaporation. Since we have assumed the film is spatially uniform, we have $\partial_{x^{\prime}}h^{\prime}=0$, and thus $\dot{h^{\prime}}=v^{\prime}_{z^{\prime}=h^{\prime}}-J^{\prime}.$ (22) The dot indicates an ordinary derivative in time. We assume a combination of constant and decaying extensional flow: $u^{\prime}=(a^{\prime}+b_{1}^{\prime}e^{-b_{2}^{\prime}t^{\prime}})x^{\prime},$ (23) where $a^{\prime}$ is a constant flow rate, and $b_{1}^{\prime}$ and $b_{2}^{\prime}$ are a flow and decay rate, respectively. ### A.2 Scalings The governing equations can be nondimensionalized using the following scalings: $c^{\prime}=c_{0}c,\quad\quad h^{\prime}=dh,\quad\quad t^{\prime}=(\ell/U)t,\quad\quad u^{\prime}=Uu,\quad\quad f^{\prime}=f_{\text{cr}}f,\quad\quad J^{\prime}=\rho UJ.$ (24) The following scalings are a result of the nondimensionalization process: $v^{\prime}=\epsilon Uv,\quad\quad a^{\prime}=(U/\ell)a,\quad\quad b_{1}^{\prime}=(U/\ell)b_{1},\quad\quad b_{2}^{\prime}=(U/\ell)b_{2}.$ (25) Dimensional quantities are denoted by primes. We follow the same scalings used in Luke et al.luke2021 to compare with their results. We take the length of the trial as $t_{s}$ and then $U=\ell/t_{s}$, where $\ell=\left(\frac{t_{s}\sigma_{0}d^{3}}{\mu}\right)^{1/4}.$ (26) Dimensional parameters used in the model are summarized in Table 5. Dimensional parameters --- Parameter \| Description \| Value \| Reference $\rho$ \| Density \| $10^{3}$ kg $\cdot$ m-3 \| Water $d$ \| Initial TF thickness \| $2-5\times 10^{-6}$ m \| Calculatedluke2021 $f_{0}^{\prime}$ \| Init. FL concentration \| $0.259-0.4$ % \| Calculatedluke2021 $v^{\prime}$ \| Evaporative thinning rate \| $0.5-25$ $\mu$m/min \| Nichols et al.nichols2005 $a^{\prime}$ \| Constant extensional flow rate \| $-0.201-1.74$ $s^{-1}$ \| Calculated $b_{1}^{\prime}$ \| Decaying extensional flow rate \| $-0.564-3.69$ $s^{-1}$ \| Calculated $b_{2}^{\prime}$ \| Decay rate \| $0.0421-12.8$ $s^{-1}$ \| Calculated $V_{w}$ \| Molar volume of water \| 1.8 $\times 10^{-5}$ m${}^{3}\cdot$ mol-1 \| Water $c_{0}$ \| Isotonic osmolarity \| $300$ mOsmol/L \| Lemp et al.lemp2011 $P_{o}$ \| Permeability of cornea \| $12.1\times 10^{-6}$ m/s \| Braun et al.braun2015 $\epsilon_{f}$ \| Naperian extinction coefficient \| $1.75\times 10^{7}$ m-1 M-1 \| Mota et al.mota1991 $\mu$ \| Viscosity \| $1.3$ $\times 10^{-3}\ \text{Pa}\cdot\text{s}$ \| Tiffanytiffany1991 $\sigma_{0}$ \| Surface tension \| $0.045$ N $\cdot$ m-1 \| Nagyogá & Tiffanynagyova1999 $\ell$ \| Characteristic length \| $0.138-0.412$ mm \| Calculated $U$ \| Characteristic velocity \| $0.0560-0.0990$ mm/s \| Calculated $t_{s}$ \| Time scale \| $1.75-6.6$ s \| Fit intervalluke2021 Table 5: The dimensional parameters used are shown. The range of estimates for thinning rates are from point measurements; this range includes the values given by our optimization. Non-dimensional parameters with typical values --- Parameter \| Description \| Expression \| Value $\epsilon$ \| Aspect ratio \| $d/\ell$ \| 0.0130 $P_{c}$ \| Permeability of cornea \| $\displaystyle P_{o}V_{w}c_{0}/(\epsilon U)$ \| 0.0653 $\phi$ \| Nondimensional Napierian extinction coefficient \| $\displaystyle\epsilon_{f}f_{\text{cr}}d$ \| 0.279 Table 6: Dimensionless parameters that arise from scaling the dimensional fluid mechanics problem. The values given are based upon the values of Table 5 and those used to generate Figure 4. FL concentration is typically reported as a percentage in the ocular literature. For a particular FL concentration $f^{\prime}$ given as a percentage, this quantity is converted to molar as $f_{M}^{\prime}$ by $f_{M}^{\prime}=\frac{\rho}{M_{w}}\frac{f^{\prime}}{100},$ (27) where $\rho$ is the density of water (Table 5) and $M_{w}$ is the molecular weight of sodium fluorescein (approximately 376 g/mol). Critical FL concentration $f_{\text{cr}}$, 0.2%, makes an 0.0053 M solution when dissolved in water. This conversion of $f_{\text{cr}}$ to molar is necessary to compute the dimensionless Napierian extinction coefficient $\phi$ (Table 6). ### A.3 Derivation of TF equations Using the scalings 24, 25, we nondimensionalize the governing equations. From the nondimensional version of Equation 23, we have that $u_{x}=a+b_{1}e^{-b_{2}t}.$ In Cartesian coordinates, conservation of mass is given by $u_{x}+v_{z}=0$. Integrating this equation over the vertical domain, we have $0=\int_{0}^{h}\left[\partial_{x}u+\partial_{z}v\right]dz=-(a+b_{1}e^{-b_{2}t})h+\partial_{t}h+J-P_{c}(c-1),$ where we’ve used the independence of $u_{x}$ from $z$. Rewriting this result as a differential equation for $h$, we have $\dot{h}=-(a+b_{1}e^{-b_{2}t})h-J+P_{c}(c-1).$ (28) Our transport equation for $c$ is (without any spatial terms): $h\dot{c}=Jc-P_{c}(c-1)c.$ (29) Multiplying Eq. (28) by $c$ and adding the result to Eq. (29), we have an ODE for the product $hc$: $\dot{(hc)}=-(a+b_{1}e^{-b_{2}t})hc.$ Separating and integrating gives $hc=A\exp\left(-at+\frac{b_{1}}{b_{2}}e^{-bt}\right),$ (30) where $A$ is an arbitrary constant of integration. Using $h(0)=c(0)=1$ and $f(0)=f_{0}$ nondimensionally, we solve for $c$ and $f$: $c=\frac{1}{h}\exp\left[-at+\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right],\quad f=\frac{f_{0}}{h}\exp\left[-at+\frac{b_{1}}{b_{2}}\left(e^{-b_{2}t}-1\right)\right].$ (31) Replacement in our ODE for $h$ gives the equations shown in the text. The equation for fluorescent intensity is $I=I_{0}\frac{1-e^{-\epsilon_{f}h^{\prime}f^{\prime}}}{\left(f^{\prime}_{\text{cr}}\right)^{2}+(f^{\prime})^{2}},$ (32) where $\epsilon_{f}$ is the molar extinction coefficient, $f_{\text{cr}}^{\prime}$ is the critical fluorescein concentration, and $I_{0}$ is a normalization factor. The nondimensional version with $f$ eliminated is given in the text.
# On The Impact Of 22Ne On The Pulsation Periods Of Carbon-Oxygen White Dwarfs With Helium Dominated Atmospheres Morgan T. Chidester School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA F.X. Timmes School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA Josiah Schwab Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA Richard H. D. Townsend Department of Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA Ebraheem Farag School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA Anne Thoul Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Ao$\hat{u}$t 19C, Bat. B5C, 4000 Liège, Belgium C. E. Fields Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements, USA Evan B. Bauer Center for Astrophysics $\|$ Harvard & Smithsonian, 60 Garden St Cambridge, MA 02138, USA Department of Physics, University of California, Santa Barbara, CA 93106, USA Michael H. Montgomery Department of Astronomy and McDonald Observatory, University of Texas, Austin, TX 78712, USA Morgan T. Chidester<EMAIL_ADDRESS> ###### Abstract We explore changes in the adiabatic low-order g-mode pulsation periods of 0.526, 0.560, and 0.729 $\mathrm{M}_{\odot}$ carbon-oxygen white dwarf models with helium-dominated envelopes due to the presence, absence, and enhancement of $\mathrm{{}^{22}Ne}$ in the interior. The observed g-mode pulsation periods of such white dwarfs are typically given to 6$-$7 significant figures of precision. Usually white dwarf models without $\mathrm{{}^{22}Ne}$ are fit to the observed periods and other properties. The root-mean-square residuals to the $\simeq$ 150$-$400 s low-order g-mode periods are typically in the range of $\sigma_{\rm rms}$ $\lesssim$ 0.3 s, for a fit precision of $\sigma_{\rm rms}/P$ $\lesssim$ 0.3%. We find average relative period shifts of $\Delta P/P$ $\simeq$ $\pm$ 0.5% for the low-order dipole and quadrupole g-mode pulsations within the observed effective temperature window, with the range of $\Delta P/P$ depending on the specific g-mode, abundance of $\mathrm{{}^{22}Ne}$, effective temperature, and mass of the white dwarf model. This finding suggests a systematic offset may be present in the fitting process of specific white dwarfs when $\mathrm{{}^{22}Ne}$ is absent. As part of the fitting processes involves adjusting the composition profiles of a white dwarf model, our study on the impact of $\mathrm{{}^{22}Ne}$ can provide new inferences on the derived interior mass fraction profiles. We encourage routinely including $\mathrm{{}^{22}Ne}$ mass fraction profiles, informed by stellar evolution models, to future generations of white dwarf model fitting processes. Stellar physics (1621); Stellar evolution (1599); Stellar pulsations (1625); White dwarf stars (1799); Non-radial pulsations (1117) ††software: MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019, http://mesa.sourceforge.net), MESASDK 20190830 (Townsend, 2019a, b), wd_builder https://github.com/jschwab/wd_builder, GYRE (Townsend & Teitler, 2013; Townsend et al., 2018, https://github.com/rhdtownsend/gyre), Montreal White Dwarf Database (Dufour et al., 2017, http://www.montrealwhitedwarfdatabase.org), matplotlib (Hunter, 2007), and NumPy (van der Walt et al., 2011). ## 1 Introduction Photons emitted from stellar surfaces and neutrinos released from stellar interiors may not directly reveal all that we want to know about the internal constitution of the stars. For example, a direct view of the chemical stratification from the core to the surface is hidden. These interior abundance profiles matter: they impact a star’s opacity, thermodynamics, nuclear energy generation, and pulsation properties. The stellar models, in turn, are used to interpret the integrated light of stellar clusters and galaxies (e.g., Alsing et al., 2020), decipher the origin of the elements (e.g., Arcones et al., 2017; Placco et al., 2020), predict the frequency of merging neutron stars and black holes (Giacobbo & Mapelli, 2018; Farmer et al., 2020; Marchant & Moriya, 2020; Abbott et al., 2020), and decipher the population(s) of exploding white dwarfs that underlay Type Ia supernova cosmology (e.g., Miles et al., 2016; Rose et al., 2020). Neutrino astronomy, in concert with stellar models, can probe the isotopic composition profiles in energy producing regions of the Sun (Borexino Collaboration et al., 2018, 2020) and nearby ($d$ $\lesssim$ 1 kpc) presupernova massive stars up to tens of hours before core-collapse (e.g., Patton et al., 2017; Simpson et al., 2019; Mukhopadhyay et al., 2020). Stellar seismology, also in concert with stellar models, can probe the elemental composition profiles in pulsating stars from the upper main-sequence (e.g., Simón-Díaz et al., 2018; Pedersen et al., 2019; Balona & Ozuyar, 2020) through the red-giant branch (e.g., Hekker & Christensen-Dalsgaard, 2017; Hon et al., 2018) to white dwarfs (WDs, e.g., Hermes et al., 2017; Giammichele et al., 2018; Córsico et al., 2019; Bell et al., 2019; Bischoff-Kim et al., 2019; Althaus et al., 2020). Most of a main-sequence star’s initial metallicity $Z$ comes from the carbon- nitrogen-oxygen (CNO) and $\mathrm{{}^{56}Fe}$ nuclei inherited from its ambient interstellar medium. All of the CNO piles up at $\mathrm{{}^{14}N}$ when H-burning on the main-sequence is completed because the $\mathrm{{}^{14}N}$($p$,$\gamma$)$\mathrm{{}^{15}O}$ reaction rate is the slowest step in the H-burning CNO cycle. During the ensuing He-burning phase, all of the 14N is converted to 22Ne by the reaction sequence $\mathrm{{}^{14}N}$($\alpha$,$\gamma$)18F(,$e^{+}\nu_{e}$)18O($\alpha$,$\gamma$)$\mathrm{{}^{22}Ne}$. The abundance of $\mathrm{{}^{22}Ne}$ when He-burning is completed is thus proportional to the initial CNO abundance of the progenitor main-sequence star. The weak reaction in this sequence powers the neutrino luminosity during He-burning (e.g., Serenelli & Fukugita, 2005; Farag et al., 2020) and marks the first time in a star’s life that the core becomes neutron rich. For zero- age main sequence (ZAMS) masses between $\simeq$ 0.5 $\mathrm{M}_{\odot}$ (Demarque & Mengel, 1971; Prada Moroni & Straniero, 2009; Gautschy, 2012) and $\simeq$ 7 $\mathrm{M}_{\odot}$ (Becker & Iben, 1979, 1980; García-Berro et al., 1997), depending on the treatment of convective boundary mixing (Weidemann, 2000; Denissenkov et al., 2013; Jones et al., 2013; Farmer et al., 2015; Lecoanet et al., 2016; Constantino et al., 2015, 2016, 2017), the $\mathrm{{}^{14}N}$($\alpha$,$\gamma$)18F(,$e^{+}\nu_{e}$)18O($\alpha$,$\gamma$)$\mathrm{{}^{22}Ne}$ reaction sequence determines the $\mathrm{{}^{22}Ne}$ content of a resulting carbon-oxygen white dwarf (CO WD). We follow the convention that $\mathrm{{}^{22}Ne}$ is the “metallicity” of the CO WD. Camisassa et al. (2016) analyze the impact of $\mathrm{{}^{22}Ne}$ on the sedimentation and pulsation properties of H-dominated atmosphere WDs (i.e., the DAV class of WD) with masses of 0.528, 0.576, 0.657, and 0.833 $\mathrm{M}_{\odot}$. These WD models result from $Z$ = 0.02 non-rotating evolutionary models that start from the ZAMS and are evolved through the core- hydrogen and core-helium burning, thermally pulsing asymptotic giant branch (AGB), and post-AGB phases. At low luminosities, $\log(L/\mathrm{L}_{\odot})$ $\lesssim$ $-4.5$, they find that $\mathrm{{}^{22}Ne}$ sedimentation delays the cooling of WDs by 0.7 to 1.2 Gyr, depending on the WD mass. They also find that $\mathrm{{}^{22}Ne}$ sedimentation induces differences in the periods that are larger than the present observational uncertainties. Giammichele et al. (2018) analyze in their supplemental material the effect of $\mathrm{{}^{22}Ne}$ on the pulsation periods of a 0.570 $\mathrm{M}_{\odot}$ template-based model for the DB WD KIC 08626021. They considered a model consisting of pure oxygen core surrounded by a pure helium envelope with the same mass and effective temperature equal to those inferred for KIC 08626021. Next, they considered a model that replaces the pure oxygen core with an oxygen-dominated core plus a trace amount of $\mathrm{{}^{22}Ne}$. They find that the model with $\mathrm{{}^{22}Ne}$ has, on average, shorter pulsation periods. This article is novel in three ways. One, we explore the impact of $\mathrm{{}^{22}Ne}$ on the adiabatic low-order g-mode pulsation periods of CO WD models with a He-dominated atmosphere (i.e., the DBV class of WD) as the models cool through the range of observed DBV effective temperatures. Two, we derive an approximation formula for the Brunt-Väisälä frequency in WDs that allows new physical insights into why the low-order g-mode pulsation periods change due to the presence, and absence, of $\mathrm{{}^{22}Ne}$. Three, we analyze how the $\mathrm{{}^{22}Ne}$ induced changes in the pulsation periods depend on the mass and temporal resolutions of the WD model. Our explorations can help inform inferences about the interior mass fraction profiles derived from fitting the observed periods of specific DBV WDs (e.g., Metcalfe et al., 2002; Fontaine & Brassard, 2002; Metcalfe, 2003; Metcalfe et al., 2003; Hermes et al., 2017; Giammichele et al., 2017, 2018; Charpinet et al., 2019; De Gerónimo et al., 2019; Bischoff-Kim et al., 2019). In Section 2 we summarize the input physics, and discuss in detail the chemical stratification, cooling properties, and g-mode pulsation periods of one DBV WD model. In Section 3 we present our results on changes to the low- order g-mode pulsation periods due to the presence, or absence, of $\mathrm{{}^{22}Ne}$ from this model. In Section 4 we study changes in the g-mode pulsation periods due to $\mathrm{{}^{22}Ne}$ from a less massive and a more massive WD model. In Section 5 we summarize and discuss our results. In Appendix A we study the robustness of our results with respect to mass and temporal resolution, and in Appendix B we discuss in more depth some of the input physics. Figure 1: Mass fraction profiles of the 0.56 $\mathrm{M}_{\odot}$DB WD resulting from the evolution of the 2.1 $\mathrm{M}_{\odot}$, $Z$=0.02, ZAMS model. ## 2 A Baseline WD Model ### 2.1 Input Physics We use MESA version r12115 (Paxton et al., 2011, 2013, 2015, 2018, 2019) to evolve a $2.1\,\mathrm{M}_{\odot}$, $Z$ = 0.02 metallicity model from the ZAMS through core H-burning and core He-burning. After winds during the thermal pulses on the AGB have reduced the H-rich envelope mass to $0.01\,\mathrm{M}_{\odot}$, the remaining hydrogen is stripped from the surface to form a young, 0.56 $\mathrm{M}_{\odot}$DB WD. This model is tuned to match the observed and inferred properties of KIC 08626021 (Bischoff-Kim et al., 2014; Giammichele et al., 2018; Timmes et al., 2018; Charpinet et al., 2019; De Gerónimo et al., 2019). Additional details of the input physics are given in Appendix B, and the MESA r12115 files to reproduce our work are available at https://doi.org/10.5281/zenodo.4338180 (catalog https://doi.org/10.5281/zenodo.4338180) ### 2.2 Mass Fraction Profiles Figure 1 shows the mass fraction $X(^{A}Z)$ profiles of the resulting 0.56 $\mathrm{M}_{\odot}$DB WD model, where $A$ is the number of nucleons and $Z$ is the number of protons. Brown boxes divide the mass fraction profiles into three regions according to their origins and uncertainties. The $X(\mathrm{{}^{12}C})$ and $X(\mathrm{{}^{16}O})$ profiles in the innermost $\simeq$ 90% by mass region are determined during core and shell He-burning. The main uncertainties in this region are the $\mathrm{{}^{12}C}$($\alpha$,$\gamma$)$\mathrm{{}^{16}O}$ reaction rate (e.g., deBoer et al., 2017), and the treatment of convective mixing boundaries during core H-and He-burning (e.g., Constantino et al., 2015, 2016, 2017). The CO and $X(\mathrm{{}^{4}He})$ profiles between $\simeq$ 1% and $\simeq$ 10% of the exterior WD mass originate from shell He-burning during the thermal pulse phase of evolution on the AGB. Most of the total He mass is held in this region. The primary uncertainties in this region are the number of thermal pulses and convective boundary layer mixing. The number of thermal pulses a model undergoes is sensitive to the mass resolution, time resolution, mass loss rate, and the treatment of convective boundaries (Iben & Renzini, 1983; Herwig, 2005; Karakas & Lattanzio, 2014). The sharp change in all the mass fractions at $\simeq$ 1% of the exterior WD mass marks the extent reached by the convective boundary during the last thermal pulse. CO profiles in this region may also be subject to other mixing processes. For example, the magnitude of the $X(\mathrm{{}^{12}C})$ “bump” is subject to the strength and duration of the thermohaline instability, which occurs when material is stable to convection according to the Ledoux criterion, but has an inverted molecular weight gradient (Baines & Gill, 1969; Brown et al., 2013; Garaud, 2018; Bauer & Bildsten, 2018). The $X(\mathrm{{}^{4}He})$ profile of the outer $\simeq$ 0.1% to 1% of the WD mass is determined by shell H-burning. All of the initial CNO mass fractions have been converted to $\mathrm{{}^{14}N}$. The main uncertainties in this region are the number of thermal pulses during the AGB phase of evolution, late or very late thermal pulses (Bloecker, 1995a, b; Blöcker, 2001), and mechanisms to remove the residual high entropy, H-rich layer to produce a DB WD from single and binary evolution (e.g., D’Antona & Mazzitelli, 1990; Althaus & Benvenuto, 1997; Parsons et al., 2016). The $X(\mathrm{{}^{22}Ne})$ profile is essentially flat and spans the inner $\simeq$ 99% by mass. As discussed in Section 1, $X(\mathrm{{}^{22}Ne})$ is created from $\mathrm{{}^{14}N}$ during He-burning. Figure 2: Evolution of baseline model’s photon luminosity $L$ and neutrino luminosity $L_{\nu}$ (left top), effective temperature $T_{\rm eff}$ and radius $R$ (left middle), central temperature $T_{c}$ and central density $\rho_{c}$ (left bottom). Time begins a few thermal timescales after the ab initio WD is released. Gray bands show the luminosity and $T_{\rm eff}$ range of currently observed DBV WD (Montreal White Dwarf Database, Dufour et al., 2017). Mass fraction profiles are shown at $T_{\rm eff}$ = 30,039 K (right top), 15,012 K (right middle), and 12,092 K (right bottom) and at the end of the evolution. Initial mass fraction profiles are shown as solid curves and the diffusing mass fraction profiles are shown as dotted curves. Figure 3: Propagation diagram for the dipole $\ell$ = 1 (top) and quadrupole $\ell$ = 2 (bottom) g-modes at $L$ = 0.01 $\mathrm{L}_{\odot}$ for the baseline WD model. The Lamb frequency ($S_{\ell}$, orange), Brunt-Väisälä frequency ($N$, blue), radial order $n$ = 1,2,3,10 eigenfrequencies (dotted black lines), nodes in the radial eigenfunction (filled circles), and g-mode period of each radial order are labeled. ### 2.3 Constructing Ab Initio White Dwarf Models Starting from a set of pre-main sequence (pre-MS) initial conditions, accurate predictions for the properties of the resulting WD model, especially the mass fraction profiles, do not exist due to the nonlinear system of equations being approximated. In addition, evolving a stellar model from the pre-MS to a WD can be resource intensive. It can thus be useful for systematic studies to build ab initio WD models (e.g., WDEC, Bischoff-Kim & Montgomery, 2018). By ab initio we mean calculations that begin with a WD model, as opposed to a WD model that is the result of a stellar evolution calculation from the pre-MS. A potential disadvantage (or advantage) of ab initio WD models is the imposed initial mass fraction profiles may not be attainable by a stellar model evolved from the pre-MS. Throughout the remainder of this article we use a new capability, wd_builder, to construct ab initio WD models in MESA of a given mass and chemical stratification. The initial structure of an ab initio WD model is approximated as an isothermal core and a radiative envelope in hydrostatic equilibrium. Here we specify an initial WD mass of 0.56 $\mathrm{M}_{\odot}$, the same WD mass as produced by the stellar evolution calculation. The imposed $X(\mathrm{{}^{4}He})$, $X(\mathrm{{}^{12}C})$, $X(\mathrm{{}^{14}N})$, $X(\mathrm{{}^{16}O})$, and $X(\mathrm{{}^{22}Ne})$ profiles are taken from the stellar evolution derived mass fraction profiles of Figure 1 and normalized to sum to unity in each cell. Henceforth we refer to this ab initio WD model as the “baseline model”. For ab initio WD models we use He-dominated, $\log$(H/He) = $-$5.0, model atmosphere tables spanning 5,000 K $\leq$ $T_{\rm eff}$ $\leq$ 40,000 K that were provided by Odette Toloza (2019, private communication) using the Koester WD atmosphere software instrument (Koester, 2010). These tabulated atmospheres for DB WDs are publicly available as a standard atmosphere option as of MESA r12115. In addition, we use five element classes for the diffusion classes – $\mathrm{{}^{4}He}$, $\mathrm{{}^{12}C}$, $\mathrm{{}^{14}N}$ $\mathrm{{}^{16}O}$, and $\mathrm{{}^{22}Ne}$. Otherwise, all of the physics implementations and modeling choices are as described in Section 2.1. The initial baseline model is then evolved with MESA. As the model is not in thermal equilibrium, there is an initial transient phase lasting a few thermal timescales that is disregarded. The thermal timescale is $\tau_{\rm th}$ $\simeq$ $E_{\rm th}/L_{\rm tot}$ $\simeq$ 0.67 Myr, where $E_{\rm th}$ is the thermal energy of the WD and $L_{\rm tot}$ is the photon plus neutrino luminosity. Specifically, we set the zero point to be 1.5 thermal timescales ($\simeq$ 1 Myr) after the transient reaches its peak luminosity. The evolution terminates when $L_{\rm tot}$ falls below $\log(L/\mathrm{L}_{\odot})$ = $-$2.5. Figure 2 shows the cooling properties of the baseline model. Plasmon neutrino emission dominates the energy loss budget at $T_{\rm eff}\gtrsim 25,000\,{\rm K}$ (e.g., Vila, 1966; Kutter & Savedoff, 1969; Winget et al., 2004; Bischoff- Kim & Montgomery, 2018). Photons leaving the WD surface begin to dominate the cooling as the electrons transition to a strongly degenerate plasma (van Horn, 1971). The luminosity becomes proportional to the enclosed mass, $L_{r}$ $\propto$ $\,M_{r}$, in this model only when $T_{\rm eff}\lesssim 20,000\,{\rm K}$ (Timmes et al., 2018). Energy transport in the interior is dominated by conduction, driven primarily by electron-ion scattering. Energy transport in the outer layers is dominated by radiation or convection associated with the partial ionization of He at $T_{\rm eff}\simeq 30,000\,{\rm K}$. Figure 2 also shows the diffusion of the initial mass fractions as the baseline WD model cools to $T_{\rm eff}$ = 30,000 K, 15,000 K and 12,138 K (corresponding to the termination at $\log(L/\mathrm{L}_{\odot})$ = $-$2.5). Element diffusion of $\mathrm{{}^{22}Ne}$ is modest for the baseline 0.56 $\mathrm{M}_{\odot}$ DB WD model. Depletion of the $\mathrm{{}^{22}Ne}$ mass fraction at $\log(1-M_{r}/M$) $\simeq$ $-1.9$ has occurred by the time the model has cooled to $T_{\rm eff}$ $\simeq$ 30,000 K. As the model cools further, the surface regions in the tail of the He-dominated layer further deplete and a small $\mathrm{{}^{22}Ne}$ bump forms and propagates inwards toward the center. The timescale for $\mathrm{{}^{22}Ne}$ to travel from near the surface to the center of this WD model is $\tau_{\rm D}\simeq 2\bar{Z}\Gamma^{1/3}\rho_{6}^{-1/2}\ {\rm Gyr}\simeq 30\ {\rm Gyr}$ (Isern et al., 1991; Bravo et al., 1992; Bildsten & Hall, 2001; Deloye & Bildsten, 2002; Camisassa et al., 2016), where $\bar{Z}$ is the mean charge of the material, $\Gamma$ is the electrostatic to thermal energy ratio, and $\rho_{6}$ is the baryon mass density in units of 106 g cm-3. Thus, the $X(\mathrm{{}^{22}Ne})$ profile does not significantly change as the 0.56 $\mathrm{M}_{\odot}$baseline model evolves to $\log(L/\mathrm{L}_{\odot})$ = $-$2.5 in $\simeq$ 350 Myr. More massive WDs show larger amounts of $\mathrm{{}^{22}Ne}$ sedimentation over the same time period (Camisassa et al., 2016). WD cooling data suggests a significant enhancement due to to $\mathrm{{}^{22}Ne}$ diffusion (Cheng et al., 2019; Bauer et al., 2020), but does not effect the baseline model until it cools to effective temperatures lower than considered here ($T_{\rm eff}$ $\lesssim$ 10,000 K). ### 2.4 Pulsation Periods of the Baseline Model Having established the structural and composition profiles of a cooling baseline WD model, we now consider the g-mode pulsation periods. Some of the material is classic (e.g., Unno et al., 1989; Fontaine & Brassard, 2008), but we also derive and verify the accuracy of an approximation formula for the Brunt-Väisälä frequency in WDs that allows physical insights into why the low- order g-mode pulsation periods change due to variations in the mass fraction of $\mathrm{{}^{22}Ne}$. This material is essential for establishing that the baseline model, before introducing any modifications to the chemical profiles, produces pulsation periods that are commensurate with the observations of DBV WDs. Figure 3 shows the propagation diagram (e.g., Unno et al., 1989) for the baseline WD model after it has cooled to $T_{\rm eff}$ = 16,045 K and dimmed to $L$ = 0.01 $\mathrm{L}_{\odot}$, within the DBV WD observation window. Adiabatic pulsation frequencies are calculated using release 5.2 of the GYRE software instrument (Townsend & Teitler, 2013; Townsend et al., 2018). For a fixed radial overtone number, the $\ell$ = 1 periods are $\sim$ $\sqrt{3}$ longer than the $\ell$ = 2 periods, due to the local dispersion relation for low-frequency g-modes $\sigma_{g}$ scaling as $\sigma_{g}^{2}\simeq\ell(\ell+1)N^{2}/(k_{r}^{2}r^{2})\ ,$ (1) where $k_{r}$ is the radial wave number. The Brunt-Väisälä frequency $N$ is $N^{2}=\frac{g^{2}\rho}{P}\frac{\chi_{T}}{\chi_{\rho}}(\nabla_{{\rm ad}}-\nabla_{T}+B)\ ,$ (2) where $g$ is the gravitational acceleration, $\rho$ is the mass density, $P$ is the pressure, $T$ is the temperature, $\chi_{T}$ is the temperature exponent $\partial({\rm ln}P)/\partial({\rm ln}\rho)\|_{T,\mu_{I}}$, $\chi_{\rho}$ is the density exponent $\partial({\rm ln}P)/\partial({\rm ln}T)\|_{\rho,\mu_{I}}$, $\nabla_{{\rm ad}}$ is the adiabatic temperature gradient, $\nabla_{T}$ is the actual temperature gradient, and $B$ accounts for composition gradients (e.g., Hansen & Kawaler, 1994; Fontaine & Brassard, 2008). Bumps in the $N$ profile of Figure 3 correspond to transitions in the $X(\mathrm{{}^{16}O})$, $X(\mathrm{{}^{12}C})$, and $X(\mathrm{{}^{4}He})$ profiles. The implementation of Equation 2 in MESA is described in Section 3 of Paxton et al. (2013). An approximation for $N^{2}$ in the interiors of WDs that yields physical insights begins by assuming $\nabla_{{\rm ad}}$ is much larger than $\nabla_{T}$ and $B$. Then $N^{2}=\frac{g^{2}\rho}{P}\frac{\chi_{T}}{\chi_{\rho}}\nabla_{{\rm ad}}\ .$ (3) In the interior of a WD the ions are ideal and dominate the temperature derivatives of an electron degenerate plasma. Substituting the pressure scale height $H$ = $P/(\rho g)$ and equation 3.110 of Hansen & Kawaler (1994) $\chi_{T}=\frac{\rho}{P}\frac{k_{B}T}{\mu_{I}m_{p}}$ (4) into Equation 3 gives $N^{2}=\frac{1}{H^{2}\chi_{\rho}}\frac{k_{B}T}{\mu_{I}m_{p}}\nabla_{{\rm ad}}\ ,$ (5) where $k_{B}$ is the Boltzmann constant, $\mu_{I}$ = 1/($\sum_{i}X_{i}/A_{i})$ is the ion mean molecular weight, and $m_{p}$ is the mass of the proton. Equation 3.90 of Hansen & Kawaler (1994) shows $\nabla_{{\rm ad}}$ = $(\Gamma_{3}-1)/\Gamma_{1}$, where $\Gamma_{1}$ is the first adiabatic index and $\Gamma_{3}$ $\rightarrow$ $k_{B}/(\mu_{I}m_{p}c_{v})$ is the third adiabatic index, where in the gas phase the ideal specific heat capacity is $c_{v}$ = $3k_{B}/(2\mu_{I}m_{p})$. The sentence beneath equation 3.112 of Hansen & Kawaler (1994) thus notes that $\Gamma_{3}-1$ = 2/3 for the ions in the gas phase ($\Gamma_{3}-1$ = 1/3 in the liquid phase). Combining these expressions, yields the approximation $N^{2}=\frac{2}{3\Gamma_{1}\chi_{\rho}H^{2}}\frac{k_{B}T}{\mu_{I}m_{p}}\ .$ (6) Figure 4: Comparison of the approximation for $N^{2}$ (blue curve) in Equation (6) and the full calculation of $N^{2}$ from MESA (green curve). Figure 5: Mass-radius relation of the baseline DB WD model at $\log(L/\mathrm{L}_{\odot})$ = $-$2.5 with key features located: the transition from $X(\mathrm{{}^{16}O})$ to $X(\mathrm{{}^{12}C})$ dominated, the rise of $X(\mathrm{{}^{4}He})$, the $X(\mathrm{{}^{12}C})$ bump, where $S_{\ell}$ $<$ $N$ occurs, the transition to $X(\mathrm{{}^{4}He})$ dominated, and where $N^{2}$ $<$ 0\. Figure 6: Period evolution of the $\ell$ = 1 (purple) and $\ell$ = 2 (green) g-modes at radial orders $n$=1,2,3,10 as the baseline model cools. Each point represents a timestep in MESA where the g-mode was calculated by GYRE. The gray band show the $T_{\rm eff}$ range of observed DBV WD. Figure 4 compares the approximation in Equation (6) with the full $N^{2}$ calculation from MESA. The difference at $r/R$ $\simeq$ 0.5 corresponds to the $X$($\mathrm{{}^{16}O}$) $\rightarrow$ $X$($\mathrm{{}^{12}C}$) transition, at $r/R$ $\simeq$ 0.8 to the $\mathrm{{}^{12}C}$ bump, and at $r/R$ $\simeq$ 0.9 to the transition to a He dominated atmosphere. Except for the outermost layers and regions where the composition gradients are significant, the agreement is sufficient to use Equation (6) as a scaling relation for building physical insights. We always use, however, the full $N^{2}$ calculation from MESA for any quantitative analysis. It is useful to reference features of the baseline model with respect to mass or radius. Figure 5 thus shows the mass-radius relation of the baseline model at $\log(L/\mathrm{L}_{\odot})$ = $-$2.5 with key transitions labeled. Figure 7: Top to Bottom: Relative period differences of the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$ and $g_{10,2}$ modes between the baseline model, PB, and a model where the $\mathrm{{}^{22}Ne}$ has been replaced with $\mathrm{{}^{14}N}$, P14N. We use the notation $g_{n,\ell}$ for a g-mode of order $n$ and degree $\ell$. Gray bands show the $T_{\rm eff}$ range of currently observed DBV WDs. Figure 8: Top to Bottom: Relative differences in the $H^{2}$, $\mu_{I}$ $\Gamma_{1}$, $\chi_{\rho}$, and $T$ contributions to $N^{2}$ in Equation 6. Subscript B represents the baseline model, and subscript 14N represents and a model where $\mathrm{{}^{22}Ne}$ has been replaced with $\mathrm{{}^{14}N}$. Figure 6 shows the low-order g-mode pulsation periods as the baseline WD model cools. The periods increase due to $N^{2}$ decreasing as the cooling progresses, per Equation 6. Higher radial orders have steeper slopes due to the periods scaling with $k_{r}$ in Equation 1. The increase in number of MESA models at $T_{\rm eff}$ $\simeq$ 30,000 K is due to the partial ionization of He, which leads to envelope convection in relatively hot DBV WDs. The change in slope at $T_{\rm eff}$ $\simeq$ 20,000 K is due to the luminosity becoming proportional to the enclosed mass, $L_{r}\propto M_{r}$, as the plasmon neutrino emission becomes insignificant. In Appendix A we show that the low-order g-mode pulsation periods of the baseline model calculated with GYRE are only weakly dependent on the mass and temporal resolution of the MESA calculations. Figure 9: Weight functions of the low-order g-modes for baseline model with $\mathrm{{}^{22}Ne}$ (black curves) and a baseline model where $\mathrm{{}^{22}Ne}$ has been replaced with $\mathrm{{}^{14}N}$ (green curves). Subpanels show the relative percent differences between the two curves. The profiles shown are when the two models have cooled to $\log(L/\mathrm{L}_{\odot})$ = $-$2.5. Nodes in the radial eigenfunctions are marked by filled circles. Figure 10: Top to Bottom: Relative period differences of the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$ and $g_{10,2}$ modes between the baseline model, PB, a zero-metallicity WD model (gray curves) where the $\mathrm{{}^{14}N}$ and $\mathrm{{}^{22}Ne}$ have been put into $\mathrm{{}^{4}He}$ and $\mathrm{{}^{12}C}$ respectively, and a super-solar metallicity model (green curves) where the $\mathrm{{}^{14}N}$ and $\mathrm{{}^{22}Ne}$ of the baseline model are doubled. Figure 11: Top to Bottom: Relative differences in the $H^{2}$, $\mu_{I}$ $\Gamma_{1}$, $\chi_{\rho}$, and $T$ contributions to $N^{2}$ in Equation 6. Subscript B represents the baseline model, and subscript Z represents the zero-metallicity models (gray curves) and super-solar metallicity models (green curves). ## 3 The Impact Of $\mathrm{{}^{22}Ne}$ Having established the cooling properties and g-mode pulsation periods of a baseline model whose mass fraction profiles are from a stellar evolution model, we now explore changes in the g-mode pulsation periods due to changes in the $\mathrm{{}^{22}Ne}$ mass fraction profile shown in Figure 1. We consider three modifications: replacing $\mathrm{{}^{22}Ne}$ with $\mathrm{{}^{14}N}$, a metal-free model, and a super-solar metallicity model. ### 3.1 Putting the 22Ne into 14N Replacing $X$($\mathrm{{}^{22}Ne}$) with $X$($\mathrm{{}^{14}N}$) is a model for the reaction sequence $\mathrm{{}^{14}N}$($\alpha$,$\gamma$)18F(,$e^{+}\nu_{e}$)18O($\alpha$,$\gamma$)$\mathrm{{}^{22}Ne}$ either physically not occurring or being ignored. Figure 7 shows the relative differences in the low-order g-mode pulsation periods from this composition change. All of the relative differences are negative, implying the pulsation periods in models that exclude $\mathrm{{}^{22}Ne}$ are longer than the corresponding pulsation periods in models that include $\mathrm{{}^{22}Ne}$. The magnitude of the relative period differences span $\simeq$ 0.25%$-$1% over the range of currently observed DBV WDs, with the $g_{1,1}$ and $g_{1,2}$ modes showing the largest differences at cooler $T_{\rm eff}$. The change in the slopes at $T_{\rm eff}$ $\simeq$ 20,000 K is due to plasmon neutrino emission becoming insignificant, and thus the luminosity becoming proportional to the enclosed mass, $L_{r}\propto M_{r}$. What drives these g-mode period changes? Replacing an isotope which has a larger mass number with an isotope which has a smaller mass number decreases $\mu_{I}$. This replacement also increases $H$ through the mechanical structure and equation of state of the CO WD. Figure 8 shows the relative differences in the $H^{2}$, $\mu_{I}$, $\Gamma_{1}$, $\chi_{\rho}$ and $T$ contributions to $N^{2}$ in Equation 6. These changes collectively determine the magnitude and sign of the period change relative to the baseline model. For this $X$($\mathrm{{}^{22}Ne}$) $\rightarrow$ $X$($\mathrm{{}^{14}N}$) model, the overall positive changes in $\mu_{I}$ and $T$ are counteracted by the negative changes from $H^{2}$, $\Gamma_{1}$, and $\chi_{\rho}$. The magnitude of the relative difference in $H^{2}$ drives the net result of a smaller $N^{2}$ and thus longer g-mode periods. The nearly uniform negative change in $H^{2}$ imply a change in the radius of the WD model. We find $(R_{B}-R_{14N})/R_{B}$ $\simeq$ $-$0.4%, meaning the $X$($\mathrm{{}^{22}Ne}$) $\rightarrow$ $X$($\mathrm{{}^{14}N}$) model has a larger radius than the model with $\mathrm{{}^{22}Ne}$. This is expected given differences in the electron fraction of a WD. Figure 9 compares the weight functions of the baseline model with $\mathrm{{}^{22}Ne}$ and the model where the $\mathrm{{}^{22}Ne}$ has been replaced with $\mathrm{{}^{14}N}$. Following Kawaler et al. (1985), the weight function is $\frac{{\rm d}\zeta}{{\rm d}r}=\frac{[C({\bf y},r)+N({\bf y},r)+G({\bf y},r)]\rho r^{2}}{\int_{r=0}^{r=R}T({\bf y},r)\rho r^{2}{\rm d}r}\ ,$ (7) where $C({\bf y},r)$ varies with the Lamb frequency, $N({\bf y},r)$ contains the Brunt-Väisälä frequency, $G({\bf y},r)$ involves the gravitational eigenfunctions, $T({\bf y},r)$ is proportional to the kinetic energy density, and ${\bf y}=(y_{1},y_{2},y_{3},y_{4})$ are the Dziembowski (1971) variables. The frequency of an adiabatic mode is then $\nu^{2}=\zeta=\int_{r=0}^{r=R}\frac{{\rm d}\zeta}{{\rm d}r}\cdot{\rm d}r\ .$ (8) The weight function for the two models is dominated by the $N({\bf y},r)$ term except for the surface layers. Figure 9 shows that the net effect of the $\mathrm{{}^{22}Ne}$ $\rightarrow$ $\mathrm{{}^{14}N}$ composition change is a shift in $\zeta$, the area under the weight function curves, towards smaller frequencies of the low-order g-modes. The subpanels in Figure 9 illustrate the relative percent differences between the weight function curves. Most of the changes in $\zeta$ occur at the CO transition region ($r/R$ $\simeq$ 0.45, see Figure 5), $\mathrm{{}^{12}C}$ bump ($r/R$ $\simeq$ 0.8 ), and at the transition to a He-dominated atmosphere ($r/R$ $\simeq$ 0.9). The changes in these regions get as large as $\sim$ 10%. We identify the dipole g-mode of radial order $n$ = 2 as being more sensitive to the location and gradient of $\mu_{I}$ at the CO transition ($r/R$ $\simeq$ 0.5) than other low-order g-modes. ### 3.2 Zero-Metallicity and Super-Solar Metallicity Replacing $X$($\mathrm{{}^{14}N}$) with $X$($\mathrm{{}^{4}He}$) and $X$($\mathrm{{}^{22}Ne}$) with $X$($\mathrm{{}^{12}C}$) is a model for ignoring the birth metallicity of the ZAMS star, CNO burning on the main- sequence, and the $\mathrm{{}^{14}N}$($\alpha$,$\gamma$)18F(,$e^{+}\nu_{e}$)18O($\alpha$,$\gamma$)$\mathrm{{}^{22}Ne}$ reaction sequence during He-burning. Most studies of the pulsation periods of observed WDs use zero-metallicity DBV WDs when deriving the interior mass fraction profiles, although see Camisassa et al. (2016) for a counterexample. Alternatively, doubling $X$($\mathrm{{}^{14}N}$) at the expense of $X$($\mathrm{{}^{4}He}$) and doubling $X$($\mathrm{{}^{22}Ne}$) at the expense of $X$($\mathrm{{}^{12}C}$) is a model for a super-solar metallicity DBV WD. Figure 10 compares the relative change in the low-order g-mode pulsation periods of the zero and super-solar metallicity models. The period differences are negative for the zero-metallicity model and positive for the super-solar metallicity model. Zero-metallicity DBV WD models have longer periods than the baseline model, which in turn has longer periods than the super-solar metallicity model. The relative period differences of the zero and super-solar metallicity models are mostly symmetric about the baseline model’s $Z$ = 0.02 metallicity. The period differences of the zero-metallicity models, averaged over the $T_{\rm eff}$ evolution, are $\Delta P(g_{1,1})$ $\simeq$ $-$0.57 s, $\Delta P(g_{1,2})$ $\simeq$ $-$0.40 s, $\Delta P(g_{2,1})$ $\simeq$ $-$0.52 s, and $\Delta P(g_{2,2})$ $\simeq$ $-$0.40 s. For the super-solar metallicity models the averaged absolute period differences are $\Delta P(g_{1,1})$ $\simeq$ 0.66 s, $\Delta P(g_{1,2})$ $\simeq$ 0.45 s, $\Delta P(g_{2,1})$ $\simeq$ 0.46 s, and $\Delta P(g_{2,2})$ $\simeq$ 0.35 s. Over the $T_{\rm eff}$ range of currently observed DBV WDs, the mean relative period change of the dipole modes is 0.57% and the maximum of relative period change is 0.88%. The relative period change of the quadrupole modes is smaller, with a mean of 0.33% and a maximum of 0.63%. Figure 11 shows the relative differences in the $H^{2}$, $\mu_{I}$, $\Gamma_{1}$, $\chi_{\rho}$ and $T$ contributions to $N^{2}$ of Equation 6 for the zero and super-solar metallicity models. These changes collectively determine the magnitude and sign of the period change relative to the baseline model. For the zero-metallicity models, the combined positive changes in $\mu_{I}$ and $T$ are counteracted by the collective negative changes from $H^{2}$, $\Gamma_{1}$, and $\chi_{\rho}$. The net change is negative, resulting in smaller $N^{2}$ and longer g-mode periods. Similar reasoning for the super-solar metallicity models leads to a net positive change, resulting in larger $N^{2}$ and smaller g-mode periods. The magnitude of the difference in $H^{2}$ drives the overall result for both metallicity cases. The nearly uniform changes in $H^{2}$ imply changes in the radii, and we find $(R_{B}-R_{Z})/R_{B}$ $\simeq$ $\pm$0.4% with zero-metallicity models having smaller radii and super-solar metallicity models having larger radii. Interrogating further the composition dependence, the top panels of Figure 12 compare the mass fraction profiles of the $X$($\mathrm{{}^{22}Ne}$) $\simeq$ 0.02 baseline and zero-metallicity at 30,000 K, 15,000 K and 12,100 K as a function of mass coordinate. Element diffusion is operating in both models. The middle panels show the relative differences in these mass fraction profiles, with the $\mathrm{{}^{22}Ne}$ and $\mathrm{{}^{14}N}$ offsets zeroed out. The C and O differences at $\log(1-M_{r}/M)$ $\simeq$ $-$0.25, from Figure 5, correspond to the C/O transition at $r/R$ $\simeq$ 0.5. The He difference at $\log(1-M_{r}/M)$ $\simeq$ $-$1.0 correlates to the rise of He at $r/R$ $\simeq$ 0.75. Similarily, the C, O and He differences at $\log(1-M_{r}/M)$ $\simeq$ $-$2.0 maps to He dominating the composition at $r/R$ $\simeq$ 0.9. These relative differences are the largest at 30,000 K, reaching $\simeq$ 7.5% for $\mathrm{{}^{16}O}$ and $\simeq$ $-$6% for $\mathrm{{}^{4}He}$. The relative differences at 15,000 K and 12,100 K have about the same magnitude, $\simeq$ 7.5% for $\mathrm{{}^{16}O}$ and $\simeq$ $-$1% for $\mathrm{{}^{4}He}$. The relative mass fraction differences span a larger range of $\log(1-M_{r}/M)$ as the models cool due to element diffusion. The bottom panels of Figure 12 show the corresponding relative difference in the $\mu_{I}$ profiles. As $\mu_{I}$ is calculated by dividing the mass fraction of a isotope by its atomic weight, the relative differences in the mass fraction profiles are reduced in the $\mu_{I}$ profiles. The $\mu_{I}$ profile for 12,100 K in terms of a mass coordinate is the same as the $\mu_{I}$ profile in Figure 11 in terms of a radial coordinate. We also computed the relative period differences between the $X$($\mathrm{{}^{22}Ne}$) $\simeq$ 0.02 baseline and zero-metallicity model with diffusion turned off to disentangle structural and diffusion effects. The results are shown in Figure 13. While there is a slight variation from the zero-metallicity gray curves shown in Figure 10, mostly in the higher order $g_{10,1}$ and $g_{10,2}$ modes, the magnitude of the relative differences remains the same. This further suggests that the period shifts are a direct consequence of the presence or absence of $\mathrm{{}^{22}Ne}$. Figure 12: Top Panels: Mass fraction profiles for 0.56 $\mathrm{M}_{\odot}$baseline (colored curves) and zero metallicity (black dashed curves) models at $T_{\rm eff}$ $\simeq$ 30,000 K, 15,000 K, and 12,100 K. Middle Panels: Relative differences in mass fraction profiles, where we have zeroed out the $\mathrm{{}^{22}Ne}$ and $\mathrm{{}^{14}N}$ offsets from $\mathrm{{}^{12}C}$ and $\mathrm{{}^{4}He}$ respectively. Bottom Panel: Relative differences in $\mu_{I}$. Figure 13: Top to Bottom: Relative period differences of the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$ and $g_{10,2}$ modes between the baseline model, PB, and the zero-metallicity WD model, PZ, with diffusion turned off. Figure 14: Mass fractions profiles for 0.52 $\mathrm{M}_{\odot}$(left column) and 0.73 $\mathrm{M}_{\odot}$(right column) ab initio DB WD models at $T_{\rm eff}$ $\simeq$ 30,000 K (top), 15,000 K (middle), and at the end of the evolution (bottom). Initial mass fraction profiles are shown as solid curves and the diffusing mass fraction profiles are shown as dotted curves. Figure 15: Relative period differences of the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$ and $g_{10,2}$ modes for 0.526 $\mathrm{M}_{\odot}$(left column) and 0.729 $\mathrm{M}_{\odot}$(right column). Differences are between the baseline model, PB, a zero-metallicity WD model (gray curves) where the $\mathrm{{}^{14}N}$ and $\mathrm{{}^{22}Ne}$ have been put into $\mathrm{{}^{4}He}$ and $\mathrm{{}^{12}C}$ respectively, and a super-solar metallicity model (green curves) where the $\mathrm{{}^{14}N}$ and $\mathrm{{}^{22}Ne}$ of the baseline model are doubled. ## 4 Trends in the period changes with the white dwarf mass Using the same physics and numerical choices as for the 0.56 $\mathrm{M}_{\odot}$ baseline model, we evolved a $Z$ = 0.02, 1.1 $\mathrm{M}_{\odot}$ ZAMS stellar model from the pre-main sequence to a 0.526 $\mathrm{M}_{\odot}$ DB WD, and a $Z$ = 0.02, 3.6 $\mathrm{M}_{\odot}$ ZAMS model to a 0.729 $\mathrm{M}_{\odot}$ DB WD. This initial to final mass mapping is similar to Table 1 of Camisassa et al. (2016). Relative to the 0.56 $\mathrm{M}_{\odot}$ baseline model, the 0.526 $\mathrm{M}_{\odot}$ WD model has a thicker He-layer and a more abbreviated extent of $X$($\mathrm{{}^{22}Ne}$). Conversely, the 0.729 $\mathrm{M}_{\odot}$ WD model has a smaller $\mathrm{{}^{12}C}$ bump, a thinner He-layer, and a more extended $X$($\mathrm{{}^{22}Ne}$) profile. These mass fraction profiles were imposed on 0.526 $\mathrm{M}_{\odot}$and 0.729 $\mathrm{M}_{\odot}$ ab initio WD models, respectively. Figure 14 shows the diffusion of these initial mass fraction profiles as the ab initio WD models cool to $T_{\rm eff}$ $\simeq$ 30,000 K, then $\simeq$ 15,000 K and finally $\simeq$ 12,000 K (corresponding to the termination at $\log(L/\mathrm{L}_{\odot})$ = $-$2.5). Element diffusion is more pronounced for the more massive 0.729 $\mathrm{M}_{\odot}$ DB WD model due to its larger surface gravity. An enhancement forms in the $X$($\mathrm{{}^{22}Ne}$) profile at $\log(1-M_{r}/M$) $\simeq$ $-2.0$ by the time the 0.729 $\mathrm{M}_{\odot}$ model has cooled to $T_{\rm eff}$ $\simeq$ 30,000 K. As the model further cools, the $X$($\mathrm{{}^{22}Ne}$) bump grows in amplitude as it propagates inwards toward the center through the He-dominated outer layers. The $X$($\mathrm{{}^{22}Ne}$) bump generates an increase in the local $N^{2}$ in the regions it propagates through from a locally larger $\mu_{I}$ and a smaller compensating $H^{2}$. The regions trailing the $X$($\mathrm{{}^{22}Ne}$) bump are depleted of $X$($\mathrm{{}^{22}Ne}$), causing a decrease in the local $N^{2}$ in these regions. We find longer low-order g-mode periods for the more massive WD, consistent with Camisassa et al. (2016). As was done for the 0.56 $\mathrm{M}_{\odot}$ baseline model, we replace $X$($\mathrm{{}^{14}N}$) with $X$($\mathrm{{}^{4}He}$) and $X$($\mathrm{{}^{22}Ne}$) with $X$($\mathrm{{}^{12}C}$) to generate a zero-metallicity ab initio DB WD model. We also double $X$($\mathrm{{}^{14}N}$) at the expense of $X$($\mathrm{{}^{4}He}$) and double $X$($\mathrm{{}^{22}Ne}$) at the expense of $X$($\mathrm{{}^{12}C}$) to generate a super-solar metallicity DB WD. Figure 15 compares the relative change in the low-order g-mode pulsation periods of the zero and super-solar metallicity 0.526 $\mathrm{M}_{\odot}$ and 0.729 $\mathrm{M}_{\odot}$ DB WD models. As for the 0.56 $\mathrm{M}_{\odot}$ baseline model, the relative period differences are mostly symmetric about the reference model’s $Z$ = 0.02 metallicity. For the 0.526 $\mathrm{M}_{\odot}$ models, over the $T_{\rm eff}$ range of currently observed DBV WDs, the mean relative period change of the dipole modes is 0.99% and the maximum of relative period change is 1.43%. The relative period change of the quadrupole modes is smaller, with a mean of 0.25% and a maximum of 0.43%. For the 0.729 $\mathrm{M}_{\odot}$ models, the mean relative period change of the dipole modes is 0.65% and the maximum of relative period change is 1.02%. The relative period change of the quadrupole modes is again smaller, with a mean of 0.40% and a maximum of 0.65%. These values are commensurate with the mean and maximum relative period changes found for the 0.56 $\mathrm{M}_{\odot}$ baseline model. There are a few trends in the relative period differences with respect to the WD mass. For the zero-metallicity $n$ = 2 and $n$ = 10 g-modes, the average relative differences in the observed Teff range increase with increasing mass. For example, as the WD mass is increased from 0.526 M⊙ to 0.560 M⊙, we find the average relative period differences increase by factors of 1.74, 1.22, 2.43, and 1.46, for the g2,1, g2,2, g10,1, and g10,2 modes, respectively. As the WD mass is further increased from 0.560 M⊙ to 0.729 M⊙, we find additional magnification factors of 1.21, 1.29, 1.21, and 1.26, for g-modes g2,1, g2,2, g10,1, and g10,2 respectively. The absence of $\mathrm{{}^{22}Ne}$ causes a greater deviation from the reference metallicity model as the WD mass increases. The g2,1 and g2,2 g-modes show a trend in the local minimum as the WD mass increases. For the 0.526 M⊙ model, the g2,1 g-mode has a local minimum at $T_{\rm eff}$ $\lessapprox$ 20,000 K. For the 0.526 M⊙ baseline model, this local minimum crosses zero at $T_{\rm eff}$ $\simeq$ 20,000 K. For the 0.729 M⊙ model, the local minimum is deeper and crosses zero at at $T_{\rm eff}$ $\simeq$ 25,000 K. These trends with mass are due to when energy lost by the cooling WD is no longer dominated by neutrino cooling. ## 5 Discussion We explored changes in the adiabatic low-order g-mode pulsation periods of 0.526, 0.560, and 0.729 $\mathrm{M}_{\odot}$ DB WD models due to the presence, absence, and enhancement of $\mathrm{{}^{22}Ne}$ as the models cool through the observed range of effective temperatures. We found mean relative period shifts of $\Delta P/P$ $\simeq$ $\pm$ 0.5% for the low-order dipole and quadrupole g-mode pulsations within the observed effective temperature window, with a range of $\Delta P/P$ that depends on the specific g-mode, mass fraction of $\mathrm{{}^{22}Ne}$, effective temperature, and mass of the WD model. Shifts in the pulsation periods due to the presence, absence, or enhancement of $X$($\mathrm{{}^{22}Ne}$) mostly arise from a competition between the pressure scale height and ion mean molecular weight. Low-mass DAV WDs, the ZZ Ceti class of stars, have pulsation periods in the 100$-$1500 s range (e.g., Vincent et al., 2020). Comparing low-mass DAV WDs produced from stellar evolution models with and without diffusion of $\mathrm{{}^{22}Ne}$, Camisassa et al. (2016) find that the $\mathrm{{}^{22}Ne}$ sedimentation induces mean period differences of $\simeq$ 3 s, reaching maximum period differences of $\simeq$ 11 s. For the more massive DAV WD models, where sedimentation of $\mathrm{{}^{22}Ne}$ is stronger, they find mean period differences of $\simeq$ 15 s between when diffusion is on and off, and a maximum period differences of $\simeq$ 47 s. Comparatively, our article focuses on DBV WD models, the evolution of the pulsation periods as the DBV WD models cool, and the impact of $\mathrm{{}^{22}Ne}$ being present, absent, or enhanced in the WD interior. Nevertheless, we conducted an experiment of turning element diffusion off in our 0.56 $\mathrm{M}_{\odot}$ baseline model. At $\rm\log(L/\mathrm{L}_{\odot})$ = $-2.5$, we find an absolute mean difference for $n$ = 1 to $n$ = 11 of $\simeq$ 16 s, with a maximum period difference at $n$ = 9 of $\simeq$ 56 s. This maximum difference equates to a $\simeq$ 7% relative difference between when diffusion is on and off. Our period changes are slightly higher than those found in Camisassa et al. (2016)’s 0.833 M⊙ model, and much larger than the differences found in their 0.576 M⊙ model. These differences could be a result of DAV versus DBV models, as DAV models have different cooling times than DBV models. In addition, Camisassa et al. (2016) computes their period differences at $\rm log(L/L_{\odot})=-2.80$ and $\rm log(L/L_{\odot})=-2.93$ for their 0.576 and 0.833 M⊙ models, respectively. These are dimmer than the $\rm\log(L/\mathrm{L}_{\odot})$ = $-2.5$ used for our calculations. Our maximum radial order is found up to 11 at this luminosity, while Camisassa et al. (2016) uses more radial orders, with a maximum radial order of 50. Giammichele et al. (2018) compares the g-mode pulsation periods of a pure oxygen core surrounded by a pure helium envelope with those from an oxygen- dominated core with $X$($\mathrm{{}^{22}Ne}$) = 0.02 surrounded by a pure helium envelope. They find including $\mathrm{{}^{22}Ne}$ yields shorter periods, with mean period differences of $\simeq$ 0.1%. We find a mean period shift that is about 5 times larger in our 0.56 $\mathrm{M}_{\odot}$baseline model. This difference may be caused by the contrast in the composition of the models, which in turn causes variances in the local mean molecular weight and pressure scale height scaling described by Equation 6. Are 1% or less period differences important? The g-mode periods of observed DBV WD are found from a Fourier analysis of the photometric light curves and are typically given to 6$-$7 significant figures of precision. Usually zero- metallicity WD models (i.e., without $\mathrm{{}^{22}Ne}$) are fit to the observed g-mode periods and other properties (e.g., $T_{\rm eff}$, $\log g$). The root-mean-square residuals to the $\simeq$ 150$-$400 s low-order g-mode periods are typically in the range $\sigma_{\rm rms}$ $\lesssim$ 0.3 s (e.g., Bischoff-Kim et al., 2014), for a fit precision of $\sigma_{\rm rms}/P$ $\lesssim$ 0.3%. Our finding of a mean relative period shift of $\Delta P/P$ $\simeq$ $\pm$ 0.5% induced by including $\mathrm{{}^{22}Ne}$ in WD models suggests a systematic offset may be present in the fitting process of specific WDs when $\mathrm{{}^{22}Ne}$ is absent. As part of the fitting process involves adjusting the composition profiles of the model WD, this study on the impact of $\mathrm{{}^{22}Ne}$ can inform inferences about the derived interior mass fraction profiles. We encourage routinely including $\mathrm{{}^{22}Ne}$ mass fraction profiles, informed by stellar evolution models, to future generations of DBV WD model fitting processes. The adiabatic low-order g-mode pulsation periods of our DB WD models depend upon simplifying assumptions in the stellar evolution calculations (e.g., convective boundary layer mixing, shellular rotation), uncertainties (e.g., mass loss history, stripping of the residual thin H layer, thickness of the He-dominated atmosphere), and unknown inherent systematics. We hypothesize that these model dependencies and systematics could mostly cancel when dividing one model result by another model result, such as when calculating the relative period shifts $\Delta P/P$. We anticipate exploring a larger range of models, similar in approach to Fields et al. (2016), to test this conjecture in future studies. The MESA project is supported by the National Science Foundation (NSF) under the Software Infrastructure for Sustained Innovation program grants (ACI-1663684, ACI-1663688, ACI-1663696). This research was also supported by the NSF under grant PHY-1430152 for the Physics Frontier Center “Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements” (JINA-CEE). A.T. is a Research Associate at the Belgian Scientific Research Fund (F.R.S-FNRS). We acknowledge useful discussions at virtual Sky House 2020. This research made extensive use of the SAO/NASA Astrophysics Data System (ADS). ## Appendix A Convergence Studies In this appendix we demonstrate that the pulsation periods of the baseline model are only weakly dependent on the details of the mass and temporal resolution of the MESA \+ GYRE calculations. A MESA parameter controlling the mass resolution is max_dq, the maximum fraction a model’s mass in one cell. That is, the minimum number of cells in a model is $N_{\rm min\ cells}$ = 1/max_dq. We use $N_{\rm min\ cells}$ = 5,000 for all the results reported. MESA can also adaptively refines its mesh based on a set of mesh functions. The maximum cell-to-cell variation of these functions is maintained around the value of the control mesh_delta_coeff. We use mesh_delta_coeff = 1 for all the results reported. Primarily as a result of these two mesh parameters, the total number of cells in the baseline model is $\simeq$ 8,000 cells. A MESA parameter controlling the time resolution is the largest change in the central temperature allowed over a timestep, delta_lgT_cntr_limit. For all the reported results, we use delta_lgT_cntr_limit = 0.001. MESA can also adaptively adjusts the timestep based on other criteria, but this setting dominates the size of every timestep as the baseline WD model cools. The total number of timesteps in the baseline model is $\simeq$ 1,000 and varies roughly linearly with delta_lgT_cntr_limit. Figure 16 shows changes in the low-order g-mode periods for different $N_{\rm min\ cells}$ as the models cool. The time resolution is held fixed at delta_lgT_cntr_limit = 0.001. Our standard $N_{\rm min\ cells}$ = 5,000 baseline model is the basis of the comparison and shown as the horizontal lines. A model with 10 times less mass resolution than our standard mass resolution, $N_{\rm min\ cells}$ = 500, induces maximum relative period changes of $\simeq$ 0.05% at $\simeq$ 30,000 K for $g_{1,1}$, $\simeq$ 0.07% at $\simeq$ 35,000 K for $g_{1,2}$, $\simeq$ 0.07% at $\simeq$ 45,000 K for $g_{2,1}$, and $\simeq$ 0.07% at $\simeq$ 45,000 K for $g_{2,2}$. A model with 5 times less mass resolution than our standard mass resolution, $N_{\rm min\ cells}$ = 1,000, reduces these maximum relative period changes by $\simeq$ 20%. A model with 5 times more mass resolution than our standard mass resolution, $N_{\rm min\ cells}$ = 25,000 causes maximum relative period changes of 0.000022% at $g_{1,1}$ to 0.028% at $g_{10,1}$. These maximum relative period changes are, respectively, a factor of $\simeq$ 20,000 to 20 smaller than the relative period change caused by including or excluding $\mathrm{{}^{22}Ne}$. Figure 17 shows changes in the low-order g-mode periods for different delta_lgT_cntr_limit as the models cool. The mass resolution is held fixed at $N_{\rm min\ cells}$ = 5,000. Our standard delta_lgT_cntr_limit = 0.001 baseline model is the basis of the comparison and shown as the horizontal lines. A model with 10 times less time resolution, delta_lgT_cntr_limit = 0.01, causes maximum relative period changes of $\simeq$ $-$0.05% at $\simeq$ 50,000 K for $g_{1,1}$, $\simeq$ 0.02% at $\simeq$ 50,000 K for $g_{1,2}$, $\simeq$ $-$0.06% at $\simeq$ 40,000 K for $g_{2,1}$, $\simeq$ $-$0.05% at $\simeq$ 45,000 K for $g_{2,2}$, $\simeq$ $-$0.25% at $\simeq$ 45,000 K for $g_{10,1}$, and $\simeq$ $-$0.25% at $\simeq$ 50,000 K for $g_{10,2}$. A model with 5 times less time resolution than our standard mass resolution, delta_lgT_cntr_limit = 0.005, reduces these maximum relative period changes by $\simeq$ 10%. A model with 5 times more time resolution, delta_lgT_cntr_limit = 0.0002, has average period changes of 0.00061 s for $g_{1,1}$, $-$0.00077 s for $g_{1,2}$, 0.0034 s for $g_{2,1}$, 0.0010 s for $g_{2,2}$, 0.0021 s for $g_{10,1}$, and 0.0014 s for $g_{10,2}$. The average period changes are a factor of $\simeq$ 1000 smaller than the average period changes caused by including or excluding $\mathrm{{}^{22}Ne}$. Figure 16: Relative differences in the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$, $g_{10,2}$ pulsation periods for different minimum mass resolutions as the baseline WD models cool. We use the notation $g_{n,\ell}$ for a g-mode of order $n$ and degree $\ell$. The minimum mass resolution of 5,000 cells, used for all the results reported, is shown by the black horizontal lines. Figure 17: Relative differences in the $g_{1,1}$, $g_{1,2}$, $g_{2,1}$, $g_{2,2}$, $g_{10,1}$, and $g_{10,2}$ pulsation period for different temporal resolutions as the baseline WD models cool. The largest change in the central temperature allowed over a timestep, delta_lgT_cntr_limit = 0.001, used for all the results reported, is shown by the black horizontal lines. ## Appendix B Input Physics Details In this appendix we briefly discuss the salient physics used in our MESA models. ### B.1 Thermodynamics The MESA r12115 equation of state (EOS) is a blend of the OPAL (Rogers & Nayfonov, 2002), SCVH (Saumon et al., 1995), PTEH Pols et al. (1995), HELM (Timmes & Swesty, 2000), and PC (Potekhin & Chabrier, 2010) EOSes. The MESA EOS also covers the late stages of WD cooling where the ions in the core crystallize (e.g., Bauer et al., 2020). WD interiors lie in the PC region of the MESA EOS, which provides a semi-analytic EOS treatment for arbitrary composition. The default in MESA version 12115 is to account for each species of ion with mass fraction greater than $10^{-3}$ when calling the PC EOS. Therefore changing the interior composition in a WD model, such as including or excluding 22Ne, self-consistently changes the thermodynamics. ### B.2 Opacities MESA r12115 divides the radiative opacity tables into two temperature regimes, low ($T$ $\lesssim$ 104 K) and high ($T$ $\gtrsim$ 104 K). For the stellar evolution calculations from the pre-MS to a WD we use the Ferguson et al. (2005) low-temperature regions, and for the high-temperature regions we use the OPAL Type I opacities (Iglesias & Rogers, 1996), smoothly transitioning to the OPAL Type II opacities (Iglesias & Rogers, 1996) starting at the end of core H-burning. In our WD models, the radiative opacities are provided by the OPAL Type 2 tables, which are functions of the hydrogen mass fraction X, metal mass fraction Z, and the C/O-enhancements. Thus for the same temperature and density, our $X$($\mathrm{{}^{22}Ne}$) $\rightarrow$ $X$($\mathrm{{}^{14}N}$) replacement in Section 3.1 does not change the radiative opacities. Our $X$($\mathrm{{}^{14}N}$) $\rightarrow$ $X$($\mathrm{{}^{4}He}$) and $X$($\mathrm{{}^{22}Ne}$) with $\rightarrow$ $X$($\mathrm{{}^{12}C}$) replacements to generate zero-metallicity ab initio DB WD in Section 3.2 decreases Z in the He-dominated envelope and increases the C enhancement in the interior. Conversely, our doubling $X$($\mathrm{{}^{14}N}$) at the expense of $X$($\mathrm{{}^{4}He}$) and doubling $X$($\mathrm{{}^{22}Ne}$) at the expense of $X$($\mathrm{{}^{12}C}$) to generate a super-solar metallicity ab initio DB WD in Section 3.2 increases Z in the He-dominated envelope and decreases the C enhancement in the interior. Electron conduction opacities are from Cassisi et al. (2007), which are the relevant opacity in the WD interior. The conduction opacities are a function of the mean atomic number $\bar{Z}$, which MESA evaluates using the full composition vector in each cell. ### B.3 Nuclear Reaction Networks We use MESA’s mesa_49.net, a nuclear reaction network that follows 49 isotopes from $\mathrm{{}^{1}H}$ to $\mathrm{{}^{34}S}$, including $\mathrm{{}^{22}Ne}$. This impact of this reaction network on properties of CO WDs from Monte Carlo stellar models is discussed by Fields et al. (2016). All forward thermonuclear reaction rates are from the JINA reaclib version V2.2 2017-10-20 (Cyburt et al., 2010). Inverse rates are calculated directly from the forward rates (those with positive $Q$-value) using detailed balance, rather than using fitted rates. The nuclear partition functions used to calculate the inverse rates are from Rauscher & Thielemann (2000). Electron screening factors for both weak and strong thermonuclear reactions are from Chugunov et al. (2007) with plasma parameters from Itoh et al. (1979). All the weak rates are based (in order of precedence) on the tabulations of Langanke & Martínez-Pinedo (2000), Oda et al. (1994), and Fuller et al. (1985). Thermal neutrino energy losses are from Itoh et al. (1996). ### B.4 Mass Loss The implementations of mass loss in MESA r12115 are based on a number of observationally and theoretically motivated prescriptions, but uncertainties remain on line-driven and dust-driven winds (Dupree, 1986; Willson, 2000; Boulangier et al., 2019). We follow the mass loss settings used by the MIST isochrones (Dotter, 2016; Choi et al., 2016), with a combination of the Reimer mass loss prescription (Reimers, 1975) with $\eta$=0.1 on the Red Giant Branch and a Blöcker mass loss prescription (Bloecker, 1995a) with $\eta$=0.5 on the AGB. ### B.5 Rotation and Magnetic Fields MESA r12115 implements the inherently 3D process of rotation by making the 1D shellular approximation (Zahn, 1992; Meynet & Maeder, 1997), where the angular velocity is constant over isobars. The transport of angular momentum and material due to rotationally induced instabilities is followed using a diffusion approximation (e.g., Endal & Sofia, 1978; Pinsonneault et al., 1989; Heger et al., 2000; Maeder & Meynet, 2003, 2004; Suijs et al., 2008) for the dynamical shear instability, secular shear instability, Eddington-Sweet circulation, Goldreich-Schubert-Fricke instability, and Spruit-Tayler dynamo. See Heger et al. (2000) for a description of the different instabilities and diffusion coefficients. Magnetic fields are implemented in MESA using the formalism of Heger et al. (2005), where a magnetic torque due to a dynamo (Spruit, 2002) allows angular momentum to be transported inside the star. The azimuthal and radial components of the magnetic field are modeled as $B_{\phi}$ $\sim$ $r\sqrt{(4\pi\rho)}\omega_{A}$ and $B_{r}$ $\sim$ $B_{\phi}/(rk)$ respectively, where $r$ is the radial coordinate, $\omega_{A}$ the Alfvén frequency, and $k$ the wavenumber. These magnetic fields provide a torque $S$ = $B_{r}B_{\phi}/(4\pi)$ which slows down the rotation rate by decreasing the amount of differential rotation (Heger et al., 2005). We initialize rotation by imposing a solid body rotation law, $\Omega/\Omega_{{\rm crit}}$ = 1.9$\times$10-4, at the ZAMS. ZAMS is defined as where the nuclear burning luminosity is 99% of the total luminosity, and the rotation rate is normalized by the surface critical rotation rate $\Omega_{crit}$ = $\sqrt{(1-L/L_{{\rm edd}})cM/R^{3}}$, where $c$ is the speed of light, $M$ is the mass of the star, $R$ the stellar radius, $L$ the luminosity and $L_{{\rm edd}}$ the Eddington luminosity. The initial magnetic field is set to $B_{r}$ = $B_{\phi}$ = 0. Effects from rotationally induced mass loss are not included. ### B.6 Element Diffusion Element diffusion is implemented in MESA r12115 following Thoul et al. (1994), and described in Section 3 of Paxton et al. (2018). All isotopes in the reaction network are categorized into classes according to their atomic masses, each of which has a representative member whose properties are used to calculate the diffusion velocities. 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# Note on the Kato property of sectorial forms Ralph Chill R. Chill, Institut für Analysis, Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany<EMAIL_ADDRESS>and Sebastian Król S. Król, Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland<EMAIL_ADDRESS> ###### Abstract. We characterise the Kato property of a sectorial form $\mathfrak{a}$, defined on a Hilbert space ${V}$, with respect to a larger Hilbert space ${H}$ in terms of two bounded, selfadjoint operators ${T}$ and ${Q}$ determined by the imaginary part of $\mathfrak{a}$ and the embedding of ${V}$ into ${H}$, respectively. As a consequence, we show that if a bounded selfadjoint operator ${T}$ on a Hilbert space ${V}$ is in the Schatten class $S_{p}({V})$ ($p\geq 1$), then the associated form $\mathfrak{a}_{T}(\cdot,\cdot):=\langle(I+i{T})\cdot,\cdot\rangle_{V}$ has the Kato property with respect to every Hilbert space ${H}$ into which ${V}$ is densely and continuously embedded. This result is in a sense sharp. Another result says that if ${T}$ and ${Q}$ commute then the form $\mathfrak{a}$ with respect to ${H}$ possesses the Kato property. ###### 1991 Mathematics Subject Classification: 46D05 The second author was partially supported by the Alexander von Humboldt Foundation and NCN grant UMO-2017/27/B/ST1/00078 ## 1\. Introduction and preliminaries Let $\mathfrak{a}:{V}\times{V}\to\mathbb{C}$ be a bounded, sectorial, coercive, sesquilinear form on a complex Hilbert space ${V}$, which is densely and continuously embedded into a second Hilbert space ${H}$. Then $\mathfrak{a}$ induces a sectorial, invertible operator ${L}_{H}$ on ${H}$, and Kato’s square root problem is to know whether the domain of ${L}_{H}^{\frac{1}{2}}$ is equal to the form domain ${V}$. If this is the case, then we say that the couple $(\mathfrak{a},{H})$ has the Kato property. In this short note we characterise the Kato property of $(\mathfrak{a},{H})$ in terms of two bounded, selfadjoint operators ${T}$, ${Q}\in{\mathcal{L}}({V})$ determined by the imaginary part of $\mathfrak{a}$ and by the embedding of ${V}$ into ${H}$, respectively. We show that the Kato property of $(\mathfrak{a},{H})$ is equivalent to the similarity of ${Q}(I+i{T})^{-1}$ to an accretive operator, or to the similarity of $(I+{Q}+i{T})(I-{Q}+i{T})^{-1}$ to a contraction; see Theorem 2.1. The established link to different characterisations known in the literature provides an interesting connection between a variety of techniques and results mainly from operator theory of bounded operators, harmonic analysis, interpolation theory, or abstract evolution equations. In particular, we show that if a bounded, selfadjoint operator ${T}$ on a Hilbert space ${V}$ is in the Schatten class $S_{p}({V})$ for some $p\geq 1$, then the associated form $\mathfrak{a}_{T}(\cdot,\cdot):=\langle(I+i{T})\cdot,\cdot\rangle_{V}$ has the Kato property with respect to every Hilbert space ${H}$ into which ${V}$ is densely and continuously embedded; see Corollary 3.2. This result is in a sense sharp; see Proposition 4.1. On the other hand, if $\mathfrak{a}$ is an arbitrary bounded, sectorial, coercive form on ${V}$, then for every nonnegative, injective operator ${Q}\in{\mathcal{L}}({V})$ which is of the form $I+P$ with $P\in S_{p}({V})$ ($p\geq 1$), the pair $(\mathfrak{a},{H}_{Q})$ has the Kato property, where ${H}_{Q}$ is the completion of ${V}$ with respect to the inner product $\langle{Q}\cdot,\cdot\rangle_{V}$; see Corollary 3.4. Another straightforward consequence of Theorem 2.1 says that for every pair $({T},{Q})$ of selfadjoint, commuting operators (with ${Q}$ being nonnegative and injective), the form $\mathfrak{a}_{T}$ has the Kato property with respect to ${H}_{Q}$; see Corollary 2.2. We conclude this introduction with some preliminaries. ### 1.1. Forms Let $\mathfrak{a}$ be a bounded, sesquilinear form on a complex Hilbert space ${V}$. Denote by $\mathfrak{a}^{}$ the adjoint form of $\mathfrak{a}$, that is, $\mathfrak{a}^{}(u,v):=\overline{\mathfrak{a}(v,u)}$ for every $u,v\in{V}$. Then we call $\displaystyle\mathfrak{s}:=\operatorname{Re}\mathfrak{a}$ $\displaystyle:=(\mathfrak{a}+\mathfrak{a}^{})/2\quad\text{ and}$ $\displaystyle\mathfrak{t}:=\operatorname{Im}\mathfrak{a}$ $\displaystyle:=(\mathfrak{a}-\mathfrak{a}^{})/2i$ the real part and the imaginary part of $\mathfrak{a}$, respectively. Note that $\mathfrak{s}=\operatorname{Re}\mathfrak{a}$ and $\mathfrak{t}=\operatorname{Im}\mathfrak{a}$ are symmetric forms on ${V}$ and $\mathfrak{a}=\mathfrak{s}+i\mathfrak{t}$. Throughout the following, we assume that $\mathfrak{a}$ is coercive in the sense that $\operatorname{Re}\mathfrak{a}(u,u)\geq\eta\,\\|u\\|_{V}^{2}$ for some $\eta>0$ and every $u\in{V}$. This means that $\mathfrak{s}=\operatorname{Re}\mathfrak{a}$ is an equivalent inner product on ${V}$, and for simplicity we assume that $\mathfrak{s}$ is equal to the inner product on ${V}$: $\mathfrak{s}(u,v)=\langle u,v\rangle_{V}$ ($u$, $v\in{V}$). We shall also assume that $\mathfrak{a}$ is sectorial, that is, there exists $\beta\geq 0$ such that (1.1) $\|\operatorname{Im}\mathfrak{a}(u,u)\|\leq\beta\,\operatorname{Re}\mathfrak{a}(u,u),\quad u\in{V}.$ Let ${H}$ be a second Hilbert space such that ${V}$ is densely and continuously embedded into ${H}$, that is, there exists a bounded, injective, linear operator $j:{V}\to{H}$ with dense range. In the sequel we identify ${V}$ with $j({V})$. The embedding $j$ induces a bounded, linear embedding ${j}{{}^{\prime}}:{H}\to{V}^{\prime}$ (where ${V}^{\prime}$ is the space of bounded, antilinear functionals on ${V}$) given by ${j}{{}^{\prime}}(u):=\langle u,\cdot\rangle_{H},\quad u\in{H}.$ Then we have the following picture: ${V}\xhookrightarrow{j\ }{H}\xhookrightarrow{{j}{{}^{\prime}}}{V}^{\prime}\text{ and }{V}\xhookrightarrow{{j}{{}^{\prime}}j}{V}^{\prime}.$ We write also $J:={j}{{}^{\prime}}j$ for the linear embedding of ${V}$ into the dual space ${V}^{\prime}$. As usual, ${V}^{\prime}$ is equipped with the inner product $\langle u,v\rangle_{{V}^{\prime}}:=\langle I_{V}u,I_{V}v\rangle_{V}$, where $I_{V}:{V}^{\prime}\rightarrow{V}$ is the Riesz isomorphism. ### 1.2. Bounded operators associated with the pair $(\mathfrak{a},{H})$ Let $(\mathfrak{a},{H})$ be given as above. We define two associated bounded, linear operators on ${V}$. In fact, by the Riesz-Fréchet representation theorem, there exist two unique selfadjoint operators ${T}={T}_{\mathfrak{a}}$, ${Q}={Q}_{H}\in{\mathcal{L}}({V})$, such that $\displaystyle\mathfrak{t}(u,v)$ $\displaystyle=\langle{T}u,v\rangle_{V}\text{ and}$ (1.2) $\displaystyle\langle u,v\rangle_{H}$ $\displaystyle=\langle{Q}u,v\rangle_{V}\text{ for every }u,\,v\in{V},$ and hence, by recalling our convention that $\mathfrak{s}=\langle\cdot,\cdot\rangle_{V}$, (1.3) $\mathfrak{a}(u,v)=\langle(I+i{T})u,v\rangle_{V}\text{ for every }u,\,v\in{V}.$ Moreover, since $\langle\cdot,\cdot\rangle_{H}$ is an inner product, ${Q}$ is nonnegative and injective. In fact, ${Q}=j^{}j$, where $j^{}:{H}\to{V}$ is the Hilbert space adjoint of $j$. Conversely, every selfadjoint operator ${T}\in{\mathcal{L}}({V})$ induces via the equality (1.3) a bounded, sesquilinear, sectorial form $\mathfrak{a}$ on ${V}$ for which $\operatorname{Re}\mathfrak{a}$ coincides with the inner product $\langle\cdot,\cdot\rangle_{V}$, and for which $\operatorname{Im}\mathfrak{a}$ is represented by ${T}$. Similarly, every nonnegative, injective operator ${Q}\in{\mathcal{L}}({V})$ induces via the equality (1.2) an inner product $\langle\cdot,\cdot\rangle_{H}:=\langle{Q}\cdot,\cdot\rangle_{V}$ on ${V}$, and thus, by taking the completion, a Hilbert space ${H}_{Q}$ into which ${V}$ is densely and continuously embedded. We say that the pair of operators $({T},{Q})$ is associated with the pair $(\mathfrak{a},{H})$, or, conversely, the pair $(\mathfrak{a},{H})$ is associated with the pair $({T},{Q})$. ### 1.3. Unbounded operators associated with the pair $(\mathfrak{a},{H})$ Given a pair $(\mathfrak{a},{H})$ as above, we define also associated closed, linear operators on ${H}$ and ${V}^{\prime}$. First, we denote by ${L}_{H}:={L}_{\mathfrak{a},{H}}$ the, in general, unbounded operator on ${H}$ given by $\displaystyle{\mathcal{D}}({L}_{H})$ $\displaystyle:=\\{u\in j({V}):\exists f\in{H}\,\forall v\in{V}\,:\,\mathfrak{a}(j^{-1}u,v)=\langle f,jv\rangle_{H}\\},$ $\displaystyle{L}_{H}u$ $\displaystyle:=f.$ Second, we denote by ${L}_{{V}^{\prime}}:={L}_{\mathfrak{a},{V}^{\prime}}$ the operator on ${V}^{\prime}$ which is given by $\displaystyle{\mathcal{D}}({L}_{{V}^{\prime}})$ $\displaystyle:=({j}{{}^{\prime}}j)({V})=J({V}),$ $\displaystyle{L}_{{V}^{\prime}}u$ $\displaystyle:=\mathfrak{a}(J^{-1}u,\cdot).$ In a similar way we define the operators ${L}_{\mathfrak{s},{H}}$ and ${L}_{\mathfrak{s},{V}^{\prime}}$ associated with the real part $\mathfrak{s}=\operatorname{Re}\mathfrak{a}$. Recall that a closed, linear operator $(A,{\mathcal{D}}(A))$ on a Banach space $X$ is called sectorial of angle $\theta\in(0,\pi)$ if $\sigma(A)\subseteq\Sigma_{\theta}:=\\{z\in\mathbb{C}:\|\arg z\|\leq\theta\\},$ and if for every $\theta^{\prime}\in(\theta,\pi)$ one has $\sup_{z\not\in\Sigma_{\theta^{\prime}}}\\|zR(z,A)\\|<\infty.$ We simply say that $A$ is sectorial if it is sectorial for some angle $\theta\in(0,\pi)$. The numerical range of a closed, linear operator $(A,{\mathcal{D}}(A))$ on a Hilbert space ${H}$ is the set $W(A):=\\{\langle Au,u\rangle_{H}:u\in{\mathcal{D}}(A),\,\\|u\\|_{H}=1\\}.$ The operator $A$ is said to be $\theta$-accretive for $\theta\in(0,\pi)$, if $W(A)\subseteq\Sigma_{\theta}$, that is, if $\|\arg\langle Au,u\rangle_{H}\|\leq\theta\text{ for every }u\in{\mathcal{D}}(A).$ If $\theta=\frac{\pi}{2}$, that is, $\operatorname{Re}\langle Au,u\rangle_{H}\geq 0$ for every $u\in{\mathcal{D}}(A)$, we say that $A$ is accretive. Both operators ${L}_{H}$ and ${L}_{{V}^{\prime}}$ defined above are sectorial for some angle $\theta\in(0,\frac{\pi}{2})$. Since $\mathfrak{a}$ is assumed to be coercive, we have $0\in\rho({L}_{H})$ and $0\in\rho({L}_{{V}^{\prime}})$, that is, both ${L}_{H}$ and ${L}_{{V}^{\prime}}$ are isomorphisms from their respective domains onto ${H}$ and ${V}^{\prime}$, respectively; see e.g. [14, Theorem 2.1, p. 58]. It is easy to check that the numerical range of ${L}_{H}$ is contained in the sector $\Sigma_{\theta}$ with $\theta=\arctan\beta$ and in particular ${L}_{H}$ is $\theta$-accretive. As a consequence, by [8, Theorem 11.13], ${L}_{H}$ admits a bounded $H^{\infty}$ functional calculus. We refer the reader to [8] or [4] for the background on fractional powers and $H^{\infty}$ functional calculus of sectorial operators. ## 2\. Characterisations of the Kato property Let $(\mathfrak{a},{H})$ be as above, that is, $\mathfrak{a}$ is a bounded, sectorial, coercive, sesquilinear form on a Hilbert space ${V}$ which embeds densely and continuously into a second Hilbert space ${H}$. Let ${L}_{H}={L}_{\mathfrak{a},{H}}$ be defined as above. We say that the couple $(\mathfrak{a},{H})$ has Kato’s property if ${\mathcal{D}}({L}_{H}^{1/2})={V}$. By the Closed Graph Theorem, if $(\mathfrak{a},{H})$ has the Kato property, then the norms $\\|{L}_{H}^{1/2}\cdot\\|_{H}$ and $\\|\cdot\\|_{{V}}$ are equivalent on ${V}$. According to Kato [6] and Lions [10], the coincidence of any two of the spaces ${\mathcal{D}}({L}_{H}^{1/2})$, ${\mathcal{D}}({L}_{H}^{1/2})$ and ${V}$ implies the coincidence of the three. Moreover, by Subsection 1.2, it is natural to say that a pair $({T},{Q})$ of selfadjoint, bounded operators on a Hilbert space ${V}$, with ${Q}$ being nonnegative and injective, has the Kato property, if the associated pair $(\mathfrak{a}_{T},{H}_{Q})$ has the Kato property, where $\mathfrak{a}_{T}(u,v)=\langle(I+i{T})u,v\rangle_{V}$ ($u$, $v\in{V}$) and ${H}_{Q}$ is the completion of $({V},\langle{Q}\cdot,\cdot\rangle_{V})$. The main result of this section is the following characterisation of the Kato property of $(\mathfrak{a},{H})$ in terms of the associated pair of bounded operators $({T},{Q})$. ###### Theorem 2.1. Let $({T},{Q})$ be the pair of operators associated with $(\mathfrak{a},{H})$ as above. Then the following assertions are equivalent: (i) $(\mathfrak{a},{H})$ has the Kato property * (ii) There exists a positive operator ${S}$ on ${V}$ such that $\langle{Q}{S}(I+i{T})u,u\rangle_{V}\in\Sigma_{\theta}\text{ for every }u\in{V}\text{ and some }\theta<\frac{\pi}{2}.$ * (ii’) There exists a positive operator ${S}$ on ${V}$ such that $\operatorname{Re}\,\langle{Q}{S}(I+i{T})u,u\rangle_{V}\geq 0\text{ for every }u\in{V}.$ * (iii) There exists a positive operator ${S}$ on ${V}$ such that $\langle{S}(I-i{T})^{-1}{Q}u,u\rangle_{V}\in\Sigma_{\theta}\text{ for every }u\in{V}\text{ and some }\theta<\frac{\pi}{2},$ that is, $(I-i{T})^{-1}{Q}$ is similar to a $\theta$-accretive operator, or, equivalently, the operator $(I-i{T})^{-1}{Q}$ has a bounded $H^{\infty}(\Sigma_{\theta})$ functional calculus. * (iv) There exists a positive operator ${S}$ on ${V}$ such that $\langle{S}{Q}(I+i{T})^{-1}u,u\rangle_{V}\in\Sigma_{\theta}\text{ for every }u\in{V}\text{ and some }\theta<\frac{\pi}{2},$ that is, ${Q}(I+i{T})^{-1}$ is similar to a $\theta$-accretive operator, or, equivalently, the operator ${Q}(I+i{T})^{-1}$ has a bounded $H^{\infty}(\Sigma_{\theta})$ functional calculus. * (v) The operator $(I-{Q}+i{T})(I+{Q}+i{T})^{-1}$ is polynomially bounded. * (vi) The operator $(I-{Q}+i{T})(I+{Q}+i{T})^{-1}$ is similar to a contraction. Recall that, if ${T}$, ${Q}\in{\mathcal{L}}({V})$ are selfadjoint operators, then ${Q}{T}$ is selfadjoint if and only if ${T}$ and ${Q}$ commute, or if and only if $\langle{Q}{T}u,u\rangle_{V}\in\mathbb{R}$ for every $u\in{V}$. Therefore, the above Theorem 2.1(ii’) gives the following sufficient condition for $(\mathfrak{a},{H})$ to have the Kato property. ###### Corollary 2.2. If ${T}$ and ${Q}$ commute, then $(\mathfrak{a},{H})$ has the Kato property. We start with auxiliary results on the operators appearing in Theorem 2.1. ###### Lemma 2.3. Let ${T}$ and ${Q}$ be selfadjoint, bounded operators on a Hilbert space ${V}$. Assume that ${Q}$ is nonnegative. Then, the operator $A:={Q}(I+i{T})^{-1}\in{\mathcal{L}}({V})$ is sectorial of angle $\theta<\frac{\pi}{2}$. ###### Proof. By a standard argument based on the Neumann series extension it is sufficient to show that $\sup_{\operatorname{Re}z\leq 0}\\|zR(z,A)\\|<\infty$. Note that for every $z\in\mathbb{C}$ with $\operatorname{Re}z\leq 0$ and every $u\in{V}$ we have $\displaystyle\langle(z+iz{T}-{Q})u,u\rangle_{V}$ $\displaystyle=(\operatorname{Re}z\,\\|u\\|_{V}^{2}-\operatorname{Im}z\,\langle{T}u,u\rangle_{V}-\langle{Q}u,u\rangle)$ $\displaystyle\phantom{=}+i\,(\operatorname{Im}z\,\\|u\\|_{V}^{2}+\operatorname{Re}z\,\langle{T}u,u\rangle_{V}),$ and therefore $\displaystyle\|\langle(z+iz{T}-{Q})u,u\rangle_{V}\|^{2}$ $\displaystyle=\|z\|^{2}\,\\|u\\|_{V}^{4}-2\operatorname{Re}z\,\\|u\\|_{V}^{2}\langle{Q}u,u\rangle_{V}+$ $\displaystyle\phantom{=}+(\|z\|^{2}\,\langle{T}u,u\rangle^{2}_{V}+2\,\operatorname{Im}z\,\langle{T}u,u\rangle_{V}\,\langle{Q}u,u\rangle_{V}+\langle{Q}u,u\rangle_{V}^{2})$ $\displaystyle\geq\|z\|^{2}\,\\|u\\|_{V}^{4},$ or, by the Cauchy-Schwarz inequality, $\\|(z+iz{T}-{Q})u\\|_{V}\geq\|z\|\,\\|u\\|_{V}.$ This inequality implies that $z+iz{T}-{Q}$ is injective and has closed range. A duality argument, using similar estimates as above, shows that $z+iz{T}-{Q}$ has dense range, and therefore $z+iz{T}-{Q}$ is invertible for every $z\in\mathbb{C}$ with $\operatorname{Re}z\leq 0$. Moreover, the above inequality shows that $\sup_{\operatorname{Re}z\leq 0}\\|z(z+iz{T}-{Q})^{-1}\\|\leq 1.$ As a consequence, $z-A=(z+iz{T}-{Q})(I+i{T})^{-1}$ is invertible for every $z\in\mathbb{C}$ with $\operatorname{Re}z\leq 0$ and $\sup_{\operatorname{Re}z\leq 0}\\|zR(z,A)\\|=\sup_{\operatorname{Re}z\leq 0}\\|(I+i{T})\,z(z+iz{T}-{Q})^{-1}\\|\leq\\|I+i{T}\\|.$ ∎ Let $A\in{\mathcal{L}}({V})$ be a bounded, sectorial operator of angle $\theta\in(0,\frac{\pi}{2})$, and let ${C}:=(I-A)(I+A)^{-1}$ be its Cayley transform. Then the equality $\displaystyle(z-1)(z-{C})^{-1}=\frac{z-1}{z+1}\,\left(\frac{z-1}{z+1}+A\right)^{-1}\,(I+A)$ shows that ${C}$ is a Ritt operator, that is, $\sigma({C})\subseteq{\mathbb{D}}\cup\\{1\\}$ (where $\mathbb{D}\subset\mathbb{C}$ is the open unit disk) and $\sup_{\|z\|>1}\\|(z-1)R(z,{C})\\|<\infty.$ From this and the preceding lemma, we obtain the following statement. ###### Lemma 2.4. The Cayley transform ${C}=(I-A)(I+A)^{-1}=(I-{Q}+i{T})(I+{Q}+i{T})^{-1}$ of the operator $A={Q}(I+i{T})^{-1}$ is a Ritt operator. Recall that a bounded operator ${C}$ on a Hilbert space ${V}$ is a Ritt operator if and only if it is power bounded and $\sup_{n\in\mathbb{N}}n\\|{C}^{n}-{C}^{n+1}\\|<\infty$; see [12]. Furthermore, a bounded operator ${C}$ on a Hilbert space is polynomially bounded if there exists a constant $M\geq 0$ such that for every polynomial $p$ one has $\\|p({C})\\|\leq M\,\sup_{\|z\|\leq 1}\|p(z)\|.$ The proof of Theorem 2.1 is a consequence of the characterisation of the Kato property by means of the boundedness of the $H^{\infty}$ functional calculus for the operator ${L}_{{V}^{\prime}}={L}_{\mathfrak{a},{V}^{\prime}}$ given by Arendt in [1, Theorem 5.5.2, p.45]. ###### Lemma 2.5. Let ${L}_{{V}^{\prime}}={L}_{\mathfrak{a},{V}^{\prime}}$ be the operator associated with $(\mathfrak{a},{H})$ as above. Then the following assertions are equivalent: * (i) $(\mathfrak{a},{H})$ has the Kato property. * (ii) ${L}_{{V}^{\prime}}$ has a bounded $H^{\infty}$ functional calculus. Moreover, if (i) or (ii) holds, then ${L}_{{V}^{\prime}}$ has a bounded $H^{\infty}(\Sigma_{\theta})$ functional calculus for every $\theta>\arctan\beta$ with $\beta$ as in (1.1). For the convenience of the reader we recall the proof of this result using our notation and with slight modifications. ###### Proof. First of all, note that the operator ${L}_{H}$ can be expressed as the operator $j^{\prime-1}{L}_{{V}^{\prime}}j^{\prime}$ with domain $\\{u\in{H}:j^{\prime}u\in{\mathcal{D}}({L}_{{V}^{\prime}}\textrm{ and }{L}_{{V}^{\prime}}j^{\prime}u\in j^{\prime}({H})\\}$. Then $({\lambda}-{L}_{H})^{-1}=j^{\prime-1}({\lambda}-{L}_{{V}^{\prime}})^{-1}j^{\prime}$ for every ${\lambda}\notin\Sigma_{\theta}$, and by the definition of the square roots via contour integrals, ${L}_{H}^{-\frac{1}{2}}=j^{\prime-1}{L}_{{V}^{\prime}}^{-\frac{1}{2}}j^{\prime}.$ (i)$\Rightarrow$(ii) Therefore, if ${\mathcal{D}}({L}_{H}^{\frac{1}{2}})={\mathcal{R}}({L}_{H}^{-\frac{1}{2}})=j({V})$, then $\displaystyle{\mathcal{D}}({L}_{{V}^{\prime}}^{\frac{1}{2}})$ $\displaystyle:={L}_{{V}^{\prime}}^{-\frac{1}{2}}({V}^{\prime})={L}_{{V}^{\prime}}^{-\frac{1}{2}}{L}_{{V}^{\prime}}(j^{\prime}j({V}))$ $\displaystyle={L}_{{V}^{\prime}}^{-\frac{1}{2}}{L}_{{V}^{\prime}}(j^{\prime}{L}_{H}^{-\frac{1}{2}}({H}))={L}_{{V}^{\prime}}^{-\frac{1}{2}}{L}_{{V}^{\prime}}({L}_{{V}^{\prime}}^{-\frac{1}{2}}j^{\prime}({H}))$ $\displaystyle=j^{\prime}({H}),$ where the last equality follows from ${L}^{\frac{1}{2}}_{{V}^{\prime}}{L}_{{V}^{\prime}}^{\frac{1}{2}}={L}_{{V}^{\prime}}$; see e.g. [8, Theorem 15.15, p.289]. By [14, Corollary 2.3, p.113], $j^{\prime}({H})=[{V}^{\prime},{\mathcal{D}}({L}_{\mathfrak{s},{V}^{\prime}})]_{\frac{1}{2}}$, where on ${\mathcal{D}}({L}_{\mathfrak{s},{V}^{\prime}})=J({V})$ we consider the graph norm of ${L}_{\mathfrak{s},{V}^{\prime}}$, that is, $\\|{L}_{\mathfrak{s},{V}^{\prime}}\cdot\\|_{{V}^{\prime}}+\\|\cdot\\|_{{V}^{\prime}}$. Since for $v\in{V}$ we have $I_{V}{L}_{{V}^{\prime}}Jv=(I+i{T})v\quad\textrm{ and }\quad I_{V}{L}_{\mathfrak{s},{V}^{\prime}}Jv=v,$ hence $\\|{L}_{{V}^{\prime}}Jv\\|_{{V}^{\prime}}=\\|(I+i{T})v\\|_{V}\quad\textrm{and }\quad\\|{L}_{{V}^{\prime}}Jv\\|_{{V}^{\prime}}=\\|v\\|_{V}.$ Consequently, the invertibility of $I+i{T}$ implies, that the graph norm of ${L}_{\mathfrak{s},{V}^{\prime}}$ is equivalent to the graph norm of ${L}_{{V}^{\prime}}={L}_{\mathfrak{a},{V}^{\prime}}$ on ${\mathcal{D}}({L}_{{V}^{\prime}})=J({V})$. Therefore, we get that $[{V}^{\prime},{\mathcal{D}}({L}_{{V}^{\prime}})]_{\frac{1}{2}}={\mathcal{D}}({L}_{{V}^{\prime}}^{\frac{1}{2}}).$ Hence, by [14, Theorem 16.3, p.532], ${L}_{{V}^{\prime}}$ has a bounded $H^{\infty}$ functional calculus. (ii)$\Rightarrow$(i) On the other hand, if ${L}_{{V}^{\prime}}$ has a bounded $H^{\infty}$ functional calculus, then as above we get ${\mathcal{D}}({L}_{{V}^{\prime}}^{\frac{1}{2}})=[{V}^{\prime},{\mathcal{D}}({L}_{{V}^{\prime}})]_{\frac{1}{2}}=j^{\prime}({H})$. Therefore, $\displaystyle{\mathcal{D}}({L}_{{H}}^{\frac{1}{2}})$ $\displaystyle:={L}_{{H}}^{-\frac{1}{2}}j^{\prime-1}(j^{\prime}({H}))={L}_{{H}}^{-\frac{1}{2}}j^{\prime-1}{L}_{{V}^{\prime}}^{-\frac{1}{2}}({V}^{\prime})$ $\displaystyle=j^{\prime-1}j^{\prime}{L}_{{H}}^{-\frac{1}{2}}j^{\prime-1}{L}_{{V}^{\prime}}^{-\frac{1}{2}}({V}^{\prime})=j^{\prime-1}{L}_{{V}^{\prime}}^{-\frac{1}{2}}{L}_{{V}^{\prime}}^{-\frac{1}{2}}({V}^{\prime})=j^{\prime-1}j^{\prime}j({V})$ $\displaystyle=j({V}).$ For the last statement about the angle of the $H^{\infty}$ functional calculus first note, that ${\mathcal{D}}({L}_{{V}^{\prime}}^{\frac{1}{2}})=[{V}^{\prime},{\mathcal{D}}({L}_{{V}^{\prime}})]_{\frac{1}{2}}=j^{\prime}({H})$ yields $L_{H}=j^{\prime-1}L_{{V}^{\prime}}^{-\frac{1}{2}}L_{{V}^{\prime}}L_{{V}^{\prime}}^{\frac{1}{2}}j^{\prime}.$ Moreover, by the Closed Graph Theorem, the operator $j^{\prime}$ is an isomorphism from $H$ onto ${\mathcal{D}}({L}_{{V}^{\prime}}^{\frac{1}{2}})=j^{\prime}(H)$ equipped with the graph norm. Since the operator $L_{H}$ is $(\arctan\beta)$-accretive, therefore it is sectorial of angle $\arctan\beta$, and consequently the operator $L_{{V}^{\prime}}$, too. Finally, for example, by [14, Theorem 16.3, p.532] (cf. [14, Remark 16.2, p.536]), $L_{{V}^{\prime}}$ has a bounded $H^{\infty}$ functional calculus in any sectorial domain $\Sigma_{\theta}$ with $\theta>\arctan\beta$. This completes the proof. ∎ ###### Proof of Theorem 2.1. Assume that $(\mathfrak{a},{H})$ has the Kato property. By Lemma 2.5, ${L}_{{V}^{\prime}}$ has a bounded $H^{\infty}(\Sigma_{\theta})$ functional calculus for every $\theta>\arctan\beta$. Fix $\theta\in(\arctan\beta,\frac{\pi}{2})$. By the characterisation of the boundedness of the $H^{\infty}$ functional calculus, [8, Theorem 11.13, p.229], ${L}_{{V}^{\prime}}$ is $\theta$-accretive with respect to an equivalent inner product $\langle\cdot,\cdot\rangle_{\theta}$ on ${V}^{\prime}$. Let $\widetilde{S}\in{\mathcal{L}}({V}^{\prime})$ be the positive operator such that $\langle\cdot,\cdot\rangle_{\theta}=\langle\widetilde{{S}}\cdot,\cdot\rangle_{{V}^{\prime}}$. Then ${S}:=I_{V}\widetilde{{S}}I_{V}^{-1}\in{\mathcal{L}}({V})$ is a positive operator on ${V}$. First, note that $I_{V}{L}_{{V}^{\prime}}Jv=(I+i{T})v$ and $I_{V}Jv={Q}v$ for every $v\in{V}$. Then, $\displaystyle\langle{L}_{{V}^{\prime}}Jv,Jv\rangle_{\theta}$ $\displaystyle=\langle\widetilde{S}{L}_{{V}^{\prime}}Jv,Jv\rangle_{{V}^{\prime}}$ $\displaystyle=\langle I_{V}\widetilde{S}I^{-1}_{V}I_{V}{L}_{{V}^{\prime}}J^{-1}v,I_{V}Jv\rangle_{V}$ $\displaystyle=\langle I_{V}{L}_{{V}^{\prime}}Jv,{S}I_{V}Jv\rangle_{V}$ $\displaystyle=\langle I_{V}Jv,{S}(I+i{T})v\rangle_{V}$ $\displaystyle=\langle(I+i{T})v,{S}{Q}v\rangle_{V}$ $\displaystyle=\langle{Q}{S}(I+i{T})v,v\rangle_{V}$ for every $v\in{V}$. Therefore, the operator ${Q}{S}(I+i{T})$ is $\theta$-accretive with respect to $\langle\cdot,\cdot\rangle_{V}$. Therefore, (i)$\Rightarrow$(ii)$\Rightarrow$(ii′). The implication (ii′)$\Rightarrow$(i) follows from a similar argument. The equivalences (ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv) follow from the following chain of equivalences which holds for every positive operator ${S}\in{\mathcal{L}}({V})$ and $\theta\in(0,\frac{\pi}{2}]$: $\displaystyle{Q}{S}(I+i{T})\text{ is }\theta\text{-accretive}$ $\displaystyle\Leftrightarrow\quad$ $\displaystyle\forall u\in{V}:\,\langle{Q}{S}(I+i{T})u,u\rangle_{V}\in\Sigma_{\theta}$ $\displaystyle\Leftrightarrow\quad$ $\displaystyle\forall u\in{V}:\,\langle{Q}{S}u,(I+i{T})^{-1}u\rangle_{V}\in\Sigma_{\theta}$ $\displaystyle\Leftrightarrow\quad$ $\displaystyle\forall u\in{V}:\,\langle u,{S}^{\frac{1}{2}}{Q}(I+i{T})^{-1}{S}^{-\frac{1}{2}}u\rangle_{V}\in\Sigma_{\theta}$ $\displaystyle\Leftrightarrow\quad$ $\displaystyle{S}^{\frac{1}{2}}(I-i{T})^{-1}{Q}{S}^{-\frac{1}{2}}\text{ is }\theta\text{-accretive}$ $\displaystyle\Leftrightarrow\quad$ $\displaystyle{S}^{\frac{1}{2}}{Q}(I+i{T})^{-1}{S}^{-\frac{1}{2}}\text{ is }\theta\text{-accretive}.$ For (iv)$\Leftrightarrow$(v), set $A:={Q}(I-i{T})^{-1}$, and note that its Cayley transform is given by ${C}:=\phi(A)=(I-{Q}+i{T})(I+{Q}+i{T})^{-1},$ where $\phi$ is the conformal map $\phi(z):=(1-z)(1+z)^{-1}$ from $\Sigma_{\frac{\pi}{2}}$ onto $\\{\|z\|<1\\}$. Moreover, for every polymomial $p$ we have $p({C})=(p\circ\phi)(A)\quad\textrm{and }\quad\sup_{z\in\Sigma_{\theta}}\|(p\circ\phi)(z)\|\leq\sup_{\|z\|<1}\|p(z)\|.$ Therefore, the boundedness of the $H^{\infty}(\Sigma_{\theta})$ functional calculus of $A$ with $\theta\leq\frac{\pi}{2}$ yields the polynomial boundedness of its Cayley transform ${C}$. For the converse, by Runge’s theorem, it is easy to see that $A$ has a bounded $\mathcal{R}(\Sigma_{\frac{\pi}{2}})$ functional calculus; here $\mathcal{R}(\Sigma_{\frac{\pi}{2}})$ stands for the algebra of rational functions with poles outside $\Sigma_{\frac{\pi}{2}}$. Then, the boundedness of the $H^{\infty}(\Sigma_{\frac{\pi}{2}})$ functional calculus follows again by an approximation argument and McIntosh’s convergence theorem [11, Section 5, Theorem]; see also [4, Proposition 3.13, p.66]. Since $A$ is $\theta$-sectorial for some $\theta<\frac{\pi}{2}$, [8, Theorem 11.13] gives (iv). Finally, for (v)$\Rightarrow$(vi), since the Cayley transform ${C}=\phi(A)$ is a Ritt operator, see Lemma 2.4, by [9, Theorem 5.1], it is similar to a contraction. The converse is a consequence of the von Neumann inequality. ∎ ###### Remark 2.6. (a) By [8, Theorem 11.13 H7)] one can show that if $(\mathfrak{a},{H})$ has the Kato property, then (iv) in Theorem 2.1 holds with $S:=\int_{\Sigma_{\pi-\theta}}A^{}e^{zA^{}}Ae^{zA}\textrm{ d}z=\int_{\Sigma_{\pi-\theta}}\|Ae^{zA}\|^{2}\textrm{ d}z,$ where $A:={Q}(I-i{T})^{-1}$ and the integral exists in the weak operator topology. (b) Note that in the case when the operator ${Q}$ is invertible on ${V}$, or equivalently, the inner products on ${H}$ and ${V}$ are equivalent, then $(\mathfrak{a},{H})$ has the Kato property simply because ${L}_{H}\in{\mathcal{L}}({H})={\mathcal{L}}({V})$. It should be pointed out, that in this case the similarity to a contraction of the operator (2.1) ${C}=(I-{Q}+i{T})(I+{Q}+i{T})^{-1}=({T}-i(I-{Q}))({T}-i(I+{Q}))^{-1},$ which is stated in Theorem 2.1 (vi), can be proved in a straightforward way. Indeed, in [3, Theorem 1] Fan proved that an operator ${C}\in{\mathcal{L}}({V})$ with $1\in\rho({C})$ is similar to a contraction if and only if it can be expressed in the form $(E-iF)(E-iG)^{-1},$ for some selfadjoint operators $E$, $F$, $G\in{\mathcal{L}}({V})$ such that $G+F$ and $G-F$ are positive with $0\in\rho(G-F)$. Therefore, in the case of ${Q}$ being invertible, the above stated expression of the operator ${C}$, that is, (2.1), satisfies these conditions. ## 3\. Kato property and triangular operators Recall that a bounded operator $\Delta$ on a Hilbert space ${V}$ is triangular if there exists a constant $M\geq 0$ such that (3.1) $\left\|\sum_{j=1}^{n}\sum_{k=1}^{j}\langle\Delta u_{j},v_{k}\rangle_{V}\right\|\leq M\sup_{\|a_{j}\|=1}\left\\|\sum_{j=1}^{n}a_{j}u_{j}\right\\|\sup_{\|a_{j}\|=1}\left\\|\sum_{j=1}^{n}a_{j}v_{j}\right\\|.$ for every $n\in\mathbb{N}$ and every $u_{1}$, $\dots$, $u_{n}$, $v_{1}$, $\dots$, $v_{n}\in{V}$. By a theorem of Kalton [5, Theorem 5.5], an operator $\Delta$ on ${V}$ is triangular if and only if $\sum_{n=1}^{\infty}\frac{s_{n}(\Delta)}{n+1}<\infty$, where $(s_{n}(\Delta))_{n\in\mathbb{N}}$ is the sequence of singular values of $\Delta$. Therefore, the Schatten-von Neumann classes are included in the class of triangular operators. We refer the reader to [5, Section 5] for basic properties of triangular operators. One interest in the class of triangular operators stems from the following perturbation theorem by Kalton [5, Theorem 7.7]. ###### Lemma 3.1. Let $A$ and ${B}$ be two sectorial operators on a Hilbert space ${H}$. Assume that ${B}$ has a bounded $H^{\infty}$ functional calculus, and that $A=(I+\Delta){B}$ for some triangular operator $\Delta$. Then $A$ has a bounded $H^{\infty}$ functional calculus, too. Combining this result with Theorem 2.1, we show that the Kato property of $(\mathfrak{a},{H})$ is preserved under certain triangular perturbations of the imaginary part of $\mathfrak{a}$, and in particular, that for every bounded, selfadjoint operators ${T}$ and ${Q}$ on a Hilbert space ${V}$ such that ${T}$ is triangular and ${Q}$ is nonnegative and injective, the pair $({T},{Q})$ has the Kato property, that is, ${Q}(I-i{T})^{-1}$ is similar to an accretive operator on ${V}$. ###### Corollary 3.2. Let $\mathfrak{a}$ and $\mathfrak{b}$ be two sectorial forms on ${V}$ with the same real parts, that is, $\operatorname{Re}\mathfrak{a}=\operatorname{Re}\mathfrak{b}$. Let the imaginary parts $\mathfrak{t}_{\mathfrak{a}}$ and $\mathfrak{t}_{\mathfrak{b}}$ of $\mathfrak{a}$ and $\mathfrak{b}$ be determined by selfadjoint operators ${T}_{\mathfrak{a}}$, ${T}_{\mathfrak{b}}\in{\mathcal{L}}({V})$, respectively. Assume that $(\mathfrak{b},{H})$ has the Kato property, and that ${T}_{\mathfrak{a}}-{T}_{\mathfrak{b}}$ is a triangular operator. Then $(\mathfrak{a},{H})$ has the Kato property, too. In particular, if ${T}_{\mathfrak{a}}$ is a triangular operator, then $(\mathfrak{a},{H})$ has the Kato property for every Hilbert space ${H}$ into which ${V}$ is densely and continuously embedded. ###### Proof of Corollary 3.2. Note that by the second resolvent equation we get $(I-i{T}_{\mathfrak{a}})^{-1}{Q}-(I-i{T}_{\mathfrak{b}})^{-1}{Q}=i(I-i{T}_{\mathfrak{a}})^{-1}({T}_{\mathfrak{a}}-{T}_{\mathfrak{b}})(I+i{T}_{\mathfrak{b}})^{-1}{Q}.$ Therefore, since the operator $i(I-i{T}_{\mathfrak{a}})^{-1}({T}_{\mathfrak{a}}-{T}_{\mathfrak{b}})$ is triangular, the claim follows from Lemma 2.3, Lemma 3.1, and Theorem 2.1 (iii). Alternatively, note that Arendt’s result, Lemma 2.5, used in the proof of Theorem 2.1, can be directly applied to get Corollary 3.2. Indeed, set $\Delta:={L}_{\mathfrak{a},{V}^{\prime}}{L}_{\mathfrak{b},{V}^{\prime}}^{-1}-I,$ so that ${L}_{\mathfrak{a},{V}^{\prime}}=(I+\Delta){L}_{\mathfrak{b},{V}^{\prime}}$. By our assumption, ${L}_{\mathfrak{b},{V}^{\prime}}$ admits a bounded $H^{\infty}$ functional calculus. We also recall that both ${L}_{\mathfrak{a},{V}^{\prime}}$ and ${L}_{\mathfrak{b},{V}^{\prime}}$ are sectorial operators. By Lemma 3.1, it is thus sufficient to show that the operator $\Delta$ is triangular. Since $\operatorname{Re}\mathfrak{a}=\operatorname{Re}\mathfrak{b}$, we get $\Delta=i\,({L}_{\operatorname{Im}\mathfrak{a},{V}^{\prime}}-{L}_{\operatorname{Im}\mathfrak{b},{V}^{\prime}})\,{L}_{\mathfrak{b},{V}^{\prime}}^{-1}$. Fix $u$ and $v$ in ${V}^{\prime}$. Then $\displaystyle\langle\Delta u,v\rangle_{{V}^{\prime}}$ $\displaystyle=\langle I_{V}\Delta u,I_{V}v\rangle_{V}$ $\displaystyle=i[({L}_{\operatorname{Im}\mathfrak{a},{V}^{\prime}}-{L}_{\operatorname{Im}\mathfrak{b},{V}^{\prime}}){L}_{\mathfrak{b},{V}^{\prime}}^{-1}u](I_{V}v)$ $\displaystyle=i\,\operatorname{Im}\mathfrak{a}\bigl{(}({L}_{\mathfrak{b},{V}^{\prime}}J)^{-1}u,\,I_{V}v\bigr{)}-i\,\operatorname{Im}\mathfrak{b}\bigl{(}({L}_{\mathfrak{b},{V}^{\prime}}J)^{-1}u,\,I_{V}v\bigr{)}$ $\displaystyle=i\langle({L}_{\mathfrak{b},{V}^{\prime}}J)^{-1}u,({T}_{\mathfrak{a}}-{T}_{\mathfrak{b}})I_{V}v\rangle_{V}.$ Since ${L}_{\mathfrak{b},{V}^{\prime}}J$ is an isomorphism from ${V}$ onto ${V}^{\prime}$, the triangularity of $\Delta$ is equivalent to the triangularity of ${T}_{\mathfrak{a}}-{T}_{\mathfrak{b}}$. For the proof of the second statement, it is sufficient to apply the one just proved for a symmetric form, that is, $\mathfrak{b}$ with $\operatorname{Im}\mathfrak{b}=0$. ∎ In an analoguous way, by combining Lemma 3.1 with Theorem 2.1 (iii), we get the following perturbation result for the real parts of forms. ###### Corollary 3.3. Let $\mathfrak{a}$ and $\mathfrak{b}$ be two sectorial forms on a space ${V}$ with the same imaginary parts, that is, $\mathfrak{t}_{\mathfrak{a}}=\mathfrak{t}_{\mathfrak{b}}$, and equivalent real parts $\mathfrak{s}_{a}$ and $\mathfrak{s}_{b}$. Let ${S}\in{\mathcal{L}}({V})$ be such that $\mathfrak{s}_{\mathfrak{a}}({S}u,v)=\mathfrak{s}_{\mathfrak{b}}(u.v)$. If ${S}-I$ is triangular and $(\mathfrak{b},{H})$ has the Kato property, then $(\mathfrak{a},{H})$ has the Kato property, too. ###### Proof. According to Theorem 2.1 (iii), if $({H},\langle\cdot,\cdot\rangle)$ is a Hilbert space such that $(\mathfrak{b},{H})$ has the Kato property, then the operator $(I-i{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}$, where ${Q}_{\mathfrak{b}}$ is a nonnegative, injective operator on $({V},\mathfrak{s}_{\mathfrak{b}})$ with $\langle u,v\rangle=\mathfrak{s}_{\mathfrak{b}}({Q}_{\mathfrak{b}}u,v)$ ($u$, $v\in{V}$), has a bounded $H^{\infty}$-functional calculus. The corresponding operator ${Q}_{\mathfrak{a}}$ for the form $\mathfrak{a}$ is equal to ${S}{Q}_{\mathfrak{b}}$ and ${T}_{\mathfrak{a}}={S}{T}_{\mathfrak{b}}$. Hence, $(I-i{T}_{\mathfrak{a}})^{-1}{Q}_{\mathfrak{a}}=(I-i{S}{T}_{\mathfrak{b}})^{-1}{S}{Q}_{\mathfrak{b}}.$ Then, since $(i{T}_{\mathfrak{b}}-i{S}{T}_{\mathfrak{b}})(I-i{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}$ is triangular and $(I-i{S}{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}-(I-i{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}=(I-i{S}{T}_{\mathfrak{b}})^{-1}(i{T}_{\mathfrak{b}}-i{S}{T}_{\mathfrak{b}})(I-i{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}$ the operator $(I-i{S}{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}$ has a bounded $H^{\infty}$ functional calculus. Moreover, note that $\displaystyle(I-i{S}{T}_{\mathfrak{b}})^{-1}{S}{Q}_{\mathfrak{b}}-(I-i{S}{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}}$ $\displaystyle=(I-i{S}{T}_{\mathfrak{b}})^{-1}({S}-I){Q}_{\mathfrak{b}}$ $\displaystyle=\Delta(I-i{S}{T}_{\mathfrak{b}})^{-1}{Q}_{\mathfrak{b}},$ where $\Delta=(I-i{S}{T}_{\mathfrak{b}})^{-1}({S}-I)(I-i{S}{T}_{\mathfrak{b}})$ is triangular. Therefore, again by Lemma 3.1, $(I-i{S}{T}_{\mathfrak{b}})^{-1}{S}{Q}_{\mathfrak{b}}$ has a bounded $H^{\infty}$ functional calculus, which completes the proof. ∎ Finally, for the sake of completeness, we state a perturbation result for the operator ${Q}$ generating the Hilbert space ${H}$ in $(\mathfrak{a},{H})$. Its proof follows directly from Lemma 3.1 and Theorem 2.1(iv). ###### Corollary 3.4. Assume that $(\mathfrak{a},{H})$ has the Kato property. Let ${Q}$ be the nonnegative, injective operator on ${V}$ associated with ${H}$. Then, $(\mathfrak{a},{H}_{\hat{{Q}}})$ has the Kato property for every Hilbert space ${H}_{\hat{{Q}}}$ with $\hat{{Q}}$ being a triangular perturbation of ${Q}$, that is, $\hat{{Q}}=(I+\Delta){Q}$ for some triangular operator $\Delta$. ## 4\. Optimality of Corollary 3.2 Below, we show that the class of triangular operators is, in a sense, the largest subclass of compact operators for which Corollary 3.2 holds. Recall that, by [5, Theorem 5.5], a compact operator ${T}$ is not triangular, if $\sum_{n\in\mathbb{N}}\frac{s_{n}({T})}{n}=\infty$, where $s_{n}({T})$ (${n\in\mathbb{N}}$) stands for the $n$-th singular value of ${T}$. ###### Proposition 4.1. Let $(a_{n})_{{n\in\mathbb{N}}}$ be a nonincreasing sequence of positive numbers with $\sum_{{n\in\mathbb{N}}}\frac{a_{n}}{n}=\infty$. Then, there exists a sectorial form $\mathfrak{a}$ such that the singular values $(s_{n}({T}_{\mathfrak{a}}))_{{n\in\mathbb{N}}}$ of the operator ${T}_{\mathfrak{a}}$ determined by the imaginary part of $\mathfrak{a}$ satisfy $s_{n}({T})\preceq a_{n}$ $({n\in\mathbb{N}})$, but not for every Hilbert space ${H}$ for which ${V}$ is densely and continuously embedded in ${H}$, the couple $(\mathfrak{a},{H})$ has the Kato property. Equivalently, there exist a selfadjoint, compact operator ${T}$ on a Hilbert space ${V}$ with $s_{n}({T})\preceq a_{n}$ $({n\in\mathbb{N}})$, and a nonnegative, injective operator ${Q}$ on ${V}$ such that ${Q}(I+i{T})^{-1}$ is not similar to an accretive operator. In order to construct an example we adapt two related results from [5] and [2]. Recall that, in [2], the sesquilinear form $\mathfrak{a}$ on a Hilbert space ${H}$ is expressed as (4.1) $\mathfrak{a}(u,v)=\langle A{S}u,{S}v\rangle_{H},\quad u,v\in{V}:={\mathcal{D}}({S}),$ where ${S}$ is a positive selfadjoint (not neccessarily bounded) operator on ${H}$, and $A$ is a bounded invertible $\theta$-accretive operator on ${H}$ for some $\theta<\pi/2$. (Here, we call the selfadjoint operator ${S}$ on ${H}$ _positive_ if $\langle{S}u,u\rangle_{H}>0$ for all $u\in{\mathcal{D}}(A)\setminus\\{0\\}$.) Then, following Kato’s terminology [6], $\mathfrak{a}$ is a regular accretive form in ${H}$. The operator ${L}_{\mathfrak{a},{H}}$ associated with the form $\mathfrak{a}$ on ${H}$ is given by ${S}A{S}$. Note that $\mathfrak{s}=\operatorname{Re}\mathfrak{a}$ is an equivalent inner product to $\langle{S}\cdot,{S}\cdot\rangle_{H}$, and in order to put it in our setting, we additionally assume that $0\in\rho({S})$. Then $\mathfrak{s}$ is a _complete_ inner product on ${V}:={\mathcal{D}}({S})$. In fact, since ${S}$ is selfadjoint, to get the completeness of this inner product, it is sufficient that ${S}$ is injective and has closed range. For the convenience of the reader we restate two auxiliary results from [5] and [2]. ###### Lemma 4.2 ([5], Theorem 8.3). Let ${H}$ be a separable Hilbert space and let $(e_{n})_{{n\in\mathbb{N}}}$ be an orthonormal basis. Let ${S}$ be the sectorial operator defined by ${S}e_{n}=2^{n}e_{n}$ $({n\in\mathbb{N}})$ with ${\mathcal{D}}({S}):=\\{x\in{H}:\sum_{n=1}^{\infty}2^{2n}\|\langle x,e_{n}\rangle_{H}\|^{2}<\infty\\}$. Suppose $K\in{\mathcal{L}}({H})$ is a non- triangular compact operator. Then, there exist bounded operators $U$ and $W$ on ${H}$ such that for every $m\in\mathbb{N}$, the operator $(I+2^{-m}WKU){S}$ fails to have a bounded $H^{\infty}$ functional calculus. ###### Lemma 4.3 ([2], Theorem 10.1). Let $A$, ${S}$, $\mathfrak{a}$ have the properties specified above. Then $(\mathfrak{a},{H})$ has the Kato property if and only if the operator $A{S}$ has a bounded $H^{\infty}$ functional calculus. ###### Lemma 4.4. Let $A$, ${S}$, $\mathfrak{a}$ have the properties specified above. Let ${T}$ and ${Q}$ be the operators associated with $\mathfrak{a}$ and ${H}$. * (i) The operator ${T}\in{\mathcal{L}}({V})$ is compact if and only if the operator $\operatorname{Im}A\in{\mathcal{L}}({H})$ is compact. Then, $s_{n}({T})\simeq s_{n}(\operatorname{Im}A)$ ($n\in\mathbb{N}$), that is, there exists $c>0$ such that $c^{-1}s_{n}({T})\leq s_{n}(\operatorname{Im}A)\leq cs_{n}({T})$ for every $n\in\mathbb{N}$. * (ii) The operator ${Q}\in{\mathcal{L}}({V})$ is compact if and only if the embedding of ${V}$ into ${H}$ is compact, if and only if ${S}^{-1}\in{\mathcal{L}}({H})$ is compact. Then, $s_{n}({Q})\simeq s_{n}({S}^{-1})\simeq s_{n}(j)$ ($n\in\mathbb{N}$) where $j$ denotes the canonical embedding of ${V}$ into ${H}$. ###### Proof. First, note that the operators ${T}$ and ${Q}$ are of the form: $\displaystyle{T}$ $\displaystyle={S}^{-1}(\operatorname{Re}A)^{-1}\operatorname{Im}A\,{S}\quad\text{and}$ $\displaystyle{Q}$ $\displaystyle={S}^{-1}(\operatorname{Re}A)^{-1}{S}^{-1}_{\|},$ where $\operatorname{Re}A$ and $\operatorname{Im}A$ denote the real and the imaginary part of $A$, and ${S}^{-1}_{\|}$ is the restriction of ${S}^{-1}\in{\mathcal{L}}({H})$ to ${V}$, considered as an operator in ${\mathcal{L}}({V},{H})$. These expressions give the first statements in (i) and (ii). The second assertion in (i) follows in a straightforward from, e.g., [13, Theorem 7.7, p. 171]. To prove the corresponding one of (ii), assume that ${S}^{-1}$ is compact with spectrum $\sigma({S}^{-1})=:\\{\mu_{n}\\}_{n\in\mathbb{N}}$, where $\mu_{n}\rightarrow 0^{+}$ as $n\rightarrow\infty$. Therefore, there exists an orthonormal system $\\{e_{n}\\}_{n\in\mathbb{N}}$ in ${H}$ such that ${S}h=\sum_{n}\mu_{n}^{-1}\langle h,e_{n}\rangle_{{H}}e_{n}$ for $h\in{\mathcal{D}}({S})=\\{h\in{H}:\sum_{n}\mu_{n}^{-2}\|(h,e_{n})\|^{2}<\infty\\}$. Let ${C}:{V}_{}\rightarrow{H}$, ${C}u:={S}^{-1}u$, $u\in{\mathcal{D}}({S})$, where ${V}_{}$ denote the Hilbert space $({\mathcal{D}}({S}),\langle{S}\cdot,{S}\cdot\rangle_{H})$. Of course, ${C}\in{\mathcal{L}}({V}_{},{H})$ and ${C}^{}{C}\in{\mathcal{L}}({V}_{\star})$ are compact. Moreover, note that ${C}^{}{C}u=\sum_{n}\mu_{n}^{2}\langle u,g_{n}\rangle_{{V}_{\star}}g_{n},\quad\quad u\in{V}_{\star},$ where $g_{n}:=\mu_{n}^{-2}e_{n}$ $(n\in\mathbb{N})$ is an orthonormal basis for ${V}_{}$. Thus, the singular values of ${C}$ are given by $s_{n}({C})=\mu_{n}$, $n\in\mathbb{N}$. Now, let $I_{S}$ denote the identity map on ${\mathcal{D}}({S})$ considered as an operator from ${V}$ onto ${V}_{}$. Therefore, we have ${S}^{-1}_{\|}={C}I_{S}$ and, by [13, Theorem 7.1, p. 171], we get $\displaystyle s_{n}({Q})$ $\displaystyle\leq\\|{S}^{-1}(\operatorname{Re}A)^{-1}\\|_{{\mathcal{L}}({H})}s_{n}({C}I_{S})=\\|{S}^{-1}(\operatorname{Re}A)^{-1}\\|_{{\mathcal{L}}({H})}s_{n}(I_{S}^{}{C}^{})$ $\displaystyle\leq\\|{S}^{-1}(\operatorname{Re}A)^{-1}\\|_{{\mathcal{L}}({H})}\\|I_{S}^{}\\|_{{\mathcal{L}}({V}_{\star},{V})}s_{n}({C}^{})$ $\displaystyle\leq\\|{S}^{-1}(\operatorname{Re}A)^{-1}\\|_{{\mathcal{L}}({H})}\\|I_{S}^{}\\|_{{\mathcal{L}}({V}_{\star},{V})}s_{n}({C})\quad\textrm{ and }$ $\displaystyle s_{n}({C})$ $\displaystyle=s_{n}((\operatorname{Re}A){S}{Q}I_{S}^{-1})$ $\displaystyle\leq\\|(\operatorname{Re}A){S}\\|_{{\mathcal{L}}({V},{H})}s_{n}({Q}I_{S}^{-1})$ $\displaystyle\leq\\|(\operatorname{Re}A){S}\\|_{{\mathcal{L}}({V},{H})}\\|I_{S}^{-1}\\|_{{\mathcal{L}}({V}_{\star},{V})}s_{n}({Q}).$ Finally, note that $s_{n}({S}^{-1})$ is equal to the $n$-th singular value of the embedding of ${V}_{}$ into ${H}$. This completes the proof. ∎ ###### Proof of Proposition 4.1. Suppose that ${H}$, ${S}$, $K$, $U$, $W$ have the properties specified above in Lemma 4.2. Fix $m\in\mathbb{N}$ such that the numerical range of the operator $A:=I+2^{-m}WKU)$ is contained in $\\{\|z-1\|<1\\}$. Then, by Lemma 4.3, the couple $(\mathfrak{a},{H})$ does not have the Kato property, where the form $\mathfrak{a}$ corresponds to the operators $A$ and ${S}$ as above. Moreover, by Lemma 4.4(i), $s_{n}({T})\simeq s_{n}(\operatorname{Im}A)\simeq s_{n}\bigl{(}2^{-m}(WKU-U^{}K^{}W^{})\bigr{)}\preceq s_{n}(K)\quad(n\in\mathbb{N}).$ This completes the proof. ∎ ## References * [1] W. Arendt, _Semigroups and evolution equations: functional calculus, regularity and kernel estimates_. In Handbook of Differential Equations (C. M. Dafermos, E. Feireisl eds.), pages 1-85, Elsevier, North Holland, 2004. * [2] P. Auscher, A. McIntosh, and A. Nahmod, _Holomorphic functional calculi of operators, quadratic estimates and interpolation_ , Indiana Univ. Math. J. 46 (1997), 375-403. * [3] K. Fan, _On similarity of operators_ ,, Adv. Math. 10 (1973), 395-400. * [4] M. Haase, The functional calculus for sectorial operators, volume 169 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2006. * [5] N. J. Kalton, _Perturbations of the $H^{\infty}$-calculus_, Collect. Math. 58 (2007), 291-325. * [6] T. Kato, _Fractional powers of dissipative operators_ , J. Math. Soc. Japan 13 (1961), 246-274. * [7] T. Kato, _Fractional powers of dissipative operators II_ , J. Math. Soc. Japan 14 (1962), 242-248. * [8] P. C. Kunstmann and L. Weis, _Maximal $L^{p}$ regularity for parabolic equations, Fourier multiplier theorems and $H^{\infty}$ functional calculus_, In Levico Lectures, Proceedings of the Autumn School on Evolution Equations and Semigroups (M. Iannelli, R. Nagel, S. Piazzera eds.), volume 69, pages 65-320, Springer, Berlin, 2004. * [9] C. Le Merdy, _The similarity problem for bounded analytic semigroups on Hilbert space_ , Semigroup Forum 56 (1998), 205-224. * [10] J.-L. Lions, _Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs_ , J. Math. Soc. Japan 14 (1962), 233-241. * [11] A. McIntosh, _Operators which have an $H_{\infty}$ functional calculus_, In Miniconference on operator theory and partial differential equations (North Ryde, 1986), volume 14 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 210-231. Austral. Nat. Univ., Canberra, 1986. * [12] B. Nagy and J. Zemánek, _A resolvent condition implying power boundedness_ , Studia Math. 134 (1999), 143-151. * [13] J. Weidmann, Linear operators in Hilbert spaces, volume 68 of Graduate Texts in Mathematics, Springer, New York, 1980. * [14] A. Yagi, Abstract parabolic evolution equations and their applications, Springer Monographs in Mathematics, Berlin, 2010.
# BOOSTR: A Dataset for Accelerator Control Systems Diana Kafkes Jason St. John Fermi National Accelerator Laboratory, Batavia, IL 60510, USA ###### Abstract BOOSTR (Booster Operation Optimization Sequential Time-Series for Reinforcement) was created to provide cycle-by-cycle time series of readings and settings from instruments and controllable devices of the Booster, Fermilab’s Rapid-Cycling Synchrotron (RCS) operating at 15 Hz. BOOSTR provides a time series from 55 device readings and settings which pertain most directly to the high-precision regulation of the Booster’s Gradient Magnet Power Supply (GMPS). To our knowledge, this is the first known dataset of accelerator device parameters made publicly available. We are releasing it in the hopes that it can be used to demonstrate aspects of artificial intelligence for advanced control systems, such as reinforcement learning and autonomous anomaly detection. ## Background & Summary Tuning and controlling particle accelerators is challenging and time consuming. Even marginal improvements to accelerator operation can translate very efficiently into improved scientific yield for an experimental particle physics program. The data released here was collected in the hopes of achieving improvement in precision for the Fermilab Booster Gradient Magnet Power Supply (GMPS) regulatory system, which is detailed below. The Fermilab Booster receives a 400 MeV proton beam from the Linear Accelerator and accelerates it to 8 GeV through synchronously raising accelerator cavity radiofrequency and instigating a controlled magnetic field to steer the beam with combined-function bending and focusing electromagnets, known as gradient magnets. These magnets are powered by the GMPS, which operates on a 15 Hz cycle between a minimum current (at injection) and a maximum current (at beam extraction). The GMPS is realized as four power supplies, evenly distributed around the Booster, and each powers one fourth of the gradient magnets. The role of the GMPS regulator is to calculate and apply small compensating offsets in the GMPS driving signal in order to improve the agreement of the resulting observed minimum and maximum currents with their set points. Without regulation, the fitted minimum of the magnetic field may vary from the set point by as much as a few percent. At beam injection, a perturbation of only a percent is enough to significantly decrease beam transfer efficiency and thereby reduce the beam flux ultimately available to the high-energy particle physics experiments run at the lab. Disturbances to the magnet current can occur due to ambient temperature changes, other nearby high-power pulsed radio-frequency systems, and electrical ground movement induced by power supplies of other particle accelerators at the complex. The current GMPS regulation involves a PID (Proportional-Integral-Derivative) control scheme (see Figure 1 for schematic). The regulator calculates estimates for the minimum and maximum currents of the offset-sinusoidal magnetic field from the previous 15 Hz cycle. These values are then used to adjust the power supply program and decrease systemic error in the next cycle’s current, such that it more closely matches the set point. Presently, the PID system achieves regulation errors corresponding to roughly 0.1% of the set value. Although some 200,000 entries populate the device database of Fermilab’s accelerator control system [1], the 55 device value time series presented here in BOOSTR [2] were collected in accordance with suggestion by Fermilab accelerator subject matter experts (SMEs). These values exhibit correlations with GMPS power supply perturbations. The full data were collected during two separate periods: from June 3, 2019 to July 11, 2019 — when the accelerator was shut down for regular maintenance — and from December 3, 2019 to April 13, 2020 — when the accelerator was shut down in response to the Covid-19 Pandemic. Data from a single day of BOOSTR was previously described in a Datasheet [3]. A proof-of-concept paper [4] (submitted to Physical Review Accelerators and Beams) used this subset of BOOSTR and demonstrated the viability of training a reinforcement learning agent to control GMPS regulation better than the existing PID system. Relative to the original Datasheet [3], this manuscript is expanded with more SME input, describes more than 100 times more data, and includes documentation of validation not presented in the original Datasheet. ## Methods #### Collection Process A data collection node was created and set to request data at 15 Hz from the Data Pool Manager of the Accelerator Control Network (ACNET) [1]. The created scheme involved front-end nodes, each managing their respective devices, replying with timestamped values at the stated rate barring differences of clock speed, input-output (I/O) lag time variations due to network traffic fluctuations, and higher-priority interruptions from competing processes on the front-end node. These inconsistencies were later addressed through a time- alignment process discussed in the Data Processing Section. The collection node stored the data in a circular buffer approximately 10 days deep. A Python script managed by a nightly cron job polled the data collection node for the most recent midnight-to-midnight 24 hours of timestamped data for each of the 55 time series identified by SMEs. A second cron-managed script did the same for relevant accelerator control events issued in the same period. These event data correspond to important cycles achieved through controlling the devices at the accelerator. Event data were requested by a separate data collection node. Each day’s data harvest was originally stored in HDF5 (Hierarchical Data Format Version 5) files. Any data instances missing from the parquet files released here were not included in the original data buffers from which this dataset was drawn. ### Data Processing Each instance was created through a concatenation of each device’s timestamp data table within every HDF5 file and then saved in parquet format. A similar procedure was undertaken for one of the accelerator control event signals polled, Event0C, as its broadcast is synchronized with the GMPS magnetic field minimum. Event0C was collected to correct a problem in the observed sampling frequency: there was an issue of the sampling of each device being nominally at 15 Hz, but in reality synchrony was demonstrably imperfect, and the time intervals between successive timestamps display varying lags. Since Event0c serves as the baseline or heartbeat of the Booster at approximately 15 Hz and is synchronized with the smoothly varying electrical current GMPS regulates, we used Event0c to time-align our data. The alignment approximates the data available to the GMPS regulator operating in real time. We used the GMPS-synchronized Event0C’s timestamp as the moment to begin forward inference, taking the value for each device time series which had the most recent corresponding timestamp. In practice, this required timestamp- sorted series for each device and finding the most recent device value, relative to Event0c timestamp, in a lookback window equal to the maximum interval between device timestamps (necessarily excluding the five month gap between our two data collection periods). This time-alignment step was run over the whole dataset in multiple parallel processes using Apache Spark. Notably, the data recorded for Event0c was missing the period from July 1 to 11, 2019. Therefore aligning on this variable discarded some of the data collected during our first period of collection. ## Data Records The data release is stored on Zenodo [2]. Each instance is a zip compressed parquet of one of the 55 aligned time series with columns corresponding to the aligned time stamp, original time stamp, difference between time stamps, and the reading/setting value [2]. The original timestamp and time difference is included to demonstrate the mechanics of our alignment process and enable a check for reproducibility. All timestamps are in Greenwich Mean Time (UTC). Our data release contains device data from each of the four gradient magnet power supplies, the GMPS PID regulator, and the Main Injector, where the beam is directed after acceleration via the Booster. Minimum and maximum current information readings and settings, the feedback and transductor gain settings, and the feed-forward start trigger are collected as part of the current PID regulation scheme. The “inhibit” value controls whether the GMPS regulator accepts settings changes for parameters other than the minimum and maximum current, such as the gain settings (any positive value will prevent changes). Additionally, $\dot{\vec{B}}$, the rate of change of the magnetic field, is recorded as a proxy for the magnetic field we are interested in regulating. Timing information derived from $\dot{\vec{B}}$ = 0 synchronizes the current PID regulator system. We acknowledge that the ACNET parameter names are by no means standardized across different particle accelerators and that they will appear especially abstruse for those well-versed in control systems who are new to working with accelerators. In Table 1, we detail explanations of each of the parameters read from devices (devices whose first letter is followed by :) and indicate whether the device setting was included in the dataset (devices whose first letter is followed by _ and appear in Setting column), since describing these corresponding pairs would be redundant. In Figures 2 and 3, we visualize metadata trends for each “nonconstant" parameter in each data collection period (see Table 2 for a list of values we considered to be virtually unchanging within the two periods) and also provide the mean and standard deviation of device readings across the two collection periods in Table 1. Furthermore, Table 1 includes dates missing in the data recorded for each reading. As a reminder the data were collected during two separate periods: from June 3, 2019 to June 30, 2019 (July is missing due to time-alignment with Event0C) and from December 3, 2019 to April 13, 2020. Finally, in Figure 4 we demonstrate the centrality of each recorded parameter with a heatmap of histogram values. Additionally, we provide the PID regulator status values B\|GMPSSC (ACNET status parameters include \|), whose 16 bits contain various motherboard states. Here we are concerned with bit 3, which indicates whether or not the GMPS regulator was on (1), and bit 7, which indicates whether the Booster is in its normal alternating current (AC) mode (1) or “coasting beam” direct current (DC) mode at constant beam energy (0). Unlike the rest of the devices, this status value is presently recorded at only 1 Hz because it was not included in our initial data node request and was relegated to an archived data node at lesser frequency. While the same time-aligning described above was applied to align B\|GMPSSC, due to the slow sampling rate, we caution the user to refer closely to the original timestamp such that they might make decisions about whether to use data when GMPS was off and to inform them of potential problems when interpolating in a region immediately before or after a status change. See Figure 5 for more details on B\|GMPSSC values. ## Technical Validation In order to verify the quality of this dataset, we pored over the electronic logbook (Elog) [5] that Fermilab Booster technicians and operators use to record changes to device settings as well as observations while in operation. We used these Elog entries to authenticate our data’s viability across timescales. First, we used the Elog to corroborate expert acknowledgement of the major spikes observed in Figures 2 and 3. These outlier changes, typically seen in the value’s mean and standard deviation, represented major changes made on that specific day, including when the Booster was switched from alternating current (AC) to direct current (DC) mode (see Figure 5) as well as when the GMPS regulator was turned off altogether. These reconciliations are presented in Table 3. Furthermore, we pinpoint changes in the AC vs. DC settings according to the Elog [5] for June 24, 2019 and March 11, 2020 in Figure 6. Here applying a bitmask reveals that a B\|GMPSSC value of 159 indicates AC mode/GMPS on, while 31 indicates DC mode/GMPS on, and 407 indicates AC mode/GMPS off. In this figure, the plotted timestamps were offset to Central Time (UTC-5) in order to align with times given in the Elog, which were not recorded in UTC. On June 24, the trace of B\|GMPSSC clearly shows GMPS regulation briefly switching off before commencing DC studies from 8:00 AM - 6:00 PM with a value of 31, then being turned back to 159. On March 11, B\|GMPSSC is at 159 before 6:00 AM, off at 407 from 6:00 - 9:50 AM, and then is set to AC mode from 9:50 AM - 12:45 PM, to DC mode from 12:45 - 3:49 PM, and back to AC mode for the rest of the day, as per the Elog [5]. The close correspondence of these changes in our data to the recorded actions and observations of Booster personnel boost our confidence in the quality and relevance of the collected dataset. Additionally, we plot settings changes on March 10 and 11 documented in the Elog [5] in Figure 7. The blip in B:ACMNPG from 6.5 to 13.5 is visible as is the slight decrease in B_VIMIN around 4:00 PM CST, which were mentioned in Table 3. ## Usage Notes BOOSTR could be used to train various control networks for accelerator regulation, to construct “digital twins” of the Fermilab Booster regulator’s control environment, or to develop anomaly detection/categorization capabilities. Please note: there are no legal or ethical ramifications of using this data as it was collected from a machine, and not collected from or representative of people. In the future, the dataset could feasibly be expanded to include more of the 200,000 available ACNET system parameters and therefore be used to control, mimic, or monitor further aspects of the particle accelerator complex. One could argue that this initial dataset might become the foundation upon which substantial fine-tuning of particle accelerators could depend. ## Code availability The preprocessing code is available here. When using BOOSTR data, the authors recommend ordering by time immediately, as the parquet files do not store the data entries sequentially [2]. ## Acknowledgements This dataset was created as part of the “Accelerator Control with Artificial Intelligence” Project conducted under the auspices of the Fermilab Laboratory Directed Research and Development Program (Project ID FNAL-LDRD-2019-027). The manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics and is registered at Fermilab as Technical Report Number FERMILAB-PUB-21-005-SCD. We are extremely grateful for Brian Schupbach and Kent Triplett for lending their Booster technical expertise, without which we could not have validated our dataset. Additionally, we would like to acknowledge Burt Holzman for guidance on getting set up in the cloud, and the help of the Databricks federal support staff, in particular Sheila Stewart. Furthermore, we would like to recognize Jovan Mitrevski and Aleksandra Ciprijanovic for useful discussions and a careful reading of this manuscript. ## Author contributions statement J.S. created the data collection script, and set up and maintained the cron jobs to record the data in HDF5 files. D.K. migrated the data from on-premise storage to the cloud, wrote the preprocessing and time-alignment code, and validated the data. Both authors reviewed this manuscript. ## Competing interests The authors declare no competing interests. ## References * [1] Cahill, K. & et. al. The fermilab accelerator control system. _ICFA Beam Dyn. Newslett._ 47, 106–124 (2008). * [2] Kafkes, D. & St. John, J. BOOSTR: A Dataset for Accelerator Control Systems (Full Release), 10.5281/zenodo.4382663 (2021). * [3] Kafkes, D. & St. John, J. BOOSTR: A Dataset for Accelerator Control Systems (Partial Release), 10.5281/zenodo.4088982 (2020). * [4] John, J. S. _et al._ Real-time artificial intelligence for accelerator control: A study at the fermilab booster (2020). 2011.07371. * [5] Hazelwood, K. J. Fermilab electronic logbook. www-bd.fnal.gov/Elog. ## Figures & Tables Figure 1: Overview of current GMPS control system [4]. Presently, a human operator specifies a target program for B:VIMIN and B:VIMAX, the GMPS compensated minimum and maximum currents respectively, via the Fermilab Accelerator Control Network that is transmitted to the GMPS control board. Table 1: Description of BOOSTR dataset parameters. Here “GMPS" denotes the Gradient Magnet Power Supplies (1-4), “MI" means main injector, “MDAT" refers to Fermilab’s Machine Data communications protocol. Device parameters that begin with B relate to the accelerator Booster, whereas device parameters that begin with I relate to the Main Injector. Parameter mean and standard deviation have been truncated to two decimal points. Parameter \| Details (Units) \| Setting \| Mean (Std) \| Missing Dates ---\|---\|---\|---\|--- B:ACMNIG \| Min AC integral feedback gain \| B_ACMNIG \| 0.75 (0) \| 2019: 6/14, 12/12 B:ACMNPG \| Min AC proportional feedback gain \| B_ACMNPG \| 9.1968 (0.75) \| 2019: 6/14, 12/12 B:ACMXIG \| Max AC integral feedback gain \| B_ACMXIG \| 0.30 (0) \| 2019: 6/14, 12/12 B:ACMXPG \| Max AC proportional feedback gain \| B_ACMXPG \| 3.00 (0) \| 2019: 6/14, 12/12 B:DCIG \| DC integral feedback gain \| B_DCIG \| 0 (0) \| 2019: 6/14, 12/12 B:DCPG \| DC proportional feedback gain \| B_DCPG \| 0.10 (1.35) \| 2019: 6/14, 12/12 B:GMPS1V \| GMPS1 output voltage (V) \| \| 81.82 (89.56) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:GMPS2V \| GMPS2 output voltage (V) \| \| 85.29 (96.00) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:GMPS3V \| GMPS3 output voltage (V) \| \| 63.67 (61.19) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:GMPS4V \| GMPS4 output voltage (V) \| \| 61.08 (65.19) \| 2019: 6/14, 12/12 B:GMPSBT \| $\partial B/\partial t$ offset trigger (s) \| B_GMPSBT \| 0 (0) \| 2019: 6/14, 12/12 B\|GMPSSC \| Binary status control of GMPS \| \| N/A (N/A) \| 2019: 6/07, 6/12, 6/14-15 \| \| \| \| and 2020: 1/18, 3/08, 3/15 B:GMPSFF \| Feedforward start trigger (s) \| B_GMPSFF \| 1.32 (1.52) \| 2019: 6/14, 12/12 B:IMAXXG \| Max transductor gain (A/V) \| B_IMAXXG \| -117.12 (1.80) \| 2019: 6/14, 12/12 B:IMAXXO \| Max transductor offset (A) \| B_IMAXXO \| 10.00 (0) \| 2019: 6/14, 12/12 B:IMINXG \| Min transductor gain (A/V) \| B_IMINXG \| -11.73 (0.23) \| 2019: 6/14, 12/12 B:IMINXO \| Min transductor offset (A) \| B_IMINXO \| 0 (0) \| 2019: 6/14, 12/12 B:IMINER \| Discrepancy from setting at min (0.1 A) \| \| 1.93 (3.86) \| 2019: 6/14, 12/12 B:IMINST \| $\partial B/\partial t$ sample off \| B_IMINST \| 0 (0) \| 2019: 6/14, 12/12 B:IPHSTC \| Phase regulator time constant \| B_IPHSTC \| 20.00 (0.01) \| 2019: 6/14, 12/12 B:LINFRQ \| 60 Hz line frequency offset (mHz) \| \| -0.44 (16.31) \| 2019: 6/03-7/11, 12/12 \| \| \| \| 12/30-31 and 2020: 1/01-06 B:NGMPS \| Number of GMPS suppliers \| \| 4.00 (0) \| 2019: 6/14, 12/12 B:PS1VGM \| GMPS1 V- to ground (V) \| \| -2.30 (23.56) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS2VGM \| GMPS2 V- to ground (V) \| \| -21.29 (27.52) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS3VGM \| GMPS3 V- to ground (V) \| \| -15.13 (14.11) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS4VGM \| GMPS4 V- to ground (V) \| \| -26.27 (17.22) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS1VGP \| GMPS1 V+ to ground (V) \| \| 52.00 (34.82) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS2VGP \| GMPS2 V+ to ground (V) \| \| 26.53 (30.53) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS3VGP \| GMPS3 V+ to ground (V) \| \| 20.17 (13.74) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:PS4VGP \| GMPS4 V+ to ground (V) \| \| 9.14 (12.75) \| 2019: 6/14, 12/12, 12/30-31 \| \| \| \| and 2020: 1/01-4/12 B:VIMAX \| Compensated max GMPS current (A) \| B_VIMAX \| 772.43 (385.98) \| 2019: 6/14, 12/12 B:VIMIN \| Compensated min GMPS current (A) \| B_VIMIN \| 83.20 (40.68) \| 2019: 6/14, B:VINHBT \| Inhibit value \| B_VINHBT \| 1.00 (0.04) \| 2019: 6/14, 12/12 B:VIPHAS \| GMPS ramp phase wrt line voltage (rad) \| B_VIPHAS \| 1.80 (0.14) \| 2019: 6/14, 12/12 I:IB \| MI lower bend current (A) \| \| 2275.71 (2571.03) \| 2019: 6/03-19, 12/12 I:MDAT40 \| MDAT measured MI current (A) \| \| 2400.57 (2576.37) \| 2019: 6/03-7/11, 12/12 I:MXIB \| Main dipole bend current (A) \| \| 2279.68 (2564.53) \| 2019: 6/03-11, 6/13-19, 12/12 Table 2: Summary of nearly constant variables in both periods of data collection. Here “nearly constant" denotes variables having a standard deviation less than $10^{-5}$ across both periods. Device \| Setting \| Constant Value ---\|---\|--- B:DCIG \| B_DCIG \| 0 B:GMPSBT \| B_GMPSBT \| 0.0003 B:IMINST \| B_IMINST \| 0 B:IMINXO \| B:IMINXO \| 0 Table 3: Summary of Booster-related electronic log (Elog) [5] entries corresponding to spikes in Figures 2 and 3. Original Central Time times are given with values in parenthesis designating UTC. Here “RF" denotes radiofrequency. Date \| Elog Entry Summary ---\|--- 6/8/2019 \| GMPS in AC mode until 8:00 AM (13:00) on 6/08, then switched off until 6/17 6/10/2019 \| GMPS was locked/tagged out for outage, West Booster gallery RF off from 8:30-10:00 (13:30-3:00) for work 6/17/2019 \| GMPS turned back on and put in AC mode 6/22/2019 \| High energy physics beam turned off at 8:00 PM (1:00 +1) (GMPS remained in AC mode) 6/24/2019 \| DC studies from 8:00 AM - 6:00 PM (13:00 - 23:00), back to AC mode 6/26/2019 \| Alternated between AC and DC mode, GMPS off for 30 min around 5:30 PM (22:30) 6/27/2019 \| GMPS in AC mode all day, but removing certain study events caused bias to creep up, eventually tripping the RF 6/28/2019 \| Alternated between AC and DC mode 12/8/2019 \| B_VIMIN adjusted, GMPS in AC mode 12/12/2019 \| AC mode, operators reset virtual machine environment locally 12/28 - 12/30/2019 \| No beam from injector, GMPS in AC mode 12/31/2019 \| Booster injection back, GMPS in AC mode 1/1/2020 \| GMPS off for 15 min around 9:30 AM (14:30), in AC mode for rest of day 2/4/2020 \| RF sparking in gallery due to reverting of RF capture settings, GMPS in AC mode 2/5/2020 \| GMPS off from 6:00 AM - 3:30 PM (11:00 - 20:30), then in AC mode for rest of day 2/6/2020 \| Lowered beam intensity to users, but GMPS was in AC mode all day 3/5/2020 \| Beam tails were large, so turned B_VIMIN down 3/10/2020 \| GMPS in AC mode all day, B:ACMNPG changed from 6.5 to 13.5, B_VIMIN decreased from 103.440 to 103.420 3/11/2020 \| GMPS in AC mode from 12:00 AM - 6:00 AM (5:00 - 11:00), off from 6:00 - 10:00 AM (11:00 - 15:00), then alternated between AC and DC mode, B_VIMIN adjusted from \| 103.418 to 103.386 3/13/2020 \| GMPS off from 9:30 AM - 11:00 AM (14:30-16:00), back on and put in AC mode 3/20/2020 \| Booster turned off on account of Covid-19 pandemic at 12:00 PM (17:00) Figure 2: Metadata variable trends for Period 1: June 3, 2019 to June 30, 2019. The graphs show the mean for each variable on the given date and shades in the standard deviation of that variable on that date. Figure 3: Metadata variable trends for Period 2: December 2, 2019 to April 13, 2020. The graphs show the mean for each variable on the given date and shades in the standard deviation of that variable on that date. Figure 4: Heatmap of histogram distributions for each reading and setting variable with equal sampling. This is only meant to characterize the centrality of each recorded value. See Fig. 2 and 3 for actual metadata value ranges. Figure 5: Daily values of B\|GMPSSC (should be interpreted as taking the mode for each day) whose bits encode relevant Booster statuses. Figure 6: Values of status B\|GMPSSC corresponding to Table 3 entries for June 24, 2019 and March 11, 2020 (timestamps were put in Central Time to align with Elog). Recall: a value of 159 indicates AC study/GMPS on, 31 indicates DC study/GMPS on, and 407 indicates AC study/GMPS off. These traces display this value at a much greater granularity than Figure 5. Figure 7: Switches corresponding to Table 3 entries for March 10, 2020 and March 11, 2020: B:ACMNPG changed from 6.5 to 13.5 and B_VIMIN decreased from 103.418 to 103.386 (timestamps were put in Central Time to align with Elog). The sudden large increase in B_VIMIN from 12:45 - 3:49 PM CST to a value off the plotted region corresponds to the DC mode observed in Figure 6.
# Bayesian Bandwidths in Semiparametric Modelling for Nonnegative Orthant Data with Diagnostics Célestin C. Kokonendji<EMAIL_ADDRESS>Sobom M. Somé <EMAIL_ADDRESS>Université Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR 6623 CNRS-UBFC, Besançon, France Université Joseph KI-ZERBO, LANIBIO, Ouagadougou, Burkina Faso Université Thomas SANKARA, UFR ST, Ouagadougou, Burkina Faso ###### Abstract Multivariate nonnegative orthant data are real vectors bounded to the left by the null vector, and they can be continuous, discrete or mixed. We first review the recent relative variability indexes for multivariate nonnegative continuous and count distributions. As a prelude, the classification of two comparable distributions having the same mean vector is done through under-, equi- and over-variability with respect to the reference distribution. Multivariate associated kernel estimators are then reviewed with new proposals that can accommodate any nonnegative orthant dataset. We focus on bandwidth matrix selections by adaptive and local Bayesian methods for semicontinuous and counting supports, respectively. We finally introduce a flexible semiparametric approach for estimating all these distributions on nonnegative supports. The corresponding estimator is directed by a given parametric part, and a nonparametric part which is a weight function to be estimated through multivariate associated kernels. A diagnostic model is also discussed to make an appropriate choice between the parametric, semiparametric and nonparametric approaches. The retention of pure nonparametric means the inconvenience of parametric part used in the modelization. Multivariate real data examples in semicontinuous setup as reliability are gradually considered to illustrate the proposed approach. Concluding remarks are made for extension to other multiple functions. ###### keywords: Associated kernel , Bayesian selector , dispersion index , model diagnostics , multivariate distribution , variation index , weighted distribution. Mathematics Subject Classification 2020: 62G07; 62G20; 62G99; 62H10; 62H12. ††journal: arXiv ## 1 Introduction The $d$-variate nonnegative orthant data on $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$ are real $d$-vectors bounded to the left by the null vector $\mathbf{0}_{d}$, and they can be continuous, discrete (e.g., count, categorial) or mixed. For simplicity, we here assume either $\mathbb{T}_{d}^{+}=[0,\infty)^{d}$ for semicontinuous or $\mathbb{T}_{d}^{+}=\mathbb{N}^{d}:=\\{0,1,2,\ldots\\}^{d}$ for counting; and, we then omit both setups of categorial and mixed which can be a mix of discrete and continuous data (e.g., [53]) or other time scales (see, e.g., [36]). Modelling such datasets of $\mathbb{T}_{d}^{+}$ need nonnegative orthant distributions which are generally not easy to handle in practical data analysis. The baseline parametric distribution (e.g., [20, 34]) for the analysis of nonnegative countinuous data is the exponential distribution (e.g., in Reliability) and that of count data is the Poisson one. However, there intrinsic assumptions of the two first moments are often not realistic for many applications. The nonparametric topic of associated kernels, which is adaptable to any support $\mathbb{T}_{d}^{+}$ of probability density or mass function (pdmf), is widely studied in very recent years. We can refer to [7, 8, 12, 13, 29, 43, 51, 52, 61, 62] for general results and more specific developments on associated kernel orthant distributions using classical cross- validation and Bayesian methods to select bandwidth matrices. Thus, a natural question of a flexible semiparametric modelling now arises for all these multivariate orthant datasets. Indeed, we first need a review of the recent relative variability indexes for multivariate semicontinuous ([31]) and count ([25]) distributions. The infinite number and complexity of multivariate parametric distributions require the study of different indexes for comparisons and discriminations between them. Simple classifications of two comparable distributions are done through under-, equi- and over-variability with respect to the reference distribution. We refer to [57] and references therein for univariate categorial data which does not yet have its multivariate version. We then survey multivariate associated kernels that can accommodate any nonnegative orthant dataset. Most useful families shall be pointed out, mainly as a product of univariate associated kernels and including properties and constructions. Also, we shall focus on bandwidth matrix selections by Bayesian methods. Finally, we have to introduce a flexible semiparametric approach for estimating multivariate nonnegative orthant distribution. Following [15] for classical kernels, the corresponding estimator shall be directed by a given parametric part, and a nonparametric part which is a weight function to be estimated through multivariate associated kernels. But what is the meaning of a diagnostic model to make an appropriate choice between the parametric, semiparametric and nonparametric approaches in this multivariate framework? Such a discussion is to highlight practical improvements on standard nonparametric methods for multivariate semicontinuous datasets, through the use of a reasonable parametric-start description. See, for instance, [27, 33, 48] for univariate count datasets. In this paper, the main goal is to introduce a family of semiparametric estimators with multivariate associated kernels for both semicontinuous and count data. They are meant to be flexible compromises between a grueling parametric and fuzzy nonparametric approaches. The rest of the paper is organized as follow. Section 2 presents a brief review of the relative variability indexes for multivariate nonnegative orthant distributions, by distinguishing the dispersion for counting and the variation for semicontinuous. Section 3 displays a short panoply of multivariate associated kernels which are useful for semicontinuous and for count datasets. Properties are reviewed with new proposals, including both appropriated Bayesian methods of bandwidths selections. In Section 4, we introduce the semiparametric kernel estimators with $d$-variate parametric start. We also investigate the corresponding diagnostic model. Section 5 is devoted to numerical illustrations, especially for uni- and multivariate semicontinuous datasets. In Section 6, we make some final remarks in order to extend to other multiple functions, as regression. Eventually, appendixes are exhibited for technical proofs and illustrations. ## 2 Relative Variability Indexes for Orthant Distributions Let $\boldsymbol{X}=(X_{1},\ldots,X_{d})^{\top}$ be a nonnegative orthant $d$-variate random vector on $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$, $d\geq 1$. We use the following notations: $\sqrt{\mathrm{var}\boldsymbol{X}}=(\sqrt{\mathrm{var}X_{1}},\ldots,\sqrt{\mathrm{var}X_{d}})^{\top}$ is the elementwise square root of the variance vector of $\boldsymbol{X}$; $\mathrm{diag}\sqrt{\mathrm{var}\boldsymbol{X}}=\mathrm{diag}_{d}(\sqrt{\mathrm{var}X_{j}})$ is the $d\times d$ diagonal matrix with diagonal entries $\sqrt{\mathrm{var}X_{j}}$ and $0$ elsewhere; and, $\mathrm{cov}\boldsymbol{X}=(\mathrm{cov}(X_{i},X_{j}))_{i,j\in\\{1,\ldots,d\\}}$ denotes the covariance matrix of $\boldsymbol{X}$ which is a $d\times d$ symmetric matrix with entries $\mathrm{cov}(X_{i},X_{j})$ such that $\mathrm{cov}(X_{i},X_{i})=\mathrm{var}X_{i}$ is the variance of $X_{i}$. Then, one has $\mathrm{cov}\boldsymbol{X}=(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})(\boldsymbol{{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\rho}}_{\boldsymbol{X}}})(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}}),$ (2.1) where $\boldsymbol{\rho}_{\boldsymbol{X}}=\boldsymbol{\rho}(\boldsymbol{X})$ is the correlation matrix of $\boldsymbol{X}$; see, e.g., [21, Eq. 2-36]. It is noteworthy that there are huge many multivariate distributions with exponential (resp. Poisson) margins. Therefore, we denote a generic $d$-variate exponential distribution by $\mathscr{E}_{d}(\boldsymbol{\mu},\boldsymbol{\rho})$, given specific positive mean vector $\boldsymbol{\mu}^{-1}:=(\mu_{1}^{-1},\ldots,\mu_{d}^{-1})^{\top}$ and correlation matrix $\boldsymbol{\rho}=(\rho_{ij})_{i,j\in\\{1,\ldots,d\\}}$. Similarly, a generic $d$-variate Poisson distribution is given by $\mathscr{P}_{d}(\boldsymbol{\mu},\boldsymbol{\rho})$, with positive mean vector $\boldsymbol{\mu}:=(\mu_{1},\ldots,\mu_{d})^{\top}$ and correlation matrix $\boldsymbol{\rho}$. See, e.g., Appendix 7.1 for more extensive exponential and Poisson models with possible behaviours in the negative correlation setup. The uncorrelated or independent $d$-variate exponential and Poisson will be written as $\mathscr{E}_{d}(\boldsymbol{\mu})$ and $\mathscr{P}_{d}(\boldsymbol{\mu})$, respectively, for $\boldsymbol{\rho}=\boldsymbol{I}_{d}$ the $d\times d$ unit matrix. Their respective $d$-variate probability density function (pdf) and probability mass function (pmf) are the product of $d$ univariate ones. According to [31] and following the recent univariate unification of the well- known (Fisher) dispersion and the (Jørgensen) variation indexes by Touré et al. [55], the relative variability index of $d$-variate nonnegative orthant distributions can be written as follows. Let $\boldsymbol{X}$ and $\boldsymbol{Y}$ be two random vectors on the same support $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$ and assume $\boldsymbol{m}:=\mathbb{E}\boldsymbol{X}=\mathbb{E}\boldsymbol{Y}$, $\boldsymbol{\Sigma}_{\boldsymbol{X}}:=\mathrm{cov}\boldsymbol{X}$ and $\mathbf{V}_{F_{\boldsymbol{Y}}}(\boldsymbol{m}):=\mathrm{cov}(\boldsymbol{Y})$ fixed, then the relative variability index of $\boldsymbol{X}$ with respect to $\boldsymbol{Y}$ is defined as the positive quantity $\mathrm{RWI}_{\boldsymbol{Y}}(\boldsymbol{X}):=\mathrm{tr}[\boldsymbol{\Sigma}_{\boldsymbol{X}}\mathbf{W}^{+}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})]\gtreqqless 1,$ (2.2) where “$\mathrm{tr}(\cdot)$” stands for the trace operator and $\mathbf{W}^{+}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})$ is the unique Moore- Penrose inverse of the associated matrix $\mathbf{W}_{F_{\boldsymbol{Y}}}(\boldsymbol{m}):=[\mathbf{V}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})]^{1/2}[\mathbf{V}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})]^{\top/2}$ to $\mathbf{V}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})$. From (2.2), $\mathrm{RWI}_{\boldsymbol{Y}}(\boldsymbol{X})\gtreqqless 1$ means the over- (equi- and under-variability) of $\boldsymbol{X}$ compared to $\boldsymbol{Y}$ is realized if $\mathrm{RWI}_{\boldsymbol{Y}}(\boldsymbol{X})>1$ ($\mathrm{RWI}_{\boldsymbol{Y}}(\boldsymbol{X})=1$ and $\mathrm{RWI}_{\boldsymbol{Y}}(\boldsymbol{X})<1$, respectively). The expression (2.2) of RWI does not appear to be very easy to handle in this general formulation on $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$, even the empirical version and interpretations. We now detail both multivariate cases of counting and of semicontinous. Their corresponding empirical versions are given in [25, 31]. ### 2.1 Relative Dispersion Indexes for Count Distributions For $\mathbb{T}_{d}^{+}=\mathbb{N}^{d}$, let $\mathbf{W}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})=\sqrt{\boldsymbol{m}}\sqrt{\boldsymbol{m}}^{\top}$ be the $d\times d$ matrix of rank 1. Then, $\boldsymbol{\Sigma}_{\boldsymbol{X}}\mathbf{W}^{+}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})$ of (2.2) is also of rank 1 and has only one positive eigenvalue, denoted by $\mathrm{GDI}(\boldsymbol{X}):=\frac{\sqrt{\mathbb{E}\boldsymbol{X}}^{\top}\,(\mathrm{cov}\boldsymbol{X})\sqrt{\mathbb{E}\boldsymbol{X}}}{\mathbb{E}\boldsymbol{X}^{\top}\mathbb{E}\boldsymbol{X}}\gtreqqless 1$ (2.3) and called generalized dispersion index of $\boldsymbol{X}$ compared to $\mathbf{Y}\sim\mathscr{P}_{d}(\mathbb{E}\mathbf{X})$ with $\mathbb{E}\mathbf{Y}=\mathbb{E}\mathbf{X}=\boldsymbol{m}$ ([25]). For $d=1$, $\mathrm{GDI}(X_{1})=\mathrm{var}X_{1}/\mathbb{E}X_{1}=\mathrm{DI}(X_{1})$ is the (Fisher) dispersion index with respect to the Poisson distribution. To derive this interpretation of GDI, we successively decompose the denominator of (2.3) as $\mathbb{E}\boldsymbol{X}^{\top}\mathbb{E}\boldsymbol{X}=\sqrt{\mathbb{E}\boldsymbol{X}}^{\top}\,(\mathrm{diag}\mathbb{E}\boldsymbol{X})\sqrt{\mathbb{E}\boldsymbol{X}}=[(\mathrm{diag}\sqrt{\mathbb{E}\boldsymbol{X}})\\!\sqrt{\mathbb{E}\boldsymbol{X}}]^{\top}(\boldsymbol{I}_{d})[(\mathrm{diag}\sqrt{\mathbb{E}\boldsymbol{X}})\\!\sqrt{\mathbb{E}\boldsymbol{X}}]$ (2.4) and the numerator of (2.3) by using also (2.1) as $\sqrt{\mathbb{E}\boldsymbol{X}}^{\top}\,(\mathrm{cov}\boldsymbol{X})\sqrt{\mathbb{E}\boldsymbol{X}}=[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\sqrt{\mathbb{E}\boldsymbol{X}}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{X}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\sqrt{\mathbb{E}\boldsymbol{X}}].$ Thus, $\mathrm{GDI}(\boldsymbol{X})$ makes it possible to compare the full variability of $\boldsymbol{X}$ (in the numerator) with respect to its expected uncorrelated Poissonian variability (in the denominator) which depends only on $\mathbb{E}\boldsymbol{X}$. In other words, the count random vector $\mathbf{X}$ is over- (equi- and under-dispersed) with respect to $\mathscr{P}_{d}(\mathbb{E}\mathbf{X})$ if $\mathrm{GDI}(\boldsymbol{X})>1$ ($\mathrm{GDI}(\boldsymbol{X})=1$ and $\mathrm{GDI}(\boldsymbol{X})<1$, respectively). This is a generalization in multivariate framework of the well- known (univariate) Fisher dispersion index by [25]. See, e.g., [4, 25] for illustrative examples. Also, we can modify $\mathrm{GDI}(\boldsymbol{X})$ to $\mathrm{MDI}(\boldsymbol{X})$, as marginal dispersion index, by replacing $\mathrm{cov}\boldsymbol{X}$ in (2.3) with $\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}}$ to obtain dispersion information only coming from the margins of $\boldsymbol{X}$. More generally, for two count random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$ on the same support $\mathbb{T}_{d}^{+}\subseteq\mathbb{N}^{d}$ with $\mathbb{E}\boldsymbol{X}=\mathbb{E}\boldsymbol{Y}$ and $\mathrm{GDI}(\boldsymbol{Y})>0$, the relative dispersion index is defined by $\mathrm{RDI}_{\boldsymbol{Y}}(\boldsymbol{X}):=\frac{\mathrm{GDI}(\boldsymbol{X})}{\mathrm{GDI}(\boldsymbol{Y})}=\frac{[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\sqrt{\mathbb{E}\boldsymbol{X}}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{X}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\sqrt{\mathbb{E}\boldsymbol{X}}]}{[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{Y}})\sqrt{\mathbb{E}\boldsymbol{Y}}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{Y}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{Y}})\sqrt{\mathbb{E}\boldsymbol{Y}}]}\gtreqqless 1;$ (2.5) i.e., the over- (equi- and under-dispersion) of $\boldsymbol{X}$ compared to $\boldsymbol{Y}$ is realized if $\mathrm{GDI}(\boldsymbol{X})>\mathrm{GDI}(\boldsymbol{Y})$ ($\mathrm{GDI}(\boldsymbol{X})=\mathrm{GDI}(\boldsymbol{Y})$ and $\mathrm{GDI}(\boldsymbol{X})<\mathrm{GDI}(\boldsymbol{Y})$, respectively). Obviously, GDI is a particular case of RDI with any general reference than $\mathscr{P}_{d}$. Consequently, many properties of GDI are easily extended to RDI. ### 2.2 Relative Variation Indexes for Semicontinuous Distributions Assuming here $\mathbb{T}_{d}^{+}=[0,\infty)^{d}$ and $\mathbf{W}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})=\boldsymbol{m}\boldsymbol{m}^{\top}$ another $d\times d$ matrix of rank 1. Then, we also have that $\boldsymbol{\Sigma}_{\boldsymbol{X}}\mathbf{W}^{+}_{F_{\boldsymbol{Y}}}(\boldsymbol{m})$ of (2.2) is of rank 1. Similar to (2.3), the generalized variation index of $\boldsymbol{X}$ compared to $\mathscr{E}_{d}(\mathbb{E}\mathbf{X})$ is defined by $\mathrm{GVI}(\boldsymbol{X}):=\frac{\mathbb{E}\boldsymbol{X}^{\top}\,(\mathrm{cov}\boldsymbol{X})\;\mathbb{E}\boldsymbol{X}}{(\mathbb{E}\boldsymbol{X}^{\top}\mathbb{E}\boldsymbol{X})^{2}}\gtreqqless 1;$ (2.6) i.e., $\mathbf{X}$ is over- (equi- and under-varied) with respect to $\mathscr{E}_{d}(\mathbb{E}\mathbf{X})$ if $\mathrm{GVI}(\boldsymbol{X})>1$ ($\mathrm{GVI}(\boldsymbol{X})=1$ and $\mathrm{GVI}(\boldsymbol{X})<1$, respectively); see [31]. Remark that when $d=1$, $\mathrm{GVI}(X_{1})=\mathrm{var}X_{1}/(\mathbb{E}X_{1})^{2}=\mathrm{VI}(X_{1})$ is the univariate (Jørgensen) variation index which is recently introduced by Abid et al. [2]. From (2.4) and using again (2.1) for rewritting the numerator of (2.6) as $\mathbb{E}\boldsymbol{X}^{\top}\,(\mathrm{cov}\boldsymbol{X})\;\mathbb{E}\boldsymbol{X}=[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\mathbb{E}\boldsymbol{X}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{X}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\mathbb{E}\boldsymbol{X}],$ $\mathrm{GVI}(\boldsymbol{X})$ of (2.6) can be interpreted as the ratio of the full variability of $\boldsymbol{X}$ with respect to its expected uncorrelated exponential $\mathscr{E}_{d}(\mathbb{E}\mathbf{X})$ variability which depends only on $\mathbb{E}\boldsymbol{X}$. Similar to $\mathrm{MDI}(\boldsymbol{X})$, we can define $\mathrm{MVI}(\boldsymbol{X})$ from $\mathrm{GVI}(\boldsymbol{X})$. See [31] for properties, numerous examples and numerical illustrations. The relative variation index is defined, for two semicontinuous random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$ on the same support $\mathbb{T}_{d}^{+}=[0,\infty)^{d}$ with $\mathbb{E}\boldsymbol{X}=\mathbb{E}\boldsymbol{Y}$ and $\mathrm{GVI}(\boldsymbol{Y})>0$, by $\mathrm{RVI}_{\boldsymbol{Y}}(\boldsymbol{X}):=\frac{\mathrm{GVI}(\boldsymbol{X})}{\mathrm{GVI}(\boldsymbol{Y})}=\frac{[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\mathbb{E}\boldsymbol{X}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{X}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{X}})\mathbb{E}\boldsymbol{X}]}{[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{Y}})\mathbb{E}\boldsymbol{Y}]^{\top}(\boldsymbol{\rho}_{\boldsymbol{Y}})\,[(\mathrm{diag}\\!\sqrt{\mathrm{var}\boldsymbol{Y}})\mathbb{E}\boldsymbol{Y}]}\gtreqqless 1;$ (2.7) i.e., the over- (equi- and under-variation) of $\boldsymbol{X}$ compared to $\boldsymbol{Y}$ is carried out if $\mathrm{GVI}(\boldsymbol{X})>\mathrm{GVI}(\boldsymbol{Y})$ ($\mathrm{GVI}(\boldsymbol{X})=\mathrm{GVI}(\boldsymbol{Y})$ and $\mathrm{GVI}(\boldsymbol{X})<\mathrm{GVI}(\boldsymbol{Y})$, respectively). Of course, RVI generalizes GVI for multivariate semicontinuous distributions. For instance, one refers to [31] for more details on its discriminating power in multivariate parametric models from two first moments. ## 3 Multivariate Orthant Associated Kernels Nonparametric techniques through associated kernels represent an alternative approach for multivariate orthant data. Let $\textbf{X}_{1},\ldots,\textbf{X}_{n}$ be independent and identically distributed (iid) nonnegative orthant $d$-variate random vectors with an unknown joint pdmf $f$ on $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$, for $d\geq 1$. Then the multivariate associated kernel estimator $\widetilde{f}_{n}$ of $f$ is expressed as $\widetilde{f}_{n}(\mathbf{x})=\frac{1}{n}\sum_{i=1}^{n}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i}),~{}~{}~{}\forall\mathbf{x}=(x_{1},\ldots,x_{d})^{\top}\in\mathbb{T}_{d}^{+},$ (3.1) where $\mathbf{H}$ is a given $d\times d$ bandwidth matrix (i.e., symmetric and positive definite) such that $\mathbf{H}\equiv\mathbf{H}_{n}\rightarrow\mathbf{0}_{\mathbf{d}}$ (the $d\times d$ null matrix) as $n\rightarrow\infty$, and $K_{\mathbf{x},\mathbf{H}}(\cdot)$ is a multivariate (orthant) associated kernel, parameterized by $\mathbf{x}$ and $\mathbf{H}$; see, e.g., [29]. More precisely, we have the following refined definition. ###### Definition 3.1 Let $\mathbb{T}_{d}^{+}$ be the support of the pdmf to be estimated, $\mathbf{x}\in\mathbb{T}_{d}^{+}$ a target vector and H a bandwidth matrix. A parameterized pdmf $\mathbf{K}_{\mathbf{x},\mathbf{H}}(\cdot)$ on support $\mathbb{S}_{\mathbf{x},\mathbf{H}}\subseteq\mathbb{T}_{d}^{+}$ is called ”multivariate orthant associated kernel” if the following conditions are satisfied: $\mathbf{x}\in\mathbb{S}_{\mathbf{x},\mathbf{H}},\;\mathbb{E}\mathcal{Z}_{\mathbf{x},\mathbf{H}}=\mathbf{x}+\mathbf{A}(\mathbf{x},\mathbf{H})\to\mathbf{x}\;and\;\mathrm{cov}\mathcal{Z}_{\mathbf{x},\mathbf{H}}=\mathbf{B}(\mathbf{x},\mathbf{H})\to\mathbf{0}_{d}^{+},$ where $\mathcal{Z}_{\mathbf{x},\mathbf{H}}$ denotes the corresponding orthant random vector with pdmf $\mathbf{K}_{\mathbf{x},\mathbf{H}}$ such that vector $\mathbf{A}(\mathbf{x},\mathbf{H})\to\mathbf{0}$ (the $d$-dimentional null vector) and positive definite matrix $\mathbf{B}(\mathbf{x},\mathbf{H})\to\mathbf{0}_{d}^{+}$ as $\mathbf{H}\to\mathbf{0}_{d}$ (the $d\times d$ null matrix), and $\mathbf{0}_{d}^{+}$ stands for a symmetric matrix with entries $u_{ij}$ for $i,j=1,\dots,d$ such that $u_{ij}\in[0,1)$. This definition exists in the univariate count case of [26, 33] and encompasses the multivariate one by [29]. The choice of the orthant associated kernel satisfying $\lim\limits_{\mathbf{H}\rightarrow\mathbf{0}_{\mathbf{d}}}\mathrm{Cov}(\mathcal{Z}_{\mathbf{x},\mathbf{H}})=\mathbf{0}_{\mathbf{d}}$ assures the convergence of its corresponding estimator named of the second order. Otherwise, the convergence of its corresponding estimator is not guarantee for $u_{ij}\in(0,1)$, a right neighborhood of $0$, in Definition 3.1 and it is said a consistent first-order smoother; see, e.g., [26] for discrete kernels. In general, $d$-under-dispersed count associated kernels are appropriated for both small and moderate sample sizes; see, e.g., [26] for univariate cases. As for the selection of the bandwidth $\mathbf{H}$, it is very crucial because it controls the degree of smoothing and the form of orientation of the kernel. As a matter of fact, a simplification can be obtained by considering a diagonal matrix $\mathbf{H}=\mathbf{diag}_{d}(h_{j})$. Since it is challenging to obtain a full multivariate orthant distribution $\mathbf{K}_{\mathbf{x},\mathbf{H}}(\cdot)$ for building a smoother, several authors suggest the product of univariate orthant associated kernels, $\mathbf{K}_{\mathbf{x},\mathbf{H}}(\cdot)=\prod_{j=1}^{d}K_{x_{j},h_{j}}(\cdot),$ (3.2) where $K_{x_{j},h_{j}}$, $j=1,\ldots,d$, belong either to the same family or to different families of univariate orthant associated kernels. The below two subsections shall be devoted to summaries of discrete and semicontinuous univariate associated kernels. Before showing some main properties of the associated kernel estimator (3.1), let us recall that the family of $d$-variate classical (symmetric) kernels $\mathbf{K}$ on $\mathbb{S}_{d}\subseteq\mathbb{R}^{d}$ (e.g., [47, 49, 60]) can be also presented as (classical) associated kernels. Indeed, from (3.1) and writting for instance $\mathbf{K}_{\mathbf{x},\mathbf{H}}(\cdot)=(\det\mathbf{H})^{-1/2}\mathbf{K}\left[\mathbf{H}^{-1/2}(\mathbf{x}-\cdot)\right]$ where “$\det$” is the determinant operator, one has $\mathbb{S}_{\mathbf{x},\mathbf{H}}=\mathbf{x}-\mathbf{H}^{-1/2}\mathbb{S}_{d}$, $\mathbf{A}(\mathbf{x},\mathbf{H})=\mathbf{0}$ and $\mathbf{B}(\mathbf{x},\mathbf{H})=\mathbf{H}^{1/2}\boldsymbol{I}_{d}\mathbf{H}^{1/2}=\mathbf{H}$. In general, one uses the classical (associated) kernels for smoothing continuous data or pdf having support $\mathbb{T}_{d}=\mathbb{R}^{d}$. The purely nonparametric estimator (3.1) with multivariate associated kernel, $\widetilde{f}_{n}$ of $f$, is generally defined up to the normalizing constant $C_{n}$. Several simulation studies (e.g., [29, Table 3.1]) are shown that $C_{n}=C_{n}(\mathbf{K},\mathbf{H})$ (depending on samples, associated kernels and bandwidths) is approximatively $1$. Without loss of generality, one here assumes $C_{n}=1$ as for all classical (associated) kernel estimators of pdf. The following proposition finally proves its mean behaviour and variability through the integrated bias and integrated variance of $\widetilde{f}_{n}$, respectively. In what follows, let us denote by $\boldsymbol{\nu}$ the reference measure (Lebesgue or counting) on the nonnegative orthant set $\mathbb{T}_{d}^{+}$ and also on any set $\mathbb{T}_{d}\subseteq\mathbb{R}^{d}$. ###### Proposition 3.2 Let $C_{n}:=\int_{\mathbb{T}_{d}}\widetilde{f}_{n}(\mathbf{x})\boldsymbol{\nu}(d\mathbf{x})=C_{n}(\mathbf{K},\mathbf{H})$. Then, for all $n\geq 1$: $\mathbb{E}(C_{n})=1+\int_{\mathbb{T}_{d}}\mathrm{Bias}\\{\widetilde{f}_{n}(\mathbf{x})\\}\boldsymbol{\nu}(d\mathbf{x})\;\;\;and\;\;\;\mathrm{var}(C_{n})=\int_{\mathbb{T}_{d}}\mathrm{var}\\{\widetilde{f}_{n}(\mathbf{x})\\}\boldsymbol{\nu}(d\mathbf{x}).$ ###### Proof 1 Let $n\geq 1$. One successively has $\mathbb{E}(C_{n})=\int_{\mathbb{T}_{d}}\left[f(\mathbf{x})+\mathbb{E}\\{\widetilde{f}_{n}(\mathbf{x})\\}-f(\mathbf{x})\right]\boldsymbol{\nu}(d\mathbf{x})=\int_{\mathbb{T}_{d}}f(\mathbf{x})\boldsymbol{\nu}(d\mathbf{x})+\int_{\mathbb{T}_{d}}\left[\mathbb{E}\\{\widetilde{f}_{n}(\mathbf{x})\\}-f(\mathbf{x})\right]\boldsymbol{\nu}(d\mathbf{x}),$ which leads to the first result because $f$ is a pdmf on $\mathbb{T}_{d}$. The second result on $\mathrm{var}(C_{n})$ is trivial. The following general result is easily deduced from Proposition 3.2. To our knwoledge, it appears to be new and interesting in the framework of the pdmf (associated) kernel estimators. ###### Corollary 3.3 If $C_{n}=1$, for all $n\geq 1$, then: $\int_{\mathbb{T}_{d}}\mathrm{Bias}\\{\widetilde{f}_{n}(\mathbf{x})\\}\boldsymbol{\nu}(d\mathbf{x})=0$ and $\int_{\mathbb{T}_{d}}\mathrm{var}\\{\widetilde{f}_{n}(\mathbf{x})\\}\boldsymbol{\nu}(d\mathbf{x})=0$. In particular, Corollary 3.3 holds for all classical (associated) kernel estimators. The two following properties on the corresponding orthant multivariate associated kernels shall be needed subsequently. (K1) There exists the second moment of $\mathbf{K}_{\mathbf{x},\mathbf{H}}$: $\mu_{j}^{2}(\mathbf{K}_{\mathbf{x},\mathbf{H}}):=\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}}\bigcap\mathbb{T}_{d}^{+}}u_{j}^{2}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{u})\boldsymbol{\nu}(d\mathbf{u})<\infty,\;\;\;\forall j=1,\ldots,d.$ (K2) There exists a real largest number $r=r(\mathbf{K}_{\mathbf{x},\mathbf{H}})>0$ and $0<c(\mathbf{x})<\infty$ such that $\|\|\mathbf{K}_{\mathbf{x},\mathbf{H}}\|\|_{2}^{2}:=\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}}\bigcap\mathbb{T}_{d}^{+}}\\{\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{u})\\}^{2}\boldsymbol{\nu}(d\mathbf{u})\leq c(\mathbf{x})(\det\mathbf{H})^{-r}.$ In fact, (K1) is a necessary condition for smoothers to have a finite variance and (K2) can be deduced from the continuous univariate cases (e.g., [24]) and also from the discrete ones (e.g., [26]). We now establish both general asymptotic behaviours of the pointwise bias and variance of the nonparametric estimator (3.1) on the nonnegative orthant set $\mathbb{T}_{d}^{+}$; its proof is given in Appendix 7.2. For that, we need the following assumptions by endowing $\mathbb{T}_{d}^{+}$ with the Euclidean norm $\|\|\cdot\|\|$ and the associated inner product $\langle\cdot,\cdot\rangle$ such that $\langle\mathbf{a},\mathbf{b}\rangle=\mathbf{a}^{\top}\mathbf{b}$. (a1) The unknown pdmf $f$ is bounded function and twice differentiable or finite difference in $\mathbb{T}_{d}^{+}$ and $\nabla f(\mathbf{x})$ and $\mathcal{H}f(\mathbf{x})$ denote respectively the gradient vector (in continuous or discrete sense, respectively) and the corresponding Hessian matrix of the function $f$ at $\mathbf{x}$. (a2) There exists a positive real number $r>0$ such that $\|\|K_{\mathbf{x},\mathbf{H}_{n}}\|\|_{2}^{2}(\det\mathbf{H}_{n})^{r}\to c_{1}(\mathbf{x})>0$ as $n\to\infty$. Note that (a2) is obviously a consequence of (K2). ###### Proposition 3.4 Under the assumption (a1) on $f$, then the estimator $\widetilde{f}_{n}$ in (3.1) of $f$ verifies $\mathrm{Bias}\\{\widetilde{f}_{n}(\mathbf{x})\\}=\left\langle\nabla f(\mathbf{x}),\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right\rangle+\frac{1}{2}\operatorname{tr}\left\\{{\cal H}f\left(\mathbf{x}\right)\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})+\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)^{\mathsf{T}}\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right]\right\\}+o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\},$ (3.3) for any $\mathbf{x}\in\mathbb{T}_{d}^{+}$. Moreover, if (a2) holds then $\mathrm{var}\\{\widetilde{f}_{n}(\mathbf{x})\\}=\frac{1}{n}f(\mathbf{x})\|\|\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}\|\|_{2}^{2}+o\left[\frac{1}{n(\det\mathbf{H}_{n})^{r}}\right].$ (3.4) For $d=1$ and according to the proof of Proposition 3.4, one can easily write $\mathbb{E}\widehat{f}_{n}(x)$ as follows: $\mathbb{E}\widehat{f}_{n}(x)=\mathbb{E}f(\mathcal{Z}_{x,h})=\sum_{k\geq 0}\frac{1}{k!}\mathbb{E}\left(\mathcal{Z}_{x,h}-\mathbb{E}\mathcal{Z}_{x,h}\right)^{k}f^{(k)}(\mathbb{E}\mathcal{Z}_{x,h}),$ where $f^{(k)}$ is the $k$th derivative or finite difference of the pdmf $f$ under the existence of the centered moment of order $k\geq 2$ of $\mathcal{Z}_{x,h}$. Concerning bandwidth matrix selections in a multivariate associated kernel estimator (3.1), one generally use the cross-validation technique (e.g., [26, 27, 28, 29, 56]). However, it is tedious and less precise. Many papers have recently proposed Bayesian approaches (e.g., [7, 8, 50, 52, 59, 61] and references therein). In particular, they have recommended local Bayesian for discrete smoothing of pmf (e.g., [7, 8, 23]) and adaptive one for continuous smoothing of pdf (e.g., [50, 52, 59]). Denote $\mathcal{M}$ the set of positive definite [diagonal] matrices [from (3.2), resp.] and let $\pi$ be a given suitable prior distribution on $\mathcal{M}$. Under the squared error loss function, the Bayes estimator of $\mathbf{H}$ is the mean of the posterior distribution. Then, the local Bayesian bandwidth at the target $\mathbf{x}\in\mathbb{T}_{d}^{+}$ takes the form $\widetilde{\mathbf{H}}(\mathbf{x}):=\int_{\mathcal{M}}\mathbf{H}\pi(\mathbf{H})\widetilde{f}_{n}(\mathbf{x})d\mathbf{H}\left[\int_{\mathcal{M}}\widetilde{f}_{n}(\mathbf{x})\pi(\mathbf{H})d\mathbf{H}\right]^{-1},\;\;\mathbf{x}\in\mathbb{T}_{d}^{+},$ (3.5) and the adaptive Bayesian bandwidth for each observation $\mathbf{X}_{i}\in\mathbb{T}_{d}^{+}$ of $\mathbf{X}$ is given by $\widetilde{\mathbf{H}}_{i}:=\int_{\mathcal{M}}\mathbf{H}_{i}\pi(\mathbf{H}_{i})\widetilde{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i})d\mathbf{H}_{i}\left[\int_{\mathcal{M}}\widetilde{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i})\pi(\mathbf{H}_{i})d\mathbf{H}_{i}\right]^{-1},\;\;i=1,\ldots,n,$ (3.6) where $\widetilde{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i})$ is the leave-one- out associated kernel estimator of $f(\mathbf{X}_{i})$ deduced from (3.1) as $\widetilde{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i}):=\frac{1}{n-1}\sum_{\ell=1,\ell\neq i}^{n}\mathbf{K}_{\mathbf{X}_{i},\mathbf{H}_{i}}(\mathbf{X}_{\ell}).$ (3.7) Note that the (global) cross-validation bandwidth matrix $\widetilde{\mathbf{H}}_{CV}$ and the global Bayesian one $\widetilde{\mathbf{H}}_{B}$ are obtained, respectively, from (3.7) as $\widetilde{\mathbf{H}}_{CV}:=\mathrm{arg}\min_{\mathbf{H}\in\mathcal{M}}\left[\int_{\mathbb{T}_{d}^{+}}\\{\widetilde{f}_{n}(\mathbf{x})\\}^{2}\boldsymbol{\nu}(d\mathbf{x})-\frac{2}{n}\sum_{i=1}^{n}\widetilde{f}_{n,\mathbf{H},-i}(\mathbf{X}_{i})\right]$ and $\widetilde{\mathbf{H}}_{B}:=\int_{\mathcal{M}}\mathbf{H}\pi(\mathbf{H})d\mathbf{H}\prod_{i=1}^{n}\widetilde{f}_{n,\mathbf{H},-i}(\mathbf{X}_{i})d\mathbf{H}\left[\int_{\mathcal{M}}\pi(\mathbf{H})d\mathbf{H}\prod_{i=1}^{n}\widetilde{f}_{n,\mathbf{H},-i}(\mathbf{X}_{i})\right]^{-1}.$ ### 3.1 Discrete Associated Kernels We only present three main and useful families of univariate discrete associated kernels for (3.2) and satisfying (K1) and (K2). ###### Example 3.5 (categorical) For fixed $c\in\\{2,3,\ldots\\}$, the number of categories and $\mathbb{T}_{1}^{+}=\\{0,1,\ldots,c-1\\}$, one defines the Dirac discrete uniform (DirDU) kernel by $K_{x,h}^{DirDU}(u)=(1-h)^{\mathds{1}_{u=x}}\left(\frac{h}{c-1}\right)^{1-\mathds{1}_{u=x}},$ for $x\in\\{0,1,\ldots,c-1\\}$, $h\in(0,1]$, with $\mathbb{S}_{x}:=\\{0,1,\ldots,c-1\\}=\mathbb{T}_{1}^{+}$, $A(x,h)=h\\{c/2-x-x/(c-1)\\}$ and $B(x,h)=h\\{c(2c-1)/6+x^{2}-xc+x^{2}/(c-1)\\}-h^{2}\\{c/2-x-x/(c-1)\\}^{2}$. It has been introduced in multivariate setup by Aitchison and Aitken [3] and investigated as a discrete associated kernel which is symmetric to the target $x$ by [26] in univariate case; see [8] for a Bayesian approach in multivariate setup. Note here that its normalized constant is always $1=C_{n}$. ###### Example 3.6 (symmetric count) For fixed $m\in\mathbb{N}$ and $\mathbb{T}_{1}^{+}\subseteq\mathbb{Z}$, the symmetric count triangular kernel is expressed as $K_{x,h}^{SCTriang}(u)=\frac{(m+1)^{h}-\|u-x\|^{h}}{P(m,h)}\mathds{1}_{\\{x,x\pm 1,\ldots,x\pm m\\}}(u),$ for $x\in\mathbb{T}_{1}^{+}$, $h>0$, with $\mathbb{S}_{x}:=\\{x,x\pm 1,\ldots,x\pm m\\}$, $P(m,h)=(2m+1)(m+1)^{h}-2\sum_{\ell=0}^{m}\ell^{h}$, $A(x,h)=0$ and $\displaystyle B(x,h)$ $\displaystyle=$ $\displaystyle\frac{1}{P(m,h)}\left\\{\frac{m(2m+1)(m+1)^{h+1}}{3}-2\sum_{\ell=0}^{m}\ell^{h+2}\right\\}$ $\displaystyle\simeq$ $\displaystyle h\left\\{\frac{m(2m^{2}+3m+1)}{3}\log(m+1)-2\sum_{\ell=1}^{m}\ell^{2}\log\ell\right\\}+O(h^{2}),$ where $\simeq$ holds for $h$ sufficiently small. It has been first proposed by Kokonendji et al. [28] and then completed in [32] with an asymmetric version for solving the problem of boundary bias in count kernel estimation. ###### Example 3.7 (standard count) Let $\mathbb{T}_{1}^{+}\subseteq\mathbb{N}$, the standard binomial kernel is defined by $K_{x,h}^{Binomial}(u)=\frac{(x+1)!}{u!(x+1-u)!}\left(\frac{x+h}{x+1}\right)^{u}\left(\frac{1-h}{x+1}\right)^{x+1-u}\mathds{1}_{\\{0,1,\ldots,x+1\\}}(u),$ for $x\in\mathbb{T}_{1}^{+}$, $h\in(0,1]$, with $\mathbb{S}_{x}:=\\{0,1,\ldots,x+1\\}$, $A(x,h)=h$ and $B(x,h)=(x+h)(1-h)/(x+1)\to x/(x+1)\in[0,1]$ as $h\to 0$. Here $B(x,h)$ tends to $x/(x+1)\in[0,1)$ when $h\to 0$ and the new Definition 3.1 holds. This first-order and under-dispersed binomial kernel is introduced in [26] which becomes very useful for smoothing count distribution through small or moderate sample size; see, e.g., [7, 8, 23] for Bayesian approaches and some references therein. In addition, we have the standard Poisson kernel where $K_{x,h}^{Poisson}$ follows the equi-dispersed Poisson distribution with mean $x+h$, $\mathbb{S}_{x}:=\mathbb{N}=:\mathbb{T}_{1}^{+}$, $A(x,h)=h$ and $B(x,h)=x+h\to x\in\mathbb{N}$ as $h\to 0$. Recently, Huang et al. [17] have introduced the Conway-Maxwell-Poisson kernel by exploiting its under-dispersed part and its second-order consistency which can be improved via the mode- dispersion approach of [37]; see also [16, Section 2.4]. ### 3.2 Semicontinuous Associated Kernels Now, we point out eight main and useful families of univariate semicontinuous associated kernels for (3.2) and satisfying (K1) and (K2). Which are gamma (G) of [9] (see also [14]), inverse gamma (Ig) (see also [42]) and log-normal 2 (LN2) by [37], inverse Gaussian (IG) and reciprocal inverse Gaussian by [46] (see also [18]), log-normal 1 (LN1) and Birnbaum-Saunders by [19] (see also [38, 41]), and Weibull (W) of [45] (see also [41]). It is noteworthy that the link between LN2 of [37] and LN1 of [19] is through changing $(x,h)$ to $(x\exp(h^{2}),2\sqrt{\log(1+h})$. Several other semicontinuous could be constructed by using the mode-dispersion technique of [37] from any semicontinuous distribution which is unimodal and having a dispersion parameter. Recently, one has the scaled inverse chi-squared kernel of [11]. Table 4.1 summarizes these eight semicontinuous univariate associated kernels with their ingredients of Definition 3.1 and order of preference (O.) obtained graphically. In fact, the heuristic classification (O.) is done through the behaviour of the shape and scale of the associated kernel around the target $x$ at the edge as well as inside; see Figure 4.1 for edge and Figure 4.2 for inside. Among these eight kernels, we thus have to recommend the five first univariate associated kernels of Table 4.1 for smoothing semicontinuous data. This approach could be improved by a given dataset; see, e.g., [35] for cumulative functions. ## 4 Semiparametric Kernel Estimation with $d$-Variate Parametric Start We investigate the semiparametric orthant kernel approach which is a compromise between the pure parametric and the nonparametric methods. This concept was proposed by Hjort and Glad [15] for continuous data, treated by Kokonendji et al. [27] for discrete univariate data and, recently, studied by Kokonendji et al. [33] with an application to radiation biodosimetry. Table 4.1: Eight semicontinuous univariate associated kernels on $\mathbb{S}_{x,h}\subseteq[0,\infty)$ and classifyed by ”O.” O. \| Name \| $K_{x,h}(u)$ \| $A(x,h)$ \| $B(x,h)$ ---\|---\|---\|---\|--- 1 \| LN2 [37] \| $(uh\sqrt{2\pi})^{-1}\exp\left(\left[\log\\{x\exp(h^{2})\\}-\log u\right]/2h^{2}\right)$ \| $x[\exp(3h^{2}\\!/2)\\!-\\!1]$ \| $x^{2}\exp(3h^{2})[\exp(h^{2})-1]$ 2 \| W [45] \| $[\Gamma(h)/x][u\Gamma(1+h)/x]^{1/h-1}\exp\left\\{-[u\Gamma(1+h)/x]^{1/h}\right\\}$ \| $0$ \| $x^{2}\\!\left[\Gamma(1\\!+\\!2h)/\Gamma^{2}(1\\!+\\!2h)\\!-\\!1\right]$ 3 \| G [9] \| $h^{-1-x/h}u^{x/h}\exp(-u/h)/\Gamma(1+x/h)$ \| $h$ \| $(x+h)h$ 4 \| BS [19] \| $(uh\sqrt{2\pi})^{-1}\\!\left[(xu)^{-1/2}\\!+\\!(x/u^{3})^{-1/2}\right]\exp\left[(2\\!-\\!u/x\\!-\\!x/u)/2h\right]$ \| $xh/2$ \| $x^{2}h(2+5h/2)/2$ 5 \| Ig [37] \| $h^{1-1/xh}u^{-1/xh}\exp(-1/uh)/\Gamma(1/xh-1)$ \| $2x^{2}h/(1-2xh)$ \| $x^{3}h/[(1-3xh)(1-2xh)^{2}]$ 6 \| RIG [46] \| $(\sqrt{2\pi uh})^{-1}\exp\left\\{[x-h][2-(x-h)/u-u/(x-h)]/2h\right\\}$ \| $0$ \| $(x-h)h$ 7 \| IG [46] \| $(\sqrt{2\pi hu^{3}})^{-1}\exp\left\\{[2-x/u-u/x)]/2hx\right\\}$ \| $0$ \| $x^{3}h$ 8 \| LN1 [19] \| $(u\sqrt{8\pi\log(1+h)})^{-1}\exp\left(-[\log u-\log x]^{2}/[8\log(1+h)]\right)$ \| $xh(h+2)$ \| $x^{2}(1+h)^{4}[(1+h)^{4}-1]$ $\Gamma(v):=\int_{0}^{\infty}s^{v-1}\exp(-s)ds$ is the classical gamma function with $v>0$. Figure 4.1: Comparative graphics of the eight univariate semicontinuous associated kernels of Table 4.1 on the edge ($x=0.3$) with $h=0.1$ and $h=0.4$. Figure 4.2: Comparative graphics of the eight univariate semicontinuous associated kernels of Table 4.1 inside ($x=2.3$) with $h=0.1$ and $h=0.4$. Without loss of generality, we here assume that any $d$-variate pdmf $f$ can be formulated (e.g., [30] for $d=1$) as $f(\mathbf{x})=w(\mathbf{x};\boldsymbol{\theta})\,p_{d}(\mathbf{x};\boldsymbol{\theta}),\;\;\;\forall\mathbf{x}\in\mathbb{T}_{d}^{+},$ (4.1) where $p_{d}(\cdot;\boldsymbol{\theta})$ is the non-singular parametric part according to a reference $d$-variate distribution with corresponding unknown parameters $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{k})^{\top}$ and $w(\cdot;\boldsymbol{\theta}):=f(\cdot)/p_{d}(\cdot;\boldsymbol{\theta})$ is the unknown orthant weight function part, to be estimated with a multivariate orthant associated kernel. The weight function at each point can be considered as the local multiplicative correction factor aimed to accommodate any pointwise departure from the reference $d$-variate distribution. However, one cannot consider the best fit of parametric models as the start distribution in this semiparametric approach. Because the corresponding weight function is close to zero and becomes a noise which is unappropriated to smooth by an associated kernel, especially for the continuous cases. Let $\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ be iid nonnegative orthant $d$-variate random vectors with unknown pdmf $f$ on $\mathbb{T}_{d}^{+}\subseteq[0,\infty)^{d}$. The semiparametric estimator of (4.1) with (3.2) is expressed as follows: $\displaystyle\widehat{f}_{n}(\mathbf{x})$ $\displaystyle=$ $\displaystyle p_{d}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i})$ (4.2) $\displaystyle=$ $\displaystyle\frac{1}{n}\sum_{i=1}^{n}\frac{p_{d}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})}{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i}),\;\;\;\mathbf{x}\in\mathbb{T}_{d}^{+},$ where $\widehat{\boldsymbol{\theta}}_{n}$ is the estimated parameter of $\boldsymbol{\theta}$. From (4.2), we then deduce the nonparametric orthant associated kernel estimate $\widetilde{w}_{n}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i})$ (4.3) of the weight function $x\mapsto w(\mathbf{x};\widehat{\boldsymbol{\theta}_{n}})$ which depends on $\widehat{\boldsymbol{\theta}}_{n}$. One can observe that Proposition 3.2 also holds for $\widehat{f}_{n}(\cdot)=p_{d}(\cdot;\widehat{\boldsymbol{\theta}}_{n})\widetilde{w}_{n}(\cdot;\widehat{\boldsymbol{\theta}}_{n})$. However, we have to prove below the analogous of Proposition 3.4. ### 4.1 Known $d$-Variate Parametric Model Let $p_{d}(\cdot;\boldsymbol{\theta}_{0})$ be a fixed orthant distribution in (4.1) with $\boldsymbol{\theta}_{0}$ known. Writing $f(\mathbf{x})=p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\,w(\mathbf{x})$, we estimate the nonparametric weight function $w$ by $\widetilde{w}_{n}(\mathbf{x})=n^{-1}\sum_{i=1}^{n}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i})/p_{d}(\mathbf{X}_{i};\boldsymbol{\theta}_{0})$ with an orthant associated kernel method, resulting in the estimator $\widehat{f}_{n}(\mathbf{x})=p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\widetilde{w}_{n}(\mathbf{x})=\frac{1}{n}\sum\limits_{i=1}^{n}\frac{p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})}{p_{d}(\mathbf{X}_{i};\boldsymbol{\theta}_{0})}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i}),\;\;\;\;\;\mathbf{x}\in\mathbb{T}_{d}^{+}.$ (4.4) The following proposition is proven in Appendix 7.2. ###### Proposition 4.1 Under the assumption (a1) on $f(\cdot)=p_{d}(\cdot;\boldsymbol{\theta}_{0})w(\cdot)$, then the estimator $\widehat{f}_{n}(\cdot)=p_{d}(\cdot;\boldsymbol{\theta}_{0})\widetilde{w}_{n}(\cdot)$ in (4.4) of $f$ satisfies $\displaystyle\mathrm{Bias}\\{\widehat{f}_{n}(\mathbf{x})\\}$ $\displaystyle=$ $\displaystyle p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\left[w(\mathbf{x})-f(\mathbf{x})\\{p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\\}^{-1}+\left\langle\nabla w(\mathbf{x}),\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right\rangle\right]$ $\displaystyle+\frac{1}{2}\;p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\left(\operatorname{tr}\left\\{{\cal H}w\left(\mathbf{x}\right)\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})+\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)^{\mathsf{T}}\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right]\right\\}\right)$ $\displaystyle+\left(1+o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\}\right),$ for any $\mathbf{x}\in\mathbb{T}_{d}^{+}$. Furthermore, if (a2) holds then one has $\mathrm{var}\\{\widehat{f}_{n}(\mathbf{x})\\}=\mathrm{var}\\{\widetilde{f}_{n}(\mathbf{x})\\}$ of (3.4). It is expected that the bias here is quite different from that of (3.3). ### 4.2 Unknown $d$-Variate Parametric Model Let us now consider the more realistic and practical semiparametric estimator $\widehat{f}_{n}(\cdot)=p_{d}(\cdot;\widehat{\boldsymbol{\theta}}_{n})\widetilde{w}_{n}(\cdot;\widehat{\boldsymbol{\theta}}_{n})$ presented in (4.2) of $f(\cdot)=p_{d}(\cdot;\boldsymbol{\theta})w(\cdot;\boldsymbol{\theta})$ in (4.1) such that the parametric estimator $\widehat{\boldsymbol{\theta}}_{n}$ of $\boldsymbol{\theta}$ can be obtained by the maximum likelihood method; see [15] for quite a general estimator of $\boldsymbol{\theta}$. In fact, if the $d$-variate parametric model $p_{d}(\cdot;\boldsymbol{\theta})$ is misspecified then this $\widehat{\boldsymbol{\theta}}_{n}$ converges in probability to the pseudotrue value $\boldsymbol{\theta}_{0}$ satisfying $\boldsymbol{\theta}_{0}:=\arg\min\limits_{\boldsymbol{\theta}}\int_{\mathbf{x}\in\mathbb{T}_{d}^{+}}f(\mathbf{x})\log[f(\mathbf{x})/p_{d}(\mathbf{x};\boldsymbol{\theta})]\boldsymbol{\nu}(d\mathbf{x})$ from the Kullback-Leibler divergence (see, e.g., [58]). By writting $p_{0}(\cdot):=p_{d}(\cdot;\boldsymbol{\theta}_{0})$ this best $d$-variate parametric approximant, but this $p_{0}(\cdot)$ is not explicitly expressible as the one in (4.4). According to [15] (see also [27]), we can represent the proposed estimator $\widehat{f}_{n}(\cdot)=p_{d}(\cdot;\widehat{\boldsymbol{\theta}}_{n})\widetilde{w}_{n}(\cdot;\widehat{\boldsymbol{\theta}}_{n})$ in (4.2) as $\widehat{f}_{n}(\mathbf{x})\doteq\frac{1}{n}\sum_{i=1}^{n}\frac{p_{0}(\mathbf{x})}{p_{0}(\mathbf{X}_{i})}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i}),\;\;\;\mathbf{x}\in\mathbb{T}_{d}^{+}.$ (4.5) Thus, the following result provides approximate bias and variance. We omit its proof since it is analogous to the one of Proposition 4.1. ###### Proposition 4.2 Let $p_{0}(\cdot):=p_{d}(\cdot;\boldsymbol{\theta}_{0})$ be the best $d$-variate approximant of the unknown pdmf $f(\cdot)=p_{d}(\cdot;\boldsymbol{\theta})w(\cdot;\boldsymbol{\theta})$ as (4.1) under the Kullback–Leibler criterion, and let $w(\cdot):=f(\cdot)/p_{0}(\cdot)$ be the corresponding $d$-variate weight function. As $n\to\infty$ and under the assumption (a1) on $f$, then the estimator $\widehat{f}_{n}(\cdot)=p_{d}(\cdot;\widehat{\boldsymbol{\theta}}_{n})\widetilde{w}_{n}(\cdot;\widehat{\boldsymbol{\theta}}_{n})$ in (4.2) of $f$ and refomulated in (4.5) satisfies $\displaystyle\mathrm{Bias}\\{\widehat{f}_{n}(\mathbf{x})\\}$ $\displaystyle=$ $\displaystyle p_{0}(\mathbf{x})\left[w(\mathbf{x})-f(\mathbf{x})\\{p_{0}(\mathbf{x})\\}^{-1}+\left\langle\nabla w(\mathbf{x}),\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right\rangle\right]$ $\displaystyle+\frac{1}{2}\;p_{0}(\mathbf{x})\left(\operatorname{tr}\left\\{{\cal H}w\left(\mathbf{x}\right)\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})+\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)^{\mathsf{T}}\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right]\right\\}\right)$ $\displaystyle+\left(1+o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\}+n^{-2}\right),$ for any $\mathbf{x}\in\mathbb{T}_{d}^{+}$. Furthermore, if (a2) holds then we have $\mathrm{var}\\{\widehat{f}_{n}(\mathbf{x})\\}=\mathrm{var}\\{\widetilde{f}_{n}(\mathbf{x})\\}$ of (3.4). Once again, the bias is different from that of (3.3). Thus, the proposed semiparametric estimator $\widehat{f}_{n}$ in (4.2) of $f$ can be shown to be better or not than the traditional nonparametric one $\widetilde{f}_{n}$ in (3.1). The following subsection provides a practical solution. ### 4.3 Model Diagnostics The estimated weight function $\widetilde{w}_{n}(\mathbf{x},\widehat{\boldsymbol{\theta}}_{n})$ given in (4.3) provides useful information for model diagnostics. The $d$-variate weight function $w(\cdot)$ is equal one if the $d$-variate parametric start model $p_{d}(\cdot;\boldsymbol{\theta}_{0})$ is indeed the true pdmf. Hjort and Glad [15] proposed to check this adequacy by examining a plot of the weight function for various potential models with pointwise confidence bands to see wether or not $w(\mathbf{x})=1$ is reasonable. See also [27, 33] for univariate count setups. In fact, without technical details here we use the model diagnostics for verifying the adequacy of the model by examining a plot of $\mathbf{x}\mapsto\widetilde{w}_{n}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})$ or $\widetilde{W}_{n}(\mathbf{x}):=\log\widetilde{w}_{n}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})=\log[\widehat{f}_{n}(\mathbf{x})/p_{d}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})]$ (4.6) for all $\mathbf{x}=\mathbf{X}_{i}$, $i=1,\ldots,n$, with a pointwise confidence band of $\pm 1.96$ for large $n$; that is to see how far away it is from zero. More precisely, for instance, $\widetilde{W}_{n}$ is $<5\%$ for pure nonparametric, it belongs to $[5\%,95\%]$ for semiparametric, and it is $>95\%$ for full parametric models. It is noteworthy that the retention of pure nonparametric means the inconvenience of parametric part considered in this approach; hence, the orthant dataset is left free. ## 5 Semicontinuous Examples of Application with Discussions For a practical implementation of our approach, we propose to use the popular multiple gamma kernels as (3.2) by selecting the adaptive Bayesian procedure of [52] to smooth $\widetilde{w}_{n}(\mathbf{x};\widehat{\boldsymbol{\theta}}_{n})$. Hence, we shall gradually consider $d$-variate semicontinuous cases with $d=1,2,3$ for real datasets. All computations and graphics have been done with the R software [44]. ### 5.1 Adaptive Bayesian Bandwidth Selection for Multiple Gamma Kernels From Table 4.1, the function $G_{x,h}(\cdot)$ is the gamma kernel [9] given on the support $\mathbb{S}_{x,h}=[0,\infty)=\mathbb{T}_{1}^{+}$ with $x\geq 0$ and $h>0$: $G_{x,h}(u)=\dfrac{u^{x/h}}{\Gamma\left(1+x/h\right)h^{1+x/h}}\exp{\left(-\dfrac{u}{h}\right)}\mathds{1}_{[0,\infty)}(u),$ where $\mathds{1}_{E}$ denotes the indicator function of any given event $E$. This gamma kernel $G_{x,h}(\cdot)$ appears to be the pdf of the gamma distribution, denoted by $\mathcal{G}(1+x/h,h)$ with shape parameter $1+x/h$ and scale parameter $h$. The multiple gamma kernel from (3.2) is written as $\mathbf{K}_{\mathbf{x},\mathbf{H}}(\cdot)=\prod_{j=1}^{d}G_{x_{j},h_{j}}(\cdot)$ with $\mathbf{H}=\mathrm{diag}_{d}\left(h_{j}\right)$. For applying (3.6) and (3.7) in the framework of semiparametric estimator $\widehat{f}_{n}$ in (4.2), we assume that each component $h_{i\ell}=h_{i\ell}(n)$, $\ell=1,\ldots,d$, of $\mathbf{H}_{i}$ has the univariate inverse gamma prior $\mathcal{I}g(\alpha,\beta_{\ell})$ distribution with same shape parameters $\alpha>0$ and, eventually, different scale parameters $\beta_{\ell}>0$ such that $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{d})^{\top}$. We here recall that the pdf of $\mathcal{I}g(\alpha,\beta_{\ell})$ with $\alpha,\beta_{\ell}>0$ is defined by $Ig_{\alpha,\beta_{\ell}}(u)=\frac{\beta_{\ell}^{\alpha}}{\Gamma(\alpha)}u^{-\alpha-1}\exp(-\beta_{\ell}/u)\mathds{1}_{(0,\infty)}(u),\;\;\ell=1,\ldots,d.$ (5.1) The mean and the variance of the prior distribution (5.1) for each component $h_{i\ell}$ of the vector $\mathbf{H}_{i}$ are given by $\beta_{\ell}/(\alpha-1)$ for $\alpha>1$, and $\beta_{\ell}^{2}/(\alpha-1)^{2}(\alpha-2)$ for $\alpha>2$, respectively. Note that for a fixed $\beta_{\ell}>0$, $\ell=1,\ldots,d$, and if $\alpha\rightarrow\infty$, then the distribution of the bandwidth vector $\mathbf{H}_{i}$ is concentrated around the null vector $\mathbf{0}$. From those considerations, the closed form of the posterior density and the Bayesian estimator of $\mathbf{H}_{i}$ are given in the following proposition which is proven in Appendix 7.2. ###### Proposition 5.1 For fixed $i\in\\{1,2,\ldots,n\\}$, consider each observation $\mathbf{X}_{i}=(X_{i1},\ldots,X_{id})^{\top}$ with its corresponding $\mathbf{H}_{i}=\mathrm{diag}_{d}\left(h_{ij}\right)$ of univariate bandwidths and defining the subset $\mathbb{I}_{i}=\left\\{k\in\\{1,\ldots,d\\}~{};X_{ik}=0\right\\}$ and its complementary set $\mathbb{I}^{c}_{i}=\left\\{\ell\in\\{1,\ldots,d\\}~{};X_{i\ell}\in(0,\infty)\right\\}$. Using the inverse gamma prior $Ig_{\alpha,\beta_{\ell}}$ of (5.1) for each component $h_{i\ell}$ of $\mathbf{H}_{i}$ in the multiple gamma estimator with $\alpha>1/2$ and $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{d})^{\top}\in(0,\infty)^{d}$, then: (i) the posterior density is the following weighted sum of inverse gamma $\displaystyle\pi(\mathbf{H}_{i}\mid\mathbf{X}_{i})$ $\displaystyle=$ $\displaystyle\frac{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}{D_{i}(\alpha,\boldsymbol{\beta})}\sum_{j=1,j\neq i}^{n}\frac{1}{p_{d}(\mathbf{X}_{j};\widehat{\boldsymbol{\theta}}_{n})}\left(\prod_{k\in\mathbb{I}_{i}}C_{jk}(\alpha,\beta_{k})\,Ig_{\alpha+1,X_{jk}+\beta_{k}}(h_{ik})\right)$ $\displaystyle\times\left(\prod_{\ell\in\mathbb{I}^{c}_{i}}A_{ij\ell}(\alpha,\beta_{\ell})\,Ig_{\alpha+1/2,B_{ij\ell}(\beta_{\ell})}(h_{i\ell})\right),$ with $A_{ij\ell}(\alpha,\beta_{\ell})=[\Gamma(\alpha+1/2)]/(\beta_{\ell}^{-\alpha}X_{i\ell}^{1/2}\sqrt{2\pi}[B_{ij\ell}(\beta_{\ell})]^{\alpha+1/2})$, $B_{ij\ell}(\beta_{\ell})=X_{i\ell}\log(X_{i\ell}/X_{j\ell})+X_{j\ell}-X_{i\ell}+\beta_{\ell}$, $C_{jk}(\alpha,\beta_{k})=[\Gamma(\alpha+1)]/[\beta_{k}^{-\alpha}(X_{jk}+\beta_{k})^{\alpha+1}]$, and $D_{i}(\alpha,\boldsymbol{\beta})=p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})\sum_{j=1,j\neq i}^{n}\left(\prod_{k\in\mathbb{I}_{i}}A_{ijk}(\alpha,\beta_{k})\right)\left(\prod_{\ell\in\mathbb{I}^{c}_{i}}B_{ij\ell}(\beta_{\ell})\right)/p_{d}(\mathbf{X}_{j};\widehat{\boldsymbol{\theta}}_{n})$; (ii) under the quadratic loss function, the Bayesian estimator $\widehat{\mathbf{\mathbf{H}}}_{i}=\mathrm{diag}_{d}\left(~{}\widehat{h}_{im}\right)$ of $\mathbf{H}_{i}$ in (4.2) is $\displaystyle\widehat{h}_{im}$ $\displaystyle=$ $\displaystyle\frac{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}{D_{i}(\alpha,\boldsymbol{\beta})}\sum_{j=1,j\neq i}^{n}\frac{1}{p_{d}(\mathbf{X}_{j};\widehat{\boldsymbol{\theta}}_{n})}\left(\prod_{k\in\mathbb{I}_{i}}C_{jk}(\alpha,\beta_{k})\right)\left(\prod_{\ell\in\mathbb{I}^{c}_{i}}A_{ij\ell}(\alpha,\beta_{\ell})\right)$ $\displaystyle\times\left(\frac{X_{jm}+\beta_{m}}{\alpha}\mathds{1}_{\\{0\\}}(X_{im})+\frac{B_{ijm}(\beta_{m})}{\alpha-1/2}\mathds{1}_{(0,\infty)}(X_{im})\right),$ for $m=1,2,\ldots,d,$ with the previous notations of $A_{ij\ell}(\alpha,\beta_{\ell})$, $B_{ijm}(\beta_{m})$, $C_{jk}(\alpha,\beta_{k})$ et $D_{i}(\alpha,\boldsymbol{\beta})$. Following Somé and Kokonendji [52] for nonparametric approach, we have to select the prior parameters $\alpha$ and $\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{d})^{\top}$ of the multiple inverse gamma of $\mathcal{I}g(\alpha,\beta_{\ell})$ in (5.1) as follows: $\alpha=\alpha_{n}=n^{2/5}>2$ and $\beta_{\ell}>0$, $\ell=1,\ldots,d$, to obtain the convergence of the variable bandwidths to zero with a rate close to that of an optimal bandwidth. For practical use, we here consider each $\beta_{\ell}=1$. ### 5.2 Semicontinuous Datasets The numerical illustrations shall be done through the following dataset of Table 5.1 which are recently used in [52] for nonpaprametric approach and only in trivariate setup as semicontinuous data. It concerns three measurements (with $n=42$) of drinking water pumps installed in the Sahel. The first variable $X_{1}$ represents the failure times (in months) and, also, it is recently used by Touré et al. [55]. The second variable $X_{2}$ refers to the distance (in kilometers) between each water pump and the repair center in the Sahel, while the third one $X_{3}$ stands for average volume (in $m^{3}$) of water per day. Table 5.1: Drinking water pumps trivariate data measured in the Sahel with $n=42$. $X_{1}:$ \| 23 \| 261 \| 87 \| 10 \| 120 \| 14 \| 62 \| 15 \| 47 \| 225 \| 71 \| 20 \| 246 \| 21 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $X_{2}:$ \| 97 \| 93 \| 94 \| 100 \| 98 \| 84 \| 96 \| 110 \| 121 \| 73 \| 90 \| 93 \| 103 \| 116 $X_{3}:$ \| 26 \| 52 \| 22 \| 39 \| 23 \| 26 \| 32 \| 17 \| 10 \| 39 \| 31 \| 42 \| 52 \| 26 $X_{1}:$ \| 19 \| 42 \| 20 \| 5 \| 12 \| 120 \| 17 \| 11 \| 3 \| 14 \| 71 \| 11 \| 5 \| 14 $X_{2}:$ \| 114 \| 82 \| 96 \| 94 \| 77 \| 91 \| 117 \| 103 \| 99 \| 113 \| 79 \| 109 \| 84 \| 118 $X_{3}:$ \| 26 \| 36 \| 43 \| 36 \| 6 \| 27 \| 15 \| 36 \| 9 \| 52 \| 11 \| 20 \| 25 \| 37 $X_{1}:$ \| 11 \| 16 \| 90 \| 1 \| 16 \| 52 \| 95 \| 10 \| 1 \| 14 \| 4 \| 7 \| 14 \| 20 $X_{2}:$ \| 98 \| 93 \| 94 \| 103 \| 109 \| 110 \| 89 \| 108 \| 101 \| 93 \| 102 \| 138 \| 103 \| 96 $X_{3}:$ \| 25 \| 18 \| 43 \| 43 \| 24 \| 38 \| 6 \| 40 \| 21 \| 34 \| 15 \| 23 \| 68 \| 37 Table 5.2 displays all empirical univariate, bivariate and trivariate variation (2.6) and dispersion (2.3) indexes from Table 5.1. Hence, each $X_{j}$, $(X_{j},X_{k})$ and $(X_{1},X_{2},X_{3})$ is over-dispersed compared to the corresponding uncorrelated Poisson distribution. But, only $(X_{1},X_{3})$ (resp. $X_{1}$) can be considered as a bivariate equi- variation (resp. univariate over-variation) with respect to the corresponding uncorrelated exponential distribution; and, other $X_{j}$, $(X_{j},X_{k})$ and $(X_{1},X_{2},X_{3})$ are under-varied. In fact, we just compute dispersion indexes for curiosity since all values in Table 5.1 are positive integers; and, we herenow omit the counting point of view in the remainder of the analysis. Table 5.2: Empirical univariate (in diagonal), bivariate (off diagonal) and trivariate (at the corner) variation and dispersion indexes. $\widehat{\mathrm{GVI}}_{3}=0.0533$ \| $X_{1}$ \| $X_{2}$ \| $X_{3}$ \| $\widehat{\mathrm{GDI}}_{3}=15.1229$ \| $X_{1}$ \| $X_{2}$ \| $X_{3}$ ---\|---\|---\|---\|---\|---\|---\|--- $X_{1}$ \| $1.9425$ \| $0.0557$ \| $1.0549$ \| $X_{1}$ \| $89.5860$ \| $14.3223$ \| $70.7096$ $X_{2}$ \| $0.0557$ \| $0.0167$ \| $0.0157$ \| $X_{2}$ \| $14.3223$ \| $1.6623$ \| $2.0884$ $X_{3}$ \| $1.0549$ \| $0.0157$ \| $0.2122$ \| $X_{3}$ \| $70.7096$ \| $2.0884$ \| $6.3192$ Thus, we are gradually investing in semiparametric approaches for three univariates, three bivariates and only one trivariate from $(X_{1},X_{2},X_{3})$ of Table 5.1. ### 5.3 Univariate Examples For each univariate semicontinuous dataset $X_{j}$, $j=1,2,3$, we have already computed the GVI in Table 5.2 which belongs in $(0.01,1.95)\ni 1$. This allows to consider our flexible semiparametric estimation $\widehat{f}_{n,j}$ with an exponential $\mathscr{E}_{1}(\mu_{j})$ as start in (4.2) and using adaptive Bayesian bandwidth in gamma kernel of Proposition 5.1. Hence, we deduce the corresponding diagnostic percent $\widetilde{W}_{n,j}$ from (4.6) for deciding an appropriate approach. In addition, we first present the univariate nonparametric estimation $\widetilde{f}_{n,j}$ with adaptive Bayesian bandwidth in gamma kernel of [50] and then propose another parametric estimation of $X_{j}$ by the standard gamma model with shape ($a_{j}$) and scale ($b_{j}$) parameters. Table 5.3 transcribes parameter maximum likelihood estimates of exponential and gamma models with diagnostic percent from Table 5.1. Figure 5.1 exhibits histogram, $\widetilde{f}_{n,j}$, $\widehat{f}_{n,j}$, exponential, gamma and diagnostic $\widetilde{W}_{n,j}$ graphs for each univariate data $X_{j}$. One can observe differences with the naked eye between $\widetilde{f}_{n,j}$ and $\widehat{f}_{n,j}$ although they are very near and with the same pace. The diagnostic $\widetilde{W}_{n,j}$ graphics lead to conclude to semiparametric approach for $X_{2}$ and to full parametric models for $X_{3}$ and slightly also for $X_{1}$. Thus, we have suggested gamma model with two parameters for improving the starting exponential model; see, e.g., [30, Table 2] for alternative parametric models. Table 5.3: Parameter estimates of models and diagnostic percents of univariate datasets. Estimate \| $\widehat{\mu}_{j}\;\;\;$ \| $\widetilde{W}_{n,j}$ ($\%$) \| $\widehat{a}_{j}\;\;\;\;$ \| $\widehat{b}_{j}\;\;\;\;$ ---\|---\|---\|---\|--- $X_{1}$ \| $0.0217$ \| $95.2381$ \| $0.7256$ \| $63.5618$ $X_{2}$ \| $0.0100$ \| $76.1905$ \| $56.9817$ \| $1.7470$ $X_{3}$ \| $0.0336$ \| $100.0000$ \| $3.7512$ \| $7.9403$ () () () () () () Figure 5.1: Comparative graphs of estimates of $X_{1}$, $X_{2}$ and $X_{3}$ with their corresponding diagnostics. ### 5.4 Bivariate and Trivariate Examples For the sake of flexibility and efficiency, we here analyse our proposed semiparametric estimation $\widehat{f}_{n}$ with an uncorrolated exponential as start in (4.2) and using adaptive Bayesian bandwidth in gamma kernel of Proposition 5.1. This concerns all bivariate and trivariate datasets from Table 5.1 for which their GVI are in $(0.01,1.06)\ni 1$ from Table 5.2. All the computation times are alsmost instantaneous. Table 5.4: Correlations, MVI, parameter estimates and diagnostic percents of bi- and trivariate cases. Dataset \| $(X_{1},X_{2})$ \| $(X_{1},X_{3})$ \| $(X_{2},X_{3})$ \| $(X_{1},X_{2},X_{3})$ ---\|---\|---\|---\|--- $\widehat{\rho}(X_{j},X_{k})$ \| $-0.3090$ \| $0.2597$ \| $0.0245$ \| $\det\widehat{\boldsymbol{\rho}}=0.8325$ $\widehat{\mathrm{MVI}}$ \| $0.0720$ \| $0.9857$ \| $0.0155$ \| $0.0634$ $(\widehat{\mu}_{j})$ \| $(0.0217,0.0100)$ \| $(0.0217,0.0336)$ \| $(0.0100,0.0336)$ \| $(0.0217,0.0100,0.0336)$ $\widetilde{W}_{n}$ ($\%$) \| $9.5238$ \| $52.3809$ \| $26.1005$ \| $0.0000$ () () () () Figure 5.2: Univariate projections of diagnostic graphs for bivariate and trivariate models. Table 5.4 reports the main numerical results of the corresponding correlations, MVI, parameter estimates and finally diagnostic percent $\widetilde{W}_{n}$ from (4.6) that we intentionally omit to represent some graphics in three or four dimensions. However, Figure 5.2 displays some representative projections of $\widetilde{W}_{n}$. From Table 5.4, the cross empirical correlations are closed to 0 and all MVI are smaller than $1$ which allow to consider uncorrelated exponential start-models. The maximum likelihood method is also used for estimating the parameters $\mu_{j}$ for getting the same results as in Table 5.3. Thus, the obtained numerical values of $\widetilde{W}_{n}$ indicate semiparametric approaches for all bivariate datasets and the purely nonparametric method for the trivariate one; see [52] for more details on this nonparametric analysis. This progressive semiparametric analysis of the trivariate dataset of Table 5.1 shows the necessity of a suitable choice of the parametric start-models, which may take into account the correlation structure. Hence, the retention of pure nonparametric means the inconvenience of parametric part used in the modelization. Note that we could consider the Marshall-Olkin exponential distributions with nonnegative correlations as start-models; but, they are singular. See Appendix 7.1 for a brief review. ## 6 Concluding Remarks In this paper, we have presented a flexible semiparametric approach for multivariate nonnegative orthant distributions. We have first recalled multivariate variability indexes GVI, MVI, RVI, GDI, MDI and RDI from RWI as a prelude to the second-order discrimination for these parametric distributions. We have then reviewed and provided new proposals to the nonparametric estimators through multivariate associated kernels; e.g., Proposition 3.2 and Corollary 3.3. Both effective adaptive and local Bayesian selectors of bandwidth matrices are suggested for semicontinuous and counting data, respectively. All these previous ingredients were finally used to develop the semiparametric modelling for multivariate nonnegative orthant distributions. Numerical illustrations have been simply done for univariate and multivariate semicontinuous datasets with the uncorrolated exponential start-models after examining GVI and MVI. The adaptive Bayesian bandwidth selection (3.6) in multiple gamma kernel (Proposition 5.1) were here required for applications. Finally, the diagnostic models have played a very interesting role for helping to the appropriate approach, even if it means improving it later. At the meantime, Kokonendji et al. [23] proposed an in-depth practical analysis of multivariate count datasets starting by multivariate (un)correlated Poisson models after reviewing GDI and RDI. They have also established an equivalent of our Proposition 5.1 for the local Bayesian bandwidth selection (3.5) by using the multiple binomial kernel from Example 3.7. As one of the many perspectives, one could consider the categorial setup with local Bayesian version of the multivariate associated kernel of Aitchison and Aitken [3] from Example 3.5 of the univariate case. At this stage of analysis, all the main foundations are now available for working in multivariate setup such as variability indexes, associated kernels, Bayesian selectors and model disgnostics. We just have to adapt them to each situation encountered. For instance, we have the semiparametric regression modelling; see, e.g., Abdous et al. [1] devoted to count explanatory variables and [48]. Also, an opportunity will be opened for hazard rate functions (e.g., [45]). The near future of other functional groups, such categorial and mixed, can now be considered with objectivity and feasibility. ## 7 Appendix ### 7.1 On a Broader $d$-Variate Parametric Models and the Marshall-Olkin Exponential According to Cuenin et al. [10], taking $p\in\\{1,2\\}$ in their multivariate Tweedie models of flexible dependence structure, another way to define the $d$-variate Poisson and exponential distributions is given by $\mathscr{P}_{d}(\boldsymbol{\Lambda})$ and $\mathscr{E}_{d}(\boldsymbol{\Lambda})$, respectively. The $d\times d$ symmetric variation matrix $\boldsymbol{\Lambda}=(\lambda_{ij})_{i,j\in\\{1,\ldots,d\\}}$ is such that $\lambda_{ij}=\lambda_{ji}\geq 0$, the mean of the corresponding marginal distribution is $\lambda_{ii}>0$, and the non-negative correlation terms satisfy $\rho_{ij}=\frac{\lambda_{ij}}{\sqrt{\lambda_{ii}\lambda_{jj}}}\in[0,\min\\{R(i,j),R(j,i)\\}),$ (7.1) with $R(i,j)=\sqrt{\lambda_{ii}/\lambda_{jj}}\,(1-\lambda_{ii}^{-1}\sum_{\ell\neq i,j}\lambda_{i\ell})\in(0,1)$. Their constructions are perfectly defined having $d(d+1)/2$ parameters as in $\mathscr{P}_{d}(\boldsymbol{\mu},\boldsymbol{\rho})$ and $\mathscr{E}_{d}(\boldsymbol{\mu},\boldsymbol{\rho})$. Moreover, we attain the exact bounds of the correlation terms in (7.1). Cuenin et al. [10] have pointed out the construction and simulation of the negative correlation structure from the positive one of (7.1) by considering the inversion method. The negativity of a correlation component is crucial for the phenomenon of under-variability in a bivariate/multivariate positive orthant model. Figure 7.1 (right) plots a limit shape of any bivariate positive orthant distribution with very strong negative correlation (in red), which is not the diagonal line of the upper bound ($+1$) of positive correlation (in blue); see, e.g., [10] for details on both bivariate orthant (i.e., continuous and count) models. Conversely, Figure 7.1 (left) represents the classic lower ($-1$) and upper ($+1$) bounds of correlations on $\mathbb{R}^{2}$ or finite support. $-3$$-2$$-1$$0$$1$$2$$3$$-3$$-2$$-1$$0$$1$$2$$3$ $0$$1$$2$$3$$4$$5$$6$$0$$1$$2$$3$$4$$5$$6$ Figure 7.1: Support of bivariate distributions with maximum correlations (positive in blue and negative in red): model on $\mathbb{R}^{2}$ (left) and also finite support; model on $\mathbb{T}_{2}^{+}\subseteq[0,\infty)^{2}$ (right), without finite support. The $d$-variate exponential $\boldsymbol{X}=(X_{1},\ldots,X_{d})^{\top}\sim\mathscr{E}_{d}(\boldsymbol{\mu},\mu_{0})$ of Marshall and Olkin [39] is built as follows. Let $Y_{1},\ldots,Y_{d}$ and $Z$ be univariate exponential random variables with parameters $\mu_{1}>0,\ldots,\mu_{d}>0$ and $\mu_{0}\geq 0$, respectively. Then, by setting $X_{j}:=Y_{j}+Z$ for $j=1,\ldots,d$, one has $\mathbb{E}X_{j}=1/(\mu_{j}+\mu_{0})=\sqrt{\mathrm{var}X_{j}}$ and $\mathrm{cov}(X_{j},X_{\ell})=\mu_{0}/\\{(\mu_{j}+\mu_{0})(\mu_{\ell}+\mu_{0})(\mu_{j}+\mu_{\ell}+\mu_{0})\\}$ for all $j\neq\ell$. Each correlation $\rho(X_{j},X_{\ell})=\mu_{0}/(\mu_{j}+\mu_{\ell}+\mu_{0})$ lies in $[0,1]$ if and only if $\mu_{0}\geq 0$. From its survival (or reliability) function $S(\mathbf{x};\boldsymbol{\mu},\mu_{0})=\exp\left(-\mu_{0}\max(x_{1},\ldots,x_{d})-\sum_{j=1}^{d}\mu_{j}x_{j}\right),$ its pdf can be written as $p_{d}(\mathbf{x};\boldsymbol{\mu},\mu_{0})=\left\\{\begin{array}[]{ll}S(\mathbf{x};\boldsymbol{\mu},\mu_{0})(\mu_{0}+\mu_{\ell})\prod\limits_{j=1,j\neq\ell}^{d}\mu_{j}&\mathrm{if}\;x_{\ell}:=\max(x_{1},\ldots,x_{d})\;\mathrm{and}\;x_{\ell}\neq x_{j},\;j\neq\ell\\\ S(\mathbf{x};\boldsymbol{\mu},\mu_{0})\mu_{0}\mu_{j_{1}}\cdots\mu_{j_{k}}&\mathrm{if}\;x_{j_{1}},\ldots,x_{j_{k}}<x_{\ell_{k+1}}=\cdots=x_{\ell_{d}}\\\ S(\mathbf{x};\boldsymbol{\mu},\mu_{0})\mu_{0}&\mathrm{if}\;x_{1}=\cdots=x_{d}>0.\end{array}\right.$ It is not absolutely continuous with respect to the Lebesgue measure in $\mathbb{T}_{d}^{+}$ and has singularities corresponding to the cases where two or more of the $x_{j}$’s are equal. Karlis [22] has proposed a maximum likelihood estimation of parameters via an EM algorithm. Finally, Kokonendji et al. [31] have calculated $\mathrm{GVI}(\boldsymbol{X})=1+\frac{\mu_{0}\sum_{j=1}^{d}(\mu_{j}+\mu_{0})^{-1}\\{\sum_{\ell\neq j}(\mu_{j}+\mu_{\ell}+\mu_{0})^{-1}(\mu_{\ell}+\mu_{0})^{-1}\\}}{\\{(\mu_{1}+\mu_{0})^{-2}+\cdots+(\mu_{d}+\mu_{0})^{-2}\\}^{2}}\geq 1\;\;(\Leftrightarrow\mu_{0}\geq 0).$ and $\mathrm{MVI}(\boldsymbol{X})=\frac{\sum_{j=1}^{d}(\mu_{j}+\mu_{0})^{-4}}{\sum_{j=1}^{d}(\mu_{j}+\mu_{0})^{-4}+2\sum_{1\leq j<\ell\leq 1}(\mu_{j}+\mu_{0})^{-2}(\mu_{\ell}+\mu_{0})^{-2}}<1.$ Hence, the Marshall-Olkin exponential model $\boldsymbol{X}\sim\mathscr{E}_{d}(\boldsymbol{\mu},\mu_{0})$ is always under- varied with respect to the MVI and over- or equi-varied with respect to GVI. If $\mu_{0}=0$ then $\mathscr{E}_{d}(\boldsymbol{\mu},\mu_{0})$ is reduced to the uncorrolated $\mathscr{E}_{d}(\boldsymbol{\mu})$ with $\mathrm{GVI}(\boldsymbol{Y})=1$. However, the assumption of non-negative correlations between components is sometimes unrealistic for some analyzes. ### 7.2 Proofs of Proposition 3.4, Proposition 4.1 and Proposition 5.1 ###### Proof 2 (Proof of Proposition 3.4) From Definition 3.1, we get (see also [29] for more details) $\displaystyle\mathbb{E}\left[\widetilde{f}_{n}(\mathbf{x})\right]-f(\mathbf{x})$ $\displaystyle=$ $\displaystyle\mathbb{E}\left[\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{X}_{j})\right]-f(\mathbf{x})=\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}_{n}}\cap\mathbb{T}_{d}^{+}}\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{u})f(\mathbf{u})\boldsymbol{\nu}(d\mathbf{u})-f(\mathbf{x})$ (7.2) $\displaystyle=$ $\displaystyle\mathbb{E}\left[f\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right)\right]-f(\mathbf{x}).$ Next, using (7.2), by a Taylor expansion of the function $f(\cdot)$ over the points ${\cal Z}_{\mathbf{x},\mathbf{H}_{n}}$ and $\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]$, we get $\displaystyle f\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right)$ $\displaystyle=$ $\displaystyle f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)+\left\langle\nabla f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right),\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\right\rangle$ (7.3) $\displaystyle+\frac{1}{2}\left\langle{\cal H}f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right),\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\right\rangle$ $\displaystyle+\left\\|{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right\\|^{2}o(1)$ $\displaystyle=$ $\displaystyle f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)+\left\langle\nabla f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right),\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\right\rangle$ $\displaystyle+\frac{1}{2}\operatorname{tr}\left[{\cal H}f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)^{\mathsf{T}}\right]$ $\displaystyle+\operatorname{tr}\left[\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}-\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)^{\mathsf{T}}\right]o(1),$ where $o(1)$ is uniform in a neighborhood of $\mathbf{x}$. Therefore, taking the expectation in both sides of (7.3) and then substituting the result in (7.2), we get $\displaystyle\mathbb{E}\left[\widetilde{f}_{n}(\mathbf{x})\right]-f(\mathbf{x})$ $\displaystyle=$ $\displaystyle f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)-f(\mathbf{x})+\frac{1}{2}\operatorname{tr}\left[{\cal H}f\left(\mathbb{E}\left[{\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right]\right)\operatorname{var}\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right)\right]$ $\displaystyle+o\left\\{\operatorname{tr}\left[\operatorname{var}\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right)\right]\right\\}$ $\displaystyle=$ $\displaystyle f\left(\mathbf{x}+\mathbf{A}\right)-f(\mathbf{x})+\frac{1}{2}\operatorname{tr}\left[{\cal H}f\left(\mathbf{x}+\mathbf{A}\right)\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]+o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\},$ where $o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\}$ is uniform in a neighborhood of $\mathbf{x}$. The second Taylor expansion of the function $f(\cdot)$ over the points $\mathbf{x}$ and $\mathbf{x}+\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)$ allows to conclude the bias (3.3). About the variance term, $f$ being bounded, we have $\mathbb{E}\left[\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{X}_{j})\right]=O(1)$. It follows that: $\displaystyle\mathrm{var}\left[\widetilde{f}_{n}(\mathbf{x})\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}\mathrm{var}\left[\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{X}_{j})\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}\left[\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}_{n}}\cap\mathbb{T}_{d}^{+}}\mathbf{K}^{2}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{u})f(\mathbf{u})\boldsymbol{\nu}(d\mathbf{u})+O(1)\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}_{n}}\cap\mathbb{T}_{d}^{+}}\mathbf{K}^{2}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{u})\begin{pmatrix}f(\mathbf{x})+\left\langle\nabla f(\mathbf{x}),\mathbf{x}-\mathbf{u}\right\rangle\\\ +\frac{1}{2}(\mathbf{x}-\mathbf{u})^{T}{\cal H}f(\mathbf{x})(\mathbf{x}-\mathbf{u})\\\ +o\left[\left(\|\|\mathbf{x}-\mathbf{u}\|\|^{2}\right)\right]\end{pmatrix}\boldsymbol{\nu}(d\mathbf{u})$ $\displaystyle=$ $\displaystyle\frac{1}{n}f(\mathbf{x})\|\|\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}\|\|_{2}^{2}+o\left[\frac{1}{n(\det\mathbf{H}_{n})^{r}}\right].$ ###### Proof 3 (Proof of Proposition 4.1) Since one has $\mathrm{Bias}[\widehat{f}_{n}(\mathbf{x})]=p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})\mathbb{E}[\widetilde{w}_{n}(\mathbf{x})]-f(\mathbf{x})$ and $\mathrm{var}[\widehat{f}_{n}(\mathbf{x})]=[p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})]^{2}\mathrm{var}[\widetilde{w}_{n}(\mathbf{x})]$, it is enough to calculate $\mathbb{E}[\widetilde{w}_{n}(\mathbf{x})]$ and $\mathrm{var}[\widetilde{w}_{n}(\mathbf{x})]$ following Proposition 3.4 applied to $\widetilde{w}_{n}(\mathbf{x})=n^{-1}\sum_{i=1}^{n}\mathbf{K}_{\mathbf{x},\mathbf{H}}(\mathbf{X}_{i})/p_{d}(\mathbf{X}_{i};\boldsymbol{\theta}_{0})$ for all $\mathbf{x}\in\mathbb{T}_{d}^{+}$. Indeed, one successively has $\displaystyle\mathbb{E}[\widetilde{w}_{n}(\mathbf{x})]$ $\displaystyle=$ $\displaystyle\mathbb{E}\left[\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{X}_{1})/p_{d}(\mathbf{X}_{1};\boldsymbol{\theta}_{0})\right]$ $\displaystyle=$ $\displaystyle\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}_{n}}\cap\mathbb{T}_{d}^{+}}\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{u})[p_{d}(\mathbf{u};\boldsymbol{\theta}_{0})]^{-1}f(\mathbf{u})\boldsymbol{\nu}(d\mathbf{u})=\mathbb{E}\left[w\left({\cal Z}_{\mathbf{x},\mathbf{H}_{n}}\right)\right]$ $\displaystyle=$ $\displaystyle w(\mathbf{x})+\left\langle\nabla w(\mathbf{x}),\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right\rangle+\frac{1}{2}\left(\operatorname{tr}\left\\{{\cal H}w\left(\mathbf{x}\right)\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})+\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)^{\mathsf{T}}\mathbf{A}\left(\mathbf{x},\mathbf{H}_{n}\right)\right]\right\\}\right)$ $\displaystyle+o\left\\{\operatorname{tr}\left[\mathbf{B}(\mathbf{x},\mathbf{H}_{n})\right]\right\\},$ which leads to the announced result of $\mathrm{Bias}[\widehat{f}_{n}(\mathbf{x})]$. As for $\mathrm{var}[\widetilde{w}_{n}(\mathbf{x})]$, one also write $\displaystyle\mathrm{var}\left[\widetilde{w}_{n}(\mathbf{x})\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}\mathrm{var}\left[\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{X}_{1})/p_{d}(\mathbf{X}_{1};\boldsymbol{\theta}_{0})\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}\left[\int_{\mathbb{S}_{\mathbf{x},\mathbf{H}_{n}}\cap\mathbb{T}_{d}^{+}}\mathbf{K}^{2}_{\mathbf{x},\mathbf{H}_{n}}(\mathbf{u})[p_{d}(\mathbf{u};\boldsymbol{\theta}_{0})]^{-2}f(\mathbf{u})\boldsymbol{\nu}(d\mathbf{u})+O(1)\right]$ $\displaystyle=$ $\displaystyle\frac{1}{n}f(\mathbf{x})[p_{d}(\mathbf{x};\boldsymbol{\theta}_{0})]^{-2}\|\|\mathbf{K}_{\mathbf{x},\mathbf{H}_{n}}\|\|_{2}^{2}+o\left[\frac{1}{n(\det\mathbf{H}_{n})^{r}}\right]$ and the desired result of $\mathrm{var}[\widehat{f}_{n}(\mathbf{x})]$ is therefore deduced. ###### Proof 4 (Proof of Proposition 5.1) We have to adapt Theorem 2.1 of Somé and Kokonendji [52] to this semiparametric estimator $\widehat{f}_{n}$ in (4.2). First, the leave-one-out associated kernel estimator (3.7) becomes $\widehat{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i}):=\frac{p_{d}(\mathbf{X}_{i};\widehat{\boldsymbol{\theta}}_{n})}{n-1}\sum_{\ell=1,\ell\neq i}^{n}\frac{1}{p_{d}(\mathbf{X}_{\ell};\widehat{\boldsymbol{\theta}}_{n})}\mathbf{K}_{\mathbf{X}_{i},\mathbf{H}_{i}}(\mathbf{X}_{\ell}).$ Then, the posterior distribution deduced from (3.6) is exppressed as $\pi(\mathbf{H}_{i}\mid\mathbf{X}_{i}):=\pi(\mathbf{H}_{i})\widehat{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i})\left[\int_{\mathcal{M}}\widehat{f}_{n,\mathbf{H}_{i},-i}(\mathbf{X}_{i})\pi(\mathbf{H}_{i})d\mathbf{H}_{i}\right]^{-1}$ and which leads to the result of Part (i) via [52, Theorem 2.1 (i)] for details. 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Theory Meth. 2018, 47, 4534–4555. ###### Contents 1. 1 Introduction 2. 2 Relative Variability Indexes for Orthant Distributions 1. 2.1 Relative Dispersion Indexes for Count Distributions 2. 2.2 Relative Variation Indexes for Semicontinuous Distributions 3. 3 Multivariate Orthant Associated Kernels 1. 3.1 Discrete Associated Kernels 2. 3.2 Semicontinuous Associated Kernels 4. 4 Semiparametric Kernel Estimation with $d$-Variate Parametric Start 1. 4.1 Known $d$-Variate Parametric Model 2. 4.2 Unknown $d$-Variate Parametric Model 3. 4.3 Model Diagnostics 5. 5 Semicontinuous Examples of Application with Discussions 1. 5.1 Adaptive Bayesian Bandwidth Selection for Multiple Gamma Kernels 2. 5.2 Semicontinuous Datasets 3. 5.3 Univariate Examples 4. 5.4 Bivariate and Trivariate Examples 6. 6 Concluding Remarks 7. 7 Appendix 1. 7.1 On a Broader $d$-Variate Parametric Models and the Marshall-Olkin Exponential 2. 7.2 Proofs of Proposition 3.4, Proposition 4.1 and Proposition 5.1
# Global Optimization of the Mean First Passage Time for Narrow Capture Problems in Elliptic Domains Jason Gilbert Electronic mail<EMAIL_ADDRESS>Department of Mathematics and Statistics, University of Saskatchewan Alexei Cheviakov Corresponding Author. Alternative English spelling: Alexey Shevyakov. Electronic mail: <EMAIL_ADDRESS>Department of Mathematics and Statistics, University of Saskatchewan ###### Abstract Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimization to seek trap arrangements that minimize average escape times. In a recent paper, the escape time expansion for a 2D elliptic domain was derived, providing the dependence of the average MFPT on sizes and locations of small internal traps. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps, and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. ## 1 Introduction The narrow capture problem, as described here, concerns the average time required for a particle undergoing Brownian motion to encounter a region within the domain, referred to as a trap, which causes its motion to cease. It is mathematically defined as a Neumann-Dirichlet Poisson problem $\begin{split}&\Delta u=-\dfrac{1}{D}\ ,\quad x\in\bar{\Omega}\ ;\qquad\bar{\Omega}=\Omega\setminus\mathop{\cup}_{j=1}^{N}\Omega_{\varepsilon_{j}}\ ;\\\\[5.0pt] &\partial_{n}u=0\ ,\quad x\in\partial\Omega\ ;\qquad u=0\ ,\quad x\in\partial\Omega_{\varepsilon_{j}}\ ,\quad j=1,\ldots,N\ ;\end{split}$ (1.1) where $\Omega\subset\mathbb{R}^{n}$, $n=2,3$, denotes the physical domain of the problem; $\\{\Omega_{\varepsilon_{j}}\\}_{j=1}^{N}$ are small volume trap domains within $\Omega$; $\bar{\Omega}$ is the domain except the traps, where the motion of particles takes place, and $\partial_{n}$ denotes the normal derivative on the boundary of the domain. The diffusivity $D$ of the medium filling $\bar{\Omega}$ is assumed constant. Problem (1.1) describes the distribution of the mean capture time, the time $u(x)$ needed for a particle to be captured by any trap, averaged over a large number of launches from the same point $x\in\bar{\Omega}$. An illustration of the problem is provided in Figure 1. Figure 1: (a) A two-dimensional narrow capture problem in the unit disk containing internal traps with absorbing boundaries $\\{\partial\Omega_{\epsilon_{j}}\\}$. (b) A three-dimensional narrow capture problem, a sample Brownian particle trajectory, leading to a capture in a trap (lowermost) denoted by purple color (color online). The boundary conditions on $\partial\Omega$ are reflective: $\partial_{n}u=0$, whereas the traps $\Omega_{\varepsilon_{j}}$ are characterized by immediate absorption of a boundary particle, which is manifested by a Dirichlet boundary conditions $u=0$ on all $\partial\Omega_{\varepsilon_{j}}$. The above generic formulation affords a variety of physical interpretations, ranging from biological to electrostatic (see, e.g., Ref. [1, 2] for an overview of applications). In this work, it will be strictly considered in terms of a particle undergoing Brownian motion [3]. In this case, the problem regards the stopping time [1, 4] of a Brownian particle, confined to a domain with a reflective boundary which contains a number of absorbing “traps”. When the path of the particle intersects the boundary of one of these traps, the particle is captured, meaning that the process of Brownian motion is stopped. This stopping time may be interpreted as the time required for the particle to leave the confining domain, thus it is often referred to as the first passage time [5, 6, 7]. As Brownian motion is an inherently random process, there is no single first passage time which accurately characterizes the process; instead, the mean first passage time (MFPT), representing the average time taken for a particle to encounter a trap based on its first observed position, is used. Interpreting the problem (1.1) accordingly, $u$ is the MFPT; $D$ is the diffusion coefficient, representing the mobility of the Brownian particle; $\Omega_{\varepsilon_{j}}$ is the portion of the domain occupied by the $j_{th}$ trap. Given the physical domain and the number and sizes of the traps, it is of interest to ask whether there is an _optimal_ arrangement of traps within the domain which minimizes the MFPT, or in other words, maximizes the rate at which Brownian particles encounter the absorbing traps. Related work dedicated to similar optimization, in the case that the traps are asymptotically small relative to the size of the domain, for various kinds of confining domains with interior or boundary traps, can be found, for example, in Refs [8, 9, 10, 11, 12] and references therein. Both (putative) globally optimal and multiple locally optimal arrangements of boundary and volume traps have been computed in various settings. In this contribution, we consider the narrow capture problem for the case of a 2D elliptical domain, elaborating on previous work on the subject for the case of a circular domain [8], and the more general case of the elliptical domain [9]. The paper is organized as follows. In Section 2 we briefly summarise the previous results which this work is based on. This includes a review of the approximations used in the case that the traps are small relative to the domain size, and are well-separated; as well as an explanation of the choice of merit function, and relevant constraints, for each case. In Section 3 the optimization method chosen in our study is described. This includes an explanation of our selection of algorithms, as well the details of their use, and some decisions made to facilitate comparison to previous studies. Specifically, we seek the optimal configuration of traps for numbers of traps $N\leq 50$, and elliptic domain eccentricities of $0$, $1/8$, $1/4$, and $1/2$. In Section 4 the results of the study are presented. These results include the optimized values of the merit functions (related to the domain-average Brownian particle capture time) for each number of traps, and each domain eccentricity; in the case of the unit disk, our results are compared to those of a previous study. Additionally, the distributions of traps for some illustrative cases are shown, and bulk measures of trap distribution are calculated for each of the optimized configurations. In Section 5, the results presented in Section 4 are discussed, and some remarks are provided regarding on the distribution of traps, the interpretation of the bulk quantities used, and some related problems. ## 2 Problem Statement The goal of this contribution is to further explore the optimal configuration of absorbing traps in terms of the asymptotic expansions for the circular and elliptical domains for which asymptotic MFPT formulae depending on trap positions are now available [8, 9]. To this end, numerical methods will be used to approximate the optimum configurations of larger numbers of traps than have previously been considered. In the case of the unit disk, the parameter space used in this study is general and does not assume any simplifying restrictions that have been imposed in previous works to reduce computation complexity. In essence, the problem at hand is to find the trap positions which minimize the spatial average of the MFPT $u(x)$, denoted by $\overline{u}_{0}$. This section will summarize the equations used to minimize this quantity, as derived in [8, 9]. Here the unit disk will be considered a special case of the elliptical domain with zero eccentricity. In either case, when the domain contains $N$ identical circular traps, each of radius $\varepsilon$, the average MFPT satisfies [9] $\overline{u}_{0}\ =\ \dfrac{\|\Omega\|}{2\pi D\nu N}\ +\ \dfrac{2\pi}{N}\textbf{e}^{T}\mathcal{G}\mathcal{A}\ ,$ (2.1) where the column vector $\mathcal{A}$ is the solution of the linear system $\left[I+2\pi\nu\left(I-E\right)\mathcal{G}\right]\mathcal{A}\ =\ \dfrac{\|\Omega\|}{2\pi DN}\textbf{e}\ .$ Here $E\equiv\textbf{e}\textbf{e}^{T}/N$, $\textbf{e}=(1,\ldots,1)^{T}$, $\nu\equiv-1/\log\varepsilon$, and the Green’s matrix $\mathcal{G}$ depends on the trap locations $\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{N}\\}$, such that $\mathcal{G}_{ij}\ =\ \left\\{\begin{array}[]{ll}G(\mathbf{x}_{i};\mathbf{x}_{j})\quad\text{if}\ i\neq j\ ,\\\\[5.0pt] R(\mathbf{x}_{i};\mathbf{x}_{j})\quad\text{if}\ i=j\ ;\end{array}\quad\text{where}\quad R(\mathbf{x}_{i};\mathbf{x}_{j})=\lim_{\mathbf{x}_{i}\to\mathbf{x}_{j}}G(\mathbf{x}_{i};\mathbf{x}_{j})\right.$ (2.2) where $G$ is the Green’s function of the domain. Examining the equation (2.1), it can be seen that the first term depends only on the combined trap size but does not depend on the trap locations. The second term explicitly depends on the trap locations is the quantity and therefore can be optimized by adjusting trap positions. The merit function subject to optimization can be chosen as $q\left(\mathbf{x}\right)\ =\ \textbf{e}^{T}\mathcal{G}\mathcal{A}\ .$ (2.3) For a value of $q$ to be a permissible solution, it is required that $\overline{u}_{0}\geq 0$, as it is a measure of time; all traps must be within the domain; no trap may be in contact with any other trap (or the two must instead be represented as a single non-circular trap). While the preceding statements are true for both the circular and the elliptical domain, the elements of the matrix $\mathcal{G}$ are populated by evaluating the Green’s function of the domain for each pair of traps, and the form of this function differs for the two cases considered here. In the case of the circular domain, the elements of the Green’s matrix $\mathcal{G}$ are computed using the Neumann Green’s function [8] $G(\mathbf{x};\mathbf{x}_{0})\ =\ \dfrac{1}{2\pi}\left(-\log\|\mathbf{x}-\mathbf{x}_{0}\|\ -\ \log\left\|\mathbf{x}\|\mathbf{x}_{0}\|-\dfrac{\mathbf{x}_{0}}{\|\mathbf{x}_{0}\|}\right\|\ +\ \dfrac{1}{2}\left(\|\mathbf{x}\|^{2}+\|\mathbf{x}_{0}\|^{2}\right)\ -\ \dfrac{3}{4}\right)\ ,$ (2.4a) and its regular part $R(\mathbf{x})\ =\ \dfrac{1}{2\pi}\left(-\log\left\|\mathbf{x}\|\mathbf{x}\|-\dfrac{\mathbf{x}}{\|\mathbf{x}\|}\right\|\ +\ \|\mathbf{x}\|^{2}\ -\ \dfrac{3}{4}\right)\ .$ (2.4b) The Green’s function for the elliptical domain, in the form of a quickly convergent series, has been derived in [9] using elliptical coordinates $(\xi,\eta)$. It has the form $\begin{split}G(\mathbf{x},\mathbf{x}_{0})\ =\ \dfrac{1}{4\|\Omega\|}\left(\|\mathbf{x}\|^{2}+\|\mathbf{x}_{0}\|^{2}\right)\ -\ \dfrac{3}{16\|\Omega\|}(a^{2}+b^{2})\ -\ \dfrac{1}{4\pi}\log\beta\ -\ \dfrac{1}{2\pi}\xi_{>}\\\\[5.0pt] \ -\ \dfrac{1}{2\pi}\sum^{\infty}_{n=0}\log\left(\prod_{j=1}^{8}\|1-\beta^{2n}z_{j}\|\right)\ ,\qquad\mathbf{x}\neq\mathbf{x}_{0}\ ,\end{split}$ (2.5) where $a$ and $b$ are the major and minor axis of the domain, respectively; the area of the domain is $\|\Omega\|=\pi ab$, the parameter $\beta=(a-b)/(a+b)$, and the values $\xi_{>}=\mathop{\hbox{\rm max}}(\xi,\xi_{0})$, $z_{1},\ldots,z_{8}$ are defined in terms of the elliptical coordinates $\xi$ and $\eta$ as follows. The Cartesian coordinates $(x,y)$ and the elliptical coordinates $(\xi,\eta)$ are related by the transformation $x=f\cosh\xi\cos\eta\ ,\quad y=f\sinh\xi\sin\eta\ ,\quad f=\sqrt{a^{2}-b^{2}}\ .$ (2.6a) For the major and minor axis of the elliptical domain, one has $a=f\cosh\xi_{b}\ ,\quad b=f\sinh\xi_{b}\ ,\quad\xi_{b}=\tanh^{-1}\left(\dfrac{b}{a}\right)=-\dfrac{1}{2}\log\beta\ .$ (2.6b) For the backward transformation, to determine $\xi(x,y)$, equation (2.6a) is solved to give $\xi=\dfrac{1}{2}\log\left(1-2s+2\sqrt{s^{2}-s}\right)\ ,\quad s\equiv\dfrac{-\mu-\sqrt{\mu^{2}+4f^{2}y^{2}}}{2f^{2}}\ ,\quad\mu\equiv x^{2}+y^{2}-f^{2}\ .$ (2.7a) In a similar fashion, $\eta(x,y)$ is found in terms of $\eta_{\star}\equiv\sin^{-1}(\sqrt{p})$ to be $\eta\ =\ \left\\{\begin{array}[]{ll}\eta_{\star}\ ,&\text{if}\ x\geq 0,\ y\geq 0\\\\[5.0pt] \pi-\eta_{\star}\ ,&\text{if}\ x<0,\ y\geq 0\\\\[5.0pt] \pi+\eta_{\star}\ ,&\text{if}\ x\leq 0,\ y<0\\\\[5.0pt] 2\pi-\eta_{\star}\ ,&\text{if}\ x>0,\ y<0\\\\[5.0pt] \end{array}\right.\ ,\quad\text{where}\quad p\equiv\dfrac{-\mu+\sqrt{\mu^{2}+4f^{2}y^{2}}}{2f^{2}}\ .$ (2.7b) Finally, the $z_{j}$-terms appearing in the infinite sum of equation (2.5) are defined via $\xi$, $\eta$, and $\xi_{b}$ as $\begin{array}[]{lll}z_{1}\equiv e^{-\|\xi-\xi_{0}\|+i(\eta-\eta_{0})}\ ,&z_{2}\equiv e^{\|\xi-\xi_{0}\|-4\xi_{b}+i(\eta-\eta_{0})}\ ,&z_{3}\equiv e^{-(\xi+\xi_{0})-2\xi_{b}+i(\eta-\eta_{0})}\ ,\\\\[4.0pt] z_{4}\equiv e^{(\xi+\xi_{0})-2\xi_{b}+i(\eta-\eta_{0})}\ ,&z_{5}\equiv e^{(\xi+\xi_{0})-4\xi_{b}+i(\eta+\eta_{0})}\ ,&z_{6}\equiv e^{-(\xi+\xi_{0})+i(\eta+\eta_{0})}\ ,\\\\[4.0pt] z_{7}\equiv e^{\|\xi-\xi_{0}\|-2\xi_{b}+i(\eta+\eta_{0})}\ ,&z_{8}\equiv e^{-\|\xi-\xi_{0}\|-2\xi_{b}+i(\eta+\eta_{0})}\ .&\\\ \end{array}$ (2.8) The regular part of the Neumann Green’s function, $R$, can be expressed in similar terms as $G$ in equation (2.5) but requires a restatement of the $z_{j}$-terms given in (2.8). It is given by $\begin{split}R(\mathbf{x}_{0})\ =\ &\dfrac{\|\mathbf{x}_{0}\|^{2}}{2\|\Omega\|}\ -\ \dfrac{3}{16\|\Omega\|}(a^{2}+b^{2})\ +\ \dfrac{1}{2\pi}\log(a+b)\ -\ \dfrac{\xi_{0}}{2\pi}\ +\ \dfrac{1}{4\pi}\log\left(\cosh^{2}\xi_{0}-\cos^{2}\eta_{0}\right)\\\\[5.0pt] &\ -\ \dfrac{1}{2\pi}\sum_{n=1}^{\infty}\log\left(1-\beta^{2n}\right)\ -\ \dfrac{1}{2\pi}\sum_{n=0}^{\infty}\log\left(\prod_{j=2}^{8}\left\|1-\beta^{2n}z^{0}_{j}\right\|\right).\end{split}\ $ (2.9a) Here $z^{0}_{j}$ denotes the limiting value of $z_{j}$, as defined in equation (2.8), as $(\xi,\eta)\to(\xi_{0},\eta_{0})$, given by $\begin{array}[]{lll}&z^{0}_{2}=\beta^{2}\ ,&z^{0}_{3}=\beta e^{-2\xi_{0}}\ ,\\\\[5.0pt] z^{0}_{4}=\beta e^{2\xi_{0}}\ ,&z^{0}_{5}=\beta^{2}e^{2(\xi_{0}+i\eta_{0})}\ ,&z^{0}_{6}=e^{2(-\xi_{0}+i\eta_{0})}\ ,\\\\[5.0pt] z^{0}_{7}=\beta e^{2i\eta_{0}}\ ,&z^{0}_{8}=\beta e^{2i\eta_{0}}\ .&\\\ \end{array}$ (2.9b) ## 3 Global optimization In this section, the methods used to find the optimum trap configurations minimizing the average MFPT (2.1) are discussed. We include a description of the general strategy for optimization, the algorithms used, and their specific implementations. For both problems the same general approach was taken to finding the optimum. This was to search iteratively, switching between global and local searches after each iteration. The global search used the particle swarm algorithm PSwarm [13], as implemented in the freely available software package OPTI [14]. The default values for the social and cognitive parameters were chosen, meaning the local optimum known by each particle tended to be as attractive as the known global optimum. These values were chosen with the intent that the parameter space would be explored as broadly as possible. For the local search, the Nelder-Mead algorithm [15], as implemented in MATLAB R2020, was used. The algorithms were chosen based on their generality and simplicity to adapt them to the current study. In addition, for the case of the elliptical domain, special care needed to be taken to ensure that the traps were well-separated. If the traps were to come into contact, or overlap with one another, the asymptotic equation (2.1) can yield non-physical values $\overline{u}_{0}<0$, (which is a common feature of asymptotic formulas that replace finite traps with “point traps” in various narrow escape and narrow capture setups). In the MFPT optimization, the traps are effectively repelled from one another, as well as from their “reflections” in the domain boundary; this is mathematically manifested in the fact that the Green’s function (2.4a) for the disk domain grows logarithmically as $\mathbf{x}\to\mathbf{x}_{0}$, as well as when $\|\mathbf{x}\|\to 1$. In particular, $q$ increases as distance between traps decreases, as traps begin to overlap, $q$ decreases extremely rapidly, appearing to the optimization algorithm to be a favourable configuration. Though this problem can be addressed by artificially assigning $q$ a very high value when an unacceptable configuration is encountered, this approach was found to significantly reduce the effectiveness of the global search, as many evaluations of $q$ would be wasted on these configurations. Instead, an optimum was first found using the iterative approach taking $\varepsilon=0$, following which a local search was carried out using these coordinates as an initial guess for a local search, and taking $\varepsilon=0.05$ in order to facilitate comparison with previous studies [9]. Defining the eccentricity of the elliptic domain according to the usual formula $\kappa\ =\ \sqrt{1-\left(\dfrac{b}{a}\right)^{2}}\ ,$ (3.1) where $a$ is the major axis and $b$ the minor, optimum configurations were computed for $N\leq 50$ for domain eccentricities of $\kappa=0$, $1/8$, $1/4$, and $1/2$. For each eccentricity value, the axes of the ellipse were chosen so that the area of the domain was fixed at $\|\Omega\|=\pi$, to allow for natural comparisons. ## 4 Optimization Results A comparison of $\overline{u}_{0}$ for the optimal configurations found for the unit disk in previous work [8] to those presented here, found using the more accurate approximation [9], shows that the newly computed optimums for the unit disk are consistently better than previous results. Though it is convention to discuss optimal configurations in terms of their interaction energy, the quantity used as the merit function, here the $\overline{u}_{0}$ is used for comparison. This serves the purpose of normalizing the results obtained using the expression for $\overline{u}_{0}$ used in previous work [8], $\overline{u}_{0}\ =\ \dfrac{\|\Omega\|}{2\pi D\nu N}\left(1+\dfrac{2\pi\nu}{N}\textbf{e}^{T}\mathcal{G}\textbf{e}\right)\ ,$ (4.1) to equation (2.1), the newly derived expression [9], $\overline{u}_{0}\ =\ \dfrac{\|\Omega\|}{2\pi D\nu N}\left(1+\dfrac{4\pi^{2}D\nu}{\|\Omega\|}\textbf{e}^{T}\mathcal{G}\mathcal{A}\right)\ .$ (4.2) Though it would be sufficient to compute only the second term for comparison, $\overline{u}_{0}$ serves as a more consistent standard for comparison. Table 1 compares $\overline{u}_{0}$ for each $N$ reported in the previous study, computed using the results found in Table 2 of Ref. [8], while Figure 2 depicts the relative difference between the two. The computed optimal values of the merit function (2.3) that correspond to putative globally optimal minima of the average MFPT (2.1), for the domain eccentricities $\kappa=0$ (circular disk), $1/8$, $1/4$, and $1/2$, are presented in Table 2 below, and are graphically shown in Figure 3. While the first three plots are nearly identical, the plot (d) for the largest eccentricity value differs significantly for small $N$, but becomes similar to the other plots for larger $N$. Plots comparing the optimal configurations of select $N$ for each of the eccentricities considered in this study are shown in Figures 4–7. Each plot shows the position of the trap within the domain, along with a visualization of a Delaunay triangulation [16] calculated using the traps as vertices, to illustrate the distribution of, and relative distance between, traps. In addition, it was of interest to see how the ring-like distribution of traps would change with the eccentricity of the domain. To visualize this change, a scaling factor was calculated for each trap such that each trap would lie on a scaled copy of the elliptic domain boundary. These scaling factors are shown in the lower subplots in Figures 4–7. For the case of $N=5$, Figure 4 can be compared to the optimal configurations presented in Ref [9], through which it can be seen that the two are qualitatively similar and exhibit the same relationship between trap distribution and domain eccentricity. In order to examine the distribution of traps in terms of their mutual distance, a Delaunay triangulation was computed to identify approximate nearest neighbours to each trap (see upper plots in Figures 4–7). In general, for a configuration of $N$ traps distributed, in some sense, “uniformly” over the elliptic domain of area $\|\Omega\|=\pi$, the average “area per trap” is given by $A(N)=\|\Omega\|/N=\pi/N$. Likening an optimal arrangement of $N$ traps to a collection of circles packed into an enclosed space, the (average) distance $\langle d\rangle$ between two neighbouring traps would be the distance between the centers of two identical circles representing the area occupied by each trap; it would be related to the area per trap as $A(N)=\pi\langle d\rangle^{2}/4$. One consequently finds that the average distance between neighbouring traps, equivalent to the diameter of one of the circles, is given by $\langle d\rangle\ =\ \sqrt{\dfrac{4\|\Omega\|}{\pi N}}\ =\ \dfrac{2}{\sqrt{N}}\ ,$ (4.3) Extending this comparison to the traps nearest the boundary, the smallest distance between a trap and the boundary was taken to be the radius of a circle surrounding the trap, and the diameter of this circle was compared to the smallest distance between two traps. This essentially provides a measure of the distance between a near-the-boundary trap and its “reflection” in the Neumann boundary. In Figure 8, for each of the four considered eccentricities of the elliptic domain, the mean pairwise distance between neighbouring traps is plotted as a function of $N$, along with minimum pairwise distance between traps, and $2\times$ minimal distance to the boundary. These are compared with the average distance formula (4.3) coming from the “area per trap” argument. It can be observed that the simple formula (4.3) may be used as a reasonable estimate of common pairwise distances between traps in an optimal configuration. The case of $\kappa=0.5$ serves as somewhat of an exception. This could either be due to a significant change in the distribution of the traps as the eccentricity increases, or it could be an artefact introduced by the calculation of the Delaunay triangulation. As mentioned in Section 3, the optimal configuration for the case $\varepsilon=0$ was used as an initial guess when used to search for the optimum in the case $\varepsilon=0.05$. Using this approach it was found that the optimal configuration for the two cases were not identical, implying that the optimal configuration significantly depends on the size of the traps. $N$ \| $\overline{u}_{0}$ \| $\overline{u}_{0}^{\prime}$ ---\|---\|--- 6 \| 0.11648 \| 0.11648 7 \| 0.09299 \| 0.09297 8 \| 0.07660 \| 0.07660 9 \| 0.06518 \| 0.06512 10 \| 0.05653 \| 0.05624 11 \| 0.04920 \| 0.04900 12 \| 0.04291 \| 0.04278 13 \| 0.03805 \| 0.03796 14 \| 0.03380 \| 0.03375 15 \| 0.03042 \| 0.03038 16 \| 0.02747 \| 0.02745 17 \| 0.02502 \| 0.02499 18 \| 0.02286 \| 0.02280 19 \| 0.02078 \| 0.02076 20 \| 0.01909 \| 0.01907 21 \| 0.01756 \| 0.01755 22 \| 0.01626 \| 0.01624 23 \| 0.01512 \| 0.01510 24 \| 0.01411 \| 0.01403 25 \| 0.01314 \| 0.01307 Table 1: Average MFPT in the unit disk for previously computed optimal configurations $\overline{u}_{0}$ ( Ref. [8], Table 2 ), compared to that of the newly computed configurations for the unit disk $\overline{u}_{0}^{\prime}$. Here it can be seen that the new values are consistently smaller, at most differing in the third significant figure. A plot of the difference between these two, relative to the previous results, can be found in Figure 2. Figure 2: Relative difference between average MFPT in the unit disk for previously computed optimal configurations $\overline{u}_{0}$, compared to that of the newly computed configurations $\overline{u}_{0}^{\prime}$ according to $(\overline{u}_{0}^{\prime}-\overline{u}_{0})/\overline{u}_{0}$. Here it can be seen that the new values are consistently smaller, at most differing in the third significant figure. A table of these values can be found in Table 1. Figure 3: The putative optimal values of the merit function (2.3) for the ellipse eccentricity values $\kappa=0$ (a), $\kappa=0.125$ (b), $\kappa=0.250$ (c), and $\kappa=0.500$ (d), as functions of the number of traps $N$. [The corresponding numerical values are presented in Table 2.] $N$ \| Merit Value ---\|--- \| $\kappa=0$ \| $\kappa=0.125$ \| $\kappa=0.25$ \| $\kappa=0.5$ 1 \| -0.0597 \| -0.0594 \| -0.0540 \| 0.1730 2 \| -0.0754 \| -0.0792 \| -0.0854 \| 0.0175 3 \| -0.0969 \| -0.0967 \| -0.0959 \| -0.0452 4 \| -0.1112 \| -0.1113 \| -0.1115 \| -0.0793 5 \| -0.1207 \| -0.1207 \| -0.1200 \| -0.1007 6 \| -0.1272 \| -0.1274 \| -0.1289 \| -0.1154 7 \| -0.1348 \| -0.1347 \| -0.1342 \| -0.1261 8 \| -0.1409 \| -0.1408 \| -0.1393 \| -0.1343 9 \| -0.1451 \| -0.1451 \| -0.1447 \| -0.1407 10 \| -0.1489 \| -0.1494 \| -0.1492 \| -0.1457 11 \| -0.1526 \| -0.1532 \| -0.1533 \| -0.1498 12 \| -0.1567 \| -0.1566 \| -0.1569 \| -0.1530 13 \| -0.1599 \| -0.1598 \| -0.1603 \| -0.1559 14 \| -0.1632 \| -0.1632 \| -0.1632 \| -0.1587 15 \| -0.1659 \| -0.1660 \| -0.1657 \| -0.1614 16 \| -0.1685 \| -0.1686 \| -0.1683 \| -0.1642 17 \| -0.1708 \| -0.1705 \| -0.1708 \| -0.1668 18 \| -0.1731 \| -0.1731 \| -0.1733 \| -0.1693 19 \| -0.1756 \| -0.1755 \| -0.1756 \| -0.1718 20 \| -0.1777 \| -0.1776 \| -0.1775 \| -0.1741 21 \| -0.1798 \| -0.1797 \| -0.1796 \| -0.1756 22 \| -0.1815 \| -0.1815 \| -0.1816 \| -0.1768 23 \| -0.1831 \| -0.1831 \| -0.1833 \| -0.1800 24 \| -0.1848 \| -0.1851 \| -0.1848 \| -0.1820 25 \| -0.1864 \| -0.1867 \| -0.1864 \| -0.1834 $N$ \| Merit Value ---\|--- \| $\kappa=0$ \| $\kappa=0.125$ \| $\kappa=0.25$ \| $\kappa=0.5$ 26 \| -0.1880 \| -0.1882 \| -0.1880 \| -0.1858 27 \| -0.1897 \| -0.1896 \| -0.1896 \| -0.1875 28 \| -0.1911 \| -0.1910 \| -0.1911 \| -0.1893 29 \| -0.1925 \| -0.1925 \| -0.1925 \| -0.1909 30 \| -0.1940 \| -0.1940 \| -0.1938 \| -0.1920 31 \| -0.1953 \| -0.1953 \| -0.1952 \| -0.1941 32 \| -0.1964 \| -0.1964 \| -0.1965 \| -0.1953 33 \| -0.1977 \| -0.1978 \| -0.1978 \| -0.1958 34 \| -0.1989 \| -0.1989 \| -0.1990 \| -0.1975 35 \| -0.2000 \| -0.2001 \| -0.2000 \| -0.1987 36 \| -0.2012 \| -0.2012 \| -0.2012 \| -0.2003 37 \| -0.2025 \| -0.2024 \| -0.2023 \| -0.2005 38 \| -0.2035 \| -0.2035 \| -0.2033 \| -0.2022 39 \| -0.2045 \| -0.2044 \| -0.2043 \| -0.2028 40 \| -0.2056 \| -0.2055 \| -0.2053 \| -0.2037 41 \| -0.2065 \| -0.2065 \| -0.2064 \| -0.2046 42 \| -0.2074 \| -0.2073 \| -0.2073 \| -0.2049 43 \| -0.2083 \| -0.2084 \| -0.2083 \| -0.2065 44 \| -0.2093 \| -0.2092 \| -0.2092 \| -0.2069 45 \| -0.2102 \| -0.2103 \| -0.2102 \| -0.2086 46 \| -0.2110 \| -0.2111 \| -0.2111 \| -0.2091 47 \| -0.2119 \| -0.2120 \| -0.2119 \| -0.2102 48 \| -0.2127 \| -0.2128 \| -0.2128 \| -0.2098 49 \| -0.2136 \| -0.2136 \| -0.2136 \| -0.2118 50 \| -0.2144 \| -0.2143 \| -0.2143 \| -0.2126 Table 2: Optimized values of the merit function (2.3), for each number of traps $N$ and eccentricity $\kappa$ considered in this study. Plots of these values are found in Figure 3. Figure 4: Plots depicting optimal trap distribution for $N=5$, comparing eccentricities of (a) $\kappa=0$, (b) $\kappa=0.125$, (c) $\kappa=0.250$, (d) $\kappa=0.500$. Upper plots show positions of traps along with a crude visualization of nearest-neighbour pairs calculated using Delaunay triangulation. Lower plots show the scaling factor applied to the domain boundary such that the scaled version would pass through each trap, numbered from 1 to $N$. Figure 5: Plots depicting optimal trap distribution for $N=10$, comparing eccentricities of (a) $\kappa=0$, (b) $\kappa=0.125$, (c) $\kappa=0.250$, (d) $\kappa=0.500$. Upper plots show positions of traps along with a crude visualization of nearest-neighbour pairs calculated using Delaunay triangulation. Lower plots show the scaling factor applied to the domain boundary such that the scaled version would pass through a given trap, numbered from 1 to $N$. Figure 6: Plots depicting optimal trap distribution for $N=25$, comparing eccentricities of (a) $\kappa=0$, (b) $\kappa=0.125$, (c) $\kappa=0.250$, (d) $\kappa=0.500$. Upper plots show positions of traps along with a crude visualization of nearest-neighbour pairs calculated using Delaunay triangulation. Lower plots show the scaling factor applied to the domain boundary such that the scaled version would pass through a given trap, numbered from 1 to $N$. Figure 7: Plots depicting optimal trap distribution for $N=40$, comparing eccentricities of (a) $\kappa=0$, (b) $\kappa=0.125$, (c) $\kappa=0.250$, (d) $\kappa=0.500$. Upper plots show positions of traps along with a crude visualization of nearest-neighbour pairs calculated using Delaunay triangulation. Lower plots show the scaling factor applied to the domain boundary such that the scaled version would pass through a given trap, numbered from 1 to $N$. Figure 8: Plots depicting local properties of trap distribution, for domain eccentricities $\kappa=0$ (a), $\kappa=0.125$ (b), $\kappa=0.250$ (c), and $\kappa=0.500$ (d). The curve entitled “Measure of Area per Trap” shows the distance $\langle d\rangle$ computed using the “area per trap” argument and the resulting formula (4.3). ## 5 Discussion At this point some interpretation of the previously stated results will be presented. This discussion will concern the putative optimal values of the average MFPT (2.1) in elliptic domains with internal traps, values of the related merit function (2.3), the positions of traps within the domain, and the bulk measures of trap distribution which were employed. To begin, the method of study will be briefly reiterated. In order to study the dependence of optimal trap configurations on both the number of traps, and the eccentricity of the elliptic domain, merit functions for the average MFPT were minimized for $N\leq 50$ and eccentricity (3.1) values of 0, 0.125, 0.25, and 0.5, while the area of the ellipse was kept constant, $\|\Omega\|=\pi$. In the search for a global optimum, an iterative approach, which switched between global and local searches, was used. The method used here was similar to that used in Ref [9], though a different algorithm was used for the local search, as well as in Ref [11] which used a different algorithm for both searches. A somewhat different approach was employed in Ref [17], which made use of numerical solutions to the Poisson problem. In the case of the unit disk, comparing the results of this study to those of a previous study [8] demonstrated that the optimums reported here are consistently an improvement on previous work. This is due to the use of a more accurate asymptotic expression [9], as well as removing the constraint that all traps be located on rings within the domain. Plots of the putative globally optimal merit function values as functions of $N$, for each eccentricity, can be found in Figure 3. From these plots it seems that eccentricity is an important factor when there are few traps, around $N<10$, but each function behaves similarly as $N$ increases. In particular, as the number of traps $N$ increases, it is natural to expect that the average MFPT $\overline{u}_{0}$ (2.1) approaches zero; the merit function $q(\mathbf{x})$ therefore must, as $N\to\infty$, approach from above the value $\|\Omega\|/(4\pi^{2}D\nu)\simeq-0.238$, which agrees with the plots in Figure 3. Examination of the positions of traps in the optimized configurations, both visually and in terms of their radial coordinates, gives the impression that the optimal configuration is one which consists of traps placed on the vertices of nested polygons. These polygons, while irregular, seem to possess some consistent structure, including being convex. (It is interesting to note that the optimal configurations of confined interacting points often take similar forms, both in two and three dimensions [18, 19, 20, 21].) Due to the geometrical symmetries of the ellipse, the optimal configurations are defined uniquely modulo the group $C^{2}\times C^{2}$ of reflections with respect to both axes, which includes the rotation by $\pi$. The numerical algorithms, however, choose a single specific representative of the equivalent putative globally optimal configurations. For example, for non-symmetric numerically optimal configuration, several traps may be found along the midline of one half (right or left) of the domain. Optimal trap configurations with the same symmetry group as the ellipse were also observed (see, e.g., Figure 5(c)). In addition to the examination of individual trap positions in each optimized configuration, quantities were calculated using the distances between neighbouring traps, defined according to a Delaunay triangulation of the trap coordinates, which served as bulk measures of the distribution of traps in each configuration. Plots of these measures, shown in Figure 8, illustrate that the mean distance between neighbouring traps tends to be close to the diameter of a circle which would occupy the average area of the domain per trap, as in equation (4.3). Additionally, the minimum distance between any two traps tends to be twice the minimum distance between a trap and the domain boundary, which supports the intuitive reasoning that for the boundary value problem (1.1) with interior traps, the Neumann boundary condition on $\partial\Omega$ “reflects” each trap, so that under the average MFPT optimization, every trap tends to “repel” from its reflection in the boundary the same way as it is repelled from other traps. In future work, it would be of interest to address two related problems. The first is to carry out similar investigations for the near-disk domains considered in Ref. [9]. Another interesting direction is the development of a scaling law which would predict the behaviour of the MFPT as the number of traps increases with their positions defined according to a specific distribution, in particular, for distributions that globally or locally minimize MFPT, or other distributions of practical significance. A similar problem, along with the dilute trap fraction limit of homogenization theory, was addressed in Refs. [22, 23] for the narrow escape problem involving boundary traps located on the surface of a sphere in three dimensions. ### Acknowledgements A.C. is thankful to NSERC of Canada for research support through the Discovery grant RGPIN-2019-05570. ## References * [1] Sidney Redner. A Guide to First-Passage Processes. Cambridge University Press, 2001. * [2] David Holcman and Zeev Schuss. The narrow escape problem. SIAM Review, 56(2):213–257, 2014. * [3] PG Saffman and M Delbrück. Brownian motion in biological membranes. Proceedings of the National Academy of Sciences, 72(8):3111–3113, 1975. * [4] Metzler Ralf, Redner Sidney, and Oshanin Gleb. First-passage Phenomena and their Applications, volume 35. World Scientific, 2014. * [5] Paul C Bressloff, Berton A Earnshaw, and Michael J Ward. 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11institutetext: Data and Web Science Group, University of Mannheim, Mannheim, Germany 11email<EMAIL_ADDRESS>22institutetext: Language Technology Lab, University of Cambridge, UK 22email<EMAIL_ADDRESS> # Evaluating Multilingual Text Encoders for Unsupervised Cross-Lingual Retrieval Robert Litschko 11 Ivan Vulić 22 Simone Paolo Ponzetto 11 Goran Glavaš 11 ###### Abstract Pretrained multilingual text encoders based on neural Transformer architectures, such as multilingual BERT (mBERT) and XLM, have achieved strong performance on a myriad of supervised language understanding tasks. Consequently, they have been adopted as a state-of-the-art paradigm for multilingual and cross-lingual representation learning and transfer, rendering cross-lingual word embeddings (CLWEs) effectively obsolete. However, questions remain to which extent this finding generalizes 1) to unsupervised settings and 2) for ad-hoc cross-lingual IR (CLIR) tasks. Therefore, in this work we present a systematic empirical study focused on the suitability of the state- of-the-art multilingual encoders for cross-lingual document and sentence retrieval tasks across a large number of language pairs. In contrast to supervised language understanding, our results indicate that for unsupervised document-level CLIR – a setup in which there are no relevance judgments for task-specific fine-tuning – the pretrained encoders fail to significantly outperform models based on CLWEs. For sentence-level CLIR, we demonstrate that state-of-the-art performance can be achieved. However, the peak performance is not met using the general-purpose multilingual text encoders ‘off-the-shelf’, but rather relying on their variants that have been further specialized for sentence understanding tasks. ###### Keywords: Cross-lingual IR Multilingual text encoders Unsupervised IR. ## 1 Introduction Cross-lingual information retrieval (CLIR) systems respond to queries in a source language by retrieving relevant documents in another, target language. Their success is typically hindered by data scarcity: they operate in challenging low-resource settings without sufficient labeled training data, i.e., human relevance judgments, to build supervised models (e.g., neural matching models for pair-wise retrieval [53, 22]). This motivates the need for robust, resource-lean and unsupervised CLIR approaches. In previous work, Litschko et al. [27] have shown that language transfer through cross-lingual embedding spaces (CLWEs) can be used to yield state-of- the-art performance in a range of unsupervised ad-hoc CLIR setups. This approach uses very weak supervision (i.e., only a bilingual dictionary spanning 1K-5K word translation pairs), or even no supervision at all, in order to learn a mapping that aligns two monolingual word embedding spaces [19, 45]. Put simply, this enables casting CLIR tasks as ’monolingual tasks in the shared (CLWE) space’: at retrieval time both queries and documents are represented as simple aggregates of their constituent CLWEs. However, this approach, by limitations of static CLWEs, cannot capture and handle polysemy in the underlying text representations. Contextual text representation models alleviate this issue [28]. They encode occurrences of the same word differently depending on its surrounding context. Such contextual representations are obtained via large models pretrained on large text collections through general objectives such as (masked) language modeling [16, 30]. Multilingual text encoders pretrained on 100+ languages, such as mBERT [16] or XLM [14], have become a de facto standard for multilingual representation learning and cross-lingual transfer in natural language processing (NLP). These models demonstrate state-of-the-art performance in a wide range of supervised language understanding and language generation tasks [36, 26], especially in zero-shot settings: a typical modus operandi is fine-tuning a pretrained multilingual encoder with task-specific data of a source language (typically English) and then using it directly in a target language. However, there are still several crucial questions remaining. First, it is unclear whether these general-purpose multilingual text encoders can be used directly for ad-hoc CLIR without any additional supervision (i.e., relevance judgments). Further, can they outperform the previous unsupervised CLIR approaches based on static CLWEs [27]? How do they perform depending on the (properties of the) language pair at hand? How can we encode useful semantic information using these models, and do different “encoding variants” (see later §3) yield different retrieval results? Are there performance differences in unsupervised sentence-level versus document-level CLIR tasks? Finally, can we boost performance by relying on sentence encoders that are specialized towards dealing with sentence-level understanding in particular? In order to address these questions, we present a systematic empirical study and profile the suitability of state-of-the-art pretrained multilingual encoders for different CLIR tasks and diverse language pairs. We evaluate two state-of-the- art general-purpose pretrained multilingual encoders, mBERT [16] and XLM [14] with a range of encoding variants, and also compare them to CLIR approaches based on static CLWEs, and specialized multilingual sentence encoders. Our key contributions can be summarized as follows: (1) We empirically validate that, without any task-specific fine-tuning, multilingual encoders such as mBERT and XLM fail to outperform CLIR approaches based on static CLWEs. Their performance also crucially depends on how one encodes semantic information with the models (e.g., treating them as sentence/document encoders directly versus averaging over constituent words and/or subwords). We also show that there is no “one-size-fits-all” approach, and the results are task- and language-pair-dependent. (2) We provide a first large-scale comparative evaluation of state-of-the art pretrained multilingual encoders on unsupervised document-level CLIR. We also empirically show that encoder models specialized for sentence-level understanding substantially outperform general-purpose models (mBERT and XLM) on sentence-level CLIR tasks. ## 2 Related Work Self-Supervised Pretraining and Transfer Learning. Recently, research on universal sentence representations and transfer learning has gained much traction. InferSent [13] transfers the encoder of a model trained on natural language inference to other tasks, while USE [8] extends this idea to a multi- task learning setting. More recent work explores self-supervised neural Transformer-based [44] models based on (causal or masked) language modeling (LM) objectives such as BERT [16], RoBERTa [30], GPT [37, 5], and XLM [14].111Note that self-supervised learning can come in different flavors depending on the training objective [10], but language modeling objectives still seem to be the most popular choice. Results on benchmarks such as GLUE [47] and SentEval [12] indicate that these models can yield impressive (sometimes human-level) performance in supervised Natural Language Understanding (NLU) and Generation (NLG) tasks. These models have become _de facto_ standard and omnipresent text representation models in NLP. In supervised monolingual IR, self-supervised LMs have been employed as contextualized word encoders [32], or fine-tuned as pointwise and pairwise rankers [33]. Multilingual Text Encoders based on the (masked) LM objectives have also been massively adopted in multilingual and cross-lingual NLP and IR applications. A multilingual extension of BERT (mBERT) is trained with a shared subword vocabulary on a single multilingual corpus obtained as concatenation of large monolingual data in 104 languages. The XLM model [14] extends this idea and proposes natively cross-lingual LM pretraining, combining causal language modeling (CLM) and translation language modeling (TLM).222In CLM, the model is trained to predict the probability of a word given the previous words in a sentence. TLM is a cross-lingual variant of standard masked LM (MLM), with the core difference that the model is given pairs of parallel sentences and allowed to attend to the aligned sentence when reconstructing a word in the current sentence. Strong performance of these models in supervised settings is confirmed across a range of tasks on multilingual benchmarks such as XGLUE [26] and XNLI [15]. However, recent work [39, 6] has indicated that these general-purpose models do not yield strong results when used as out-of-the-box text encoders in an unsupervised transfer learning setup. We further investigate these preliminaries, and confirm this finding also for unsupervised ad-hoc CLIR tasks. Multilingual text encoders have already found applications in document-level CLIR. Jiang et al. [22] use mBERT as a matching model by feeding pairs of English queries and foreign language documents. MacAvaney et al. [31] use mBERT in a zero-shot setting, where they train a retrieval model on top of mBERT on English relevance data and apply it on a different language. However, prior work has not investigated unsupervised CLIR setups, and a systematic comparative study focused on the suitability of the multilingual text encoders for diverse ad-hoc CLIR tasks and language pairs is still lacking. Specialized Multilingual Sentence Encoders. An extensive body of work focuses on inducing multilingual encoders that capture sentence meaning. In [2], the multilingual encoder of a sequence-to-sequence model is shared across languages and optimized to be language-agnostic, whereas Guo et al. [20] rely on a dual Transformer-based encoder architectures instead (with tied/shared parameters) to represent parallel sentences. Rather than optimizing for translation performance directly, their approach minimizes the cosine distance between parallel sentences. A ranking softmax loss is used to classify the correct (i.e., aligned) sentence in the other language from negative samples (i.e., non-aligned sentences). In [50], this approach is extended by using a bidirectional dual encoder and adding an additive margin softmax function, which serves to push away non-translation-pairs in the shared embedding space. The dual-encoder approach is now widely adopted [20, 51, 18, 39, 56], and yields state-of-the-art multilingual sentence encoders which excel in sentence-level NLU tasks. Other recent approaches propose input space normalization, and re-aligning mBERT and XLM with parallel data [56, 6], or using a teacher-student framework where a student model is trained to imitate the output of the teacher network while preserving high similarity of translation pairs [39]. In [51], authors combine multi-task learning with a translation bridging task to train a universal sentence encoder. We benchmark a series of representative sentence encoders; their brief descriptions are provided in §3.3. CLIR Evaluation and Application. The cross-lingual ability of mBERT and XLM has been investigated by probing and analyzing their internals [23], as well as in terms of downstream performance [34, 49]. In CLIR, these models as well as dedicated multilingual sentence encoders have been evaluated on tasks such as cross-lingual question-answer retrieval [51], bitext mining [58, 59], and semantic textual similarity (STS) [21, 25]. Yet, the models have been primarily evaluated on sentence-level retrieval, while classic ad-hoc (unsupervised) document-level CLIR has not been in focus. Further, previous work has not provided a large-scale comparative study across diverse language pairs and with different model variants, nor has tried to understand and analyze the differences between sentence-level and document-level tasks. In this work, we aim to fill these gaps. ## 3 Multilingual Text Encoders We now provide an overview of all the multilingual models in our comparison. For completeness, we first briefly describe static CLWEs (§3.1). We then discuss mBERT and XLM as representative general-purpose multilingual text encoders trained with LM objectives (§3.2), as well as specialized multilingual sentence encoders later in §3.3. ### 3.1 CLIR with (Static) Cross-lingual Word Embeddings In a standard CLIR setup, we assume a query $Q_{L_{1}}$ issued in a source language $L_{1}$, and a document collection comprising $N$ documents $D_{i,L_{2}}$, $i=1,\ldots,N$ in a target language $L_{2}$. Let $d=\\{t_{1},t_{2},\dots,t_{\|D\|}\\}\in D$ be a document consisting of $\|D\|$ terms $t_{i}$. A typical approach to CLIR with static CLWEs is to represent queries and documents as vectors $\overrightarrow{Q},\overrightarrow{D}\in\mathbb{R}^{d}$ in a $d$-dimensional shared embedding space [46, 27]. Each term is represented independently and obtained by performing a lookup on a pre-computed static embedding table $\overrightarrow{t_{i}}=emb\left(t_{i}\right)$. There exist a range of methods for inducing shared embedding spaces with different levels of supervision, such as parallel sentences, comparable documents, small bilingual dictionaries, or even methods without any supervision [41]. Given the shared CLWE space, both query and document representations are obtained as aggregations of their term embeddings. We follow Litschko et al. [27] and represent documents as the weighted sum of their terms’ vectors, where each term’s weight corresponds to its inverse document frequency (idf) : $\overrightarrow{d}=\sum_{i=1}^{N_{d}}{\mathit{idf}(t^{d}_{i})\cdot\overrightarrow{t^{d}_{i}}}$. During retrieval documents are ranked according to the cosine similarity to the query. ### 3.2 Multilingual (Transformer-Based) Language Models: mBERT and XLM Massively multilingual pretrained neural language models such as mBERT and XLM can be used as a dynamic embedding layer to produce contextualized word representations, since they share a common input space on the subword level (e.g. word-pieces, byte-pair-encodings) across all languages. Let us assume that a term (i.e., a word-level token) is tokenized into a sequence of $K$ subword tokens ($K\geq 1$; for simplicity, we assume that the subwords are word-pieces (wp)): $t_{i}=\big{\\{}\textit{wp}_{i,k}\big{\\}}^{K}_{k=1}$. The multilingual encoder then produces contextualized subword embeddings for the term’s $K$ constituent subwords $\overrightarrow{wp_{i,k}}$, $k=1,\ldots,K$, and we can aggregate these subword embeddings to obtain the representation of the term $t_{i}$: $\overrightarrow{t_{i}}=\psi\left(\\{\overrightarrow{wp_{i,k}}\\}^{K}_{k=1}\right)$, where the function $\psi()$ is the aggregation function over the $K$ constituent subword embeddings. Once these term embeddings $\overrightarrow{t_{i}}$ are obtained, we follow the same CLIR setup as with CLWEs in §3.1. Static Word Embeddings from Multilingual Transformers. We first use multilingual transformers (mBERT and XLM) in two different ways to induce static word embedding spaces for all languages. In a simpler variant, we feed terms into the encoders in isolation (ISO), that is, without providing any surrounding context for the terms. This effectively constructs a static word embedding table similar to what is done in §3.1, and allows the CLIR model (§3.1) to operate at a non-contextual word level. An empirical CLIR comparison between ISO and CLIR operating on CLWEs [27] then effectively quantifies how well multilingual encoders (mBERT and XLM) encode word-level representations. In a more elaborate variant we do leverage the contexts in which the terms appear, and construct average-over-contexts embeddings (AOC). For each term $t$ we collect a number of sentences $s_{i}\in\mathcal{S}_{t}$ in which the term occurs. We use the full set of Wikipedia sentences $\mathcal{S}$ to sample sets of contexts $\mathcal{S}_{t}$ for vocabulary terms. For a given sentence $s_{i}$ let $j$ denote the position of $t$’s first occurrence. We then transform $s_{i}$ with mBERT or XLM as the encoder, $enc(s_{i})$, and extract the contextualized embedding of $t$ via mean-pooling, i.e., by averaging embeddings of its constituent subwords, $\psi\left(\\{\overrightarrow{wp_{j,k}}\\}^{K}_{k=1}\right)=1/K\cdot\sum_{k=1}^{K}{\overrightarrow{wp_{j,k}}}$. For each vocabulary term, we obtain $N_{t}=min(\|\mathcal{S}_{t}\|,\tau)$ contextualized vectors, with $\|\mathcal{S}_{t}\|$ as the number of Wikipedia sentences containing $t$ and $\tau$ as the maximal number of sentence samples for a term. The final static embedding of $t$ is then simply the average over the $N_{t}$ contextualized vectors. The obtained static AOC and ISO embeddings, despite being induced with multilingual encoders, however, did not appear to be well-aligned across languages [29, 6]. We evaluated the static ISO and AOC embeddings induced for different languages with multilingual encoders (mBERT and XLM), on the bilingual lexicon induction (BLI) task [19]. We observed poor BLI performance, suggesting that further projection-based alignment of respective monolingual ISO and AOC spaces is required. To this end, we use the the standard Procrustes method [43, 1] to align the embedding spaces of two languages, with bilingual dictionaries from [19] as the supervision guiding the alignment. Concretely, for each language pair in our experiments we project the AOC (ISO) embeddings of the source language to the AOC (ISO) space of the target language. Direct Text Embedding with Multilingual Transformers. In both AOC and ISO, we use the multilingual (contextual) encoders to obtain the static embeddings for word types (i.e., terms): we can then leverage in exactly the same ad-hoc retrieval setup (§3.1) in which CLWEs had previously been evaluated [27]. In an arguably more straightforward approach, we also use pretrained multilingual Transformers (i.e., mBERT or XLM) to directly encode the whole input text (SEMB). We encode the input text by averaging the contextualized representations of all terms in the text (we again compute the weighted average, where the terms’ IDF scores are used as weights, see §3.1). For SEMB, we take the contextualized representation of each term $t_{i}$ to be the contextualized representation of its first subword token, i.e., $\overrightarrow{t_{i}}=\psi\left(\\{\overrightarrow{wp_{i,k}}\\}^{K}_{k=1}\right)=\overrightarrow{wp_{i,1}}.$333In our initial experiments taking the vector of the first term’s subword consistently outperformed averaging vectors of all its subwords. ### 3.3 Specialized Multilingual Sentence Encoders Off-the-shelf pretrained multilingual Transformers such as mBERT and XLM have been shown to produce poor sentence embeddings yielding sub-par performance in unsupervised text similarity tasks; therefore, in order to be successful in semantic text (sentences or paragraph) comparisons, they first need to be fine-tuned on text matching (typically sentence matching) datasets [39, 6, 57]. Such encoders specialized for semantic similarity are supposed to encode sentence meaning more accurately, supporting tasks that require unsupervised (ad-hoc) semantic text matching. In contrast to mBERT and XLM, which contextualize (sub)word representations, these models directly produce a semantic embedding of the input text. We provide a brief overview of the models included in our comparative evaluation. Language Agnostic SEntence Representations (LASER) [2] adopts a standard sequence-to-sequence architecture typical for neural machine translation (MT). It is trained on 223M parallel sentences covering 93 languages. The encoder is a multi-layered bidirectional LSTM and the decoder is a single-layer unidirectional LSTM. The 1024-dimensional sentence embedding is produced by max-pooling over the outputs of encoder’s last layer. The decoder then takes the sentence embedding as additional input as each decoding step. The decoder- to-encoder attention and language identifiers on the encoder side are deliberately omitted, so that all relevant information gets ‘crammed’ into the fixed-sized sentence embedding produced by the encoder. In our experiments, we directly use the output of the encoder to represent both queries and documents. Multilingual Universal Sentence Encoder (m-USE) is a general purpose sentence embedding model for transfer learning and semantic text retrieval tasks [51]. It relies on a standard dual-encoder neural framework [9, 52] with shared weights, trained in a multi-task setting with an additional translation bridging task. For more details, we refer the reader to the original work. There are two pretrained m-USE instances available – we opt for the 3-layer Transformer encoder with average-pooling. Language-agnostic BERT Sentence Embeddings (LaBSE) [18] is another neural dual-encoder framework, also trained with parallel data. Unlike in LASER and m-USE, where the encoders are trained from scratch on parallel data, LaBSE training starts from a pretrained mBERT instance (i.e., a 12-layer Transformer network pretrained on the concatenated corpora of 100+ languages). In addition to the multi-task training objective of m-USE, LaBSE additionally uses standard self-supervised objectives used in pretraining of mBERT and XLM: masked and translation language modelling (MLM and TLM, see §2). For further model details, we refer the reader to the original work. DISTIL [39] is a teacher-student framework for injecting the knowledge obtained through specialization for semantic similarity from a specialized monolingual transformer (e.g., BERT) into a non-specialized multilingual transformer (e.g., mBERT). It first specializes for semantic similarity a monolingual (English) teacher encoder $M$ using the available semantic sentence-matching datasets for supervision. In the second, knowledge distillation step a pretrained multilingual student encoder $\widehat{M}$ is trained to mimic the output of the teacher model. For a given batch of sentence-translation pairs $\mathcal{B}=\\{(s_{j},t_{j})\\}$, the teacher- student distillation training minimizes the following loss: $\mathcal{J}(\mathcal{B})=\frac{1}{\|\mathcal{B}\|}\sum_{j\in\mathcal{B}}\left[\left(M(s_{j})-\widehat{M}(s_{j})\right)^{2}+\left(M(s_{j})-\widehat{M}(t_{j})\right)^{2}\right].$ The teacher model $M$ is Sentence-BERT [38], BERT specialized for embedding sentence meaning on semantic text similarity [7] and natural language inference [48] datasets. The teacher network only encodes English sentences $s_{i}$. The student model $\widehat{M}$ is then trained to produce for both $s_{j}$ and $t_{j}$ the same representation that $M$ produces for $s_{j}$. We benchmark different DISTIL models in our CLIR experiments, with the student $\widehat{M}$ initialized with different multilingual transformers. ## 4 Experimental Setup Evaluation Data. We follow the experimental setup of Litschko et al. [27], and compare the models from §3 on language pairs comprising five languages: English (EN), German (DE), Italian (IT), Finnish (FI) and Russian (RU). For document-level retrieval we run experiments for the following nine language pairs: EN-{FI, DE, IT, RU}, DE-{FI, IT, RU}, FI-{IT, RU}. We use the 2003 portion of the CLEF benchmark [4],444http://catalog.elra.info/en- us/repository/browse/ELRA-E0008/ with 60 queries per language pair. The document collection sizes are 17K (RU), 55K (FI), 158K (IT), and 295K (DE). For sentence-level retrieval, also following [27], for each language pair we sample from Europarl [24] 1K source language sentences as queries and 100K target language sentences as the “document collection”.555Russian is not included in Europarl and we therefore exclude it from sentence-level experiments. Further, since some multilingual encoders have not seen Finnish data in pretraining, we additionally report the results over a subset of language pairs that do not involve Finnish. Baseline Models. In order to establish whether multilingual encoders outperform CLWEs in a fair comparison, we compare their performance against the strongest CLWE-based CLIR model from the recent comparative study [27], dubbed Proc-B. Proc-B induces a bilingual CLWE space from pretrained monolingual fastText embeddings666https://fasttext.cc/docs/en/pretrained- vectors.html using the linear projection computed as the solution of the Procrustes problem given the dictionary of word-translation pairs. Compared to simple Procrustes mapping, Proc-B iteratively (1) augments the word translation dictionary by finding mutual nearest neighbours and (2) induces a new projection matrix using the augmented dictionary. The final bilingual CLWE space is then plugged into the CLIR model from §3.1. Our document-level retrieval SEMB models do not get to see the whole document but only the first $128$ word-piece tokens. For a more direct comparison, we therefore additionally evaluate the Proc-B baseline (Proc-BLEN) which is exposed to exactly the same amount of document text as the multilingual XLM encoder (i.e., the leading document text corresponding to first $128$ word- piece tokens) Finally, we compare CLIR models based on multilingual Transformers to a baseline relying on machine translation baseline (MT-IR). In MT-IR, 1) we translate the query to the document language using Google Translate and then 2) perform monolingual retrieval using a standard Query Likelihood Model [35] with Dirichlet smoothing [55]. Model Details. For all multilingual encoders we experiment with different input sequence lengths: $64$, $128$, $256$ subword tokens. For AOC we collect (at most) $\tau=60$ contexts for each vocabulary term: for a term not present at all in Wikipedia, we fall back to the ISO embedding of that term. We also investigate the impact of $\tau$ in §5.3. For purely self-supervised models (SEMB, ISO, AOC) we independently evaluate representations from different Transformer layers (cf. §5.3). For comparability, for ISO and AOC – methods that effectively induce static word embeddings using multilingual contextual encoders – we opt for exactly the same term vocabularies used by the Proc-B baseline, namely the top 100K most frequent terms from respective monolingual fastText vocabularies. We additionally experiment with three different instances of the DISTIL model: (i) $\text{DISTIL}_{\text{XLM-R}}$ initializes the student model with the pretrained XLM-R transformer [11]; $\text{DISTIL}_{\text{USE}}$ instantiates the student as the pretrained m-USE instance [51]; whereas $\text{DISTIL}_{\text{DistilmBERT}}$ distils the knowledge from the Sentence-BERT teacher into a multilingual version of DistilBERT [42], a 6-layer transformer pre-distilled from mBERT.777Working with mBERT directly instead of its distilled version led to similar scores, while increasing running times. For SEMB models we scale embeddings of special tokens (sequence start and end tokens, e.g., [CLS] and [SEP] for mBERT) with the mean IDF value of input terms. ## 5 Results and Discussion ### 5.1 Document-Level Cross-lingual Retrieval Table 1: Document-level CLIR results (Mean Average Precision, MAP). Bold: best model for each language-pair. : difference in performance w.r.t. Proc-B significant at $p=0.05$, computed via paired two-tailed t-test with Bonferroni correction. \| EN-FI \| EN-IT \| EN-RU \| EN-DE \| DE-FI \| DE-IT \| DE-RU \| FI-IT \| FI-RU \| AVG \| w/o FI ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Baselines \| \| \| \| \| \| \| \| \| \| \| MT-IR \| .276 \| .428 \| .383 \| .263 \| .332 \| .431 \| .238 \| .406 \| .261 \| .335 \| .349 Proc-B \| .258 \| .265 \| .166 \| .288 \| .294 \| .230 \| .155 \| .151 \| .136 \| .216 \| .227 $\text{Proc-B}_{\text{LEN}}$ \| .165 \| .232 \| .176 \| .194 \| .207 \| .186 \| .192 \| .126 \| .154 \| .181 \| .196 Models based on multilingual Transformers $\text{SEMB}_{\text{XLM}}$ \| .199 \| .187* \| .183 \| .126* \| .156* \| .166* \| .228 \| .186* \| .139 \| .174 \| .178 $\text{SEMB}_{\text{mBERT}}$ \| .145* \| .146* \| .167 \| .107* \| .151* \| .116* \| .149* \| .117 \| .128* \| .136 \| .137 1-12[.4pt/1pt] $\text{AOC}_{\text{XLM}}$ \| .168 \| .261 \| .208 \| .206* \| .183 \| .190 \| .162 \| .123 \| .099 \| .178 \| .206 $\text{AOC}_{\text{mBERT}}$ \| .172* \| .209* \| .167 \| .193* \| .131* \| .143* \| .143 \| .104 \| .132 \| .155 \| .171 1-12[.4pt/1pt] $\text{ISO}_{\text{XLM}}$ \| .058* \| .159* \| .050* \| .096* \| .026* \| .077* \| .035* \| .050* \| .055* \| .067 \| .083 $\text{ISO}_{\text{mBERT}}$ \| .075* \| .209 \| .096* \| .157* \| .061* \| .107* \| .025* \| .051* \| .014* \| .088 \| .119 Similarity-specialized sentence encoders (with parallel data supervision) \| \| \| $\text{DISTIL}_{\text{XLM-R}}$ \| .216 \| .190* \| .179 \| .114* \| .237 \| .181 \| .173 \| .166 \| .138 \| .177 \| .167 $\text{DISTIL}_{\text{USE}}$ \| .141* \| .346* \| .182 \| .258 \| .139* \| .324* \| .179 \| .104 \| .111 \| .198 \| .258 $\text{DISTIL}_{\text{DistilmBERT}}$ \| .294 \| .290* \| .313 \| .247* \| .300 \| .267* \| .284 \| .221* \| .302* \| .280 \| .280 1-12[.4pt/1pt] LaBSE \| .180* \| .175* \| .128 \| .059* \| .178* \| .160* \| .113* \| .126 \| .149 \| .141 \| .127 LASER \| .142 \| .134* \| .076 \| .046* \| .163* \| .140* \| .065* \| .144 \| .107 \| .113 \| .094 m-USE \| .109* \| .328* \| .214 \| .230* \| .107* \| .294* \| .204 \| .073 \| .090 \| .183 \| .254 We show the performance (MAP) of multilingual encoders on document-level CLIR tasks in Table 1. The first main finding is that none of the self-supervised models (mBERT and XLM in ISO, AOC, and SEMB variants) outperforms the CLWE baseline Proc-B. However, the full Proc-B baseline has, unlike mBERT and XLM variants, been exposed to the full content of the documents. A fairer comparison, against Proc-BLEN, which has also been exposed only to the first $128$ tokens, reveals that SEMB and AOC variants come reasonably close, albeit still do not outperform Proc-BLEN. This suggests that the document-level retrieval could benefit from encoders able to encode longer portions of text, e.g., [3, 54]. For document-level CLIR, however, these models would first have to be ported to multilingual setups. While SEMB and AOC variants exhibit similar performance, ISO variants perform much worse. The direct comparison between ISO and AOC demonstrates the importance of contextual information and seemingly limited usability of multilingual encoders as word encoders, if no context is available. Similarity-specialized multilingual encoders, which rely on pretraining with parallel data, yield mixed results. Three models, $\text{DISTIL}_{\text{DistilmBERT}}$, $\text{DISTIL}_{\text{USE}}$ and m-USE, generally outperform the Proc-B baseline888As expected, m-USE and $\text{DISTIL}_{\text{USE}}$ perform poorly on language pairs involving Finnish, as they have not been trained on any Finnish data. LASER is the only encoder trained on parallel data that does not beat the Proc-B baseline. We believe this is because (a) LASER’s recurrent encoder provides text embeddings of lower quality than Transformer-based encoders of m-USE and DISTIL variants and (b) it has not been subdued to any self-supervised pretraining like DISTIL models. Even the best-performing CLIR model based on a multilingual encoder ($\text{DISTIL}_{\text{DistilmBERT}}$) overall falls behind the MT-based baseline (MT-IR). However, the performance of MT-IR crucially depends on the quality of MT for the concrete language pair: for language pairs with weaker MT (e.g., FI-RU, EN-FI, FI-RU, DE-RU), $\text{DISTIL}_{\text{DistilmBERT}}$ can substantially outperform MT-IR (e.g., 9 MAP points for FI-RU and DE-RU); the gap in favor of MT-IR is, as expected, largest for most similar language pairs, for which also the most reliable MT systems exist (EN-IT, EN-DE). In other words, the feasibility and robustness of a strong MT-IR CLIR model seems to diminish with more distant language pairs and lower-resource language pairs. We plan to investigate this conjecture in more detail in future work. The variation in results with similarity-specialized sentence encoders indicates that: (a) despite their seemingly similar high-level architectures typically based on dual-encoder networks [8], it is important to carefully choose a sentence encoder in document-level retrieval, and (b) there is an inherent mismatch between the granularity of information encoded by the current state-of-the-art text representation models and the document-level CLIR task. ### 5.2 Sentence-Level Cross-Lingual Retrieval Table 2: Sentence-level CLIR results (MAP). Bold: best model for each language-pair. : difference in performance with respect to Proc-B, significant at $p=0.05$, computed via paired two-tailed t-test with Bonferroni correction. \| EN-FI \| EN-IT \| EN-DE \| DE-FI \| DE-IT \| FI-IT \| AVG \| w/o FI ---\|---\|---\|---\|---\|---\|---\|---\|--- Baselines \| \| \| \| \| \| \| \| MT-IR \| .639 \| .783 \| .712 \| .520 \| .676 \| .686 \| .669 \| .723 Proc-B \| .143 \| .523 \| .415 \| .162 \| .342 \| .137 \| .287 \| .427 Models based on multilingual Transformers $\text{SEMB}_{\text{XLM}}$ \| .309 \| .677* \| .465 \| .391* \| .495* \| .346* \| .447 \| .545 $\text{SEMB}_{\text{mBERT}}$ \| .199* \| .570 \| .355 \| .231* \| .481* \| .353* \| .365 \| .469 1-9[.4pt/1pt] $\text{AOC}_{\text{XLM}}$ \| .099 \| .527 \| .274* \| .102* \| .282 \| .070* \| .226 \| .361 $\text{AOC}_{\text{mBERT}}$ \| .095* \| .433* \| .274* \| .088* \| .230* \| .059* \| .197 \| .312 1-9[.4pt/1pt] $\text{ISO}_{\text{XLM}}$ \| .016* \| .178* \| .053* \| .006* \| .017* \| .002* \| .045 \| .082 $\text{ISO}_{\text{mBERT}}$ \| .010* \| .141* \| .087* \| .005* \| .017* \| .000* \| .043 \| .082 Similarity-specialized sentence encoders (with parallel data supervision) $\text{DISTIL}_{\text{XLM-R}}$ \| .935* \| .944* \| .943* \| .911* \| .919* \| .914* \| .928 \| .935 $\text{DISTIL}_{\text{USE}}$ \| .084* \| .960* \| .952* \| .137 \| .920* \| .072* \| .521 \| .944 $\text{DISTIL}_{\text{DistilmBERT}}$ \| .817* \| .902* \| .902* \| .810* \| .842* \| .793* \| .844 \| .882 1-9[.4pt/1pt] LaBSE \| .971* \| .972* \| .964* \| .948* \| .954* \| .951* \| .960 \| .963 LASER \| .974* \| .976* \| .969* \| .967* \| .965* \| .961* \| .969 \| .970 m-USE \| .079* \| .951* \| .929* \| .086* \| .886* \| .039* \| .495 \| .922 We show the sentence-level CLIR performance in Table 2. Unlike in the document-level CLIR task, self-supervised SEMB variants here manage to outperform Proc-B. The better relative SEMB performance than in document-level retrieval is somewhat expected: sentences are much shorter than documents (i.e., typically shorter than the maximal sequence length of $128$ word pieces). All purely self-supervised mBERT and XLM variants, however, perform worse than the translation-based baseline. Multilingual encoders specialized with parallel data excel in sentence-level CLIR, all of them substantially outperforming the competitive MT-IR baseline. This however, does not come as much of a surprise, since these models (a) have been trained using parallel data, and (b) have been optimized exactly on the sentence similarity task. In other words, in the context of the cross-lingual sentence-level task, these models are effectively supervised models. The effect of supervision is most strongly pronounced for LASER, which was, by being also trained on parallel data from Europarl, effectively subdued to in- domain training. We note that at the same time LASER was the weakest model from this group on average in the document-level CLIR task. ### 5.3 Further Analysis We further investigate three aspects that may impact CLIR performance of multilingual encoders: (1) layer(s) from which we take vector representations, (2) number of contexts used in AOC variants, and (3) sequence length in document-level CLIR. Layer Selection. All multilingual encoders have multiple layers and one may select (sub)word representations for CLIR at the output of any of them. Figure 1 shows the impact of taking subword representations after each layer for self-supervised mBERT and XLM variants. We find that the optimal layer differs across the encoding strategies (AOC, ISO, and SEMB) and tasks (document-level vs. sentence-level CLIR). ISO, where we feed the terms into encoders without any context, seems to do best if we take the representations from lowest layers. This makes intuitive sense, as the parameters of higher Transformer layers encode compositional rather than lexical semantics [17, 40]. For AOC and SEMB, where both models obtain representations by contextualizing (sub)words in a sentence, we get the best performance for higher layers – the optimal layers for document-level retrieval (L9/L12 for mBERT, and L15 for XLM) seem to be higher than for sentence-level retrieval (L9 for mBERT and L12/L11 for XLM). Figure 1: CLIR performance of mBERT and XLM as a function of the Transformer layer from which we obtain the representations. Results (averaged over all language pairs) shown for all three encoding strategies (SEMB, AOC, ISO). Number of Contexts in AOC. We construct AOC term embeddings by averaging contextualized representations of the same term obtained from different Wikipedia contexts. This raises an obvious question of a sufficient number of contexts needed for a reliable (static) term embedding. Figure 2 shows the AOC results depending on the number of contexts used to induce the term vectors (cf. $\tau$ in §3). The AOC performance seems to plateau rather early – at around 30 and 40 contexts for mBERT and XLM, respectively. Encoding more than 60 contexts (as we do in our main experiments) would therefore bring only negligible performance gains. Figure 2: CLIR performance of AOC variants (mBERT and XLM) w.r.t. the number of contexts used to obtain the term embeddings. Input Sequence Length. Multilingual encoders have a limited input length and they, unlike CLIR models operating on static embeddings (Proc-B, as well as our AOC and ISO variants), effectively truncate long documents. In our main experiments we truncated the documents to first $128$ word pieces. Now we quantify (Table 3) if and to which extent this has a detrimental effect on document-level CLIR performance. Somewhat counterintuitively, encoding a longer chunk of documents ($256$ word pieces) yields a minor performance deterioration (compared to the length of $128$) for all multilingual encoders. We suspect that this is a combination of two effects: (1) it is more difficult to semantically accurately encode a longer portion of text, leading to semantically less precise embeddings of $256$-token sequences; and (2) for documents in which the query-relevant content is not within the first $128$ tokens, that content might often also appear beyond the first $256$ tokens, rendering the increase in input length inconsequential to the recognition of such documents as relevant. Table 3: Document CLIR results w.r.t. the input text length. Scores averaged over all language pairs not involving Finnish. Length \| $\text{SEMB}_{\text{mBERT}}$ \| $\text{SEMB}_{\text{XLM}}$ \| $\text{DIST}_{\text{use}}$ \| $\text{DIST}_{\text{XLM-R}}$ \| $\text{DIST}_{\text{DmBERT}}$ \| mUSE \| LaBSE \| LASER ---\|---\|---\|---\|---\|---\|---\|---\|--- 64 \| .104 \| .128 \| .235 \| .167 \| .237 \| .254 \| .127 \| .089 128 \| .137 \| .178 \| .258 \| .162 \| .280 \| .247 \| .125 \| .068 256 \| .117 \| .158 \| .230 \| .146 \| .250 \| .197 \| .096 \| .027 ## 6 Conclusion Pretrained multilingual (mostly Transformer-based) encoders have been shown to be widely useful in natural language understanding (NLU) tasks, when fine- tuned in supervised settings with some task-specific data; their utility as general-purpose text encoders in unsupervised multilingual settings, such as the ad-hoc cross-lingual IR, has been much less investigated. In this work, we systematically validated the suitability of a wide spectrum of cutting-edge multilingual encoders for document- and sentence-level CLIR across several language pairs. Our study encompassed purely self-supervised multilingual encoders, mBERT and XLM, as well as the multilingual encoders that have been specialized for semantic text matching on semantic similarity datasets and parallel data. Opposing the main findings from supervised NLU tasks, we have demonstrated that self-supervised multilingual encoders (mBERT and XLM), without exposure to any further supervision, in most settings fail to outperform CLIR models based on cross-lingual word embeddings (CLWEs). Semantically-specialized multilingual sentence encoders, on the other hand, do outperform CLWEs, but the gains are pronounced only in the sentence retrieval task. 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# Personalised Recommendations in Mental Health Apps: The Impact of Autonomy and Data Sharing Svenja Pieritz Telefonica Alpha, Spain<EMAIL_ADDRESS>, Mohammed Khwaja Imperial College London, UK<EMAIL_ADDRESS>, A. Aldo Faisal Imperial College London, UK<EMAIL_ADDRESS>and Aleksandar Matic Koa Health, Spain<EMAIL_ADDRESS> (2021) ###### Abstract. The recent growth of digital interventions for mental well-being prompts a call-to-arms to explore the delivery of personalised recommendations from a user’s perspective. In a randomised placebo study with a two-way factorial design, we analysed the difference between an autonomous user experience as opposed to personalised guidance, with respect to both users’ preference and their actual usage of a mental well-being app. Furthermore, we explored users’ preference in sharing their data for receiving personalised recommendations, by juxtaposing questionnaires and mobile sensor data. Interestingly, self- reported results indicate the preference for personalised guidance, whereas behavioural data suggests that a blend of autonomous choice and recommended activities results in higher engagement. Additionally, although users reported a strong preference of filling out questionnaires instead of sharing their mobile data, the data source did not have any impact on the actual app use. We discuss the implications of our findings and provide takeaways for designers of mental well-being applications. User Perception; Personalisation; Recommender Systems; Personality Traits ††copyright: acmcopyright††journalyear: 2021††copyright: acmcopyright††conference: CHI Conference on Human Factors in Computing Systems; May 8–13, 2021; Yokohama, Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI ’21), May 8–13, 2021, Yokohama, Japan††price: 15.00††doi: 10.1145/3411764.3445523††isbn: 978-1-4503-8096-6/21/05††ccs: Human-centered computing Human computer interaction (HCI)††ccs: Human-centered computing Empirical studies in ubiquitous and mobile computing ## 1\. Introduction Digital mental well-being interventions present the promise to mitigate the global shortage of mental healthcare professionals in a cost-effective and scalable manner (Tal and Torous, 2017). Their emergence has been accelerated by the experiences of the COVID-19 pandemic (Organization et al., 2020) and its impact on mental health (Pfefferbaum and North, 2020). The growth of the digital mental health space has been paralleled by the rapid increase in research and development of new interventions and content. Benefits of rich content are indisputable, yet a vast amount of choices can also misfire—which is known as a paradox of choice (Schwartz, 2004). For this reason, we are witnessing the advent of recommender systems also in digital mental health platforms (Khwaja et al., 2019a). Although personalised recommendations represent an important aid with respect to the choice overload and moreover in improving the intervention effectiveness, delivering those recommendations entails two main challenges. Firstly, how to balance autonomy and personalised guidance has become an important topic in the design of personalised technologies. Secondly, data sharing concerns are undetachable from the automatic personalisation models. Both challenges have a very specific relevance when it comes to digital health applications (Burr et al., 2020). While users’ autonomy is one of the common principles in designing digital experiences (Peters et al., 2018), patients in traditional doctor-patient settings typically expect (and often prefer) to “be told what to do” rather than to “do what they want”. This raises an ethical tension between ensuring the safety of patients and respecting their right to autonomy (Burr et al., 2020). In addition, data privacy, like autonomy, is another central theme in personalised technologies—especially for digital services that rely on behavioural signals and sensitive mental health data to personalise interventions. There are a myriad of associated challenges including unintended data leakages, lack of users’ technical literacy, the need of finding an appropriate balance between using less privacy-invasive monitoring and providing more tailored interventions to improve health outcomes, and so on. We empirically investigate the multifaceted challenges of autonomy and data sharing in mental health applications from the point-of-view of users. The importance of understanding the user’s perspective stems from the fact that user disengagement represents one of the key challenges towards an improved effectiveness of digital mental health interventions (Makin et al., 2017; Chikersal et al., 2020; Karapanos, 2015; Eysenbach, 2005). Similar to pharmacological therapies, no matter how personalised and efficient a digital intervention is, a user’s adherence is a pre-requisite to receive the desired benefits. As the content in mental health applications is growing, we are likely at the dawn of expansion of personalised recommender systems (Khwaja et al., 2019a). Therefore, the question on how to design the user experience of delivering personalised recommendations deserves an important place in Human Computer Interaction (HCI) research. Our objective is to inform digital user experience designers on how to best promote users’ engagement when providing diverse digital mental health content. To this end, we explore users’ declared preference as well as their actual app usage with respect to: 1) a primarily autonomous versus a primarily guided user experience, 2) data to be shared in order to receive recommendations. Specifically, we address the following research questions: * • Do users prefer an autonomous or guided experience in a mental health app? * • Does receiving an autonomous versus guided experience impact the actual app use? * • To power a recommendation system, do users prefer to share smartphone data or to self-report their personality traits? * • Does sharing smartphone data as opposed to self-reporting personality traits influence the actual app use? We used a commercially available mental health application that includes more than 100 activities (i.e. interventions) and delivered it to $N=218$ participants. In a two-factor factorial design experiment, we randomly assigned half of the participants to a guided user experience and the other half to an autonomous selection of mental well-being app content. Independently, we assigned half of the participants to a self-reported way of capturing a user model and half to a consent form for sharing smartphone data–that could be used to infer the same user model. We used the Big Five personality traits (Donnellan et al., 2006; Goldberg et al., 2006) as a user model, as personality has been widely used to personalise digital health solutions (Halko and Kientz, 2010) and they can be inferred passively with smartphone sensing data (Chittaranjan et al., 2011; Wang et al., 2018; Khwaja et al., 2019b). The participants were primed that they will receive personalised recommendations that are based on the data that they agreed to share. However, in reality, the recommendations were random. We opted for a random placebo experimental design, based on priming, to reduce the dependency on the recommendation system accuracy that may not always be uniform for all users, thus representing a confounding factor. Having four randomised groups allowed us to delve into the relative differences in the actual app usage and users’ declared preferences as a function of the two factors—autonomy and data sharing. The choices put forth to mental health intervention designers are not trivial, especially in light of ethical tensions related to paternalistic design choices (Floridi, 2016) or the possible risks arising from increasingly sensitive data streams. Yet, both design choices are important to tackle in order to unlock the value of personalised technology (Floridi et al., 2018). This paper deepens understanding of users’ preference and their actual app usage as a consequence of the app design choices, and contributes to the related debates in the HCI community and beyond. ## 2\. Background and Related Work Blom and Monk defined personalisation as ”a process that increases personal relevance” (Blom, 2000). Personalisation has gained significant attention in digital services, since providing targeted user experience has been shown to increase acceptance (Jorstad et al., 2005). Particularly in health applications, personalisation was shown to increase not only engagement but also effectiveness and ultimately well-being. Noar et al (Noar et al., 2007) conducted a meta-analysis of 57 studies that used tailored messages to deliver health behaviour change interventions (for smoking cessation, diet, cancer screening, etc.), and concluded that personalised health interventions are more effective than generic ones. Zanker et al (Zanker et al., 2019) argued that personalisation can impact a range of outcomes including user engagement, app behaviours, and adoption rates. Recent studies have also found that personalisation of digital health apps can significantly improve health outcomes (Madeira et al., 2018; Chawla and Davis, 2013), however, the manner in which personalisation is delivered to the users and how they perceived it can be even more important than the extent to which a service is really personalised (Li, 2016). Our work builds on the previous literature by further exploring the topic of delivering personalised recommendations in digital mental health from the users’ perspective. We explored both users’ preference as well as how their engagement with the app are impacted by a) different ways of providing personalised recommendation—by giving users more or less autonomy in choosing the app content, and b) different ways of sharing the data required for delivering personalisation. Our study highlights the importance of autonomy and data privacy in the design of digital mental health services and provides key takeaways for user experience design. ### 2.1. Autonomy Autonomy has been an important focus in HCI, and specifically in persuasive technologies. Rughinis et al. (Rughiniş et al., 2015) decoupled five dimensions of autonomy in the context of health and well-being apps including: (1) degree of control that the user has; (2) degree of functional personalisation; (3) degree of truthfulness and reliability of the information in the app; (4) users’ understanding of the goal-pursuit and (5) promotion of moral values by what the app recommends. Embedding autonomy in the design of digital services impacts not only motivation and user experience but also psychological well-being. For this reason, Peters et al (Peters et al., 2018) included autonomy as one of the three key principles in “designing for well- being” (in addition to competence and relatedness), using Self Determination Theory (Ryan and Deci, 2000) as the basis for their approach. For instance, game designers have long explored the concept of autonomy and showed that the perceived autonomy in video games contributes to game enjoyment and also short-term well-being (Ryan et al., 2006). While autonomy leads to improved well-being and engagement (in addition to being ethically recommended (Pullman, 1999)), providing a range of choices may act as a demotivating factor (Schwartz, 2004). Besides, providing more guidance with tailored interventions can lead to improved effectiveness of the intervention. Hence, designers of personalised applications face conflicting requirements. In this study, we set to explore how the degree of autonomy impacts the users’ subjective preference, as well as their engagement with a mental health application. ### 2.2. Data Privacy Data privacy and related topics—including but not limited to transparency, control and data governance—have been extensively discussed over the past decade due to rapid technological expansion. These topics gained even more prominence after the introduction of the EU’s General Data Protection Regulation (GDPR) (Voigt and Von dem Bussche, 2017). The HCI community has promptly focused their efforts on understanding how these topics may impact interaction with digital services. Providing personalised recommendations typically relies on using sensitive information streams and past studies indicate that users’ attitude towards sharing potentially sensitive data was shown to be very conservative (Jamal et al., 2013). For mobile health apps specifically, Peng et al (Peng et al., 2016) conducted six focus groups and five individual interviews with 44 participants to compare what users value the most in these kinds of apps. While participants valued the benefits of personalisation, the authors found that they were strongly hesitant to share personal information for receiving these benefits. In another study, HCI researchers conducted a “Wizard of Oz” study to investigate whether the benefits of receiving highly personalised services—Ads in particular—offsets concerns related to sharing personal data (Matic et al., 2017). Interestingly, the study showed that participants’ concerns were less pronounced when an actual benefit of sharing the data was clearly visible. However, the users’ concerns on how the system inferred the user model (concretely users’ personality) remained strongly highlighted in semi-structured interviews. On a related topic, a recent study (Kim et al., 2020) explored how users perceived automatic personality detection using a mixed-methods approach. They conducted a survey (with 89 participants) to understand which data streams users were willing to share, and afterwards developed a machine learning model (with the preferred data from 32 participants) to predict personality traits. Subsequently, they interviewed 9 participants to understand how users perceived the personality prediction model after seeing the prediction results. They observed that participants’ opinions on data sharing were mixed and suggested that transparency can help in addressing users’ concerns such as trust and privacy. In our randomised placebo study, we primed participants that the selection of recommended activities in a mental health app was personalised to their personal data. The goal was to explore if the benefits of having a personalised experience will outpower their concerns about sharing the data. The success of placebo effect was evaluated and confirmed by including a control group in the experiment. We additionally contributed to the existing literature by comparing the actual app engagement and the user’s preference towards data sharing. Data privacy and autonomy were emphasised as key topics in the ethics of digital well-being (Floridi et al., 2018). To the best of our knowledge our work is the first that thoroughly explores how these two elements impact users’ actual app usage and self-declared preferences in a digital mental health app. ## 3\. Methods To understand users’ preferences and the usage of a mobile mental health app in the context of delivering recommendations, we used Foundations111https://foundations.koahealth.com/. Foundations was a suitable platform for our study as it contains a large library with numerous intervention activities. In this section, we detail the methodology applied in this experiment. ### 3.1. Mental Health App Figure 1. (a) Open library with all activities. Some activities are locked and dependent on the completion of others. (b) Recommended activities at the bottom of the home screen. All activities shown to users are randomly generated. [Two screenshots of the smartphone app-Foundations](a) Screenshot of the smartphone app-Foundations displaying colourful circles with activity titles corresponding to three different categories, namely “Covid-19: Staying resilient in times of crisis”, “Relaxation” and “Working with thoughts”. The top bar consists of free buttons (Modules, Programmes, Activities). The activities button is activated. The bottom menu shows three icons for home, library and journey. The library icon is activated. (b) Screenshot of the smartphone app-Foundations with a single column design and two sections. The upper section consists of the description of the “Working with thoughts” programme and a link to the programme overview. The lower section shows the heading “Other activities for you” followed by two circles with activity titles on them. The bottom menu shows three icons for home, library and journey. The home icon is activated. Foundations is an evidence-based digital mental health platform designed to improve users’ resilience and decrease their stress levels. At the time of this study, the version of the app incorporated 10 modules with 102 intervention activities in total. Each activity has a specific format—such as simple blog posts, relaxation audios, interactive journaling and games—to help users relax, sleep better, boost their self confidence, think positively, and similar. The app provides an open library with some activities locked in the beginning (Figure 1 (a)). Upon completion of each activity, users are asked to rate their experience using a thumbs up or thumbs down icon. The home screen contains a section called ”Other activities for you” that shows a recommendation of two activities at a time (Figure 1 (b)). In our study, these recommendations were random i.e. not personalised (although presented so), which guaranteed that all the users have received the same experience. Automatic recommendations may work better for specific groups of users which would have biased the results of our study. ### 3.2. Study Design To determine how the way of data sharing and the autonomy of the user experience impact both users’ preferences and the actual usage of a mental well-being app, we designed a study consisting of three parts: (1) Onboarding questionnaire, followed by (2) the app usage for seven days with daily reminders, and finally (3) an exit questionnaire (Figure 2). As the goal was to investigate the effect of the two variables, ”autonomy” and ”preferred way of data sharing”, we designed a two-factor factorial experiment. A two-factor factorial design is an experimental design in which data is collected for all possible combinations of the levels of the two factors of interest (Mukerjee and Wu, 2007). In our case, each factor has two levels. For the preferred way of data sharing, the two levels are (1) selecting mobile sensing data and (2) completing a questionnaire, for building a personalisation model. For the former level, half of the users (randomly selected) were asked to select smartphone data streams that can be used to automatically infer their personality. The other half of the users received the 20-item personality questionnaire (Donnellan et al., 2006) to complete. We defined two different user experiences that we refer to as ”the degree of autonomy”, namely (1) receiving a primarily guided user experience with the option to choose other activities out of an open library, and (2) receiving a primarily autonomous user experience with the option to use recommended activities on the home screen. Overall, this led to 4 experimental groups that can be combined according to the variables they have in common. The combination of two groups along one identical variable is referred to as a cluster. For example, the two groups that receive an autonomous user experience—but differ in the way of data sharing—are combined and referred to as the autonomous cluster. This design allows for one group per each permutation of the two variables, which enables an analysis of all conditions separately, as well as combined. For an effect size (Cohen’s d) of 1, statistical power of 95% and significance level of 0.05, the estimated sample size to produce a meaningful statistical significance with the Mann-Whitney test is 30. Thus, we set the criteria to have at least 30 samples in each group. Figure 2. Experimental Design [Flowchart of the experimental design]Flowchart visualising the experimental design with 10 elements and flow links. The start state is “Introduction”. The end state is “Exit questionnaire”. Details provided in section 3.2. The groups differed in the onboarding questionnaire and in daily reminders during the app usage. The primary purpose of the onboarding questionnaire was to give the user the impression that the collected data will be the base for receiving personalised recommended activities in the app. However, this questionnaire was solely used for priming, and no actual personalisation was occurring in the app. All the activities participants found in the recommendation section of the app were randomly selected, as described in section 3.1. The onboarding questionnaire consisted of the data sharing (smartphone modalities or questionnaire) and directions on app usage (autonomous or guided). Upon completion of the questionnaire, all participants received instructions on how to install Foundations and were asked to complete at least one activity a day for one week. Daily reminders were sent according to the degree of autonomy. These reminders consisted of either a daily recommended activity for participants in the guided cluster, or a general reminder to use the app for those in the autonomous cluster. The daily recommended activities were selected from the most popular activities in the app’s library. After seven days, all users completed the exit questionnaire—which was identical for all groups. Since we did not use the users’ data to personalise recommendations in the app, we included an additional control group to verify whether or not the priming was successful. The control group filled out a control questionnaire to match the workload to the other groups but this group did not receive any priming on personalisation. In summary, this design resulted in having five groups: * • Questionnaire-Guided (QG): Personality questionnaire + daily email with activity recommendation + priming that the email recommendations are based on the reported personality * • Data-Guided (DG): Data modality selection + daily email with activity recommendation + priming that the email recommendations are based on the automatically inferred personality * • Questionnaire-Autonomous (QA): Personality questionnaire + daily email as a general reminder to complete one activity + priming that recommendations on the home screen are based on the reported personality * • Data-Autonomous (DA): Data modality selection + daily email as a general reminder to complete one activity + priming that recommendations on the home screen are based on the automatically inferred personality * • Control (C): Control questionnaire + daily email as a general reminder to complete one activity Our study was approved by the internal ethics board. As the whole set of intervention activities in Foundations has been recently evaluated in a Randomised Control trial (Catuara-Solarz et al., 2021) and demonstrated an overall improvement in users’ overall well-being, no harm was expected to be introduced by a deception study that recommends users with the most popular activities. ### 3.3. Data Collection The onboarding and exit questionnaires were created using the Typeform 222https://www.typeform.com/ survey collection tool. We designed five variations of the onboarding questionnaire for each of the five groups defined in Section 3.2. In each of these questionnaires, participants were presented with a consent form explaining details on the data collection and purpose of the study—in compliance with the EU General Data Protection Regulation (GDPR). For users in questionnaire cluster, a 7-point Likert scale (1 strongly disagree to 7 strongly agree) was used for the personality questionnaire. Users in the data cluster were provided with 10 different smartphone sensing data categories and asked to select at least 4 that could be sampled from their smartphones. The rationale for introducing the data choice was to resemble the choice that users have in real-world applications. Android and iOS give users the possibility to opt-out from specific data streams. Moreover, in Europe—where we conducted the experiments—this is a strict regulatory requirement as per the GDPR. We selected the 10 most common sensing modalities that have been used in the previous literature to predict personality traits (Mønsted et al., 2018; de Montjoye et al., 2013; Chittaranjan et al., 2011, 2013; Wang et al., 2018; Khwaja et al., 2019b). The 10 options included: * • Time spent with different applications (App time) * • Geographical location (Location) * • Number of steps walked (Steps) * • Noise in the environment (Noise) * • Bluetooth and WiFi data (Bluetooth/Wifi) * • Battery level (Battery) * • Ambient Light in the environment (Light) * • Call history (without phone numbers) (Calls) * • Frequency of social network usage (Social network) * • Phone lock/unlock data and screen usage (Un(Lock)) We asked users to select at least 4 options out of 10 and explained that selecting more options leads to a higher accuracy in inferring personality. After the onboarding, users were asked to use Foundations for a week. App usage logs consisting of activities completed, time taken per activity etc. were recorded for each user during the study. Upon using the app for an entire week, the users were presented with an exit questionnaire. This questionnaire had four sections asking users (Ex1) about their overall experience of the mental health app and their perspectives on personalisation of the app (Ex2) if they prefer to have autonomy in selecting activities or have the app select the right activity for them, (Ex3) if they prefer to complete a personality questionnaire or provide smartphone sensing data and their privacy preferences regarding the same. Based on the Technology Acceptance Model (Lee et al., 2003), the first set of questions (Ex1) was defined to understand how users perceived the app in general. The second (Ex2) and third (Ex3) set of questions were related to the users’ preference to be guided vs to have autonomy, as well as sharing the data through a questionnaire or by providing their mobile sensing data. Ex3 also included questions related to privacy concerns (a recent study that explored personality profiling by a chatbot indicated that participants generally regarded personality as sensitive data that they would be reluctant to share (Völkel et al., 2020)). Ex1 \- Ex3 were delivered as a 7-point Likert scale or a multiple choice (select ’X’ or ’Y’). Additionally, we had two free text questions where the users could give suggestions on the how the app could be improved and more personalised to them (Ex4). Subsequently, we presented the participants with demographic questions - gender, age 333We asked range of age rather than exact number, education level and the continent of residence. The exit questionnaire concluded with a text block that debriefed the participants. ### 3.4. Participants and Inclusion Criteria Table 1. Demographics of participants Demographic \| Particular \| Complete \| QG \| DG \| QA \| DA \| C ---\|---\|---\|---\|---\|---\|---\|--- Size of Population \| 218 \| 45 \| 52 \| 40 \| 40 \| 41 Gender \| Female \| 113 \| 24 \| 25 \| 21 \| 17 \| 26 \| Male \| 105 \| 21 \| 27 \| 19 \| 23 \| 15 Age \| 15-19# \| 6 \| - \| 1 \| 1 \| 2 \| 2 \| 20-24 \| 26 \| 3 \| 3 \| 8 \| 5 \| 7 \| 25-29 \| 23 \| 6 \| 3 \| 4 \| 7 \| 3 \| 30-34 \| 26 \| 3 \| 9 \| 4 \| 6 \| 4 \| 35-39 \| 25 \| 7 \| 7 \| 3 \| 4 \| 4 \| 40-44 \| 31 \| 6 \| 9 \| 5 \| 3 \| 8 \| 45-49 \| 17 \| 4 \| 6 \| 3 \| 2 \| 2 \| 50-54 \| 29 \| 7 \| 9 \| 5 \| 4 \| 4 \| 55-59 \| 11 \| 4 \| 2 \| - \| 3 \| 2 \| 60+ \| 24 \| 5 \| 3 \| 7 \| 4 \| 5 Education \| Secondary School \| 69 \| 14 \| 19 \| 13 \| 14 \| 9 \| Bachelor’s Degree \| 92 \| 17 \| 22 \| 17 \| 16 \| 20 \| Master’s Degree \| 30 \| 9 \| 5 \| 4 \| 6 \| 6 \| Ph.D. or higher \| 8 \| 1 \| 2 \| 2 \| - \| 3 \| Trade School \| 15 \| 1 \| 4 \| 4 \| 4 \| 2 \| Prefer not to say \| 4 \| 3 \| - \| - \| - \| 1 QG = Questionnaire-Guided, DG = Data-Guided, G3 = Questionnaire-Autonomous, G4 = Data-Autonomous, C = Control #The minimum age of participants is 18. We provided this age option to maintain uniformity with the other age ranges The participants in our study were recruited through an external agency that operates in Europe. The inclusion criteria included a high proficiency in English and the minimal age of 18. We also required a minimum of 30 participants in each group and gender balance. In early July 2020, the recruitment agency sent an invite for the study through their internal mailing list and all the participants completed the study by the end of July 2020. All participants were recruited from Europe. Through the recruitment agency, we provided a monetary incentive to all participants who completed the study. Users were instructed that successful completion and receiving the incentive requires completing the onboarding questionnaire, installation and use of the mental health app for 1 week, and completing the exit questionnaire. Users were reminded each day that skipping any of the steps would result in their disqualification from the study. 700 participants were registered for the study and were randomly assigned to one of the five groups. Based on the group allocation, they were asked to complete the corresponding onboarding questionnaire. All 700 users completed the onboarding questionnaire and were then instructed to install the app on their smartphones. Out of the 700 users, 353 participants installed the mental health app. For one week after installing the app, users received daily reminders to use the app and to engage with at least one activity per day. Using the app for 4 or more days qualified the users for the last stage of the study. We chose a threshold of 4 days, as anytime less than this would be insufficient to explore the app well. 241 participants fulfilled this criteria and were directed to the exit questionnaire. Finally, 218 users completed the exit questionnaire and this population was used for our analyses. The demographics of the participants are provided in Table 1. Having more than 40 participants in each group exceeded the minimum number of completes required in each group. The demographic distribution indicates that the sample involved a diverse population. ### 3.5. Statistical Methods To report statistics, we use the guidelines laid out in (Habibzadeh, 2017). For normally distributed data, we report mean ($M$) and standard deviation ($SD$) and for data that deviated from the normal distribution, we report the median value ($Mdn$) and interquartile range ($IQR$). Interquartile range is defined as difference between the upper quartile (75 percentile) and lower quartlie (25 percentile). In order to compare the differences in two distributions, we use the Mann-Whitney U test (also known as the Wilcoxon rank sum test) (McKnight and Najab, 2010). The Mann-Whitney U test is non- parametrised and works well for comparing distributions that are non-normal, as opposed to the parametric Student’s t‐test. Additionally, when comparing three or more distributions, we use the Kruskal–Wallis test (the non- parametric equivalent of the one-way ANOVA) (McKight and Najab, 2010). Although the experimental design would have allowed us to conduct ANOVAs (or Kruskall-Wallis tests) to look at differences between all 5 conditions, we decided not to use this statistical method because our research questions focused on degree of autonomy and data sharing separately rather than combined. The literature provided no base to hypothesise that any of those combinations could lead to significantly different preferences or behaviours and we did not want to make many pairwise comparisons only for the sake of obtaining more comparisons. Data processing was performed with the Python programming language. All statistical tests (except the power analysis) were conducted using the SciPy library (Jones et al., 2001) while data visualisation plots were generated using the Matplotlib library (Hunter, 2007). The power analysis was conducted in Microsoft Excel, using the Mann-Whitney power function $MW\\_POWER$ from the Real Statistics library (Zaiontz, 2020). ## 4\. Results ### 4.1. Experimental validity We first tested whether the inclusion criteria and randomisation were executed according to our design. Major demographic characteristics as well as the total number of participants, were correctly balanced across the groups (Table 1). To probe the additional motivation to use the app beyond the monetary incentive, we asked participates to rate the extent to which they wanted to reduce the amount of stress levels on a Likert scale 1 to 7. The median score of 6 (IQR = 2) suggested a generally high interest in reducing stress levels. A Kruskal-Wallis test showed no significant difference among the five groups (H(4) = 2.34, p ¿ .05), which indicates that the randomisation across the groups was correctly applied and that the stress level was not expected to act as a confounding factor when comparing results across the groups. Table 2. Summary of Results \| \| Autonomous vs. Guided \| Questionnaire vs. Data ---\|---\|---\|--- In-App Behaviours \| Completed Activities \| Significantly more completed activities in the autonomy cluster \| No significant difference in number of completed activities \| Ratio of Recommended versus Chosen Activities \| Autonomous cluster: 25% Guided cluster: 60% \| - \| Session Duration \| No significant difference in session duration \| No significant difference in session duration \| Activity Ratings \| Significantly higher ratings of activities in the autonomy cluster \| No significant difference in activity ratings Declarative Data \| Preference \| All users preferred to have a more guided user experience \| All users preferred to complete a personality questionnaire. \| Privacy Preference \| - \| All users agreed that providing mobile data had more privacy risks Onboarding Behaviours \| Completion time \| - \| No significant difference in completion time Participants were informed that they were going to receive recommeneded activities personalised for them. However, in reality, the recommended selection of activities (both those sent daily and those included within the app) were random. Therefore, the success of our priming strategy was a prerequisite for exploring the perception and effects of personalised recommendations. Unlike other domains—such as shopping items, music, movies, etc.—where people are typically well aware of what constitutes a personalised recommendation, there is a low level of understanding of meaningful symptoms and personal characteristics when it comes to the personalisation of interventions. To this end, we compared the response to the statement “I believe that activities were personalised for me” (provided at the end of the study in the Exit questionnaire) which was rated on a scale 1-7. We compared the ratings between the personalisation cluster (QG, DG, QA and DA) and the control group; and the former rated the perceived personalisation significantly higher (U = 2725.5, p ¡ .05). Despite the fact that the activity recommendations were not based on the Big Five personality traits, the participants believed so–indicating that the priming was successful. The results from our experiment are summarised in Table 2 and explained in detail in the following sections. ### 4.2. Guided vs autonomous user experience We compare the app usage behaviours and self-reported preferences between the guided (QG+DG) and the autonomous clusters (QA+DA). #### 4.2.1. App usage behaviours The number of completed activities considers only those activities that the user both started and finished. Figure 3 (a) shows that the number of activities completed by users in the autonomous cluster (Mdn = 19, IQR = 22.5) was significantly higher than those in the guided cluster (Mdn = 7, IQR = 3), U = 1427, p ¡ .001. We also observed that the ratio of recommended activities from the home screen vs. voluntary chosen activities from the library amounts to 25% for the autonomous cluster. While the ratio of recommended activities from the email reminders vs. activities from the library made up 60% in the guided cluster. Figure 3. Differences between the guided and autonomous clusters for (a) average number of completed activities, (b) median session duration per user and (c) proportion of good (1) ratings [Boxplot-figures comparing app usage behaviours]The three boxplot-figures compare the average number of completed activities, median session duration per user and the ratio of good vs bad ratings for the autonomy and guided cluster. Mean values and interquartile range are provided in section 4.2.1. The figure shows the significant difference between the two clusters for the number of completed activities and the ratio of good vs bad ratings. The median session duration per user was not significantly different between the autonomy and guided cluster. The autonomy cluster has higher median values in all three comparisons. Subsequently, we investigated how the degree of autonomy impacted the session duration–defined as the median number of seconds for which a user was actively using the app before closing it. We observed that there was no statistical difference between autonomous (Mdn = 184 seconds, IQR = 363.2 seconds) and guided (Mdn = 158 seconds, IQR = 280.4 seconds) clusters, U = 3346, p ¿ .05 (Figure 3 (b)) The design of the Foundations provides a simple format for rating each activity, namely the users are asked to rate each activity upon its completion with either a thumbs up or thumbs down. We binary coded these ratings as 1 and 0 respectively and calculated the proportion of good (1) ratings per user–number of good ratings/(number of good+bad ratings)–which resulted in a value between 0 and 1. Figure 3 (c) shows that the proportion of good ratings of users in the autonomous cluster (Mdn = 1, IQR = 0.1) was significantly higher than in the guided cluster (Mdn = 0.85, IQR = 0.2), U = 3047, p ¡ .01. #### 4.2.2. Self-reported preference on autonomy After using the app for a week, we asked users to rate if: A1. They would like the mental health app to choose an activity/intervention for them (guided) and A2. They would like to choose an activity/intervention for themselves (autonomous). In general, users agreed more strongly that the app should provide an activity to them (Mdn = 5, IQR = 2), as opposed to them having autonomy to select their own activities (Mdn = 4, IQR = 2). The Mann-Whitney U test confirms that there is a statistical significance in their preference between the two ($U=17051.0,p<.001$). When asked to directly compare the two options, 77.9% of the users preferred to have an activity provided to them by the mental health app. Subsequently, we compared the preference for the guided and autonomous clusters separately. The percentage of users that preferred to have an activity suggested directly by the app was similar across the guided (78.4%) and autonomous clusters (77.8%). Next, we assessed the difference in average ratings between A1 and A2 within each cluster. For both the guided and autonomous clusters, users rated A1 higher than A2 with statistical significance ($U=2931.5,p<.001$ and $U=2623.5,p<.01$ respectively). This shows that, irrespective of receiving a guided or autonomous experience, all users preferred to have an app that suggests interventions for them instead of selecting activities solely on their own. ### 4.3. Questionnaire vs data selection We compare the app usage behaviours and self-reported preferences between the questionnaire (QG+QA) and data selection clusters (DG+DA) #### 4.3.1. App usage behaviours Similar to the comparison described in Section 4.2.1, we compared the number of completed activities, median session duration per user and proportion of good ratings between the questionnaire and data selection clusters. Using Mann Whitney U tests, we found no significant difference for any of these metrics (Supplementary Figure 1). #### 4.3.2. Onboarding behaviours We aimed to explore whether the way of data sharing (completing the personality questionnaire vs selecting the data modalities) is related to the time taken to complete the onboarding questionnaire. To do this, we compared the completion time for the questionnaire cluster against the data selection cluster. While the median time taken to complete the onboarding questionnaire was greater for the questionnaire cluster (Mdn = 142 seconds, IQR = 82 seconds) than the data selection cluster (Mdn = 102 seconds, IQR = 68 seconds), the Mann Whitney U test indicated that there was no significant difference between the two distributions ($U=1799.5,p>.05$). The number of screens and the priming text in the onboarding questionnaires were comparable for the two clusters. The major difference in the two was the personality questionnaire versus the smartphone sensing data selections. Hence, it can be concluded that there is no significant difference between the time taken to complete the 20-item personality questionnaire and the time needed to select a subset of a list of smartphone sensing data modalities, in an onboarding process. Figure 4. Proportions of users from the data sharing cluster that preferred to provide different data modalities. Column names correspond to the data modalities described in Section 3.3 [Bar chart of the percentage of users that chose to provide each data modality]The figure shows a bar chart visualising the percentage of users that chose to provide each data modality. The bars correspond to the data modalities described in Section 3.3. and are displayed in ascending order. The most and least selected data modalities are described in 4.3.2. In addition, we also explored the data categories that the users in the data selection cluster were most willing to provide. Figure 4 shows the proportion of users that provided a particular data modality. The error bars in the figure represent the standard deviation of the proportions obtained individually from DG and DA. The users were least willing to provide 1. call history (25.0%), 2. bluetooth and wifi data (26.1%) and 3. noise in the environment sampled from the microphone (34.0%). As expected, these are data modalities that have the largest privacy and security concerns across both users and technologists (Elkhodr et al., 2012; Mayer et al., 2016; Sipior et al., 2014). Additionally, the data modalities that users are most willing to provide are 1. battery level (72.8%), 2. number of steps walked (71.7%) and 3. time spent on different applications (68.5%). #### 4.3.3. Self-reported preference on data sharing Users were asked to rate from 1 to 7: D1. If they were willing to complete a 5-10 min personality questionnaire (with up to 50 questions) to receive personalised recommendations for activities and D2. If they were willing to provide personal sensing data (e.g., GPS location) from their smartphone to receive personalised recommendations for activities. A Mann-Whitney U test confirmed with statistical significance ($U=11568,p<.001$) that users were more willing to complete a personality questionnaire (Mdn = 6, IQR = 2), than provide their smartphone sensing data for personalisation (Mdn = 4, IQR = 3). The users were also asked to select if D3 They would rather prefer to complete a personality questionnaire or provide their smartphone data. 90.4% of the 218 users said they would prefer to complete a personality questionnaire to have a personalised app experience. Next, we compared the preferences for the questionnaire and data selection clusters. For D3, The percentage of users that preferred to complete the personality questionnaire instead of providing data is notably high across both the clusters (questionnaire: 92.9% and data selection: 85.9%). We also assessed the difference in ratings between D1 and D2 within each cluster. Using Mann-Whitney U tests, we observed that users in both clusters rated D1 higher than D2, with statistical significance ($U=1915,p<.001$ for the questionnaire cluster and $U=2112.5,p<.001$ for the data selection cluster). This indicates that all users—irrespective of the way of data sharing—preferred to complete the personality questionnaire over providing their smartphone data. #### 4.3.4. Self-reported preference on privacy risks An additional objective was to investigate if there was a difference in how users viewed privacy risks between completing a personality questionnaire and providing their smartphone data. We asked users to rate: Pr1. If they believed that filling out personality questionnaires for personalisation has potential privacy and data protection risks and Pr2. If they believed that providing a mental health app with their smartphone’s sensing data for personalisation has potential privacy and data protection risks. All users believed that completing a personality questionnaire had less privacy risks (Mdn = 4, SD = 2) compared to providing sensing data from their smartphones (Mdn = 5, SD = 3). The difference between the two questions was statistically significant, $U=21118.5,p<.05$. Within the two clusters, we also found a similar trend. Both clusters rated Pr2 higher than Pr1 with statistical significance ($U=1206.5,p<.01$ for the questionnaire cluster and $U=2106.5,p<.05$ for the data selection cluster). ## 5\. Discussion In this study, we explored how (1) the degree of autonomy in the user experience, and (2) the data to be shared impact users’ preferences and app behaviours in a mental health app. In the following, we discuss the results and highlight the main takeaways. ### 5.1. Asymmetry between in-app behaviours and preference for the degree of autonomy The balance between autonomy and guidance is a critical topic in personalised recommender systems, and when it comes to the area of digital mental health it has a peculiar importance. In a traditional setting, for the selection of the right intervention, autonomy is secondary to the expertise of the medical professional. However, in digital experiences, autonomy was shown to be an essential design criterion to create engagement (Peters et al., 2018). Our results highlight the challenge of finding the right balance between the two and shed light on the contrast between users’ preferences and their actual behaviour in the app. This together provides a set of practical takeaways for user experience designers that we discuss in the following. Our findings demonstrated that the difference in the degree of autonomy could influence subsequent behaviours in a mental health mobile application. We showed significant between-group differences in user behaviours, although all participants used the same application. Since there was no actual personalisation in the app, our results are independent of the accuracy of a recommendation system and solely ascribed to the perceived degree of autonomy in the user experience. Our results challenge the popular notion that the more personalised or guided, the better an app is perceived by users. We witnessed that a primarily autonomous experience led to the greatest engagement i.e. the highest number of completed activities and best ratings. Contrary to expectations, the most guided and tailored experience appeared to discourage users’ exploration and spontaneous app use. However, when asked about the subjective preference after the study had been completed, a significantly higher number of users expressed their preference for more guidance instead of autonomy. This finding shows a discrepancy between behavioural and declarative data. Our results confirm that the preferences communicated by the user do not necessarily result in quantitatively improved engagement metrics. This emphasises the importance of cautiously interpreting user research results and combining them with quantitative data, when possible, throughout the process of designing personalised user experiences. Interestingly, several answers to the free-text question Do you have any suggestions on how Foundations could be more personalised for you? referred to reminders, for instance: “Have daily reminders to help with routine”, “Maybe a reminder to be set daily” and “I like receiving the daily reminders. I have an 18 month old, so maybe you could set the reminder to come back on later, like a snooze button?“. This may inspire a potential solution for an experience design that is in-between autonomy and guidance e.g. a combination of an autonomous navigation and more frequent notifications suggesting personalised content. This can result in providing more guidance without negatively impacting the users’ perceived or actual agency. In reality, none of the two clusters of users were exposed to an extreme choice between autonomy or guidance. The imposed content consumption, primarily in an autonomous versus primarily in a guided way, was clearly reflected in the actual app use—the guided cluster completed a significantly higher number of recommended activities than the autonomous cluster. However, the total number of completed activities was three times higher in the autonomous cluster. As efficacy and engagement are key pillars of digital intervention design (Murray et al., 2016), our results can be utilised by designers to optimise for these metrics. In line with our findings, the interaction in mental health apps could be designed in a similar way to popular entertainment applications such as Spotify or Netflix. Specifically, the interaction design may directly encourage autonomous navigation while providing an easy access to recommended and personalised content, thus mitigating choice overload. Moreover, different trade-offs can be made between engagement and efficacy. If the success of a specific digital therapy does not critically depend on a volume of the app use but on a targeted engagement with certain interventions, the user experience can be more guided. On the other hand, autonomous interaction designs would be more suitable to encourage a higher frequency of the app use when critical for the therapy success (e.g. meditation techniques are supposed to be practiced more regularly for optimal results). Our results are aligned with the autonomy advocates (Ryan & Deci (Deci and Ryan, 2012), Peters (Peters et al., 2018)), however our findings additionally underline an important space for utilising the advantages of increasingly sophisticated recommender systems that ultimately can optimise for both efficacy and engagement. ### 5.2. Users prefer questionnaires but app engagement is unaffected Personality traits have been used as a foundation for personalising digital health applications (Halko and Kientz, 2010) and for providing personalised activity recommendations that can improve mental well-being (Khwaja et al., 2019a). Personality traits can be obtained using questionnaires (Donnellan et al., 2006; Goldberg et al., 2006) or inferred using machine learning models. The latter has given rise to the field of automatic personality detection. Studies in this field have shown that personality can be detected from Facebook, Twitter or Instagram usage (Ferwerda and Tkalcic, 2018a, b; Skowron et al., 2016; Hall and Caton, 2017), gaming behaviour (Yee et al., 2011), music preferences (Nave et al., 2018) and smartphone sensing data (Chittaranjan et al., 2011, 2013; de Montjoye et al., 2013; Wang et al., 2018; Khwaja et al., 2019b). All of these studies are based on the premise that digital behaviour data—captured passively—can be used to infer a user’s personality traits automatically with machine learning, without requiring them to answer long questionnaires. However, none of these studies explored users’ preferences in obtaining such data to infer a user’s personality passively, especially to personalise features in a real-world application. Our work set out to answer this important question, in the context of obtaining smartphone sensing data to personalise user experience in a mental health app. Our results indicate that an overwhelming majority of the users prefer to complete a personality questionnaire over providing their mobile sensing data, irrespective of whether they completed the personality questionnaire before using the app or were asked to provide their smartphone data. These results are consistent with related studies showing users’ improved comprehension of algorithms by using ”white-box” explanations (Cheng et al., 2019). Users have predominantly perceived that their smartphone sensing data entails more privacy risks than completing a personality questionnaire. This can be attributed to trust and privacy concerns with the collection of any kind of digital data (Gilbert et al., 2010; Dwyer et al., 2007). Despite the fact that smartphone sensing was perceived as obtrusive, there was no difference in app behaviour between users who completed a personality questionnaire and those who opted to provide mobile sensing data. Additionally, results from the onboarding process indicate that there is no significant difference between the time taken to complete the data consent process and the time taken to complete the 20 item personality questionnaire (Donnellan et al., 2006). Expectedly, users were less willing to provide more invasive data such as call history, Bluetooth data and noise from the microphone. This can have a significant impact on the accuracy of personality prediction models. Recent studies have indicated that call history data (de Montjoye et al., 2013), Bluetooth data (Staiano et al., 2012) and noise data from microphone (Khwaja et al., 2019b; Wang et al., 2018) are strong predictors of personality traits. Should collecting mobile sensing data not be leveraged to provide other benefits to users than personality modeling for personalising the user experience, the app designers may consider avoiding the collection of smartphone data altogether. Users appear to have a strong preference towards completing a questionnaire instead and although automatic personality modelling is supposed to reduce the end user effort, it does not bring an added value in this context. This was further echoed by the users’ answers to the free-text question Do you have any suggestions on how Foundations could be more personalised for you? including “An in depth questionnaire“, “Maybe a regular opt-in questionnaire so you let the app know whether your conditions or state of mind is changing“ and “I think it could be more personalised by asking more about the persons life, work, family and friends.“. This suggests that users may be willing to provide even more personal information than personality as long as they consciously and directly provide it and the app becomes more tailored to their needs as a result. As additionally suggested by the users, momentary information represents an opportunity for personalising the experience even further. In this regard, the Ecological Momentary Assessment (EMA) (Shiffman et al., 2008) has been a widely used method that prompts users (via smartphone notifications) at different times during the day to report how they feel, what they are doing, where they are, and similar. Recent studies have shown that behaviour and mood data collected via mobile EMAs is related to mental health and health outcomes such as sleep (Wang et al., 2014). Thus, data gathered from EMA surveys can point out the opportune moments to provide personalised interventions. Ultimately, the decision on gathering user models through passive sources or questionnaires requires practitioners to make a trade-off between the required amount of information, model accuracy, users’ privacy concerns and a potential survey fatigue (Porter et al., 2004). ### 5.3. Limitations Our study required us to make several trade-offs in the experimental design, which we discuss in the following. Firstly, having identical app versions for all groups was an asset for our experimental design, although it also represented a limitation at the same time. On the one hand, it enabled us to control the perceptional aspect. On the other, having more advanced versions would have allowed us to explore the interaction between perceived accuracy and perception of personalisation, which could make the results more generalisable. Secondly, we did not personalise the app according to each user’s actual personality which may prompt a question whether the deception of personalisation will impact the users’ trust in the app and result in a lower app usage. However, an alternative solution of providing actual personalisation would have entailed a new set of challenges. In particular, the quality of recommendations is rarely uniform and frequently biased towards specific user profiles. This issue would have been difficult or even impossible to control for. Instead, by providing random recommendations based on the most popular activities, we reduced the impact of this issue. We recognise that there is no ideal experimental design in this regard and that it entails trade-offs. However, 25% of the completed activities in the autonomous group were recommended, which indicates that the choice of the most popular activities was appropriate. Furthermore, the recommendations were perceived as personalised, as tested between the personalisation cluster with the control group (Section 4.1). Thirdly, we did not collect smartphone data from participants in the data group. As detailed in Section 3.3, we asked users to provide us access to their preferred data streams as a base for personalisation. However, in order not to increase the complexity of the study, we opted to use such data consent forms only as priming. Collecting smartphone sensing data would have given us an opportunity to do a more detailed behavioural analysis and further our findings. Lastly, all of our participants were recruited in Europe, which may have introduced a cultural bias and reduced the generalisability of our findings. ## 6\. Conclusion In this study, we investigated how the degree of autonomy in the user experience and different ways of data sharing affect both users’ preference and the actual usage of a mental well-being app. We conducted a randomised placebo study with a two-factor factorial design consisting of an onboarding questionnaire, app usage over seven days, and an exit questionnaire. Our results revealed an asymmetry between what users declared as their preference for autonomy (versus guidance) and how they used the app in reality. The analysis of in-app behaviours showed that a primarily autonomous design with the option to access content recommendations kept users more engaged with the app than a primarily guided experience design. However, when asked in the form of questionnaires, the majority of participants declared their preference for a more guided experience. The analysis of qualitative data suggested a potential compromise between different experience designs to satisfy both engagement metrics and subjective user preferences. Personalising the user experience typically requires personal data to be shared, which may impact the manner in which the app will be used. However, when analysing the actual app use, we found no impact of the data source on how users interacted with the app. Interestingly, the time taken for completing a personality questionnaire was comparable to the duration of completing a form to obtain consent for the usage of smartphone data. Yet, users indicated a strong preference for completing a personality questionnaire over providing their mobile sensing data (to infer personality). As mental health applications are becoming increasingly important and rich in content, our study provides key design takeaways on delivering personalised recommendations, to ultimately improve both engagement and efficacy of interventions. ## Acknowledgements We would like to thank Emily Stott and Jordan Drewitt for their feedback and support. This work has been supported from funding awarded by the European Union’s Horizon 2020 research and innovation programme, under the Marie Sklodowska-Curie grant agreement no. 722561. ## References (1) * Blom (2000) Jan Blom. 2000. Personalization: a taxonomy. In _CHI’00 extended abstracts on Human factors in computing systems_. 313–314. * Burr et al. (2020) Christopher Burr, Mariarosaria Taddeo, and Luciano Floridi. 2020. The ethics of digital well-being: A thematic review. _Science and engineering ethics_ (2020), 1–31. * Catuara-Solarz et al. 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# Motif Identification using CNN-based Pairwise Subsequence Alignment Score Prediction ††thanks: Identify applicable funding agency here. If none, delete this. Ethan Moyer School of Biomedical Engineering, Science and Health Systems Drexel University Philadelphia, PA https://orcid.org/0000-0002-8023-3810 Anup Das College of Engineering Drexel University Philadelphia, PA https://orcid.org/0000-0002-5673-2636 ###### Abstract A common problem in bioinformatics is related to identifying gene regulatory regions marked by relatively high frequencies of motifs, or deoxyribonucleic acid sequences that often code for transcription and enhancer proteins. Predicting alignment scores between subsequence k-mers and a given motif enables the identification of candidate regulatory regions in a gene, which correspond to the transcription of these proteins. We propose a one- dimensional (1-D) Convolution Neural Network trained on k-mer formatted sequences interspaced with the given motif pattern to predict pairwise alignment scores between the consensus motif and subsequence k-mers. Our model consists of fifteen layers with three rounds of a one-dimensional convolution layer, a batch normalization layer, a dense layer, and a 1-D maximum pooling layer. We train the model using mean squared error loss on four different data sets each with a different motif pattern randomly inserted in DNA sequences: the first three data sets have zero, one, and two mutations applied on each inserted motif, and the fourth data set represents the inserted motif as a position-specific probability matrix. We use a novel proposed metric in order to evaluate the model’s performance, $S_{\alpha}$, which is based on the Jaccard Index. We use 10-fold cross validation to evaluate out model. Using $S_{\alpha}$, we measure the accuracy of the model by identifying the 15 highest-scoring 15-mer indices of the predicted scores that agree with that of the actual scores within a selected $\alpha$ region. For the best performing data set, our results indicate on average 99.3% of the top 15 motifs were identified correctly within a one base pair stride ($\alpha=1$) in the out of sample data. To the best of our knowledge, this is a novel approach that illustrates how data formatted in an intelligent way can be extrapolated using machine learning. ###### Index Terms: Motif Finding, Convolution Neural Network, Pairwise Sequence Alignment ## I Introduction Measuring the similarity of two sequences is a well known problem called sequence alignment. This topic includes a vast category of methods for identifying regions of high similarity in biological sequences, such as those in deoxyribonucleic Acid (DNA), ribonucleic acid (RNA), and protein [7]. Specifically, DNA pairwise sequence alignment (PSA) methods are concerned with finding the best arrangement of two DNA sequences. Some historically notable dynamic programming PSA methods are the Needleman-Wunsch (NW) algorithm for global alignment [1] and Smith-Waterman (SW) algorithm for local alignment [2]. The main difference between global and local alignment is related to the difference in length of the two sequences: global alignment attempts to find the highest-scoring end-to-end alignment between two sequences of approximately the same length, and local alignment searches for local regions of high similarity between two sequences with different lengths [8]. Figure 1 shows this difference between local and global DNA alignment with two sequences aligned in a 5’ (i.e. five prime) to 3’ direction. In molecular biology, this orientation refers to the directionality of the carbon backbone in DNA. The top subfigure displays global alignment where a query sequence is aligned end-to-end with a reference. The bottom subfigure displays local alignment where a short query sequence is most optimally aligned with a longer reference sequence. This latter alignment displays how the query sequence is approximately equal to a subsequence of the reference sequence. Figure 1: Local vs. Global Alignment. In general, DNA is composed of a permutation of the four nucleotides [adenine (A), thymine (T), cytosine (C), guanine (G)] and an ambiguous base (N). In this way, local alignment methods recognize approximate subsequence matches of a query sequence with respect to a given reference sequence. One common paradigm utilizing local alignment is to examine similarities between a query sequence and specific k-long subsequences in a given gene, known as k-mers, found within the reference sequence. Traditional local alignment algorithms calculate these scores between the query sequence and each k-mer in the reference sequence. The aim of this research is to identify where the most likely subsequence matches of the query sequence occur in each reference sequence using machine learning methods. One such type of query sequence that is of high biological significance is a sequence motif, which are short reoccurring subsequences of DNA [5]. Therefore, this research follows the ability of machine learning methods to gauge the relative enrichment of various representations of motifs (or motif patterns) in independent reference sequences. More specifically, the efficacy of identifying motif enrichment in sequences is explored using a one-dimensional (1-D) convolution neural network (CNN). Four different data sets are generated, each with a different motif pattern randomly inserted in approximately 10,000 reference sequences: the first three data sets have zero, one, and two mutations applied on each inserted motif, and the fourth data set represents the inserted motif as a position-specific probability matrix (PPM). In this data structure, each nucleotide position corresponds to a frequency of nucleotides [22]. These distinct motif patterns help display how the CNN model can recognize both subsequence matches with exact, inexact, and probabilistic motifs. Each sample in a given data set consists of artificial sequences enriched with a given motif pattern at a frequency between five and fifteen occurrences per 1,000 base pairs (bp). These samples are split into 986 overlapping 15-mers with a corresponding calculated local alignment score from the BioPython Aligner [20]. These sores are then predicted using a CNN with 10-fold cross validation. In order to measure the performance of the model, the average out of sample mean squared error (MSE), R2, and accuracy scores are reported. While the MSE of the model trained on each data set is not representative of the model’s effectiveness, the Jaccard Index and $S_{\alpha}$, a novel modified version of the Jaccard Index, are better suited to capture accuracy of the model. The standard MSE is not suitable for this problem because it inherently only displays differences between predicted and actual values. Since our aim is to locate those highest-scoring 15-mers, we need a metric that determines at which positions they occur and with what accuracy (see subsection V-A). This new metric, $S_{\alpha}$, measures the degree of similarity between two sets where each pair of elements can be different by at most $\alpha$. Because of the plateauing nature of this metric as seen in each data set and the risks involved in increasing alpha, only $S_{0}$ to $S_{5}$ are reported. In implementing this new metric, the accuracy of the model increases dramatically across all four data sets compared to the Jaccard Index. This indicates that while the model is not able to precisely identify the highest- scoring k-mers exactly, it is able to accurately identify their local region. As expected, the model’s accuracy is far higher for the data sets with relatively simple inserted motif patterns–non-probabilistic consensus motifs–compared to that of the data set with more complex inserted motif patterns, such as consensus PPM. ## II Background Clusters of motifs across a genome strongly correlate to a gene regulatory regions [18]. These regions are especially important for motif enrichment analysis, where known motifs are identified in the regulatory sequence of a gene in order to determine which proteins (transcription factors and enhancers) control its transcription [6] [19]. Motif enrichment analysis is only relevant given that the regulatory region of a gene is known, otherwise the sequence under study may be from a non-coding region of an organism’s genome or an untranslated region of a gene [9]. Given that the regulatory region of a gene is unknown, one frequently used approach to identifying it is to first locate sequences enriched with highly conserved motifs. Fortunately, many motifs that have been discovered are common amongst genes serving a similar role across organisms, such as a negative regulatory region for eukaryotes [10]. Finding these conserved motifs may facilitate the identification of the regulatory regions in a gene. For that reason, identifying the exact or relative positions of a given motif in a gene or sequence is a relevant inquiry in the process for classifying candidate regulatory regions of a gene. A software toolkit known as MEME Suit includes three different methods for motif-sequence searching [23]: FIMO (Find Individual Motif Occurrences) [21], GLAM2SCAN (Gapped Local Alignment of Motifs SCAN) [24], and MAST (Motif Alignment and Search Tool) [25]. FIMO focuses on scanning both DNA and protein sequences for a given motif represented as PPM. This software tool calculates the log-likelihood ratio score, p-value, and q-value (false discovery rate) for each subsequence position in a sequence database [21]. Typically, GLAM2SCAN performs a Waterman-Eggert local alignment between motifs found by GLAM2, its companion motif-finding algorithm, and a sequence database. These local alignment scores are generated from an aligner programmed with position specific residue scores, deletion scores, and insertion scores returned from the GLAM2 algorithm. The $n$ highest alignments are returned to the user [24]. MAST locates the highest-scoring $n$ subsequences with respect to a motif described as a position-specific score matrix. Using the QFAST algorithm, MAST calculates the p-value of a group of motif matches. This is accomplished by first finding the p-value of each match (position p-value’) and normalizing it for the length of the motif (’sequence p-value’). Then each of these normalized p-values are multiplied together to find the statistical significance across all located motifs in the database (’combined p-value’) [25]. ## III Data Analysis & Curation A single data set contains approximately 10,000 randomly generated DNA sequences, each 1,000 bp long. The number of samples vary slightly from one to another due to some inconsistencies that are removed in prepossessing. A 15-mer motif is inserted into each sample anywhere from five to fifteen times. Four separate data sets of this structure are created where a different motif pattern is inserted randomly into each sequence. The first three data sets have zero, one, and two mutations applied on each inserted motif. These mutations are applied in order to determine whether the proposed model has the potential to identify consensus motifs and non-exact consensus motifs across many sequences. Since motifs mostly exist as profiles where each base pair position corresponds to a frequency table of nucleotides, the fourth data set is created where the inserted motifs are based off of a PPM [11]. Equation 1 is used to calculate the PPM indicated by matrix $M$ given a set of candidate motifs, or sequences that are thought to be from the same motif PPM. This equation counts the number of occurrences of each nucleotide in set $\gamma$ for each nucleotide position across all motifs, where $\gamma=\\{A,T,C,G\\}$; $I=\\{0,1\\}$ represents an indicator function, where $I(x=\gamma)$ is 1 if $x=\gamma$ and 0 otherwise; $i{\displaystyle\in}$ (1, …, L), where L is the length of each motif; and $j{\displaystyle\in}(1,...,N)$, where N is the number of motifs. $M_{\alpha,k}=\frac{1}{N}\sum^{N}_{i=1}I(X_{i,j}=\gamma)$ (1) In order to apply Equation 1 on candidate motifs, the DNA sequence data must be formatted as nucleotide position counts shown in Figure 2. This figure illustrates the conversion of a list of candidate motifs to matrix $M_{counts}$ and then to $PPM$ using Equation 1. While Figure 2 displays this process for five 10-mers, the fourth data sets in this work relies on profiles built from ten 15-mers. TACAGAGTTG CCATAGGCGT TGAACGCTAC ACGGACGATA CGAATTTACG $\downarrow$ $M_{counts}$ = A 1 1 3 3 2 1 0 2 1 1 T 2 0 0 1 1 1 1 2 2 1 C 2 2 1 0 1 1 1 1 1 1 G 0 2 1 1 1 2 3 0 1 2 $\downarrow$ $PPM$ = A 0.2 0.2 0.6 0.6 0.4 0.2 0.0 0.4 0.2 0.2 T 0.4 0.0 0.0 0.2 0.2 0.2 0.2 0.4 0.4 0.2 C 0.4 0.4 0.2 0.0 0.2 0.2 0.2 0.2 0.2 0.2 G 0.0 0.4 0.2 0.2 0.2 0.4 0.6 0.0 0.2 0.4 Figure 2: The conversion of five candidate subsequence motifs to PPM using Equation 1. ## IV Feature & Output Selection In order to format the sequence data into a structure that is both recognizable and meaningful to a CNN, we first split each sequence into a list of overlapping 15-mers. Next, we generate a one-hot encoding for each nucleotide in the 15-mers. The resulting feature set is composed of 60 values. Figure 3 displays this process using a small subsequence example formatted as 4-mers. Figure 3: DNA subsequence k-mer formatting by one-hot encoding nucleotides. To obtain the target values, each of these 15-mers are pairwise aligned with the consensus motif for the given data set motif pattern using the SW algorithm. Given two sequences, $a$ of length $n$ and $b$ of length $m$, this algorithm begins by defining an $n+1$ by $m+1$ matrix $H$. The first column and first row are assigned $0$, and the following recurrence relation is applied to assign the rest of the values in $H$. $H(i,j)=max\begin{cases}H(i-1,j-1)+\sigma(a_{i},b_{j})\\\ H(i,j-1)+W\\\ H(i-1,j)+W\\\ 0\end{cases}$ where W is a gap score and $\sigma$ is a score matrix such that $\sigma(a_{i},b_{j})=\begin{cases}+1&\quad\text{if }a_{i}=b_{j}\\\ -2&\quad\text{if }a_{i}\neq b_{j}\end{cases}$ In the case when $a_{i}=b_{j}$, $\sigma$ returns a match score of $+1$, and in the case when $a_{i}\neq b_{j}$, $\sigma$ returns a mismatch score of $-2$. The gap score, $W$, is assigned $-2.5$. The match, mismatch, and gap score can be configured for different alignments. These parameters are used because they are the most optimal for this type of local alignment [4]. Once $H$ is assigned its values, the best alignment is obtained by finding the maximum value in $H$ and tracing back the matrix elements that led up to this maximum. In this way, the maximum value in $H$ defines the optimal path in $H$ for the best alignment between sequences $a$ and $b$ [2]. The calculated alignment scores are normalized based on the maximum alignment score in each sample. ## V Methods ### V-A CNN Model Evaluation Although the MSE loss function is effective at penalizing large differences between predicted and target values, such as outliers in the data, it does not successfully represent the predictive power of the model given the scope of the problem [14]. In the data, the target value from each sample ranges from zero to one. This range already generates an inherently small MSE. Even when the MSE for each sample is normalized, the metric is overshadowed by the overwhelming majority of the predicted values that were approximately equal to the global mean of each sample. In other words, the MSE as a metric does not capture the correct information pertaining to the five to fifteen inserted motif patterns in each sample due to a large unequal distribution of such scores that deviate from the global mean. This problem is analogous to that of an unequal class distribution in a classification problem. The goal of the model is to score the CNN based on its ability to locate the 15 highest-scoring 15-mers, because we inserted a motif pattern at most 15 times into a single sample. Since this network deals with continuous values instead of discrete classes, initially we cannot be certain of the 15-mer to which a 15-mer score at any index $i$ corresponds. However, a higher scoring 15-mer has a greater probability of corresponding to that of a motif, whereas the lower scoring 15-mers carry little information. This is due to the fact that each score in the data is generated from a local alignment between 15-mer and the given consensus motif. In this way, only the highest 15-scoring 15-mers are of interest. As previously mentioned, we indicate that there is an unequal distribution between the number of scores corresponding to that of each inserted motif and the global mean of each sample. Using these observations, we rationalize that we only have to examine the 15 highest- scoring indices. This generality that the 15 highest-scoring idicies correspond to the inserted motif patterns is further supported by the notion that probability of observing a random 15-mer exactly equal or similar to the inserted motifs is relatively low. Thus, the indices of the predicted 15 highest-scoring 15-mer inherently hold information about the position of possible inserted motif patterns because it is at these indices at which the local alignment is conducted. Due to the low likelihood of observing a false positive (when a 15-mer is identified as a motif but in all actuality is not one), we create a one-to-one correspondence between the indices of the actual motif indices and that of the predicted motifs using high local alignment scores. The accuracy of this one-to-one correspondence can be measured using the Jaccard Index given in Equation 2. $J(A,B)=\frac{\|A\cap B\|}{\|A\cup B\|}$ (2) We propose a more generalized index, $S_{\alpha}$, in Equation 3 which measures the similarity of two sets with an allowed margin of error of $\alpha$. Because of the high locality of local alignment score predictions and due to the fact that the highest-scoring 15-mers can still be found from examining the immediate region of a prediction, this margin of error serves as a heuristic for motif identification. In this metric, two items are considered identical if they are no more than $\alpha$ away from each other. In the scope of this work, sets $A$ and $B$ contain the indices of the 15 highest-scoring 15-mers of the actual data and predicted data, respectively. When $\alpha=0$, $S_{0}(A,B)$ in Equation 2 is identical to $J(A,B)$ in Equation 3. Conversely, as $\alpha$ increases, the allowed distance between indices in sets $A$ and $B$ increases. For example, when $\alpha=2$, a predicted 15-mer index $i$ and actual 15-mer index $i+2$ are considered the same. $J(A,B\mid\alpha)=S_{\alpha}(A,B)=\frac{\|\bigcup\limits_{\mu=0}^{\alpha}A\cap\\{x+\mu\mid x\in B\\}\|}{\|A\cup B\|}$ (3) The following process is an algorithm to calculate a modified version of the Jaccard Index. Using the $argsort$ function in NumPy, we examine the indices that order both the actual outputs and the predicted outputs. In looping through the each of the top $n$ indices of the predicted outputs, we count the number of them which are contained in the list of indices of the actual outputs. The process returns the score as count over the maximum possible value, which in this case is $n$. This is implemented in Algorithm 1 Algorithm 1 Measuring Jaccard Index with stride $\alpha$ 1:procedure $s_{\alpha}$ 2: $\textit{n}\leftarrow\text{number of highest-scoring k-mers to analyze}$ 3: $\textit{score}\leftarrow 0$ 4: $\textit{act\\_outputs}\leftarrow\text{actual outputs}$ 5: $\textit{pred\\_outputs}\leftarrow\text{outputs from CNN}$ 6: $\textit{act\\_indxs}\leftarrow\text{indices that would sort }\textit{act\\_outputs}$ 7: $\textit{pred\\_indxs}\leftarrow\text{indices that would sort }\textit{pred\\_outputs}$ 8: _outerloop_ : 9: for $i$ := 1 to $n$ do 10: $\textit{pred\\_indx}\leftarrow\textit{pred\\_indxs(i)}$. 11: for $j$ := 0 to $\alpha$ do 12: if $\textit{pred\\_indxs}\in\textit{act\\_indxs}-j$ then 13: $score\leftarrow score+1$. 14: goto _outerloop_. 15: if $\textit{pred\\_indxs}\in\textit{act\\_indxs}+j$ then 16: $score\leftarrow score+1$. 17: goto _outerloop_. 18: $normalized\\_score\leftarrow score/n$. ## VI Results Each of the four data sets is characterized by 10,000 samples where each sample contains a sequence that is 1,000 bp in length. In each sample, a motif pattern is inserted randomly anywhere from five to fifteen times. The first three data sets include inserted motif patterns with zero, one, and two mutations. The fourth data set includes an inserted motif pattern represented based on a PPM. Each data set is evaluated using out of sample data generated from 10-fold cross validation based on eight metrics: MSE, R2, and $S_{0}$-$S_{5}$. Table I: CNN Results. The average out of sample MSE, R2, and $S_{0}$-$S_{5}$ for each data set. A fifth analysis is conducted with another data set using a motif representation similar to that of the fourth data set with the MafK transcription factor from the BATCH1 regulatory gene [26]. This motif is a 15-mer with a less conserved consensus sequence compared to that of the former four data sets. While this data set did not perform as well as the other four data sets with a $S_{9}$ of 45.3%, this analysis brought to light the consideration of the aligner scoring matrix as another hyperparameter to this work. As it turns out, the performance of the model varies greatly with the chosen match score, mismatch score penalty, and gap score penalty for the currently implemented alignment method. For instance, the $S_{9}$ varies from 33.7% to 52.6% with different scoring hyperparameters. The former result is derived from an aligner with a match score of +2.0, mismatch score penalty of -3.0, and gap score penalty of -3.5, whereas the latter result is derived from an aligner with a match score of +2.0, mismatch score penalty of -4.0, and gap score penalty of -4.5. It is currently unclear what aligner hyperparameters are most optimal for this more complex data set and the original four data sets explored in the work. Although there is evidence to suggest that aligner scoring matrices vary with the type of inserted motif pattern, it is unclear whether the most optimal hyperparameters change from motif to motif. One possible interpretation of the dependence of the model’s chosen evaluation metric, $S_{\alpha}$, on the aligner hyperparameters is related to the fact that the CNN predicts alignment scores that are normalized within each sample. Therefore, the farther these highest-scoring scores are from the global mean, the more likely that the proposed metric will be able to recognize inserted motifs. Conversely, when analyzing a data set with a less conserved motif consensus sequence, such as that of the MafK transcription factor, the alignment scores are closer to the global mean of each sample. This in turn makes recognizing the indices of the highest-scoring segments more challenging. It follows that the aligner hyperparameters which capitalize on increasing this difference are most favorable for all motifs, regardless of pattern. ### VI-A Convolution Neural Network (CNN) Architecture CNN is a class of deep learning models which can infer patterns based on data formatted as a grid structure, such as a set of prices over time for stock or a grid representation of pixels in an image (add reference for these architectures). These Artificial Neural Netowrk (ANNs) use a linear mathematical operation called convolution in at least one of their layers [3]. The convolution operation is commonly identified by the following two equations: $s(t)=\int x(a)w(t-a)da$ (4) $s(t)=(xw)(t)$ (5) Equation 4 explicitly denotes the equation for convolution, whereas Equation 5 displays how an asterisk can be used to for the linear operation. In both equations, $x$ is referred to as the input. Typically, this is formatted as a multidimensional array, or a tensor, that matches the size and dimensions of the data. The second argument is $w$, representing a kernel, which stores parameters for the model also formatted as a tensor. This argument is adapted throughout the training process of the model. The output of both functions, $s$, is called the feature map of the convolution layer. This is what is fed into the next layer of the network [3]. Hidden layers are generated from applying a kernel, or filter, of weights over the receptive field of the inputs. More specifically, the hidden layer is computed based off of the filter weights and the input layer as it strides across the feature space [28]. This operation can either compress or expand input space depending on the applied kernel [29]. This paradigm is followed by rounds of activations, normalizations, and pooling [29]. The model typically ends with a fully connected layer to compute its outputs [28]. The proposed model is represented in Figure 4 [cite my paper]. Figure 4: CNN model. (create better caption) The model is marked by three rounds of a 1-D convolution layer, a batch normalization layer, a dense layer, and a 1-D maximum pooling layer. After these 12 layers, the model finishes off with a 50% dropout layer, a flattened layer, and finally a fully connected layer corresponding to the 986 alignment scores for each sample [13] [12]. The model described above is ran on all four data sets for 100 epochs with a batch size of 80 and compiled with the Adam optimizer (learning rate=0.001, beta 1=0.9, beta 2=0.999, epsilon=1e-07). Of the 10,000 samples in each dataset, 80% is reserved for training the network and the remaining 20% is used for validation after each epoch. For its loss function, the model relies on Mean Squared Error (MSE), which is calculated between predicted values ($y_{pred}$) and target values ($y_{act}$) with the following formula in Equation 6: $MSE(y_{pred},y_{act})=\frac{1}{n}\sum_{i=1}^{n}(y_{pred,i}-y_{act,i})$ (6) ## VII Discussion As displayed in this work, deep learning models, such as a CNN, have the capacity to recognize and predict the positions of an inserted motif with great accuracy. Furthermore, data structures can be devised to take advantage of unequal class distributions in regression problems as highlighted by the design of k-mer data representation in this work and the incorporation of $S_{\alpha}$ as a novel evaluation metric. In analyzing the results in Table I, there is a characteristic pattern between the accuracy metrics across each data set. For instance, in comparing $S_{0}$-$S_{5}$ for the first data set with zero mutations applied on each inserted motif, the score monotonically increases with an increasing $\alpha$. This is evident for the three other data sets as well. With respect to this particular trend, it is expected that as $\alpha$ increases, the score will also increase since $\alpha$ relates directly to the allowed margin of error, making $S_{\alpha}$ less conservative. Additionally, the model’s accuracy is far higher for the data sets with relatively simple inserted motif patterns, such as nonmutated and mutated consensus motifs, compared to that of the fourth data set with a PPM motif pattern. This relationship can be explained by the process by which the scores for each 15-mer are calculated. For a given 15-mer, a score is computed based on its local alignment with a given consensus motif. For the first data set, these local alignment scores generated are derived from each inserted motif, whereas in the latter three data sets, the scores are not necessarily derived from each data set’s consensus motif since the motif patterns support variable inserted motif. In all data sets, the largest increase in $S_{\alpha}$ appears to be between the $S_{0}$ and $S_{1}$. After this point, change in $S_{\alpha}$ plateaus after a given $\alpha$. With the consideration that the likelihood of observing a false positive is relatively low, this indicates that the addition of stride $\alpha$ is well-advised. This is the case because the increase in $\alpha$ only influences $S_{\alpha}$ up to a certain point. It is expected that as $\alpha\xrightarrow{}\beta$, where $\beta$ is the maximum $\alpha$ on either side of a given motif index, $S_{\alpha}\xrightarrow{}1$ because every single $n$ indices will be covered by the stride ${\alpha}$. In the case that $S_{\alpha}\xrightarrow{}1$, the certainty for each identified motif decreases with increasing $S_{\alpha}$ regardless; however, the absence of this limit in the data indicates that the certainty of the identified motifs does not decreases dramatically from $S_{0}$ to $S_{5}$. Furthermore, the presence of a plateauing $S_{\alpha}$ supports the thought that a decrease in the certainty of an identified motif is negligible. This analysis can be drawn further in noticing that the point at which $S_{\alpha}$ plateaus increases as the complexity of the motif pattern increases. In the case of a more complex motif pattern, such as either of the PPMs, a greater $\alpha$ is required to fully encapsulate accuracy of the model’s predictions. Even then, the certainty of such motif identification with increasing $\alpha$ decreases. In subsection V-A, we draw a one to one correspondence between the actual motif indices and that of the predicted motifs by only examining the indices of the 15 highest-scoring 15-mers in both the actual scores and predicted scores. This is not a strong one-to-one correspondence because the number of inserted motifs actually varies randomly from five to fifteen times sample to sample. By design, this is a confounding variable When $S_{\alpha}$ is applied on a sample with five inserted motifs, the returned score is predicted to be an underestimate of the model’s prediction. This is due to the fact that this function only examines the highest 15-scoring indices for each sample. In the case of five inserted motifs, there would be ten 15-mers identified as high- scoring motifs, when in reality these are random 15-mers in the sequence. 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# Current algebras on $S^{3}$ of complex Lie algebras Tosiaki Kori Department of Mathematics Graduate School of Science and Engineering Waseda University, Tokyo 169-8555, Japan email<EMAIL_ADDRESS> ###### Abstract Let $\mathcal{L}$ be the space of spinors on $S^{3}$ that are the restrictions to $S^{3}$ of the Laurent polynomial type harmonic spinors on $\mathbf{C}^{2}$. $\mathcal{L}$ becomes an associative algebra. For a simple Lie algebra $\mathfrak{g}$ the real Lie algebra $\mathcal{L}\mathfrak{g}$ generated by $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}$ is called $\mathfrak{g}$-current algebra. The real part $\mathcal{K}$ of $\mathcal{L}$ becomes a commutative subalgebra of $\mathcal{L}$. For the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}\,$, $\mathcal{K}\mathfrak{h}=\mathcal{K}\otimes_{\mathbf{R}}\mathfrak{h}$ becomes a Cartan subalgebra of $\mathcal{L}\mathfrak{g}$. We investigate the adjoint representation of $\mathcal{K}\mathfrak{h}$ and find that the set of non-zero weights corresponds bijectively to the root space of $\mathfrak{g}$. Let $\mathfrak{g}=\mathfrak{h}+\mathfrak{e}+\mathfrak{f}$ be the standard triangular decomposition of $\mathfrak{g}$, and let $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{h}$, $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{e}$ and $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{f}$ generate respectively the Lie subalgebras $\mathcal{L}\mathfrak{h}$, $\mathcal{L}\mathfrak{e}$ and $\mathcal{L}\mathfrak{f}$ of $\mathcal{L}\mathfrak{g}$. Then we have the triangular decomposition $\mathcal{L}\mathfrak{g}=\mathcal{L}\mathfrak{h}+\mathcal{L}\mathfrak{e}+\mathcal{L}\mathfrak{f}\,$, that is also associated with the weight space decomposition of $\mathcal{L}\mathfrak{g}$. With the aid of the basic vector fields on $S^{3}$ that arise from the infinitesimal representation of $SO(3)$ we introduce a triple of 2-cocycles $\\{c_{k};\,k=0,1,2\,\\}$ on the Lie algebra $\mathcal{L}\mathfrak{g}$. Then we have the central extenstion $\mathcal{L}\mathfrak{g}\oplus\oplus_{k=0}^{2}\mathbf{C}a_{k}$ associated to the 2-cocycles $\\{c_{k}\\}_{k=0,1,2}$. Adjoining a derivation coming from the radial vector field $\mathbf{n}$ on $S^{3}$ we obtain the second central extension $\widehat{\mathfrak{g}}=\mathcal{L}\mathfrak{g}\oplus\oplus_{k=0}^{2}\mathbf{C}a_{k}\oplus\mathbf{C}n$. The root space decomposition and the Chevalley generators of $\widehat{\mathfrak{g}}\,$will be given. Mathematics Subject Classification. 17B65, 17B67, 22E67, 81R10, 81R25, 15B33. Key Words Infinite dimensional Lie algebras, Current algebras, Kac-Moody Lie algebra, Spinor analysis. ## 1 Introduction Any affine Kac-Moody algebra of untwisted type can be realized in terms of a central extension of the loop algebra of a semisimple Lie algebra, [K]. Let $L=\mathbf{C}[t,t^{-1}]$ be the algebra of Laurent polynomials in $t$. Given a semisimple Lie algebra $\mathfrak{g}$ the loop algebra $L\mathfrak{g}=L\otimes_{\mathbf{C}}\mathfrak{g}$ is an infinite dimensional complex Lie algebra with the bracket $[\,,\,]$ defined by $[P\otimes x,\,Q\otimes y]=PQ\otimes\,[x,y]\,,\quad P,Q\in L,\,x,y\in\mathfrak{g}.$ We define a 2-cocycle on the algebra $L$ by the formula: $c_{o}(P,Q)=\frac{1}{2\pi}\int_{S^{1}}\,\frac{dP}{dt}(t)\cdot Q(t)\,dt.$ By virtue of the non-degenerate symmetric bilinear form $(\cdot\|\cdot)$ on $\mathfrak{g}$ we extend the 2-cocycle $c_{o}$ to a 2-cocycle $c$ on the Lie algebra $L\mathfrak{g}\,$: $c(P\otimes x,\,Q\otimes y)\,=\,(x\|y)c_{0}(P,Q)\,.$ Let $L\mathfrak{g}\oplus\mathbf{C}a$ be the extension of $L\mathfrak{g}$ by a 1-dimensional center associated to the cocycle $c$. The Euler derivation $t\frac{d}{dt}$ acts on $L\mathfrak{g}\oplus\mathbf{C}a$ as an outer derivation and kills $c$. Then adjoining the derivation $d$ to $L\mathfrak{g}\oplus\mathbf{C}a$ we have the Lie algebra: $\widehat{\mathfrak{g}}=L\mathfrak{g}\oplus\mathbf{C}a\oplus\mathbf{C}d.$ We follow this procedure to have central extensions of current algebras on $S^{3}$. We introduce the algebra of Laurent polynomial type harmonic spinors on $S^{3}$. It is called the algebra of current on $S^{3}$ and is denoted by $\mathcal{L}\,$. It plays the same role as the algebra $L$ of Laurent polynomials does for the loop algebra. The current algebra of $\mathfrak{g}$ on $S^{3}$ is the real Lie algebra $\mathcal{L}\mathfrak{g}$ that is generated by $\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}$ . Then we shall introduce a triple of 2-cocycles on $\mathcal{L}\,$, and extend them to 2-cocycles on the current algebra $\mathcal{L}\mathfrak{g}$. For this purpose we prepare in section 2 a rather long introduction to our previous results on analysis of harmonic spinors on $\mathbf{C}^{2}$, [F, G-M, Ko1, Ko2, Ko3] and [K-I], that is, we develop some parallel results as in classical analysis; the separation of variable method for Dirichlet problem, Fourier expansion by the eigenfuctions of Laplacian, Cauchy integral formula for holomorphic functions and Laurent expansion of meromorphic functions etc.. For example, the Dirac operator on spinors corresponds to the Cauchy-Riemann operator on complex functions. Let $\Delta=\mathbf{H}^{2}$ be the 4-dimensional spinor space, that is, an irreducible representation of the complex Clifford algebra $\,{\rm Clif}^{c}_{4}=End(\Delta)$. The algebraic basis of ${\rm Clif}^{c}_{4}$ is given by the Dirac matrices: $\gamma_{k}\,=\,\left(\begin{array}[]{cc}0&-i\sigma_{k}\\\ i\sigma_{k}&0\end{array}\right)\,,\,k=1,2,3$, and $\gamma_{4}\,=\,\left(\begin{array}[]{cc}0&-I\\\ -I&0\end{array}\right)\,.$ Where $\sigma_{k}$ are Pauli matrices. Let $S=\mathbf{R}^{4}\times\Delta$ be the spinor bundle. The Dirac operator is defined by the following formula: $\mathcal{D}=\,-\,\frac{\partial}{\partial x_{1}}\gamma_{4}\,-\,\frac{\partial}{\partial x_{2}}\gamma_{3}\,-\,\frac{\partial}{\partial x_{3}}\gamma_{2}\,-\,\frac{\partial}{\partial x_{4}}\gamma_{1}\,:\,C^{\infty}(\mathbf{R}^{4},S)\longrightarrow\,C^{\infty}(\mathbf{R}^{4},S)\,.$ Let $S^{\pm}=\mathbf{R}^{4}\times\Delta^{\pm}$ be the ( even and odd ) half spinor bundle corresponding to the decomposition $\Delta=\Delta^{+}\oplus\Delta^{-}$: $\Delta^{\pm}\simeq\mathbf{H}$. The half spinor Dirac operator $D=\mathcal{D}\|S^{+}$ has the polar decomposition: $D=\gamma_{+}\left(\frac{\partial}{\partial n}-\partial\\!\\!\\!/\right)$ with the tangential (nonchiral) component $\partial\\!\\!\\!/\,$ on $S^{3}\subset\mathbf{R}^{4}$. The tangential Dirac operator $\partial\\!\\!\\!/$ on $S^{3}$ is a self adjoint elliptic differential operator. The eigenvalues of $\partial\\!\\!\\!/$ are $\\{\frac{m}{2},\,\,-\frac{m+3}{2}\,;\,m=0,1,\cdots\\}$ with multiplicity $(m+1)(m+2)$. We have an explicitly written polynomial formula of eigenspinors $\left\\{\phi^{+(m,l,k)},\,\phi^{-(m,l,k)}\right\\}_{0\leq l\leq m,\,0\leq k\leq m+1}$ corresponding to the eigenvalues $\frac{m}{2}$ and $-\frac{m+3}{2}\,$ respectively that give rise to a complete orthonormal system in $L^{2}(S^{3},S^{+})$, [Ko1, Ko2]. A spinor $\phi$ on a domain $G\subset\mathbf{C}^{2}$ is called a harmonic spinor on $G$ if $D\phi=0$. Each $\phi^{+(m,l,k)}$ is extended to a harmonic spinor on $\mathbf{C}^{2}$, while each $\phi^{-(m,l,k)}$ is extended to a harmonic spinor on $\mathbf{C}^{2}\setminus\\{0\\}$ that is regular at infinity. Every harmonic spinor $\varphi$ on $\mathbf{C}^{2}\setminus\\{0\\}$ has a Laurent expansion by the basis $\phi^{\pm(m,l,k)}$: $\varphi(z)=\sum_{m,l,k}\,C_{+(m,l,k)}\phi^{+(m,l,k)}(z)+\sum_{m,l,k}\,C_{-(m,l,k)}\phi^{-(m,l,k)}(z).$ The set of spinors of Laurent polynomial type is denoted by $\mathbf{C}[\phi^{\pm}]$. Let $\mathbf{H}$ be the algebra of quaternion numbers. We look an even spinor also as a $\mathbf{H}$-valued smooth function: $C^{\infty}(S^{3},S^{+})=C^{\infty}(S^{3},\mathbf{H})$, so that the space of spinors $C^{\infty}(S^{3},S^{+})$ is endowed with a multiplication rule: $\phi_{1}\cdot\phi_{2}\,=\,\begin{pmatrix}\,u_{1}u_{2}-\bar{v}_{1}v_{2}\,\\\\[5.69046pt] \,v_{1}u_{2}+\bar{u}_{1}v_{2}\,\end{pmatrix}\,,\quad\mbox{ for }\,\phi_{i}=\begin{pmatrix}\,u_{i}\\\\[5.69046pt] \,v_{i}\end{pmatrix},\,i=1,2\,.$ (1.1) Let $\mathcal{L}=\mathbf{C}[\phi^{\pm}]\|S^{3}$ be the space of spinors on $S^{3}$ that are obtained by restricting the spinors of Laurent polynomial type. $\mathcal{L}$ becomes an associative subalgebra of $C^{\infty}(S^{3},S^{+})$ that is called the algebra of current on $S^{3}$. In section 3 we introduce the 2-cocycles on $C^{\infty}(S^{3},S^{+})$. Let $\\{\theta_{0},\,\theta_{1},\,\theta_{2}\,\\}$ be the basis of vector fields on $S^{3}$ coming from the infinitesimal representation of $SO(3)$. Our 2-cocycles on $C^{\infty}(S^{3},S^{+})$ are defined as follows. We put $\Theta_{k}\phi=\,\frac{1}{2}\,\left(\begin{array}[]{c}\,\theta_{k}\,u\\\\[8.5359pt] \,\theta_{k}\,v\end{array}\right),\,k=0,1,2,\quad\mbox{ for }\,\phi=\begin{pmatrix}u\\\ v\end{pmatrix}.$ We introduce the following three non-trivial real valued 2-cocycles $c_{k},\,k=0,1,2$, on $C^{\infty}(S^{3},S^{+})$ : $c_{k}(\phi_{1},\phi_{2})\,=\,\,\frac{1}{2\pi^{2}}\int_{S^{3}}\,\,tr\,(\,\Theta_{k}\phi_{1}\cdot\phi_{2}\,)\,d\sigma,\qquad\forall\phi_{1}\,,\,\phi_{2}\in\,C^{\infty}(S^{3},S^{+})\,.$ Since each $\Theta_{k}\,,k=0,1,2$, preserves $\mathcal{L}$, the 2-cocycles $c_{k},\,k=0,1,2$, restrict to $\mathcal{L}$. Hitherto we prepared the spaces $C^{\infty}(S^{3},\,S^{+})$ and $\mathcal{L}$ that will play the role of coefficients of current algebras. These are complex algebras. On the other hand $C^{\infty}(S^{3},S^{+})\simeq C^{\infty}(S^{3},\mathbf{H})$ has a $\mathbf{H}$-module structure, while our basic interest is on the real Lie algebra $\mathcal{L}=\mathbf{C}[\phi^{\pm}]\|S^{3}\,$. In such a way it is frequent that we deal with the fields $\mathbf{H}$, $\mathbf{C}$ and $\mathbf{R}$ in one formula. So to prove a steady point of view for our subjects we shall introduce here the concept of quaternion Lie algebras, [Kq]. First we note that a quaternion module $V=\mathbf{H}\otimes_{\mathbf{C}}V_{o}=V_{o}+JV_{o}$, $V_{o}$ being a $\mathbf{C}$-module, has two involutions $\sigma$ and $\tau$: $\sigma(u+Jv)=u-Jv\,,\quad\tau(u+Jv)=\overline{u}+J\overline{v}\,,\quad u,v\in V_{o}\,.$ A quaternion Lie algebra $\,\mathfrak{q}$ is a real submodule of a quaternion module $V$ that is endowed with a real Lie algebra structure compatible with the involutions $\sigma$ and $\tau$: $\displaystyle\sigma\mathfrak{q}\,\subset\mathfrak{q}\,,$ $\displaystyle\sigma[x\,,y]\,=[\sigma x\,,\sigma y]\,,\quad\tau[x\,,y]\,=[\tau x\,,\tau y]\,\quad\mbox{ for }\,x,y\in\mathfrak{q}.$ For a complex Lie algebra $\mathfrak{g}$ the quaternification of $\mathfrak{g}$ is a quaternion Lie algebra $\mathfrak{g}^{q}$ that is generated ( as a real Lie algebra ) by $\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{g}$. For example, $\mathfrak{so}^{\ast}(2n)=\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{so}(n,\mathbf{C})$ is the quaternification of $\mathfrak{so}(n,\mathbf{C})$. $\mathfrak{sl}(n,\mathbf{H})$ is the quaternification of $\mathfrak{sl}(n,\mathbf{C})$ though $\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{sl}(n,\mathbf{C})$ is not a Lie algebra. The algebra of current $\mathcal{L}$ is a quaternion Lie algebra. In fact $\mathcal{L}$ is a real submodule of $C^{\infty}(S^{3},\,\mathbf{H})$ that is invariant under the involutions $\sigma$ and $\tau$. The associative algebra $\mathcal{L}$ has the following four commutative subalgebras: $\\{\phi\in\mathcal{L};\,\sigma{\phi}=\pm\phi,\,\tau\phi=\pm\phi\,\\}.$ The real part $\mathcal{K}\,=\\{\phi\in\mathcal{L};\,\sigma{\phi}=\phi,\,\tau\phi=\phi\,\\}$ plays an important role. $\mathcal{K}$ is a commutative normal subalgebra of $\mathcal{L}$, and satisfies the condition $[\mathcal{K},\,\mathcal{L}]=0$. Let $\mathfrak{g}$ be a simple Lie algebra that we suppose to be a subalgebra of $\mathfrak{sl}(n,\mathbf{C})$. Let $\mathcal{L}\mathfrak{g}$ be the quaternion Lie algebra generated by $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}$ with the Lie bracket defined by $[\phi_{1}\otimes X_{1}\,,\,\phi_{2}\otimes X_{2}\,]\,=\,(\phi_{1}\cdot\phi_{2})\,X_{1}X_{2}\,-\,(\phi_{2}\cdot\phi_{1})\,X_{2}X_{1}$ (1.2) for $\phi_{1},\,\phi_{2}\in\mathcal{L},\,X_{1},X_{2}\in\mathfrak{g}\,.$ Here the right hand side is the bracket of the tensor product of the associative algebra $\mathcal{L}$ and the matrix algebra $\mathfrak{g}$. $\mathcal{L}\mathfrak{g}$ is called the $\,\mathfrak{g}$-current algebra. Let $\mathfrak{h}$ be the Cartan subalgebra of $\mathfrak{g}$. $\mathfrak{g}$ has the standard triangular decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{e}\,+\,\mathfrak{f}\,$. Let $\mathcal{L}\mathfrak{h}$, $\mathcal{L}\mathfrak{e}$ and $\mathcal{L}\mathfrak{f}$ be the Lie subalgebras of $\mathcal{L}\mathfrak{g}$ generated by $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{h}$, $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{e}$ and $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{f}$ respectively. Let $\mathcal{K}\mathfrak{h}=\mathcal{K}\otimes_{\mathbf{R}}\mathfrak{h}$. We find that $\mathcal{K}\mathfrak{h}$ is a Cartan subalgebra of $\mathcal{L}\mathfrak{g}$. It extends the adjoint representation $ad_{\mathfrak{h}}:\mathfrak{h}\longrightarrow End_{\mathbf{C}}(\mathfrak{g})$ to the adjoint representation $ad_{\mathcal{K}\mathfrak{h}}:\mathcal{K}\mathfrak{h}\longrightarrow End_{\mathcal{L}}(\mathcal{L}\mathfrak{g})$. The associated weight space decomposition of $\mathcal{L}\mathfrak{g}$ with respect to $\mathcal{K}\mathfrak{h}$ will be given. We find that the space of non-zero weights of $\mathcal{L}\mathfrak{g}$ corresponds bijectively to the root space of $\mathfrak{g}$. Let $\mathfrak{g}_{\lambda}$ be the root space of root $\lambda$ and let $\Phi^{\pm}$ be the set of positive (respectively negative ) roots of $\mathfrak{g}$. Then we have $\mathcal{L}\mathfrak{e}\,=\sum_{\lambda\in\Phi^{+}}\mathcal{L}\otimes_{\mathbf{R}}\mathfrak{g}_{\lambda},\quad\mathcal{L}\mathfrak{f}\,=\sum_{\lambda\in\Phi^{-}}\mathcal{L}\otimes_{\mathbf{R}}\mathfrak{g}_{\lambda}.$ Hence $\mathcal{L}\mathfrak{e}=\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{e}$ and $\mathcal{L}\mathfrak{f}=\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{f}$. $\,\mathcal{L}\mathfrak{h}$ has the weight $0$: $[\mathcal{K}\mathfrak{h},\,\mathcal{L}\mathfrak{h}\,]=0\,$. Accordingly we have the triangular decomposition of the $\mathfrak{g}$-current algebra: $\mathcal{L}\mathfrak{g}=\mathcal{L}\mathfrak{h}\,+\,\mathcal{L}\mathfrak{e}\,+\,\mathcal{L}\mathfrak{f}\,,\quad\mbox{direct sum}\,.$ We discuss in section 5 our central subject to give the central extension of $\mathfrak{g}$-current algebra. We extend each 2-cocycle $\,c_{k}\,,\,k=0,1,2\,$ on $\mathcal{L}$ to a 2-cocycle on $\mathcal{L}\mathfrak{g}\,$ by the formula $c_{k}(\phi\otimes X,\,\psi\otimes Y)\,=\,(X\|Y)\,c_{k}(\phi,\psi),\quad\phi,\,\psi\in\mathcal{L}\,,\,X,Y\in\mathfrak{g},$ (1.3) where $(X\|Y)=Trace(XY)$ is the Killing form of $\mathfrak{g}$. Then we have the associated central extension: $\,\mathcal{L}\mathfrak{g}(a)=\mathcal{L}\mathfrak{g}\,\oplus(\oplus_{k=0}^{2}\mathbf{C}a_{k}),$ which is a quaternion Lie algebra. The radial vector field $\mathbf{n}$ on $\mathbf{C}^{2}\setminus 0$ acts on $\mathcal{L}\mathfrak{g}(a)$ as an outer derivation. Then, adjoining the derivation $\mathbf{n}$, we have the second central extension: $\widehat{\mathfrak{g}}=\mathcal{L}\mathfrak{g}(a)\oplus\mathbf{C}n.$ We shall investigate the root space decomposition of $\,\widehat{\mathfrak{g}}\,$. For a root $\alpha\in\Phi$, let $\mathfrak{g}_{\alpha}=\\{x\in\mathfrak{g};\,[\,h,\,x\,]=\alpha(h)x\,,\,\forall h\in\mathfrak{h}\,\\}$ denote the root space of $\alpha$. Put $\widehat{\mathfrak{h}}\,=\,\,\mathfrak{h}\,\oplus(\oplus_{k=0}^{2}\mathbf{C}a_{k})\oplus(\mathbf{C}\,n)\,$ $\widehat{\mathfrak{h}}$ is a commutative subalgebra of $\widehat{\mathfrak{g}}\,$ and $\,\widehat{\mathfrak{g}}$ is decomposed into a direct sum of the simultaneous eigenspaces of $ad\,(\hat{h})$, $\,\hat{h}\in\widehat{\mathfrak{h}}\,$, and $\Phi\subset\mathfrak{h}^{\ast}$ is regarded as a subset of $\,\widehat{\mathfrak{h}}^{\ast}$. We introduce $\Lambda_{k}\in\widehat{\mathfrak{h}}^{\ast};\,k=0,1,2$ as the dual elements of $a_{k};\,k=0,1,2$, and $\delta\in\widehat{\mathfrak{h}}^{\ast}$ as the dual element of $n$. Then $\alpha_{1},\,\cdots\,,\alpha_{l},\,\delta,\,\Lambda_{0},\Lambda_{1},\Lambda_{2}$ give a basis of $\widehat{\mathfrak{h}}^{\ast}$. The set of simple root are $\widehat{\Phi}=\left\\{\frac{m}{2}\delta+\alpha\,;\quad\alpha\in\Phi\,,\,m\in\mathbf{Z}\,\right\\}\bigcup\left\\{\frac{m}{2}\delta;\quad m\in\mathbf{Z}\,\right\\}\,.$ $\widehat{\mathfrak{g}}$ has the weight space decomposition: $\widehat{\mathfrak{g}}\,=\,\oplus_{m\in\mathbf{Z}}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,\oplus\,\,\oplus_{\alpha\in\Phi,\,m\in\mathbf{Z}}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,.$ Each weight space is given as follows. $\displaystyle\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,$ $\displaystyle=$ $\displaystyle\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{g}_{\alpha}\,,\quad\mbox{ for $\alpha\neq 0$ and and $m\in\mathbf{Z}$},\,,$ $\displaystyle\widehat{\mathfrak{g}}_{0\delta}$ $\displaystyle=$ $\displaystyle(\,\mathcal{L}[0]\mathfrak{h}\,)\oplus(\oplus_{k=0}^{2}\mathbf{C}a_{k})\oplus(\mathbf{C}n)\,\supset\,\widehat{\mathfrak{h}}\,,$ $\displaystyle\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}$ $\displaystyle=$ $\displaystyle\,\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{h}\,\,,\quad\mbox{for $0\neq m\in\mathbf{Z}$ . }\,$ Where $\mathcal{L}[m]$ is the subspace of $\mathcal{L}=\mathbf{C}[\phi^{\pm}]\|S^{3}$ constituting of those elements $\phi\in\mathbf{C}[\phi^{\pm}]$ that are of homogeneous degree $m$: $\phi(z)=\|z\|^{m}\phi(\frac{z}{\|z\|})$. $\mathcal{L}[0]\mathfrak{h}$ is the Lie subalgebra generated by $\mathcal{L}[0]\otimes_{\mathbf{C}}\mathfrak{h}$. In our previous paper [K-I] we dealt with central extensions of $S^{3}\mathbf{H}\otimes_{\mathbf{C}}U(\mathfrak{g})$, where $U(\mathfrak{g})$ is the universal enveloping algebra of a simple algebra $\mathfrak{g}$. In [K-I] we called $\mathbf{H}\otimes_{\mathbf{C}}U(\mathfrak{g})$ the quaternification of $\mathfrak{g}$. But it is too big to consider as an adequate object to be studied as a quaternification of a Lie algebra. So we present here new definitions of a quaternion Lie algebra and a quaternification. This article contains many arguments, proofs and calculations that are parallel to those in [K-I], but we prefer to repeat them so that the readers need not refer to our old descriptions and can understand the theory as an unified one. ## 2 Preliminaries on spinor analysis on $S^{3}\subset\mathbf{C}^{2}$ Here we prepare a fairly long preliminary of spinor analysis on $\mathbf{R}^{4}$ because I think various subjects belonging to quaternion analysis or detailed properties of harmonic spinors of the Dirac operator on $\mathbf{R}^{4}$ are not so familiar to the readers. We refer to [F, Ko1] for the exposition on Dirac operators on $\mathbf{R}^{4}$ and to [D-S-Sc, G-M, Ko2] for the function theory of harmonic spinors. Subsections 2.1, 2.2, 2.3 are to remember the theory of harmonic spinors. ### 2.1 Spinors and the Dirac operator on $\mathbf{R}^{4}$ Let $\mathbf{K}$ be the field $\mathbf{R}$ or $\mathbf{C}$. Let $V$ be a $\mathbf{K}$-vector space equipped with a quadratic form $q$ over the field $\mathbf{K}$. The Clifford algebra $C_{\mathbf{K}}(V,q)$ is a $\mathbf{K}$-algebra which contains $V$ as a sub-vector space and is generated by the elements of $V$ subject to the relations $v_{1}v_{2}+v_{2}v_{1}=2q(v_{1},v_{2})\,,$ for $v_{1},\,v_{2}\in V$. In the sequel we denote ${\rm Clif}_{n}=C_{\mathbf{R}}(\mathbf{R}^{n},\,-x_{1}^{2}-\cdots-x_{n}^{2})$ and ${\rm Clif}_{n}^{c}\,=C_{\mathbf{C}}(\mathbf{C}^{n},z_{1}^{2}+\cdots+z_{n}^{2})$. It holds ${\rm Clif}_{n}^{c}={\rm Clif}_{n}\otimes_{\mathbf{R}}\mathbf{C}$. We have an important isomorphism: ${\rm Clif}_{n+2}^{c}={\rm Clif}_{n}^{c}\otimes_{\mathbf{C}}\mathbf{C}(2)\,\,.$ (2.1) Here $\mathbf{K}(m)$ denotes the algebra of $m\times m$-matrices with entries in the field $\,\mathbf{K}$. The left multiplication of $\mathbf{H}$ yields an endomorphism of $\mathbf{H}$; $\mathbf{H}\simeq End_{\mathbf{H}}\mathbf{H}\simeq\mathbf{C}(2)$. Then the corresponding matrices to $i\,,j\,,k\in\mathbf{H}\,$ are given by $i\sigma_{3},\,i\sigma_{2},\,i\sigma_{1}$. Where the Pauli matrices are $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\\[5.69046pt] 1&0\end{array}\right)\,,\,\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\\[5.69046pt] i&0\end{array}\right)\,,\,\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\\[5.69046pt] 0&-1\end{array}\right)\,.$ The relations $\sigma_{i}^{2}=-1$, $i=1,2,3$, and $\sigma_{1}\sigma_{3}+\sigma_{3}\sigma_{1}=0$ shows that $\\{\sigma_{1},\,\sigma_{3}\\}$ generate ${\rm Clif}^{c}_{2}$, so that ${\rm Clif}^{c}_{2}=\mathbf{H}$. Let $\Delta=\mathbf{C}^{2}\otimes_{\mathbf{C}}\mathbf{C}^{2}$ be the vector space of complex 4-spinors that gives the spinor representation of Clifford algebra ${\rm Clif}^{c}_{4}\,$: ${\rm Clif}^{c}_{4}={\rm End}_{\mathbf{C}}(\Delta)=\mathbf{C}(4)$. So $\,{\rm Clif}_{4}^{c}$ is generated by the following Dirac matrices: $\gamma_{k}\,=\,\left(\begin{array}[]{cc}0&-i\sigma_{k}\\\ i\sigma_{k}&0\end{array}\right)\,,\quad k=1,2,3,\quad\gamma_{4}\,=\,\left(\begin{array}[]{cc}0&-I\\\ -I&0\end{array}\right)\,.$ The set $\left\\{\gamma_{p},\quad\gamma_{p}\gamma_{q},\quad\gamma_{p}\gamma_{q}\gamma_{r},\quad\gamma_{p}\gamma_{q}\gamma_{r}\gamma_{s}\,;\quad 1\leq p,q,r,s\leq 4\,\right\\}$ (2.2) gives a 16-dimensional basis of the representation ${\rm Clif}^{c}_{4}\,\simeq\,{\rm End}_{\mathbf{C}}(\Delta)\,$ with the following relations: $\gamma_{p}\gamma_{q}+\gamma_{q}\gamma_{p}=2\delta_{pq}\,.$ The representation $\Delta$ decomposes into irreducible representations $\Delta^{\pm}=\mathbf{C}^{2}$ of $\,{\rm Spin}(4)$. Let $S=\mathbf{C}^{2}\times\Delta$ be the trivial spinor bundle on $\mathbf{C}^{2}$. The corresponding bundle $S^{+}=\mathbf{C}^{2}\times\Delta^{+}$ ( respectively $S^{-}=\mathbf{C}^{2}\times\Delta^{-}$ ) is called the even ( respectively odd ) half spinor bundle and the sections are called even ( respectively odd ) spinors. On the other hand, since ${\rm Clif}^{c}_{4}=\mathbf{H}(2)\otimes_{\mathbf{R}}\mathbf{C}$ and $\Delta=\mathbf{H}^{2}=\mathbf{H}\oplus\mathbf{H}$, we may look an even spinor on $M\subset\mathbf{R}^{4}$ as a $\mathbf{H}$ valued smooth function: $C^{\infty}(M,\mathbf{H})\,=\,C^{\infty}(M,S^{+})$. We feel free to use the alternative notation to write a spinor: $C^{\infty}(M,\mathbf{H})\,\ni\,u+jv\,\longleftrightarrow\,\left(\begin{array}[]{c}u\\\ v\end{array}\right)\,\in\,C^{\infty}(M,S^{+})\,.$ (2.3) The Dirac operator is defined by $\mathcal{D}=c\circ d\,:\,C^{\infty}(M,S)\,\longrightarrow\,C^{\infty}(M,S)\,.$ where $d:S\rightarrow T^{}\mathbf{C}^{2}\otimes S\simeq T\mathbf{C}^{2}\otimes S$ is the covariant derivative which is the exterior differential in this case, and $c:T\mathbf{C}^{2}\otimes S\rightarrow S$ is the bundle homomorphism coming from the Clifford multiplication. With respect to the Dirac matrices $\\{\gamma_{j}\\}_{j=1,2,3,4}\,$, (2.2), the Dirac operator has the expression: $\mathcal{D}=\,-\,\frac{\partial}{\partial x_{1}}\gamma_{4}\,-\,\frac{\partial}{\partial x_{2}}\gamma_{3}\,-\,\frac{\partial}{\partial x_{3}}\gamma_{2}\,-\,\frac{\partial}{\partial x_{4}}\gamma_{1}\,.$ By means of the decomposition $S=S^{+}\oplus S^{-}$ the Dirac operator has the chiral decomposition: $\mathcal{D}=\begin{pmatrix}0&D^{\dagger}\\\ D&0\end{pmatrix}:C^{\infty}(\mathbf{C}^{2},S^{+}\oplus S^{-})\rightarrow C^{\infty}(\mathbf{C}^{2},S^{+}\oplus S^{-}).$ If we adopt the notation $\frac{\partial}{\partial z_{1}}=\frac{\partial}{\partial x_{1}}-i\frac{\partial}{\partial x_{2}}\,,\quad\frac{\partial}{\partial z_{2}}=\frac{\partial}{\partial x_{3}}-i\frac{\partial}{\partial x_{4}}\,,$ $D$ and $D^{\dagger}$ have the following coordinate expressions; $D=\begin{pmatrix}\frac{\partial}{\partial z_{1}}&-\frac{\partial}{\partial\bar{z_{2}}}\\\ \\\ \frac{\partial}{\partial z_{2}}&\frac{\partial}{\partial\bar{z_{1}}}\end{pmatrix},\quad D^{\dagger}=\begin{pmatrix}\frac{\partial}{\partial\bar{z_{1}}}&\frac{\partial}{\partial\bar{z_{2}}}\\\ \\\ -\frac{\partial}{\partial z_{2}}&\frac{\partial}{\partial z_{1}}\end{pmatrix}.$ ### 2.2 Harmonic spinors #### 2.2.1 harmonic polynomials on $S^{3}\subset\mathbf{C}^{2}$ The right action of $SU(2)$ on $\mathbf{C}^{2}$ is written by $R_{g}z\,=\,\left(\begin{array}[]{c}\,az_{1}-b\overline{z}_{2}\\\ az_{2}+b\overline{z}_{1}\end{array}\right),\quad g=\left(\begin{array}[]{cc}a&-\overline{b}\\\ b&\overline{a}\end{array}\right)\in SU(2),\,z=\left(\begin{array}[]{c}z_{1}\\\ z_{2}\end{array}\right)\in\mathbf{C}^{2}.$ Then the infinitesimal action of $su(2)$ on $\mathbf{C}^{2}$ is $((dR_{e})X)F=\frac{d}{dt}\|_{t=0}R_{\exp\,tX}F\,,\quad X\in su(2)\,.$ It yields the following basis of vector fields $(\theta_{0},\theta_{1},\theta_{2})$ on $\\{\|z\|=1\\}\simeq S^{3}$ : $\theta_{1}=\frac{1}{\sqrt{-1}}dR(\sigma_{2})\,,\,\,\theta_{2}=\frac{1}{\sqrt{-1}}dR(\sigma_{1})\,,\,\,\theta_{0}=-\frac{1}{\sqrt{-1}}dR(\sigma_{3})\,.$ (2.4) We prefer often the following basis $(e_{+},e_{-},\theta)\,$ given by $\theta_{0}=\sqrt{-1}\theta\,,\quad\theta_{1}=e_{+}+e_{-}\,,\quad\theta_{2}=\sqrt{-1}(e_{+}-e_{-})\,.$ (2.5) The local coordinate expression of these vector fields becomes: $\displaystyle e_{+}$ $\displaystyle=$ $\displaystyle- z_{2}\frac{\partial}{\partial\bar{z_{1}}}+z_{1}\frac{\partial}{\partial\bar{z_{2}}},\quad e_{-}=-\bar{z_{2}}\frac{\partial}{\partial z_{1}}+\bar{z_{1}}\frac{\partial}{\partial z_{2}}$ (2.6) $\displaystyle\theta$ $\displaystyle=$ $\displaystyle\left(z_{1}\frac{\partial}{\partial z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}-\bar{z_{1}}\frac{\partial}{\partial\bar{z_{1}}}-\bar{z_{2}}\frac{\partial}{\partial\bar{z_{2}}}\right)\,,$ (2.7) and the following commutation relations hold; $[\theta,e_{+}]=2e_{+},\quad[\theta,e_{-}]=-2e_{-},\quad[e_{+},e_{-}]=-\theta.$ The dual basis are given by the differential 1-forms: $\displaystyle\theta_{0}^{\ast}$ $\displaystyle=$ $\displaystyle\frac{1}{2\sqrt{-1}\|z\|^{2}}(\overline{z}_{1}dz_{1}+\overline{z}_{2}dz_{2}-z_{1}d\overline{z}_{1}-z_{2}d\overline{z}_{2}),$ $\displaystyle\theta_{1}^{\ast}$ $\displaystyle=$ $\displaystyle\frac{1}{2\|z\|^{2}}(e_{+}^{\ast}+e_{-}^{\ast})\,,\qquad\theta_{2}^{\ast}=\frac{1}{2\sqrt{-1}\|z\|^{2}}(e_{+}^{\ast}-e_{-}^{\ast})\,,$ where $e_{+}^{\ast}=(-\overline{z}_{2}d\overline{z}_{1}+\overline{z}_{1}d\overline{z}_{2})\,,\quad e_{-}^{\ast}=(-z_{2}dz_{1}+z_{1}dz_{2})\,,$ where we wrote the formulae extended to $\mathbf{C}^{2}\setminus\\{0\\}$. $\theta^{\ast}_{k}$, $k=0,1,2$, are real 1-forms: $\,\overline{\theta}_{k}^{\ast}=\theta_{k}^{\ast}\,$. It holds that $\theta_{j}^{\ast}(\theta_{k})=\delta_{jk}\,$ for $\,j,k=0,1,2\,$. The integrability condition becomes $\frac{\sqrt{-1}}{2}d\theta_{0}^{\ast}=\theta_{1}^{\ast}\wedge\theta_{2}^{\ast}\,,\quad\frac{\sqrt{-1}}{2}d\theta_{1}^{\ast}=\theta_{2}^{\ast}\wedge\theta_{0}^{\ast}\,,\quad\frac{\sqrt{-1}}{2}d\theta_{2}^{\ast}=\theta_{0}^{\ast}\wedge\theta_{1}^{\ast}\,,$ (2.8) and $\theta_{0}^{\ast}\wedge\theta_{1}^{\ast}\wedge\theta_{2}^{\ast}=d\sigma_{S^{3}}\,$ is the volume form on $S^{3}$. In the following we denote a function $f(z,\bar{z})$ of variables $z,\bar{z}\,$ simply by $f(z)$. ###### Definition 2.1. For $m=0,1,2,\cdots$, and $l,k=0,1,\cdots,m$, we define the monomials: $\displaystyle v^{k}_{(l,m-l)}$ $\displaystyle=$ $\displaystyle(e_{-})^{k}z^{l}_{1}z^{m-l}_{2}.$ (2.9) $\displaystyle w^{k}_{(l,m-l)}$ $\displaystyle=$ $\displaystyle(-1)^{k}\frac{l!}{(m-k)!}\,v^{m-l}_{(k,m-k)}\,.$ (2.10) We note that the monomials $v^{k}_{(l,m-l)}$ in (2.9) come naturally from the calculations of the right action of $SU(2)$ on $\mathbf{C}^{2}$, so as the monomials $w^{k}_{(l,m-l)}$ are obtained by the left action of $SU(2)$ on $\mathbf{C}^{2}\setminus\\{0\\}$, [Ko0, Ko1]. ###### Proposition 2.2. 1. 1. $v^{k}_{(l,m-l)}$ are harmonic polynomials on $\mathbf{C}^{2}$; $\Delta v^{k}_{(l,m-l)}=0\,$, where $\Delta=\frac{\partial^{2}}{\partial z_{1}\partial\bar{z}_{1}}+\frac{\partial^{2}}{\partial z_{2}\partial\bar{z}_{2}}$. 2. 2. $\left\\{\,\frac{1}{\sqrt{2}\pi}v^{k}_{(l,m-l)}\,;\,m=0,1,\cdots,\,0\leq k,l\leq m\,\right\\}$ forms a $L^{2}(S^{3})$-complete orthonormal system of the space of harmonic polynomials. The similar assertions hold for $\left\\{\,\frac{1}{\sqrt{2}\pi}w^{k}_{(l,m-l)}\,;\,m=0,1,\cdots,\,0\leq k,l\leq m\,\right\\}$. 3. 3. For each pair $(m,l)$, $0\leq l\leq m$, the subspace $H_{(m,l)}=\\{v^{k}_{(l,m-l)}\,;0\leq k\leq m+1\\}$ gives a $(m+1)$-dimensional right representation of $su(2)$ with the highest weight $\frac{m}{2}$. 4. 4. For each pair $(m,l)$, $0\leq l\leq m$, the subspace $H^{{\dagger}}_{(m,l)}=\\{w^{k}_{(l,m-l)}\,;0\leq k\leq m+1\\}$ gives a $(m+1)$-dimensional left representation of $su(2)$ with the highest weight $\frac{m}{2}$. ###### Proposition 2.3. The set of harmonic polynomials on $S^{3}$ form a graded algebra. The proof will be found in the proof of Theorem 3.2 of the next section. #### 2.2.2 Harmonic spinors on $S^{3}\subset\mathbf{C}^{2}$ The radial vector field is defined by $\frac{\partial}{\partial n}=\frac{1}{2\|z\|}(\nu+\bar{\nu}),\qquad\nu=z_{1}\frac{\partial}{\partial z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}.$ We shall denote by $\gamma$ the Clifford multiplication of the radial vector $\frac{\partial}{\partial n}\,$. The multiplication $\gamma$ changes the chirality: $\gamma=\gamma_{+}\oplus\gamma_{-}:S^{+}\oplus S^{-}\longrightarrow S^{-}\oplus S^{+}$, and $\gamma^{2}=1$. ###### Proposition 2.4. [Ko1] The Dirac operators $D$ and $D^{\dagger}$ have the following polar decompositions: $\displaystyle D$ $\displaystyle=\gamma_{+}\left(\frac{\partial}{\partial n}-\partial\\!\\!\\!/\right)\,:$ $\displaystyle\,S^{+}\longrightarrow\,S^{-},$ $\displaystyle D^{\dagger}$ $\displaystyle=\left(\frac{\partial}{\partial n}+\partial\\!\\!\\!/+\frac{3}{2\|z\|}\right)\gamma_{-}\,\,:$ $\displaystyle\,S^{-}\longrightarrow\,S^{+}\,,$ where the non-chiral Dirac operator $\partial\\!\\!\\!/$ is given by $\partial\\!\\!\\!/=-\left[\sum^{3}_{i=1}\left(\frac{1}{\|z\|}\theta_{i}\right)\cdot\nabla_{\frac{1}{\|z\|}\theta_{i}}\right]=\frac{1}{\|z\|}\begin{pmatrix}-\frac{1}{2}\theta&\,e_{+}\\\\[5.69046pt] -e_{-}&\,\frac{1}{2}\theta\end{pmatrix}.$ $\partial\\!\\!\\!/$ restricted on $S^{3}=\\{\|z\|=1\\}$ is called the tangential Dirac operator: $\partial\\!\\!\\!/\|S^{3}:C^{\infty}(S^{3},S^{+})\longrightarrow C^{\infty}(S^{3},S^{+})$ The tangential Dirac operator on $S^{3}$ is a self adjoint elliptic differential operator. Now we introduce a basis of the space of even harmonic spinors by the following formula. ###### Definition 2.5. For $m=0,1,2,\cdots;l=0,1,\cdots,m$ and $k=0,1,\cdots,m+1$, we put $\displaystyle\phi^{+(m,l,k)}(z)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{(m+1-k)!}{k!l!(m-l)!}}\begin{pmatrix}kv^{k-1}_{(l,m-l)}\\\ \\\ -v^{k}_{(l,m-l)}\end{pmatrix},$ $\displaystyle\phi^{-(m,l,k)}(z)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{(m+1-k)!}{k!l!(m-l)!}}\left(\frac{1}{\|z\|^{2}}\right)^{m+2}\begin{pmatrix}w^{k}_{(m+1-l,l)}\\\ \\\ w^{k}_{(m-l,l+1)}\end{pmatrix}.$ (2.11) We have the following ###### Proposition 2.6. [Ko1] 1. 1. $\phi^{+(m,l,k)}$ is a harmonic spinor on $\mathbf{C}^{2}$ and $\phi^{-(m,l,k)}$ is a harmonic spinor on $\mathbf{C}^{2}\backslash\\{0\\}$ that is regular at infinity. 2. 2. On $S^{3}=\\{\|z\|=1\\}$ we have: $\partial\\!\\!\\!/\phi^{+(m,l,k)}=\frac{m}{2}\phi^{+(m,l,k)}\,,\qquad\partial\\!\\!\\!/\phi^{-(m,l,k)}=-\frac{m+3}{2}\phi^{-(m,l,k)}\,.$ 3. 3. The eigenvalues of $\,\partial\\!\\!\\!/$ are $\frac{m}{2}\,,\quad-\frac{m+3}{2}\,;\quad m=0,1,\cdots,$ and the multiplicity of each eigenvalue is equal to $(m+1)(m+2)$. 4. 4. The set of eigenspinors $\left\\{\frac{1}{\sqrt{2}\pi}\phi^{+(m,l,k)},\quad\frac{1}{\sqrt{2}\pi}\phi^{-(m,l,k)}\,;\quad m=0,1,\cdots,\,0\leq l\leq m,\,0\leq k\leq m+1\right\\}$ forms a complete orthonormal system of $L^{2}(S^{3},S^{+})$. The Cauchy kernel ( fundamental solution ) of the half Dirac operator $D:C^{\infty}(\mathbf{C}^{2},\,S^{+})\longrightarrow C^{\infty}(\mathbf{C}^{2},\,S^{-})$ is given by $K^{{\dagger}}(z,\zeta)=\frac{1}{\|\zeta-z\|^{3}}\gamma_{-}(\zeta-z)\,:C^{\infty}(\mathbf{C}^{2},\,S^{+})\longrightarrow C^{\infty}(\mathbf{C}^{2},\,S^{-}),\quad\|z-c\|<\|\zeta-c\|\,.$ We have the following integral representation of spinors: ###### Theorem 2.7. [Ko1] Let $G$ be a domain of $\mathbf{C}^{2}$ and let $\varphi\in C^{\infty}(\overline{G},\,S^{+})$. We have $\varphi(z)=-\frac{1}{2\pi^{2}}\int_{G}\,K^{{\dagger}}(z,\zeta)D\varphi(\zeta)dv(\zeta)+\frac{1}{2\pi^{2}}\int_{\partial G}\,K^{{\dagger}}(z,\zeta)(\gamma_{+}\varphi)(\zeta)d\sigma(\zeta)\,,\quad z\in G.$ The Cauchy kernel has the following eigenfunction expansion by the basis $\phi^{(\pm(m,l,k)}(z-c)$: ###### Theorem 2.8. [Ko1, Ko2] For $\|z-c\|<\|\zeta-c\|\,$, $K^{{\dagger}}(z,\zeta)\cdot\gamma_{+}(\zeta-c)=\sum_{\,m,l,k\,}\,\|\zeta-c\|^{-(2m+3)}\overline{\phi^{+(m,l,k)}(\zeta-c)}\otimes\phi^{+(m,l,k)}(z-c)\,.$ That is, the Cauchy kernel and the Bergman kernel on the 4-disc $\|z\|\leq 1$ coincide. ## 3 2-cocycles on the space of spinors over $S^{3}$ We shall introduce a triple of 2-cocycles on the space of smooth spinors on $S^{3}$, then on the space of Laurent polynomial type harmonic spinors. We shall further introduce a 2-cocycle coming from the radial derivation of spinors. ### 3.1 Algebra of Laurent polynomial type harmonic spinors on $S^{3}$ #### 3.1.1 The space of even spinors $\Delta^{+}$ is isomorphic to the quaternion vector space $\mathbf{H}$, and we have an identification $C^{\infty}(S^{3},\,S^{+})\simeq C^{\infty}(S^{3},\mathbf{H})$, (2.3). Hence the multiplication of two even spinors is defined by $\phi_{1}\cdot\phi_{2}\,=\,\,\left(\begin{array}[]{c}u_{1}u_{2}-\overline{v}_{1}v_{2}\\\ v_{1}u_{2}+\overline{u}_{1}v_{2}\end{array}\right)\,,$ (3.1) for $\phi=\left(\begin{array}[]{c}u_{i}\\\ v_{i}\end{array}\right)$, $i=1,2$. It corresponds to the quaternion multiplication: $(u_{1}+jv_{1})(u_{2}+jv_{2})=(u_{1}u_{2}-\overline{v}_{1}v_{2})+j(v_{1}u_{2}+\overline{u}_{1}v_{2}).$ With this multiplication the $\mathbf{C}$-vector space $C^{\infty}(S^{3},\,S^{+})$ becomes an associative $\mathbf{R}$-algebra. We have the Laurent expansion of harmonic spinors, that is, a harmonic spinor $\varphi$ on $\mathbf{C}^{2}\setminus\\{0\\}$ has an expansion by the basic spinors $\\{\,\phi^{\pm(m,l,k)}\\}_{m,l,k}\,$: $\varphi(z)=\sum_{m,l,k}\,C_{+(m,l,k)}\phi^{+(m,l,k)}(z)+\sum_{m,l,k}\,C_{-(m,l,k)}\phi^{-(m,l,k)}(z),$ (3.2) which is uniformly convergent on any compact subset of $\mathbf{C}^{2}\setminus\\{0\\}$. The coefficients $C_{\pm(m,l,k)}$ are given by the formula: $C_{\pm(m,l,k)}=\,\frac{1}{2\pi^{2}}\int_{S^{3}}\,\langle\varphi,\,\phi^{\pm(m,l,k)}\rangle\,d\sigma,$ where $\langle\,,\,\rangle$ is the inner product of $S^{+}$. We have $\int_{S^{3}}\,tr\,\varphi\,d\sigma=4\pi^{2}Re.C_{+(0,0,1)},$ (3.3) $Re.$ designates the real part. ###### Definition 3.1. We call the series (3.2) a spinor of Laurent polynomial type if only finitely many coefficients $C_{-(m,l,k)}$ are non-zero . The space of spinors of Laurent polynomial type is denoted by $\mathbf{C}[\phi^{\pm}]$. ###### Theorem 3.2. The restriction of $\,\mathbf{C}[\phi^{\pm}\,]$ to $S^{3}$ is an associative subalgebra of $C^{\infty}(S^{3},\,S^{+})=C^{\infty}(S^{3},\mathbf{H})$ generated by the spinors: $\displaystyle I$ $\displaystyle=\phi^{+(0,0,1)}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\quad J$ $\displaystyle=-\,\phi^{+(0,0,0)}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right),$ $\displaystyle\kappa$ $\displaystyle=\phi^{+(1,0,1)}=\left(\begin{array}[]{c}z_{2}\\\ -\overline{z}_{1}\end{array}\right),\quad\mu$ $\displaystyle=\phi^{-(0,0,0)}=\left(\begin{array}[]{c}z_{2}\\\ \overline{z}_{1}.\end{array}\right).$ ###### Proof. In Lemma 4.1 of [Ko0] we proved the product formula for the harmonic polynomials $v^{k}_{(a.b)}\,$: $v^{k_{1}}_{(a_{1},b_{1})}v^{k_{2}}_{(a_{2},b_{2})}=\sum_{j=0}^{a_{1}+a_{2}+b_{1}+b_{2}}C_{j}\|z\|^{2j}\,v^{k_{1}+k_{2}-j}_{(a_{1}+a_{2}-j,\,b_{1}+b_{2}-j)}\,,$ (3.6) for some rational numbers $C_{j}=C_{j}(a_{1},a_{2},b_{1},b_{2},k_{1},k_{2})$. Let $k=k_{1}+k_{2}$, $a=a_{1}+a_{2}$ and $b=b_{1}+b_{2}$. The above product formula yields the fact that, restricted to $S^{3}$, the harmonic polynomial $v^{k}_{(a,b)}$ is equal to a constant multiple of $\,v^{k_{1}}_{(a_{1},b_{1})}\cdot v^{k_{2}}_{(a_{2},b_{2})}$ modulo a linear combination of polynomials $v^{k-j}_{(a-j,b-j)}\,$, $1\leq j\leq min(k,a,b)$. Hence the set of harmonic polynomials form a graded algebra. On the other hand we see that a spinor of the form $\left(\begin{array}[]{c}v^{k}_{(l,m-l)}\\\ 0\end{array}\right)$ or $\left(\begin{array}[]{c}0\\\ v^{k+1}_{(l,m-l)}\end{array}\right)$ is written by a linear combinations of $\phi^{+(m,l,k+1)}$ and $\phi^{-(m-1,k,l)}$. Therefore we find that any product of two spinors $\phi^{\pm(m_{1},l_{1},k_{1})}\cdot\phi^{\pm(m_{2},l_{2},k_{2})}$ is written as a linear combination of $\phi^{\pm(m_{1}+m_{2}-n,\cdot,\cdot)}$, $1\leq n\leq m_{1}+m_{2}$. Therefore $\mathbf{C}[\phi^{\pm}]\|_{S^{3}}$ becomes an associative algebra. Moreover $\phi^{\pm(m,l,k)}$ is written by a linear combination of the products $\,\phi^{\pm(m_{1},l_{1},k_{1})}\cdot\phi^{\pm(m_{2},l_{2},k_{2})}$ for $0\leq m_{1}+m_{2}\leq m-1\,$, $0\leq l_{1}+l_{2}\leq l$ and $0\leq k_{1}+k_{2}\leq k$ . Hence we find that the algebra $\mathbf{C}[\phi^{\pm}]\|_{S^{3}}$ is graded and is generated by the four spinors $I=\phi^{+(0,0,1)}\,,\,J=-\phi^{+(0,0,0)}\,,\,\kappa=\phi^{+(1,0,1)}\,,\,\mu=\phi^{-(0,0,0)}\,$ ∎ Examples $\displaystyle\phi^{+(1,1,1)}$ $\displaystyle=$ $\displaystyle\,-\kappa\,J\,=\left(\begin{array}[]{c}z_{1}\\\ \overline{z}_{2}\end{array}\right)\,,\quad\phi^{-(0,0,1)}=-\mu J=\left(\begin{array}[]{c}-z_{1}\\\ \overline{z}_{2}\end{array}\right)\,$ $\displaystyle\phi^{+(1,0,0)}$ $\displaystyle=$ $\displaystyle\sqrt{2}\left(\begin{array}[]{c}0\\\ -z_{2}\end{array}\right)=\frac{1}{\sqrt{2}}J(\kappa+\mu).$ $\displaystyle\phi^{-(1,1,1)}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\mu\cdot\left(-\kappa+\mu+J(\lambda+\nu)\right)=\left(\begin{array}[]{c}\|z_{2}\|^{2}-\|z_{1}\|^{2}\\\ 2\overline{z}_{1}\overline{z}_{2}\end{array}\right)\,,\qquad\mbox{ for }\,\|z\|=1\,.$ We must note that $\mathbf{C}[\,\phi^{\pm}\,]$ over $\mathbf{C}^{2}\setminus\\{0\\}$ is not an algebra because in the formula (3.6) $\|z\|\neq 1$ out of $S^{3}$. ###### Corollary 3.3. Let $\tau$, $\sigma$ be the involutions on $C^{\infty}(S^{3},S^{+})$ defined by $\tau\phi=\left(\begin{array}[]{c}\overline{u}\\\ \overline{v}\end{array}\right)\,,\quad\sigma\phi=\left(\begin{array}[]{c}u\\\ -v\end{array}\right)\,\quad\mbox{ for }\,\phi=\left(\begin{array}[]{c}u\\\ v\end{array}\right)\,.$ Then the involutions $\tau$ and $\sigma$ are homomorphisms of $\mathbf{R}$-algebra $\,\mathbf{C}[\phi^{\pm}]\|_{S^{3}}$. In fact, since $\left(\begin{array}[]{c}v^{k}_{(l,m-l)}\\\ 0\end{array}\right)$ and $\left(\begin{array}[]{c}0\\\ v^{k+1}_{(l,m-l)}\end{array}\right)$ are written by linear combinations of $\phi^{+(m,l,k+1)}$ and $\phi^{-(m-1,k,l)}$, it certainly follows that $\sigma\phi^{\pm(m-1,k,l)}\in\mathbf{C}[\phi^{\pm}]$. By virtue of the property $\,\overline{v}^{k}_{m,l}=(-1)^{m-l-k}\frac{k!}{(m-k)!}v^{m-k}_{m,m-l}\,$, $\,\tau$ is also a homomorphism: $\,\sigma\phi_{1}\cdot\sigma\phi_{2}=\sigma(\phi_{1}\cdot\phi_{2})\,,\quad\tau\phi_{1}\cdot\tau\phi_{2}=\tau(\phi_{1}\cdot\phi_{2})\,.$ #### 3.1.2 Now we introduce the following $\mathbf{R}$-bilinear bracket on $C^{\infty}(S^{3},S^{+})$: $\left[\phi_{1},\,\phi_{2}\,\right]\,=\,\bigl{[}\,\begin{pmatrix}u_{1}\\\ v_{1}\end{pmatrix},\,\begin{pmatrix}u_{2}\\\ v_{2}\end{pmatrix}\,\bigr{]}\,=\,\begin{pmatrix}v_{1}\bar{v}_{2}-\bar{v}_{1}v_{2}\,\\\ \,(u_{2}-\bar{u}_{2})v_{1}-(u_{1}-\bar{u}_{1})v_{2}\,\end{pmatrix},$ (3.10) for even spinors $\phi_{1}=\begin{pmatrix}u_{1}\\\ v_{1}\end{pmatrix}$ and $\phi_{2}=\begin{pmatrix}u_{2}\\\ v_{2}\end{pmatrix}$. From Theorem 3.2, Corollary3.3 and (3.10) we have the following ###### Proposition 3.4. $\left(\,C^{\infty}(S^{3},S^{+}),\,[\,,\,\,]\,\right)\,$ is a quaternion Lie algebra. $\left(\,\mathbf{C}[\phi^{\pm}]\|_{S^{3}}\,,\,[\,,\,]\right)\,$ is a quaternion Lie subalgebra of $(C^{\infty}(S^{3},S^{+}),\,[\,,\,]\,)$. ### 3.2 2-cocycles on $C^{\infty}(S^{3},S^{+})$. Let $\phi\,,\,\psi\,\in C^{\infty}(S^{3},S^{+})$. We define the trace of a spinor $\phi=\begin{pmatrix}u\\\ v\end{pmatrix}$ by the formula: $tr\,\phi\,=\,u+\overline{u}.$ It is invariant by the involutions $\sigma$ and $\tau$. Evidently we have $tr\,[\phi,\psi]=0$. In the following we introduce three 2-cocycles on $C^{\infty}(S^{3},S^{+})$ that come from the base vector fields $\theta_{k}\,;\,k=0,1,2$, on $S^{3}$, (2.5). ###### Definition 3.5. For a $\varphi=\left(\begin{array}[]{c}u\\\ v\end{array}\right)\in C^{\infty}(S^{3},S^{+})$ , we put $\Theta_{k}\,\varphi\,=\,\frac{1}{2}\,\left(\begin{array}[]{c}\,\theta_{k}\,u\\\\[8.5359pt] \,\theta_{k}\,v\end{array}\right),\qquad k=0,1,2.$ Note that $\theta_{k}$ is a real vector field: $\,\theta_{k}=\overline{\theta}_{k}$, so is $\Theta_{k}$. ###### Lemma 3.6. For any $k=0,1,2$, and $\phi,\,\psi\in C^{\infty}(S^{3},S^{+})$, we have $\displaystyle\Theta_{k}\,(\phi\cdot\psi\,)\,$ $\displaystyle=$ $\displaystyle\,(\Theta_{k}\,\phi)\cdot\,\psi\,+\,\phi\cdot\,(\Theta_{k}\,\psi)\,.$ (3.11) $\displaystyle\int_{S^{3}}\,\Theta_{k}\,\varphi\,d\sigma\,$ $\displaystyle=$ $\displaystyle\,0.$ (3.12) ###### Proof. The first equation follows from the fact: $\overline{\theta}_{k}=\theta_{k}$. The second assertion follows from the fact $\int_{S^{3}}\,\theta_{k}f\,d\sigma\,=\,0\,,$ (3.13) for any function $f$ on $S^{3}$. This is proved as follows. We consider the 2-form $\beta=f\theta_{1}^{\ast}\wedge\theta_{2}^{\ast}$. By virtue of the integrable condition (2.8) we have $d\beta=(\theta_{0}f)\,\theta_{0}^{\ast}\wedge\theta_{1}^{\ast}\wedge\theta_{2}^{\ast}=\theta_{0}f\,d\sigma\,.$ Hence $0=\int_{S^{3}}\,d\beta\,=\,\int_{S^{3}}\theta_{0}f\,d\sigma.$ Similarly for the integral of $\theta_{k}f$, $k=1,2$. ∎ ###### Remark 3.7. The formula (3.13) is an evident fact if we recognize the invariance under the action of $SO(4)$ of each $\theta_{k}$ and the volume form $d\sigma$ . This is pointed to me by Professor T. Iwai of Kyoto University. ###### Definition 3.8. For $\phi_{1}$ and $\phi_{2}\in C^{\infty}(S^{3},S^{+})$, we put $c_{k}(\phi_{1},\phi_{2})\,=\,\,\frac{1}{2\pi^{2}}\int_{S^{3}}\,\,tr\,(\,\Theta_{k}\phi_{1}\cdot\phi_{2}\,)\,d\sigma\,,\quad k=0,1,2\,.$ ###### Proposition 3.9. 1. 1. For each $k=0,1,2$, $c_{k}$ defines a 2-cocycle on the $\mathbf{R}$-algebra $C^{\infty}(S^{3},S^{+})\,$. That is, $c_{k}$ satisfies the equations: $\displaystyle c_{k}(\phi_{1}\,,\,\phi_{2})\,=\,-\,c_{k}(\phi_{2}\,,\,\phi_{1})\,,$ (3.14) $\displaystyle c_{k}(\phi_{1}\cdot\phi_{2}\,,\,\phi_{3})+c_{k}(\phi_{2}\cdot\phi_{3}\,,\,\phi_{1}\,)+c_{k}(\phi_{3}\cdot\phi_{1}\,,\,\phi_{2}\,)=0,$ (3.15) for any $\phi_{1},\,\phi_{2},\,\phi_{3}\in C^{\infty}(S^{3},S^{+})$. 2. 2. For each $k=0,1,2$, $c_{k}$ defines a 2-cocycle on the real Lie algebra $C^{\infty}(S^{3},S^{+})\,$. That is, $c_{k}$ satisfies the equations: $\displaystyle c_{k}(\phi_{1}\,,\,\phi_{2})\,=\,-\,c_{k}(\phi_{2}\,,\,\phi_{1})\,,$ (3.16) $\displaystyle c_{k}(\,[\phi_{1}\,,\phi_{2}]\,,\,\phi_{3})+c_{k}(\,[\phi_{2}\,,\phi_{3}]\,,\,\phi_{1}\,)+c_{k}(\,[\phi_{3}\,,\phi_{1}]\,,\,\phi_{2}\,)=0,$ (3.17) for any $\phi_{1},\,\phi_{2},\,\phi_{3}\in C^{\infty}(S^{3},S^{+})$. 3. 3. $c_{k}$ is a non-trivial 2-cocycle, that is, there is no 1-cochain $b$ such that $c_{k}(\phi_{1}\,,\phi_{2})=b(\,[\phi_{1},\,\phi_{2}]\,)$. 4. 4. Each $c_{k}$ is invariant under the involutions $\sigma$ and $\tau$. Each 2-cocycle $c_{k}\,$, $\,k=0,1,2$, restricts to the space $\mathbf{C}[\phi^{\pm}]\|S^{3}$. ###### Proof. Evidently each $c_{k}$ is $\mathbf{R}$-bilinear ( It is not $\mathbf{C}$-bilinear ). By the formula (3.12) and the Leibnitz rule (3.19) we have $\displaystyle 0$ $\displaystyle=$ $\displaystyle\,\int_{S^{3}}\,\,tr\,(\,\Theta_{k}\,(\phi_{1}\cdot\phi_{2})\,)\,d\sigma\,=\,\int_{S^{3}}\,tr\,\left(\,\Theta_{k}\,\phi_{1}\,\cdot\phi_{2}\,\right)d\sigma\,+\,\int_{S^{3}}\,tr\,\left(\,\phi_{1}\cdot\,\Theta_{k}\,\phi_{2}\,\right)d\sigma$ Hence $\,c_{k}(\phi_{1}\,,\,\phi_{2}\,)\,+\,c_{k}(\,\phi_{2}\,,\,\phi_{1}\,)\,=0\,$. The following calculation proves (3.15). $\displaystyle c_{k}(\phi_{1}\cdot\phi_{2}\,,\,\phi_{3})$ $\displaystyle=$ $\displaystyle\,\int_{S^{3}}\,\,tr\,(\,\Theta_{k}(\,\phi_{1}\cdot\phi_{2}\,)\cdot\,\phi_{3}\,)\,d\sigma$ $\displaystyle=$ $\displaystyle\,\int_{S^{3}}\,\,tr\,(\,\Theta_{k}\phi_{1}\cdot\phi_{2}\cdot\phi_{3}\,)d\sigma\,+\,\,\int_{S^{3}}\,\,tr\,(\,\Theta_{k}\phi_{2}\cdot\,\phi_{3}\,\cdot\phi_{1}\,)d\sigma$ $\displaystyle=$ $\displaystyle\,c_{k}(\phi_{1}\,,\,\phi_{2}\cdot\phi_{3}\,)+c_{k}(\phi_{2}\,,\,\phi_{3}\cdot\phi_{1}\,)=\,-c_{k}(\phi_{2}\cdot\phi_{3}\,,\phi_{1}\,)\,-\,c_{k}(\phi_{3}\cdot\phi_{1}\,,\,\phi_{2}).$ Suppose now that $c_{0}$ is the coboundary of a 1-cochain $b:\,C^{\infty}(S^{3},S^{+})\longrightarrow\mathbf{C}$. Then $c_{0}(\phi_{1},\,\phi_{2})=(\delta\,b)(\phi_{1},\,\phi_{2})\,=\,b(\,[\phi_{1},\,\phi_{2}]\,)$ for any $\phi_{1},\phi_{2}\in C^{\infty}(S^{3},S^{+})$. Take $\phi_{1}=\,\frac{1}{\sqrt{2}}\phi^{+(1,1,2)}\,=\left(\begin{array}[]{c}-\overline{z}_{2}\\\ 0\end{array}\right)$ and $\,\phi_{2}=\frac{1}{2}(\phi^{+(1,0,1)}+\phi^{-(0,0,0)})=\left(\begin{array}[]{c}z_{2}\\\ 0\end{array}\right)$ . Then $[\,\phi_{1},\,\phi_{2}\,]=0$, so $(\delta b)(\phi_{1},\phi_{2})=0$. But $c_{0}(\phi_{1},\phi_{2})=\frac{1}{2}\,$. Therefore $c_{0}$ can not be a coboundary. For $\phi_{1}$ and $\phi_{3}=\phi^{+(1,0,2)}=\sqrt{2}\left(\begin{array}[]{c}\overline{z}_{1}\\\ 0\end{array}\right)$, we have $[\phi_{1},\phi_{3}]=0$ and $c_{1}(\phi_{1},\phi_{3})=-\frac{1}{\sqrt{2}}$. So $c_{1}$ can not be a coboundary by the same reason as above. Similarly for $c_{2}$. ∎ Examples 1. 1. $c_{0}(\phi^{\pm(m,l,k)},\,\phi^{\pm(p,q,r)})=0,\quad c_{0}(\,\phi^{+(1,1,2)}\,,\,\sqrt{-1}(\phi^{+(1,0,1)}+\phi^{-(0,0,0)})\,)\,=\sqrt{2}\,.$ 2. 2. Let $\kappa=\phi^{+(1,0,1)}=\left(\begin{array}[]{c}z_{2}\\\ -\overline{z}_{1}\end{array}\right),\quad\kappa_{\ast}=\,\frac{-\sqrt{-1}}{\sqrt{2}}(\phi^{-(0,0,0)}-\phi^{+(1,1,2)}-\phi^{+(1,0,1)})=\sqrt{-1}\left(\begin{array}[]{c}\overline{z}_{2}\\\ \overline{z}_{1}\end{array}\right).$ Then $(\Theta_{0}\,\kappa)\cdot\kappa_{\ast}\,=\,-\frac{1}{2}\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$ and $c_{0}(\kappa,\,\kappa_{\ast})=\frac{1}{2\pi^{2}}\int_{S^{3}}\,tr\,[(\Theta_{0}\kappa)\cdot\kappa_{\ast}\,]\,d\sigma_{3}=-1.$ Similarly $c_{1}(\kappa,\,\kappa_{\ast})=c_{2}(\kappa,\,\kappa_{\ast})=0$ ### 3.3 Radial derivative on $C^{\infty}(S^{3},S^{+})$ We define the following operator $\mathbf{n}$ on $C^{\infty}(S^{3})$: $\mathbf{n}\,f(z)=\|z\|\frac{\partial}{\partial n}f(z)\,=\,\frac{1}{2}(\nu+\bar{\nu})f(z)\,.$ (3.18) Here we consider the radial derivative of a function on $\mathbf{C}^{2}$ and then restrict it to $S^{3}=\\{\|z\|=1\\}$. For an even spinor $\varphi=\left(\begin{array}[]{c}u\\\ v\end{array}\right)\in C^{\infty}(S^{3},S^{+})$, we put $\mathbf{n}\,\varphi\,=\,\left(\begin{array}[]{c}\mathbf{n}\,u\\\\[5.69046pt] \mathbf{n}\,v\end{array}\right).$ ###### Proposition 3.10. 1. 1. $\mathbf{n}(\phi_{1}\cdot\phi_{2})=(\mathbf{n}\phi_{1})\cdot\phi_{2}+\phi_{1}\cdot(\mathbf{n}\phi_{2})\,.$ (3.19) $\mathbf{n}\,[\,\phi_{1}\,,\,\phi_{2}\,]\,=\,[\,\mathbf{n}\phi_{1}\,,\,\phi_{2}\,]\,+\,[\,\phi_{1}\,,\,\mathbf{n}\phi_{2}\,]\,.$ (3.20) 2. 2. $\mathbf{n}\phi^{+(m,l,k)}=\frac{m}{2}\,\phi^{+(m,l,k)},\quad\mathbf{n}\phi^{-(m,l,k)}=\,-\frac{m+3}{2}\,\phi^{-(m,l,k)}.$ (3.21) 3. 3. If $\varphi$ is a spinor of Laurent polynomial type: $\varphi(z)=\sum_{m,l,k}\,C_{+(m,l,k)}\phi^{+(m,l,k)}(z)+\sum_{m,l,k}\,C_{-(m,l,k)}\phi^{-(m,l,k)}(z).$ then $\mathbf{n}\varphi$ is also a spinor of Laurent polynomial type and we have $\int_{S^{3}}\,tr\,(\,\mathbf{n}\,\varphi\,)\,d\sigma\,=\,0\,.$ (3.22) ###### Proof. The formula (3.21) follows from the definition (2.5). The last assertion follows from (3.3) and the fact that the coefficient of $\phi^{+(0,0,1)}$ in the Laurent expansion of $\mathbf{n}\varphi$ vanishes. ∎ Therefore the derivations $\Theta_{k}$, $\,k=0,1,2$, and $\mathbf{n}$ act on the space of Laurent polynomial type harmonic spinors $\mathbf{C}[\phi^{\pm}]\|S^{3}$. ###### Proposition 3.11. $c_{k}(\,\mathbf{n}\phi_{1}\,,\phi_{2}\,)\,+\,c_{k}(\,\phi_{1}\,,\mathbf{n}\phi_{2}\,)\,=\,0\,\quad k=0,1,2\,.$ (3.23) ###### Proof. Since $\,\theta_{0}\,\mathbf{n}\,=\,(\nu-\bar{\nu})(\nu+\bar{\nu})=\nu^{2}-\bar{\nu}^{2}=\,\mathbf{n}\,\theta_{0}\,$, we have $\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{S^{3}}\,tr\,(\,\mathbf{n}(\Theta_{0}\phi_{1}\cdot\phi_{2})\,)\,d\sigma=\int_{S^{3}}\,tr\,(\,(\mathbf{n}\Theta_{0}\phi_{1})\cdot\phi_{2}+\Theta_{0}\phi_{1}\cdot\mathbf{n}\phi_{2}\,)\,d\sigma$ $\displaystyle=$ $\displaystyle\int_{S^{3}}\,tr\,((\Theta_{0}\,\mathbf{n}\,\phi_{1})\cdot\phi_{2}\,)\,d\sigma+\int_{S^{3}}\,tr\,(\Theta_{0}\phi_{1}\cdot\mathbf{n}\phi_{2}\,)\,d\sigma$ $\displaystyle=$ $\displaystyle c_{0}(\mathbf{n}\phi_{1},\phi_{2})+c_{0}(\phi_{1},\mathbf{n}\phi_{2})\,.$ The others are proved similarly. ∎ ### 3.4 Homogeneous decomposition of $\mathbf{C}[\phi^{\pm}]$ Let $\mathbf{C}[\phi^{\pm};\,N]$ be the subspace of $\mathbf{C}[\phi^{\pm}]$ consisting of those elements that are of homogeneous degree $N$: $\varphi(z)=\|z\|^{N}\varphi(\frac{z}{\|z\|})$. $\mathbf{C}[\phi^{\pm};\,N]$ is spanned by the spinors $\varphi=\phi_{1}\cdots\phi_{n}$ such that each $\phi_{i}$ is equal to $\phi_{i}=\phi^{+(m_{i},l_{i},k_{i})}$ or $\,\phi_{i}=\phi^{-(m_{i},l_{i},k_{i})}$ , where $m_{i}\geq 0$ and $0\leq l_{i}\leq m_{i}+1,\,0\leq k_{i}\leq m_{i}+2$, and such that $N=\sum_{i:\,\phi_{i}=\phi^{+(m_{i},l_{i},k_{i})}}\,m_{i}\,\,-\,\sum_{i:\,\phi_{i}=\phi^{-(m_{i},l_{i},k_{i})}}\,(m_{i}+3).$ It holds that $\mathbf{n}\varphi=\frac{N}{2}\varphi$, so the eigenvalues of $\mathbf{n}$ on $\mathbf{C}[\phi^{\pm}]$ are $\left\\{\frac{N}{2};\,N\in\mathbf{Z}\,\right\\}$ and $\mathbf{C}[\phi^{\pm};\,N]$ is the space of eigenspinors for the eigenvalue $\frac{N}{2}$. Example $\phi=\phi^{+(2,00)}\cdot\phi^{-(0,00)}\in\mathbf{C}[\phi^{\pm};-1]\,,\quad\mbox{ and }\,\mathbf{n}\phi=-\frac{1}{2}\phi\,.$ We note that $-\frac{1}{2}$ is not an eigenvalue of $\partial\\!\\!\\!/$. We have the eigenspace decomposition of the radial derivative $\mathbf{n}$: $\mathbf{C}[\phi^{\pm}]\,=\,\bigoplus_{N\in\mathbf{Z}}\,\mathbf{C}[\phi^{\pm};\,N]\,$ (3.24) The radial derivation $\mathbf{n}$ acts on $\mathbf{C}[\phi^{\pm}]\,$ and preserves the homogeneous degree. ## 4 $\mathfrak{g}$-current algebras on $S^{3}$ ### 4.1 Algebra of current $\mathcal{L}$ on $S^{3}$ ###### Definition 4.1. We denote $\,\mathcal{L}\,=\,\mathbf{C}[\phi^{\pm}]\|S^{3}\,$ and call $\mathcal{L}$ the algebra of current on $S^{3}\,$. By virtue of Theorem 3.2 $\,\mathcal{L}$ is an associative $\mathbf{C}$-algebra generated by the spinors $I=\phi^{+(0,0,1)},\,J=-\phi^{+(0,0,0)},\,\kappa=\phi^{+(1,0,1)},\,\mu=\phi^{-(0,0,0)}\,.$ We have given the definition of a quaternion Lie algebra in the introduction. It is a real submodule of a quaternion module that is endowed with a real Lie algebra structure compatible with the involutions $\sigma$ and $\tau$, (1). ###### Proposition 4.2. $\mathcal{L}$ is a quaternion Lie algebra with the induced bracket: $[\phi_{1},\phi_{2}]=\phi_{1}\cdot\phi_{2}-\phi_{2}\cdot\phi_{1}\,,\quad\phi_{1},\,\phi_{2}\in\mathcal{L}\,.$ (4.1) In particular $\mathcal{L}$ is invariant under the involutions $\sigma$ and $\tau$. $\mathcal{L}$ is also invariant under the derivations $\Theta_{k}$, $k=0,1,2$, and the radial derivation: $\Theta_{k}\phi\in\mathcal{L}\,,\quad\mathbf{n}\phi\in\mathcal{L}\quad\mbox{ for }\,\,\forall\phi\in\mathcal{L}\,.$ (4.2) ###### Proof. We have already seen these properties in section 3. ∎ The quaternion Lie algebra $\mathcal{L}$ has the following subalgebras. $\displaystyle\mathcal{L}^{r}_{0}=$ $\displaystyle\\{\phi\in\mathcal{L}\,;\,\sigma\phi=\phi,\,\tau\phi=\phi\,\\}\,,\quad$ $\displaystyle\mathcal{L}^{0}_{r}\,=\\{\phi\in\mathcal{L}:\,\sigma\phi=\,-\phi,\,\tau\phi=\phi\,\\}\,,$ $\displaystyle\mathcal{L}^{i}_{0}\,=$ $\displaystyle\\{\phi\in\mathcal{L}\,;\,\tau\phi=-\phi\,,\sigma\phi=\phi\,\\}\,,\quad$ $\displaystyle\mathcal{L}^{0}_{i}\,=\\{\phi\in\mathcal{L}\,;\,\tau\phi=-\phi\,,\sigma\phi=-\phi\,\\}.$ $\phi=\left(\begin{array}[]{c}u\\\ v\end{array}\right)\in\mathcal{L}^{r}_{0}$ if $u$ is real and $v=0$. $\phi\in\mathcal{L}^{0}_{r}$ if $v$ is real and $u=0$. $\phi\in\mathcal{L}^{i}_{0}$ if $u$ is pure imaginary and $v=0$, and $\phi\in\mathcal{L}^{0}_{i}$ if $u=0$ and $v$ is pure imaginary. For $\phi_{k}=\left(\begin{array}[]{c}u_{k}\\\ v_{k}\end{array}\right)\,\in\mathcal{L}^{r}_{0}+\mathcal{L}^{0}_{r}$, $k=1,2$, we have $\phi_{1}\cdot\phi_{2}\,=\,\left(\begin{array}[]{c}u_{1}u_{2}-v_{1}v_{2}\\\ v_{1}u_{2}+u_{1}v_{2}\end{array}\right)=\phi_{2}\phi_{1}\,.$ (4.3) Hence $\mathcal{L}^{r}_{0}\,+\,\mathcal{L}^{0}_{r}\,,\,\mathcal{L}^{r}_{0}\,$ and $\mathcal{L}^{0}_{r}\,$ are commutative Lie subalgebras of $\mathcal{L}\,$. Similarly $\mathcal{L}^{i}_{0}\,$ and $\mathcal{L}^{0}_{i}\,$ are commutative subalgebras and the following relations hold; $[\mathcal{L}^{0}_{r}\,,\mathcal{L}^{0}_{i}]=\,\mathcal{L}^{i}_{0}\,,\quad[\mathcal{L}^{0}_{i}\,,\mathcal{L}^{i}_{0}]\,=\,\mathcal{L}^{0}_{r}\,.\quad[\mathcal{L}^{i}_{0}\,,\mathcal{L}^{0}_{r}]\,=\,\mathcal{L}^{0}_{i}\,.$ These are proved by a calculation of the Lie bracket (4.1). For example, for $\phi=\left(\begin{array}[]{c}0\\\ t\end{array}\right)\in\mathcal{L}^{0}_{r}$ and $\psi=\left(\begin{array}[]{c}\sqrt{-1}u\\\ 0\end{array}\right)\in\mathcal{L}^{i}_{0}$, we have $[\phi,\psi]=\left(\begin{array}[]{c}0\\\ 2\sqrt{-1}tu\end{array}\right)\in\mathcal{L}^{0}_{i}\,$. The others follow similarly. ###### Definition 4.3. We put $\mathcal{K}\,=\,\mathcal{L}^{r}_{0}\,,\qquad\mathcal{K}^{\bot}=\mathcal{L}^{0}_{r}\,+\,\mathcal{L}^{0}_{i}\,+\,\mathcal{L}^{i}_{0}\,.$ (4.4) ###### Proposition 4.4. 1. 1. $\mathcal{K}$ is a commutative subalgebra of the associative algebra $\mathcal{L}$. We have $N(\mathcal{K})=\mathcal{K}\,,$ (4.5) where $N(\mathcal{K})$ is the normalizer of $\mathcal{K}$: $\,N(\mathcal{K})=\,\\{\psi\in\mathcal{L}\,;\quad\phi\psi\in\mathcal{K}\,,\quad\forall\phi\in\mathcal{K}\,\\}$. 2. 2. $\,\mathcal{K}^{\bot}$ is an ideal of $\mathcal{L}$ complementary to $\mathcal{K}$, and we have $\,\mathcal{K}\cdot\mathcal{K}^{\bot}=\mathcal{K}^{\bot}\cdot\mathcal{K}\,=\,\mathcal{K}^{\bot}\,.$ (4.6) 3. 3. The quaternion Lie algebra $\mathcal{L}$ is decomposed into $\,\mathcal{L}\,=\,\mathcal{K}\,+\,\mathcal{K}^{\bot}\,,\quad\mbox{ direct sum.}$ It holds that $[\mathcal{K}\,,\,\mathcal{L}\,]\,=\,0\,,\qquad[\,\mathcal{L}\,,\,\mathcal{L}\,]\,=\,\mathcal{K}^{\bot}\,.$ (4.7) The proof follows by direct calculations of the multiplication of spinors (3.1) and the Lie bracket (4.1). Examples $\displaystyle\frac{1}{2}(\phi^{+(1,1,1)}-\phi^{-(0,0,1)})+\frac{1}{\sqrt{2}}\phi^{+(1,0,2)}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}z_{1}+\overline{z}_{1}\\\ 0\end{array}\right)\in\mathcal{K}$ $\displaystyle(\phi^{-(1,0,0)}+\phi^{-(1,0,2)})-J\cdot(\phi^{-(1,1,0)}+\phi^{-(1,1,2)})$ $\displaystyle=$ $\displaystyle\sqrt{2}\left(\begin{array}[]{c}z_{1}^{2}+\overline{z}_{1}^{2}+z_{2}^{2}+\overline{z}_{2}^{2}\\\ 0\end{array}\right)\,\in\,\mathcal{K}$ $\displaystyle\phi^{-(0,0,1)}+\frac{1}{\sqrt{2}}(\phi^{+(0,0,0)}\phi^{+(1,1,0)}-\phi^{+(1,0,0)})$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\ z_{2}+\overline{z}_{2}\end{array}\right)\in\mathcal{L}^{0}_{r}$ $\displaystyle-\phi^{+(1,1,1)}-\phi^{-(0,0,1)}+\sqrt{2}\phi^{+(1,0,0)}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\ 2(z_{2}-\overline{z}_{2})\end{array}\right)\in\mathcal{L}^{0}_{i}\,.$ $\displaystyle\phi^{+(1,1,1)}-\phi^{-(0,0,1)}-\sqrt{2}\phi^{+(1,0,2)}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}2(z_{1}-\overline{z}_{1})\\\ 0\end{array}\right)\in\mathcal{L}^{i}_{0}.$ ### 4.2 $\mathfrak{g}-$Current algebras on $S^{3}$ and its subalgebras $C^{\infty}(S^{3},\,\mathfrak{gl}(n,\mathbf{H})\,)=C^{\infty}(S^{3},\,\mathbf{H})\otimes_{\mathbf{C}}\mathfrak{gl}(n,\mathbf{C})$ becomes a quaternion Lie algebra with the Lie bracket defined by $[\,\phi_{1}\otimes X_{1}\,,\,\phi_{2}\otimes X_{2}\,]=(\phi_{1}\cdot\phi_{2})\otimes X_{1}X_{2}\,-(\phi_{2}\cdot\phi_{1})\otimes X_{2}X_{1}\,,$ (4.13) for $\phi_{1},\phi_{2}\in C^{\infty}(S^{3},\,\mathbf{H}),\,X_{1},X_{2}\in\mathfrak{gl}(n,\mathbf{C})$. In (4.13) the right hand side is in the tensor product of the associative algebra $C^{\infty}(S^{3},\,\mathbf{H})\simeq C^{\infty}(S^{3},\,S^{+})$ and the matrix algebra $\mathfrak{gl}(n,\mathbf{C})$. Let $(\,\mathfrak{g}\,,\,[\,\,,\,\,]\,)$ be a complex Lie algebra, that we suppose to be a subalgebra of $\mathfrak{gl}(n,\mathbf{C})$ for some $n$. Then $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}\,$ becomes a $\mathbf{C}$-submodule of the $\mathbf{H}$-module $C^{\infty}(S^{3},\,\mathbf{H})\otimes_{\mathbf{C}}\mathfrak{gl}(n,\mathbf{C})=C^{\infty}(S^{3},\,\mathfrak{gl}(n,\mathbf{H})\,)$. The involutions $\sigma$ and $\tau$ on $\,\mathcal{L}$ are extended to $\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}\,$ by $\sigma(\phi\otimes X)=\sigma(\phi)\otimes X\,$ and $\tau(\phi\otimes X)=\tau(\phi)\otimes X\,$ respectively for $\phi\in\mathcal{L}$ and $X\in\mathfrak{g}\,$. Thus $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}\,$ endowed with the bracket (4.13) generates a quaternion Lie algebra. ###### Definition 4.5. The quaternion Lie algebra generated by $\left(\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}\,,\,[\,\,,\,\,]\,\right)$ is called $\mathfrak{g}$-current algebra, and is denoted by $\mathcal{L}\mathfrak{g}$. As the following examples show $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}$ is not necessarily a Lie algebra so that the Lie algebra $\mathcal{L}\mathfrak{g}$ is defined as that which is generated by $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}$ in the Lie algebra $C^{\infty}(S^{3},\,\mathfrak{gl}(n,\mathbf{H})\,)$. Examples: The following elements are in $\,\mathcal{L}\mathfrak{g}\,\ominus\,(\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g})\,$. 1. 1. $\sqrt{-1}(X_{1}X_{2}+X_{2}X_{1})\,\in\mathcal{L}\mathfrak{g}\,,\quad\mbox{ for }\forall X_{1},\,X_{2}\in\mathfrak{g}\,.$ In fact we have $\mathcal{L}\mathfrak{g}\,\ni\,[\,J\otimes X_{1}\,,\,\sqrt{-1}J\otimes X_{2}\,]\,=\,\sqrt{-1}I\otimes(X_{1}X_{2}+X_{2}X_{1})\,.$ Here we saw that the right hand-side calculated in $C^{\infty}(S^{3},\,\mathfrak{gl}(n,\mathbf{H})\,)$ gives the left-hand side element in $\mathcal{L}\mathfrak{g}\,$. 2. 2. $\sqrt{-1}J\otimes(X_{1}X_{2}+X_{2}X_{1})\,\in\mathcal{L}\mathfrak{g}\,,\quad\mbox{ for }\forall X_{1},\,X_{2}\in\mathfrak{g}\,.$ In fact $\mathcal{L}\mathfrak{g}\,\ni\,[\,J\otimes X_{1}\,,\,\sqrt{-1}I\otimes X_{2}\,]\,=\,\sqrt{-1}J\otimes(X_{1}X_{2}+X_{2}X_{1})\,.$ 3. 3. $(z_{1}-\overline{z}_{1})(z_{2}+\overline{z}_{2})J\otimes(X_{1}X_{2}+X_{2}X_{1})\,\in\mathcal{L}\mathfrak{g}\,,\quad\mbox{ for }\forall X_{1},\,X_{2}\in\mathfrak{g}\,.$ In fact, let $\phi_{1}=\left(\begin{array}[]{c}z_{1}+\overline{z}_{1}\\\ z_{2}+\overline{z}_{2}\end{array}\right)$ and $\phi_{2}=\left(\begin{array}[]{c}z_{1}-\overline{z}_{1}\\\ 0\end{array}\right)$. Then $\displaystyle\mathcal{L}\mathfrak{g}\,\ni\,$ $\displaystyle[\,\phi_{1}\otimes X_{1},\,\phi_{2}\otimes X_{2}\,]-\,(z_{1}^{2}-\overline{z}_{1}^{2})I\otimes[X_{1},X_{2}]\,$ $\displaystyle=(z_{1}-\overline{z}_{1})(z_{2}+\overline{z}_{2})J\otimes(X_{1}X_{2}+X_{2}X_{1}).$ ### 4.3 Quaternification and $\mathfrak{g}$-current algebras Remember that the quaternification of a complex Lie algebra $\mathfrak{g}$ is the quaternion Lie algebra $\mathfrak{g}^{q}$ generaetd by $\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{g}=\mathfrak{g}+J\mathfrak{g}$. The latter is not a Lie algebra in general. Since $I=\phi^{+(0,0,1)}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$ and $J=-\phi^{+(0,0,0)}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$ are in $\,\mathcal{L}$, $\mathfrak{g}^{q}$ is a subspace of $\mathcal{L}\mathfrak{g}$. We have the following relations: $S^{3}\mathfrak{g}^{q}\supset S^{3}\mathfrak{g}+J\,(S^{3}\mathfrak{g})\supset\mathcal{L}\mathfrak{g}\,\supset\mathfrak{g}^{q},$ where $S^{3}\mathfrak{g}+J\,(S^{3}\mathfrak{g})$ is not necessarily a Lie algebra in general and $S^{3}\mathfrak{g}^{q}=S^{3}\mathbf{H}\otimes\mathfrak{g}^{q}$ is the Lie algebra with bracket (4.13). The following examples show the case where both $S^{3}\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{g}$ and $\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{g}$ become Lie algebras. Examples 1. 1. $\mathfrak{gl}(n,\mathbf{H})=\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{gl}(n,\mathbf{C})\subset\mathcal{L}\mathfrak{gl}(n,\mathbf{C})\subset S^{3}\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{gl}(n,\mathbf{C})\,=S^{3}\mathfrak{gl}(n,\mathbf{H})$ 2. 2. $so^{\ast}(2n)=\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{so}(n,\mathbf{C})\,\subset\,\mathcal{L}\mathfrak{so}(n,\mathbf{C})\,\subset S^{3}\mathbf{H}\otimes\mathfrak{so}(n,\mathbf{C})=S^{3}\mathfrak{so}^{\ast}(2n)$ 3. 3. $\mathfrak{sp}(2n)=\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{u}(n)\,\subset\mathcal{L}\mathfrak{u}(n)\,\subset\,S^{3}\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{u}(n)\,=\,S^{3}\mathfrak{sp}(2n).$ In general $\mathbf{H}\otimes_{\mathbf{C}}\mathfrak{g}$ is not a Lie algebra. We know that the quaternification of $\mathfrak{sl}(n,\mathbf{C})$ is $\mathfrak{sl}(n,\mathbf{H})$, [Kq], and it is not contained in $\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{sl}(n,\mathbf{C})$ as is seen by the following example: Let $\\{h_{i}=E_{ii}-E_{i+1\,i+1};\,1\leq i\leq n-1\,,\quad E_{ij},\,i\neq j\,\\}$ be the basis of $\mathfrak{g}=\mathfrak{sl}(n,\mathbf{C})$. Then $[\sqrt{-1}Jh_{1},Jh_{2}\,]=-2\sqrt{-1}E_{22}\,\in\mathfrak{g}^{q}\subset\mathcal{L}\mathfrak{g}$ but not in $\mathcal{L}\otimes\mathfrak{g}$. ### 4.4 Root space decomposition of $\mathfrak{g}$-current algebras #### 4.4.1 Let $\mathfrak{g}$ be a simple Lie algebra with Cartan matrix $A=\left(c_{ij}\right)$. Let $\mathfrak{h}$ be a Cartan subalgebra, $\Phi$ the corresponding root system. Let $\Pi=\\{\alpha_{i};\,i=1,\cdots,l=\dim\,\mathfrak{h}\\}\subset\mathfrak{h}^{\ast}$ be the set of simple roots and $\\{h_{i}=\alpha_{i}^{\vee}\,;\,i=1,\cdots,l\,\\}\subset\mathfrak{h}$ be the set of simple coroots. The Cartan matrix $A=(\,c_{ij}\,)_{i,j=1,\cdots,r}$ is given by $c_{ij}=\left\langle\alpha_{i}^{\vee},\,\alpha_{j}\right\rangle$. $\alpha(h)$ is real if $h\in\mathfrak{h}$ is real. Let $\,\mathfrak{g}_{\alpha}=\\{\xi\in\mathfrak{g}\,;\,ad(h)\xi\,=\,\alpha(h)\xi,\quad\forall h\in\mathfrak{h}\\}$ be the root space of $\alpha\in\Phi$. Then $\dim_{\mathbf{C}}\,\mathfrak{g}_{\alpha}=1$. Let $\Phi_{\pm}$ be the set of positive ( respectively negative ) roots of $\mathfrak{g}$ and put $\mathfrak{e}=\sum_{\alpha\in\Phi_{+}}\,(\mathfrak{g})_{\alpha}\,,\quad\mathfrak{f}=\sum_{\alpha\in\Phi_{-}}\,(\mathfrak{g})_{\alpha}\,.$ Fix a standard set of generators $\,h_{i}\in\mathfrak{h}\,,\,e_{i}\in\mathfrak{g}_{\alpha_{i}}$, $f_{i}\in\mathfrak{g}_{-\alpha_{i}}$. $\mathfrak{g}$ is generated by $X\,=\,\\{e_{i},\,f_{i},\,h_{i}\,;\,i=1,\cdots,l\,\\}$, and these generators satisfy the relations: $[\,h_{i},\,h_{j}\,]\,=\,0\,,\quad[\,e_{i}\,,\,f_{j}\,]\,=\,\delta_{ij}h_{i}\,,\quad[\,h_{i}\,,\,e_{j}\,]\,=\,c_{ji}e_{j}\,,\quad[\,h_{i}\,,\,f_{j}\,]\,=\,-\,c_{ji}f_{j}\,.$ (4.14) This is a presentation of $\mathfrak{g}$ by generators and relations which depend only on the root system $\Phi$. The triangular decomposition of the simple Lie algebra $\mathfrak{g}$ becomes $\mathfrak{g}=\mathfrak{f}+\mathfrak{h}+\mathfrak{e}$, ( direct sum ) with the space of positive root vectors $\mathfrak{e}$ and the space of negative root vectors $\mathfrak{f}$. $\mathfrak{g}$ is considered as a quaternion Lie subalgebra of the $\mathfrak{g}$-current algebra $\,\mathcal{L}\mathfrak{g}\,$; $\displaystyle i\,:\,\mathfrak{g}\,\ni\,X\,\longrightarrow\,\phi^{+(0,0,1)}\otimes X\,\in\,\mathcal{L}\mathfrak{g}\,,$ (4.15) $\displaystyle\left[\phi^{+(0,0,1)}\otimes X,\,\phi^{+(0,0,1)}\otimes Y\right]_{\mathcal{L}\mathfrak{g}}=\left[X,Y\right]_{\mathfrak{g}}\,.$ We adopt the following abbreviated notations: For $\phi_{i}\in\mathcal{L}$. $\,x_{i}\in\mathfrak{g}\,$, $i=1,\cdots,t$, we put $\displaystyle x_{12\cdots t}$ $\displaystyle=$ $\displaystyle[\,x_{1},\,[\,x_{2},\,[\,\cdots\,\cdots\,x_{t}\,]\,]\cdots\,]\,,$ $\displaystyle\,\phi_{12\cdots t}\ast x_{12\cdots t}$ $\displaystyle=$ $\displaystyle[\phi_{1}\otimes x_{1},\,[\phi_{2}\otimes x_{2},\,[\,\cdots\,\cdots\,,\,\phi_{t}\otimes x_{t}\,]\,]\cdots\,]\,.$ (4.16) Every element of $\mathcal{L}\mathfrak{g}$ is expressed as a linear combination of $\,\phi_{12\cdots t}\ast x_{12\cdots t}$’s. We have a projection from $\mathcal{L}\mathfrak{g}$ to $\mathfrak{g}$ that extends the correspondence: $\pi:\,\mathcal{L}\mathfrak{g}\ni\,\phi_{12\cdots t}\ast x_{12\cdots t}\,\longrightarrow\,x_{12\cdots t}\,\in\mathfrak{g}.$ (4.17) It is obtained by letting all $\phi_{i}$’s in (4.16) equal to $\phi^{+(0,0,1)}$. #### 4.4.2 The adjoint representation $ad_{\mathcal{K}\mathfrak{h}}:\,\mathcal{K}\mathfrak{h}\longrightarrow\,End(\mathcal{L}\mathfrak{g})$ We shall investigate the triangular decomposition of $\mathfrak{g}$-current algebra $\mathcal{L}\mathfrak{g}$. ###### Definition 4.6. Let $\mathcal{L}\,\mathfrak{h}$, $\,\mathcal{L}\,\mathfrak{e}$ and $\mathcal{L}\,\mathfrak{f}$ respectively be the Lie subalgebras of the $\mathfrak{g}$-current algebra $\mathcal{L}\mathfrak{g}$ that are generated by $\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{h}\,$, $\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{e}\,$ and $\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{f}\,$ respectively. Let $\mathcal{K}\,\mathfrak{h}$ and $\,\mathcal{K}^{\bot}\,\mathfrak{h}$ be the Lie subalgebras of $\mathcal{L}\,\mathfrak{g}\,$ generated by $\mathcal{K}\otimes_{\mathbf{R}}\mathfrak{h}\,$ and $\mathcal{K}^{\bot}\otimes_{\mathbf{R}}\mathfrak{h}$ respectively. $\mathcal{L}\mathfrak{e}$ consists of linear combinations of elements of the form $\phi_{12\cdots t}\ast e_{12\cdots t}$ for $\phi_{j}\in\mathcal{L}$ and $e_{j}\in\mathfrak{e}$, $j=1,2,\cdots,\,t$. Similarly $\,\mathcal{L}\mathfrak{f}\,$ is generated by $\phi_{12\cdots t}\ast f_{12\cdots t}$ with $\phi_{j}\in\mathcal{L}\,$ and $\,f_{j}\in\mathfrak{f}$, $j=1,2,\cdots,\,t\,$. Later we shall see that $\mathcal{L}\mathfrak{e}=\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{e}$, viewed as a real Lie algebra. This is a crucial fact in our investigation. ###### Lemma 4.7. 1. 1. $\mathfrak{h}\,\subset\,\mathcal{K}\,\mathfrak{h}\,.\qquad\mathcal{L}\mathfrak{h}=\mathcal{K}\mathfrak{h}+\mathcal{K}^{\bot}\mathfrak{h}\,.$ 2. 2. $\mathcal{K}\mathfrak{h}\,=\,\mathcal{K}\otimes_{\mathbf{R}}\mathfrak{h}\,.$ $\mathcal{K}\,\mathfrak{h}$ is a commutative subalgebra of $\mathcal{L}\,\mathfrak{g}\,$, and $N(\mathcal{K}\mathfrak{h})\,=\,\mathcal{K}\mathfrak{h}\,$. That is, $\mathcal{K}\mathfrak{h}$ is a Cartan subalgebra of $\mathcal{L}\mathfrak{g}\,$, where $N(\mathcal{K}\mathfrak{h})=\\{\,\xi\in\mathcal{L}\mathfrak{g};\,[\kappa,\xi]\in\mathcal{K}\mathfrak{h},\,\forall\kappa\in\mathcal{K}\mathfrak{h}\,\\}$ is the normalizer of $\mathcal{K}\mathfrak{h}$. 3. 3. $[\,\mathcal{K}\,\mathfrak{h}\,,\,\mathcal{L}\mathfrak{h}\,]\,=\,0\,,\quad[\,\mathcal{K}\,\mathfrak{h}\,,\,\mathcal{L}\,\mathfrak{e}\,]\,=\,\mathcal{L}\mathfrak{e}\,,\quad[\,\mathcal{K}\,\mathfrak{h}\,,\,\mathcal{L}\,\mathfrak{f}\,]\,=\,\mathcal{L}\,\mathfrak{f}\,.$ 4. 4. $[\,\mathcal{L}\mathfrak{h}\,,\,\mathcal{L}\mathfrak{h}\,]\,=\mathcal{K}^{\bot}\mathfrak{h}\,.$ (4.18) ###### Proof. Let $\phi_{i}\in\mathcal{K}$ and $h_{i}\in\mathfrak{h}$, $i=1,2$. Since $\phi_{1}\phi_{2}\,=\,\phi_{2}\phi_{1}$ from (4.3), We have $[\phi_{1}\otimes h_{1}\,,\,\phi_{2}\otimes h_{2}\,]\,=\,(\phi_{1}\phi_{2})\,[h_{1},h_{2}]=0\,$. So $\mathcal{K}\mathfrak{h}=\mathcal{K}\otimes_{\mathbf{R}}\mathfrak{h}$, and $\mathcal{K}\mathfrak{h}$ is a commutative Lie algebra. Now the first assertions follow from the definitions; $\phi^{+(0,0,1)}\otimes\mathfrak{h}\subset\mathcal{K}\mathfrak{h}$. We shall prove $N(\mathcal{K}\mathfrak{h})\,=\,\mathcal{K}\mathfrak{h}\,$. Let $\psi\otimes x\in(\mathcal{L}\otimes\mathfrak{g})\cap\,N(\mathcal{K}\mathfrak{h})$. Then $[\phi\otimes h,\,\psi\otimes x]=(\phi\psi)\otimes[h,x]\in\mathcal{K}\otimes\mathfrak{h}$ for any $\phi\in\mathcal{K}$ and $h\in\mathfrak{h}$. Then $\phi\psi\in\mathcal{K}$ for all $\phi\in\mathcal{K}$, so (4.5) implies $\psi\in\mathcal{K}$. And $[h,x]\in\mathfrak{h}$ for all $h\in\mathfrak{h}$. $\mathfrak{h}$ being a Cartan subalgebra it follows $x\in\mathfrak{h}$. Hence $\psi\otimes x\in\mathcal{K}\mathfrak{h}$. $N(\mathcal{K}\mathfrak{h})$ being generated by $(\mathcal{L}\otimes\mathfrak{g})\cap\,N(\mathcal{K}\mathfrak{h})$, it follows $N(\mathcal{K}\mathfrak{h})\,=\mathcal{K}\mathfrak{h}$. We proceed to the proof of the 3rd assertion. Let $\phi\otimes h\in\mathcal{K}\otimes\mathfrak{h}$ and $\psi\otimes h^{\prime}\in\mathcal{L}\otimes\mathfrak{h}$ with $\phi\in\mathcal{K}\,,\psi\in\mathcal{L}$ and $h\,,\,h^{\prime}\in\mathfrak{h}$. By virtue of (4.7 ) we have $[\phi\otimes h,\,\psi\otimes h^{\prime}]=(\phi\psi)\otimes\,[h.h^{\prime}]=0\,$. Jacobi identity yields $[\phi\otimes h,\,[\psi_{1}\otimes h_{1},\,\psi_{2}\otimes h_{2}]\,]=0$ for $\psi_{i}\in\mathcal{L},\,h_{i}\in\mathfrak{h}$, $i=1,2$, and $[\phi\otimes h,\,\psi_{12\cdots t}\ast h_{12\cdots t}\,]=0$. Hence $[\,\mathcal{K}\,\mathfrak{h}\,,\mathcal{L}\mathfrak{h}\,]=0$ . Let $\psi\otimes e_{j}\in\mathcal{L}\otimes\mathfrak{e}$. We have $[\phi\otimes h_{i}\,,\,\psi\otimes e_{j}\,]\,=\,(\phi\psi)\otimes[h_{i},e_{j}]\,=(\phi\psi)\otimes c_{ji}e_{j}\,\in\mathcal{L}\mathfrak{e}\,$. The similar argument with Jacobi identity yields $[\phi\otimes h_{i},\,\psi_{j_{1}\cdots j_{t}}\ast e_{j_{1}\cdots j_{t}}\,]\,=\,\left(c_{j_{1}i}+\cdots c_{j_{t}i}\right)(\phi\psi_{j_{1}}\psi_{j_{2}}\cdots\psi_{j_{t}})\otimes e_{j_{1}\cdots j_{t}}\in\mathcal{L}\mathfrak{e}\,.$ (4.19) So we have $[\phi\otimes h_{i}\,,\,\mathcal{L}\mathfrak{e}\,]\subset\mathcal{L}\mathfrak{e}$, hence $[\,\mathcal{K}\mathfrak{h},\,\mathcal{L}\mathfrak{e}\,]\subset\mathcal{L}\mathfrak{e}$. Similarly $\,[\,\mathcal{K}\,\mathfrak{h},\,\mathcal{L}\mathfrak{f}\,]\subset\mathcal{L}\mathfrak{f}$. Conversely any element $\psi_{j_{1}\cdots j_{t}}\ast e_{j_{1}\cdots j_{t}}\,\in\,\mathcal{L}\mathfrak{e}$ satisfies the relation (4.19) for all $\phi\otimes h\in\mathcal{K}\mathfrak{h}$ with non-zero $(c_{j_{1}i}+\cdots c_{j_{t}i})$ hence $[\,\mathcal{K}\mathfrak{h}\,,\,\mathcal{L}\mathfrak{e}\,]\,=\,\mathcal{L}\mathfrak{e}\,$. Similarly $[\,\mathcal{K}\mathfrak{h}\,,\,\mathcal{L}\mathfrak{f}\,]\,=\,\mathcal{L}\mathfrak{f}\,$. Finally we shall prove the assertion; $[\,\mathcal{L}\mathfrak{h}\,,\,\mathcal{L}\mathfrak{h}\,]\,=\mathcal{K}^{\bot}\mathfrak{h}\,$. Let $\psi_{1},\,\psi_{2}\in\mathcal{L}$ and $h_{1},\,h_{2}\in\mathfrak{h}$. We have $[\psi_{1}\otimes h_{1}\,,\psi_{2}\otimes h_{2}\,]\,=\,(\psi_{1}\psi_{2})\otimes h_{1}h_{2}-\,(\psi_{2}\psi_{1})\otimes h_{2}h_{1}\,=\,[\psi_{1},\psi_{2}]\otimes h_{2}h_{1}\,$. Here the right hand side is the multiplication of matrices with coefficients in $\mathcal{L}$. While the left hand side is in $\mathcal{L}\mathfrak{h}$. The relation (4.7) implies $[\mathcal{L}\otimes\mathfrak{h},\,\mathcal{L}\otimes\mathfrak{h}]\,\subset\mathcal{K}^{\bot}\mathfrak{h}$ and $[\,\mathcal{L}\mathfrak{h}\,,\,\mathcal{L}\mathfrak{h}\,]\,=\mathcal{K}^{\bot}\mathfrak{h}\,$. ∎ The examples at the end of subsection 4.2 testify to the assertion 4. We note that $\mathcal{L}\mathfrak{g}$ is not a soluble Lie algebra. We regard the associative algebra $\mathcal{L}$ as a $\mathcal{K}$-module, and we regard $\mathcal{L}$ as the coefficient ring of $\mathcal{L}\mathfrak{g}$. $\mathcal{K}\mathfrak{h}$ being a Cartan subalgebra of $\mathcal{L}\mathfrak{g}$, we consider the adjoint representation $ad_{\mathcal{K}\mathfrak{h}}\,\in End_{\mathcal{K}}(\mathcal{L}\mathfrak{g})\,$ and its weight space decomposition. The adjoint representation $ad_{\mathcal{K}\mathfrak{h}}$ is written as follows: $\displaystyle ad_{\phi\otimes h}\,(\psi\otimes x)$ $\displaystyle=$ $\displaystyle\,(\phi\,\psi)\,\otimes ad_{h}x\,,$ (4.20) $\displaystyle ad_{\phi\otimes h}(\psi_{1\cdots m}\ast x_{1\cdots m})$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\phi\,[\psi_{1}\otimes x_{1},[\psi_{2}\otimes x_{2},\,\cdots[\psi_{i}\otimes ad_{h}x_{i},[\psi_{i+1}\otimes x_{i+1},\cdots,\psi_{m}\otimes x_{m}]\cdots],$ for $\phi\otimes h\in\mathcal{K}\mathfrak{h}\,$ and $\psi\otimes x,\,\psi_{1\cdots m}\ast x_{1\cdots m}\in\mathcal{L}\mathfrak{g}\,$. Let $Hom_{\mathbf{\mathcal{K}}}(\,\mathcal{K}\mathfrak{h}\,,\,\mathcal{L})\,=\,\\{\lambda\,:\mathcal{K}\mathfrak{h}\,\longrightarrow\mathcal{L}\,,\quad\lambda(\kappa)=\alpha(h)\,\phi\quad\mbox{for $\forall\kappa=\phi\otimes h$}\\},$ (4.21) with $\alpha\in\mathfrak{h}^{\ast}=Hom(\mathfrak{h},\,\mathbf{C})$ defined by $\alpha(h)\circ\pi=\pi_{o}\circ\lambda(i\,h)\,,\,\forall h\in\mathfrak{h}$. Where $i:\mathfrak{h}\hookrightarrow\mathcal{K}\mathfrak{h}$ is the embedding (4.15), $\,\pi:\mathcal{L}\mathfrak{g}\longrightarrow\mathfrak{g}$ is the projection (4.17) and $\pi_{o}:\mathcal{L}\longrightarrow\mathbf{C}$ the projection to the constant term of a Laurent polynomial type spinor. In the above we regard $\alpha(h)\in End_{\mathbf{C}}(\mathfrak{g})$ by the multiplication. Similarly we regard $\lambda(\kappa)\in End_{\mathcal{L}}(\mathcal{L}\mathfrak{g})$ as the multiplication of $\lambda(\kappa)=\alpha(h)\phi\in\mathcal{L}$, which is not necessarily in $\mathcal{K}$. For each $\lambda\in Hom_{\mathcal{K}}(\,\mathcal{K}\,\mathfrak{h}\,,\,\mathcal{L})$, we put $(\mathcal{L}\mathfrak{g})_{\lambda}=\\{\xi\in\mathcal{L}\mathfrak{g}\,;\quad ad_{\kappa}\,\xi\,=\,\lambda(\kappa)\,\xi\,,\quad\forall\kappa\in\mathcal{K}\mathfrak{h}\\}\,.$ (4.22) $\lambda\in Hom_{\mathcal{K}}(\,\mathcal{K}\mathfrak{h}\,,\,\mathcal{L})$ is called a weight whenever $(\mathcal{L}\mathfrak{g})_{\lambda}\neq 0$. $(\mathcal{L}\mathfrak{g})_{\lambda}$ is called the weight space of weight $\lambda$. The set of the non-zero weights is denoted by ${\Phi}_{\mathcal{L}}=\,\\{\lambda\in\,Hom(\,\mathcal{K}\mathfrak{h}\,,\,\mathcal{L})\,;\,\lambda\neq 0\,\\}\,.$ Then $\mathcal{L}\mathfrak{g}$ is the direct sum of the weight spaces: $\mathcal{L}\mathfrak{g}\,=\,(\mathcal{L}\mathfrak{g})_{0}\,\oplus_{\lambda\in\Phi_{\mathcal{L}}}\,(\mathcal{L}\mathfrak{g})_{\lambda}\,.$ (4.23) We have $ad_{\kappa}\,[\,\xi_{1}\,,\,\xi_{2}\,]\,=\,[\,ad_{\kappa}\xi_{1}\,,\,\xi_{2}\,]\,+\,[\,\xi_{1}\,,\,ad_{\kappa}\xi_{2}\,]\,,$ (4.24) for all $\kappa\in\mathcal{K}\mathfrak{h}\,$, $\xi_{i}\in\mathcal{L}\mathfrak{g}$, $i=1,2$. This follows inductively from the definition of $ad_{\mathcal{K}\mathfrak{h}}$; (4.20). Therefore it holds that if $\xi,\,\eta\in\,\mathcal{L}\mathfrak{g}$ are weight vectors of weights $\lambda,\,\mu$ then $[\xi\,,\,\eta]$ is a weight vector of weight $\lambda+\mu\,$: $[\,(\mathcal{L}\mathfrak{g})_{\lambda}\,,\,(\mathcal{L}\mathfrak{g})_{\mu}\,]\subset\,(\mathcal{L}\mathfrak{g})_{\lambda+\mu}\,.$ (4.25) ###### Proposition 4.8. The adjoint representation $ad_{\mathfrak{h}}$ of $\mathfrak{g}$ extends to the adjoint representation $ad_{\mathcal{K}\mathfrak{h}}$ of $\mathcal{L}\mathfrak{g}$ . ###### Proof. $\phi^{+(0,0,1)}\in\mathcal{K}$ and the abbreviation $\phi^{+(0,0,1)}\otimes\mathfrak{h}\,\simeq\mathfrak{h}$ imply the embedding $i:\,\mathfrak{h}\longrightarrow\,\mathcal{K}\mathfrak{h}$. The adjoint representation $ad_{\mathcal{K}\mathfrak{h}}$ restricts to the adjoint representation of $\mathfrak{h}$ on $\mathfrak{g}$ if we take $\phi=\psi=\phi^{+(0,0,1)}$ in (4.20). Then we have $ad_{h}\circ\pi\,=\,\pi\circ ad_{ih}\,,\quad\forall h\in\mathfrak{h}.$ (4.26) Conversely we see from (4.20) that the action of the representation $ad_{\mathcal{K}\mathfrak{h}}$ on $\mathcal{L}\mathfrak{g}$ comes from $ad_{\mathfrak{h}}\in End(\mathfrak{g})$. If $ad_{h}y=0$ for $h\in\mathfrak{h}$ and $y\in\mathfrak{g}$ then $ad_{\phi\otimes h}\psi\otimes y=0$ for all $\phi\in\mathcal{K}$ and $\psi\in\mathcal{L}$ . In fact, since $[\mathcal{K},\mathcal{L}]=0$ we have $[\phi\otimes h,\,\psi\otimes y]=(\phi\cdot\psi)\otimes[h,y]=0$. ∎ ###### Proposition 4.9. 1. 1. The root spaces of the adjoint representation $ad_{\mathcal{K}\mathfrak{h}}$ on $\mathcal{L}\mathfrak{g}$ and that of $ad_{\mathfrak{h}}$ on $\mathfrak{g}$ correspond bijectively: $\Phi_{\mathcal{L}}\simeq\Phi$. 2. 2. For $\lambda\in\Phi$ it holds that $(\mathcal{L}\mathfrak{g})_{0}\,=\,(\mathcal{L}\mathfrak{h})\,,\quad(\mathcal{L}\mathfrak{g})_{\lambda}\,=\,\mathcal{L}\otimes\,\mathfrak{g}_{\lambda}.$ (4.27) 3. 3. $\mathcal{L}\mathfrak{g}$ is the direct sum of the weight spaces: $\mathcal{L}\mathfrak{g}\,=\,\mathcal{K}\mathfrak{h}\,\oplus\,\mathcal{K}^{\bot}\mathfrak{h}\,\oplus\,\oplus_{\lambda\in\Phi}\,(\mathcal{L}\otimes\,\mathfrak{g}_{\lambda})\,.$ (4.28) ###### Proof. Let $\lambda\in\Phi_{\mathcal{L}}\,$. There exists a weight vector $\xi\in\mathcal{L}\mathfrak{g}$ with the weight $\lambda$: $\,[\phi\otimes h\,,\,\xi\,]\,=\,\lambda(\phi\otimes h)\xi\,$ for any $\phi\otimes h\in\mathcal{K}\mathfrak{h}\,$. We define $\check{\lambda}\in Hom(\,\mathfrak{h}\,,\mathbf{C}\,)$ by the formula $\check{\lambda}(h)=\lambda(\phi^{+(0,0,1)}\otimes h)$. Then $\check{\lambda}$ becomes a root of the representation $ad_{\mathfrak{h}}$ on $\mathfrak{g}$: $\,[h,x]=[\,\phi^{+(0,0,1)}\otimes h\,,\,\phi^{+(0,0,1)}\otimes x\,]\,=\,\check{\lambda}(h)x\,$. Conversely let $\xi=\psi_{1\cdots m}\ast x_{1\cdots m}\in\mathcal{L}\mathfrak{g}$. We suppose that each $x_{i}\in\mathfrak{g}$ is a weight vector with root $\beta_{i}\in\Phi$, $i=1,\cdots,m$. General elements of $\mathcal{L}\mathfrak{g}$ are linear combinations of such vectors. It follows from (4.20) that $ad_{\phi\otimes h}\xi=\,\left(\Sigma_{i=1}^{m}\beta_{i}(h)\phi\right)\,\xi\,,\quad\forall\phi\otimes h\in\mathcal{K}\mathfrak{h}.$ Hence $\Sigma_{i=1}^{m}\beta_{i}(h)\phi\in\Phi_{\mathcal{L}}$, and $\xi$ is a weight vector of $ad_{\phi\otimes h}$. The relation extends linearly to $\mathcal{L}\mathfrak{g}$. Thus we have proved the first assertion. From (4.20) we have $\mathcal{L}\otimes\mathfrak{g}_{\alpha}\,\subset\,(\mathcal{L}\mathfrak{g})_{\alpha}\,$ for any $\alpha\in\Phi$. Lemma 4.7 shows that $\mathcal{L}\mathfrak{h}\,\subset\,(\mathcal{L}\mathfrak{g})_{0}$. Then (4.19) yields that $\phi_{i_{1}i_{2}\cdots i_{t}}\otimes e_{i_{1}i_{2}\cdots i_{t}}$ and $\phi_{i_{1}i_{2}\cdots i_{t}}\otimes f_{i_{1}i_{2}\cdots i_{t}}$ are weight vectors. Thus all Lie products of generators $\left\\{\,\phi\otimes e_{i}\,,\,\phi\otimes f_{i}\,,\,\phi\otimes h_{i}\,;\,\phi\in\mathcal{L}\,,\,i=1,\cdots,l\,\,\right\\}$ are weight vectors. Since every element of $\mathcal{L}\mathfrak{g}$ is a linear combination of products of these weight vectors we deduce from (4.23) and the fact $\Phi\simeq\Phi_{\mathcal{L}}$ that $\mathcal{L}\mathfrak{g}\,=\,(\mathcal{L}\mathfrak{g})_{0}\,\oplus\,\oplus_{\alpha\in\Phi}(\mathcal{L}\mathfrak{g})_{\alpha}\,.$ (4.29) Now the simple roots $\alpha_{1},\cdots,\alpha_{l}\in\Phi$ are linearly independent, so the only monomials which have weight $\alpha_{j}$ are the weight vectors of $\mathcal{L}\otimes\mathfrak{g}_{\alpha_{j}}$. We conclude $\,(\mathcal{L}\mathfrak{g})_{\alpha_{j}}\,=\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}_{\alpha_{j}}\,.$ (4.30) Hence $\,(\mathcal{L}\mathfrak{g})_{\alpha}\,=\,\mathcal{L}\otimes_{\mathbf{C}}\mathfrak{g}_{\alpha}\,$ for all $\alpha\in\Phi$. Therefore (4.29) becomes $\mathcal{L}\mathfrak{g}\,=\,(\mathcal{L}\mathfrak{g})_{0}\,\oplus\,\oplus_{\alpha\in\Phi}(\mathcal{L}\otimes\mathfrak{g}_{\alpha})\,.$ (4.31) Now we shall prove $(\mathcal{L}\mathfrak{g})_{0}=\mathcal{L}\mathfrak{h}\,$. We regard $\mathcal{L}\mathfrak{g}$ as a $\mathcal{K}\mathfrak{h}$-module. Hence $\mathcal{L}\mathfrak{h}$ is a $\mathcal{K}\mathfrak{h}$-submodule. $\mathcal{L}\mathfrak{h}$ is contained in $(\mathcal{L}\mathfrak{g})_{0}$ by Lemma 4.7. If $\,\mathcal{L}\mathfrak{h}\neq(\mathcal{L}\mathfrak{g})_{0}\,$ the $\mathcal{K}\mathfrak{h}$-module $(\mathcal{L}\mathfrak{g})_{0}/\mathcal{L}\mathfrak{h}$ will have a 1-dimensional submodule $M/\mathcal{L}\mathfrak{h}$ on which $\mathcal{K}\mathfrak{h}$ acts with weight $0$. That is, $[\,\mathcal{K}\mathfrak{h},\,M/\mathcal{L}\mathfrak{h}\,]=0$. Then $[\,\mathcal{K}\mathfrak{h},\,M\,]\subset\,\mathcal{L}\mathfrak{h}\,$ and $M$ is a $\mathcal{K}\mathfrak{h}$-submodule of $\mathcal{L}\mathfrak{h}$. That is a contradiction. ∎ We know that any weight $\lambda\in\Phi$ is of the form $\sum_{i=1}^{l}\,k_{i}\alpha_{i}$, $k_{i}\in\mathbf{Z}$. Moreover a non-zero weight $\lambda$ has the form $\lambda=\sum_{i=1}^{l}k_{i}\alpha_{i},\,k_{i}\in\mathbf{Z}$, with all $k_{i}\geq 0$ or all $k_{i}\leq 0$. Therefore $\displaystyle\mathcal{L}\mathfrak{e}$ $\displaystyle=$ $\displaystyle\sum_{\lambda\in\Phi^{+}}\mathcal{L}\otimes_{\mathbf{R}}\mathfrak{g}_{\lambda}$ (4.32) $\displaystyle\mathcal{L}\mathfrak{f}$ $\displaystyle=$ $\displaystyle\sum_{\lambda\in\Phi^{-}}\mathcal{L}\otimes_{\mathbf{R}}\mathfrak{g}_{\lambda}$ (4.33) From the above discussion we have the following ###### Theorem 4.10. The $\mathfrak{g}$-current algebra $\mathcal{L}\mathfrak{g}$ has the following triangular decomposition $\displaystyle\mathcal{L}\mathfrak{g}$ $\displaystyle=$ $\displaystyle\,\mathcal{L}\mathfrak{e}\,\oplus\,\mathcal{L}\mathfrak{h}\,\oplus\,\mathcal{L}\mathfrak{f}\,.$ $\displaystyle\mathcal{L}\mathfrak{e}\,$ $\displaystyle=$ $\displaystyle\,\mathcal{L}\otimes\,\mathfrak{e}\,,\quad\mathcal{L}\mathfrak{f}\,=\,\mathcal{L}\otimes\,\mathfrak{f}\,,$ $\displaystyle\mathcal{L}\mathfrak{h}\,$ $\displaystyle=$ $\displaystyle\,\mathcal{K}\mathfrak{h}\,\oplus\,\mathcal{K}^{\bot}\mathfrak{h}\,.$ ###### Corollary 4.11. $\mathcal{L}\mathfrak{g}\ominus\,(\,\mathcal{L}\otimes\mathfrak{g}\,)\,=\,\mathcal{K}^{\bot}\mathfrak{h}\,.$ (4.34) ## 5 Central extensions of the $\mathfrak{g}$-current algebra ### 5.1 Central extensions of the $\mathfrak{g}$-current algebra $\mathcal{L}\mathfrak{g}$ Let $(V,\,[\,\cdot\,,\,\cdot\,]_{V}\,)$ be a quaternion Lie algebra. A central extension of $(V,\,[\,\cdot\,,\,\cdot\,]_{V}\,)$ is a quaternion Lie algebra $(W,\,[\,\cdot\,,\,\cdot\,]_{W}\,)$ such that $W=V\oplus Z$ ( direct sum ) and $Z$ is contained in the center of $W$; $\,Z\,\subset\\{w\in W\,:\,[w,x]_{W}=0\,,\forall x\in W\\}\,,$ and such that $\,[\,\cdot\,,\,\cdot\,]_{W}$ restricts to $\,[\,\cdot\,,\,\cdot\,]_{V}$. Let $\mathfrak{g}$ be a simple Lie algebra and let $\mathcal{L}\mathfrak{g}$ be the $\mathfrak{g}$-current algebra. We write the invariant bilinear form ( Killing form ) on $\mathfrak{g}$ by $(x\|y)=\,Trace\,(xy).$ We have $(xy\|z)=(yz\|x)$. In Proposition 3.9 we introduced 2-cocycles $\\{c_{k}\,;\,k=0,1,2\,\\}$ on the space of current $\mathcal{L}$. We extend them to the 2-cocycles on the $\mathfrak{g}$-current algebra $\mathcal{L}\mathfrak{g}$ by $c_{k}(\,\phi_{1}\otimes x\,,\,\phi_{2}\otimes y\,)=\,(x\|y)\,c_{k}(\phi_{1},\phi_{2})\,,\quad k=0,1,2,$ (5.1) for $\phi_{1},\,\phi_{2}\in\mathcal{L}$ and $x,\,y\in\mathfrak{g}$. Associated to the the 2-cocycles $c_{k}$, $k=0,1,2$, we have the central extensions of $\mathcal{L}\mathfrak{g}\,$. ###### Theorem 5.1. Let $a_{k}$, $k=0,1,2$, be three indefinite numbers. Put $\mathcal{L}\mathfrak{g}(a)\,=\,\mathcal{L}\mathfrak{g}\oplus(\oplus_{k=0,1,2}\mathbf{C}a_{k})\,.$ (5.2) We endow $\mathcal{L}\mathfrak{g}(a)\,$ with the following bracket: $\displaystyle[\,\phi\otimes x\,,\,\psi\otimes y\,]^{a}$ $\displaystyle=$ $\displaystyle[\phi\otimes x\,,\,\psi\otimes y\,]+(x\|y)\,\sum_{k=0}^{2}\,c_{k}(\phi,\psi)\,a_{k}\,,$ $\displaystyle[\,a_{k}\,,\,\phi\otimes x\,]^{a}$ $\displaystyle=$ $\displaystyle 0\,,\,\,k=0,1,2,$ (5.3) for $\phi\otimes x\,,\,\psi\otimes y\in\mathcal{L}\otimes\mathfrak{g}$. The bracket is extended to $\mathcal{L}\mathfrak{g}$. The conjugation automorphism $\sigma$ is extended to $\mathcal{L}\mathfrak{g}(a)$ by $\,\sigma a_{k}=a_{k}$, $k=0,1,2$. We shall further complete the central extension of the current algebra $\mathcal{L}\mathfrak{g}$ by adjoining a derivation coming from the radial vector field $\mathbf{n}$ on $S^{3}$. ###### Lemma 5.2. The derivation $\mathbf{n}$ on $S^{3}\mathbf{H}$ restricts to the outer derivation on $\mathcal{L}$. $\mathbf{n}$ is extended to an outer derivation of the Lie algebra $\mathcal{L}\mathfrak{g}$ by $\,\mathbf{n}\,(\phi\otimes x\,)\,=\,(\,\mathbf{n}\phi\,)\otimes x,\qquad\,\phi\in\mathcal{L}\,,\,x\in\mathfrak{g}\,.$ (5.4) Then $\mathbf{n}$ acts on $\mathcal{L}\mathfrak{g}(a)$ by killing the $a_{k}$’s. From Propositions 3.10 and 3.11 we have $\displaystyle\,[\,\mathbf{n}(\phi_{1}\otimes x_{1})\,,\,\phi_{2}\otimes x_{2}\,]^{a}\,+\,[\,\phi_{1}\otimes x_{1}\,,\,\mathbf{n}(\phi_{2}\otimes x_{2})\,]^{a}$ $\displaystyle\,=\,(\mathbf{n}\phi_{1}\cdot\phi_{2})\otimes\,x_{1}x_{2}\,-\,(\phi_{2}\cdot\mathbf{n}\phi_{1})\otimes\,x_{2}x_{1}\,\,+\,(\phi_{1}\cdot\mathbf{n}\phi_{2})\otimes\,x_{1}x_{2}\,\,-\,(\mathbf{n}\phi_{2}\cdot\phi_{1})\otimes\,x_{2}x_{1}\,$ $\displaystyle+\,(x_{1}\|x_{2})\sum_{k}\left(\,{c}_{k}(\mathbf{n}\phi_{1},\,\phi_{2})\,+\,c_{k}(\phi_{1},\,\mathbf{n}\phi_{2})\right)a_{k}\,$ $\displaystyle\,=\,\mathbf{n}(\phi_{1}\cdot\phi_{2})\otimes x_{1}x_{2}\,-\,\mathbf{n}(\phi_{2}\cdot\phi_{1})\otimes x_{2}x_{1}\,=\mathbf{n}\left(\,[\,\phi_{1}\otimes x_{1}\,,\,\phi_{2}\otimes x_{2}\,]^{a}\,\,\right)\,.$ (5.5) Hence $\mathbf{n}$ is a derivation that acts on the Lie algebra $\mathcal{L}\mathfrak{g}(a)\,$. ###### Theorem 5.3. Let $a_{k}$, $k=0,1,2$, and $\,{\rm n}$ be the above indefinite elements. We consider the $\mathbf{C}$-vector space: $\widehat{\mathfrak{g}\,}\,=\,\mathcal{L}\mathfrak{g}\oplus(\oplus_{k=0}^{2}\,\mathbf{C}\,a_{k})\oplus(\mathbf{C}\,{\rm n})\,.$ (5.6) We endow $\,\widehat{\mathfrak{g}}\,$ with the following bracket extended to $\,\widehat{\mathfrak{g}}\,$: $\displaystyle[\,\phi\otimes x\,,\,\psi\otimes y\,]_{\widehat{\mathfrak{g}}}\,=\,[\,\phi\otimes x\,,\,\psi\otimes y\,]^{a}$ $\displaystyle\quad=\,[\,\phi\otimes x\,,\,\psi\otimes y\,]\,+\,(x\|y)\,\sum_{k=0}^{2}\,c_{k}(\phi,\psi)\,a_{k}\,,$ (5.7) $\displaystyle\,[\,a_{k}\,,\,\phi\otimes x\,]_{\widehat{\mathfrak{g}}}\,=0\,,\quad[\,{\rm n}\,,\,\phi\otimes x\,]_{\widehat{\mathfrak{g}}}=\,\mathbf{n}\phi\otimes x\,,$ (5.8) $\displaystyle[\,{\rm n}\,,\,a_{k}\,]_{\widehat{\mathfrak{g}}}\,=0,\quad k=0,1,2\,,$ for $x,y\in\mathfrak{g}$ and $\phi,\,\psi\,\in\,\mathcal{L}\,$. The involution $\sigma$ is extended to $\widehat{\mathfrak{g}}\,$ by $\sigma(\,\phi\otimes x)=\sigma\phi\otimes x\,,\quad\sigma a_{k}=0\,,\quad\sigma{\rm n}\,={\rm n}\,.$ Then we get a quaternion Lie algebra $\left(\,\widehat{\mathfrak{g}}\,,\,[\,\cdot,\cdot\,]_{\widehat{\mathfrak{g}}}\,\right)$. ###### Proof. We write simply $[\,,\,]$ instead of $[\,,\,]_{\widehat{\mathfrak{g}}}\,$. It is enough to prove the following Jacobi identity: $[\,[\,{\rm n}\,,\,\phi_{1}\otimes x_{1}\,]\,,\,\phi_{2}\otimes x_{2}\,]+[\,[\phi_{1}\otimes x_{1},\phi_{2}\otimes x_{2}\,]\,,\,{\rm n}\,]\,+\,[\,[\phi_{2}\otimes x_{2},\,{\rm n}\,],\,\phi_{1}\otimes x_{1}\,]=0.$ From the defining equation (5.8) and the equation (5.5), the sum of the 1st and 3rd terms is equal to $[\,[\,n,\,\phi_{1}\otimes x_{1}]\,,\,\phi_{2}\otimes x_{2}\,]\,+\,[\,\phi_{1}\otimes x_{1}\,,\,[n\,\,\phi_{2}\otimes x_{2}]\,]\,=\,\mathbf{n}\left(\,[\,\phi_{1}\otimes x_{1}\,,\,\phi_{2}\otimes x_{2}\,]\,\,\right)\,,$ which is equal to $\,-\,[\,[\phi_{1}\otimes x_{1},\phi_{2}\otimes x_{2}\,]\,,\,{\rm n}\,]$. ∎ ###### Proposition 5.4. The centralizer of $n\in\,\widehat{\mathfrak{g}}\,$ is given by $(\,\mathcal{L}[0]\,\otimes_{\mathbf{C}}\mathfrak{g}\,)\,\oplus\,(\oplus_{k}\mathbf{C}a_{k}\,)\oplus\mathbf{C}{\rm n}\,.$ Here $\,\mathcal{L}[0]$ is the subspace in $\mathcal{L}$ generated by $\phi_{1}\cdots\phi_{n}$ with $\phi_{i}$ being $\phi_{i}=\phi^{\pm(m_{i},l_{i},k_{i})}$ such that $\sum_{i;\,\phi_{i}=\phi^{+(m_{i},l_{i},k_{i})}}\,m_{i}-\sum_{i;\,\phi_{i}=\phi^{-(m_{i},l_{i},k_{i})}}\,(m_{i}+3)=0\,.$ $\,\mathcal{L}[0]\mathfrak{g}\,$ is the subalgebra of $\widehat{\mathfrak{g}}\,$ generated by $\,\mathcal{L}[0]\,\otimes_{\mathbf{C}}\mathfrak{g}\,$. ###### Definition 5.5. We call the quaternion Lie algebra $\,\widehat{\mathfrak{g}}\,$ the affine current algebra over $\mathfrak{g}$ : $\widehat{\mathfrak{g}\,}\,=\,\mathcal{L}\mathfrak{g}\oplus(\oplus_{k=0}^{2}\,\mathbf{C}\,a_{k})\oplus(\mathbf{C}\,{\rm n})\,.$ (5.9) ### 5.2 Root space decomposition of the current algebra $\,\widehat{\mathfrak{g}}\,$ Let $\widehat{\mathfrak{g}\,}\,=\,\mathcal{L}\mathfrak{g}\oplus(\oplus_{k=0}^{2}\,\mathbf{C}\,a_{k})\oplus(\mathbf{C}\,{\rm n})\,$ be the affine current algebra over $\mathfrak{g}$, Definition 5.5. Let $\widehat{\mathfrak{h}}\,=\,\mathfrak{h}\oplus(\oplus_{k}\mathbf{C}a_{k})\oplus(\mathbf{C}{\rm n}\,)\,.$ (5.10) Where we applied the identification $\mathfrak{h}\ni h\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\phi^{+(0,0,1)}\otimes h\in\mathcal{L}\mathfrak{g}$. $\widehat{\mathfrak{h}}\,$ is a commutative subalgebra of $\widehat{\mathfrak{g}}$. From the discussion in previous sections, in particular by virtue of Theorem 4.10, Corollary 4.11, (4.27) and (5.6) , we know that any element $\xi\,\in\widehat{\mathfrak{g}}$ is written in the form: $\displaystyle\xi$ $\displaystyle=$ $\displaystyle\,x\,+\,\sum p_{j}a_{j}+q{\rm n}\,,\quad x\in\mathcal{L}\mathfrak{g}\,,\quad p_{j},\,q\in\mathbf{C}\,,\,\,j=0,1,2\,,\,$ (5.11) $\displaystyle x$ $\displaystyle=$ $\displaystyle\,y+\,\sum_{\alpha\in\Phi}\,\varphi_{\alpha}\otimes x_{\alpha}\,,\quad\varphi_{\alpha}\in\mathcal{L}\,,\quad x_{\alpha}\in\mathfrak{g}_{\alpha}\,,$ $\displaystyle y$ $\displaystyle=$ $\displaystyle\kappa+z\in\mathcal{L}\mathfrak{h}\,,\quad\kappa\in\mathcal{K}\mathfrak{h}\,,\quad z\in\mathcal{K}^{\bot}\mathfrak{h}\,$ Any element of $\,\widehat{\mathfrak{h}}\,$ is written in the form $\,\hat{h}=\phi^{+(0,0,1)}\otimes h+\sum s_{k}a_{k}+\,t\,{\rm n}\,,\quad h\in\mathfrak{h}\,,\,s_{k},\,t\in\mathbf{C}\,.$ From Lemma 4.7 we have $[\phi\otimes h\,,y\,]=0$ for any $\phi\in\mathcal{K}$, $h\in\mathfrak{h}$ and $y\in\mathcal{L}\mathfrak{h}$, in particular $[\phi^{+(0,0,1)}\otimes h\,,y\,]=0$. So we see that the adjoint action of $\,\hat{h}=h+\sum s_{i}a_{i}+t{\rm n}\in\widehat{\mathfrak{h}}\,$ on $\,\xi=y+\,\sum_{\alpha}\,\varphi_{\alpha}\otimes x_{\alpha}+\sum p_{j}a_{j}+q{\rm n}\in\widehat{\mathfrak{g}}\,$ becomes $ad(\hat{h})(\xi)\,=\,\sum_{\alpha}\,\alpha(h)\varphi_{\alpha}\otimes x_{\alpha}\,+\,t\,\sum_{\alpha}\,(\mathbf{n}\,\varphi_{\alpha})\otimes x_{\alpha}\,+\,t\,[{\rm n}\,,\,y]\,.$ (5.12) Let $\widehat{\mathfrak{h}}^{\ast}$ be the dual space of $\widehat{\mathfrak{h}}$: $\widehat{\mathfrak{h}}^{\ast}\,=\,Hom_{\mathbf{C}}(\widehat{\mathfrak{h}}\,,\mathbf{C})\,.$ An element $\alpha$ of the dual space $\mathfrak{h}^{}$ of $\mathfrak{h}$ is regarded as a element of $\,\widehat{\mathfrak{h}}^{\,\ast}$ by putting $\left\langle\,\alpha,a_{k}\,\right\rangle=\left\langle\,\alpha,{\rm n}\,\right\rangle=0,\quad k=0,1,2\,.$ So $\Phi\subset\mathfrak{h}^{}$ is seen to be a subset of $\,\widehat{\mathfrak{h}}^{\,}$. We define $\delta\,,\,\Lambda_{k}\,\in\widehat{\mathfrak{h}}^{\,}$, $k=0,1,2$, by $\displaystyle\left\langle\delta,\alpha_{i}^{\vee}\,\right\rangle$ $\displaystyle=\left\langle\,\Lambda_{k},\alpha_{i}^{\vee}\,\right\rangle=0,$ $\displaystyle\left\langle\,\delta,a_{k}\,\right\rangle$ $\displaystyle=0\,,\qquad\left\langle\,\delta,{\rm n}\,\right\rangle=1\,,$ (5.13) $\displaystyle\left\langle\,\Lambda_{k},a_{k}\,\right\rangle$ $\displaystyle=1\,,\qquad\left\langle\,\Lambda_{k},{\rm n}\,\right\rangle=0\,,\quad 1\leqq i\leqq l,\,\,k=0,1,2\,.$ Then $\alpha_{1},\,\cdots\,,\alpha_{l},\,\delta,\,\Lambda_{0},\,\Lambda_{1},\,\Lambda_{2}\,$ give the basis of $\widehat{\mathfrak{h}}^{\ast}$. We shall investigate the decomposition of $\,\widehat{\mathfrak{g}}\,$ into a direct sum of the simultaneous eigenspaces of $ad\,(\hat{h})\,$, $\hat{h}\in\widehat{\mathfrak{h}}\,$. For a 1-dimensional representation $\lambda\in\widehat{\mathfrak{h}}^{\ast}$ we put $\widehat{\mathfrak{g}}_{\lambda}\,=\,\left\\{\xi\in\widehat{\mathfrak{g}}\,;\quad\,[\,\hat{h},\,\xi\,]_{\widehat{\mathfrak{g}}}\,=\,\langle\lambda,\hat{h}\rangle\,\xi\quad\mbox{ for }\,\forall\hat{h}\in\,\widehat{\mathfrak{h}}\,\right\\}.$ (5.14) $\lambda$ is called a root of the representation $\left(\,\widehat{\mathfrak{g}}\,,\,ad(\widehat{\mathfrak{h}}\,)\right)$ if $\lambda\neq 0$ and $\,\widehat{\mathfrak{g}}_{\lambda}\neq 0$. $\,\widehat{\mathfrak{g}}_{\lambda}$ is called the root space of $\lambda\,$. Let $\widehat{\Phi}$ be the set of roots: $\widehat{\Phi}=\left\\{\lambda=\alpha+\sum_{j=0}^{2}n_{j}\Lambda_{j}\,+\,k_{0}\delta\in\widehat{\mathfrak{h}}^{\ast}\,;\,\alpha=\sum_{i=1}^{l}\,k_{i}\alpha_{i}\in\Phi,\,k_{i},\,n_{j}\in\mathbf{Z},\,0\leq i\leq l,\,j=0,1,2\,\right\\}.$ The set $\widehat{\Pi}=\\{\,\alpha_{1},\cdots,\alpha_{l},\,\Lambda_{0},\Lambda_{1},\Lambda_{2},\,\delta\,\\}$ forms a fundamental basis of $\,\widehat{\Phi}\,$. Thus we have $\widehat{\mathfrak{g}}\,=\,\widehat{\mathfrak{g}}_{0}\,\oplus\,\left(\oplus_{\lambda\in\widehat{\Phi}}\,\widehat{\mathfrak{g}}_{\lambda}\,\right)\,\,.$ (5.15) We investigate the root spaces $\,\widehat{\mathfrak{g}}_{\lambda}\,$ for $(i)\,\lambda=\alpha+k\delta,\,0\neq\alpha\in\Phi\,,\quad(ii)\,\lambda=k\delta,\quad k\neq 0,\quad(iii)\,\lambda=0\delta\,\quad\mbox{and }\,(iv)\,\lambda=0\,.$ We may assume that the weight vector $\xi\in\widehat{\mathfrak{g}}$ of each weight $\lambda$ takes the form $\xi=y+\sum_{\alpha\in\Phi}\varphi_{\alpha}\otimes x_{\alpha}$ because others do not contribute to give weight, see (5.12). Let $x\in\mathfrak{g}_{\alpha}$ for $\alpha\in\Phi$, $\alpha\neq 0$, and let $\varphi\in\mathcal{L}[m]$ for $m\in\mathbf{Z}$, that is, $\varphi$ is $m$-homogeneous, (3.24). From (5.12) we have $\displaystyle[\,\phi\otimes h,\,\varphi\otimes x\,]_{\widehat{\mathfrak{g}}}$ $\displaystyle=$ $\displaystyle(\phi\,\varphi)\otimes[\,h,\,x\,]\,=\left\langle\alpha,h\right\rangle\varphi\otimes x,$ $\displaystyle[\,\mathbf{n},\,\varphi\otimes x\,]_{\widehat{\mathfrak{g}}}$ $\displaystyle=$ $\displaystyle\frac{m}{2}\varphi\otimes x,$ for any $\phi\otimes h\in\mathcal{K}\mathfrak{h}$. That is, $[\,\hat{h}\,,\varphi\otimes x]_{\widehat{\mathfrak{g}}}=\left\langle\frac{m}{2}\delta+\alpha\,,\,\hat{h}\,\right\rangle(\varphi\otimes x)\,,$ for every $\hat{h}\in\widehat{\mathfrak{h}}$, Then we see $\mathcal{L}[m]\otimes\mathfrak{g}_{\alpha}\,\subset\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}$. Now let $y\in\mathcal{L}\mathfrak{h}$. It is written by a linear combination of terms of the form $y^{\prime}=\phi_{i_{1}i_{2}\cdots i_{t}}\otimes h_{i_{1}i_{2}\,\cdots i_{t}}$ with $h_{j}\in\mathfrak{h}$ and $\phi_{j}\in\mathcal{L}[m_{j}\,]$, $j=i_{1},\cdots,i_{t}$, so that $\mathbf{n}y^{\prime}=(\frac{1}{2}\,\sum_{k=1}^{t}\,m_{k}\,\,)\phi_{i_{1}i_{2}\cdots i_{t}}\otimes h_{i_{1}i_{2}\,\cdots i_{t}}\,,$ and we find that $y^{\prime}\in\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}$ with $m=\sum_{k=1}^{t}\,m_{k}\in\mathbf{Z}$. Hence $\mathcal{L}\mathfrak{h}\,\subset\,\,\widehat{\mathfrak{g}}_{0\delta}\oplus\oplus_{m\neq 0}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,,$ with $\,\widehat{\mathfrak{g}}_{0\delta}=\mathcal{L}[0]\otimes\mathfrak{h}\,$, and $\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}=\mathcal{L}[m]\otimes\mathfrak{h}\,$. ###### Proposition 5.6. We have the following relations: 1. 1. $\left[\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,,\,\widehat{\mathfrak{g}}_{\frac{n}{2}\delta+\beta}\,\right]_{\widehat{\mathfrak{g}}}\,\subset\,\widehat{\mathfrak{g}}_{\frac{m+n}{2}\delta+\alpha+\beta}\,\,,$ (5.16) for $\alpha,\,\beta\in\Phi$ and for $m,n\in\mathbf{Z}$. 2. 2. $\left[\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,,\,\widehat{\mathfrak{g}}_{\frac{n}{2}\delta}\,\right]_{\widehat{\mathfrak{g}}}\,\subset\,\widehat{\mathfrak{g}}_{\frac{m+n}{2}\delta}\,,$ (5.17) for $m,n\in\mathbf{Z}$. The Proposition is proved by a standard argument using the properties of Lie bracket. ###### Theorem 5.7. 1. 1. $\displaystyle\widehat{\Pi}$ $\displaystyle=$ $\displaystyle\left\\{\frac{m}{2}\,\delta+\alpha\,;\quad\alpha\in\Pi\,,\,m\in\mathbf{Z}\,\right\\}$ (5.18) $\displaystyle\bigcup\left\\{\frac{m}{2}\,\delta\,;\quad m\in\mathbf{Z}\,\right\\}\,.$ is a base of $\,\widehat{\Phi}$. 2. 2. For $\alpha\in\Phi$, $\alpha\neq 0$ and $m\in\mathbf{Z}$, we have $\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,=\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{g}_{\alpha}\,.$ (5.19) 3. 3. $\displaystyle\,\widehat{\mathfrak{g}}_{0\delta}$ $\displaystyle=$ $\displaystyle\mathcal{L}[0]\otimes_{\mathbf{C}}\mathfrak{h}\,\supset\widehat{\mathfrak{h}},$ (5.20) $\displaystyle\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}$ $\displaystyle=$ $\displaystyle\,\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{h}\,,\quad\mbox{for $0\neq m\in\mathbf{Z}$ . }\,$ (5.21) 4. 4. $\widehat{\mathfrak{g}}$ has the following decomposition: $\widehat{\mathfrak{g}}\,=\,\widehat{\mathfrak{g}}_{0\delta}\oplus\left(\oplus_{0\neq m\in\mathbf{Z}}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,\right)\oplus\,\,\left(\oplus_{\alpha\in\Phi,\,m\in\mathbf{Z}}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,\right)$ (5.22) ###### Proof. First we prove the second assertion. We have already proved $\mathcal{L}[m]\otimes\mathfrak{g}_{\alpha}\,\subset\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}$. Conversely, for $m\in\mathbf{Z}$ and $\xi\in\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}$, we shall show that $\xi$ has the form $\,\phi\otimes x\,$ with $\phi\in\mathcal{L}[m]$ and $x\in\mathfrak{g}_{\alpha}\,$. Let $\xi=\psi\otimes x\,+\sum\,p_{k}a_{k}+q{\rm n}$. Then $\displaystyle[\hat{h},\xi]_{\widehat{\mathfrak{g}}}=[\,\phi^{+(0,0,1)}\otimes h+\sum s_{k}a_{k}+t{\rm n}\,,\,\psi\otimes x\,+\sum\,p_{k}a_{k}+q{\rm n}\,]_{\widehat{\mathfrak{g}}}=\,\psi\otimes[\,h\,,\,x\,]$ $\displaystyle\qquad+\,t(\,\sum_{n\in\mathbf{Z}}\,\frac{n}{2}\psi_{n}\,\otimes x\,)$ for any $\hat{h}=\phi^{+(0,0,1)}\otimes h+\sum s_{k}a_{k}+tn\in\widehat{\mathfrak{h}}\,$, where $\psi=\sum_{n}\psi_{n}$ is the homogeneous decomposition of $\psi$. From the assumption we have $\displaystyle[\,\hat{h},\xi\,]_{\widehat{\mathfrak{g}}}\,$ $\displaystyle=$ $\displaystyle\,\langle\,\frac{m}{2}\delta+\alpha\,,\,\hat{h}\,\rangle\,\xi\,$ $\displaystyle=$ $\displaystyle<\alpha,\,h>\psi\otimes x\,+(\frac{m}{2}t+<\alpha,\,h>)(\sum p_{k}a_{k}+q{\rm n})\,$ $\displaystyle\quad+\,\frac{m}{2}t\,(\sum_{k}\,\psi_{k})\otimes x.$ Comparing the above two equations we have $p_{k}=q=0$, and $\psi_{k}=0$ for all $k$ except for $k=m$. Therefore $\psi\in\mathcal{L}[m]$. We also have $[\hat{h},\xi]_{\widehat{\mathfrak{g}}}=\psi\otimes[h,x]=\langle\alpha,\,h\rangle\,\psi\otimes x$ for any $\hat{h}=\phi^{+(0,0,1)}\otimes h+\sum s_{k}a_{k}+td\in\widehat{\mathfrak{h}}$. Hence $x$ has weight $\alpha$ and $\xi=\psi_{m}\otimes x\in\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,$. We have proved $\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}=\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{g}_{\alpha}\,.$ Now we shall show $\mathcal{L}\mathfrak{h}\,\supset\,\,\widehat{\mathfrak{g}}_{0\delta}\oplus\oplus_{m\neq 0}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,.$ where $\,\widehat{\mathfrak{g}}_{0\delta}=\mathcal{L}[0]\otimes\mathfrak{h}\,$, and $\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}=\mathcal{L}[m]\otimes\mathfrak{h}\,$. The converse implication has been proved before, so both sides coincide. Let $\xi=\in\widehat{\mathfrak{g}}_{0\delta}\oplus\oplus_{m\neq 0}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta}\,$ which we may assume to be the form $\xi=y+\sum\,p_{k}a_{k}+q{\rm n}$. It satisfies $[\,\hat{h},\xi\,]_{\widehat{\mathfrak{g}}}\,=\,\langle\,\frac{m}{2}\delta\,,\,\hat{h}\,\rangle\,\xi\,,\quad\forall\widehat{h}\in\widehat{\mathfrak{h}}\,,$ for $m=0$ or $m\neq 0$. From (5.12) we find $\xi=y\in\mathcal{L}[m]\mathfrak{h}$. The above discussion yields the first and the fourth assertions. ∎ ###### Corollary 5.8. $\oplus_{\Phi\ni\alpha\neq 0}\,\widehat{\mathfrak{g}}_{\frac{m}{2}\delta+\alpha}\,=\,\mathcal{L}[m]\otimes_{\mathbf{C}}\mathfrak{g}.$ ### 5.3 Chevalley generators of $\,\widehat{\mathfrak{g}}$ By the natural embedding of $\mathfrak{g}$ in $\widehat{\mathfrak{g}}$ we have the vectors $\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle\phi^{+(0,0,1)}\otimes h_{i}\,\in\widehat{\mathfrak{h}},\,$ $\displaystyle e_{i}$ $\displaystyle=$ $\displaystyle\phi^{+(0,0,1)}\otimes e_{i}\,\in\widehat{\mathfrak{g}}_{0\delta+\alpha_{i}},\quad f_{i}=\phi^{+(0,0,1)}\otimes f_{i}\,\in\widehat{\mathfrak{g}}_{0\delta-\alpha_{i}},\qquad i=1,\cdots,l\,.$ Then $\displaystyle\left[e_{i}\,,f_{j}\,\right]_{\widehat{\mathfrak{g}}}$ $\displaystyle=$ $\displaystyle\,\delta_{ij}\,h_{i}\,,$ $\displaystyle\left[h_{i}\,,e_{j}\,\right]_{\widehat{\mathfrak{g}}}$ $\displaystyle=$ $\displaystyle\,a_{ij}\,e_{j},\quad\left[h_{i}\,,f_{j}\,\right]_{\widehat{\mathfrak{g}}}=\,-a_{ij}\,f_{j}\,,\quad 1\leq i,j\leq l.$ (5.23) We have obtained a part of generators of $\widehat{\mathfrak{g}}$ that come naturally from $\mathfrak{g}$. We want to augment these generators to the Chevalley generators of $\widehat{\mathfrak{g}}$. We take the following set of generators of the algebra $\mathcal{L}$: $\displaystyle I$ $\displaystyle=\phi^{+(0,0,1)}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),\,\qquad J$ $\displaystyle=\phi^{+(0,0,0)}=\left(\begin{array}[]{c}0\\\ -1\end{array}\right)\,,$ (5.28) $\displaystyle\kappa$ $\displaystyle=\phi^{+(1,0,1)}\,=\,\left(\begin{array}[]{c}z_{2}\\\ -\overline{z}_{1}\end{array}\right),\qquad\lambda$ $\displaystyle=\,\phi^{-(0,0,0)}=\left(\begin{array}[]{c}z_{2}\\\ \overline{z}_{1}\end{array}\right).$ (5.33) We put $\displaystyle\kappa_{\ast}$ $\displaystyle=$ $\displaystyle\,\frac{-\sqrt{-1}}{\sqrt{2}}\phi^{+(1,1,2)}+\frac{\sqrt{-1}}{2}(\phi^{-(0,0,0)}-\phi^{+(1,0,1)})\,=\,\sqrt{-1}\left(\begin{array}[]{c}\overline{z}_{2}\\\ \overline{z}_{1}\end{array}\right)\,$ $\displaystyle\lambda_{\ast}$ $\displaystyle=$ $\displaystyle\,\frac{-\sqrt{-1}}{\sqrt{2}}\phi^{+(1,1,2)}-\,\frac{\sqrt{-1}}{2}(\phi^{-(0,0,0)}-\phi^{+(1,0,1)})\,=\,\sqrt{-1}\left(\begin{array}[]{c}\overline{z}_{2}\\\ -\overline{z}_{1}\end{array}\right)\,$ ###### Lemma 5.9. 1. 1. $\kappa\,\in\mathcal{L}[1]\,,\qquad\,\lambda\,\in\mathcal{L}[-3\,]\,.$ (5.36) 2. 2. $\displaystyle\,c_{0}(\kappa,\kappa_{\ast})\,$ $\displaystyle=-1\,,\quad c_{1}(\kappa,\kappa_{\ast})=c_{2}(\kappa,\kappa_{\ast})=0,$ (5.37) $\displaystyle c_{0}(\lambda,\lambda_{\ast})$ $\displaystyle=-1\,,\quad\,c_{1}(\lambda,\lambda_{\ast})=c_{2}(\lambda,\lambda_{\ast})=0\,.$ (5.38) Let $\theta$ be the highest root of $\mathfrak{g}$ and suppose that $e_{\theta}\in\mathfrak{g}_{\theta}$ and $f_{\theta}\in\mathfrak{g}_{-\theta}$ satisfy the relations $[e_{\theta}\,,\,f_{\theta}]\,=\,h_{\theta}$ and $(e_{\theta}\|f_{\theta})=1$. We introduce the following vectors of $\,\widehat{\mathfrak{g}}\,$; $\displaystyle f_{J}$ $\displaystyle=J\otimes f_{\theta}\,\in\widehat{\mathfrak{g}}_{0\delta-\theta}\,,\quad$ $\displaystyle e_{J}$ $\displaystyle=(-J)\otimes e_{\theta}\,\in\widehat{\mathfrak{g}}_{0\delta+\theta}\,,$ (5.39) $\displaystyle f_{\kappa}$ $\displaystyle=\kappa\otimes f_{\theta}\,\in\widehat{\mathfrak{g}}_{\frac{1}{2}\delta-\theta}\,,\quad$ $\displaystyle e_{\kappa}$ $\displaystyle=\kappa_{\ast}\otimes e_{\theta}\,\in\widehat{\mathfrak{g}}_{-\frac{3}{2}\delta+\theta}\oplus\widehat{\mathfrak{g}}_{\frac{1}{2}\delta+\theta}\,,$ (5.40) $\displaystyle f_{\lambda}$ $\displaystyle=\lambda\otimes f_{\theta}\,\in\widehat{\mathfrak{g}}_{-\frac{3}{2}\delta-\theta}\,,\quad$ $\displaystyle e_{\lambda}$ $\displaystyle=\lambda_{\ast}\otimes e_{\theta}\,\in\widehat{\mathfrak{g}}_{-\frac{3}{2}\delta+\theta}\oplus\widehat{\mathfrak{g}}_{\frac{1}{2}\delta+\theta}\,.$ (5.41) Then we have the generators of $\mathcal{L}\mathfrak{g}\oplus\,\oplus_{k=0}^{2}\mathbf{C}a_{k}$ that are given by the following triples: $\displaystyle\left(\,\widehat{e}_{i},\widehat{f}_{i},h_{i}\right)\quad i=1,2,\cdots,l,$ $\displaystyle\left(\widehat{e}_{\lambda},\widehat{f}_{\lambda},h_{\theta}\right),\quad\left(\widehat{e}_{\kappa},\widehat{f}_{\kappa},h_{\theta}\,\right),\quad\,\left(\widehat{e}_{J},\widehat{f}_{J},h_{\theta}\right)\,\,.$ (5.42) These triples satisfy the following relations. ###### Proposition 5.10. 1. 1. $\left[\,e_{\pi}\,,\,f_{i}\,\right]_{\widehat{\mathfrak{g}}}=\,\left[\,f_{\pi}\,,\,e_{i}\,\right]_{\widehat{\mathfrak{g}}}=0\,,\quad\mbox{for }\,1\leq i\leq l,\,\mbox{ and }\,\pi=J,\,\kappa,\,\lambda\,.$ (5.43) 2. 2. $\left[\,e_{J}\,,\,f_{J}\,\right]_{\widehat{\mathfrak{g}}}=\,\widehat{h}_{\theta}\,,$ (5.44) 3. 3. $\quad\left[\,e_{\lambda}\,,\,f_{\lambda}\,\right]_{\widehat{\mathfrak{g}}}=\sqrt{-1}\,\widehat{h}_{\theta}-a_{0},\quad\left[\,e_{\kappa}\,,\,f_{\kappa}\,\right]_{\widehat{\mathfrak{g}}}=\sqrt{-1}\,\widehat{h}_{\theta}\,-a_{0}\,.$ (5.45) Adding the element $n$ to these generators of $\mathcal{L}\mathfrak{g}\oplus\,\oplus_{k=0}^{2}\mathbf{C}a_{k}$ we have obtained the Chevalley generators of $\widehat{\mathfrak{g}}$. ## References [A] J.F.Adams, Lectures on Lie Groups, W.A.Benjamin, Inc. 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# Distilling Interpretable Models into Human-Readable Code Walker Ravina∗, Ethan Sterling, Olexiy Oryeshko, Nathan Bell, Honglei Zhuang, Xuanhui Wang, Yonghui Wu, Alexander Grushetsky Google, Mountain View, CA, USA walkerravina, esterling, olexiy, nathanbell, hlz, xuanhui, yonghui, <EMAIL_ADDRESS> ###### Abstract. The goal of model distillation is to faithfully transfer teacher model knowledge to a model which is faster, more generalizable, more interpretable, or possesses other desirable characteristics. Human-readability is an important and desirable standard for machine-learned model interpretability. Readable models are transparent and can be reviewed, manipulated, and deployed like traditional source code. As a result, such models can be improved outside the context of machine learning and manually edited if desired. Given that directly training such models is difficult, we propose to train interpretable models using conventional methods, and then distill them into concise, human- readable code. The proposed distillation methodology approximates a model’s univariate numerical functions with piecewise-linear curves in a localized manner. The resulting curve model representations are accurate, concise, human-readable, and well-regularized by construction. We describe a piecewise-linear curve- fitting algorithm that produces high-quality results efficiently and reliably across a broad range of use cases. We demonstrate the effectiveness of the overall distillation technique and our curve-fitting algorithm using four datasets across the tasks of classification, regression, and ranking. Model distillation; human readable; piecewise-linear curves ∗Corresponding author ††ccs: Computing methodologies Machine learning approaches ## 1\. Introduction Interpretable models are critical for high-stakes decision-making scenarios (Rudin, 2018) such as guiding bail or parole decisions, assessing loan eligibility, and guiding medical treatment decisions. In these cases, the explanation of a model’s output (_e.g_. individual feature contributions) should be examinable and understandable, to ensure transparency, accountability, and fairness of the outcomes. To achieve intrinsic interpretability, univariate functions are widely used in interpretable models. In the classic Generalized Additive Models (GAMs) (Hastie and Tibshirani, 1986), the model is a sum of univariate shape functions, $M=f_{0}+f_{1}(x_{1})+f_{2}(x_{2})+f_{3}(x_{3})+\dots+f_{n}(x_{n}).$ where $x_{i}$’s are $n$ features and $f_{i}$’s are the shape functions. Such a model is simple but often less accurate than a model with feature interactions. Recently, Lou _et al_. (Lou et al., 2013) showed that adding a limited number of pairwise feature interactions allows GAM-style additive models to capture a significant fraction of the accuracy of a fully- interacting model. In many cases of interest, such feature interactions are intuitively captured with products of univariate functions, $g_{1}(c_{1})\cdot f_{1}(x_{1})+g_{2}(c_{2})\cdot f_{2}(x_{2})+\dots,$ or products of groups of features, $(g_{1,1}(c_{1})+g_{1,2}(c_{2}))\cdot f_{1}(x_{1})+(g_{2,1}(c_{1})+g_{2,2}(c_{2}))\cdot f_{2}(x_{2})+\dots,$ where the magnitude of one function (_i.e_. $f_{i}$) is modulated by a function (_i.e_. $g_{i}$ or $g_{i,j}$) of another ”context” feature (_i.e_. $c_{i}$) (Zhuang et al., 2021). In other cases, the interaction amongst features is adequately approximated by additive models of univariate functions nested within univariate functions, $\displaystyle f(x_{1},x_{2},x_{3})\approx$ $\displaystyle\ g_{1}(f_{1,1}(x_{1})+f_{1,2}(x_{2})+f_{1,3}(x_{3}))\ +$ $\displaystyle\ g_{2}(f_{2,1}(x_{1})+f_{2,2}(x_{2})+f_{2,3}(x_{3}))+\dots,$ where the outer function $g_{i}$ captures nonlinear behavior (Chen et al., 2018). Indeed, the Kolmogorov–Arnold representation theorem (Kolmogorov, 1957; Wikipedia, 2020c) guarantees that every continuous multivariate function of $n$ inputs can be represented as a sum of $2n$ such terms, $f(x_{1},\dots,x_{n})=\sum_{i=0}^{2n}g_{i}\left(\sum_{j=1}^{n}f_{i,j}(x_{j})\right).$ In practice a single outer function is often sufficient, yielding an interpretable model. Figure 1. Shape plots for numerical features from models learned on the COMPAS dataset. The GAM forest model is shown in blue dots, while its distillation into the two-segment curve model is shown in orange lines (both map to the left Y axis). Cumulative distribution functions of the corresponding signals are shown in grey (right Y axis). In the classic GAM models, splines are used as shape functions (Hastie and Tibshirani, 1986). Another commonly used shape function is piecewise-linear functions (Wikipedia, 2020f). These representations contain a small number of variables (_e.g_. knots) and thus are concise and human-readable. However, directly optimizing such representations often yields less accurate models than alternative model representations. For example, Lou _et al_. (Lou et al., 2012) showed that learning spline GAMs is less accurate than learning bagged boosted decision forest GAMs. Our experiments show similar results for directly optimizing GAMs composed of piecewise-linear curves using Stochastic Gradient Descent (SGD) methods. Broadly speaking, the model representations using decision forest GAMs have the advantage during model optimization, but the resultant models are not human-readable. This is the case even when there exists a simpler model with a concise, human-readable form that provides comparable accuracy. Inspired by the model distillation work in which relatively small decision forests or neural networks can be distilled from much larger ensembles, but not trained directly from data, to match the accuracy of complex models (Buciluundefined et al., 2006; Hinton et al., 2015), we propose to distill interpretable models into readable representations in a separate process after model optimization. This decouples the initial, learned model representation from the final, published model representation. For example, the proposed distillation methodology can be applied to additive models trained using bagged boosted decision trees (Lou et al., 2012), as well as additive neural nets (Agarwal et al., 2020; Zhuang et al., 2021). In this paper, we describe a technique for distilling models composed of univariate components into human readable representations, in particular, the piecewise-linear curves described in Section 2.2. The output of our distillation technique is illustrated in Listing 1 and Figure 1, which show textual and graphical representations of piecewise-linear curves obtained by applying our approach to a decision forest GAM trained on the COMPAS dataset (described in Section 2.1). The distilled model is a concise representation of the decision forest GAM model and is converted to human-readable source code. Listing 1: Code for a distilled COMPAS model ⬇ score = sum([ PWLCurve("age", [(18, 3.13), (21, 0.5914), (46, -0.7206)], fx="log"), PWLCurve("priors_count", [(0, -0.8415), (1, -0.4452), (38, 2.146)], fx="log1p"), PWLCurve("length_of_stay", [(0, -0.1855), (3, -0.04099), (4, 0.2443)], fx="log1p"), EnumCurve("c_charge_degree", {1: 0.0198, 2: -0.0384}), ## ... other features ... ]) From here on, we will use ”curves” to refer to piecewise-linear curves, ”curve models” to refer to models where each component is a curve, and ”code” to refer to the textual representations of curve models or curves. The rest of this paper is structured as follows. After presenting the preliminaries in Section 2, we elaborate on the benefits of using curve models in Section 3. We then describe the localized distillation process in Section 4 and piecewise-linear approximation algorithm, sometimes referred to as segmented regression (Wikipedia, 2020f), for creating curve models in Section 5. Lastly, we present experimental results on for datasets: COMPAS, FICO, MSLR-WEB30K, and CWS in Section 6 and conclude the paper in Section 7. ## 2\. Preliminaries Throughout the paper, we will use the data sets used in this paper as concrete examples to explain our methods. Thus, we first describe them in this section. We also give the formal definition of piecewise-linear-curves in this section. ### 2.1. Data Sets We used the following four datasets to represent different settings: classification, regression, and ranking. The first three are publicly available. * • The COMPAS dataset111https://github.com/propublica/compas-analysis is the result of a ProPublica investigation (Angwin et al., 2016) into possible racial bias of the proprietary COMPAS model score for defendants in Broward county, Florida. The dataset has been studied extensively in the context of bias, fairness, and interpretability (Tan et al., 2018; Dressel and Farid, 2018; Kleinberg, 2018; Chouldechova, 2017). Labels are binary and indicate whether recidivism occurred for an individual within a time period. We use area under the receiver operating characteristic curve (AUC-ROC) to measure classifier accuracy. COMPAS has 6 features and four of them are used as examples in this paper: age, priors_count, length_of_stay, and c_charge_degree. * • The FICO dataset (FIC, 2018) is composed of real-world anonymized credit applications along with risk scores. Labels are a risk score for an individual. We use root mean square error (RMSE) to measure regressor accuracy. FICO has 24 features and we use two features as examples in our paper: MSinceMostRecentDelq, Months Since Most Recent Delinquency; PercentTradesWBalance, Percent Trades with Balance. * • The MSLR-WEB30K dataset (Qin and Liu, 2013) is a widely used learning-to-rank benchmark dataset. Labels are per document relevance judgements. We use normalized discounted cumulative gain at $k=5$ (NDCG@5) to measure ranker accuracy. MSLR-WEB30K is significantly larger both in number of features (136) and number of training examples (~2,000,000 per cross validation fold). We use it to compare our curve approximation algorithm to pwlf (Jekel and Venter, 2019), a publicly available alternative, on the basis of accuracy, robustness and efficiency. We use two features as examples in our paper: feature_0011, Body stream length; feature_0128, Inlink number. * • The Chrome Web Store (CWS) dataset is a private and anonymized dataset originating from Chrome Web Store logs. Each query corresponds to a visit to the Chrome Web Store. The items within each query were the ones shown to the user. Labels correspond to user actions such as clicking, installing, or no action whatsoever. We again use NDCG@5 to measure ranker accuracy. A similar, but distinct datset from the Chrome Web Store was previously studied by Zhaung _et al_. (Zhuang et al., 2021). Unlike in that previous work, in this instance we do not utilize query level ”context” features, instead using only 14 item level features. The queries are also distinct. In each case, we distill a decision forest GAM and evaluate the accuracy of the distilled curve models. The COMPAS and FICO datasets represent high-stakes domains (Rudin, 2018) in which the benefits of curve models, discussed below, are particularly compelling. FICO, MSLR-WEB30K, and CWS have been previously studied in the context of interpretability (Agarwal et al., 2020; Zhuang et al., 2021; Lou et al., 2013; Chen et al., 2018). Furthermore, the results from MSLR-WEB30K demonstrate that the accuracy of this approach is not limited to small datasets. ### 2.2. Piecewise-Linear Curves A piecewise linear curve (PWLCurve) is defined by a list of control points $S=[(x_{k},y_{k})]_{k=1}^{K}$ through which the curve must pass. Between control points, output $y$ values are determined by performing linear interpolation between neighboring control points. Beyond the leftmost or rightmost control points, output values are capped to the $y_{k}$-value of the neighboring control point. More formally, assuming $x_{k}$’s are ordered, _i.e_. $x_{k}<x_{k+1}$, the definition of a piecewise linear curve can be described as: $PWL(x;S)=\begin{cases}y_{1}&\text{if }x<x_{1},\\\ \frac{y_{k+1}-y_{k}}{x_{k+1}-x_{k}}(x-x_{k})+y_{k}&\text{if }x_{k}\leq x\leq x_{k+1},\\\ y_{K}&\text{if }x>x_{K}.\end{cases}$ In most cases of interest 5 or 6 control points, defining 4 or 5 interior segments, is sufficient to capture the desired behavior. We allow for an optional $x$-transformation, specified with the fx argument, to fit curves to data with different scales. When an $x$-transformation is present it is applied to the input value and $x$-values of all the control points, and then linear interpolation is performed in the transformed space. We support identity (default), log, log1p and symlog1p transformations. Here symlog1p is defined as sgn(x) * log1p(abs(x)) and is suitable for highly- variable features that take on both positive and negative values. Univariate categorical functions are represented by EnumCurve, which directly maps input values to outputs using a discrete mapping. ## 3\. Background & Motivation Interpretable models are critical for high-stakes decisions (Rudin, 2018) and provide many advantages over more complex model structures (Caruana et al., 2015; Du et al., 2019). In this section we explain how distilling interpretable models into curve models reinforces these benefits and addresses a variety of real-world engineering challenges. Here, one underlying theme is that distilling models into human-readable source code _reduces a novel machine learning problem to an established software engineering problem with an abundance of existing solutions_. ### 3.1. Greater Transparency A model is transparent if it provides a textual or graphical representation that enables its behavior to be understood comprehensively (Ustun and Rudin, 2014). One way in which the proposed method provides greater transparency is by simplifying graphical depictions of a model while retaining its essential characteristics. It is often argued, implicitly or explicitly, that the shape plots of an interpretable model are an _exact description_ of the model and therefore provide a reliable way to understand the model. While this claim is narrowly true, it is misleading in general. Unless given specific guidance, humans will naturally discount certain fine-grained details of the plots when developing an understanding of the model. By distilling interpretable models to a concise representation, we discard extraneous characteristics and reduce the mental effort necessary to understand the model. For example, it is not immediately obvious what understanding an individual should derive from the shape plots of the feature_0011 (body stream length), and feature_0128 (inlink number) features in the initially-learned MSLR-WEB30K model, shown in Figure 2. Indeed, different individuals may derive qualitatively different understandings from these graphical depictions. However, given the additional knowledge that the distilled curve model represented by the overlaid curves in Figure 2 has nearly identical accuracy, an observer can make much stronger inferences about the model’s essential characteristics. Interpretability can be increased even further by imposing monotonicity constraints. We discuss the effect of such constraints in Section 6.4. Figure 2. Shape plots for a GAM forest model (in blue dots) and 5 segment curve distillation (in orange lines) for the MSLR-WEB30K dataset. Clearly when distillation yields a simpler model with comparable accuracy we would say the distillation process has succeeded. However, instances where distillation yields a model with inferior accuracy warrant further investigation because the apparent ”failure” can often be attributed to essential characteristics of the teacher model that were not successfully transferred to the student _because they violate a prescribed notion of human- interpretability_. We examine one representative example of this phenomenon in Section 6.2. While a complete discussion of this principle is beyond the scope of this paper, we note that the idea can be viewed as an extension of the use of structural constraints to define “interpretable” models, just now applied to the structure of individual functions in the model. Under this policy, if the accuracy of a candidate model cannot be reproduced using a predefined class of expressive, “human-scale” functions (_e.g_. curves with a small number of truncated control points) its transparency would be called into question. ### 3.2. Constructive Regularization The proposed method can also be viewed as a post-hoc regularization process that is completely compatible with, and complementary to, optimization-based regularization techniques (_e.g_. L1/L2 penalties or monotonicity constraints). In the context of regularization, our emphasis on conciseness is aligned with the minimum description length principle (Wikipedia, 2020d) for model selection. Ustun and Rudin (Ustun and Rudin, 2014) applied similar reasoning to motivate linear models with small, integer-valued weights. The constrained description length of curves provides limited capacity for capturing idiosyncratic behavior. As a result, curve distillation successfully removes aberrations from teacher model functions. This regularization effect can be seen in Figure 1 and Figure 2. The fewer segments the greater the effect. To find the most concise curve model we can repeatedly apply the proposed method with decreasing number of control points. Naturally, the optimality of this approach is subject to the limitations of our localized distillation methodology (see Section 4) and curve approximation algorithm (see Section 5). While it is difficult to directly compare models with different functional representations, comparing the length and readability of their corresponding code is instructive. One practical advantage of curve-based regularization is that regularity is enforced by construction and the complexity of individual curves is readily apparent and quantifiable. Therefore, organizations that adopt curve models can set objective guidelines about model complexity that developers can anticipate when submitting model candidates for approval. Such guidelines can specify the maximum number of curve segments, maximum number of significant digits per curve control point, or monotonicity of the curve. Similar to the use of nothing-up-my-sleeve numbers in cryptography (Wikipedia, 2020e), curve models enable developers to preemptively address suspicions about potential weaknesses and constructively prove the robustness of a given model candidate. In general, standardizing development around curve models is a straightforward way for organizations to systematically enforce best practices, defend against common mistakes and pitfalls, and expedite model verification and approval. The accessible, readable nature of curve models enables organization members beyond engineers (_e.g_. executives, product managers, etc.) to participate in this approval process. ### 3.3. Readable, Editable Code Curve model code can be read, reviewed, merged, and versioned like conventional source code. An example model for the COMPAS dataset is shown in Listing 1. One can understand how a curve model would behave under novel or extremal conditions by mentally “evaluating” the model under hypothetical “what if?” scenarios without the need for additional tools. Subjecting models to a traditional source code review process facilitates a more rigorous examination of the model’s characteristics and greater accountability than is possible with non-readable models. Indeed, conducting “model review” through source code review ensures that the candidate model itself - not some separate, potentially inconsistent description or artifact of the model or how it was trained - is the subject of review. In the event that undesirable model behavior is discovered, the model’s code may be directly edited to correct such issues. For example, in the case of the COMPAS model a user may wish to deliberately cap the contribution of features such as priors_count and length_of_stay features for legitimate policy reasons not captured by classification metrics such as AUC-ROC. The contribution of other features can be entirely removed. Agarwal _et al_. (Agarwal et al., 2020) discussed how such an approach of training with biased features and then removing them can potentially be better than simply training without biased features. This approach can prevent the model from extracting bias through other features which are correlated with biased ones. Model transparency is essential in the context of high-stakes decisions (Rudin, 2018) arising in criminal justice, finance, health care, and other areas. Providing the complete source of the model in simple, portable, human- readable code makes the models transparent. Compared to human-readable models produced by CORELS (Angelino et al., 2017), which are expressed in universally-understandable if-then language, curve models sacrifice accessibility for greater expressiveness and general-purpose application. ### 3.4. Collaborative Model Development Curve distillation is compatible with any algorithm or modeling technique that results in univariate functions. In the experiments section we apply the proposed technique to decision forest GAMs on several datasets. Previous work (Zhuang et al., 2021) applied the proposed technique to GAMs learned via neural networks, as well as similar neural networks with limited interactions via multiplicative pairs. Organizing collaborative development around curve models enables engineers to apply a plurality of different tools, techniques, or platforms to optimize components of a (potentially large-scale) model. Engineers are free to choose a modeling approach that maximizes their productivity, similarly to how engineers use multiple IDEs, code formatters, or linters to collaboratively develop software. Curve distillation can be viewed as a “format conversion” tool that translates an arbitrary and potentially exotic model representation into a fixed, agreed-upon vocabulary of human-readable building blocks. ### 3.5. Straightforward Deployment Curve models are fast-to-evaluate and straightforward to deploy. Since evaluation requires minimal computation - just a handful of floating point operations per curve - curve models are well-suited for performance-critical applications. Curves are a portable, platform-agnostic representation that can be natively supported in a variety of languages or systems with little effort. For example, Listing 2 shows a C++ implementation of a COMPAS model with 2 segments. In general, curve models are straightforward to deploy because they offer a multitude of integration options. Curves can be embedded in configuration files, passed via CGI parameters, manually embedded into complex applications in a piecemeal fashion, systematically translated to a target representation, or evaluated by existing runtime systems with a few incremental extensions. Listing 2: A COMPAS model as a C++ function ⬇ double COMPAS(double age, double priors_count, double length_of_stay, int charge_degree, // ... other features ... ) { static auto age_curve = PWLCurve({{18, 3.13}, {21, 0.5914}, {46, -0.7206}}, "log"); static auto priors_count_curve = PWLCurve( {{0, -0.8415}, {1, -0.4452}, {38, 2.146}},"log1p"); static auto length_of_stay_curve = PWLCurve( {{0, -0.1855}, {3, -0.04099}, {4, 0.2443}}, "log1p"); static auto charge_degree_curve = EnumCurve({{1, 0.0198}, {2, -0.0384}}); // ... other features ... return (age_curve.Eval(age) + priors_count_curve.Eval(priors_count) + length_of_stay_curve.Eval(length_of_stay) + charge_degree_curve.Eval(charge_degree) + // ... other features ... ); } ## 4\. Localized Distillation Our distillation process takes two inputs: a teacher model containing one or more univariate functions, and a representative dataset (generally the training data). Our method differs from conventional distillation techniques in that we (1) distill each univariate function in isolation and (2) optimize for mean squared error (MSE) when approximating each univariate function. Specifically, each univariate function in the teacher model is evaluated on the dataset to produce representative $(x,y)$ example pairs. For discrete categorical features we create a mapping where each unique $x$ is mapped to the mean $y$. For numerical features, we produce a PWLCurve using the approximation algorithm described in Section 5. If the teacher model contains univariate functions nested within other univariate functions we replace the source functions in a bottom-up fashion. Otherwise, all non-nested functions can be approximated in parallel. The final model is constructed by replacing each original univariate function with its PWLCurve approximation. Conventionally, model distillation involves a global optimization using the same (or at least similar) objective to the original teacher model training. This objective may differ from a point-wise MSE objective. For example, ranking objectives often have pair-wise definitions. Why then do we advocate a localized optimization using a MSE objective in all circumstances? The primary answer is that, in the context of interpretable models, there is substantial value in maintaining a strong one-for-one correspondence between each source function and target function. Notably, this allows us to visualize each shape function in the teacher model against its corresponding curve replacement. Additionally, we can attribute distillation failures - data instances where the curve model is less accurate than the teacher model - to specific univariate functions, and to take remedial actions. For example, in the Figure 5 we can immediately tell that the shape function of $x_{1}$ was not well- approximated by a curve. In the experiments section we show that the meaningful behavior of nearly all shape functions can be accurately captured by curves with three to five segments. Furthermore, when the meaningful behavior is not captured, it is generally due to inherently non-interpretable behavior being lost. While a global optimization approach (_i.e_. optimizing the parameters of all curves in the target model simultaneously) using a problem-specific metric might produce a more accurate result, it is computationally more expensive and would lack the same one-to-one correspondence with the teacher model, making distillation failures more difficult to diagnose. Furthermore, if higher accuracy is desired, the output of the proposed distillation process can be used to initialize a global optimization of the curve model’s parameters. ## 5\. Piecewise-Linear Curve Approximation Given a univariate numerical function $f(x)\rightarrow y$, our goal is to produce a PWLCurve $c(x)\rightarrow y$ that faithfully approximates $f(x)$ by minimizing the $MSE(c(x),f(x))$ over sample data. Clearly, the accuracy of the overall distillation method depends critically on the accuracy of the individual curve approximations - _i.e_. how much metric loss is incurred when each $c(x)$ is substituted for the corresponding $f(x)$ in the trained model. Additionally, the practical success of the methodology also depends on the robustness and efficiency of the approximation algorithm. To enable systematic use of curve distillation in model training pipelines, the approximation algorithm must run with minimal configuration. Complex hyperparameters pose a significant barrier to entry. We have designed pwlfit, our piecewise linear approximation algorithm, so that in practice users only need to consider the num_segments and mono (monotonicity) parameters. While num_segments=5 segments and mono=False is sufficient to get high accuracy (as demonstrated by our experiments), it is desirable to investigate whether the model can be further simplified with fewer segments or with monotonicity restrictions. To facilitate such investigations it is important that distillation runs quickly (_e.g_. less than 1 second per function) which enables interactive analysis via Jupyter notebooks (Kluyver et al., 2016) or other tools. These practical considerations have informed various decisions in the design of pwlfit. In particular, we prefer an algorithm which quickly and reliably yields high accuracy results with minimal configuration to one which sacrifices either of these practical considerations for marginal gains in accuracy. In this section we will describe the salient characteristics and noteworthy features of pwlfit. We invite interested readers to consult the publicly- available source code of pwlfit (Sterling and Ravina, 2019), for additional details. ### 5.1. Algorithm Given a list of $(x,y,weight)$ points and a desired number of segments $k$, we search for a PWLCurve to minimize mean squared error, MSE. A PWLCurve with $k$ segments is characterized by its $k+1$ control points – a set of $x$-knots and their corresponding $y$-knots. Given only the $x$-knots, we can solve a linear least squares expression for the optimal $y$-knots and the resulting error. Since we don’t know the correct $x$-knots, we search through the space of possible $x$-knots and solve a least squares expression at each step to calculate the error.222pwlf (Jekel and Venter, 2019) implements a similar approach. We will compare with it in our experiments. #### 5.1.1. Initial Downsampling For performance, we randomly downsample large datasets to approximately one million points before fitting. We downsample to reduce the cost of sorting, which dominates the runtime for large data. This downsampling imposes a negligible quality loss. To further reduce runtime, we discretize the search space for $x$-knots. We choose num_samples $x$-values from the data, spaced equally by cumulative weight, and search over the combinations of $x$-knots from that sampled set of candidates. Using the default 100 samples, our candidates are the $x$-values at $(0\%,1.01\%,\dots,98.9\%,100\%)$ of the cumulative weight. #### 5.1.2. Knot Discretization For data with many repeated $x$-values, some of our candidates will be duplicates. For example, $55\%$ of the values in the length_of_stay feature in the COMPAS data set are 0 or 1. In such cases, we iteratively resample at higher rates (such as $0\%,0.505\%,1.01\%$, etc.) until we collect a suitable number of distinct candidates, never exceeding the specified num_samples parameter. #### 5.1.3. Condensation To minimize the cost of each linear least squares step, we condense the data using a novel technique described in Appendix B. Given num_samples candidate knots, we condense the full data into two synthetic points per adjacent pair of candidates, for a total of 2 * (num_samples - 1) synthetic points. For any function that’s linear between each adjacent pair of candidate $x$-knots, which is guaranteed by our choice of discrete candidate $x$-knots, these condensed points perfectly recreate the loss of that function over the full data set. We run our linear least squares solver on the condensed points instead of the full data set, which reduces our cost per solve from ${\mathcal{O}(\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\footnotesize{\@listingGroup{ltx_lst_identifier}{num\textunderscore points}}}}}})$ to ${\mathcal{O}(\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\footnotesize{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0.0,0.27,0.13}{n}um\textunderscore samples}}}}}})$. This is purely a performance optimization, with no quality implications. Figure 3. Candidate $x$-knots (red vertical lines) and derived condensed points (pink large dots) on the age piece of a COMPAS GAM forest model (blue dots). For visual clarity, this illustration considers only five $x$-knot candidates. #### 5.1.4. Global Optimization via Greedy Search After discretization, the solution space consists of ${{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\footnotesize{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0.0,0.27,0.13}{n}um\textunderscore samples}}}}}}\choose{\text{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\footnotesize{\@listingGroup{ltx_lst_keyword}{\color[rgb]{0.0,0.27,0.13}{n}um\textunderscore segments}}}}}}+1}$ $x$-knot combinations, which is still too large for an exhaustive search. To make the search tractable we use a greedy search heuristic that optimizes one $x$-knot at a time. Specifically, at each step of the process we evaluate the error associated with each candidate $x$-knot, and keep the candidate that yields the least error. With this approach, we optimize in two stages. We begin with a single $x$-knot as our solution, and greedily add the best remaining candidate $x$-knot until our solution consists of (num_segments + 1) $x$-knots. Then we cycle through our solution, removing one $x$-knot at a time and replacing that $x$-knot with the best remaining candidate $x$-knot, which could be the same $x$-knot that we just removed. We continue this cycle of iterative improvements until our solution converges, or until we’ve exceeded the maximum number of iterations (defaulting to 10 iterations). #### 5.1.5. Slope Constraints & Monotonicity pwlfit can impose a minimum and/or maximum slope on the solution via bounded least squares. Instead of solving the least squares expression directly for the $y$-knots, we solve it for the deltas between adjacent $y$-knots. Then we impose a min/max slope by bounding the deltas. Slope restrictions can be used to limit the spikiness of curves, but we primarily use them to impose monotonicity. For example, specifying min_slope=0 restricts to monotonically non-decreasing functions while max_slope=0 restricts to monotonically non- increasing functions. Specifying a min_slope greater than 0 or a max_slope less than 0 restricts to strictly increasing or decreasing functions, respectively. pwlfit can deduce the direction of monotonicity by applying isotonic regression (Wikipedia, 2020b) to the condensed points. We fit an increasing and a decreasing isotonic regression, and use the direction that minimizes mean squared error. The user can override this behavior by specifying the direction explicitly or by disabling monotonicity entirely. #### 5.1.6. Input Transformations pwlfit can also interpolate in a transformed $x$-coordinate space instead of the original space, as a simple form of feature engineering. pwlfit transforms the $x$-values before learning the curve. Specifically, pwlfit will choose a candidate $x$-transformation, fx, among log, log1p, or symlog1p based on the range of the $x$-values and then proceed with that transformation if it increases the Pearson correlation between fx and $y$ by a noticeable amount over the identity transformation. Alternatively, the user can specify any strictly increasing 1D transform or specify the identity transform to disable transformation. ## 6\. Experiments ### 6.1. Distillation Accuracy Table 1 and Figure 4(a) show the results obtained from experiments on the different datasets. A complete set of results can be found in Table 2 in Appendix A. The results of applying our distillation technique with our piecewise-linear approximation algorithm are presented as pwlfit. We present results from using various numbers of segments with and without a monotonicity restriction and otherwise default parameters. In all cases we truncated the control points to four significant digits. We also present several additional reference points to provide context. * • SGD: We directly learn the curves with the Adadelta(Zeiler, 2012) optimizer. We initialize the $y$ values of the control points as zeros. For the $x$ values of the control points we use the quantiles for numerical features (_e.g_. 0%, 50%, 100% for a three point, two segment curve) or all unique values for categorical features. We then apply Adadelta to optimize the $y$ values. Simultaneously optimizing $x$ and $y$ values was also attempted, but the results were always worse than optimizing $y$ values alone. * • NAM: Neural Additive Models (NAMs) (Agarwal et al., 2020) is another method for learning interpretable models proposed by Agarwal _et al_. We present their result for reference where applicable. * • Interacting forest: We train a bagged, boosted decision forest allowing feature interactions to demonstrate the accuracy of a non-interpretable, high- complexity ”black box” model. * • GAM forest: We train a bagged boosted decision forest GAM by restricting each tree to use only one feature. This model is also the source model for our distillation technique. * • pwlf: We apply our distillation technique using an alternative piecewise- linear approximation algorithm pwlf(Jekel and Venter, 2019). On each dataset we used five fold cross validation and present the metric mean and sample standard deviation across folds. We used three different metrics to evaluate accuracy: AUC-ROC, RMSE, and NDCG@5 for the three different tasks of classification, regression, and ranking. Further details on our experimentation setup can be found in Appendix A and further details on the datasets, labels, and metrics can be found in Preliminaries 2.1. Table 1. Metrics across datasets. For AUC-ROC and NDCG@5 higher is better, and for RMSE lower is better. MSLR-WEB30K and CWS were not used by the NAM paper and are omitted from that row. Metric values are the mean from five fold cross validation $\pm$ the sample standard deviation. Model \| COMPAS (AUC-ROC) \| FICO (RMSE) \| MSLR-WEB30K (NDCG@5) \| CWS (NDCG@5) ---\|---\|---\|---\|--- Interacting forest \| $0.742\pm 0.011$ \| $3.128\pm 0.092$ \| $0.485\pm 0.002$ \| $0.461\pm 0.003$ GAM forest \| $0.741\pm 0.013$ \| $3.495\pm 0.109$ \| $0.442\pm 0.002$ \| $0.460\pm 0.004$ NAM \| $0.741\pm 0.009$ \| $3.490\pm 0.081$ \| \| pwlfit num_segments=5, mono=False \| $0.743\pm 0.012$ \| $3.494\pm 0.096$ \| $0.441\pm 0.002$ \| $0.454\pm 0.002$ pwlfit num_segments=5, mono=True \| $0.743\pm 0.013$ \| $3.693\pm 0.101$ \| $0.437\pm 0.003$ \| $0.452\pm 0.003$ pwlf num_segments=5 \| $0.743\pm 0.012$ \| $3.503\pm 0.096$ \| $0.433\pm 0.003$ \| $0.454\pm 0.003$ SGD num_segments=5 \| $0.741\pm 0.010$ \| $3.643\pm 0.097$ \| $0.405\pm 0.002$ \| $0.448\pm 0.004$ SGD num_segments=20 \| $0.738\pm 0.011$ \| $3.499\pm 0.117$ \| $0.419\pm 0.003$ \| $0.455\pm 0.003$ (a) Top level metrics across datasets. For AUC-ROC and NDCG@5 higher is better, for RMSE lower is better. (b) Per fit metrics for MSLR-WEB30K Fold 1. Note that for 4 segments pwlf has extreme outliers for RMSE vs the source submodel. Figure 4. Comparisons of different methods across the 4 datasets. Our results show that applying our distillation technique with 4-5 segments with pwlfit produces models which are as accurate as both the source GAM forest and NAM models for all datasets except CWS where a small gap remains. We investigate this accuracy gap in detail in Section 6.2 below. In the case of the COMPAS dataset these models are as accurate as full complexity models. Applying our technique with pwlf produces competitive results, albeit less accurate on the MSLR-WEB30K dataset. By contrast, the results show that learning curves directly via SGD is less general. On the FICO and CWS datasets more segments are required to achieve accuracy comparable to the GAM forest models. On the MSLR-WEB30K dataset the accuracy is inferior even with many more segments. The consistent accuracy of applying our distillation approach with pwlfit on these four datasets and three separate tasks (classification, regression, learning to rank) demonstrates that the process is not sensitive to either the specific data or the top level objective being used. ### 6.2. Distillation Failures In Section 3.1 we explained how distillation yielding a model with inferior accuracy warrants further investigation because the purported ”failure” can often be attributed to essential yet non-interpretable characteristics of the teacher model not transferring to the student model. The accuracy gap observed on the CWS dataset is an example of this phenomenon. Figure 5 shows the worst two fits from the CWS dataset. The plots have been redacted to maintain the privacy of the dataset. For each plot it is clear that the original teacher submodel had some non-interpretable behavior which was lost during distillation. This is most evident for feature $x_{1}$, the worst offender, where the output is highly erratic. If the original teacher submodel is not distilled for these two features then the accuracy gap between the original teacher model and 5 segment non-monotonic distillation drops from 0.0059 to 0.003 (_i.e_. ~50% of the gap is recovered). To identify the above two failures we applied the following method. * • Begin with the original teacher model. For each submodel compute the metric delta against the teacher model from distilling only that submodel and no others. * • Perform the above on each cross validation fold using the validation set and average the metric deltas across folds. * • Sort the features by their associated metric delta to determine the worst distillations. Figure 5. The worst curve distillations from the CWS dataset using 5 segments. ### 6.3. Efficiency & Robustness The experiments of the previous section showed that pwlfit more accurately distills the source model across datsets than pwlf. We also found on the MSLR- WEB30K dataset that pwlfit is more efficient and robust than pwlf. Figure 4(b) shows per fit metrics from the first fold of the MSLR-WEB30K dataset as the number of segments varies without monotonicity. The top plot shows the time in seconds, as measured on a ThinkStation P520, to fit each of the 136 submodels of the source GAM forest. We find that pwlfit is faster in the average case as the number of segments increases, and has a narrower distribution. The bottom plot shows the RMSE of each fit against the 136 submodels of the source GAM forest. We again find that pwlfit performs favorably in the average case with a narrower runtime distribution. It’s worth noting that pwlf by default does not perform any downsampling. For the MSLR-WEB30K dataset running pwlf without any downsampling was prohibitively expensive. For all of our experiments we ran pwlf with a pre- processing downsample to a random 1000 examples. We found this to be a fair point for balancing speed and quality when comparing to pwlfit. It is of course possible with both algorithms to modify the number of samples used to strike a different trade-off between run time and accuracy. ### 6.4. Monotonicity As discussed in Section 5, pwlfit can fit monotonic curves with automatic direction detection. Figure 4(a) compares curve models fit with and without monotonicity constraints (automatically inferring the direction) across datasets. For the COMPAS dataset monotonic and non monotonic models are comparably accurate, while for FICO, MSLR-WEB30K, and CWS, non-monotonic models are more accurate. Monotonicity with respect to appropriate features is desirable for interpretable models. In these cases a monotonic model may be preferable to a non-monotonic one, even if it is less accurate. For example, Figure 6 compares monotonic and non-monotonic 5 segment curve models on the FICO dataset for the MSinceMostRecentDelq, and PercentTradesWBalance features. Given the semantic meaning of these features, it is desirable from a transparency and incentives standpoint for the model output to be monotonic with respect to each of them. Figure 6. Curve distillations on the FICO dataset using 5 segments without monotonicity (orange) and with monotonicity (green). ## 7\. Conclusion We have introduced a novel method for distilling interpretable models into human-readable code using piecewise-linear curves and demonstrated its efficacy on four datasets. We have shown that curve models match or outperform the accuracy achieved by other additive models. On smaller datasets, curve models match the accuracy of more complex models, like interacting decision forests. Our localized distillation methodology is applicable to any model containing univariate numerical functions and is straightforward to implement using the publicly-available pwlfit(Sterling and Ravina, 2019) library. We have explained how curve model distillation reinforces interpretability and addresses a variety of real-world engineering challenges. Curve models are 1) transparent, 2) well-regularized, 3) easy to analyze for presence of biases or other fairness issues, and 4) can be directly edited or improved outside the context of machine learning to fix the aforementioned fairness issues. Distilling models into human-readable code allows one to address novel machine learning problems using well-established software engineering methods. Curve models can be improved by multiple contributors in parallel, reviewed, and made to systematically follow best practices. Curve models are well-suited for production applications, since they can be natively supported in many languages, are easy to deploy, and fast to evaluate. ###### Acknowledgements. We thank Vytenis Sakenas, Jaime Fernandez del Rio, Benoit Zhong, and Petr Mitrichev for their support and providing the algorithms and optimization infrastructure used in our experiments. We also thank Paul Heymann, Diego Federici, Mike Bendersky, Paul Haahr, and Petr Mitrichev for their helpful feedback and detailed reviews. Lastly, we thank Xinyu Qian, Janelle Lee, Po Hu, and Chary Chen for preparing the CWS data set for our experiments. ## References * (1) * FIC (2018) 2018\. FICO Explainable Machine Learning Challenge. https://community.fico.com/s/explainable-machine-learning-challenge. * Agarwal et al. (2020) Rishabh Agarwal, Nicholas Frosst, Xuezhou Zhang, Rich Caruana, and Geoffrey E. Hinton. 2020. 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In _Proceedings of the 14th ACM International Conference on Web Search and Data Mining_ _(WSDM ’21)_. to appear. ## Appendix A Experimental Details Table 2. Metrics across datasets. For AUC-ROC and NDCG@5 higher is better, and for RMSE lower is better. MSLR-WEB30K and CWS were not used by the NAM paper and are omitted from that row. Metric values are the mean from five fold cross validation $\pm$ the sample standard deviation. Model \| COMPAS (AUC-ROC) \| FICO (RMSE) \| MSLR-WEB30K (NDCG@5) \| CWS (NDCG@5) ---\|---\|---\|---\|--- Interacting forest \| $0.742\pm 0.011$ \| $3.128\pm 0.092$ \| $0.485\pm 0.002$ \| $0.461\pm 0.003$ GAM forest \| $0.741\pm 0.013$ \| $3.495\pm 0.109$ \| $0.442\pm 0.002$ \| $0.460\pm 0.004$ NAM \| $0.741\pm 0.009$ \| $3.490\pm 0.081$ \| \| pwlfit num_segments=1, mono=False \| $0.741\pm 0.008$ \| $3.781\pm 0.105$ \| $0.432\pm 0.003$ \| $0.449\pm 0.003$ pwlfit num_segments=1, mono=True \| $0.741\pm 0.008$ \| $3.780\pm 0.105$ \| $0.432\pm 0.003$ \| $0.450\pm 0.003$ pwlfit num_segments=2, mono=False \| $0.742\pm 0.011$ \| $3.617\pm 0.101$ \| $0.438\pm 0.002$ \| $0.452\pm 0.003$ pwlfit num_segments=2, mono=True \| $0.742\pm 0.011$ \| $3.713\pm 0.103$ \| $0.435\pm 0.003$ \| $0.451\pm 0.003$ pwlfit num_segments=3, mono=False \| $0.743\pm 0.010$ \| $3.536\pm 0.099$ \| $0.440\pm 0.002$ \| $0.453\pm 0.002$ pwlfit num_segments=3, mono=True \| $0.743\pm 0.011$ \| $3.691\pm 0.100$ \| $0.437\pm 0.002$ \| $0.451\pm 0.003$ pwlfit num_segments=4, mono=False \| $0.742\pm 0.012$ \| $3.505\pm 0.094$ \| $0.441\pm 0.002$ \| $0.454\pm 0.003$ pwlfit num_segments=4, mono=True \| $0.742\pm 0.012$ \| $3.691\pm 0.101$ \| $0.437\pm 0.003$ \| $0.452\pm 0.004$ pwlfit num_segments=5, mono=False \| $0.743\pm 0.012$ \| $3.494\pm 0.096$ \| $0.441\pm 0.002$ \| $0.454\pm 0.002$ pwlfit num_segments=5, mono=True \| $0.743\pm 0.013$ \| $3.693\pm 0.101$ \| $0.437\pm 0.003$ \| $0.452\pm 0.003$ pwlf num_segments=2 \| $0.742\pm 0.014$ \| $3.728\pm 0.101$ \| $0.428\pm 0.004$ \| $0.451\pm 0.003$ pwlf num_segments=3 \| $0.743\pm 0.012$ \| $3.565\pm 0.108$ \| $0.434\pm 0.004$ \| $0.453\pm 0.003$ pwlf num_segments=4 \| $0.744\pm 0.012$ \| $3.498\pm 0.099$ \| $0.436\pm 0.004$ \| $0.453\pm 0.003$ pwlf num_segments=5 \| $0.743\pm 0.012$ \| $3.503\pm 0.096$ \| $0.433\pm 0.003$ \| $0.454\pm 0.003$ SGD num_segments=1 \| $0.728\pm 0.008$ \| $4.349\pm 0.059$ \| $0.352\pm 0.002$ \| $0.435\pm 0.003$ SGD num_segments=2 \| $0.734\pm 0.007$ \| $4.028\pm 0.095$ \| $0.382\pm 0.001$ \| $0.443\pm 0.004$ SGD num_segments=3 \| $0.742\pm 0.008$ \| $3.887\pm 0.080$ \| $0.394\pm 0.002$ \| $0.449\pm 0.003$ SGD num_segments=4 \| $0.742\pm 0.010$ \| $3.742\pm 0.110$ \| $0.403\pm 0.003$ \| $0.449\pm 0.004$ SGD num_segments=5 \| $0.741\pm 0.010$ \| $3.643\pm 0.097$ \| $0.405\pm 0.002$ \| $0.448\pm 0.004$ SGD num_segments=6 \| $0.742\pm 0.010$ \| $3.583\pm 0.105$ \| $0.408\pm 0.002$ \| $0.452\pm 0.002$ SGD num_segments=7 \| $0.741\pm 0.010$ \| $3.604\pm 0.098$ \| $0.408\pm 0.002$ \| $0.452\pm 0.003$ SGD num_segments=8 \| $0.741\pm 0.010$ \| $3.561\pm 0.111$ \| $0.411\pm 0.003$ \| $0.449\pm 0.004$ SGD num_segments=9 \| $0.741\pm 0.011$ \| $3.522\pm 0.103$ \| $0.414\pm 0.002$ \| $0.449\pm 0.003$ SGD num_segments=10 \| $0.741\pm 0.010$ \| $3.544\pm 0.117$ \| $0.415\pm 0.002$ \| $0.452\pm 0.005$ SGD num_segments=15 \| $0.740\pm 0.011$ \| $3.499\pm 0.101$ \| $0.419\pm 0.003$ \| $0.454\pm 0.003$ SGD num_segments=20 \| $0.738\pm 0.011$ \| $3.499\pm 0.117$ \| $0.419\pm 0.003$ \| $0.455\pm 0.003$ ### A.1. Cross Validation We performed 5-fold cross validation on all datasets. * • COMPAS & FICO: The datasets were split into 5 equal parts. Each part was used once as a test set (20%) with the remaining parts as the training set (80%). We used the same random folds as in the NAM paper (Agarwal et al., 2020). No validation set was used given the small size of the data. Instead we used out of bag evaluation wherever a validation set would be used (see below). * • MSLR-WEB30K: We used the predefined folds and partitions from the original dataset. For each fold it allocates 60% for training 20% for validation and 20% for testing. * • CWS: We used a dataset of 60,000 queries and 2,690,439 items with an average of ~44 items per query. The dataset was split into 5 equal parts. Each part was used once as a test set. Of the remaining parts 80% was used as training and 20% as validation. Overall this resulted in 64% for training, 16% for validation and 20% for test for each fold. ### A.2. Ensemble Learning For both SGD and tree models, we trained ensembles with 56 bags using a bag fraction of $\frac{7}{8}$ to produce the random subsets. For MSLR-WEB30K and CWS, queries were randomly divided into bags. For the other datasets, individual examples were randomly divided into bags. When applying our distillation technique, we distilled the ensembles into a single PWLCurve per feature. When learning the curves directly via SGD, we averaged the learned $y$-coordinate values across bags to obtain the final model. ### A.3. Loss Functions We trained SGD and tree models using log-loss for the COMPAS dataset, mean squared error (MSE) for the FICO dataset, and ranking loss (ApproxNDCG (Qin et al., 2010)) for the MSLR-WEB30K and CWS datasets. ### A.4. Hyper-parameters For the COMPAS, and FICO datasets hyper-parameters were tuned using out of bag evaluation on the training set of the first fold. For MSLR-WEB30K and CWS, we used the validation sets of the first fold. * • SGD: We tuned the batch size in {128, 256, 512, 1024, 4096}. We used the Adadelta (Zeiler, 2012) optimizer and tuned a sufficient maximum number of steps for convergence. No other parameters were tuned. * • Interacting forest: We trained depth 5 trees using an internal boosted forest algorithm. We tuned a sufficient maximum number of steps for convergence. No other parameters were tuned. * • GAM forest: We trained depth 3 trees restricted to using a single feature with an internal boosted forest algorithm. We tuned a sufficient maximum number of steps for convergence. No other parameters were tuned. In all cases, we trained models for the tuned maximum number of steps and then truncated models after training. Truncation used a confidence-based truncation algorithm which attempts to select the earliest step for which no later step provides a confident win. This algorithm was run on the validation set if present or otherwise utilized out of bag evaluation. ### A.5. Code The GitHub repository for pwlfit (Sterling and Ravina, 2019) contains several Jupyter notebooks applying our distillation technique and performing the analyses shown in this paper. Please reference the v0.2.0 release to get the accompanying data files and appropriate version of the Jupyter notebooks. ## Appendix B Linear Condense Linear condensing is a data optimization designed to reduce the runtime complexity of our piecewise-linear curve fitting. ### B.1. Motivation/Overview pwlfit picks a set of candidate $x$-knots and searches through combinations of those $x$-knots. For each combination considered, it solves a linear least squares expression for the ideal $y$-knots, calculates the resulting squared error, and prefers the combination that yields the lowest error. Each solve is linear in the size of input, which is slow for large data. We could downsample to save compute at the cost of accuracy. Instead, we introduce a technique to save compute at no cost in accuracy. We condense the data into $\mathcal{O}(\\#candidates)$ synthetic points. These synthetic points perfectly recreate the true squared error over the full data for every PWLCurve that will be considered. We then optimize over the synthetic points instead of the real points. This is possible because we know the candidate $x$-knots ahead of time. A PWLCurve defined on those $x$-knots will always be linear between any adjacent $x$-knots in the set of candidates. As we show in the theorem, we can condense arbitrarily many points down to two points such that linear fits are the same on those two points as on the full set. In the corollary, we apply this process separately between each pair of candidate $x$-knots, producing two points between each pair. Together, the squared error of such a PWLCurve is the same on those synthetic points as it is on the full data set. (Up to a constant that we safely ignore because it’s the same for each PWLCurve.) ### B.2. Definitions For convenience, we take standard definitions and specialize them for weighted 2D points. ###### Definition B.1. Let a ‘point’ refer to a real-valued triple of the form $(x,y,weight)$ with $weight>0$. ###### Definition B.2. Let a ‘line’ refer to a function of the form $f(x)=mx+b$ for $m,b,x\in\mathbb{R}$. ###### Definition B.3. For any function $f:\mathbb{R}\to\mathbb{R}$ and finite point set $P$, define the squared error $SE(f,P)$ as the sum of $(f(x)-y)^{2}\cdot weight$ for each point in $P$. If $P$ is empty, we consider the squared error to be $0$. ###### Definition B.4. For any finite point set $P$, define the ‘best fit line’ $bestfitline(P)$ as the line $L$ that minimizes $SE(L,P)$. In the degenerate case where multiple lines minimize $SE$, let the best fit line be the solution with zero slope, and if multiple solutions have zero slope, let the best fit line be the solution with a zero $y$-intercept. There are two degenerate cases that require tie-breaking. If the point set is empty, every line has the same squared error, so our definition chooses $f(x)=0$ as the best fit line. If the point set is nonempty but all its points have the same $x$, then any line with the correct value at $x$ will minimize the squared error, so our definition choose the horizontal line. ### B.3. Theorem ###### Theorem B.5. Given a set of points $P$, we can construct a set $P^{\prime}$ of two or fewer points such that 1\. $min_{x}(P)<=min_{x}(P^{\prime})<=max_{x}(P^{\prime})<=max_{x}(P)$, and 2\. For any line $L$, $SE(L,P)=SE(L,P^{\prime})+SE(bestfitline(P),P)$. ###### Remark. These properties are desirable because (2) allows us to compute the squared error of $M$ lines over a data set of $N$ points in $\mathcal{O}(N+M)$ instead of the naive $\mathcal{O}(NM)$, and (1) allows us to extend this property from lines to a useful class of piecewise-linear curves in the corollary. Note that the points in $P^{\prime}$ are constructed, rather than chosen from $P$. The construction of $P^{\prime}$ is implemented in pwlfit (Sterling and Ravina, 2019) as linear_condense.linear_condense. ###### Proof. Let $X$, $Y$, and $W$ represent the $x,y$, and $weight$ values of $P$, respectively. We dismiss the trivial case where $P$ is empty; in that case, an empty $P^{\prime}$ satisfies the requirements. Likewise, we dismiss the case where $min(X)=max(X)$ since $P^{\prime}=\\{Centroid(P)\\}$ fulfills our desired properties. With those cases resolved, we assume for the rest of this proof that $min(X)<max(X)$. #### B.3.1. Reframe the Coordinate System To begin, we reframe the coordinate system such that the origin is the centroid of $P$ and $y=0$ is the best fit line. (This simplifies the math.) We ensure that the shift of coordinates is reversible and preserves the squared error. $Centroid(P)=(X\cdot W/sum(W),Y\cdot W/sum(W))$. We translate the coordinate frame by this centroid so that, under the new coordinates, $Centroid(P)=(0,0)$. After translation, $X\cdot W=0$ and $Y\cdot W=0$. Additionally, we skew the coordinate system by the slope of the best fit line: we replace $Y$ with $Y-X\cdot slope(bestfitline(P))$. With the centroid at the origin, the slope of the best fit line is $Covariance(X,Y,W)/Variance(X,W)$ = $(XY\cdot W)/(XX\cdot W)$. After skewing this slope to 0, $XY\cdot W$ = 0. Under the new coordinate frame, $SE(bestfitline(P),P)=SE(y=0,P)=Y^{2}\cdot W$. We will determine $P^{\prime}$ under this new coordinate system. Afterwards, we can easily convert $P^{\prime}$ back to the original coordinate system by reversing the skew and the translation. #### B.3.2. Squared Error of an arbitrary line We will express $SE(line,P)$ as $SE(bestfitline(P),P)$ plus leftover terms. From that, we will derive a $P^{\prime}$ such that $SE(line,P^{\prime})$ equals those leftover terms. For an arbitrary line $y=mx+b$, $SE(y=mx+b,P)=(mX+b-Y)^{2}\cdot W=(m^{2}X^{2}+2bmX-2mXY+b^{2}-2bY+Y^{2})\cdot W.$ In our coordinate frame, $X\cdot W=0$, $Y\cdot W=0$, and $XY\cdot W=0$. So $SE(y=mx+b,P)=(m^{2}X^{2}+b^{2}+Y^{2})\cdot W.$ $Y^{2}\cdot W=SE(bestfitline(P),P)$. Therefore, $SE(y=mx+b,P)=m^{2}X^{2}\cdot W+b^{2}\cdot W+SE(bestfitline(P),P).$ $m^{2}X^{2}\cdot W+b^{2}\cdot W=SE(y=mx+b,P)-SE(bestfitline(P),P).$ #### B.3.3. Squared error over $P^{\prime}$ $SE(y=mx+b,P^{\prime})=SE(y=mx+b,P)-SE(bestfitline(P),P)$ for all lines $y=mx+b$ $\iff$ $(mX^{\prime}+b-Y^{\prime})^{2}\cdot W^{\prime}=m^{2}X^{2}\cdot W+b^{2}\cdot W$ for all lines $y=mx+b$. The above equation can be viewed as a quadratic polynomial in the two variables $m$ and $b$. To hold for all values of $m$ and $b$, the coefficients of each $m^{c}b^{d}$ must be equal on both sides of the equation. Then the equation holds iff: 1\. $X^{\prime 2}\cdot W^{\prime}=X^{2}\cdot W$, and 2\. $X^{\prime}\cdot W^{\prime}=0$, and 3\. $sum(W)=sum(W^{\prime})$, and 4\. $Y^{\prime}\cdot W^{\prime}=0$, and 5\. $Y^{\prime 2}\cdot W^{\prime}=0$, and 6\. $X^{\prime}Y^{\prime}\cdot W^{\prime}=0$. (5) $\iff$ $Y^{\prime}=0$, which also guarantees (4) and (6). We will use 1-3 to derive a satisfactory $X^{\prime}$ and $W^{\prime}$. #### B.3.4. Deriving $X^{\prime}$ and $W^{\prime}$ We’ve determined that $Y^{\prime}=0$. Let $X^{\prime}:=(x_{1},x_{2})$ and $W^{\prime}:=(w_{1},w_{2})$. Without loss of generality, let $x_{1}$ ¡= $x_{2}$. Then, to satisfy our squared error expression, it’s necessary and sufficient that: 1\. $x_{1}^{2}w_{1}+x_{2}^{2}w_{2}=X^{2}\cdot W$, and 2\. $x_{1}w_{1}+x_{2}w_{2}=0$, and 3\. $w_{1}+w_{2}=sum(W)$. Because we have three equations in four unknowns, we cannot directly solve for $x_{1},x_{2},w_{1},w_{2}.$ To produce a fourth equation, we choose the constraint that $x_{1}/x_{2}$ = $min(X)/max(X)$. This choice will simplify the math, and will ensure that $min(X)<=x_{1}<=x_{2}<=max(X)$. With this fourth equation, we solve the simultaneous equations to produce: $x_{1}=-stddev(X,W)\sqrt{-min(X)/max(X)}$ $x_{2}=stddev(X,W)\sqrt{max(X)/-min(X)}$. $w_{1}=sum(W)\cdot max(X)/(max(X)-min(X))$ $w_{2}=sum(W)\cdot-min(X)/(max(X)-min(X))$. Note that, because the centroid is zero, $min(X)<0<max(X)$, so these expressions are all defined. (The denominators are never 0 and values beneath the square roots are never negative.) $P^{\prime}={(x_{1},0,w_{1}),(x_{2},0,w_{2})}$ satisfies our requirements. #### B.3.5. Verify that $min(X)<=x_{1}<=x_{2}<=max(X)$ We wanted $P^{\prime}$ to satisfy the the squared error expression, which it does, and also have its x-values bounded by the x-values of $P$, which we prove now. Let $\mu:=E(X,W)$, the expected value of $X$ weighted by $W$, which is equivalent to the x-value of $Centroid(P)$. By the Bhatia–Davis inequality (Wikipedia, 2020a), $stddev(X,W)^{2}<=(\mu-min(X))(max(X)-\mu)$. (This inequality is equivalent to the observation that the standard deviation of a distribution is maximized when all the xs are at the extremes – i.e. equal to min(X) or max(X).) Since $\mu$ is zero for $P$, $stddev(X,W)^{2}<=-min(X)max(X)$. $x_{1}^{2}=stddev(X,W)^{2}\cdot(-min(X)/max(X))<=-min(X)max(X)\cdot(-min(x)/max(X))=min(X)^{2}.$ $x_{1}<0$ and $min(X)<0$, so $x_{1}^{2}<=min(X)^{2}\implies min(X)<=x_{1}$. The proof that $x_{2}<=max(X)$ is similar. Therefore $min(X)<=x_{1}<=x_{2}<=max(X)$, as desired. ∎ ### B.4. Corollary ###### Corollary B.6. Given a set of points $P$ and a set of x-knots $K$, we can construct a set of points $P^{\prime}$ with $\|P^{\prime}\|<=2(\|K\|-1)$ such that, for any PWLCurve $C$ whose x-knots are elements of $K$, $SE(C,P)=SE(C,P^{\prime})+c$, where $c$ is a constant determined exclusively by $P$ and $K$ that’s the same for every $C$. Note that the points in $P^{\prime}$ are constructed, rather than chosen from $P$. The construction of $P^{\prime}$ is implemented in pwlfit (Sterling and Ravina, 2019) as linear_condense.condense_around_knots. #### B.4.1. Preprocess $P$ by clamping Let $k:=\|K\|$, and consider $K$ in sorted order. Piecewise-linear curves are constant for input values that exceed the range of their x-knots, so $C$ is constant for $x<=min(K)=K[0]$ and for $x>=max(K)=K[k-1]$. Therefore we can clamp the x-values of $P$ to $[K[0],K[k-1]]$ without altering $SE(C,P)$. We do so as a preprocess. #### B.4.2. Partition $P$ by $K$ To construct $P^{\prime}$ from $P$, we first partition $P$ by $K$ into $k+1$ disjoint pieces labeled $P_{0}$, $P_{1}$, …, $P_{k}$. \- $P_{0}$ := $\\{x\in P\|x<K[0]\\}$. \- $P_{i}$ := $\\{x\in P\|K[i-1]<=x<K[i]\\}$ for $1<=i<=k-2$. \- $P_{k-1}$ := $\\{x\in P\|K[k-2]<=x<=K[k-1]\\}$. \- $P_{k}$ := $\\{x\in P\|K[k-1]<x\\}$. Because we clamped $P$, $P_{0}$ and $P_{k}$ are empty. Therefore $\bigcup_{i=1}^{k-1}P_{i}=P$. A PWLCurve is linear between consecutive control points, so $C$ is linear over each $P_{i}$. #### B.4.3. Convert each partition into two points From the theorem, for each $P_{i}$, we can produce a two-point set $P_{i}^{\prime}$ with $min_{x}(P_{i})<=min_{x}(P_{i}^{\prime})<=max_{x}(P_{i}^{\prime})<=max_{x}(P_{i})$, such that for any line $L$, $SE(L,P_{i})=SE(L,P_{i}^{\prime})+SE(bestfitline(P_{i}),P_{i})$. $C$ is linear over each $P_{i}$, so $SE(C,P_{i})=SE(C,P_{i}^{\prime})+SE(bestfitline(P_{i}),P_{i})$. #### B.4.4. Recombine partitions Let $P^{\prime}:=\bigcup_{i=1}^{k-1}P_{i}^{\prime}$. Each $P_{i}^{\prime}$ consists of two points, so $P^{\prime}$ consists of $2(\|K\|-1)$ points. $\displaystyle SE(C,P)$ $\displaystyle=\sum_{i=1}^{k-1}SE(C,P_{i})$ $\displaystyle={\sum_{i=1}^{k-1}(SE(C,P_{i}^{\prime})+SE(bestfitline(P_{i}),P_{i}))}$ $\displaystyle={SE(C,P^{\prime})+\sum_{i=1}^{k-1}SE(bestfitline(P_{i}),P_{i})}.$ $\sum_{i=1}^{k-1}SE(bestfitline(P_{i}),P_{i})$ is determined by $P$ and $K$, and is therefore the same for every $C$. Therefore we’ve proven the corollary.
# Templates of generic geographic information for answering where-questions Ehsan Hamzei, Stephan Winter and Martin Tomkoa Correspondence concerning this article should be addressed to Ehsan Hamzei, Department of Infrastructure Engineering, The University of Melbourne, Parkville, VIC 3010, Australia; Email<EMAIL_ADDRESS>aDepartment of Infrastructure Engineering, The University of Melbourne, Parkville, Australia ###### Abstract In everyday communication, where-questions are answered by place descriptions. To answer where-questions automatically, computers should be able to generate relevant place descriptions that satisfy inquirers’ information needs. Human- generated answers to where-questions constructed based on a few anchor places that characterize the location of inquired places. The challenge for automatically generating such relevant responses stems from selecting relevant anchor places. In this paper, we present templates that allow to characterize the human-generated answers and to imitate their structure. These templates are patterns of generic geographic information derived and encoded from the largest available machine comprehension dataset, MS MARCO v2.1. In our approach, the toponyms in the questions and answers of the dataset are encoded into sequences of generic information. Next, sequence prediction methods are used to model the relation between the generic information in the questions and their answers. Finally, we evaluate the performance of predicting templates for answers to where-questions. ###### keywords: question answering; notion of place; scale; prominence ††articletype: ARTICLE TEMPLATE ## 1 Introduction Consider the following question and its corresponding answer, taken from the Microsoft Machine Comprehension (MS MARCO) dataset v2.1 (Nguyen ., 2016): Question: Where is Putney Bridge? Answer: Putney Bridge is a bridge crossing of the River Thames in west London. This where-question is answered using a _place description_ – a description that characterizes the location of interest (Putney Bridge) based on a few anchor places (River Thames and London). Place descriptions, however, are not the only way to answer where-questions. Where-questions can also be answered via other representations such as maps or sketches (Church ., 2010). Invariant to the chosen representation, the answers localize the place in question based on its spatial relationships with chosen anchor places (Couclelis ., 1987). Hence, answering where-questions poses the following challenges no matter what representation is used: * • Generating informative answers – i.e., the answer should complete the inquirers’ gap of knowledge in a way that obvious or already-known responses should be avoided and useful and necessary information are included (Shanon, 1983). In the example, obvious, inadequate or undetailed answers such as on Earth or in the UK or over a river are avoided by the responder. * • Answering the question in a cognitively efficient manner (D. Wilson Sperber, 2002) – e.g., producing short and straightforward place descriptions (Hamzei, Li ., 2019) and personalized map labelling strategies in map visualizations (JA. Wilson, 2018). In the example, the responder excludes unnecessary information such as the nearby theaters and restaurants to keep the answer as simple and relevant as possible. * • Determining the level of granularity of answers – e.g., a suitable zoom level for maps (Ballatore, 2019) and referring to places of suitable granularity in place descriptions (Hamzei, Winter Tomko, 2019). In our example, the name of the roads and streets that are connected to the bridge are neglected in the answer based on the judgement of the responder for the relevant scale level. * • Selecting places that can be assumed to be known by the inquirer – e.g., labelling the places known to inquirers in maps (Suomela ., 2009) and referring to them in place descriptions as anchors. In the example, the location of _River Thames_ and _London_ are assumed to be known to the inquirer. Where these challenges are met by an answer, the communication succeeds. Addressing these challenges is a necessary step towards answering where- questions. To understand and imitate human selectivity in choosing anchor places, we investigate and characterize human-generated answers to where- questions. The results of our research are applied for generating answers to where-questions as natural language responses (place descriptions). Selecting relevant anchor places is an essential part of generating place descriptions that succeed in answering where-questions. Moreover, information about anchor places can be used in static maps to be visualized in a proper context frame (Ballatore, 2019). Current geographic question answering systems are often focused on coordinate retrieval as answers to where-questions (e.g., Luque ., 2006; Stadler ., 2012). While coordinates are useful for communication between location-based services to perform spatial analysis or visualization (Jiang Yao, 2006), it is not necessarily a relevant response to inquirers without a proper map visualization. Yet, a characterization of relevant anchor places to localize a place in question is still missing. In this paper, we study human-generated answers to where-questions to inform the properties of such answers and to devise and test a method to imitate their structures in machine-generated responses. To achieve these goals, the information in a where-question and its answer is modelled as an ordered set of places that are mentioned in their content. Then the properties of places in questions and corresponding answers are derived and further investigated. This model forms a template (i.e., an ordered set of place properties) that enables computers to learn and imitate human answering behaviour. In other words, place properties are utilized to understand why a set of places are chosen as anchors to localize the place in question and how this selectivity can be imitated by computers. The properties that are used in the templates are generic geographic information that describe the shared meaning of places in form of generic types from a finite set of categories. Referring to the example above, the place in question is a bridge which is localized by referring to the river it goes over and the city it belongs to. Here, the template captures the structure of the answer as relationships between bridges and rivers, and bridges and cities. ### 1.1 Background: Geographic Question Answering Geographic Question Answering (GeoQA) is defined as methods and algorithms that help inquirers to satisfy their information need by deriving answers to their geographic questions. In GeoQA, answering geographic questions can be based on diverse information sources such as textual information (Mishra ., 2010; Ferrés Rodríguez, 2006), geodatabases (Chen ., 2013), and spatially- enabled knowledge bases (Ferrés Rodríguez, 2010). GeoQA (and in general QA) architectures typically resolve three tasks: (a) question classification and intent analysis, (b) finding relevant sources, and (c) extracting answers from the sources (Ferrés Rodríguez, 2006). The classification of the questions (Hamzei, Li ., 2019; Mohasseb ., 2018) enables GeoQA to coarsely identify the intent and purpose of asking questions (e.g., localization, or navigation). Next, the questions are translated into formal representations such as database queries or even just a vector representation of extracted keywords (Punjani ., 2018). Using the representations, the information sources can be searched or queried to look up the possible answers (Zheng ., 2019). Finally, the factoid answers are retrieved from the sources – e.g., a sentence in a Web document, a cell in a selected table, or a node in a graph knowledge base (Sun ., 2018). In recent years, several GeoQA studies were conducted for answering geographic questions (Stadler ., 2012), creating knowledge bases from unstructured data (Mai ., 2018), and relaxing unanswerable questions (Mai ., 2020). Focusing on answering geographic questions, previous studies provide solutions to retrieve responses from knowledge bases (Stadler ., 2012) and documents (Buscaldi ., 2006; Luque ., 2006). GeoQA studies are mostly focused on what/which questions about geographic places (e.g., Scheider ., 2020; Vahedi ., 2016). In answering where-questions, the task is either simplified into retrieving stored coordinates (Luque ., 2006; Stadler ., 2012), or selecting a part of text without explicit adaptation to the question (Buscaldi ., 2006). When answering where-questions, the answer extraction step is particularly challenging. Without a well-designed approach to imitate human answering behavior, the extracted answers can easily be over-specified and consequently uninterpretable for the inquirer, or under-specified and thus obvious and uninformative to the inquirer (Shanon, 1983). Hence, the challenge is to provide relevant answers by selecting proper set of anchor places to localize the place in question. ### 1.2 Rationale and Research Gap To enable computers to provide responses with similar qualities to human- generated answers, the responses need to be relevant. An answer is relevant if its positive cognitive effects to inquirers are large and the processing effort to achieve the effect is small (D. Wilson Sperber, 2002). In other words, answers should be informative enough and as straightforward as possible. Assuming human-generated answers are relevant responses, machine- generated responses should imitate the selectivity in human-generated answer to provide useful pieces of information and avoid unnecessary ones. Generating such relevant responses is the major prerequisite of intelligent GeoQA as defined by Winter (2009). Generic information captures shared meaning of geographic places. While generic geographic information is not used in QA, it has been used to investigate and characterize place descriptions (Richter ., 2013; Edwardes Purves, 2007), route descriptions (Raubal Winter, 2002), and regions (Tomko Purves, 2008). This research hypothesizes that, at least in the English language, generic geographic information can be used to characterize human answering behavior and ultimately to generate templates for answering where-questions. We approach this hypothesis by addressing three sub-hypotheses. ###### Sub-hypothesis 1 (Characteristics of the answers). Human-generated answers to where-questions have special characteristics that can be described and characterized in terms of generic geographic information such as type, scale, and prominence; ###### Sub-hypothesis 2 (Relation between where-questions and their answers). There is a strong relationship between generic information in the content of where-questions and their answers which can be used to characterize human answering behavior; ###### Sub-hypothesis 3 (Generating answers to where-questions). If Hypotheses 1 and 2 hold, the characteristics of human-generated answers and the relation between the questions and their answers can be used to generate templates to answer to where-questions. To investigate the hypotheses, the following research questions will be addressed: 1. 1. How can the characterizing patterns of the human-generated answers be derived? 2. 2. How does generic geographic information in where-questions relate to the generic information in their human-generated answers? 3. 3. How can the templates be generated to imitate the structure of human-generated answers? By addressing the research questions, we contribute: * • A generalized approach to investigate human answering behavior to where- questions using generic geographic information; * • An investigation of the human-generated answers to where-question asked in Web search, using patterns of type, scale and prominence of places. ## 2 Methodology To investigate the hypotheses, we propose a generalized approach of _specific- generic translation_. Next, a method using specific-generic translation is devised to investigate the QA in interaction of people with a general-purpose search engine. Other QA scenarios (e.g., human-human dialogue, human-social bot interaction) may require different design of specific-generic translations. ### 2.1 Specific-Generic Translation Figure 1 shows the proposed approach to derive characterizing patterns in human-generated answers and generating template to answer where-questions. The approach includes two stages: (1) _learning generic patterns_ where the objective is to investigate and to characterize human answering behavior into a machine learning model, and (2) _answering questions_ where the model is used to generate answers. The novelty of the approach is in the encoding of questions and answers into their generic meaning and to model the relation between questions and answers in their generic forms. Later, the model is used to generate generic forms of answers to where-questions. Finally, the answer is constructed by decoding generic form (e.g., type) of the answer into its specific representation (i.e., toponym). Figure 1: Specific-generic translation approach The specific-generic translation approach involves the following steps: 1. 1. Selecting a set of generic information classes (e.g., place type, scale, prominence, functions and access) based on the context of QA and availability of data; 2. 2. Defining a schema for each selected generic information class; 3. 3. Designing an information extraction approach to encode the questions and answers into generic forms (Generic Encoder in Figure 1); 4. 4. Evaluating how effective each generic class is in capturing the relation between the questions and their human-generated answers (Generalized Patterns in Figure 1). The results of evaluation also provide insights about human answering behavior in the context of the QA problem. 5. 5. Training a predictive model that can learn generalized patterns of human- generated answers (Predictive Model in Figure 1); 6. 6. Defining a decoding approach to map generic forms of answers into specific (toponym) representation (Generic Decoder in Figure 1). This step can be followed by additional steps such as natural language generation to be used in real-world applications. In this paper, we discussed the results of the first five steps for question answering in a Web search scenario in details. The last step is only demonstrated using examples. ### 2.2 Type, Scale and Prominence (TSP) Encoding Based on the specific-generic translation, TSP encoding is proposed to investigate where-questions constructed only with toponyms. The generic forms which are used to investigate these questions and their answers are _type_ , _scale_ and _prominence_ of the toponyms. We first introduce our terminology before discussing the details of the proposed TSP encoding method. #### 2.2.1 Definitions The investigated types of where-questions are defined as: * • Toponym-Based Where-Question (TWQ): A toponym-based where-question is a geographical where-question that is generated completely using toponyms. For example, _Where is Beverly Hills?_ is a toponym-based where-question, while _Where is Clueless filmed?_ (without toponym) and _Where can I buy furniture?_ (affordance, buying furniture) do not belong to this type. * • Simple Where-Question (SWQ): Simple where-questions are a sub-class of TWQs that contains only one toponym in their body (e.g., _Where is Beverly Hills?_). * • Detailed Where-Question (DWQ): Detailed where-questions are a sub-class of TWQs with more than one toponym in their content (e.g., _Where is Beverly Hills, California?_). In DWQs, contextual details are provided in the content of the questions that shows what the inquirer already know – e.g., Beverly Hills is located in California. We use _type_ , _scale_ and _prominence_ , defined as: * • Type: A type (e.g., restaurant, mountain) is a reference to a group of places with similar characteristics (e.g., affordance, or physical properties). Type defines similar places and differentiates dissimilar ones, sometimes in a hierarchical or taxonomical manner. Here, the relation between a specific reference to a place (unambiguous toponym) and its type is considered as a one-to-one relation. * • Scale: Scale is defined as a finite hierarchically-organized ordinal set of levels grounded in the human understanding and differentiation of places based on their size and their relationships (i.e., containment). The relation between scale and an unambiguous toponym is considered as one-to-one. Due to the specific context of the QA scenario, very fine levels of scale of geographic entities, such as room-level or object-level, can be neglected here, while in everyday human-human communication these levels of scale may have a more important role. * • Prominence: Prominence is a measure of how well-known a place is. In this research, prominence is characterized by a finite ordinal set of levels. While prominence of places is subjective and differs from person to person based on their experience, here prominence is considered as an absolute and objective measure to rank places, established through a proxy measure defined later. This approach enables to avoid individual experiential biases and is supported by the evidence of success in day to day communication in which the absolute prominence evaluation is adapted between hearers and speakers. Type, scale and prominence are used to characterize place descriptions (Richter ., 2013; Edwardes Purves, 2007). These geographic concepts can be used to capture different kinds of relationships among places. These relationships can be used to understand the relation between where-questions and their answers. For example, such relationships between rivers and seas (flows to), and cities and countries (part of) can be captured using place type. Considering the relation between where-questions and their answers, containment (different levels) and nearness (a same level) can be captured through differences among scale levels. Finally, prominence is a measure to check whether the answer is interpretable by the inquirers – i.e., more prominent places are expected to be better known by inquires. Finally, aspects of human-generated answers which are investigated in this paper are defined below: * • Content: The content of an answer is a collection of distinct information units that are presented to satisfy the inquirer’s information need. Content can be generic (e.g., type) or specific (e.g., toponym). Content is the most important aspect of the answers, in a way that the difference between correct and incorrect responses are completely based on their content. * • Style: The style of an answer is the way that the content is presented. Style directly influences the perception of naturalness of the response. Referring to the introductory example, …Putney Bridge is a bridge crossing of the River Thames in west London and …Putney Bridge is a bridge in west London which goes over River Thames are two different styles of answers (with same content) to the question. Here, the former is preferred by the responder. #### 2.2.2 TSP Sequences In TSP encoding, we use a sequence representation to model generic/specific information in the questions and answers. A sequence is defined as an ordered set of items (here, references to generic/specific geographic information). We first model questions and their corresponding answers as sequences of toponyms (specific representation). Then, these toponym sequences are encoded into type, scale and prominence sequences by translating each specific toponym into its corresponding generic type, scale and prominence reference. Referring to the introductory example, the specific representations (toponym sequences) and the encoded type sequences (an example of a generic sequence) of question and answer are presented below: * • Toponym sequences: [Putney Bridge] [River Thames, London] * • Type sequences: [bridge] [river, city] Here, the _content_ refers to the information items in the sequences, and their order defines the _style_ in which the information is presented. ## 3 Implementation Figure 2 shows the proposed workflow111Additional details of implementation are presented in the supplementary material (Section 1) to investigate TWQs and their answers. Here, we detail the dataset, extraction, encoding, generic patterns and prediction. A complete implementation of the proposed TSP encoding approach also includes decoding from generic to specific information. Here, the decoding step is demonstrated through examples, and a fully automated implementation remains out of scope of this paper. Figure 2: The proposed implementation approach ### 3.1 Data The questions in MS MARCO v2.1 (Nguyen ., 2016) are categorized into five categories using tags: (1) _numeric_ , (2) _entity_ , (3) _location_ , (4) _person_ , and (5) _description_ (Nguyen ., 2016). Geographic questions can thus be easily extracted using the predefined _location_ tag. The dataset contains over one million records divided into _training_ , _development_ and _testing_ subsets. Each record in the dataset includes a _question_ , _human-generated answer(s)_ (except for records in the _test_ dataset, where the answers are deliberately excluded), a _tag_ , and a _set of documents_ retrieved by the Microsoft Bing search engine222More information about the dataset can be found in: https://github.com/dfcf93/MSMARCO. The ‘location questions’ in MS MARCO (56,721 question-answer pairs) include 36,939 geographic questions, and the remainder are questions about fictional, mystic and other non-geographic places (Hamzei, Li ., 2019). Among the geographic questions, 13,195 pairs of questions and answers are geographic where-questions (Hamzei, Li ., 2019). There are several reasons to choose MS MARCO for this study considering other available datasets such as SQuAD (Rajpurkar ., 2016): * • MS MARCO is the largest available QA dataset; * • The questions are labelled and geographic questions can be easily extracted; * • All questions are asked in a specific real-world scenario (i.e., Web search); * • Inquirers pose questions to resolve their information needs while in some datasets such as SQuAD, questions are made from documents. In other words, questions in SQuAD is more about what a document can answer rather than what actual inquirers want to know. * • The answers are provided using an open form strategy. The answerers can utilize suggested documents (one or more) and their own knowledge to answer a question. Hence, the answers are not directly extracted as a single span of a document. ### 3.2 Extraction We first extract the questions labelled as _location_ and starting with a _where_ -token. Next, the text is the toponyms inside the questions and answers are identified using Named Entity Recognition (NER) and gazetteer lookups using both OSM Nominatim and Geonames gazetteers. Here, the Stanford NLP toolkit is used to extract named entities (Finkel ., 2005). In this step, if a compound noun phrase is tagged as location, first the compound noun is checked by gazetteer lookup; if it is not identified, then its constituent simple nouns are checked. If a compound or simple noun phrase is found in both gazetteers, it is stored as a toponym. For the extracted toponyms, we retain only records for which (1) the OSM and Geonames records have same name, and (2) the Geonames’ point-based coordinates are inside the region-based representation of their corresponding OSM records. The toponym disambiguation is then undertaken based on the _minimum spatial context_ heuristic (Leidner ., 2003). We use bounding boxes to determine which combination of the geographic locations satisfy the minimum spatial extent condition. In cases of duplicate places in GeoNames which lead to the same bounding boxes, the combination with more specific place types is selected. For example, populate place (PPL) is a place type in GeoNames which could be a village, city, state and even country. Hence, administrative divisions (e.g., state) are chosen over the populated places. Finally, if the toponym ambiguity still exists, we use importance value to select the more prominent combination. More sophisticated heuristics in toponym disambiguation (e.g., Wang ., 2010; Lieberman Samet, 2012) are not used due to reliance on significant assumptions – e.g., the relation between place types in the toponyms, city-country relation. These heuristics constrain the relationships between type, scale and prominence of resolved places in the text. This may impact the results of this study and lead to stronger associations based on type, scale and prominence between toponyms in questions and answers. Here, to present fair results, we avoid using these disambiguation methods. ### 3.3 Encoding The gazetteers’ attributes for the extracted toponyms have been used as proxies to capture type, scale and prominence of the toponyms in questions and answers. Using these proxies, the sequence representations for each question- answer pair are encoded into TSP sequences. Type: The Geonames type schema333https://www.geonames.org/export/codes.html has been used without modification to encode generic place types. This schema contains 667 unique types of geographic features, covering both natural and man-made geographic types. Scale: To capture scale, we have extended the schema from Richter . (2013). This schema contains seven levels of granularity: (1) furniture, (2) room, (3) building, (4) street, (5) district, (6) city, and (7) country. We have extended the coarse levels of granularity by adding _county_ , _state_ , _country_ , and _continent_ , and removing the _furniture_ and _room_ levels from the schema. OSM records include an attribute related to the OSM definition of scale (i.e., place_rank444https://wiki.openstreetmap.org/wiki/Nominatim/Development_overview), a number between 0 to 30. We convert the extracted gazetteers’ records into the appropriate scale level based on a look-up table that maps OSM scale levels into the proposed scale schema (see the supplementary material, Section 1.1). Prominence: To capture prominence, the _importance_ attribute in the extracted OSM Nominatim record is used. The OSM importance value is estimated using Wikipedia importance score (Thalhammer Rettinger, 2016) with some minor tweaks555https://lists.openstreetmap.org/pipermail/geocoding/2013-August/000916.html. The value is defined between 0 and 1, and it is designed to be used for ranking search results. We translate these values into seven finite levels of prominence, derived by _natural breaks_ classification (Jenks, 1967) of the frequency spectrum of the values. ### 3.4 Distribution Analysis and Rule Mining Distribution analysis and rule mining techniques are used to extract and investigate patterns in the human-generated answers and the relation between the questions and their answers. Distributions of type, scale and prominence sequences are used to compare the questions and answers. To derive patterns in the questions and their answers, association rule mining, a-priori algorithm (Agrawal Srikant, 1994), is used. The strength of the extracted rules are evaluated using the standard measures – i.e., _support_ , _confidence_ , and _lift_. Support defines how frequently an association rule is observed in the whole dataset, and confidence determines how often the rule is true. Lift is a measure to evaluate the importance of the rules – i.e., lift greater than one shows positive and strong dependency among the elements of the extracted rule. This part of the method is devised to test the first and second hypotheses. ### 3.5 Prediction The input for the prediction is an encoded sequence of TWQs, and the output is the generic sequence of their corresponding answers. The problem can then be formulated as a sequence prediction from concatenated generic sequences for the questions and their answers, where a part of a sequence is known, and the rest is predicted. Table 1 shows the sequence prediction methods which are used in this study. We used and extended an open-source toolkit for sequence analysis (Fournier-Viger ., 2016) to implement the prediction methods. These classic methods are divided into probabilistic (Cleary Witten, 1984; Pitkow Pirolli, 1999; Padmanabhan Mogul, 1996) and non-probabilistic categories (Ziv Lempel, 1978; Laird Saul, 1994; Gueniche ., 2013, 2015). The probabilistic methods are based a graph representation of conditional probabilities (Cleary Witten, 1984) or Markov chain’s transition probability matrix (Pitkow Pirolli, 1999; Padmanabhan Mogul, 1996) of the sequence elements. The non-probabilistic methods compress the sequences in a lossy (Ziv Lempel, 1978; Laird Saul, 1994) or lossless approaches (Gueniche ., 2013, 2015) into tree-based (Gueniche ., 2013, 2015) or graph-based (Laird Saul, 1994) data structures (for a review of sequence prediction methods see Tax . (2020)). The structure of sequence and the relation between prior elements in the sequence to their succeeding elements are trained into a model. The model is then tested on an unseen part of data using K-fold cross validation (K=10). We considered two baseline methods to evaluate the performance of the sequence prediction methods: (1) random sequence generation and (2) most frequent pattern. The random generation baseline only utilizes the schema of type, scale and prominence without any information about the distributions of values in the answers. The most frequent patterns baseline predicts templates of answers using the schema and the distribution of generic references in the answers. The difference between the prediction performances of random generation and the most frequent patterns shows the impacts of using the distribution of generic values in generating templates of answers (see hypothesis 1). The sequence prediction methods also consider the relation between generic values in the questions to their answers. Consequently, the improvement in generating the templates compared to the most frequent patterns baseline is related to the association between generic values of questions and their answers (hypothesis 2). Table 1: Sequence prediction methods Method \| Publication \| Year ---\|---\|--- Lampel-Ziv 1978 (LZ78) \| (Ziv Lempel, 1978) \| 1978 First order Markov Chains (Mark1) \| (Cleary Witten, 1984) \| 1984 Transition Directed Acyclic Graph (TDAG) \| (Laird Saul, 1994) \| 1994 Dependency Graph (DG) \| (Padmanabhan Mogul, 1996) \| 1996 All-k-Order Markov Chains (AKOM) \| (Pitkow Pirolli, 1999) \| 1999 Compact Prediction Tree (CPT) \| (Gueniche ., 2013) \| 2013 Compact Prediction Tree Plus (CPT+) \| (Gueniche ., 2015) \| 2015 In prediction, each generic form of questions is used to predict the same generic form of their answers. In addition, we have devised an approach to predict one of the generic forms of an answer using all generic forms (i.e., type, scale and prominence) of its corresponding question. Algorithm 1 shows the process to use all three type/scale/prominence sequences to predict a generic form of the answers in each generic class. Here, each combination of type, scale and prominence values are mapped to a unique code. Using these codes, a new sequence is generated for each question/answer to capture type, scale and prominence together. Next, these sequences are used to predict the generic form of answers. Finally, a reverse mapping is used to decode these sequences into type, scale and prominence sequences. Algorithm 1 Training and prediction based on type-scale-prominence together 1:procedure $\mathbf{TSP\\_Prediction}$($type$, $scale$, $prominence$) 2: generate a code for each unique combination of $type$-$scale$-$prominence$ ($TSP$) 3: create encoded sequences based on generated $TSP$ codes 4: train a model to predict $TSP$ in answers based on $TSP$ in the questions 5: for every $question$ do 6: given a $question$ ($TSP$); predict the $answer$ ($TSP$) 7: decode the predicted $answer$ ($TSP$) to $answer$ ($type$/$scale$/$prominence$) 8: if multiple predictions are allowed then 9: avoid counting duplicate decoded values for $type$/$scale$/$prominence$ 10: end if 11: end for 12:end procedure 13: ## 4 Results ### 4.1 Extraction and Encoding The assessment of toponym extraction, finding TWQs, and categorizing the questions into SWQs and DWQs are presented in Table 2. Here, average precision and recall of the extraction results are calculated using manually annotated data (5% of TBWQs and their answers). For the task of finding TWQs in the dataset, the _false negatives_ (TWQs that have not been extracted) are not investigated, hence the recall is unknown. As shown in Table 2, 6,274 TWQs and their answers are found in the dataset. The TWQs are approximately 11.1% of the _location questions_ of the dataset. For evaluation, 5% of extracted TWQs (314 questions) are investigated and the precision of extraction is 91.7% – i.e., 288 of 314 extracted questions are TWQs. Using the 288 TWQs, the precision and recall of extracting toponyms and classifying the questions to SWQs and DWQs are presented in Table 2. Table 2: Extraction evaluation Extraction \| #Extracted \| #Investigated \| Precision \| Recall ---\|---\|---\|---\|--- TWQs \| 6274 \| 314 (5%) \| 91.7% (288 out of 314) \| – SWQs \| 3285 \| 121 out of 288 \| 89.4% \| 90.2% DWQs \| 2989 \| 167 out of 288 \| 92.7% \| 92.1% Toponyms \| 22307 \| 1133666unique toponyms extracted from the sampled questions and answers \| 88.6% \| 90.8% Table 3 shows the number of records that are completely encoded for question- answer pairs in type, scale and prominence sequences. Here, if even the information for one place (which is mentioned either in the question or its answer) is missing, the question and its answer are not used to extract patterns or test the predictability of generating generic form the answer. As shown in the table, the encoding into scale and prominence is not always possible due to incompleteness of attribute information (i.e., _place_rank_ and _importance_) in OSM Nominatim. Table 3: Encoding results Encoding \| #TWQs \| #SWQs \| #DWQs ---\|---\|---\|--- Type sequences \| 6,274 \| 3,285 \| 2,989 Scale sequences \| 3,936 \| 1,985 \| 1,951 Prominence sequences \| 6,051 \| 3,098 \| 2,953 ### 4.2 Distributions The distribution of TWQs777A detailed comparison of SWQs and DWQs is presented in the supplementary material (Section 2) and their answers based on type, scale and prominence are shown in Figures 3, 4 and 5. Figure 3 shows that the diversity of types in the questions is higher than in the answers. While administrative divisions are more frequent than other generic types in both questions and answers, they are more dominant in the answers. Figure 3: Distribution of place types in the questions and in the answers. Figure 4 shows the scale in the answers is systematically one level coarser than in the questions. In addition, the distribution shows that city-level and state-level scales are frequently observed in the questions, while the answers mostly contain references of county and country levels of scale. The results further show that the coarsest level of scale (i.e., continent level) is rarely observed in the answers. This observation shows an answer at the continent level would be under-specified in most cases, and therefore uninformative. Figure 4: Distribution of levels of scale in all toponym-based where questions and answers. The distributions of prominence levels in questions and answers are similar to the distributions by scale (Figure 5). In the questions, we observe a bi-modal distribution of levels of prominence in the content of questions. The distribution of prominence in the answers, however, shows that higher levels are dominant. In contrast to the distributions by scale, the most prominent level is dominant in the answers. Hence, people tend to refer to well-known places in their answers. Unlike with scale, the highest levels of prominence do not necessarily lead to obvious or irrelevant answers888A detailed analysis of sequence distributions is available in supplementary material (Section 3). Figure 5: Distribution of prominence levels in the questions versus answers ### 4.3 Extracted Rules To test Hypotheses 1 and 2 (see Section 1.4), we extract strong rules in the encoded pairs of questions-answers through association rule mining. The association rules extracted from the answers can be used to describe how answers are constructed in detail (Hypothesis 1). The relationship between the content of the questions and their answers can thus also be further investigated (Hypothesis 2). Tables 4-6 show the top five extracted rules (based on _frequency_ /_support_) for type, scale and prominence, respectively. In the tables, the values starting with _Q-_ relate to the contents of the questions and the values starting with _A-_ to the content of the corresponding answers. As shown in the tables, some rules describe the structure of answers (e.g., {A-ADM1, A-ADM2} =>{A-PCLI}) while the others describe the relationships between questions and answers (e.g., {Q-ADM2} =>{A-PCLI}). Table 4: Extracted rules from type sequences Rank \| rule \| support \| confidence \| lift \| frequency ---\|---\|---\|---\|---\|--- Simple where-questions 1 \| {A-ADM2} =>{A-ADM1} \| 0.15 \| 0.52 \| 1.28 \| 478 2 \| {Q-ADM1} =>{A-PCLI} \| 0.08 \| 0.74 \| 1.69 \| 259 3 \| {A-ADM1,A-ADM2} =>{A-PCLI} \| 0.08 \| 0.54 \| 1.24 \| 259 4 \| {Q-ADM2} =>{A-PCLI} \| 0.06 \| 0.52 \| 1.21 \| 188 5 \| {Q-PPL,A-ADM2} =>{A-PCLI} \| 0.04 \| 0.54 \| 1.23 \| 112 Detailed where-questions 1 \| {Q-ADM1} =>{A-ADM2} \| 0.57 \| 0.76 \| 1.12 \| 1701 2 \| {Q-ADM1} =>{A-PCLI} \| 0.38 \| 0.50 \| 1.13 \| 1126 3 \| {A-PCLI} =>{A-ADM2} \| 0.35 \| 0.79 \| 1.17 \| 1053 4 \| {A-ADM2,Q-ADM1} =>{A-PCLI} \| 0.31 \| 0.54 \| 1.21 \| 916 5 \| {Q-PPL} =>{A-ADM2} \| 0.22 \| 0.78 \| 1.15 \| 656 Table 5: Extracted rules from scale sequences Rank \| Rule \| support \| confidence \| lift \| frequency ---\|---\|---\|---\|---\|--- Simple where-questions 1 \| {Q-6} =>{A-9} \| 0.21 \| 0.55 \| 1.01 \| 417 2 \| {Q-6} =>{A-8} \| 0.20 \| 0.54 \| 1.24 \| 404 3 \| {A-7} =>{A-9} \| 0.16 \| 0.56 \| 1.73 \| 307 4 \| {Q-6} =>{A-7} \| 0.15 \| 0.54 \| 1.37 \| 295 5 \| {A-7} =>{A-8} \| 0.15 \| 0.54 \| 1.25 \| 293 Detailed where-questions 1 \| {Q-8} =>{A-7} \| 0.65 \| 0.80 \| 1.08 \| 1277 2 \| {Q-6} =>{Q-8} \| 0.49 \| 0.81 \| 0.99 \| 952 3 \| {Q-6} =>{A-7} \| 0.48 \| 0.80 \| 1.07 \| 940 4 \| {A-9} =>{Q-8} \| 0.45 \| 0.87 \| 1.07 \| 887 5 \| {A-7,Q-6} =>{Q-8} \| 0.42 \| 0.88 \| 1.07 \| 823 Table 6: Extracted rules from prominence sequences Rank \| Rule \| support \| confidence \| lift \| frequency ---\|---\|---\|---\|---\|--- Simple where-questions 1 \| {A-4} =>{A-7} \| 0.14 \| 0.54 \| 1.09 \| 425 2 \| {A-5} =>{A-7} \| 0.13 \| 0.50 \| 1.02 \| 417 3 \| {Q-3} =>{A-7} \| 0.12 \| 0.52 \| 1.05 \| 382 4 \| {Q-6} =>{A-7} \| 0.08 \| 0.58 \| 1.18 \| 260 5 \| {Q-4} =>{A-7} \| 0.08 \| 0.53 \| 1.07 \| 250 Detailed where-questions 1 \| {Q-6} =>{A-4} \| 0.32 \| 0.56 \| 1.19 \| 957 2 \| {Q-6} =>{A-7} \| 0.30 \| 0.51 \| 1.06 \| 884 3 \| {A-4} =>{A-7} \| 0.25 \| 0.54 \| 1.11 \| 742 4 \| {Q-3} =>{Q-6} \| 0.24 \| 0.54 \| 0.93 \| 695 5 \| {Q-3} =>{A-7} \| 0.22 \| 0.51 \| 1.04 \| 650 Table 4 shows the dominant role of administrative divisions in the human- generated answers. Association rules extracted based on the scale (Table 5) show the _greater-than_ and _between_ levels of the answers to SWQs and DWQs. The top five patterns of answers are mostly constructed with references to the highest level of prominence (_A-7_). This shows the major impact of prominence in human answering behavior to where-questions – i.e., people refer to prominent places in answering where-questions. Tables 4, 5 and 6 show that stronger association rules with higher support are extracted from DWQs in comparison to SWQs. The rules show strong associations between antecedent and consequent parts of the extracted rules with lift value greater than one. The results show that stronger rules with higher confidence and support are extracted using scale in comparison to type and prominence. The tables only present the extracted rules with highest frequency and support. These tables show how a small set of generic rules describes a large proportion of data in the MS MARCO dataset. Sorting rules by confidence or lift will change the order of the rules. For example, the maximum lift (equal to 8.93) in the extracted rules belongs to {Q-6, Q-9} =>{A-8} for detailed- where questions using scale. The frequency of this rule is 43, and it describes the relevant scale level (between minimum and maximum levels of the question) for detailed where-questions. The maximum confidence is 0.93 for detailed where-questions encoded by type. This association rule is {Q-PPLA2, Q-ADM1} =>{A-ADM2} with a frequency 109. This rule describes that populated places in detailed where-questions are mostly localized by referring to counties they belong to. ### 4.4 Predicting the Generic Form of Answers We test the predictability of the generic sequence of an answer given the generic sequence of the corresponding question. We investigate different prediction scenarios, including (1) the same generic class prediction (e.g., predicting type sequence of answers using type sequence of questions), and (2) prediction of one generic class using all generic classes (e.g., predicting type sequence of answers using type/scale/prominence sequences of questions, see Algorithm 1). We assess the prediction accuracy of content and content-and-style of the answers (defined in Section 2.2.1). Referring to the introductory example, if type sequence of the answer is predicted as [river, city] then it is captured as a correct prediction for both content and content-and-style. The other permutation of this sequence (i.e., [city, river]) is considered as a correct prediction of content and incorrect prediction for content-and-style. Evidently, any other type sequence is an incorrect prediction for both content and content-and-style scenarios. Each prediction scenario is applied over all questions, SWQs and DWQs to investigate the impacts of question types on the prediction accuracy. Each scenario is tested using all six sequence prediction methods and is compared with the two baseline approaches (i.e., random generation and most frequent patterns). Only the best prediction performances among the six sequence prediction methods are presented. The best performance is the maximum prediction accuracy achieved by one of the methods for a prediction scenario. We also test prediction accuracy when multiple predictions are allowed – i.e., top-k predictions for $k$ from one to five. In top-k predictions, $k$ unique sequences are predicted for each answer and if one of the sequences matches with the generic form of the answer then the prediction is successful. Table 7 shows the best performances in predicting type sequences of answers.The prediction accuracy based of TSP sequences is noticeably higher than that of predictions using only type sequences. This shows a complementary role of scale and prominence in predicting type sequence of the answers. Contrasting DWQs and SWQs shows that extra details in DWQs are useful for prediction of the generic form of answers. In addition, we observe how subjectivity in style of answers and flexibility of language to convey information lead to noticeable less accuracy in prediction of content-and- style of answers in comparison to prediction of content. This observation is related to the flexibility of natural language, in which the same meaning can be presented in different ways. Finally, the number of predictions ($k$ in the table) shows that the accuracy dramatically increases in the case of multiple predictions. Table 7: Prediction accuracy for type sequences #Predictions (k) \| Content \| Content and Style ---\|---\|--- \| Type $\rightarrow$ Type \| TSP $\rightarrow$ Type \| Type $\rightarrow$ Type \| TSP $\rightarrow$ Type All questions 1 \| 45.2 \| 55.7 \| 29.0 \| 40.7 2 \| 68.9 \| 77.1 \| 44.6 \| 60.5 3 \| 80.2 \| 83.3 \| 57.8 \| 73.3 4 \| 83.6 \| 84.7 \| 64.0 \| 76.1 5 \| 84.4 \| 85.5 \| 68.3 \| 77.4 Simple where-questions 1 \| 39.5 \| 47.5 \| 14.2 \| 27.4 2 \| 60.8 \| 69.4 \| 32.7 \| 48.5 3 \| 73.2 \| 75.8 \| 48.2 \| 63.4 4 \| 77.2 \| 77.5 \| 58.1 \| 66.2 5 \| 78.5 \| 78.2 \| 63.1 \| 67.0 Detailed where-questions 1 \| 59.1 \| 67.3 \| 47.1 \| 59.6 2 \| 80.4 \| 88.7 \| 61.3 \| 76.3 3 \| 84.0 \| 91.2 \| 65.9 \| 84.4 4 \| 88.0 \| 91.3 \| 73.6 \| 86.4 5 \| 88.5 \| 92.1 \| 75.6 \| 87.1 Tables 8 and 9 show that compared to type sequence prediction, the TSP sequences contribute less effectively in predicting the prominence and scale sequences – i.e., only slightly improve the prediction accuracy. When considering multiple predictions, TSP sequences lead to worse results than prominence sequences or scale sequences alone. This can be explained by overfitting to specific patterns in the training dataset. Here, overfitting is observed because the schema of types is more than 20 times larger than the scale and prominence schemas. Hence, using type in prediction of scale or prominence leads to very detailed patterns that are not generalizable enough and decrease the prediction accuracy on unseen data. Finally, scale is the most predictable, and prominence is the least predictable generic class. Similar to the observations based on type prediction performances, DWQs are more predictable than SWQs based on scale and prominence. Table 8: Prediction accuracy for scale sequences #Predictions (k) \| Content \| Content and Style ---\|---\|--- \| Scale $\rightarrow$ Scale \| TSP $\rightarrow$ Scale \| Scale $\rightarrow$ Scale \| TSP $\rightarrow$ Scale All questions 1 \| 55.0 \| 56.7 \| 38.2 \| 42.2 2 \| 79.4 \| 79.2 \| 61.0 \| 62.8 3 \| 91.6 \| 86.1 \| 79.0 \| 76.0 4 \| 96.3 \| 88.7 \| 92.0 \| 81.9 5 \| 98.0 \| 89.3 \| 96.0 \| 83.5 Simple where-questions 1 \| 48.5 \| 49.5 \| 20.4 \| 28.6 2 \| 79.6 \| 71.8 \| 49.1 \| 49.8 3 \| 89.9 \| 78.3 \| 71.9 \| 67.0 4 \| 95.6 \| 81.8 \| 90.3 \| 74.0 5 \| 97.5 \| 82.6 \| 94.9 \| 75.5 Detailed where-questions 1 \| 69.6 \| 68.2 \| 59.8 \| 60.6 2 \| 88.4 \| 89.6 \| 78.4 \| 77.3 3 \| 95.8 \| 93.3 \| 88.6 \| 87.1 4 \| 97.5 \| 95.2 \| 94.8 \| 92.1 5 \| 98.6 \| 95.2 \| 97.0 \| 92.7 Table 9: Prediction accuracy of prominence sequences #Predictions (k) \| Content \| Content and Style ---\|---\|--- \| Prominence $\rightarrow$ Prominence \| TSP $\rightarrow$ Prominence \| Prominence $\rightarrow$ Prominence \| TSP $\rightarrow$ Prominence All questions 1 \| 50.8 \| 53.0 \| 19.9 \| 30.7 2 \| 74.1 \| 73.4 \| 39.2 \| 49.1 3 \| 85.0 \| 81.9 \| 61.6 \| 66.4 4 \| 92.1 \| 86.7 \| 79.2 \| 77.1 5 \| 96.1 \| 88.6 \| 89.2 \| 81.8 Simple where-questions 1 \| 45.4 \| 45.6 \| 14.3 \| 19.5 2 \| 75.4 \| 69.4 \| 34.9 \| 39.1 3 \| 84.7 \| 77.0 \| 54.5 \| 56.9 4 \| 91.3 \| 80.5 \| 73.7 \| 68.2 5 \| 95.6 \| 81.9 \| 87.9 \| 72.7 Detailed where-questions 1 \| 53.3 \| 58.2 \| 26.9 \| 43.8 2 \| 75.1 \| 80.0 \| 50.9 \| 60.4 3 \| 86.0 \| 88.9 \| 70.9 \| 78.4 4 \| 93.1 \| 93.0 \| 82.5 \| 87.0 5 \| 96.8 \| 95.4 \| 91.9 \| 92.6 Table 10 shows the improvement of accuracy in best prediction performances compared to two baselines – i.e., random generator, and most frequent pattern(s). The minimum improvement is +18.3% in prediction of type sequences of answers using type sequences of questions in comparison to the most frequent pattern(s). This observation shows that strong patterns exist in the distributions of answers and consequently, the baseline method performs well in prediction of type sequences of answers. The strongest improvement is +61.6% when comparing the best predictive performance of type sequences using type/scale/prominence sequences together, compared to the random baseline. This is because of the large number of distinct types in type schema that lead to false predictions for the random baseline. The accuracy improvements illustrate the strong relationship between the generic content of questions and generic content of their answers. Table 10: Accuracy improvement using sequence prediction compared to the baselines Prediction Scenario \| Random \| Most Frequent Pattern(s) ---\|---\|--- Type $\rightarrow$ Type \| +48.9% \| +18.3% Scale $\rightarrow$ Scale \| +58.1% \| +27.6% Prominence $\rightarrow$ Prominence \| +39.2% \| +30.4% TSP $\rightarrow$ Type \| +61.6% \| +31.0% TSP $\rightarrow$ Scale \| +54.1% \| +23.6% TSP $\rightarrow$ Prominence \| +42.3% \| +33.5% Overall \| +50.7% \| +27.4% To compare the sequence prediction methods, we used the difference between the prediction accuracy of each method to the best performance achieved by all methods for each prediction scenario. Table 11 shows the root mean square error (RMSE) for each sequence prediction method. The RMSE shows how well- performed a method is in comparison to other methods. If the RMSE of a method is lower than others, the prediction accuracy of the method is higher than the others. The prediction scenarios in Table 11 are simplified groups of actual predictions. For example, prediction scenario of scale is related to predicting scale sequences of answers using (1) scale sequence of questions or (2) type/scale/prominence sequences of questions. As shown in Table 11, in all scenarios the CPT method is the best performing method and TDAG performs worst based on the RMSE values. The results suggest that CPT is the best method to construct predictive models to predict the generic form of answers. Table 11: RMSE of sequence prediction methods Prediction Scenario \| LZ78 \| Mark1 \| TDAG \| DG \| AKOM \| CPT \| CPT+ ---\|---\|---\|---\|---\|---\|---\|--- Type \| 7.4% \| 15.2% \| 21.8% \| 13.4% \| 17.3% \| 7.1% \| 12.9% Scale \| 9.8% \| 12.5% \| 17.9% \| 10.6% \| 14.3% \| 5.7% \| 11.8% Prominence \| 8.7% \| 13.3% \| 19.2% \| 9.9% \| 15.2% \| 4.9% \| 11.5% Content \| 8.9% \| 15.5% \| 22.7% \| 9.1% \| 17.4% \| 1.9% \| 10.2% Content and Style \| 8.6% \| 11.7% \| 16.1% \| 13.2% \| 13.7% \| 8.2% \| 13.8% ## 5 Demonstration: From Generic to Specific Translating generic encoding of answer to specific form (e.g., type sequence to toponym sequence) is the last phase in the proposed approach. Our approach to the generic-to-specific translation problem is grounded in the following assumption: _places mentioned in the questions have relationships to places referred to in their answers, and these relations can be found in a knowledge base_. In addition, the specific form of questions and generic form of answers are available through encoding and prediction, respectively. Based on this assumption and the available information, the specific form of answer can be derived using a SPARQL query template (Query 1). While the _structure_ of a suitable knowledge base for this purpose has been studied before by Chen . (2018), no such knowledge base is yet available with the definitions of type, scale and prominence as used in this study. Hence, the translation is only demonstrated here using the introductory example999More examples are provided in the supplementary material (Section 4). We have used DBPedia and Geonames as sources to demonstrate how SPARQL queries can be used to find specific forms of answers. Considering the information stored in DBPedia and Geonames, this demonstration is limited to type sequences of the answers because the prominence and scale are not available in the place ontology of these knowledge bases. Even the type schema used in DBPedia is different from the Geonames’ type schema, and consequently in the following example, mapping to the DBPedia type schema is done manually. ⬇ PREFIX [KNOWLEDGE BASE] SELECT distinct ?question ?answer WHERE { VALUES ?question [SPECIFIC] . ?answer a [GENERIC] . {?question ?r ?answer} UNION {?answer ?r ?question} . } Query 1: SPARQL template Referring to the introductory example, the where-question and its answer is modelled as follows: * • specific representation (question): [Putney Bridge]; * • TSP encoding (question): type sequence [BDG], scale sequence [4], prominence sequence [3]; * • TSP encoding (answer): type sequence [STM, ADM2], scale sequence [6, 6], prominence sequence [6, 7]; * • specific representation (answer): [River Thames, London] The SPARQL queries for finding the specific forms of answers are presented in Queries 2 and 3 using DBPedia and Geonames ontologies. The results of these queries are shown in Table 12. Using DBPedia, the generic forms are correctly translated into River Thames and London. However, the generic to specific translation using Geonames is partially successful. In Geonames, places are conceptualized as points and it supports only containment. This example shows that point-based conceptualization of places is not sufficient for generic to specific translation and more diverse support of spatial relationships can be useful to find the correct specific forms. ⬇ PREFIX dbo: <http://dbpedia.org/ontology/> SELECT distinct ?q1 ?a1 ?a2 WHERE { VALUES ?q1 {<http://dbpedia.org/resource/Putney_Bridge>} ?a1 a dbo:PopulatedPlace . {?a1 ?r1 ?q1} UNION {?q1 ?r1 ?a1} . ?a2 a dbo:River . {?a2 ?r2 ?q1} UNION {?q1 ?r2 ?a2} . } Query 2: SPARQL query of the example (DBPedia) ⬇ PREFIX gn: <http://www.geonames.org/ontology#> SELECT distinct ?q1 ?a1 ?a2 WHERE { VALUES ?q1 {<http://sws.geonames.org/6619925/>} ?a1 gn:featureCode gn:A.ADM2 . {?a1 ?r1 ?q1} UNION {?q1 ?r1 ?a1} . ?a2 gn:featureCode gn:H.STM . {?a2 ?r2 ?q1} UNION {?q1 ?r2 ?a2} . } Query 3: SPARQL query of the example (Geonames) Table 12: SPARQL results to find specific form of the answer Knowledge Base \| Q1 \| A1 \| A2 ---\|---\|---\|--- DBPedia \| Putney Bridge \| London \| River Thames Geonames \| Putney Bridger \| London \| – ## 6 Discussion The results of the proposed method shows how generic information can be used to characterize and imitate human answering behavior to generate templates for answering the questions. While the results are limited to the human-search engine interaction, the proposed methodology (specific-generic translation) is flexibly defined to be applicable to other QA scenarios as well. We have used type, scale and prominence as generic classes to investigate MS MARCO dataset. We have compared their potentials in describing human answering behavior and their performances in predicting the generic forms of the answers. As a result, two major observations are reported. First, while strong patterns for each generic class have been observed, we find that scale is the most predictive class. This is because where-questions are a specific subset of spatial questions and scale directly captures inherent spatial aspect of places. Meanwhile, in the notions of type and prominence, other aspects of places contribute as well – e.g., functional and physical aspects. In addition, scale is a generic class that captures hierarchical relationships between places, and previous studies show that these relationships are the basis for answering where-questions (Shanon, 1983). Moreover, we have observed that type is performing better than prominence in both characterizing and predicting human-generated answers. This observation is highly influenced by the proxies used to capture type and prominence. Second, when comparing SWQs and DWQs, our investigation shows that the generic templates to answer to DWQs, compared to SWQs, can be generated more accurately. We find stronger rules and patterns in the answers to DWQs than in answers to SWQs. This is because DWQs contain richer details which helps narrowing down the list of possible relevant answers. To illustrate this point, two examples are provided (1) _Where in California is Beverly Hills?_ and (2) _Where is Beverly Hills?_ In the first question, the list of possible relevant answers is narrowed down to _Los Angeles County_ because the inquirer already knows it is in _California_. For the latter, respondents are free to subjectively guess the state of the inquirer’s geographic knowledge and provide answers such as _Los Angeles County_ , _California_ , and _United States_. Theoretical limitations: The instruction for specific-generic approach is devised in a flexible manner to be usable in different GeoQA scenarios. However, utilizing the approach needs a careful design (e.g., selecting appropriate list of generic classes) to fit for a particular scenario. The proposed TSP encoding is limited to the QA scenario of general Web search and may not be suitable for other QA scenarios such as human interaction with autonomous cars. In short, the theoretical limitations of this study are: 1. 1. The generic-to-specific translation approach is only focused on where- questions, and other types of geographic questions are neglected. 2. 2. The proposed approach is focused only on the questions and their relationship with the answers, when no other contextual information about inquirers is available. 3. 3. The approach is designed with an exclusive focus on toponyms while qualitative spatial relationships have an important role in answering where questions. 4. 4. The additional impacts of qualitative spatial relationships (e.g., _in southern part of_) as modifiers of scale are neglected in the TSP encoding. Results limitations: There are some limitations to the implementation presented in this study: 1. 1. The biases of the MS MARCO dataset directly influence our results. The data are extracted from the Microsoft Bing search engine, and hence the results are necessarily biased to the questions asked by users of this search engine. In addition, the sampling approach used when extracting MS MARCO questions from the MS Bing query logs may have a direct and unquantifiable impact on the generality of the results. 2. 2. The results are influenced by the geographic biases and incompleteness of data in Geonames and OSM Nominatim. The bias and incompleteness of gazetteers are well-documented by Acheson . (2017). 3. 3. The bias in the proxies that have been used to capture the TSP encoding also have an impact on the results. Despite these limitations, the identified patterns align well with everyday experience and provide a grounding for answering where-questions. ## 7 Conclusions Generating responses with a similar quality to human-generated answers is a challenge to current search engines and QA systems. In particular, where- questions are hard to answer because the responses can sometimes be either vague or obvious to the inquirers. To avoid generating ambiguous or obvious responses or retrieving unnecessary information as a part of the answers, a proper set of anchor places must be identified to localize the place in question. The assumption that answers to where-questions can be found completely, without any further modification, inside a textual document or as a node or its properties in a knowledge base may not hold in general. Consequently, we introduced here an approach to generate templates to answer where-questions based on relevant pieces of information. The approach is based on the automatic extraction of patterns of generic geographic forms from human-generated QA. These captured in predictive models and are used to generate templates of answers similar human-generated responses. Three generic classes (i.e., type, scale and prominence) are used to investigate the properties of the anchor places in human-generated answers. We have used questions and answers from MS MARCO v2.1, an extensive dataset constructed from questions submitted to a general-purpose search engine. Using distribution analysis and rule mining techniques, we have identified the characteristics and recurrent patterns in the questions and their answers (Hypotheses 1 and 2). We have then applied sequence prediction methods to generate the generic forms for answers based on the generic forms of the corresponding questions (Hypothesis 3). We have also briefly sketched an approach how such generic forms may help with the generation of the appropriate answers, based on the information available in the knowledge bases. The results show that the prediction of answer structures based on scale is more precise, compared to predictions relying on type and prominence. The rules extracted based on scale have higher support and confidence than the rules extracted from type or prominence. We also observe how the type of questions (i.e., SWQs vs. DWQs) influence the strength of the extracted rules and lead to noticeable differences in prediction performances. Finally, we compared different sequence prediction methods and find that CPT (Gueniche ., 2013) is the best performing approach in all scenarios. However, the results of this study are limited to human interaction with a general purpose search engine. Consequently, an important future direction of this research is to investigate other corpora of QA related to different scenarios – e.g., human- human dialogue. We have also observed that the neglect of qualitative spatial relationships in our encoding and prediction mechanism may present a a major theoretical shortcoming of the proposed specific-generic translation. Consequently, developing a more sophisticated encoding is necessary to extract a deeper understanding of the human answering behavior of where-questions. Developing an automatic approach to decode generic forms of answers into specific representations (i.e., toponyms) is a necessary step to complete the process of the specific-generic translation approach. Available information in documents or knowledge bases can be used to derive the specific representations. Another important future direction is to investigate how the proposed approach can be combined with current personalization methods, in order to adapt answers to specific inquirers and their context. Finally, investigation of other types of where-questions (i.e., where-questions with generic references) and their human-generated answers using specific-generic translation remains as a future work. ## 8 Data and Codes Availability Statement This study makes use of a third-party data source, MS MARCO v2.1 (Nguyen ., 2016). The dataset is freely available under a proprietary agreement for non- commercial use101010http://www.msmarco.org/dataset.aspx. The computational workflow of this publication is implemented in Java and R. The implementation is available under the MIT License111111https://opensource.org/licenses/MIT and accessible in GitHub: https://github.com/hamzeiehsan/Template-for- answering-where-questions. ## Acknowledgments The support by the Australian Research Council grant DP170100109 is acknowledged. We also thank the anonymous reviewers for their helpful comments that improve the quality of this paper. ## References * Acheson . (2017) ACHESON2017309Acheson, E., Sabbata, SD. Purves, RS. 2017\. A quantitative analysis of global gazetteers: Patterns of coverage for common feature types A quantitative analysis of global gazetteers: Patterns of coverage for common feature types. 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LABEL:firstpage–LABEL:lastpage # Patterns formed in a thin film with spatially homogeneous and non- homogeneous Derjaguin disjoining pressure A.S.ALSHAIKHI$\,{}^{1}$ M.GRINFELD$\,{}^{1,2}$ S.K.WILSON$\,{}^{1}$ $\,{}^{1}$ Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH,UK $\,{}^{2}$ email: <EMAIL_ADDRESS> (20 December 2020; resubmitted 11 June 2021; 2000) ###### Abstract We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady state solutions on the wettability contrast in that case. ###### keywords: thin films, disjoining pressure, non-homogeneous substrates, pattern formation ###### 2020 Mathematics Subject Classification: 7 ††volume: 000 4K35 (Primary); 35B32 (Secondary) ## 1 Introduction Thin liquid films on solid substrates occur in many natural situations. For example, they appear in tear films in the eye which protect the cornea [6] or in streams of lava from a volcanic eruption [19]. Moreover, thin liquid films occur in many technological applications, such as coatings [22] and lubricants (e.g. oil films which lubricate the piston in a car engine [41], drying paint layers [18], and in the manufacture of microelectronic devices [21]). For extensive reviews of thin-film flow see, for example, Oron et al. [28] and Craster and Matar [10]. As these liquid films are thin, the Navier–Stokes equation governing their flow can be reduced to a single degenerate fourth-order quasi-linear parabolic partial differential equation (PDE) usually known as a thin-film equation. In many applications a choice of a disjoining pressure, which we denote by $\Pi$, must be made. Such a term describes the action of surface forces on the film [37]. In different situations, different forms of disjoining pressure are appropriate; these may incorporate long-range van der Waals forces and/or various types of short-range interaction terms such as Born repulsion; inclusion of a particular type of interaction can have significant effects on the wettability of the surface and the evolution of the film film, sometimes leading to dewetting phenomena, i.e. the rupture of the film and the appearance of dry spots. (Here and subsequently by “wettability” of the surface we mean the ability of a solid surface to reduce the surface tension of a liquid on contact with it such that it spreads over the surface and wets it.) Witelski and Bernoff [43] were one of the first authors to analyse mathematically the rupture of three-dimensional thin films. In particular, considering a disjoining pressure of the form $\Pi=-1/(3h^{3})$ (we use the sign convention adopted in Honisch et al. [17]), they analysed planar and axisymmetric equilibrium solutions on a finite domain. They showed that a disjoining pressure of this form leads to finite-time rupture singularities, that is, the film thickness approaches zero in finite time at a point (or a line or a ring) in the domain. In a related more recent paper, Ji and Witelski [20] considered a different choice of disjoining pressure and investigated the finite-time rupture solutions in a model of thin film of liquid with evaporative effects. They observed different types of finite-time singularities due to the non-conservative terms in the model. In particular, they showed that the inclusion of non-conservative term can prevent the disjoining pressure from causing finite-time rupture. A pioneering theoretical study of a thin-film equation with a disjoining pressure term given by a combination of negative powers of the thin film thickness is that by Bertozzi et al. [4], who studied the formation of dewetting patterns and the rupture of thin liquid films due to long-range attractive and short-range Born repulsive forces, and determined the structure of the bifurcation diagram for steady state solutions, both with and without the repulsive term. Aiming to quantify the temporal coarsening in a thin film, Glasner and Witelski [15] examined two coarsening mechanisms that arise in dewetting films: mass exchange that influences the breakdown of individual droplets and spatial motion that results in droplet rupture as well as merging events. They provided a simple model with a disjoining pressure which combines the effects of both short- and long-range forces acting on the film. Kitavtsev et al. [23] analysed the long-time dynamics of dewetting in a thin-film equation by using a disjoining pressure similar to that used by Bertozzi et al. [4]. They applied centre manifold theory to derive and analysed an ordinary differentual equation model for the dynamics of dewetting. The recent article by Witelski [42] presents a review of the various stages of dewetting for a film of liquid spreading on a hydrophobic substrate. Different types of behaviour of the film are observed depending on the form of the disjoining pressure: finite-time singularities, self-similar solutions and coarsening. In particular, he divides the evolution of dewetting processes into three phases: an initial linear instability that leads to finite-time rupture (short time dynamics), which is followed by the propagation of dewetting rims and instabilities of liquid ridges (intermediate time dynamics), and the eventual formation of quasi-steady droplets (long time dynamics). Most of the previous studies of thin liquid films focussed on films on homogeneous substrates. However, thin liquid films on non-homogeneous chemically patterned substrates are also of interest. These have many practical applications, such as in the construction of microfluidic devices and creating soft materials with a particular pattern [30]. Chemically patterned substrates are an efficient way to obtain microstructures of different shapes by using different types of substrate patterning [34]. Chemical modification of substrates can also be used to avoid spontaneous breakup of thin films, which is often highly undesirable, as, for example, in printing technology [5, 1]. Due to their many applications briefly described above, films on non- homogeneous substrates have been the object of a number of previous theoretical studies which motivate the present work. For example, Konnur et al. [24] found that in the case of an isolated circular patch with wetting properties different from that of the rest of the substrate that chemical non- homogeneity of the substrate can greatly accelerate the growth of surface instabilities. Sharma et al. [35] studied instabilities of a liquid film on a substrate containing a single heterogeneous patch and a substrate with stripes of alternating less and more wettable regions. The main concern of that paper was to investigate how substrate patterns are reproduced in the liquid film, and to determine the best conditions for templating. Thiele et al. [39] performed a bifurcation analysis using the continuation software package AUTO [11] to study dewetting on a chemically patterned substrate by solving a thin-film equation with a disjoining pressure, using the wettability contrast as a control parameter. The wettability contrast measures the degree of heterogeneity of the substrate; it is introduced and defined rigorously in (45) in Section 5. Honisch et al. [17] modelled an experiment in which organic molecules were deposited on a chemically non- homogeneous silicon oxide substrates with gold stripes and discuss the redistribution of the liquid following deposition. In a recent paper, Liu and Witelski [27] studied thin films on chemically heterogeneous substrates. They claim that in some applications such as digital microfluidics, substrates with alternate hydrophilic and hydrophobic rectangular areas are better described by a piecewise constant function than by a sinusoidal one. Therefore, in contrast to other studies, including the present one, they study substrates with wettability characteristics described by such a function. Based on the structure of the bifurcation diagram, they divide the steady-state solutions into six distinct but connected branches and show that the only unstable branch corresponds to confined droplets, while the rest of the branches are stable. In the present work, we build on the work of Thiele et al. [39] and Honisch et al. [17]. Part of our motivation is to clarify and explain rigorously some of the numerical results reported in these papers. In the sinusoidally striped non-homogeneous substrate case, we offer a justification for using a form of the disjoining pressure that differs from the one used in these two papers. A detailed plan of the paper is given in the last paragraph of Section 2. ## 2 Problem Statement Denoting the thickness of the thin liquid film by $z=h(x,y,t)$, where $(x,y,z)$ are the usual Cartesian coordinates and $t$ is time, Honisch et al. [17] consider the thin-film equation $h_{t}=\nabla\cdot\left\\{Q(h)\nabla P(h,x,y)\right\\},\qquad t>0,\qquad(x,y)\in\mathbb{R}^{2},$ (1) where $Q(h)=h^{3}/(3\eta)$ is the mobility coefficient with $\eta$ being the dynamic viscosity. The generalized pressure $P(h,x,y)$ is given by $P(h,x,y)=-\gamma\Delta h-\Pi(h,x,y),$ (2) where $\gamma$ is the coefficient of surface tension and we follow [17] in taking the Derjaguin disjoining pressure $\Pi(h,x,y)$ in the spatially homogeneous case to be of the form $\Pi(h,x,y)=-\frac{A}{h^{3}}+\frac{B}{h^{6}}$ (3) suggested, for example, by Pismen [29]. Here $A$ and $B$ are positive parameters that measure the relative contributions of the long-range forces (the $1/h^{3}$ term) and the short-range ones (the $1/h^{6}$ term). However, we will see that both of these constants can be scaled out of the mathematical problem. Equation (3) uses the exponent $-3$ for the long-range forces and $-6$ for the short-range forces as in Honisch et al. [17]. Other choices include the pairs of long- and short-range exponents $(-2,-3)$, $(-3,-4)$ and $(-3,-9)$ discussed by [4, 33, 36]. In terms of the classification of Bertozzi et al. [4, Definition 1], the choice $(-3,-6)$ is admissible (as are all the other pairs above), and falls in their region II; we expect that choosing other admissible exponent pairs will give qualitatively the same results as those obtained here. In what follows, we study thin films on both homogeneous and non-homogeneous substrates. In the non-homogeneous case, we will modify (3) by assuming that the Derjaguin pressure term $\Pi$ changes periodically in the $x$-direction with period $L$. The appropriate forms of $\Pi$ in the non-homogeneous case are discussed in Section 5. Hence, in order better to understand solutions of (1), we study its one- dimensional version, $h_{t}=(Q(h)P(h,x)_{x})_{x},\quad\quad 0<x<L.$ (4) We start by characterising steady state solutions of (4) subject to periodic boundary conditions at $x=0$ and $x=L$. In other words, we seek steady state solutions $h(x)$ of (4), satisfying the non-local boundary value problem $\gamma h_{xx}+\frac{B}{h^{6}}-\frac{A}{h^{3}}-\frac{1}{L}\int_{0}^{L}\left[\frac{B}{h^{6}}-\frac{A}{h^{3}}\right]\,\hbox{d}{x}=0,\;\;0<x<L,$ (5) subject to the constraint $\frac{1}{L}\int_{0}^{L}h(x)\,\hbox{d}{x}=h^{},$ (6) where the constant $h^{}\,(>0)$ denotes the (scaled) average film thickness, and the periodic boundary conditions $h(0)=h(L),\quad h_{x}(0)=h_{x}(L).$ (7) Now we non-dimensionalise. Setting $H=\left(\frac{B}{A}\right)^{1/3},\;\;h=H\tilde{h},\quad\hbox{and}\quad x=L\tilde{x},$ (8) in (5) and removing the tildes, we obtain $\epsilon^{2}h_{xx}+f(h)-\int_{0}^{1}f(h)\,\hbox{d}{x}=0,\;\;0<x<1,$ (9) where $f(h)=\frac{1}{h^{6}}-\frac{1}{h^{3}},$ (10) and $\epsilon^{2}=\frac{\gamma B^{4/3}}{L^{2}A^{7/3}},$ (11) subject to the periodic boundary conditions $h(0)=h(1),\quad h_{x}(0)=h_{x}(1),$ (12) and the volume constraint $\int_{0}^{1}h(x)\,\hbox{d}{x}=\bar{h},$ (13) where $\bar{h}=\frac{h^{}A^{1/3}}{B^{1/3}}.$ (14) Note that the problem (9)–(14) is very similar to the corresponding steady state problem for the Cahn–Hilliard equation considered as a bifurcation problem in the parameters $\bar{h}$ and $\epsilon$ by Eilbeck et al. [12]. The boundary conditions considered in that work were the physically natural double Neumann conditions. The periodic boundary conditions (12) in the present problem slightly change the analysis, but our general approach in characterising different bifurcation regimes still follows that of Eilbeck et al. [12], though the correct interpretation of the limit as $\epsilon\to 0^{+}$ is that now we let the surface tension $\gamma$ go to zero. In particular, we perform a Liapunov–Schmidt reduction to determine the local behaviour close to bifurcation points and then use AUTO (in the present work we use the AUTO-07p version [11]) to explore the global structure of branches of steady state solutions both for the spatially homogeneous case and for the spatially non-homogeneous case in the case of an $x$-periodically patterned substrate. We first investigate the homogeneous case and, having elucidated the structure of the bifurcations of non-trivial solutions from the constant solution $h=\bar{h}$ in that case in Sections 3 and 4, we study forced rotational ($O(2)$) symmetry breaking in the non-homogeneous case in Section 5. In Appendix A, we present a general result about $O(2)$ symmetry breaking in the spatially non-homogeneous case. It shows that in the spatially non-homogeneous case, only two steady state solutions remain from the orbit of solutions of (9)–(14) induced by its $O(2)$ invariance. We concentrate on the simplest steady state solutions of (9)–(14), as by a result of Laugesen and Pugh [25, Theorem 1] only such solutions, that is, constant solutions and those having only one extremum point, are linearly stable in the homogeneous case. For information about dynamics of one-dimensional thin-film equations the reader should also consult Zhang [46]. In what follows, we use $\\|\cdot\\|_{2}$ to denote $L^{2}([0,1])$ norms. ## 3 Liapunov–Schmidt Reduction in the Spatially Homogeneous Case We start by performing an analysis of the dependence of the global structure of branches of steady state solutions of the problem in the spatially homogeneous case given by (9)–(14) on the parameters $\bar{h}$ and $\epsilon$. In what follows, we do not indicate explicitly the dependence of the operators on the parameters $\bar{h}$ and $\epsilon$, and all of the calculations are performed for a fixed value of $\bar{h}$ and close to a bifurcation point $\epsilon=\epsilon_{k}$ for $k=1,2,3,\ldots$ defined below. We set $v=h-\bar{h}$, so that $v=v(x)$ has zero mean, and rewrite (9) as $G(v)=0,$ (15) where $G(v)=\epsilon^{2}v_{xx}+f(v+\bar{h})-\int_{0}^{1}f(v(x)+\bar{h})\,\hbox{d}{x}.$ (16) If we set $H=\left\\{w\in C(0,1)\,:\,\int_{0}^{1}w(x)\,\hbox{d}{x}=0\right\\},$ (17) where $G$ is an operator from $D(G)\subset H\rightarrow H$, then $D(G)$ is given by $D(G)=\left\\{v\in C^{2}(0,1)\,:\,v(0)=v(1),\,v_{x}(0)=v_{x}(1),\,\int_{0}^{1}v(x)\,\hbox{d}{x}=0\right\\}.$ (18) The linearisation of $G$ at $v$ applied to $w$ is defined by $dG(v)w=\lim_{\tau\to 0}\frac{G(v+\tau w)-G(v)}{\tau}.$ (19) We denote $dG(0)$ by $S$, so that $S$ applied to $w$ is given by $Sw=\epsilon^{2}w_{xx}+f^{\prime}(\bar{h})w.$ (20) To locate the bifurcation points, we have to find the nontrivial solutions of the equation $Sw=0$ subject to $w(0)=w(1),\quad\quad w_{x}(0)=w_{x}(1).$ (21) The kernel of $S$ is non-empty and two dimensional when $\epsilon=\epsilon_{k}=\frac{\sqrt{f^{\prime}(\bar{h})}}{2k\pi}\quad\textrm{for}\quad k=1,2,3,\ldots,$ (22) and is spanned by $\cos(2k\pi x)$ and $\sin(2k\pi x)$. That these values of $\epsilon$ correspond to bifurcation points follows from two theorems of Vanderbauwhede [40, Theorems 2 and 3]. In a neighbourhood of a bifurcation point $(\epsilon_{k},0)$ in $(\epsilon,v)$ space, solutions of $G(v)=0$ on $H$ are in one-to-one correspondence with solutions of the reduced system of equations on $\mathbb{R}^{2}$, $g_{1}(x,y,\epsilon)=0,\quad g_{2}(x,y,\epsilon)=0,$ (23) for some functions $g_{1}$ and $g_{2}$ to be obtained through the Liapunov–Schmidt reduction [16]. To set up the Liapunov–Schmidt reduction, we decompose $D(G)$ and $H$ as follows: $D(G)=\hbox{ker}\,S\oplus M$ (24) and $H=N\oplus\hbox{range}\,S.$ (25) Since $S$ is self-adjoint with respect to the $L^{2}$-inner product denoted by $\langle\cdot,\cdot\rangle$, we can choose $M=N=\hbox{span}\,\left\\{\cos(2kx),\sin(2kx)\right\\},$ (26) and denote the above basis for $M$ by $\left\\{w_{1},w_{2}\right\\}$ and for $N$ by $\left\\{w_{1}^{},w_{2}^{}\right\\}$. We also denote the projection of $H$ onto $\hbox{range}\,S$ by $E$. Since the present problem is invariant with respect to the group $O(2)$, the functions $g_{1}$ and $g_{2}$ must have the form $g_{1}(x,y,\epsilon)=xp(x^{2}+y^{2},\epsilon),\quad g_{2}(x,y,\epsilon)=yp(x^{2}+y^{2},\epsilon),$ (27) for some function $p(\cdot,\cdot)$ [9], which means that in order to determine the bifurcation structure, the only terms that need to be computed are $g_{1,x\epsilon}$ and $g_{1,xxx}$, as these immediately give $g_{2,y\epsilon}$ and $g_{2,yyy}$ and all of the other second and third partial derivatives of $g_{1}$ and $g_{2}$ are identically zero. Following Golubitsky and Schaeffer [16], we have $\displaystyle g_{1,x\epsilon}$ $\displaystyle=$ $\displaystyle\langle w_{1}^{},dG_{\epsilon}(w_{1})-d^{2}G(w_{1},S^{-1}EG_{\epsilon}(0))\rangle,$ (28) $\displaystyle g_{1,xxx}$ $\displaystyle=$ $\displaystyle\langle w_{1}^{},d^{3}G(w_{1},w_{1},w_{1})-3d^{2}G(w_{1},S^{-1}E[d^{2}G(w_{1},w_{1})])\rangle,$ (29) where $d^{r}G(z_{1},\ldots,z_{r})=\left.\frac{\partial^{r}}{\partial t_{1}\ldots\partial t_{r}}G(t_{1}z_{1}+\ldots+t_{r}z_{r})\right\|_{t_{1}=\ldots=t_{r}=0}\quad\textrm{for}\quad r=1,2,3,\ldots,$ (30) and we choose $w_{1}=w_{1}^{}=\cos(2k\pi x),$ (31) where $w_{1}\in\hbox{ker}\ S$ and $w_{1}^{}\in(\hbox{range}\ S)^{\perp}$. In particular, from (30) we have $\displaystyle d^{2}G(z_{1},z_{2})$ $\displaystyle=$ $\displaystyle\left.\frac{\partial^{2}}{\partial t_{1}\partial t_{2}}G(t_{1}z_{1}+t_{2}z_{2})\right\|_{t_{1}=t_{2}=0}$ (32) $\displaystyle=$ $\displaystyle\frac{\partial^{2}}{\partial t_{1}\partial t_{2}}\Big{[}\epsilon_{k}(t_{1}z_{1,xx}+t_{2}z_{2,xx})+f(t_{1}z_{1}+t_{2}z_{2}+\bar{h})$ $\displaystyle-$ $\displaystyle\left.\int_{0}^{1}f(t_{1}z_{1}+t_{2}z_{2}+\bar{h})\,\hbox{d}{x}\Big{]}\right\|_{t_{1}=t_{2}=0}$ $\displaystyle=$ $\displaystyle f^{\prime\prime}(\bar{h})z_{1}z_{2}-\int_{0}^{1}f^{\prime\prime}(\bar{h})z_{1}z_{2}\,\hbox{d}{x},$ and so $\displaystyle d^{2}G(\cos(2k\pi x),\cos(2k\pi x))$ $\displaystyle=$ $\displaystyle f^{\prime\prime}(\bar{h})\cos^{2}(2k\pi x)-\int_{0}^{1}f^{\prime\prime}(\bar{h})\cos^{2}(2k\pi x)\,\hbox{d}{x}$ (33) $\displaystyle=$ $\displaystyle f^{\prime\prime}(\bar{h})\cos^{2}(2k\pi x)-\frac{1}{2}f^{\prime\prime}(\bar{h}).$ To obtain $S^{-1}E[d^{2}G(w_{1},w_{1})]$, which we denote by $R(x)$, so that $SR=E[d^{2}G(w_{1},w_{1})]$, we use the definition of $\epsilon_{k}$ given in (22) and solve the second order ordinary differential equation satisfied by $R(x)$, $R_{xx}+4k^{2}\pi^{2}R=4k^{2}\pi^{2}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})}\cos^{2}(2k\pi x)-2k^{2}\pi^{2}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})},$ (34) subject to $R(0)=R(1)\quad\textrm{and}\quad R_{x}(0)=R_{x}(1)$ (35) in $\hbox{ker}\,S$. We obtain $R(x)=S^{-1}E[d^{2}G(w_{1},w_{1})]=-\frac{1}{6}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})}\cos(4k\pi x),$ (36) and hence $\displaystyle d^{2}G(w_{1},S^{-1}E[d^{2}G(w_{1},w_{1})])$ $\displaystyle=$ $\displaystyle d^{2}G\left(\cos(2k\pi x),-\frac{1}{6}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})}\cos(4k\pi x)\right)$ (37) $\displaystyle=$ $\displaystyle f^{\prime\prime}(\bar{h})\cos(2k\pi x)\left(-\frac{1}{6}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})}\cos(4k\pi x)\right)$ $\displaystyle-\int_{0}^{1}f^{\prime\prime}(\bar{h})\cos(2k\pi x)\left(-\frac{1}{6}\frac{f^{\prime\prime}(\bar{h})}{f^{\prime}(\bar{h})}\cos(4k\pi x)\right)\hbox{d}{x}$ $\displaystyle=$ $\displaystyle-\frac{1}{6}\frac{[f^{\prime\prime}(\bar{h})]^{2}}{f^{\prime}(\bar{h})}\cos(2k\pi x)\cos(4k\pi x).$ In addition, from (30) we have $\displaystyle d^{3}G(z_{1},z_{2},z_{3})$ $\displaystyle=$ $\displaystyle\left.\frac{\partial^{3}}{\partial t_{1}\partial t_{2}\partial t_{3}}G(t_{1}z_{1}+t_{2}z_{2}+t_{3}z_{3})\right\|_{t_{1}=t_{2}=t_{3}=0}$ (38) $\displaystyle=$ $\displaystyle f^{\prime\prime\prime}(\bar{h})z_{1}z_{2}z_{3}-\int_{0}^{1}f^{\prime\prime\prime}(\bar{h})z_{1}z_{2}z_{3}\,\hbox{d}{x},$ and therefore $\displaystyle d^{3}G(\cos(2k\pi x),\cos(2k\pi x),\cos(2k\pi x))$ $\displaystyle=$ $\displaystyle f^{\prime\prime\prime}(\bar{h})\cos^{3}(2k\pi x)-\int_{0}^{1}f^{\prime\prime\prime}(\bar{h})\cos^{3}(2k\pi x)\,\hbox{d}{x}$ (39) $\displaystyle=$ $\displaystyle f^{\prime\prime\prime}(\bar{h})\cos^{3}(2k\pi x).$ Putting all of the information in (37) and (39) into (29) we obtain $\displaystyle g_{1,xxx}$ $\displaystyle=$ $\displaystyle\langle w_{1}^{},d^{3}G(w_{1},w_{1},w_{1})-3d^{2}G(w_{1},S^{-1}E[d^{2}G(w_{1},w_{1})])\rangle$ (40) $\displaystyle=$ $\displaystyle\int_{0}^{1}\cos(2k\pi x)\left[f^{\prime\prime\prime}(\bar{h})\cos^{3}(2k\pi x)-3\left(-\frac{1}{6}\frac{[f^{\prime\prime}(\bar{h})]^{2}}{f^{\prime}(\bar{h})}\cos(2k\pi x)\cos(4k\pi x)\right)\right]\,\hbox{d}{x}$ $\displaystyle=$ $\displaystyle\frac{3}{8}f^{\prime\prime\prime}(\bar{h})+\frac{1}{8}\frac{[f^{\prime\prime}(\bar{h})]^{2}}{f^{\prime}(\bar{h})}.$ In addition, $G_{\epsilon}(v)=v_{xx}$, so that $G_{\epsilon}(0)=0$ at $v=0$, and hence we have $d^{2}G(w_{k},S^{-1}EG_{\epsilon}(0))=0.$ (41) Furthermore, since $dG_{\epsilon}(w)=w_{xx}$, from (28) we obtain $\displaystyle g_{1,x\epsilon}$ $\displaystyle=$ $\displaystyle\langle w_{1}^{},dG_{\epsilon}(w_{1})-d^{2}G(w_{1},S^{-1}EG_{\epsilon}(0))\rangle$ (42) $\displaystyle=$ $\displaystyle\int_{0}^{1}\cos(2k\pi x)\left(-4\pi^{2}k^{2}\cos(2k\pi x)\right)\,\hbox{d}{x}$ $\displaystyle=$ $\displaystyle-2k^{2}\pi^{2}.$ Referring to (27) and the argument following that equation, the above analysis shows that as long as $f^{\prime}(\bar{h})>0$ at $\epsilon=\epsilon_{k}$ a circle of equilibria bifurcates from the constant solution $h\equiv\bar{h}$. The direction of bifurcation is locally determined by the sign of $g_{1,xxx}$ given by (40). Hence, using $1/\epsilon$ as the bifurcation parameter, the bifurcation of nontrivial equilibria is supercritical if $g_{1,xxx}$ is negative and subcritical if it is positive. By finding the values of $\bar{h}$ where $g_{1,xxx}$ given by (40) with $f(h)$ given by (10) is zero, we finally obtain the following proposition: ###### Proposition 1 Bifurcations of nontrivial solutions from the constant solution $h=\bar{h}$ of the problem $(\ref{nl})$–$(\ref{mc})$ are supercritical if $1.289<\bar{h}<1.747$ and subcritical if $1.259<\bar{h}<1.289$ or if $\bar{h}>1.747$. definition ###### Proof 3.1 The constant solution $h\equiv\bar{h}$ will lose stability as $\epsilon$ is decreased only if $f^{\prime}(\bar{h})>0$. i.e. if $-6/{\bar{h}}^{7}+3/{\bar{h}}^{4}>0$, for $\bar{h}>2^{1/3}\approx 1.259$. From (40) we have that $g_{1,xxx}=\frac{57\bar{h}^{6}-426\bar{h}^{3}+651}{2\bar{h}^{9}(\bar{h}^{3}-2)},$ (43) so that $g_{1,xxx}<0$ if $\bar{h}\in(1.289,1747)$ giving the result. For $\bar{h}\leqslant 2^{1/3}$ there are no bifurcations from the constant solution $h=\bar{h}$. Furthermore, we have the following proposition: ###### Proposition 1 The problem $(\ref{nl})$–$(\ref{mc})$ has no nontrivial solutions when $\bar{h}\leq 1$. ###### Proof 3.2 Assume that such a nontrivial solution exists. Then, since $\bar{h}\leq 1$, its global minimum, achieved at some point $x_{0}\in(0,1)$, must be less than 1. (We may take the point $x_{0}$ to be an interior point by translation invariance.) But then $\epsilon^{2}h_{xx}(x_{0})=\int_{0}^{1}f(h)\,\hbox{d}{x}-f(h(x_{0}))<0,$ (44) so the point $x_{0}$ cannot be a minimum. ## 4 Two-Parameter Continuation of Solutions in the Spatially Homogeneous Case To describe the change in the global structure of branches of steady state solutions of the problem (9)–(14) as $\bar{h}$ and $\epsilon$ are varied, we use AUTO [11] and our results are summarised in Figure 1. As Figure 1 shows, a curve of saddle-node (SN) bifurcations which originates from $\bar{h}\approx 1.289$ at $1/\epsilon\approx 23.432$ satisfies $\bar{h}\rightarrow 1^{+}$ as $1/\epsilon\rightarrow\infty$, while a curve of SN bifurcations which originates from $\bar{h}\approx 1.747$, $1/\epsilon\approx 13.998$ satisfies $\bar{h}\rightarrow\infty$ as $1/\epsilon\rightarrow\infty$. Figure 1: The global structure of branches of steady state solutions with a unique maximum, including both saddle-node (SN) (shown with dash-dotted curves) and pitchfork (PF) bifurcation branches (shown with solid curves). The nucleation regime $1<\bar{h}<2^{1/3}\approx 1.259$ (Regime I), the metastable regime $2^{1/3}<\bar{h}<1.289$ and $\bar{h}>1.747$ (Regime II), and the unstable regime $1.289<\bar{h}<1.747$ (Regime III) are also indicated. Figure 1 identifies three different bifurcation regimes, denoted by I, II and III, with differing bifurcation behaviour occurring in the different regimes, namely (using the terminology of [12] in the context of the Cahn-Hilliard equation): • a “nucleation” regime for $1<\bar{h}<2^{1/3}\approx 1.259$ (Regime I), * • a “metastable” regime for $2^{1/3}<\bar{h}<1.289$ and $\bar{h}>1.747$ (Regime II), and * • an “unstable” regime for $1.289<\bar{h}<1.747$ (Regime III). In Regime I, the constant solution $h(x)\equiv\bar{h}$ is linearly stable which follows from analysing the spectrum of the operator $S$ for $f^{\prime}(\bar{h})<0$ in (20) and (21), but under sufficiently large perturbations the system will evolve to a non-constant steady state solution. See [25] for an extensive discussion of the stability analysis of steady state solutions to thin-film equations. In Regime II, as $\epsilon$ is decreased the constant solution $h(x)\equiv\bar{h}$ loses stability through a subcritical bifurcation. In Regime III, as $\epsilon$ is decreased, the constant solution $h(x)\equiv\bar{h}$ loses stability through a supercritical bifurcation. ## 5 The Spatially Non-Homogeneous Case Honisch et al. [17] chose the Derjaguin disjoining pressure $\Pi(h,x,y)$ to be of the form $\Pi(h,x,y)=\left(\frac{1}{h^{6}}-\frac{1}{h^{3}}\right)(1+\rho\,G(x,y)),$ (45) where the function $G(x,y)$ models the non-homogeneity of the substrate and the parameter $\rho$, which can be either positive or negative, is called the “wettability contrast”. Following Honisch et al. [17], in the remainder of the present work we consider the specific case $G(x,y)=\sin\left(2\pi x\right):=G(x),$ (46) corresponding to an $x$-periodically patterned (i.e. striped) substrate. There are, however, some difficulties in accepting (45) as a physically realistic form of the disjoining pressure for a non-homogeneous substrate. The problems arise because the two terms in (45) represent rather different physical effects. Specifically, since the $1/h^{6}$ term models the short- range interaction amongst the molecules of the liquid and the $1/h^{3}$ term models the long-range interaction, assuming that both terms reflect the patterning in the substrate in exactly the same way through their dependence on the same function $G(x,y)$ does not seem very plausible. Moreover, there are other studies which assume that the wettability of the substrate is incorporated in either the short-range interaction term (see, for example, Thiele and Knobloch [38] and Bertozzi et al. [4]) or the long-range interaction term (see, for example, Ajaev et al. [2] and Sharma et al. [35]), but not both simultaneously. Hence in what follows we will consider the two cases $\Pi(h,x)=\Pi_{\rm LR}(h,x)$ and $\Pi(h,x)=\Pi_{\rm SR}(h,x)$, where LR stands for “long range” and SR stands for “short range” where $\Pi_{\rm LR}(h,x)=\frac{1}{h^{6}}-\frac{1}{h^{3}}(1+\rho G(x))$ (47) and $\Pi_{\rm SR}(h,x)=\frac{1}{h^{6}}(1+\rho G(x))-\frac{1}{h^{3}},$ (48) in both of which $G(x)$ is given by (46) and $\rho$ is the wettability contrast. For small wettability contrast, $\|\rho\|\ll 1$, we do not expect there to be significant differences between the influence of $\Pi_{\rm LR}$ or $\Pi_{\rm SR}$ on the bifurcation diagrams, as these results depend only on the nature of the bifurcation in the homogeneous case $\rho=0$ and on the symmetry groups under which the equations are invariant. To see this, consider the spatially non-homogeneous version of (9), that is, the boundary value problem $\epsilon^{2}h_{xx}+f(h,x)-\int_{0}^{1}f(h,x)\,\hbox{d}{x}=0,\;\;0<x<1,$ (49) where now $f(h,x)=\Pi_{\rm LR}(h,x)\quad\hbox{or}\quad f(h,x)=\Pi_{\rm SR}(h,x).$ (50) subject to the periodic boundary conditions and the volume constraint, $h(0)=h(1),\quad h_{x}(0)=h_{x}(1),\quad\hbox{and}\quad\int_{0}^{1}h(x)\,\hbox{d}{x}=\bar{h}.$ (51) Seeking an asymptotic solution to (49)–(51) in the form $h(x)=\bar{h}+\rho h_{1}(x)+O(\rho^{2})$ in the limit $\rho\to 0$, by substituting this anzatz into (49) we find that in the case of $\Pi_{\rm LR}(h,x)$ we have $h_{1}(x)=\frac{S\bar{h}^{4}\sin\left(2\pi x\right)}{4\pi^{2}\bar{h}^{7}\epsilon^{2}-3\bar{h}^{3}+6},$ (52) while in the case of $\Pi_{\rm SR}(h,x)$ the corresponding result is $h_{1}(x)=\frac{\bar{h}\sin\left(2\pi x\right)}{4\pi^{2}\bar{h}^{7}\epsilon^{2}-3\bar{h}^{3}+6}.$ (53) For non-zero values of $\rho$, in both the $\Pi_{\rm LR}$ and $\Pi_{\rm SR}$ cases, the changes in the bifurcation diagrams obtained in the homogeneous case ($\rho=0$) are an example of forced symmetry breaking (see, for example, Chillingworth [8]), which we discuss further in Appendix A. More precisely, we show there that when $\rho\neq 0$, out of the entire $O(2)$ orbit only two equilibria are left after symmetry breaking. Figure 2: Bifurcation diagrams of solutions with a unique maximum, showing $\\|h-\bar{h}\\|_{2}$ plotted as a function of $1/\epsilon$ when the disjoining pressure is $\Pi_{\rm LR}$ for $\rho=0$ (dashed curves), $\rho=0.005$ (dotted curves) and $\rho=0.05$ (solid curves) for (a) $\bar{h}=1.24$, (b) $\bar{h}=1.3$, and (c) $\bar{h}=2$, corresponding to Regimes I, III, and II, respectively. Figure 2 shows how for small wettability contrast $\|\rho\|\ll 1$, the resulting spatial non-homogeneity introduces imperfections [16] in the bifurcation diagrams of the homogeneous case $\rho=0$ discussed in Section 4. It presents bifurcation diagrams in Regimes I, II and III when the disjoining pressure $\Pi_{\rm LR}$ is given by (47) for a range of small values of $\rho$ together with the corresponding diagrams in the case $\rho=0$. The corresponding bifurcation diagrams when the disjoining pressure $\Pi_{\rm SR}$ is given by (48) are very similar and hence are not shown here. For large wettability contrast, specifically for $\|\rho\|\geq 1$, significant differences between the two forms of the disjoining pressure are to be expected. When using $\Pi_{\rm LR}$, one expects global existence of positive solutions for all values of $\|\rho\|$; see, for example, Wu and Zheng [44]. On the other hand, when using $\Pi_{\rm SR}$, there is the possibility of rupture of the liquid film for $\|\rho\|\geq 1$; see, for example, [4, 44], which means in this case we do not expect positive solutions for sufficiently large values of $\|\rho\|$. Figure 3: Bifurcation diagram for steady state solutions with a unique maximum showing $\\|h-\bar{h}\\|_{2}$ plotted as a function of $\rho$ when the disjoining pressure is $\Pi_{\rm LR}$, $\bar{h}=3$ and $1/\epsilon=50$. The leading-order dependence of $\\|h-\bar{h}\\|_{2}$ on $\rho$ as $\rho\to 0$, given by (52), is shown with dashed lines. In Figure 3 we plot the branches of the positive solutions of (49)–(51) with a unique maximum when the disjoining pressure is $\Pi_{\rm LR}$ for $\bar{h}=3$ and $1/\epsilon=50$, so that when $\rho=0$, we are in Regime II above the curve of pitchfork (PF) bifurcations from the constant solution shown in Figure 1. The work of Bertozzi et al. [4] and of Wu and Zheng [44], shows that strictly positive solutions exist for all values of $\|\rho\|$, i.e. beyond the range $\rho\in[-2,2]$ for which we have performed the numerical calculations. Figure 4 shows that the situation is different when the disjoining pressure is $\Pi_{\rm SR}$ (with the same values of $\bar{h}$ and $\epsilon$). At $\|\rho\|=1$, the upper branches of solutions disappear due to rupture of the film, so that at some point $x_{0}\in[0,1]$ we have $h(x_{0})=0$ and a strictly positive solution no longer exists, while the other two branches coalesce at a saddle node bifurcation at $\|\rho\|\approx 5.8$. Note that in Figures 3 and 4, the non-trivial “solution” on the axis $\rho=0$ is, in fact, a whole $O(2)$-symmetric orbit of solutions predicted by the analysis leading to Figure 1. Figure 4: Bifurcation diagram showing $\\|h-\bar{h}\\|_{2}$ plotted as a function of $\rho$ when the disjoining pressure is $\Pi_{\rm SR}$, for $\bar{h}=3$ and $1/\epsilon=50$. The leading order dependence of $\\|h-\bar{h}\\|_{2}$ on $\rho$ as $\rho\to 0$, given by (53), is shown with dashed lines. Note that the upper branches of solutions cannot be extended beyond $\|\rho\|=1$ (indicated by filled circles). Note that when the disjoining pressure is $\Pi_{\rm SR}$, given by (48), we were unable to use AUTO to continue branches of solutions beyond the rupture of the film at $\|\rho\|=1$. Figure 5: Solutions $h(x)$ when the disjoining pressure is $\Pi_{\rm SR}$ for $\bar{h}=2$ and $1/\epsilon=30$ for $\rho=0$, 0.97, 0.98, 0.99 and 1, denoted by “1”, “2”, “3”, “4” and “5”, respectively, showing convergence of strictly positive solutions to a non-strictly positive one as $\rho\to 1^{-}$. Figure 5 shows the film approaching rupture as $\rho\to 1^{-}$ at the point where the coefficient of the short-range term disappears when $\rho=1$, i.e. when $1+\sin(2\pi x)=0$ and hence at $x=3/4$. These numerical results are consistent with the theoretical arguments of Bertozzi et al. [4]. Investigation of the possible leading-order balance in (9) when the disjoining pressure is $\Pi_{\rm SR}$ and $\rho=1$ in the limit $x\to 3/4$ reveals that $h(x)=O\left((x-3/4)^{2/3}\right)$; continuing the analysis to higher order yields $h=(2\pi^{2})^{\frac{1}{3}}\left(x-\frac{3}{4}\right)^{\frac{2}{3}}-\frac{4\left(4\pi^{10}\right)^{\frac{1}{3}}\epsilon^{2}}{27}\left(x-\frac{3}{4}\right)^{\frac{4}{3}}+O\left(\left(x-\frac{3}{4}\right)^{2}\right).$ (54) Figure 6 shows the detail of the excellent agreement between the solution $h(x)$ when $\rho=1$ and the two-term asymptotic solution given by (54) close to $x=3/4$. Figure 6: Detail near $x=3/4$ of the solution $h(x)$ shown with solid line when the disjoining pressure is $\Pi_{\rm SR}$ and $\rho=1$, and the two-term asymptotic solution given by (54) shown with dashed lines for $\bar{h}=2$ and $\epsilon=1/30$. Figures 7 and 8 show the multiplicity of solutions with a unique maximum for the disjoining pressures $\Pi_{\rm LR}$ and $\Pi_{\rm SR}$, respectively, in the $(1/\epsilon,\rho)$-plane in the three regimes shown in Figure 1. Figure 7: Multiplicity of strictly positive solutions with a unique maximum in the $(1/\epsilon,\rho)$-plane when the disjoining pressure is $\Pi_{\rm LR}$ for (a) $\bar{h}=1.24$ (Regime I), (b) $\bar{h}=1.3$ (Regime III), and (c) $\bar{h}=2$ (Regime II). Figure 8: Multiplicity of strictly positive solutions with a unique maximum in the $(1/\epsilon,\rho)$-plane when the disjoining pressure is $\Pi_{\rm SR}$ for (a) $\bar{h}=1.24$ (Regime I), (b) $\bar{h}=1.3$ (Regime III), and (c) $\bar{h}=2$ (Regime II). In Figure 8 the horizontal dashed line at $\rho=1$ indicates rupture of the film and loss of a smooth strictly positive solution, showing that there are regions in parameter space where no such solutions exist. Let us explain in detail the interpretation of Figure 7(c); all of the other parts of Figure 7 and Figure 8 are interpreted in a similar way. Figure 9: Bifurcation diagram of steady state solutions with $\bar{h}=2$ (Regime II) for $\rho=0$ (dashed curves) and $\rho=0.005$ (solid curves) indicating the different branches of steady state solutions. Each curve in Figure 7(c) is a curve of saddle-node bifurcations. Similar to Figure 2(c), Figure 9 shows the bifurcation diagram of steady state solutions with $\bar{h}=2$ (Regime II) for $\rho=0$ and $\rho=0.005$. As $1/\epsilon$ is increased, we pass successively through saddle-node bifurcations with the multiplicity of the steady state solutions changing from 1 to 3 to 5 and then back to 3 again. In Figure 10, we plot the five steady state solutions with a unique maximum indicated in Figure 9 by (i)–(v). Figure 10: Steady state solutions on the five branches of solutions indicated in Figure 9 by (i)–(v). ## 6 Conclusions In the present work we have investigated the steady state solutions of the thin-film evolution equation (1) both in the spatially homogeneous case (9)–(12) and in the spatially non-homogeneous case for two choices of spatially non-homogeneous Derjaguin disjoining pressure given by (47) and (48). We have given a physical motivation for our choices of the disjoining pressure. For reasons explained in the last paragraph of Section 2, we concentrated on branches of solutions with a unique maximum. In the spatially homogeneous case (9)–(14), we used the Liapunov-Schmidt reduction of an equation invariant under the action of the $O(2)$ symmetry group to obtain local bifurcation results and to determine the dependence of the direction and nature of bifurcation in bifurcation parameter $1/\epsilon$ on the average film thickness $\bar{h}$; our results on the existence of three different bifurcation regimes, (namely nucleation, metastable, and unstable) are summarised in Propositions 1 and 1 and in Figure 1 obtained using AUTO. In the spatially non-homogeneous case (49)–(51), we clarified the $O(2)$ symmetry breaking phenomenon (see Appendix A) and presented imperfect bifurcation diagrams in Figure 2 and global bifurcation diagrams using the wettability contrast $\rho$ as a bifurcation parameter for fixed $\epsilon$ and $\bar{h}$ in Figures 3 and 4. Let us discuss Figure 3 in more detail; for convenience, it is reproduced in Figure 11 with labelling added to the different branches of strictly positive steady state solutions with a unique maximum. Below we explain what these different labels mean. Figure 11: A sketch of the bifurcation diagram plotted in Figure 3 with the different solution branches labelled. Our results can be described using the language of global compact attractors. For more information on attractors in dissipative infinite-dimensional dynamical systems please see the monograph of Hale [14] and Wu and Zheng [44] for applications in the thin-film context. In systems such as (4), the global compact attractor of the PDE is the union of equilibria and their unstable manifolds. In Figures 12 and 13 we sketch the global attractor of (4) for $\epsilon=1/50$ and $\bar{h}=3$, the values used to plot Figure 3. For these values of the parameters the attractor is two-dimensional and we sketch its projection onto a plane. Figure 12: Sketch of the global attractor for $\rho=0$. The circle represents the $O(2)$ orbit of steady state solutions and $O$ represents the constant solution $h(x)=\bar{h}$. Figure 13: Sketch of the global attractor for small non-zero values of $\|\rho\|$. The points $A$, $B$, $C$ correspond to the steady state solutions labelled in Figure 11. When $\rho=0$, for $1/\epsilon=50$, the attractor is two-dimensional; the constant solution $h\equiv\bar{h}$ denoted by $O$ has a two-dimensional unstable manifold and $X$ corresponds to a whole $O(2)$ orbit of steady state solutions. A sketch of the attractor in this case is shown in Figure 12. When $\|\rho\|$ takes small positive values, only two steady state solutions, denoted by $A$ and $C$ remain from the entire $O(2)$ orbit, as discussed in Appendix A, while the constant solution continues to $B$ without change of stability. The resulting attractor is sketched in Figure 13. Increasing $\|\rho\|$ causes the steady state solutions $B$ and $C$ to coalesce in a saddle node bifurcation, so that the attractor degenerates to a single asymptotically stable steady state solution. It would be interesting to understand why this collision of steady state solution branches occurs. We have also explored the geometry of film rupture which occurs as $\rho\to 1^{-}$ when the disjoining pressure is given by $\Pi_{\rm SR}$; this phenomenon is shown in detail in Figures 5 and 6. Finally, in Figures 7 and 8, we showed the results of a two-parameter continuation study in the $(1/\epsilon,\rho)$ plane, showing how the multiplicity of positive steady state solutions changes as parameters are varied, and, in particular, indicating in the case of disjoining pressure $\Pi_{\rm SR}$ shown in Figure 8 regions in parameter space where no such solutions exist. We conjecture that in these regions the solution of the unsteady problem with any positive initial condition converges to a weak solution of the thin-film equation with regions in which $h(x)=0$, i.e. solutions with film rupture. For a discussion of such (weak) solutions of thin-film equations in the homogeneous case the reader is referred to the work of Laugesen and Pugh [26]. In the case of disjoining pressure $\Pi_{\rm SR}$, we could not use the AUTO-07p version of AUTO to continue branches of solutions beyond rupture. It would be an interesting project to develop such a capability for this powerful and versatile piece of software. Figures 8(b) and 8(c) provide numerical evidence for the existence of a curve of saddle-node bifurcations converging to the point $(0,1)$ in the $(1/\epsilon,\rho)$ plane; an explanation for this feature of the global bifurcation diagrams requires further study. To summarise: our study was primarily motivated by the work of Honisch et al. [17]. While we have clarified the mathematical properties of (9)–(12) and (49)–(51), so that the structure of bifurcations in Figure 3(a) of that paper for non-zero values of $\rho$ is now understood, many of their other numerical findings are still to be explored mathematically. For example, the stability of ridge solutions shown in their Figure 5 in the context of the full two- dimensional problem of a substrate with periodic wettability stripes. There is clearly much work still to be done on heterogeneous substrates with more complex wettability geometry. A final remark that might be of interest to the reader is that the solutions of equations (49)–(51), the steady state solutions of (4), a degenerate quasi- linear fourth-order PDE, can also be thought of as the steady state solutions of a much simpler (Rubinstein-Sternberg type [31]) second-order semi-linear non-local equation, $h_{t}=\gamma h_{xx}+\Pi(h,x)-\frac{1}{L}\int_{0}^{L}\Pi(h,x)\,\hbox{d}{x},\quad 0<x<L.$ (55) It would be interesting to compare the dynamics of (4) and (55), for example using the spectral comparison principles of Bates and Fife [3]. For other work on non-local reaction-diffusion equations such as (55), please see Budd et al. [7] and the review of Freitas [13]. We are grateful to Prof. U. Thiele (University of Münster) for clarifications concerning the work of Honisch et al. [17] and for sharing with us the AUTO codes used in that work which formed the basis of our continuation analysis. We are also grateful to the two anonymous referees whose remarks helped us to improve significantly the readability of the present work. ## Appendix A $O(2)$ Symmetry Breaking by Spatial Non-homogeneity In this Appendix, we present an argument that shows that when the wettability contrast is present, i.e. when $\rho\neq 0$, the breaking of the $O(2)$ symmetry which equation (49) with the periodic boundary conditions (51) has for $\rho=0$ , leaves only two steady state solutions. This is, in principle, a known result (see, for example, Chillingworth [8]), but, since we are not aware of an easily accessible reference, we give the details here. As before, we set $G(x)=\sin(2\pi x)$. We provide the proof for $\Pi_{\rm SR}$ given by (48), the proof for $\Pi_{\rm LR}$ given by (47) is similar. For the case of $\Pi_{\rm SR}$, let us rewrite the boundary value problem (49) in the form $\epsilon^{2}h_{xx}+f_{1}(h)+\rho f_{2}(h)G(x)-\int_{0}^{1}[f_{1}(h)+\rho f_{2}(h)G(x)]\,\hbox{d}{x}=0,\;\;0<x<1,$ (56) where $f_{1}(h)=\frac{1}{h^{6}}-\frac{1}{h^{3}},$ (57) and $f_{2}(h)=\frac{1}{h^{6}},$ (58) i.e. we separate the spatially homogeneous and spatially non-homogeneous components of the disjoining pressure. Equation (56) is subject to the periodic boundary conditions (51). Suppose that when $\rho=0$ there is an orbit of steady state solutions, i.e. a continuous closed curve of solutions $h_{0,s}(x)$, parameterised by $s\in\mathbb{R}/[0,1]$, such that $h_{0,s}(x):=h_{0}(x+s)$, for some function $h_{0}(x)$, i.e. all these solutions are related by translation. We aim to understand what remains of this orbit for small non-zero $\rho$. Fix $s\in\mathbb{R}/[0,1]$. We write $h(x)=h_{0,s}(x)+\rho h_{1}(x)+O(\rho^{2}).$ (59) Substituting this expansion into (56) and collecting the $O(\rho)$ terms, we have $\displaystyle\epsilon^{2}h_{1,xx}$ $\displaystyle+$ $\displaystyle(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))h_{1}-\int_{0}^{1}\left[f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s})\right]h_{1}\,\hbox{d}{x}$ (60) $\displaystyle=$ $\displaystyle- f_{2}(h_{0,s})G+\int_{0}^{1}f_{2}(h_{0,s})G\,\hbox{d}{x},$ where, just like $h_{0,s}(x)$, $h_{1}(x)$ also satisfies the periodic boundary conditions (51). Now set $Ku:=\epsilon^{2}u_{1,xx}+(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u-\int_{0}^{1}[f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s})]u\,\hbox{d}{x},$ (61) and let $D(K)$, the domain of $K$, be $D(K)=\left\\{f\in C^{2}\left(\left[0,1\right]\right)\;\|\,f(0)=f(1),f^{\prime}(0)=f^{\prime}(1)\right\\}.$ (62) The operator $K$ is self-adjoint with respect to the $L^{2}([0,1])$ inner product. Invoking the Fredholm Alternative [32, Theorem 7.26], we conclude that (60) has $1$-periodic solutions if and only if the right-hand side of (60) is orthogonal in $L^{2}([0,1])$ to the solutions of $Ku=0$. Next, we show that $u:=h_{0,s}^{\prime}$ solves $Ku=0$. Indeed, by differentiating (56) with $\rho=0$ with respect to $x$, we find that $u$ solves the equation $\epsilon^{2}u_{xx}+(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u=0.$ (63) Integrating this equation over the interval $[0,1]$, we have that $\int_{0}^{1}(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u\,\hbox{d}{x}=0.$ (64) Hence $\begin{split}0&~{}=\epsilon^{2}u_{xx}+(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u\\\ &~{}=\epsilon^{2}u_{xx}+(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u+\int_{0}^{1}(f_{1}^{\prime}(h_{0,s})+f_{2}^{\prime}(h_{0,s}))u\,\hbox{d}{x}\\\ &~{}=Ku.\end{split}$ (65) Also note that as $h_{0,s}(x)$ satisfies periodic boundary conditions, $\int_{0}^{1}h_{0,s}^{\prime}(x)\hbox{d}{x}=0.$ (66) Hence the solvability condition for (60) is $\int_{0}^{1}h_{0,s}^{\prime}(r)\left[-f_{2}(h_{0,s})G+\int_{0}^{1}f_{2}(h_{0,s})G\,\hbox{d}{x}\right]\hbox{d}{r}=0.$ (67) By (66), this condition reduces to $\int_{0}^{1}f_{2}(h_{0,s})h_{0,s}^{\prime}G\,\hbox{d}{x}=0.$ (68) Now recall that $h_{0,s}(x)=h_{0}(x+s)$, so if we write $F(x+s)=f_{2}(h_{0}(x+s))h_{0}^{\prime}(x+s)$, the function $F(\cdot)$ is $1$-periodic in $x$ with zero mean. 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11institutetext: High Energy Physics and Astrophysics Laboratory, Department of Physics, Faculty of Sciences SEMLALIA, Cadi Ayyad University, P.O.B. 2390, Marrakesh, Morocco. # Giant dipole resonance in Sm isotopes within TDHF method A. Ait Ben Mennana _e-mail:_<EMAIL_ADDRESS>M. Oulne _e-mail:_ <EMAIL_ADDRESS> (Received: date / Revised version: date) ###### Abstract In this work, we have studied the isovector giant dipole resonance (IVGDR) in even-even Sm isotopes within time-dependent Hartree-Fock (TDHF) with four Skyrme forces SLy6, SVbas, SLy5 and UNEDF1. The approach we have followed is somewhat similar to the one we did in our previous work in the region of Neodymium (Nd, Z=60) [Physica Scripta (2020)]. We have calculated the dipole strength of ${}^{128-164}\text{Sm}$, and compared with the available experimental data. An overall agreement between them is obtained. The dipole strength in neutron-deficient ${}^{128-142}\text{Sm}$ and in neutron-rich ${}^{156-164}\text{Sm}$ isotopes are predicted. Shape phase transition as well as shape coexistence in Sm isotopes are also investigated in the light of IVGDR. In addition, the correlation between the quadrupole deformation parameter $\beta_{2}$ and the splitting $\Delta E/\bar{E}_{m}$ of the giant dipole resonance (GDR) spectra is studied. The results confirm that $\Delta E/\bar{E}_{m}$ is proportional to quadrupole deformation $\beta_{2}$. ## 1 Introduction Giant resonances (GRs) represent an excellent example of collective modes of many,if not all, particles in the nucleus harakeh2001 . GRs are of particular importance because they currently provide the most reliable information about the bulk behavior of the nuclear many-body system. The so-called isovector giant diople resonance (IVGDR) is the oldest and best known of giant resonances. This is due to high selectivity for isovector $E_{1}$ in photo- absorption experiments. Several attempts of theoretical description of GDR have been made using the liquid drop model. Among them, Goldhaber and Teller (GT) interpreted it as collective vibrations of the protons moving against the neutrons in the nucleus with the centroid energy of the form $E_{c}\propto A^{-1/6}$goldhaber1948 . Somewhat later, Steinwedel and Jensen (SJ) interpreted it as a vibration of proton fluid against neutron fluid with a fixed surface where the centroid energy has the form $E_{c}\propto A^{-1/3}$ speth1981 . The experimental data are adjusted by a combination of these two berman1975 : in light nuclei, the data follow the law $A^{-1/6}$, while the dependence $A^{-1/3}$ becomes more and more dominant for increasing values of A. Since its first observation bothe1937 , it has been much studied both experimentally (see for example Refs.carlos1971 ; carlos1974 ; berman1975 ; donaldson2018 ) and theoretically (see for example Refs.goeke1982 ; maruhn2005 ; reinhard2008 ; yoshida2011 ; benmenana2020 ). The GDR spectra of nucleus can predict its shape (spherical, prolate, oblate, triaxial). It has a single peak for heavier spherical nuclei while in light nuclei it is split into several fragments harakeh2001 . In deformed nuclei, the GDR strength is split in two components corresponding to oscillations of neutrons versus protons along and perpendicular to the symmetry axis speth1991 ; harakeh2001 . Several microscopic approaches have been employed to study GDRs in deformed nuclei such as Separable Random-Phase-Approximation (SRPA) reinhard2008 ; reinhard2007c , time-dependent Skyrme-Hartree-Fock method maruhn2005 ; fracasso2012 , Relativistic Quasi-particle Random Phase Approximation (RQRPA) ring2009 and Extended Quantum Molecular Dynamics (EQMD) wang2017 . Experimentally, the GDR is induced by various ways such as photoabsorption carlos1971 ; carlos1974 ; Masur2006 inelastic scattering donaldson2018 ; ramakrishnan1996 ,$\gamma$-decay gundlach1990 . The time-dependent Hartree-Fock (TDHF) dirac1930 method has been employed in many works to investigate GRs in nuclei. It provides a good approximation for GR. Early, TDHF calculations concentrated on giant monopole resonance (GMR)blocki1979 ; chomaz1987 because they require only a spherical one- dimensional code. In the last few years with the increase in computer power, large scale TDHF calculations become possible with no assumptions on the spatial symmetry of the systemmaruhn2005 ; maruhn2006 ; stevenson2004 . Such calculations are performed by codes using a fully three dimensional (3D) Cartesian grid in coordinate space sky3d . In our previous work benmenana2020 , TDHF method provided an accurate description of the GDR in ${}^{124-160}\text{Nd}$ isotopes. Four Skyrme forces were used in this work. We obtained an overall agreement with experiment with slight advantage for SLy6 CHABANAT1998 . In this paper, we aim to study another even-even isotopic chain namely ${}^{128-164}\text{Sm}$ with four Skyrme forces SLy6 CHABANAT1998 , SLy5 CHABANAT1998 , SVbasreinhard2009 and UNEDF1 kortelainen2012 . The first three forces were used in our previous work benmenana2020 and gave acceptable results for GDR in Nd isotopes. The new Skyrme force UNEDF1 provided also satisfactory results in. Many previous experimental and theoretical works have studied the isotopic chain of Samarium Sm (Z = 62). From the experimental point of view one can see for example Ref.carlos1974 ) and from the theoretical one Refs.yoshida2011 ; wang2017 . Besides the study of GDR, many works (Refs.yoshida2011 ; ring2009 ; tao2013 ) studied the so-called pygmy dipole resonance (PDR) which correspond to low- energy E1 strength in nuclei with a pronounced neutron exces. The pygmy mode is regarded as vibration of the weakly bound neutron skin of the neutron-rich nucleus against the isospin-symmetric core composed of neutrons and protons paar2007 . In Ref. yoshida2011 , the authors studied PDR in some spherical nuclei such as ${}^{144}\text{Sm}$ and deformed ones such as ${}^{152-154}\text{Sm}$. For spherical nuclei, they found a concentration of the E1 strength in low-energy between 8 and 10 MeV, whereas for deformed nuclei the dipole strength is fragmented into low-energy states. They also showed that the nuclear deformation increases the low-lying strength E1 at E $<$ 10 MeV. The PDR mode is out of our current work in which we aim at a description of the GDR which lie at a high excitation energy range of $\sim$ 10-20 MeV. In this paper, the TDHF approximation negele1982 has been applied to study the GDR and shape evolution in even-even Sm (Z=62) isotopes from mass number A=128 to A=164. This study is done with SKY3D code sky3d which uses a fully three dimensional (3D) Cartesian grid in coordinate space with no spatial symmetry restrictions and includes all time-odd terms. Consequently, it is possible to study both spherical and deformed system within the limitation of mean field theory. Due to the open-shell nature of these nuclei, pairing and deformation properties must be taken into account in this study. Firstly, a static calculation gives some properties of the ground-state of the nucleus like root mean square (r.m.s), $\beta_{2}$, $\gamma$. In dynamic calculation, the ground-state of the nucleus is boosted by imposing a dipole excitation to obtain the GDR spectra and some of its properties (resonance energies, width). The paper is organized as follows: in Sec.2, we give a brief description of TDHF method and the GDR in deformed nuclei. In Sec.3, we present details of the numerical calculations. Our results and discussion are presented in Sec.4. Finally, Sec.5 gives the summary. ## 2 Time-Dependent Hartree-Fock method (TDHF) to giant resonances ### 2.1 TDHF method The time-dependent Hartree-Fock (TDHF) approximation has been extensively discussed in several references engel1975 ; kerman1976 ; koonin1977 . A brief introduction of the TDHF method is presented as follows. The TDHF is a self-consistent mean field (SCMF) theory which was proposed by Dirac in 1930 dirac1930 . It generalizes the static hartree-Fock (HF) and has been very successful in describing the dynamic properties of nuclei such as for example, giant resonances maruhn2005 ; stevenson2004 ; reinhard2007 ; blocki1979 and Heavy-ion collisions simenel2018 ; maruhn2006 . The TDHF equations are determined from the variation of Dirac action ${}S\equiv S_{t_{0},t_{1}}[\psi]=\int_{t_{0}}^{t_{1}}dt\bra{\psi(t)}\bigg{(}i\hbar\frac{d}{dt}-\hat{H}\bigg{)}\ket{\psi(t)},$ (1) where $\ket{\psi}$ is the Slater determinant, $t_{0}$ and $t_{1}$ define the time interval, where the action S is stationary between the fixed endpoints $t_{0}$ and $t_{1}$, and $\hat{H}$ is the Hamiltonian of the system. The energy of the system is defined as $E=\bra{\psi}\hat{H}\ket{\psi}$, and we have ${}\bra{\psi}\frac{d}{dt}\ket{\psi}=\sum_{i=1}^{N}\bra{\varphi_{i}}\frac{d}{dt}\ket{\varphi_{i}},$ (2) where $\ket{\varphi_{i}}$ are the occupied single-particle states. The action S can be expressed as $\displaystyle{}S$ $\displaystyle=$ $\displaystyle\int_{t_{0}}^{t_{1}}dt\bigg{(}i\hbar\sum_{i=1}^{N}\bra{\varphi_{i}}\frac{d}{dt}\ket{\varphi_{i}}-E[\varphi_{i}]\bigg{)}$ (3) $\displaystyle=$ $\displaystyle\int_{t_{0}}^{t_{1}}dt\bigg{(}i\hbar\sum_{i=1}^{N}\int dx\,\varphi_{i}^{}(x,t)\frac{d}{dt}\varphi_{i}(x,t)-E[\varphi_{i}]\bigg{)}$ The variation of the action S with respect to the wave functions $\varphi_{i}^{}$ reads ${}\frac{\delta S}{\delta\varphi_{i}^{}(x,t)}=0,$ (4) for each $i=1....N$, $t_{0}\leq t\leq{t_{1}}$ and for all $x$. More details can be found for example in Refs. kerman1976 ; simenel2012 . We finally get the TDHF equation ${}i\hbar\frac{\partial}{\partial t}\varphi_{i}(t)=\hat{h}[\rho(t)]\varphi_{i}(t)\quad\text{for}\quad 1\leq i\leq\text{N}.$ (5) where $\hat{h}$ is the single-particle Hartree-Fock Hamiltonian. The TDHF equations (5) are solved iteratively by a small time step $\Delta t$ during which we assume that the Hamiltonian remains constant. To conserve the total energy E, it is necessary to apply a symmetric algorithm by time reversal, and therefore to estimate the Hamiltonian at time $t+\frac{\Delta t}{2}$ to evolve the system between time $t\;\text{and}\;t+\Delta t$ flocard1978 ; bonche1976 ${}\ket{\varphi(t+\Delta t)}\simeq e^{-i\frac{\Delta t}{\hbar}\hat{h}(t+\frac{\Delta t}{2})}\ket{\varphi(t)}.$ (6) ### 2.2 Giant dipole resonance in deformed nuclei In deformed axially symmetric nuclei, one of the most spectacular properties of the GDR is its splitting into two components associated to vibrations of neutrons against protons along (K=0) and perpendicularly to (K=1) the symmetry axis. Therefore, the GDR strength represents a superposition of two resonances with energies $E_{i}\sim R_{i}^{-1}\sim A^{-1/3}$ speth1981 where R is the nuclear radius, and even three resonances in the case of asymmetric nuclei. This splitting has been observed experimentally carlos1974 ; berman1975 ; Masur2006 ; donaldson2018 and treated theoretically by different models maruhn2005 ; reinhard2008 ; yoshida2011 . For the axially symmetric prolate nuclei, the GDR spectra present two peaks where the low-energy $E_{z}$ corresponds to the oscillations along the major axis of symmetry and the high- energy $E_{x}=E_{y}$ corresponds to the oscillations along transverse minor axes of the nuclear ellipsoid, due to $E\sim R^{-1}$. For an oblate nucleus, it is the opposite situation to the prolate case. For triaxial nuclei, the oscillations along three axes are different ,i.e., $E_{x}\neq E_{y}\neq E_{z}$. For spherical nuclei, the vibrations along three axes degenerate and their energies coincide $E_{x}=E_{y}=E_{z}$. ## 3 Details of Calculations In this work, the GDR in even-even ${}^{128-164}\text{Sm}$ isotopes has been studied by using the code Sky3D (v1.1) sky3d . This code solves the HF as well as TDHF equations for Skyrme interactions SKYRME1958 . Calculations were performed with four Skyrme functional: SLy6 CHABANAT1998 , SLy5CHABANAT1998 , SVbasreinhard2009 , UNEDF1 kortelainen2012 . These Skyrme forces are widely used for the ground state properties (binding energies, radii…) and dynamics (as giant resonances) of nuclei including deformed ones. In particular they provide a reasonable description of the GDR: SLy6maruhn2005 ; reinhard2008 , SVbasreinhard2009 , SLy5fracasso2012 and UNEDF1 kortelainen2012 . The parameters set of these functionals used in this study is shown in Table 1. Table 1: Parameters ($t,x$) of the Skyrme forces used in this work. Parameters \| UNEDF1 \| SVbas \| SLy6 \| SLy5 ---\|---\|---\|---\|--- $t_{0}$ (MeV.fm3) \| -2078.328 \| -1879.640 \| -2479.500 \| -2484.880 $t_{1}$ (MeV.fm5) \| 239.401 \| 313.749 \| -1762.880 \| 483.130 $t_{2}$ (MeV.fm5) \| 1574.243 \| 112.676 \| -448.610 \| -549.400 $t_{3}$ (MeV.fm3+3σ) \| 14263.646 \| 12527.389 \| 13673.000 \| 13763.000 $x_{0}$ \| 0.054 \| 0.258 \| 0.825 \| 0.778 $x_{1}$ \| -5.078 \| -0.381 \| -0.465 \| -0.328 $x_{2}$ \| -1.366 \| -2.823 \| -1.000 \| -1.000 $x_{3}$ \| -0.161 \| 0.123 \| 1.355 \| 1.267 $\sigma$ \| 0.270 \| 0.300 \| 0.166 \| 0.166 W0 (MeV.fm5) \| 76.736 \| 124.633 \| 122.000 \| 126.000 A first step of calculation concerns a static calculation which allows to determine the ground state for a given nucleus. This state is obtained by solving the static HF + BCS equations (8) in a three-dimensional (3D) Cartesian mesh with a damped gradient iteration method on an equidistant grid and without symmetry restrictions sky3d . ${}\hat{h}\psi_{i}(x)=\epsilon_{i}\psi_{i}(x)\quad\text{for}\quad i=1,....,A,$ (7) where $\hat{h}$ is the single-particle Hamiltonien, and $\epsilon_{i}$ is the single-particle energy of the state $\psi_{i}(x)$ with $x=(\vec{r},\sigma,\tau)$. We used a cubic box with size a = 24 fm and a grid spacing of $\Delta x$ = 1.00 fm in each direction. In SKY3D code sky3d , the static HF + BCS equations (7) are solved iteratively until a convergence is obtained ,i.e., when for example the sum of the single-particle energy fluctuations becomes less than a certain value determined at the beginning of the static calculation. In this study we take as a convergence value $10^{-5}$ which is sufficient for heavy nuclei (for more details see Ref. sky3d . The pairing is treated in the static calculation, which allows to calculate the pairing energy $\displaystyle{}E_{pair}=\frac{1}{4}\sum_{q\in\\{p,n\\}}V_{pair,q}\int d^{3}r\|\xi_{q}\|^{2}F(r)$ (8) where the pairing density $\xi_{q}$ reads sky3d $\displaystyle{}\xi(\vec{r})=\sum_{\alpha\in\\{p,n\\}}\sum_{s}u_{\alpha}v_{\alpha}\|\psi_{\alpha}(\vec{r},s)\|^{2}$ (9) where $v_{\alpha}$, $u_{\alpha}=\sqrt{1-v_{\alpha}^{2}}$ are the occupation and non-occupation amplitude of single-particle state $\psi_{\alpha}$ , respectively, and the function $F=1$ or $F=1-\rho/\rho_{0}$ gives a pure $\delta$-interaction (DI), also called volume pairing (VDI) where $\rho_{0}\rightarrow\infty$ or density dependent $\delta$-interaction (DDDI), respectively, while $\rho_{0}=0.16$ fm-3 is the saturation density. $V_{P,N}$ represents the pairing strength which is obtained from the force definition in the SKY3D code sky3d . In dynamic calculations, the ground-state wave function obtained by the static calculations is excited by an instantaneous initial dipole boost operator in order to put the nucleus in the dipole mode maruhn2005 ; simenel2009 ; stevenson2008 . ${}\varphi_{\alpha}^{(g.s)}(r)\longrightarrow\varphi_{\alpha}(r,t=0)=\exp(ib\hat{D})\varphi_{\alpha}^{(g.s)}(r),$ (10) where $\varphi_{\alpha}^{(g.s)}(r)$ represents the ground-state of nucleus before the boost, b is the boost amplitude of the studied mode , and $\hat{D}$ the associated operator. In our case, $\hat{D}$ represents the isovector dipole operator defined as $\displaystyle{}\hat{D}$ $\displaystyle=$ $\displaystyle\frac{NZ}{A}\bigg{(}\frac{1}{Z}\sum_{p=1}^{Z}\vec{z}_{p}-\frac{1}{N}\sum_{n=1}^{N}\vec{z}_{n}\bigg{)}$ (11) $\displaystyle=$ $\displaystyle\frac{NZ}{A}\bigg{(}\vec{R}_{Z}-\vec{R}_{N}\bigg{)},$ where $\vec{R}_{Z}$ (resp. $\vec{R}_{N}$ ) measures the proton (resp. neutron) average position on the z axis. The spectral distribution of the isovector dipole strength is obtained by applying a boost (10) with a small value of the amplitude of the boost b to stay well in the linear regime of the excitation. For a long enough time, the dipole moment $\hat{D}=\bra{\psi(t)}\hat{D}\ket{\psi(t)}$ is recorded along the dynamical evolution. Finally, the dipole strength $S_{D}(\omega)$ can be obtained by performing the Fourier transform $D(\omega)$ of the signal $\hat{D}(t)$, defined as ring1980 $\displaystyle{}S_{D}(\omega)$ $\displaystyle=$ $\displaystyle\sum_{\nu}\delta(E-E_{\nu})\big{\|}\bra{\nu}\hat{D}\ket{0}\big{\|}^{2}.$ (12) Some filtering is necessary to avoid artifacts in the spectra obtained by catting the signal at a certain final time, in order to the signal vanishes at the end of the simulation time. In practice we use windowing in the time domain by damping the signal $D(t)$ at the final time with $cos\big{(}\frac{\pi t}{2T_{fin}}\big{)}^{n}$ sky3d . ${}D(t)\longrightarrow D_{fil}=D(t).cos\bigg{(}\frac{\pi t}{2T_{fin}}\bigg{)}^{n},$ (13) where n represents the strength of filtering and $T_{fin}$ is the final time of the simulation. More details can be founded in Refs. sky3d ; reinhard2006 . In this work, all dynamic calculations were performed in a cubic space with 24 x 24 x 24 fm3 according to the three directions (x, y, z) and a grid spacing of 1 fm. We chose nt= 4000 as a number of time steps to be run, and dt = 0.2 fm/c is the time step, so Tf = 800 fm/c is the final time of simulation. Pairing is frozen in the dynamic calculation ,i.e., the BCS occupation numbers are frozen at their initial values during time evolution. ## 4 Results and Discussion In this section we present our numerical results of static calculations concerning some properties of the ground-state, and dynamic calculations concerning some properties of the GDR for ${}^{128-164}\text{Sm}$ nuclei. ### 4.1 Ground-state properties The isotopic chain of Sm (Z=62) studied in this work displays a transition from spherical, when neutron number N is close to magic number N = 82, to the axially deformed shapes when N increases or decreases carlos1974 ; meng2005 ; wang2017 ; naz2018 . Among the properties of the ground-state of nuclei, there are the deformation parameters $\beta_{2}$ and $\gamma$ which give an idea on the shape of the nucleus ring1980 ; takigawa2017 . These deformation parameters are defined as follows sky3d ${}\beta=\sqrt{a_{0}^{2}+2a_{2}^{2}}\qquad,\quad\gamma=atan\bigg{(}\frac{\sqrt{2}a_{2}}{a_{0}}\bigg{)}$ (14) ${}a_{m}=\frac{4\pi}{5}\frac{Q_{2m}}{AR^{2}}\qquad,\quad R=1.2A^{1/3}(fm),$ (15) where $Q_{2m}$ is the quadrupole moment defined as ${}Q_{2m}=\int\rho(\vec{r})r^{2}Y_{2m}(\theta,\varphi)d\vec{r}$ (16) The deformation parameters ($\beta$,$\gamma$) often called Bohr-Mottelson parameters are treated as a probe to select the ground-state of all nuclei in this article. Table 2 displays the numerical results obtained for the deformation parameters ($\beta_{2}$,$\gamma$) based on Eq. (14) of ${}^{128-164}\text{Sm}$ isotopes with four Skyrme forces, including the available experimental data from Ref.raman2001 and the HFB calculations based on the D1S Gogny force HFB for comparison. Fig. 1 shows the variation of $\beta_{2}$ as a function of neutrons number N. Table 2: The deformation parameters ($\beta_{2}$,$\gamma$) calculated with UNEDF1, SVbas, SLy6, and SLy5 are compared with the experimental data are from Ref.raman2001 , and data from Ref.HFB . Nuclei \| UNEDF1 \| SVbas \| SLy6 \| SLy5 \| HFB$\\_$Gogny.HFB \| Exp. raman2001 ---\|---\|---\|---\|---\|---\|--- ${}^{128}\text{Sm}$ \| (0.406; $0.0^{\circ}$) \| (0.398; $4.8^{\circ}$) \| (0.402; $8.6^{\circ}$) \| (0.401; $7.6^{\circ}$) \| (0.398; $8.0^{\circ}$) \| —– ${}^{130}\text{Sm}$ \| (0.393; $0.1^{\circ}$) \| (0.377; $0.0^{\circ}$) \| (0.381; $0.0^{\circ}$) \| (0.381; $0.0^{\circ}$) \| (0.377; $0.0^{\circ}$) \| —– ${}^{132}\text{Sm}$ \| (0.388; $0.0^{\circ}$) \| (0.374; $0.0^{\circ}$) \| (0.371; $0.0^{\circ}$) \| (0.382; $0.0^{\circ}$) \| (0.380; $0.0^{\circ}$) \| —– ${}^{134}\text{Sm}$ \| (0.377; $0.0^{\circ}$) \| (0.399; $0.0^{\circ}$) \| (0.308; $14.8^{\circ}$) \| (0.314; $12.2^{\circ}$) \| (0.436; $0.0^{\circ}$) \| 0.366 ${}^{136}\text{Sm}$ \| (0.260; $21.3^{\circ}$) \| (0.252; $22.5^{\circ}$) \| (0.261; $22.4^{\circ}$) \| (0.263; $21.9^{\circ}$) \| (0.252; $22.0^{\circ}$) \| 0.293 ${}^{138}\text{Sm}$ \| (0.205; $27.5^{\circ}$) \| (0.207; $26.1^{\circ}$) \| (0.228; $25.6^{\circ}$) \| (0.227; $25.2^{\circ}$) \| (0.183; $25.0^{\circ}$) \| 0.208 ${}^{140}\text{Sm}$ \| (0.026; $14.8^{\circ}$) \| (0.113; $0.0^{\circ}$) \| (0.181; $27.7^{\circ}$) \| (0.181; $27.6^{\circ}$) \| (0.147; $35.0^{\circ}$) \| —– ${}^{142}\text{Sm}$ \| (0.000; $20.6^{\circ}$) \| (0.000; $14.4^{\circ}$) \| (0.001; $0.0^{\circ}$) \| (0.003; $17.0^{\circ}$) \| (0.000; $0.0^{\circ}$) \| —– ${}^{144}\text{Sm}$ \| (0.001; $4.0^{\circ}$) \| (0.000; $8.5^{\circ}$) \| (0.000; $12.7^{\circ}$) \| (0.000; $1.5^{\circ}$) \| (0.000; $0.0^{\circ}$) \| 0.087 ${}^{146}\text{Sm}$ \| (0.014; $58.0^{\circ}$) \| (0.052; $1.2^{\circ}$) \| (0.063; $0.0^{\circ}$) \| (0.064; $0.7^{\circ}$) \| (0.045; $2.0^{\circ}$) \| —– ${}^{148}\text{Sm}$ \| (0.128; $0.0^{\circ}$) \| (0.151; $0.2^{\circ}$) \| (0.167; $3.6^{\circ}$) \| (0.162; $0.0^{\circ}$) \| (0.167; $0.0^{\circ}$) \| 0.142 ${}^{150}\text{Sm}$ \| (0.211; $0.0^{\circ}$) \| (0.220; $0.0^{\circ}$) \| (0.225; $0.0^{\circ}$) \| (0.223; $0.0^{\circ}$) \| (0.204; $0.0^{\circ}$) \| 0.193 ${}^{152}\text{Sm}$ \| (0.302; $0.0^{\circ}$) \| (0.306; $0.0^{\circ}$) \| (0.305; $0.0^{\circ}$) \| (0.302; $0.0^{\circ}$) \| (0.273; $0.0^{\circ}$) \| 0.306 ${}^{154}\text{Sm}$ \| (0.335; $0.0^{\circ}$) \| (0.337; $0.0^{\circ}$) \| (0.341; $0.0^{\circ}$) \| (0.338; $0.0^{\circ}$) \| ((0.347; $0.0^{\circ}$) \| 0.341 ${}^{156}\text{Sm}$ \| (0.349; $0.0^{\circ}$) \| (0.348; $0.0^{\circ}$) \| (0.350; $0.0^{\circ}$) \| (0.349; $0.0^{\circ}$) \| (0.336; $0.0^{\circ}$) \| —– ${}^{158}\text{Sm}$ \| (0.357; $0.0^{\circ}$) \| (0.356; $0.0^{\circ}$) \| (0.362; $0.0^{\circ}$) \| (0.363; $0.0^{\circ}$) \| (0.351; $0.0^{\circ}$) \| —– ${}^{160}\text{Sm}$ \| (0.361; $0.0^{\circ}$) \| (0.360; $0.0^{\circ}$) \| (0.368; $0.0^{\circ}$) \| (0.366; $0.0^{\circ}$) \| (0.361; $0.0^{\circ}$) \| —– ${}^{162}\text{Sm}$ \| (0.365; $0.0^{\circ}$) \| (0.362; $0.0^{\circ}$) \| (0.369; $0.0^{\circ}$) \| (0.367; $0.0^{\circ}$) \| (0.360; $0.0^{\circ}$) \| —– ${}^{164}\text{Sm}$ \| (0.367; $0.0^{\circ}$) \| (0.363; $0.0^{\circ}$) \| (0.373; $0.0^{\circ}$) \| (0.369; $0.0^{\circ}$) \| (0.360; $0.0^{\circ}$) \| —– Figure 1: The Quadrupole deformation parameter $\beta_{2}$ of ${}^{124-160}\text{Nd}$ isotopes as function of their neutron number N. The experimental data are from Ref. raman2001 . From Fig.1, we can see the $\beta_{2}$ values of our calculations are generally close to experimental ones raman2001 . On the other hand, there is an agreement between our calculations and HFB theory based on the D1S Gogny forceHFB . In the vicinity of the region where N = 82, the $\beta_{2}$ values show minima ($\beta_{2}\simeq 0$) as expected because all nuclei with the magic number N=82 are spherical. For the ${}^{140}\text{Sm}$ nucleus, we find different results between the four Skyrme forces in this study. For the Skyrme forces SLy6 and SLy5, ${}^{140}\text{Sm}$ has a triaxial shape ($\gamma\simeq 28.0^{\circ}$). It has a prolate shape for SV-bas ($\gamma=0.0^{\circ}$), and has an approximate spherical form for UNEDF1 force ($\beta_{2}\simeq 0.026$). For comparison, Calculations by Möller et al.moller2008 , based on the finite- range droplet model, predicted the ground state of ${}^{140}\text{Sm}$ nucleus to be triaxial ($\gamma=30.0^{\circ}$). In table 2, the ($\beta_{2}$,$\gamma$) values obtained in this work as well as those of HFB theory based on the D1S Gogny forceHFB and avialable experimental data raman2001 show a shape transition from spherical ${}^{144}\text{Sm}$ (N=82) to deformed shape below and above the magic neutron number N=82. For ${}^{128-144}\text{Sm}$ isotopes below N = 82, the isotopic chains exhibit a transition from prolate ($\gamma=0.0^{\circ}$) to spherical shape ($\beta_{2}\simeq 0.000$) passing through triaxial form ($22.0^{\circ}\leq\gamma\leq 28.0^{\circ}$) for ${}^{136-140}\text{Sm}$ isotopes, and for neutron number higher than N = 82, both the experimental and theoretical results show that the prolate deformation increases gradually and then saturates at a value which closes to $\beta_{2}\simeq$ 0.368. ### 4.2 Giant dipole resonance in ${}^{128-164}\text{Sm}$ nuclei Based on the TDHF ground states for ${}^{128-164}\text{Sm}$ isotopes obtained in static calculations, we perform dynamic calculation such as GDR in this work to obtain some of its properties as we will see later. #### 4.2.1 The time evolution of the dipole moment $D_{m}(t)$ The dipole moment $D_{m}(t)$ defined by Eq. (11) allows to predict the collective motions of nucleons along the three directions x, y and z. The time evolution of $D^{i}_{m}(t)$ where i denotes x, y and z of ${}^{138}\text{Sm}$, ${}^{144}\text{Sm}$ and ${}^{154}\text{Sm}$ is plotted in Fig. 2. We note that the collective motion of nucleons in GDR is done generally along two axes. The oscillation frequency $\omega_{i}$ is related to the nuclear radius $R_{i}$ by $\omega_{i}\propto R_{i}^{-1}$ where i$\in${x,y,z}. Fig. 2(a) shows the time evolution of dipole moment for ${}^{144}\text{Sm}$ and ${}^{154}\text{Sm}$. Figure 2: The dipole moment $D_{m}(t)$ as function of the simulation time t(fm/c) calculated with the Skyrme force SLy6 for ${}^{138}\text{Sm}$, ${}^{144}\text{Sm}$ and ${}^{154}\text{Sm}$. For the ${}^{144}\text{Sm}$ nucleus, the three components $D^{x}_{m}(t)$, $D^{y}_{m}(t)$ and $D^{z}_{m}(t)$ are identical ,i.e., the oscillation frequencies along the three axes are equal ( $\omega_{x}=\omega_{y}=\omega_{z}$) which confirms that this nucleus has a spherical shape as we predicted in static calculations ($\beta_{2}\simeq 0.000$). For the ${}^{154}\text{Sm}$ nucleus, the $D^{x}_{m}(t)$ and $D^{y}_{m}(t)$ values are identical and differ from the values of $D^{z}_{m}(t)$ ,i.e., the oscillation frequencies along the symmetry z-axis $\omega_{z}$ are lower than that along the two other axes x and y which they are equal $\omega_{x}=\omega_{y}$. This confirms that ${}^{154}\text{Sm}$ has a prolate shape because $\omega_{z}<\omega_{x}=\omega_{y}$Masur2006 which is consistent with our static calculations ($\gamma=0.0^{\circ}$). We point out that we found almost the same situation for the prolate nuclei namely ${}^{130-134}\text{Sm}$ and ${}^{148-164}\text{Sm}$. In Fig. 2(b), the values of the three components $D^{i}_{m}(t)$ are different from each other in the case of the ${}^{138}\text{Sm}$ nucleus. We notice that the oscillation frequencies $\omega_{i}$ along the three axes are different from each other $\omega_{x}\neq\omega_{y}\neq\omega_{z}$ which confirms that this nucleus has a triaxial shape as we predicted in static calculations ($\gamma\simeq 25.0^{\circ}$). The same situation occurs for ${}^{136}\text{Sm}$. We note also that the time evolution of dipole moment $D_{m}(t)$ is almost the same for the others Skyrme forces (SLy5, UNEDF1, SVbas) with an exception for some nuclei as ${}^{140}\text{Sm}$. The periodicity of the three components $D^{i}_{m}(t)$ allows the excitation energies $E_{i}$ to be estimated for the oscillations along each of the three axes. For ${}^{144}\text{Sm}$, we obtain, for $D^{x}_{m}(t)$, $D^{y}_{m}(t)$ and $D^{z}_{m}(t)$, the same period T $\simeq$ 84.3 fm/c giving an excitation energy $E_{x}=E_{y}=E_{z}\simeq$ 14.70 MeV. This value is slightly lower than the experimental one $E_{GDR}^{exp.}$=15.3$\pm$ 0.1 carlos1974 . The table 3 shows the excitation energies for ${}^{138}\text{Sm}$ and ${}^{154}\text{Sm}$ nuclei with Skyrme force SLy6. Table 3: The excitation energies along the three axes for ${}^{138}\text{Sm}$ and ${}^{154}\text{Sm}$ with Sly6, obtained from the time evolution of $D^{i}_{m}(t)$. Nuclei \| \| $E_{x}$(MeV) \| \| \| $E_{y}$(MeV) \| \| \| $E_{z}$(MeV) ---\|---\|---\|---\|---\|---\|---\|---\|--- ${}^{138}\text{Sm}$ \| \| 14.75 \| \| \| 16.52 \| \| \| 13.40 ${}^{154}\text{Sm}$ \| \| 15.62 \| \| \| 15.62 \| \| \| 11.90 #### 4.2.2 GDR Spectrum The calculation of the Fourier transform of the isovector signal D(t) allows to obtain the GDR energy spectrum. The spectral strength S(E) (12) is simply the imaginary part of the Fourier transform of D(t). Figs. 3 \- 6 display the GDR spectra in ${}^{128-164}\text{Sm}$ isotopes calculated with the four Skyrme forces, compared with the available experimental data carlos1974 . It needs to be pointed out that the experimental data for Sm isotopes from A=128 to A=142, and from A=152 to A=160, and ${}^{146}\text{Sm}$ are not yet available. The calculated GDR spectra in ${}^{144-154}\text{Sm}$ isotopes together with the available experimental data carlos1974 are shown in Fig.3. It can be seen that all four Skyrme forces give generally acceptable agreement with the experiment with a slight down-shift of the order of 0.5 MeV for SLy5, SLy6 in the case of the spherical nucleus ${}^{144}\text{Sm}$ and the weakly deformed ${}^{148-150}\text{Sm}$ nuclei , and slight up-shift ($\sim$ 0.5 MeV) for SVbas force. The agreement is better for deformed ${}^{152-154}\text{Sm}$ nuclei , where all Skyrme forces produce the deformation splitting, in which rare-earth nuclei as Samarium (Sm) with neutron number N$\approx$90 show an example of shape transitionscarlos1974 ; maruhn2005 ; benmenana2020 . For ${}^{144}\text{Sm}$ (N=82), its GDR strength has a single-humped shape. The vibrations along the three axes degenerate ,i.e., they are the same resonance energy $E_{i}$ ($E_{x}=E_{y}=E_{z}$), which confirms that this nucleus is spherical due to the relation $E_{i}\propto R_{i}^{-1}$ where i$\in${x,y,z} speth1981 . For ${}^{148}\text{Sm}$ and ${}^{150}\text{Sm}$ nuclei, the two resonance peaks move away slightly from each other but the total GDR presents one peak, so they are also weakly deformed nuclei with prolate shape. For ${}^{152}\text{Sm}$ and ${}^{154}\text{Sm}$ nuclei, the total GDR splits into two distinct peaks which confirms that these nuclei are strongly deformed with prolate shape since the oscillations along the major axis (K=0 mode) are characterized by lower frequencies than the oscillations perpendicular to this axis (K=1 mode) speth1991 . The isotope ${}^{146}\text{Sm}$ for which we do not have experimental data, SLy6, Sly5 and SVbas give a weakly deformed nucleus ($\beta_{2}\simeq$0.06) where the resonance peaks along the major and the minor axis are very close together, whereas UNEDF1 gives an approximate spherical nucleus ($\beta_{2}\simeq$0.01). Calculations in Ref. naz2018 , based on the self- consistent Relativistic-Hartree-Bogoliubov (RHB) formalism, predicted a shape coexistence for ${}^{146}\text{Sm}$. In order to verify the shape coexistence wood1992 ; heyde2011 in ${}^{146}\text{Sm}$ nucleus, we redid the static calculations several times from different initial deformations with SLy6 force. In all cases, we obtained two minima (prolate and oblate) whose their properties are displayed in Table 4. We can see that the difference in energy between these two minima is around $\Delta$(B.E)$\simeq$ 0.07 MeV. This is a clear indication of a shape coexistence in ${}^{146}\text{Sm}$ nucleus. According to the value of deformation parameter $\gamma$, this competition of shape is between oblate ($\gamma=60^{\circ}$) and prolate ($\gamma=0^{\circ}$) shape, but the deformation is very weak ($\beta_{2}\simeq 0.05$) in both cases. Fig.4 shows the calculated GDR spectra corresponding to two minima (prolate, oblate). It confirms this suggestion: the upper panel (Fig.4(a)) shows an oblate shape for ${}^{146}\text{Sm}$ due to oscillations along the shorter axis (K=0 mode) which are characterized by higher energies than the oscillations along the axis ($\mid$K$\mid$=1 mode) perpendicular to it, while the lower panel (Fig.4(b)) shows a prolate shape for this nucleus. In both cases, the deformation splitting $\Delta$E between the two peaks is too small which confirms that this nucleus is very weakly deformed. Figure 3: (Color online) GDR spectra in the chain of ${}^{144-154}\text{Sm}$ calculated with SLy6, SLy5, SVbas and UNEDF1. The solid(red), dashed(green) and dotted-dashed(blue) lines denote the dipole strengths: total, along the long axis and the short axis (multiplied by 2) respectively. The calculated strength total is compared with the experimental data carlos1974 depicted by black solid squares. Table 4: The ground-state properties of two minima for ${}^{146}\text{Sm}$ nucleus. Properties \| \| Prolate minimum \| \| \| Oblate minimum \| \| \| ---\|---\|---\|---\|---\|---\|---\|---\|--- Binding energy (B.E) \| \| -1999.73 MeV \| \| \| -1999.66 MeV \| \| \| Root mean square (r.m.s) \| \| 4.970 fm \| \| \| 4.969 fm \| \| \| Quadrupole deformation $\beta_{2}$ \| \| 0.063 \| \| \| 0.048 \| \| \| Deformation parameter $\gamma$ \| \| $0^{\circ}$ \| \| \| $60^{\circ}$ \| \| \| Figure 4: (Color online) The calculated GDR spectra for ${}^{146}\text{Sm}$ with the Skyrme force SLy6. , Fig.5 shows the GDR strength in neutron-deficient ${}^{128-142}\text{Sm}$ isotopes. We can see that the deformation decreases gradually from the well deformed nucleus ${}^{128}\text{Sm}$ ($\beta_{2}\simeq$0.4) to the approximate spherical one ${}^{142}\text{Sm}$ ($\beta_{2}\simeq$0.0) ,i.e., when the neutron number N increases and closes to the magic number N=82. We note that all Skyrme forces in this work give almost the same GDR spectra except for ${}^{140}\text{Sm}$. According to the GDR strength along the three axes, the ${}^{128}\text{Sm}$ nucleus is weakly triaxial with SLy6, SLy5 and SVbas whereas it has a prolate shape with UNEDF1. For the ${}^{130-132}\text{Sm}$ isotopes, all the four Skyrme forces predict a prolate shape for them. For ${}^{134}\text{Sm}$, SVbas and UNEDF1 predict a prolate shape, while SLy5 and SLy6 give a weak triaxial shape. For ${}^{136-138}\text{Sm}$ isotopes, we can see that the oscillations along the three axes correspond to different resonance energies $E_{i}$ ($E_{x}\neq E_{y}\neq E_{z}$), which shows that these nuclei are deformed with triaxial shape. The four Skyrme forces give different results for ${}^{140}\text{Sm}$ as displayed in Fig.5. The SLy family (SLy5 and SLy6) predict a triaxial shape, SVbas predicts a prolate shape while UNEDF1 gives an approximate spherical shape. For ${}^{142}\text{Sm}$, all Skyrme forces predict a spherical shape where the GDR strengths along the three axes are identical ,i.e., ($E_{x}=E_{y}=E_{z}$). Figure 5: (Color online) GDR spectra in the isotopic chain ${}^{128-142}\text{Sm}$ calculated with SLy6, SLy5, SVbas and UNEDF1. The solid(red), dashed(green) and dotted-dashed(blue) lines denote the dipole strengths: total, along the long axis and the short axis(multiplied by 2 except ${}^{136-140}\text{Sm}$) respectively. The dotted (magenta) line denotes the strength along the third middle axis in the case of the triaxial nuclei ${}^{136-140}\text{Sm}$. Fig.6 shows the GDR strength in neutron-rich ${}^{156-164}\text{Sm}$ isotopes. We can see that all Skyrme forces provide quite similar results. From ${}^{156}\text{Sm}$ (N=94) to ${}^{164}\text{Sm}$ (N=102), the deformation gradually gets broader, and their GDRs acquire a pronounced double-humped shape. Therefore, these nuclei are strongly deformed with prolate shape since the oscillations energies along the longer axis (z-axis) are lower than those of oscillations along the short axis (x and y axes) ,i.e., $E_{z}<E_{x}=E_{y}$. Figure 6: (color online) The GDR spectra in the isotopic chain ${}^{156-164}\text{Sm}$ calculated with SLy6, SLy5, SVbas and UNEDF1. The solid(red), dashed(green) and dotted-dashed(magenta) lines denote the dipole strengths: total, along the long axis and the short axis(multiplied by 2) respectively. In order to compare the results between different Skyrme forces under consideration, we plot their GDR spectra into one figure, together with experimental data. Fig.7 shows the GDR strength in ${}^{144}\text{Sm}$, and ${}^{154}\text{Sm}$ calculated by the four Skyrme forces as well as the experimental data from Ref.carlos1974 . It can be seen there is a dependence of the GDR spectra on various Skyrme forces. We note a small shift of the average peak position of $\sim$ 1 MeV between these forces. The peak position of energy obtained with the Skyrme force SVbas is located highest among these four Skyrme forces. For the spherical nucleus ${}^{144}\text{Sm}$, the Skyrme force UNEDF1 reproduces well the shape and the peak among the four Skyrme forces. The agreement is less perfect with other forces. The SLy5 and SLy6 forces give very similar results, the strength exhibits a slight downshift while a slight upshift with SVbas functional. For the deformed nucleus ${}^{154}\text{Sm}$, there is an excellent agreement between the different functionals and the experiment, with a slight upshift for the K=1 mode for SVbas force. We can explain this dependence y the fact that it is linked to certain basic characteristics and nuclear properties of the Skyrme forces as shown in Table 5. The isovector effective mass $m_{1}^{}/m$ is related to the sum rule enhancement factor $\kappa$ by $m_{1}^{}/m=1/(1+\kappa)$ berman1975 , i.e., the larger isovector effective mass corresponds to the lighter value of the enhancement factor. We can easily see that the increase of the factor $\kappa$ (i.e., low isovector effective mass $m_{1}^{}/m$) causes the GDR strength to shift towards the higher energy region, as indicated in Ref.nesterenko2006 for the GDR in ${}^{154}\text{Sm}$, ${}^{238}\text{U}$ and ${}^{154}\text{No}$, and in Ref.oishi2016 for ${}^{174}\text{Yb}$. For example, the large collective shift in SVbas can be related to a very high enhancement factor $\kappa$=0.4 compared to other Skyrme forces. In addition to the dependence with the enhancement factor $\kappa$, Fig.7 also shows a connection between GDR energy and symmetry energy $a_{sym}$. The peak energy of the GDR moves towards the higher energy region when $a_{sym}$ decreases, as pointed in Ref.stone2007 for the GDR in doubly magic ${}^{208}\text{Pb}$, and in our previous work for Nd isotopes benmenana2020 . Figure 7: (Color online) The calculated GDR spectra ${}^{144}\text{Sm}$ and ${}^{154}\text{Sm}$ with Skyrme forces UNEDF1, SLy6, SLy5 and SVbas for . the experimental data carlos1974 are depicted by triangle. Table 5: The sum rule enhancement factor $\kappa$, isovector effective mass $m_{1}^{}/m=1/(1+\kappa)$, and symmetry energy $a_{sym}$ for the Skyrme forces under consideration. Forces \| \| $m_{1}^{}/m$ \| \| \| $\kappa$ \| \| \| $a_{sym}(MeV)$ ---\|---\|---\|---\|---\|---\|---\|---\|--- SLy6 CHABANAT1998 \| \| 0.80 \| \| \| 0.25 \| \| \| 31.96 SLy5 CHABANAT1998 \| \| 0.80 \| \| \| 0.25 \| \| \| 32.03 UNEDF1 kortelainen2012 \| \| $\simeq$1.00 \| \| \| 0.001 \| \| \| 28.98 SVbas reinhard2009 \| \| 0.715 \| \| \| 0.4 \| \| \| 30.00 #### 4.2.3 Relation between deformation splitting $\Delta E$ and quadrupole deformation $\beta_{2}$ As we mentioned above, the GDR strength splits into two peaks for deformed nuclei. Each peak corresponds to a resonance energy $E_{i}$ of GDR. We denoted by $E_{1}$ and $E_{2}$ the energies corresponding to K=0 and K=1 modes respectively. The total resonance energy of giant resonance is defined by the formula garg2018 ${}E_{m}=\frac{\int_{0}^{+\infty}S(E)EdE}{\int_{0}^{+\infty}S(E)dE},$ (17) where S(E) (12) is the strength function of giant resonance. In Table 6 , the resonance energies E1 and E2 of ${}^{128-164}\text{Sm}$ nuclei are presented, including the available experimental data from Ref.carlos1974 . From this table, we can see an overall agreement between our results and the experimental data, with a slightly advantage for the Sly6 functional. For instance, the result of the semi-spherical ${}^{144}\text{Sm}$ gives $E_{GDR}^{SLy6}$=15.05 MeV which is very close to $E_{GDR}^{Exp.}$=(15.30 $\pm$ 0.10) MeV. Also for deformed nuclei as ${}^{152}\text{Sm}$ and ${}^{154}\text{Sm}$, the results ($E_{1},E_{2}$) with SLy6 are very close to those of experiment. Table 6: The resonance energy centroids $E_{1}$ and $E_{2}$ of ${}^{128-164}\text{Sm}$ corresponding to oscillation along the major axis (K=0) and the minor axis (K=1) respectively. The experimental data are from ref. carlos1974 . \| UNEDF1 \| SVBas \| SLy5 \| SLy6 \| Exp.carlos1971 ---\|---\|---\|---\|---\|--- Nuclei \| E1 \| E2 \| E1 \| E2 \| E1 \| E2 \| E1 \| E2 \| E1 \| E2 ${}^{\textbf{128}}\textbf{Sm}$ \| 13.36 \| 17.79 \| 13.36 \| 17.75 \| 12.54 \| 16.61 \| 12.76 \| 16.90 \| — \| — ${}^{\textbf{130}}\textbf{Sm}$ \| 13.32 \| 17.68 \| 13.46 \| 17.64 \| 12.58 \| 16.49 \| 12.82 \| 16.76 \| — \| — ${}^{\textbf{132}}\textbf{Sm}$ \| 13.15 \| 17.59 \| 13.47 \| 17.55 \| 12.63 \| 16.46 \| 12.89 \| 16.69 \| — \| — ${}^{\textbf{134}}\textbf{Sm}$ \| 13.22 \| 17.43 \| 13.22 \| 17.60 \| 12.96 \| 16.13 \| 13.22 \| 16.35 \| — \| — ${}^{\textbf{136}}\textbf{Sm}$ \| 14.00 \| 16.84 \| 14.20 \| 16.91 \| 13.27 \| 15.87 \| 13.50 \| 16.11 \| — \| — ${}^{\textbf{138}}\textbf{Sm}$ \| 14.34 \| 16.51 \| 14.47 \| 16.66 \| 13.48 \| 15.70 \| 13.70 \| 15.95 \| — \| — ${}^{\textbf{140}}\textbf{Sm}$ \| 15.42 \| 15.73 \| 14.93 \| 16.25 \| 13.73 \| 15.50 \| 13.95 \| 15.72 \| — \| — ${}^{\textbf{142}}\textbf{Sm}$ \| 15.59 \| 15.59 \| 15.78 \| 15.78 \| 14.80 \| 14.83 \| 15.03 \| 15.04 \| — \| — ${}^{\textbf{144}}\textbf{Sm}$ \| 15.57 \| 15.57 \| 15.79 \| 15.79 \| 14.84 \| 14.84 \| 15.05 \| 15.05 \| 15.30$\pm$ 0.1 \| — ${}^{\textbf{146}}\textbf{Sm}$ \| 15.27 \| 15.45 \| 15.45 \| 15.85 \| 14.15 \| 14.95 \| 14.34 \| 15.16 \| — \| — ${}^{\textbf{148}}\textbf{Sm}$ \| 14.07 \| 15.69 \| 14.25 \| 16.15 \| 13.34 \| 15.24 \| 14.29 \| 15.47 \| 14.80$\pm$ 0.1 \| — ${}^{\textbf{150}}\textbf{Sm}$ \| 13.40 \| 15.86 \| 13.62 \| 16.31 \| 12.91 \| 15.38 \| 13.06 \| 15.59 \| 14.60$\pm$ 0.1 \| — ${}^{\textbf{152}}\textbf{Sm}$ \| 12.80 \| 16.07 \| 13.14 \| 16.65 \| 12.46 \| 15.62 \| 12.60 \| 15.82 \| 12.45$\pm$ 0.1 \| 15.85$\pm$ 0.1 ${}^{\textbf{154}}\textbf{Sm}$ \| 12.53 \| 16.06 \| 12.93 \| 16.98 \| 12.23 \| 15.70 \| 12.37 \| 15.91 \| 12.35$\pm$ 0.1 \| 16.10$\pm$ 0.1 ${}^{\textbf{156}}\textbf{Sm}$ \| 12.36 \| 15.97 \| 12.80 \| 16.55 \| 12.12 \| 15.64 \| 12.26 \| 15.82 \| — \| — ${}^{\textbf{158}}\textbf{Sm}$ \| 12.22 \| 15.84 \| 12.69 \| 16.47 \| 12.01 \| 15.60 \| 12.15 \| 15.77 \| — \| — ${}^{\textbf{160}}\textbf{Sm}$ \| 12.08 \| 15.69 \| 12.35 \| 16.37 \| 11.87 \| 15.51 \| 12.07 \| 15.69 \| — \| — ${}^{\textbf{162}}\textbf{Sm}$ \| 11.96 \| 15.53 \| 12.52 \| 16.26 \| 11.87 \| 15.40 \| 11.99 \| 15.57 \| — \| — ${}^{\textbf{164}}\textbf{Sm}$ \| 11.84 \| 15.37 \| 12.44 \| 16.14 \| 11.82 \| 15.33 \| 11.95 \| 15.50 \| — \| — Fig. 8 displays the resonance energies ($E_{1}$, $E_{2}$) evolution as function of the neutron number N from ${}^{128}\text{Sm}$ (N=66) to ${}^{164}\text{Sm}$ (N=102). We can see for all the four Skyrme forces that the resonance energy $E_{1}$ along the major axis (k=0 mode) increases with the neutron number N (i.e., mass number A) until the region around N=82 (magic number) and then trends to decreases. The opposite happens for the resonance energy $E_{2}$, i.e., it decreases with the increasing of N until N=82 , and then gradually increases. We can clearly see that the SLy6 reproduces the experimental data best among the four Skyrme forces. It was shown to provide a satisfying description of the GDR for spherical and deformed nuclei nesterenko2008 ; reinhard2008 . The SVbas functional gives somewhat high values of $E_{1}$ and $E_{2}$ among the other forces due to its large enhancement factor $\kappa$ ($\kappa$=0.4) as we discussed above. Figure 8: (Color online) The peak positions $E_{1}$ and $E_{2}$ of GDR in ${}^{128-164}\text{Sm}$ along major axis (square symbol) and minor axis (circle symbol) respectively. The experimental data are depicted by black square ($E_{1}$) and circle ($E_{2}$). In Fig. 9, we plotted the evolution of the GDR-splitting value $\Delta E=E_{2}-E_{1}$ as a function of the neutron number N. It can be easily seen for all the four Skyrme forces, that the GDR splitting $\Delta$E decreases gradually with the increase of N and then increases. It takes the minimum value $\Delta E$=0 at N=82 (magic Number ) which corresponds to the spherical nucleus ${}^{144}\text{Sm}$ and achieves a maximum for strongly deformed nuclei as ${}^{164}\text{Sm}$. Such a result confirms that the splitting of GDR is related to the deformation structure of nuclei. Figure 9: (Color online) The GDR-splitting $\Delta E$ as a function of the neutron number N for ${}^{128-164}\text{Sm}$ nuclei calculated with SLy6, SVbas, SLy5 and UNEDF1. Since the GDR-splitting is caused by the deformation, it is possible to relate the nuclear deformation parameter $\beta_{2}$ with the ratio $\Delta E/\bar{E}$, where $\bar{E}$ is the mean resonance energy. Fig. 10 displays the correlation between the quadrupole deformation $\beta_{2}$ and $\Delta E/\bar{E}$ for ${}^{128-164}\text{Sm}$ nuclei calculated with the Skyrme forces under consideration. We can see for all of the four Skyrme forces that there is an almost linear relationship between $\Delta E/\bar{E}$ and $\beta_{2}$, i.e., ${}\Delta E/\bar{E}\simeq a.\beta_{2},$ (18) where a is a parameter depending slightly on the Skyrme force. This fact confirms that the size of the GDR-splitting is proportional to the quadrupole deformation parameter $\beta_{2}$. The relation (18) was already studied in Refs. okamoto1958 ; ring2009 ; benmenana2020 . Figure 10: (Color online) The correlation between the deformation parameter $\beta_{2}$ and the ratio $\Delta E/\bar{E}$. circles denote the data in the Sm isotopes and lines are the fitting results. ## 5 Conclusion The isovector giant dipole resonance (IVGDR) has been investigated in the isotopic chain of Samarium (Sm). The study covers even-even Sm isotopes from ${}^{128}\text{Sm}$ to ${}^{164}\text{Sm}$. The investigations have been done within the framework of time dependent Hartree-Fock (TDHF) method based on the Skyrme functional. The calculations were performed with Four Skyrme forces: SLy6, SLy5, SVbas and UNEDF1. In static calculations, some properties of ground state like the deformation parameters ($\beta_{2},\gamma$) have been calculated by using SKY3D code sky3d . In dynamic calculations, the dipole moment Dm(t) and the strength of GDR are calculated and compared with the available experimental data carlos1974 . The results obtained showed that TDHF method can reproduce the shape and the peak of the GDR spectra. All four Skyrme forces generally reproduce the average position of the GDR strength with a small shift depending on the used Skyrme force. The agreement is better with the SLy6 force among these Skyrme forces. The GDR strengths in ${}^{128-142}\text{Sm}$, ${}^{146}\text{Sm}$ and ${}^{156-164}\text{Sm}$ nuclei are also predicted in this work. Finally, some properties of GDR ($\bar{E}$, $E_{1}$, $E_{2}$, $\Delta E$ ) have been calculated with the four Skyrme forces. The results with SLy6 were very close to the experimental data compared to the other forces. A correlation between the ratio $\Delta E/\bar{E}$ and the quadrupole deformation parameter $\beta_{2}$ was found. For all Skyrme forces, we have found the relation $\Delta E/\bar{E}=a.\beta_{2}+b$ with the value of b is negligible. 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PASA 2024 # Resolving VLBI correlator ambiguity in the time delay model improves precision of geodetic measurements O. Titov1 A. Melnikov2 and Y. Lopez3 1Geoscience Australia, Canberra, Australia 2Institute of Applied Astronomy of Russian Academy of Science, Saint-Petersburg, Russia 3University of Tasmania, Hobart, Australia ###### Abstract The modern Very Long Baseline Interferometry (VLBI) relativistic delay model, as documented in the IERS Conventions refers to the time epoch when the signal passes one of two stations of an interferometer baseline (selected arbitrarily from the pair of stations and called the “reference station”, or "station 1"). This model consists of the previous correlation procedure used before the year 2002. However, since 2002 a new correlation procedure that produces the VLBI group delays referring to the time epoch of signal passage at the geocenter has been used. A corresponding correction to the conventional VLBI model delay has to be introduced. However, this correction has not been thoroughly presented in peer reviewed journals, and different approaches are used at the correlators to calculate the final group delays officially published in the IVS database. This may cause an inconsistency up to 6 ps for ground-based VLBI experiments between the group delay obtained by the correlator and the geometrical model delay from the IERS Conventions used in data analysis software. Moreover, a miscalculation of the signal arrival moment to the "reference station" could result a larger modelling error (up to 50 ps). The paper presents the justification of the correction due to transition between two epochs elaborated from the Lorentz transformation, and the approach to model the uncertainty of the calculation of the signal arrival moment. The both changes are particularly essential for upcoming broadband technology geodetic VLBI observations. ###### doi: 10.1017/pas.2024.xxx ###### keywords: IVS – broadband Very Long Baseline Interferometry (VLBI) – relativity – Geodesy – Lorentz transformation – reference radio sources ## 1 INTRODUCTION The Very Long Baseline Interferometry (VLBI) technique measures the difference between the arrival times of a signal from a distant radio source at two radio telescopes (Schuh & Behrend (2012)). The signal is recorded at each radio telescope together with time marks from independent hydrogen masers. Due to separation of the radio telescopes by a few hundred or thousand kilometres, the plain wave front passes first telescope earlier then the second one. This difference in the arrival time of the signal at both radio telescopes is known as time delay, and the frequency shift due to the relative motion of the telescopes around the geocentre is known as delay rate. The time delay and delay rate are found during cross-correlation of the two independent records. There are two types of correlators (XF and FX) based on the order of the mathematical operations – cross-correlation (X) and Fourier transformation (F). Baseline-based correlators are designed as XF type correlators, and station-based correlators are FX type correlators. For the baseline-based XF-type MarkIII correlator used before 2002, the observables referred to the position of one of the two stations (station 1). For the station-based FX-type MarkIV correlator all observables for all baselines at one single multi-baseline scan are referred to the geocentre as a common reference point. As 1 ps precision is required for the time delay calculation, all first-order and second-order effects of special relativity should be taken into account. One of the goals of the International VLBI Service activities is to achieve 1-mm accuracy from the analysis of routine geodetic VLBI sessions. The accuracy of the daily scale factor improved dramatically in 2002 when the MarkIII correlator was replaced by MarkIV correlator (Titov & Krásná (2018)). However, so far this value varies about 3-4 mm despite technological developments since 2002. One possible reason for the lack of improvement in accuracy is the inconsistency between the VLBI observable group delays and the relativistic delay model developed in 1980s-90s, published in the IERS Conventions 2010 (Petit & Luzum (2010)). The transition from the MarkIII to MarkIV correlator was not followed by any changes in the IERS Conventions model that still refers to the epoch of the wavefront passage of station 1. Thus, it remains consistent with the XF-type correlators. To make the output delay of the FX- correlator consistent with the IERS Conventions 2010 model (XF-type), an additional correction needs to be applied. Unfortunately, this correction has not been officially presented in explicit form. This conversion difference was called “subtle” (Whitney (2000)); however, it reaches 20 ns, which is quite significant. Corey (2000) developed a simple geometric approach under assumption of the finiteness of the speed of light to obtain this correction, but his final equation comprised a major term only, while several minor terms were not included. In this paper, it is emphasised that the relativistic correction due to the change of the reference epoch definition should be derived from the Lorenz transformation to secure the 1-ps accuracy. Therefore, the final equation of the recommended group delay should include some minor relativistic terms due to coupling of the barycentric and geocentric velocities of the radiotelescopes to be added to the version by Corey (2000). A detailed development of the correction based on the Lorenz transformation is given in Appendix. This correction is essential from the theoretical point of view; however, its impact on the geodetic results is negligible for ground-based baselines (less than 1 mm). A more serious problem is caused by the uncertainty in the signal arrival time as calculated by the correlators, even if the problem of the epoch calculation is fixed. Within the adopted procedure, for a single multi-station scan this time is common for all stations and is usually rounded to integer number of seconds. Meanwhile, for a multi-station scan, the factual signal arrival time is individual for each station, the output group delay is converted to the time common for all stations within one scan using a reasonable polynomial approximation. Therefore, the final output delay for each baseline is referred to the common time of scan. Theoretically, this output delay should be perfectly consistent to the delay at the time of the signal arrival to the "reference station" of each baseline. However, this is not guaranteed due to the uncertainty of the reference epoch definition (discussed in the Appendix) and hidden numerical issues during the polynomial approximation. To estimate an additional correction, the standard parametrical model should be extended. For each scan we have a time of the signal arrival (common for $N$ stations) and a set of $N(N-1)/2$ time delays for all baselines. Instead of seeking for $N(N-1)/2$ errors in the delays themselves, it would be easier to treat the signal arrival time as the parameter to be updated assuming the delays are errorless. A possible approach to model this type inconsistency is presented analytically in Finkelstein, Kreinovich, & Pandey (1983). A second order term in Equation (18) may be generalised at 1-ps accuracy in the form $\displaystyle\delta{\tau_{12}}=\frac{(\boldsymbol{b}\cdot\boldsymbol{s})}{c^{2}}\frac{(({\epsilon\boldsymbol{w_{1}}+\boldsymbol{w_{2}})\cdot\boldsymbol{s})}}{1+\epsilon}$ (1) The case $\epsilon=0$ corresponds to the selection of the reference clock at station 1, and the case $\epsilon=\infty$ corresponds to the selection of the reference clock at station 2. The relativistic group delay model from the IERS Conventions has an intrinsic assumption that $\epsilon=0$. A violation of this assumption results in a small deviation of the $\epsilon$ from zero. For a small value of $\epsilon$ it could be parametrized with the partial derivative $\displaystyle\frac{\partial\delta{\tau_{12}}}{\partial\epsilon}=\frac{(\boldsymbol{b}\cdot\boldsymbol{s})(({\boldsymbol{w_{1}}-\boldsymbol{w_{2}})\cdot\boldsymbol{s})}}{c^{2}}$ (2) By its analytical representation, this new parameter $\epsilon$ should be referred to the group of parameters to model the clock instability (offset, rate, 2nd derivative, etc). In total, $(N-1)$ parameters should be added to the traditional procedure of the VLBI delay modelling. The parameters $\epsilon$ could be estimated with Equation (2) by the least squares individually for each VLBI station clock except to the clock at the network "reference station" that is assumed to be errorless. Then for two arbitrary stations (i and j) the corresponding delay is calculated as follows $\displaystyle\delta{\tau_{ij}}=(\epsilon_{i}-\epsilon_{j})\frac{(\boldsymbol{b_{ij}}\cdot\boldsymbol{s})((\boldsymbol{w_{i}}-\boldsymbol{w_{j}})\cdot\boldsymbol{s})}{c^{2}}$ (3) ## 2 DATA ANALYSIS Figure 1: Contribution of the three third order terms from Eq (11) for baselines KOKEE12M - GGAO12M (7405 km) (top) and WETTZ13S - KOKEE12M (10358 km) (bottom). The second term in Equation (18) (of the Appendix) is the diurnal variation of the Earth scale’s factor that replaces the diurnal aberration applied for the traditional astronomical observations. This is the only term due to the Earth’s rotation implemented by the FX-correlator software developers (in accordance to Corey (2000)). However, a more accurate approach based on the Lorenz transformation (12) reveals additional minor terms in Equation (18) due to coupling of the two velocities $V$ and $w_{2}$. The first term in Equation (18) is the coordinate term due to the transformation from the barycentric to the geocentric reference frame, and it could be ignored for the scope of this paper. Figure 2: Systematic group delay for baseline KOKEE12M - GGAO12M (7405 km) in accordance with (3). We used one of the recent VGOS experiments (VT9290, 17-Oct-2019) for more detailed analysis. This 24-hour experiment included five radio telescopes (WETT13S, ONSA13NE, ONSA13SW, GGAO12M and KOKEE12M) equipped with the broad band receivers. Observations were performed in four bands with dual linear polarisation (3000-3480 MHz, 5240-5740 MHz, 6360-6840 MHz and 10200-10680 MHz) (Alef et al. (2019)). Fig 1 shows the contribution of the three “missed” terms in Equation (18) to the total delay for two baselines: KOKEE12M - GGAO12M (7405.4 km) and KOKEE12M - WETTZ13S (10357.6 km). As expected, the correction on Fig 1 is essential for long baselines (up to 6 ps). Standard geodetic VLBI observations operated in two frequencies, 2.3 GHz (S-band) and 8.4 GHz (X-band), are not sensitive to the effect of the time of signal arrival. Therefore, we used the new broad band VLBI data (VGOS project) to estimate the parameter $\epsilon$. Due to the higher sample rate and the broader bandwidth of the recorded data the formal accuracy of the VGOS geodetic results is better than for standard S/X observations by an order of magnitude. The VGOS data files were processed using the OCCAM software (Titov, Tesmer, & Boehm (2004)) (version 6.3) in two modes. A first solution produces a standard set of parameters for estimating - (i) corrections to the positions of radio telescopes in the ITRF2014 frame (Altamimi et al. (2016)), (ii) Earth orientation parameters, (iii) wet troposphere delay and two gradients, (iv) three parameters to model the clock instability for each station except the reference one (clock offset, clock rate and second derivative), and (v) corrections to the ICRF3 positions of several radio sources that expose a high level of astrometric instability in the past. A second solution was used to estimate the parameter $\epsilon$ for all stations except for the reference one. Estimates of the parameter $\epsilon$ for six VGOS stations operating during 2019 are shown in Table 1. About half of the estimates are statistically significant. This means that typically, the time of the radio wave arrival to the reference station is not calculated by the correlator with sufficient accuracy. The resulting group delay calculated by Equation (3) for baseline GGAO12M - KOKEE12M at the same session (17-Oct-2019, MJD = 58744) is shown in Fig 2. We selected this baseline because for both stations in this experiment the estimates of $\epsilon$ are larger than usual ($-0.885\cdot 10^{-3}$ for GGAO12M and $0.892\cdot 10^{-3}$ for KOKEE12M). The range of the peak-to-peak variations is about 80 ps. This results in additional, hidden, source of systematic error for all other parameters. ### 2.1 ANALYSIS OF ASTROMETRIC RESULTS A comprehensive analysis of geodetic parameters is beyond of the scope of this paper. Herewith we discuss only effect of the additional parameter on the astrometric positions of two well-known reference radio sources, namely 0552+398 and 1156+295. Both sources were observed in twenty 24-h broadband VLBI experiments during 2019, with a large number of observations. As a result, their formal positional errors for both components are less than 50 $\mu$as for almost all experiments. Therefore, statistical investigation of the astrometrical results would demonstrate the advantage of the new VLBI technology application and the effect of the inclusion of the additional modelling parameter. #### 2.1.1 Radio source 0552+398 Radio source 0552+398 is one of the most actively observed radio sources by geodetic VLBI since 1979 due to its strong flux density and good astrometric stability. It was included to the list of reference radio sources of ICRF1, (Ma et al. (1998)), ICRF2 (Fey et al. (2015)) and ICRF3 (Charlot et al. (2020)). It was also treated as a ’stable’ one after independent verification by Feissel-Vernier (2003). The source 0552+398 has no apparent structure at S- and X-bands images. However, its imaging at higher frequencies (24 and 43 GHz) discloses a sub-milliarcsec jet in the east direction from the core (Charlot et al. (2010)). Recently, a second component was revealed by Bolotin et al. (2019) from the analysis of the broadband observations during the CONT17 campaign. Figure 3: Daily corrections to the ICRF3 coordinates of the radio source 0552+398 (up: right ascension bottom: declination). Black circles - standard solution, white circles - solution included the parameter $\epsilon$. While the daily estimates of the corrections to the declination component in Fig 3 vary around the original ICRF3 catalogue position within 0.6 mas, the estimates of the correction to right ascension (RA = 05h 55m 30s.80561207) (Charlot et al. (2020)) show a non-zero offset of approximately 0.2 mas. The available post-ICRF3 catalogues (e.g. the celestial reference frame solution aus2020a published by International VLBI Service (IVS)) including S/X observations during 2019-2020 do not detect any essential offset of the 0552+398 positions with respect to the ICRF3 catalogue coordinates. This potentially indicates that the jet observed at high frequencies (24 and 43 GHz) is also essential for frequencies between 2 and 11 GHz, even though it is not detected on the S/X images. We conclude that the broadband VLBI observations are more sensitive to the sub-milliarcsec structure than the traditional S/X VLBI observations as also hinted by Bolotin et al. (2019). Therefore, the positions of the reference radio sources observed by the new broadband technology are not necessary to be coincided with the S/X data positions. Figure 4: Difference between two solution corrections for the radio source 0552+398 (up: right ascension bottom: declination). #### 2.1.2 Radio source 1156+295 Radio source 1156+295 has actively monitored for the last 30 years over a wide range of frequencies. Despite its extended structure in S- and X-bands with an elongated jet in the north direction (e.g. Kellermann et al. (1998)), the radio source 1156+295 demonstrates a moderate range of astrometric instability. At the same time, no structure was reported in 24 GHz and 43 GHz (Charlot et al. (2010)). Therefore, it was selected as one of the defining reference sources in the second ICRF realization (ICRF2) (Fey et al. (2015)), although, not included to the list of the ICRF3 reference sources. Our analysis of the broadband VLBI results highlights a higher range of astrometric instability in declination than right ascension time series (Fig 5) during 2019, presumably, induced by the jet oriented to the north direction. The average declination component is shifted approximately 0.2 mas south with respect to the ICRF3 catalogue position. The difference between the two sets of daily estimates in Fig 4 and 6 does not reveal any noticeable astrometric signature due to inclusion of $\epsilon$ to the list of estimated parameters. For both sources the peak-to-peak variations do not exceed 0.25 mas in both components. Therefore, for radio sources 0552+398 and 1156+295, the inclusion of new parameter does not change the source position estimates essentially. However, for rare observed radio sources this difference may cause a substantial change in the final catalogue positions. Figure 5: Daily corrections to the ICRF3 coordinates of the radio source 1156+295 (up: right ascension bottom: declination). Black circles - standard solution, white circles - solution included the parameter $\epsilon$. Figure 6: Difference between two solution corrections for the radio source 1156+295 (up: right ascension bottom: declination). ## 3 DISCUSSION AND CONCLUSION The transition from the XF-type to FX-type correlators for processing geodetic VLBI data requires a corresponding revision of the relativistic group delay in the IERS Conventions to secure a match between the correlator output and the theoretical model. Alternatively, a special correction needs to be done at the final step of the post-correlation data processing. In Equation (18) we show in the four last terms the relativistic correction due to the time transformation from the epoch of the geocenter to the epoch of station 1. This correction is derived from the modified version of the Lorenz transformation in Equation (12). Missing of the three minor terms in Equation (18) can lead to a discrepancy of the group delay model at a level of 6 ps for long baselines. This is, in particular, pertinent for the intensive experiments for rapid estimation of Universal Time because a typical observational network consists of 2 or 3 radio telescopes separated by a long baseline (> 7000 km). We would like to recommend this equation be applied for the post-processing analysis of VLBI data at the modern FX-correlators. Another effect, though may not be directly linked to the first one, is the uncertainty of the time of signal registration for each telescope as measured by the local clock (hydrogen maser) at the reference station and extrapolated during the process of correlation. This effect also refers to the difference of the geocentric velocities of the both radio telescopes, but it could be introduced as the extension of the clock instability model. The additional parameter describes how far the actual time of the signal arrival deviates from the time presented in the VLBI data file. Our analysis of broadband VLBI data over 2019 reveals that the parameter is statistically significant in many cases (Table 1). The corresponding systematic effect is up to 100 ps in time delays, and up to 0.25 mas in estimates of daily radio source positions. It is not yet clear whether the source structure effect directly links to the problem of precisely determining the time of the signal arrival to the radio telescopes. The algorithm of the numerical calculation of the signal arrival time always relies on the assumption that the phase reference point of the target source is the same for all frequency bands. However, with a broadband receiver, we may have four different phase reference points at the four frequency bands. Therefore, four signals in each band may arrive to the receiver at four different times even from a point-like radio source. A standard calibration may not compensate this inconsistency perfectly, mostly due to the non-linear behaviour of the phase during the fringe-fitting process. In addition, an extended radio source may have four different phase reference points at four frequencies referring to the celestial reference frame. Thus, the actual differences between the signal arrival times for four frequency bands could change unpredictably. As a result, the signal arrival time presented in the broadband VLBI data file as a single value has some level of uncertainty making the additional parameter $\epsilon$ feasible for routine application using Equation (3). While it was not essential for the traditional S/X VLBI observations, the broadband VLBI observations are more accurate, and, more advanced parametrical model should be used to match these observations. ###### Acknowledgements. We are thankful to Sergei Kopeikin (University of Missouri-Columbia), Slava Turyshev (JPL), James Anderson (GFZ), and Igor Surkis (IAA RAS) for fruitful discussions on the theoretical aspects and the technical details of the correlation process. Also we thank the PASA Editor-in-Chief, and the anonymous referee for their constructive comments and suggestions which have significantly improved the clarity of the paper. This paper is published with the permission of the CEO, Geoscience Australia. We used the International VLBI Service (IVS) products available electronically at http://ivscc.bkg.bund.de/products-data/products.html. ## References * Alef et al. (2019) Alef W., Anderson J. M., Bernhart S., de Vicente P., González García J., Haas R., La Porta L., et al., 2019, evga.conf, 24, 107 * Altamimi et al. (2016) Altamimi Z., Rebischung P., Métivier L., Collilieux X., 2016, JGRB, 121, 6109. doi:10.1002/2016JB013098 * Bolotin et al. (2019) Bolotin S., Baver K., Bolotina O., Gipson J., Gordon D., Le Bail K., MacMillan D., 2019, evga.conf, 24, 224 * Charlot et al. (2010) Charlot P., Boboltz D. A., Fey A. L., Fomalont E. B., Geldzahler B. J., Gordon D., Jacobs C. S., et al., 2010, AJ, 139, 1713. doi:10.1088/0004-6256/139/5/1713 * Charlot et al. (2020) Charlot P., Jacobs C. S., Gordon D., Lambert S., de Witt A., Böhm J., Fey A. L., et al., 2020, Astron.Astroph. 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Springer, Cham, 19, arXiv:1808.06769 * Titov & Krásná (2018) Titov O., Krásná H., 2018, Astron.Astroph., 610, A36. doi:10.1051/0004-6361/201731901 * Whitney (2000) Whitney, A. R., 2000, International VLBI Service for Geodesy and Astrometry 2000 General Meeting Proceedings, 187. * Will (1971) Will C. M., 1971, ApJ, 163, 611. doi:10.1086/150804 * Will (1992) Will C. M., 1992, PhRvD, 45, 403. doi:10.1103/PhysRevD.45.403 ## Appendix A DEVELOPMENT OF THE RELATIVISTIC GROUP DELAY MODELS FOR THE EPOCH OF GEOCENTER AND FOR THE EPOCH OF THE REFERENCE STATION ### A.1 THE CONVENTIONAL GEOCENTRIC DELAY MODEL The equation for the relativistic group delay model has been developed in the 1980s-90s (e.g. Hellings (1986), Kopeikin (1990), Klioner (1991), Soffel et al. (1991) to approximate the observed VLBI data at the 1-ps level of accuracy. The conventional group delay model was finally adopted Petit & Luzum (2010) $\tau_{g}=\frac{-\frac{(\boldsymbol{b}\cdot\boldsymbol{s})}{\textrm{c}}\Big{(}1-\frac{2GM}{c^{2}R}-\frac{\|\boldsymbol{V}\|^{2}}{2\textrm{c}^{2}}-\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{\textrm{c}^{2}}\Big{)}-\frac{(\boldsymbol{b}\cdot\boldsymbol{V})}{\textrm{c}^{2}}\Big{(}1+\frac{(\boldsymbol{s}\cdot\boldsymbol{V})}{2\textrm{c}}\Big{)}}{1+\frac{(\boldsymbol{s}\cdot(\boldsymbol{V}+{w_{2}}))}{\textrm{c}}}$ (4) where $\boldsymbol{b}$ is the vector of baseline $\boldsymbol{b}=\boldsymbol{r}_{2}-\boldsymbol{r}_{1}$, $\boldsymbol{s}$ is the barycentric unit vector of the radio source, $\boldsymbol{V}$ is the barycentric velocity of the geocenter, $\boldsymbol{w_{2}}$ is the geocentric velocity of station 2, $c$ is the speed of light, $G$ is the gravitational constant, $M$ is the mass of the Sun, $R$ is the geocentric distance to the Sun, and ($\cdot$) is the dot-product operator of two vectors. The reference epoch is the UTC epoch of the wavefront passage at the reference station. In accordance with the assumption, station 1 is treated as the reference station, the geocentric velocity of station 2 is presented in (1) explicitly. A modern revision (e.g. Soffel, Kopeikin, & Han (2017)) is to add some smaller terms (less than 1 ps), but the analytical model (1) is still valid for the analysis of VLBI data. ### A.2 LORENTZ TRANSFORMATION The radio signal is received by two radio telescopes on the surface of the rotating Earth, and their coordinates are presented in the Geocentric Celestial Reference System (GCRS) comoving with the Earth. Positions of reference radio sources emitting the signals are in the Barycentric Celestial Reference System (BCRS). So, a detailed transformation of the coordinates from BCRS to GCRS is traditionally based on the metric tensor of the Solar System at the first and second post-Newtonian level (e.g. Hellings (1986), Kopeikin (1990), Klioner (1991), Soffel, Kopeikin, & Han (2017)). However, many lower order effects are not observable, therefore, a simplified approach could be developed for the relativistic model delay. The conventional Lorenz transformation is given by $\displaystyle\boldsymbol{x^{\prime}}=$ $\displaystyle\boldsymbol{x}+(\gamma-1)\frac{(\boldsymbol{V}\cdot\boldsymbol{x})\boldsymbol{V}}{\|\boldsymbol{V}\|^{2}}-\gamma\boldsymbol{V}t$ (5) $\displaystyle t^{\prime}=$ $\displaystyle\gamma\Bigg{(}t-\frac{(\boldsymbol{V}\cdot\boldsymbol{x})}{\textrm{c}^{2}}\Bigg{)}.$ where $\gamma=\bigg{(}\sqrt{1-\frac{\|\boldsymbol{V}\|^{2}}{\textrm{c}^{2}}}\bigg{)}^{-1}$ is the Lorentz "gamma-factor" (Mansouri & Sexl (1977), Will (1992)). It should be noted that this factor here is not the parameter $\gamma$ of the Post- Newtonian formalism (PPN) used in general relativity Will (1971). Transformation (5) links the geocentric reference system $S^{\prime}(x^{\prime},t^{\prime})$ that is moving with velocity $\boldsymbol{V}$ around the Solar System Barycentre (SSB) and the barycentric reference system $S(x,t)$ located at the SSB. It could be shown (Titov & Krásná (2019)) that the time delay derived from (5) may be presented in the form $\tau_{g_{0}}=\frac{-\frac{(\boldsymbol{b}\cdot\boldsymbol{s})}{\textrm{c}}\Big{(}1-\frac{\|\boldsymbol{V}\|^{2}}{2\textrm{c}^{2}}\Big{)}-\frac{(\boldsymbol{b}\cdot\boldsymbol{V})}{\textrm{c}^{2}}\Big{(}1+\frac{(\boldsymbol{s}\cdot\boldsymbol{V})}{2\textrm{c}}\Big{)}}{1+\frac{(\boldsymbol{s}\cdot\boldsymbol{V})}{\textrm{c}}}$ (6) Whether an astronomical instrument with a reference clock were placed in the Earth’s geocenter and the Solar gravitation were ignored, the equation (6) would be applied to reduction of the geodetic VLBI data. However, further complications will be discussed in two next subsections. #### A.2.1 SPACE AND TIME TRANSFORMATION INCLUDING GRAVITATIONAL POTENTIAL The relativistic model (6) does not include the term proportional to the Solar gravitational potential $\frac{2U}{c^{2}}$, where $U=\frac{GM}{R}$, and few terms with the geocentric velocity $\boldsymbol{w_{2}}$ presented in (4). Hellings (1986) showed that the former term appears due to the Solar gravitational field (in the Schwarzschild metric) at the Earth geocentre. Therefore, Hellings (1986) has developed new equations for the relationships between intervals of physical distance and time, measured in a moving reference geocentric frame, and the intervals, given in the barycentric coordinate system including the gravitational field of the Sun is given by $\displaystyle\boldsymbol{x^{\prime}}=$ $\displaystyle(1+\frac{2U}{c^{2}})\boldsymbol{x}-(1+\frac{2U}{c^{2}})(\gamma-1)\frac{(\boldsymbol{V}\cdot\boldsymbol{x})\boldsymbol{V}}{\|\boldsymbol{V}\|^{2}}-$ (7) $\displaystyle-(1-\frac{2U}{c^{2}})\gamma\boldsymbol{V}t$ $\displaystyle t^{\prime}=$ $\displaystyle\gamma\Bigg{(}(1-\frac{2U}{c^{2}})t-(1+\frac{2U}{c^{2}})\frac{(\boldsymbol{V}\cdot\boldsymbol{x})}{\textrm{c}^{2}}\Bigg{)}.$ Transformation (7) reduces to the Lorenz transformation (5) if the Solar potential $U=0$. The corresponding equation for the relativistic group delay includes the Solar gravitational potential at the geocenter of the Earth. $\tau_{g_{U}}=\frac{-\frac{(\boldsymbol{b}\cdot\boldsymbol{s})}{\textrm{c}}\Big{(}1-\frac{2U}{c^{2}}-\frac{\|\boldsymbol{V}\|^{2}}{2\textrm{c}^{2}}\Big{)}-\frac{(\boldsymbol{b}\cdot\boldsymbol{V})}{\textrm{c}^{2}}\Big{(}1+\frac{(\boldsymbol{s}\cdot\boldsymbol{V})}{2\textrm{c}}\Big{)}}{1+\frac{(\boldsymbol{s}\cdot\boldsymbol{V})}{\textrm{c}}}$ (8) Titov & Girdiuk (2015) showed that the term proportional to $\frac{2U}{c^{2}}$ in (8) could be unified with the general relativity effect of the gravitational delay. Therefore, we will not include it into further analysis; however, we discuss it here as it is a part of the conventional geometric part of the relativistic delay model (Petit & Luzum (2010)). #### A.2.2 LORENZ TRANSFORMATION REFERRING TO THE EPOCH OF FIRST STATION Physical clocks (hydrogen masers) used for VLBI observations are located at the Earth surface rather than at the geocenter. As two clocks separated by a long baseline are involved for a routine observational experiment, one of them should be selected as "reference" clock. This choice is completely arbitrary, though, once it is made, the geocentric velocity of the second ("no reference") clock appears explicitly in the analytical equations. The standard approach is to consider a difference between barycentric coordinates of two radio telescopes, $\boldsymbol{r_{1}}(t_{1})$ and $\boldsymbol{r_{2}}(t_{2})$, measured at the two epochs ${t_{1}}$ and ${t_{2}}$, to expand the vector $\boldsymbol{r_{2}}(t_{2})$ as follows $\displaystyle\boldsymbol{r_{2}}(t_{2})=\boldsymbol{r_{2}}(t_{1})+\boldsymbol{w_{2}(t_{1})}(t_{2}-t_{1}),$ (9) where $\boldsymbol{w_{2}}=\boldsymbol{w_{2}(t_{1})}$ is the geocentric velocity of the second station at epoch $t_{1}$. Denoting $\boldsymbol{B(t_{1})}$ a difference between two barycentric vectors at the same epoch $\boldsymbol{B}=\boldsymbol{B(t_{1})}=\boldsymbol{r_{2}(t_{1})}-\boldsymbol{r_{1}(t_{1})}$ one could get for the time difference $(t_{2}-t_{1})$ $\displaystyle c(t_{2}-t_{1})=-(\boldsymbol{B}\cdot\boldsymbol{s})-(\boldsymbol{w_{2}\cdot\boldsymbol{s}})(t_{2}-t_{1})$ (10) It should be noted here that $\boldsymbol{B}$ is a formal three-component vector rather than a meaningful physical value, though it links to the physical distance between two terrestrial positions of radio telescopes on the Earth at ${t_{1}}$. Eq (10) could be obtained by alternative way. Let’s introduce of a new geocentric reference frame $S^{\prime\prime}=S^{\prime\prime}(x^{\prime\prime},t^{\prime\prime})$ with the reference epoch referred to station 1 in a such way that two geocentric reference frames $S^{\prime\prime}$ and $S^{\prime}$ are linked by new transformation $\displaystyle\boldsymbol{x"}=$ $\displaystyle\boldsymbol{x^{\prime}}$ (11) $\displaystyle t"=$ $\displaystyle t^{\prime}-\frac{(\boldsymbol{w_{2}}\cdot\boldsymbol{x^{\prime}})}{\textrm{c}^{2}}$ Transformation (11) could be easily combined with the Lorentz transformation (5) $\displaystyle\boldsymbol{x"}=$ $\displaystyle\boldsymbol{x}+(\gamma-1)\frac{(\boldsymbol{V}\cdot\boldsymbol{x})\boldsymbol{V}}{\|\boldsymbol{V}\|^{2}}-\gamma\boldsymbol{V}t$ (12) $\displaystyle t"=$ $\displaystyle\gamma\Bigg{(}t-\frac{(\boldsymbol{V}\cdot\boldsymbol{x})}{\textrm{c}^{2}}\Bigg{)}-\frac{(\boldsymbol{w_{2}}\cdot\boldsymbol{x})}{\textrm{c}^{2}}-$ $\displaystyle-(\gamma-1)\frac{(\boldsymbol{V}\cdot\boldsymbol{x})(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{{c^{2}}\cdot\|\boldsymbol{V}\|^{2}}+\gamma\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})t}{\boldsymbol{c}^{2}}.$ It is obvious that the transformations (11) and (12) are pertinent only for an individual pair of two radio telescopes equipped with their own high precision clocks, one of which is a reference clock and the second clock is moving with instantaneous velocity $\boldsymbol{w_{2}}$ . The transformation (12) is fully consistent with the special relativity postulates and reflects the situation when the position of the reference clock is not at the reference frame origin (geocentre). For a classical astronomic instrument the reference frame origin and position of the reference clock are referred to the same topocentric position of the instrument on the Earth surface. In this scenario, the geocentric velocity of the instrument is simply added to the barycentric velocity in the formulae of the Lorenz transformation, i.e. the velocity $\boldsymbol{V}$ is replaced by the sum $\boldsymbol{V}+\boldsymbol{w_{2}}$ in (5) followed by a substantial change in (6). From the observational point of view this results in the appearance of the classical diurnal aberration effect. In geodetic VLBI, there is no the diurnal aberration effect at all. Instead of that, as it will be shown later, the geocentric velocity $\boldsymbol{w_{2}}$ contributes to the diurnal variation of the scale factor with magnitude up to 20 ns (or 6 meters in the linear scale) for a standard baseline of 6000 km in length and a geocentric velocity of 300 m/s. For calculating the group delay from (12), one needs to develop the corresponding velocity transformation. As both reference frames $S^{\prime\prime}$ and $S^{\prime}$ are geocentric, the time component is only changed due to transition from (5) to (12). Traditionally, authors proceed to the equation of the relativistic time delay (6) consistent with the XF-type correlator directly (e.g. Hellings 1986, Kopeikin 1990, Soffel et al 2017). Therefore, these two transformations (5) and (11) merge together and the difference between the delays (4) and (6) is lost. However, for the FX-type correlators, this procedure must be separated into two steps to provide a proper relativistic conversion between observables produced by the XF and FX correlators. To elaborate equation (4) (without the $\frac{2U}{c^{2}}$ term) from transformation (12), let’s consider the velocity transformation ${\boldsymbol{v_{x}"}}=\frac{\boldsymbol{dx"}}{dt"}$ $\displaystyle{v_{x}"}=\frac{\boldsymbol{dx}+(\gamma-1)\frac{(\boldsymbol{V}\cdot\boldsymbol{dx})\boldsymbol{V}}{\|\boldsymbol{V}\|^{2}}-\gamma\boldsymbol{V}dt}{\gamma\Bigg{(}(1+\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{\boldsymbol{c}^{2}})dt-\frac{(\boldsymbol{V}\cdot\boldsymbol{dx})}{\textrm{c}^{2}}\Bigg{)}-\frac{(\boldsymbol{w_{2}}\cdot\boldsymbol{dx})}{\textrm{c}^{2}}-\frac{(\boldsymbol{V}\cdot\boldsymbol{dx})(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{2{c}^{4}}}$ (13) or, denoting ${\boldsymbol{v_{x}}}=\frac{\boldsymbol{dx}}{dt}$ within the 1 ps level of accuracy $\displaystyle{v_{x}"}=\frac{\boldsymbol{v_{x}}+\frac{(\boldsymbol{V}\cdot\boldsymbol{v_{x}})\boldsymbol{V}}{2c^{2}}-\boldsymbol{V}}{\Bigg{(}1+\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{\boldsymbol{c}^{2}}-\frac{(\boldsymbol{V}\cdot\boldsymbol{v_{x}})}{\textrm{c}^{2}}\Bigg{)}\Bigg{(}1+\frac{\|\boldsymbol{V}\|^{2}}{2c^{2}}\Bigg{)}-\frac{(\boldsymbol{w_{2}}\cdot{\boldsymbol{v_{x}}})}{\textrm{c}^{2}}}$ (14) Now apply for a standard transition to the radio source vector $c\boldsymbol{s}=-\boldsymbol{v_{x}}$ $\displaystyle{s"}=\frac{{s}+\frac{(\boldsymbol{V}\cdot\boldsymbol{s})\boldsymbol{V}}{2c^{2}}+\frac{\boldsymbol{V}}{c}}{\Bigg{(}1+\frac{(\boldsymbol{V}\cdot\boldsymbol{s})}{c}+\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{c^{2}}\Bigg{)}\Bigg{(}1+\frac{\|\boldsymbol{V}\|^{2}}{2c^{2}}\Bigg{)}+\frac{(\boldsymbol{w_{2}}\cdot{s})}{\textrm{c}}}$ (15) and, after reduction of negligible terms, $\displaystyle{s"}=\frac{{s}+\frac{(\boldsymbol{V}\cdot\boldsymbol{s})\boldsymbol{V}}{2c^{2}}+\frac{\boldsymbol{V}}{c}}{1+\frac{(\boldsymbol{V}\cdot\boldsymbol{s})}{c}+\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{c^{2}}+\frac{\|\boldsymbol{V}\|^{2}}{2c^{2}}+\frac{(\boldsymbol{w_{2}}\cdot{s})}{\textrm{c}}}$ (16) This equation could be converted to the form consistent with the conventional group delay model at 1-ps level after inclusion of the Solar gravitation term (8) $\displaystyle{s"}=\frac{{s}\Bigg{(}1-\frac{2U}{c^{2}}-\frac{\|\boldsymbol{V}\|^{2}}{2c^{2}}-\frac{(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}{c^{2}}\Bigg{)}+\frac{\boldsymbol{V}}{c}\Bigg{(}1+\frac{(\boldsymbol{V}\cdot\boldsymbol{s})}{2c}\Bigg{)}}{1+\frac{((\boldsymbol{V}+\boldsymbol{w_{2}})\cdot\boldsymbol{s})}{c}}$ (17) Development of the time delay from (17) as $\tau=-\frac{(\boldsymbol{b}\cdot\boldsymbol{s"})}{c}$ provides the conventional group delay model (4). Now it is obvious that this model is based on the modification of the Lorentz transformation (12) in which the transformation of time is presented in a non-standard way because our reference clocks are physically located at the Earth surface rather than at the geocenter. Eq (6) misses the terms including the velocity of the second radio telescope in (4). At the 1 ps level of accuracy this difference $\delta\tau=\tau_{g}-\tau_{g_{0}}$ comprises five terms $\displaystyle\delta\tau=\frac{2(\boldsymbol{b}\cdot\boldsymbol{s}){U}}{c^{3}}+\frac{(\boldsymbol{b}\cdot\boldsymbol{s}){(\boldsymbol{w_{2}}\cdot\boldsymbol{s})}}{c^{2}}+$ (18) $\displaystyle+\frac{(\boldsymbol{b}\cdot\boldsymbol{s}){(\boldsymbol{V}\cdot\boldsymbol{w_{2}})}}{c^{3}}+\frac{(\boldsymbol{b}\cdot\boldsymbol{V}){(\boldsymbol{w_{2}}\cdot\boldsymbol{s})}}{c^{3}}-$ $\displaystyle-\frac{2(\boldsymbol{b}\cdot\boldsymbol{s}){(\boldsymbol{V}\cdot\boldsymbol{s})}{(\boldsymbol{w_{2}}\cdot\boldsymbol{s})}}{c^{3}}$ Table 1: Estimates of the parameter $\epsilon$ (units $10^{-3}$) for six VGOS stations in 2019) MJD \| GGAO12M \| KOKEE12M \| ONSA13NE \| ONSA13SW \| RAEGYEB \| WESTFORD ---\|---\|---\|---\|---\|---\|--- 58534.2499 \| $0.015\pm 0.049$ \| $0.055\pm 0.045$ \| $0.164\pm 0.053$ \| ————– \| $0.127\pm 0.051$ \| $-0.135\pm 0.047$ 58547.2499 \| $-0.186\pm 0.049$ \| $0.265\pm 0.043$ \| $0.258\pm 0.053$ \| ————– \| $0.173\pm 0.053$ \| $-0.337\pm 0.049$ 58561.2499 \| $-0.425\pm 0.081$ \| $0.051\pm 0.060$ \| $0.243\pm 0.077$ \| ————– \| $0.106\pm 0.068$ \| $-0.150\pm 0.060$ 58575.2500 \| $0.016\pm 0.054$ \| $0.078\pm 0.047$ \| $0.117\pm 0.057$ \| ————– \| $0.312\pm 0.054$ \| $-0.042\pm 0.064$ 58589.2497 \| $-0.252\pm 0.061$ \| $0.262\pm 0.050$ \| $0.233\pm 0.055$ \| ————– \| ————– \| $-0.278\pm 0.057$ 58603.2497 \| $-0.330\pm 0.062$ \| $0.290\pm 0.051$ \| $0.221\pm 0.058$ \| ————– \| ————– \| $-0.333\pm 0.061$ 58617.2497 \| $-0.109\pm 0.058$ \| $0.117\pm 0.051$ \| $0.038\pm 0.065$ \| ————– \| ————– \| $-0.097\pm 0.062$ 58632.2494 \| $-0.563\pm 0.122$ \| ————– \| $-0.202\pm 0.074$ \| ————– \| ————– \| ————– 58659.2497 \| $0.049\pm 0.076$ \| $0.058\pm 0.061$ \| $0.166\pm 0.067$ \| ————– \| ————– \| $-0.178\pm 0.069$ 58673.2499 \| $-0.085\pm 0.180$ \| ————– \| $-0.068\pm 0.107$ \| ————– \| ————– \| ————– 58687.2496 \| $0.015\pm 0.080$ \| $-0.068\pm 0.091$ \| $0.008\pm 0.077$ \| $0.001\pm 0.076$ \| ————– \| $0.024\pm 0.066$ 58701.2499 \| $-0.068\pm 0.068$ \| $0.069\pm 0.059$ \| $0.255\pm 0.074$ \| $0.223\pm 0.074$ \| $0.155\pm 0.072$ \| $-0.237\pm 0.060$ 58715.2498 \| $-0.282\pm 0.050$ \| \| $0.066\pm 0.055$ \| $0.068\pm 0.054$ \| $0.145\pm 0.059$ \| $0.025\pm 0.049$ 58732.2499 \| $-0.160\pm 0.047$ \| $0.099\pm 0.041$ \| $-0.002\pm 0.048$ \| $-0.037\pm 0.048$ \| $0.056\pm 0.050$ \| $-0.025\pm 0.039$ 58743.2499 \| $-0.206\pm 0.045$ \| $0.128\pm 0.038$ \| $0.204\pm 0.046$ \| $0.153\pm 0.045$ \| $0.311\pm 0.046$ \| $-0.124\pm 0.036$ 58757.2497 \| $-0.171\pm 0.053$ \| $0.147\pm 0.042$ \| $0.173\pm 0.048$ \| $0.226\pm 0.050$ \| ————– \| $-0.145\pm 0.044$ 58774.2493 \| $-0.885\pm 0.071$ \| $0.892\pm 0.079$ \| $-0.220\pm 0.072$ \| $-0.128\pm 0.073$ \| ————– \| ————– 58785.2496 \| $-0.214\pm 0.051$ \| $0.194\pm 0.031$ \| $0.153\pm 0.038$ \| $0.143\pm 0.038$ \| ————– \| ————– 58802.2498 \| $0.106\pm 0.062$ \| $-0.034\pm 0.035$ \| $-0.007\pm 0.047$ \| $-0.004\pm 0.048$ \| ————– \| ————– 58813.2500 \| $-0.135\pm 0.046$ \| $0.150\pm 0.036$ \| $0.206\pm 0.048$ \| $0.195\pm 0.048$ \| ————– \| $-0.189\pm 0.044$ 58827.2497 \| $-0.211\pm 0.056$ \| ————– \| $0.182\pm 0.042$ \| $0.183\pm 0.042$ \| ————– \| $-0.135\pm 0.041$ 58844.2499 \| ————– \| $0.472\pm 0.073$ \| $-0.018\pm 0.074$ \| $-0.148\pm 0.074$ \| ————– \| $-0.223\pm 0.049$ 58858.2498 \| $-0.072\pm 0.066$ \| ————– \| $-0.123\pm 0.051$ \| $-0.058\pm 0.052$ \| ————– \| $0.048\pm 0.047$
# Blocked and Hierarchical Disentangled Representation From Information Theory Perspective Ziwen Liu University of Chinese Academy of Sciences <EMAIL_ADDRESS>Mingqiang Li Information Science Academy of China Electronics Technology Group Corporation <EMAIL_ADDRESS>Congying Han University of Chinese Academy of Sciences <EMAIL_ADDRESS> ###### Abstract We propose a novel and theoretical model, blocked and hierarchical variational autoencoder (BHiVAE), to get better-disentangled representation. It is well known that information theory has an excellent explanatory meaning for the network, so we start to solve the disentanglement problem from the perspective of information theory. BHiVAE mainly comes from the information bottleneck theory and information maximization principle. Our main idea is that (1) Neurons block not only one neuron node is used to represent attribute, which can contain enough information; (2) Create a hierarchical structure with different attributes on different layers, so that we can segment the information within each layer to ensure that the final representation is disentangled. Furthermore, we present supervised and unsupervised BHiVAE, respectively, where the difference is mainly reflected in the separation of information between different blocks. In supervised BHiVAE, we utilize the label information as the standard to separate blocks. In unsupervised BHiVAE, without extra information, we use the Total Correlation (TC) measure to achieve independence, and we design a new prior distribution of the latent space to guide the representation learning. It also exhibits excellent disentanglement results in experiments and superior classification accuracy in representation learning. ## 1 Introduction ##### Disentanglement Representation Learning an interpretable and disentangled representation of data to reflect the semantic meaning is what machine learning always pursues [5, 6, 8, 27]. Disentangled representation is defined in [5] as:_a representation where a change in one dimension corresponds to a change in one factor of variation, while being relatively invariant to changes in other factors._ As far as our understanding is concerned, the fact that different dimensions do not affect each other means probabilistically independent. As popular generative models, Variational Autoencoder (VAE) [15] and Generative Adversarial Networks(GAN) [11] have been applied in disentanglement. For example, InfoGAN [8], based on the GAN model, maximizes the mutual information between the small subset of the latent variables and the observations which makes the latent variables contain more information about the real data, hence increases the interpretability of the latent representation. Based on InfoGAN, FineGAN [18, 30] creates a hierarchical architecture that assigns the background, object shape, and object appearance to different hierarchy to generate images of fine-grained object categories. And VAE model, derived from autoencoder [1] is also widely applied to representation learning, VAEs have been demonstrated their unique power to constrain representations disentanglement. For example, $\beta$-VAE [12], $\beta$-TCVAE [7], FactorVAE [14] and so on [10] are able to get more disentangled representation. ##### Information Theory Information Theory has been proposed by Shannon in 1948 [28], which came from communication research. Mutual information is the fundamental metric for measuring the relationship about information between random variables. In representation learning, it has been applied widely [3, 8, 13, 25], with graph network [26, 34], and gets some explanatory meaning on machine learning [29]. We can conclude the application as two ideas: The first one is Information Maximization Principle(InfoMax) [4, 19], which enforces representation to preserve more information about the input data through the transformers (CNN, GNN); some works [8, 13, 35] regularize their original model with InfoMax term to get more informative and interpretable model. The other one is the Information Bottleneck(IB) theory [29, 32, 33]. It analyzes the process of information transmission and the loss through the networks. IB theory considers the network process as a Markov chain and uses the Data Processing Inequality (DPI) [9] to explain the variation of information in deep networks. In 2015, Variational Information Bottleneck (VIB) method [2] offers a variational form of supervised IB theory. Also, IB theory has been revealed a unique ability [36] to explain how and why VAEs models design this architecture. With this knowledge of disentanglement and information, we initiate our model, blocked and hierarchical variational autoencoder (BHiVAE), completely from information theory perspective to get better interpretability and controllability. In BHiVAE, because of the neural network’s different ability to extract features with different net depth, we locate data factors into different layers. Furthermore, the weak expressiveness of single-neuron pushes us to use neuron blocks to represent features. We also discuss the supervised and unsupervised version model. In the supervised model, we utilize the label to separate the representation from feature information. In the unsupervised model, we give out a unique prior distribution to better meet our model and use additional discriminators to split information. Of course we give enough experiments in MNIST [17], CelebA [20] and dSprite [23] datasets to show the great performance in disentanglement. In summary, our work mainly makes the following contributions: * • We approach the disentanglement problem for the first time entirely from an information theory perspective. Most previous works on disentanglement have been based on existing models and modified to fit the framework for solving entanglement problems. * • We present Blocked and Hierarchical Variational Autoencoder (BHiVAE) in both supervised and unsupervised cases. In the supervised case, we utilize the known feature information to guide the representation learning in each hierarchy; in the unsupervised case, we propose a novel distribution-based method to meet our neural block set. * • We perform experiments thoroughly on several public datasets, MNIST, dSprites and CelebA, comparing with VAE, $\beta$-VAE, FactorVAE, $\beta$-TCVAE, and Guided-VAE in several classic metrics. From the results, our method BHiVAE shows an excellent performance considering all the indicators together. ## 2 Related Work In order to get disentangled representation, some previous work has made a significant contribution to it. Based on VAE, $\beta$-VAE [12] adds a coefficient weight to the KL-divergence term of the VAE loss and get a more disentangled representation. Mostly there is a significant advantage in that it trains more stably than InfoGAN. However, $\beta$-VAE sacrifices the reconstruction result at the same time. $\beta$-TCVAE [7] and FactorVAE [14] explored this issue in more detail and found TC term is the immediate causes to promote disentanglement. Guided VAE [10] also gives out a model using different strategies in supervised and unsupervised situations to get disentanglement representation. It uses additional discriminator to guide the representation learning and learn the knowledge about latent geometric transformation and principal components. This idea of using different methods with different supervised information inspires us. FineGAN [30] based on InfoGAN, generates the background, object shape, and object appearance images respectively in different hierarchies, then combines these three images into true image. In FineGAN, what helps the disentanglement is the mutual information between the latent codes and each factor. And MixNMatch [18], developed from FineGAN, becomes a conditional generative model that learns disentangled representation and encodes different features from real image and then uses additional discriminators to match the representation to the prior distribution given by FineGAN model. Previous works have made simple corrections to $\beta$-VAE or GAN model, adding some useful terms for solving disentanglement. In our work, we fully consider the disentanglement problem from information theory and then establish the BHiVAE model. Information theory and optimal coding theory [9, 36] have shown that longer code can express more information. So in our model, instead of using only one dimension node to represent a ground-truth factor as in previous work, we choose multiple neural nodes to do so. In the meantime, different ground-truth factors of data contain different levels of information, and the depth of the neural network affects the depth of information extracted, so a hierarchical architecture is used in our model for extracting different factor features at different layers. Therefore, in order to satisfy the requirement of disentanglement representation, i.e., the irrelevance between representation neural blocks, We only need to minimize the mutual information between blocks of the same layer due to characteristics of hierarchical architecture. (a) Encoder part (b) Decoder part Figure 1: Architecture of Hierarchical VAE model: Encoder part in the left- side and decoder in the right-side. ## 3 Proposed Method We propose our model motivated by IB theory and VAEs, like $\beta$-VAE, Factor-VAE, $\beta$-TCVAE, Guided-VAE, and FineGAN. Therefore, in this section, we first introduce the IB theory and VAEs models, and then we present our detailed model architecture and discuss supervised and unsupervised BHiVAE methods. ### 3.1 Information Theory and VAEs IB theory aims to learn a representation $Z$ that maximizes the compression of informaiton in real data $X$ while maximizing the expression of target $Y$. So we can describe it as: $\displaystyle\min I(X;Z)-\beta I(Z;Y)$ (1) the target $Y$ is the attribute information under supervision, and is equal to $X$ under unsupervision [36]. In the case of supervised IB theory [2], we can get the upper bound: $\displaystyle I_{\phi}(X;Z)-\beta I_{\theta}(Z;Y)\leq$ $\displaystyle\mathbb{E}_{p_{D}(x)}[D_{KL}(q_{\phi}(z\|x)\\|p(z))]$ $\displaystyle-\beta\mathbb{E}_{p(x,y)}[q_{\phi}(z\|x)\log p_{\theta}(y\|z)]$ (2) The first term represents the KL divergence between the posterior $q_{\phi}(z\|x)$ and the prior distribution $p(z)$; and absolutely, the second term equals cross-entropy loss of label prediction. And in the case of unsupevised IB theory, the we can rewrite the objective Eq. (1) as: $\displaystyle\min I_{\phi}(X;Z)-\beta I_{\theta}(Z;X)$ (3) Unsupervised IB theory seems like generalization of VAEs model, with an encoder to learn representation and a decoder to reconstruct. $\beta$-VAE [12] is actually the upper bound of it: $\displaystyle\mathcal{L}_{\beta-VAE}=$ $\displaystyle\mathbb{E}_{p(x)}[D_{KL}(q_{\phi}(z\|x)\\|p(z))$ $\displaystyle-\beta\mathbb{E}_{q_{\phi}(z\|x)}[\log(p_{\theta}(x\|z))]]$ (4) FactorVAE [14] and $\beta$-TCVAE [7] just add more weight on the TC term $\mathbb{E}_{q(z)}[\log\frac{q(z)}{\tilde{q}(z)}]$, which express the dependence across dimensions of variable in information theory, where $\tilde{q}(z)=\prod_{i=1}^{n}q(z_{i})$. We build our BHiVAE model upon above works and models. We focus on information transmission and loss through the whole network, and then achieve it through different methods. ### 3.2 BHiVAE Now let us present our detailed model architecture. As shown in Fig 1, feed data $X$ into the encoder (parameterized as $\phi$), and in the first layer, we get the latent representation $z^{1}$, be divided into two parts $s^{1}$ and $h^{1}$. The part $s^{1}$ is the final representation part, which corresponds to feature $y^{1}$, and $h^{1}$ is the input of next layer’s encoder to get latent representation $z^{2}$. Then through three similar network processes, we can get three representation parts $s^{1},s^{2},s^{3}$, which are disentangled, and get the part $c^{3}$ in the last layer, that contains information other than the above attributes of the data. All of them make up the whole representation $z=(s^{1};s^{2};s^{3};c^{3})$. The representation of each part is then mapped to the same space by a different decoder (all parameterized as $\theta$) and finally concatenated together to reconstruct the raw data, which is shown in Fig 1(b). For the problem we discussed, we need to get the final disentangled representation $z$, i.e., we need the independence between each representation part $s^{1},s^{2},s^{3}$, and $c^{3}$. (a) Unsupervised (b) Supervised Figure 2: Different methods for constraining information segmentation between $s^{i}$ and $z^{i}$. Then we can separate the whole problem into two sub-problem in $i$-th layer, so the input is $h^{i-1}$(where $h^{0}=x$): * (1) Information flow $h^{i-1}\rightarrow s^{i}\rightarrow y^{i}$: Encode the upper layer’s output $h^{i-1}$ to representation $z^{i}$, with one part $s^{i}$ containing sufficient information about one feature factor $y^{i}$; * (2) Information separation of $s^{i}$ and $h^{i}$: Eliminate the information about $s^{i}$ in $h^{i}$ while requiring $s^{i}$ only to contain label $y^{i}$ information. The first subproblem can be regarded as IB problem, the goal is to learn a representation of $s^{i}$, i.e. maximally expressive about feature $y^{i}$ while minimally informative about input data $h^{i-1}$. So it can described as: $\displaystyle\min I(h^{i-1};s^{i})-\beta I(s^{i};y^{i})$ (5) To satisfy the second subproblem is a complex issue, and it requires different methods to achieve it with different known conditions. So we will introduce these in follow conditions in detail. In summary, our representation is designed to enhance the internal correlation of each block while reducing the relationships between them to achieve the desired disentanglement goal. #### 3.2.1 Supervised BHiVAE In supervised case, we denote the input of $i$-th layer as $h^{i-1}$ ($h^{0}=x$). Given the $i$-th layer label $y^{i}$, we require the representation part $s^{i}$ to predict the feature correctly while being as compressed as possible. So the objective in $i$-th ($i=1,2,3$) layer can be described as with information measure: $\displaystyle\mathcal{L}_{sup}^{class}(i)=I(h^{i-1};s^{i})-\beta I(s^{i};y^{i})$ (6) We can get a upper bound of it: $\displaystyle\mathcal{L}_{sup}^{class}(i)$ $\displaystyle=I(h^{i-1};s^{i})-\beta I(s^{i};y^{i})$ $\displaystyle\leq\mathbb{E}_{p(h^{i-1})}[D_{KL}(q_{\phi}(s^{i}\|h^{i-1})\\|p(s))]$ $\displaystyle-\beta\mathbb{E}_{p(z^{i-1},y^{i})}[\mathbb{E}_{q_{\phi}(s^{i}\|h^{i-1})}[\log p_{\theta}(y^{i}\|s^{i})]]$ $\displaystyle\triangleq\mathcal{L}_{sup}^{class_{up}}(i)$ (7) So we need one more classifier $\mathcal{C}_{i}$ in Fig 2(b) to predict $y^{i}$ with $s^{i}$. For the second requirement, since $s^{i}$ is completely informative about $y^{i}$ which constrained in first subproblem, the elimination of information about $y^{i}$ is required for $h^{i}$: $\displaystyle\mathcal{L}_{info}^{sup}(i)$ $\displaystyle=I(h^{i},y^{i})$ $\displaystyle=H(y^{i})-H(y^{i}\|h^{i})$ (8) $H(y^{i})$ is a constant, so minimizing $\mathcal{L}_{info}^{sup}(i)$ is equal to minimize: $\displaystyle\mathcal{L}_{info}^{sup_{e}}(i)=-H(y^{i}\|h^{i})$ (9) This is like a principle of maximum entropy, just requiring $h^{i}$ can’t predict the factor feature $y^{i}$ at all, i.e. the probability predicted by $h^{i}$ of each category is $\frac{1}{n_{i}}$ ($n_{i}$ denotes the number of $i$-th feature categories). And $h^{i}$ shares the classifier $\mathcal{C}_{i}$ with $s^{i}$ as Fig 2(b) shows. So in our supervised model, we can get the total objective as: $\displaystyle\min\\{\mathcal{L}^{sup}$ $\displaystyle=\sum_{i=1}^{n}\mathcal{L}_{class}^{sup}(i)+\gamma\mathcal{L}_{info}^{sup_{e}}(i)\\}$ (10) where $\beta$ and $\gamma$ in the objective are hyper-parameter. The objective (10) satisfies two requirement we need, and deal with the second subproblem with a novel approach. #### 3.2.2 Unsupervised BHiVAE In the unsupervised case, we know nothing about the data source, so we can only use reconstruction to constrain the representation. However, only reconstruction is not enough for disentanglement problem [21], so we try to use an unique representation prior distribution to guide the representation learning. We know that all disentanglement models of the VAE series match the posterior distribution $q_{\phi}(z\|x)$ to standard normal distribution prior $\mathcal{N}(0,I)$, and they can get disentanglement representation in each dimension because of the independence across $\mathcal{N}(0,I)$. For meeting our neural block representation set, we set the prior distribution $p(z)$ as $\mathcal{N}(0,\Sigma)$, where $\Sigma$ is a block diagonal symmetric matrix. Of course, the dimension of each block corresponds to the segmentation of each hidden layer. In the unsupervised model, the target is reconstruction, so we can decompose Eq. (5) as: $\displaystyle\min$ $\displaystyle I(h^{i-1};s^{i})-\beta I(s^{i};x)$ $\displaystyle\leq\mathbb{E}_{p(h^{i-1})}[D_{KL}(q(z^{i}\|h^{i-1})\\|p(z))]$ (11) $\displaystyle-D_{KL}(q_{\phi}(z^{i})\\|p(z))$ (12) $\displaystyle-\beta[\mathbb{E}_{p(h^{i-1},y^{i})}[\mathbb{E}_{q_{\phi}(s^{i}\|h^{i-1})}[\log p_{\theta}(x\|s^{i})]]$ (13) $\displaystyle- D_{KL}(q_{\phi}(z^{i-1})\\|p_{D}(x))]$ (14) The first two terms are meant to constrain the capacity of representation $z^{i}$, and the last two reinforce the reconstruction. VAEs model use (11) and (13) to achieve, and adversarial autoencoder [22] use the KL divergence (12) between the posterior distribution $q_{\phi}(z^{i})$ and prior $p(z)$ to constrain the capacity of representation and get better representation. In our model, we also minimize the KL divergence between the posterior distribution $q_{\phi}(z^{i})$ and prior $\mathcal{N}(0,\Sigma)$, i.e., $D_{KL}(q_{\phi}(z^{i})\\|\mathcal{N}(0,\Sigma))\rightarrow 0$. And we choose the determinstic encoder, so we get the objective: $\displaystyle\mathcal{L}_{recon}^{uns}=$ $\displaystyle D_{KL}(q_{\phi}(z^{i})\\|\mathcal{N}(0,\Sigma))$ $\displaystyle-\beta\mathbb{E}_{p(h^{i-1})}[\mathbb{E}_{q_{\phi}(s^{i}\|h^{i-1})}[\log p_{\theta}(x\|s^{i})]]$ (15) We use a discriminator at the top of Fig 2(a) to estimate and optimize $D_{KL}(p_{\phi}(h^{i})\\|\mathcal{N}(0,\Sigma))$. Unlike the supervised case, we adopt a different method to satisfy the information separation requirement. When $s^{i}$ and $h^{i}$ are independent in probability, the mutual information between them comes to zero, i.e., no shared information between $s^{i}$ and $h^{i}$. Here we apply an alternative definition of mutual information, Total Correlation (TC) penalty [14, 37], which is a popular measure of dependence for multiple random variables. $KL(q(z)\\|q(\tilde{z}))$ where $q(\tilde{z})=\prod^{d}_{j=1}q(z_{j})$ is typical TC form, and in our case, we use the form $KL(p(z^{i})\\|p(h^{i})p(s^{i}))=I(h^{i};s^{i})$. So we can get the information separation objective as: $\displaystyle\mathcal{L}_{info}^{uns}(i)$ $\displaystyle=I(h^{i};s^{i})$ (16) $\displaystyle=KL(p(z^{i})\\|p(h^{i})p(s^{i}))$ (17) In practice, $KL$ term is intractable to compute. The multiplication of marginal distributions $p(h^{i})p(s^{i})$ is not analytically computable, so we take a sampling approach to simulate it. After getting the a batch of representations $\\{z^{i}_{j}=(s^{i}_{j};h^{i}_{j})\\}_{j=1}^{N}$ in $i$-th layer, we randomly permute across the batch for $\\{s^{i}_{j}\\}_{j=1}^{N}$ and $\\{h^{i}_{j}\\}_{j=1}^{N}$ to generate sample batch under distribution $p(z^{i})p(s^{i})$. But direct estimating density ratio $\frac{p(z^{i})}{p(h^{i})p(s^{i})}$ is often impossible. Thus, with random samples, we conduct a density ratio method [24, 31]: use an additional classifier $D(x)$ that distinguishes between samples from the two distributions, at the bottom of Fig 2(a): $\displaystyle\mathcal{L}_{info}^{uns}(i)$ $\displaystyle=KL(p(z^{i})\\|p(h^{i})p(s^{i}))$ $\displaystyle=TC(z^{i})$ $\displaystyle=\mathbb{E}_{q(z)}[\log\frac{p(z^{i})}{p(h^{i})p(s^{i})}]$ $\displaystyle\approx\mathbb{E}_{q(z)}[\log\frac{D(z^{i})}{1-D(z^{i})}]$ (18) In summary, the total objective under unsupervision is: $\displaystyle\max\\{\mathcal{L}^{unsup}=\sum_{i=1}^{n}(\mathcal{L}_{recon}^{sup}+\gamma\mathcal{L}_{info}^{sup}(i))\\}$ (19) ## 4 Experiments In this section, we present our results in quantitative and qualitative experiments. We also perform experiments comparing with $\beta$-VAE, FactorVAE, and $\beta$-TCVAE in several classic metrics. Here are datasets used in our experiments: MNIST [17]: handwriting digital $(28\times 28\times 1)$ images with 60000 train samples and 10000 test samples; dSprites [23]: 737280 2D shapes $(64\times 64\times 1)$ images procedurally generated from 6 ground truth independent latent factors: shapes (heart,oval and square), x-postion (32 values), y-position (32 values), scale (6 values) and rotation (40 values); CelebA (cropped version) [20]: 202599 celebrity face $(64\times 64\times 3)$ images with 5 landmark locations, 40 binary attributes annotations. In the following, we perform several qualitative and quantitative experiments on these datasets and show some results comparison in both unsupervised and supervised cases. We demonstrated the ability of our model to disentangle in the unsupervised case. Besides, we also show the representation learned in the supervised case. (a) Layer1 with KL=0.61 (b) Layer2 with KL=0.49 (c) Layer3 with KL=0.11 Figure 3: Scatter distribution VS. Prior distribution: Scatter plot of three layers representation $\\{s^{i}\\}_{i=1}^{3}$; and (C) visualizes the known category information with different colors. (a) $\beta$-VAE (b) FactorVAE (c) Guided-VAE (d) BHiVAE Figure 4: Traversal images on MNIST: In (a), (b) and (c), the images in $i$-th row are generated by changing $z^{i}$ from -3 to 3; and we change $\\{s^{1},s^{2},s^{3},c^{3}\\}$ from (-3,-3) to (3,3), then generate the images in each row. ### 4.1 Training Details When training BHiVAE model, we need the encoder and decoder (Fig 1) both in supervised and unsupervised cases. On the CelabA dataset, we build our network with both a convolutional layer and a fully connected layer. On the MNIST and dSprites datasets, the datasets are both $64\times 64$ binary images, so we design our network to consist entirely of fully connected layers. In evaluating the experimental results, we use the Z-differ [12], SAP [16], and MIG [7] metrics to measure the quality of the disentangled representation, and observe the images generated by the traversal representation. Moreover, we use some pre-trained classifiers on attribute features to analyze the model according to the classification accuracy. ### 4.2 Unsupervised BHiVAE In the unsupervised case, as introduced in the previous section, the most significant novel idea is we use a different prior $\mathcal{N}(0,\Sigma)$ to guide the representation learning. Additionally, we need another one to estimate the KL divergence (18). Therefore, two extra discriminators are needed for BHiVAE in Fig 2(a). Actually, because we aim to get $D_{KL}(q_{\phi}(z^{i})\\|p(z))=0$, the latent representation $\\{z_{j}^{i}\\}_{j=1}^{N}$ can be considered as generated from true distribution, while prior and permuted ’presentations’ $\\{z^{i-perm}_{j}\\}_{j=1}^{N}$ can both be considered as false. Therefore, we can simplify the network to contain only one discriminator to score these three distributions. We want to reinforce the relationship within $s^{i}$ to retain the information and then decrease the dependency between $s^{i}$ and $h^{i}$ to separate information, so in our unsupervised experiments, we use this prior $\mathcal{N}(0,\Sigma)$, where $\displaystyle\scriptsize\Sigma=\begin{bmatrix}1&0.5&0&\cdots&0\\\ 0.5&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&1\end{bmatrix}$ First, we use some experiments to demonstrate the feasibility and validity of this prior setting. We train the model on the dSprites dataset first, with setting the dimension of representation $z$ to 14 ($d(z)=14$), where $d(s^{i})=2,i=1,2,3$ and $d(c^{3})=8$. Then we get a representation in each layer of the 1000 test images, while the three subfigures in Fig 3 shows a scatter plot of each layer representation respectively, and the curves in these figures both are the contour of the block target distribution $p(s)\sim\mathcal{N}(0,\begin{bmatrix}1&0.5\\\ 0.5&1\end{bmatrix})$. And it is shown in Fig 3 that in the first and second layer, the distribution of $s^{1}$ and $s^{2}$ do not sufficiently match the prior $p(s)$, but as the layer forward, the KL divergence between $q_{\phi}(s^{i})$ and $p(s)$ keep decresing, and the scatter plot of $s^{i}$ fits the prior distribution more closely. In the model, we train the encoder globally, so the front layer’s representation learning can be influenced by the change of deeper representation and then yields larger KL divergence than the next layer. Even more surprisingly, in Fig 3(c), we find that in the third layer, visualizing the ’Shape’ attribute of dSprites dataset, there is an apparent clustering effect (the different colors denote different categories). This result proves our hypothesis about the deep network’s ability: the deeper network is, the more detailed information it extracts. And it almost matches the prior perfectly. Fig 3(c) also gives us a better traversal way. In previous works, because only one dimension represents the attribute, they can simply change the representation from $a$ to $b$ ($a$ and $b$ both are constant). However, this does not fit our model, so the direction of the category transformation in Fig 3(c) inspires us to traverse the data along the diagonal line ($y=x$). Our block prior $p(s)$ also supports that (because the prior distribution’s major axis is the diagonal line too). We perform several experiments under above architecture setting and traversal way to show the disentanglement quality on MNIST datasets. The disentanglement quantitative results of comparing with $\beta$-VAE [12], FactorVAE [14] and Guided-VAE [10] are presented in Fig 4. Here, considering the dimension of the representation and the number of parameters, other works’ bottleneck size is set to 12, i.e., $d(z)=12$. This setting helps reduce the impact of differences in complexity between model frameworks. However, for a better comparison, we only select seven dimensions that change more regularly. In our model, we change the three-block representation $\\{s^{i}\\}_{i=1}^{3}$ and then the rest representation $c^{3}$ changes according to two dimensions as a whole, i.e., $c^{3}=(c^{3}_{1:2},c^{3}_{3:4},c^{3}_{5:6},c^{3}_{7:8})$. And Fig 4 shows that $\beta$-VAE hardly ever gets a worthwhile disentangled representation, but FactorVAE appears to attribute change as representation varies. Moreover, Fig 4(c) and Fig 4(d) both show great disentangled images, with $h_{1}$ changing in Guided-VAE and $s^{1}$ changing in BHiVAE, the handwriting is getting thicker, and $h_{3},s^{2}$ control the angle of inclination. These all demonstrate the model capabilities of our model. \| Z-diff $\uparrow$ \| SAP $\uparrow$ \| MIG $\uparrow$ ---\|---\|---\|--- VAE[15] \| 67.1 \| 0.14 \| 0.23 $\beta$-VAE[12]($\beta$=6) \| 97.3 \| 0.17 \| 0.41 FactorVAE[14]($\gamma$=7) \| 98.4 \| 0.19 \| 0.44 $\beta$-TCVAE[7]($\alpha$=1,$\beta$=8,$\gamma$=2) \| 96.5 \| 0.41 \| 0.49 Guided-VAE[10] \| 99.2 \| 0.4320 \| 0.57 BHiVAE(Ours)($\beta$=10, $\gamma$=3) \| 99.0 \| 0.4312 \| 0.61 Table 1: Disentanglement Scores: Z-diff score, SAP score, MIG score on the dSprites dataset in the unsupervised case. The bold note the best results and blue is the second best result. We then progress to the traversal experiments on the dSprites dataset. This dataset has clear attributes distinctions, and these allow us to better observe the disentangled representation. In these experiments, BHiVAE learns a 10-dimensional representation $z=(s^{1},s^{2},s^{3},c^{3}_{1:2},c^{3}_{3,4})$ and 8-dimensional $z=(z_{1},z_{2},\dots,z_{8})$ in other works. We present the experiments results in Fig 5 of reconstruction and traversal results. The first and second rows in four figure represent original and reconstruction images respectively. In Fig 5(d), it shows that our first three variables $s^{1},s^{2},s^{3}$ have learned the attribute characteristics (Scale, Orientation, and Position) of the data. Moreover, we perform two quantitive experiments comparing with previous works and present our results in Table 1 and Table 2. The experiments are all based on the same experiment setting in Fig 4. (a) $\beta$-VAE (b) FactorVAE (c) GuidedVAE (d) BHiVAE(Ours) Figure 5: Traversal images on dsprites: Images in first and second row of each figure are original and reconstruction images respectively. And others rows correspond the traversal images. First, we compare BHiVAE with previous models with Z-differ Score [12], SAP Score [16] and MIG Score [7] and present the results in Table 1. It is clear that our model BHiVAE is at the top and that the MIG metric is better than other popular models. The high value of the Z-diff score indicates that learned disentangled representation has less variance on the attributes of generated data as corresponding dimension changing, while SAP measures the degree of coupling between data factors and representations. Additionally MIG metric uses mutual information to measure the correlation between the data factor and learned disentangled representation, and our work is just modeled from the perspective of mutual information, which makes us performs best on the MIG score. Not only that, but we also perform transferability experiments by conducting classification tasks on the generated representation. Here we set the representation dimensions to be the same in all models. First, we have learned a pre-trained model to obtain the representation $z$ and a pre-trained classifier to predict MNIST image label from representation. We compare the classification accuracy in Table 2 with different dimension settings. \| $d_{z}=10\uparrow$ \| $d_{z}=16\uparrow$ \| $d_{z}=32\uparrow$ ---\|---\|---\|--- VAE[15] \| 97.21%$\pm$0.42 \| 96.62% $\pm$ 0.51 \| 96.41%$\pm$0.22 $\beta$-VAE[12]($\beta$=6) \| 94.32% $\pm$0.48 \| 95.22%$\pm$0.36 \| 94.78%$\pm$0.53 FactorVAE[14]($\gamma$=7) \| 93.7%$\pm$0.07 \| 94.62%$\pm$0.12 \| 93.69%$\pm$ 0.26 $\beta$-TCVAE[7]($\alpha$=1,$\beta$=8,$\gamma$=2) \| 98.4%$\pm$0.04 \| 98.6%$\pm$0.05 \| 98.9%$\pm$0.11 Guided-VAE[10] \| 98.2%$\pm$0.08 \| 98.2%$\pm$0.07 \| 98.40% $\pm$0.08 BHiVAE(Ours)($\beta$=10, $\gamma$=3) \| 98.2%$\pm$0.09 \| 98.7%$\pm$0.10 \| 99.0%$\pm$0.05 Table 2: Accuracy of representation under unsupervised case: The bold note the best results and blue is the second best result. Our model appears not higher accuracy than FactorVAE and Guided-VAE in the case of $d_{z}=10$. That block representation setting causes a small number of factors it learns. However, as $d(z)$ is increased, our representation can learn more attribute factors of data, and then the classification accuracy can also be improved. ### 4.3 Supervised BHiVAE In the supervised case, we still did the qualitative and quantitative experiments to evaluate our model. The same as the unsupervised case, overall autoencoder is required, and then we need a classifier to satisfy the segmentation of information at each level, as shown in Fig 2(b). And we set the dimension of representation $z$ as 12 ($d(z)=12,d(c^{3})=6$, and $d(s^{i})=2,i=1,2,3$). Figure 6: Traversal Results Comparison on CelebA: The first column is the traversal change of Gender, the second column is the change of Black Hair, the first row is from Guided-VAE [10], the second row is ours, following the procedure of Guided-VAE. We first perform several experiments comparing with Guided-VAE [10] in two attributes(Gender and Black Hair) and present the results in Fig 6. When changing each attribute $s^{i}\in\\{s^{1},s^{2},s^{3}\\}$, we keep other attributes representations and content representation $c^{3}$ unchanged. We use the third layer representation $s^{3}$ to control gender attribute, while the first layers correspond to the black hair and bale, respectively. In the supervised case, compared to Guided-VAE, we use multiple dimensions to control an attribute while Guided-VAE uses only one dimension, which may lead to insufficient information to control the traversal results. And Fig 6 shows that our model has a broader range of control over attributes, especially reflected in the range of hair from pure white to pure black. Besides, our quantitative experiment is to first pre-train the BHiVAE model and three attribute classifiers of the representation and then get the representationS of the training set, traversing the three representation blocks a,b,c from $(-3,3)$ to $(3,3)$ along with the diagonal($y=x$). Fig 7 shows that all three attributes have a transformation threshold in the corresponding representation blocks. Figure 7: The classifier result used to determine if the property is available. We traverse the Black Hair ($s^{1}$), Bale ($s^{2}$) and Gender ($s^{3}$) attributes. Figure 8: Comparison of the accuracy of Block and Single setting model for Black Hair attribute. ### 4.4 Block Nodes VS. Single Node In the previous experiments, we are all making judgments about how well the representation is disentangled and did not prove that the block setting is beneficial, so we set up the following comparison experiments for this problem. For the comparison experiment here, we set the dimension of the model representation $z$ to 10, 16, and 32. Then in the comparison experiment, we just changed the dimension of representation $s^{1}$ (black hair) in the first layer to 1, and therefore the dimension of $c^{3}$ is changed to 5, 11, and 27 accordingly. First we pre-train these two models under the same conditions and learn a binary classifier that predicts the black hair attributes with representation $z$. It is shown in Fig 8 that Block is better than Single in every dimension setting, and the accuracy of them has increased with increasing representation dimension. It could be that there is still some information about black hair in other representation parts of the model, and then the increasing dimension will allow more information about black hair to be preserved, getting better prediction accuracy. ## 5 Conclusion and Future Work We propose a new model, blocked and hierarchical variational autoencoder, for thinking about and solving disentanglement problems entirely through the perspective of information theory. We innovatively propose a blocked disentangled representation and hierarchical architecture. 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# Weak deflection angle by electrically and magnetically charged black holes from nonlinear electrodynamics Qi-Ming<EMAIL_ADDRESS>Li<EMAIL_ADDRESS>and Yu- Xiao<EMAIL_ADDRESS>corresponding author 1Institute of Physics, Shannxi University of Technology, Hanzhong 723000, China 2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou University, Lanzhou 730000, China 3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China ###### Abstract Nonlinear electrodynamic (NLED) theories are well-motivated for their extensions to classical electrodynamics in the strong field regime, and have been extensively investigated in seeking for regular black hole solutions. In this paper, we focus on two spherically symmetric and static black hole solutions based on two types of NLED models: the Euler-Heisenberg NLED model and the Bronnikov NLED model, and calculate the weak deflection angle of light by these two black holes with the help of the Gauss-Bonnet theorem. We investigate the effects of the one-loop corrections to quantum electrodynamics on the deflection angle and analyse the behavior of the deflection angle by a regular magnetically charged black hole. It is found that the weak deflection angle of the electrically charged Einstein-Euler-Heisenberg black hole increases with the one-loop corrections and the regular magnetically charged black hole based on the Bronnikov NLED model has a smaller deflection angle than the singular one. Besides, we also calculate the deflection angle of light by the geodesic method for verification. In addition, we discuss the effects of a cold non-magnetized plasma on the deflection angle and find that the deflection angle increases with the plasma parameter. ###### pacs: 95.30.Sf, 98.62.Sb, 97.60.Lf ## I Introduction To solve the divergence of the self energy of a point-like charge, Born and Infeld generalized Maxwell’s theory and proposed the Born-Infeld electrodynamics Born1934 . However, this theory did not attract much attention until its reemergence at the low energy scale of some string theories. Afterwards, Heisenberg and Euler introduced a new extension to the standard electromagnetic theory (known as Euler-Heisenberg (EH) electrodynamics) Heisenberg1936 , which takes into account the one-loop corrections to quantum electrodynamics (QED) and can explain the vacuum polarization in QED. As extensions to Born-Infeld and EH electrodynamics, nonlinear electrodynamic (NLED) models have been studied in different aspects since then. For instance, NLED models can be used to explain the inflation of the universe in the early times Salcedo2000 ; Camara2004 . Some types of NLED models can depict the accelerated expansion of the universe instead of dark energy and remove the Big Bang singularity Elizalde2003 ; Novello2004 ; Vollick2008 ; Kruglov2015 . In addition, in recent years, NLED models attract much more attention for their ability in seeking regular black hole solutions. The first regular black hole model was proposed by Bardeen Bardeen1968 . However, this regular black hole was obtained without a specified source associated to its line element. Remarkably, in 1998, Ay$\acute{\text{o}}$n-Beato and Garc$\acute{\text{{\i}}}$a found that NLED model minimally coupled to general relativity (GR) can be a possible source generating such a regular black hole solution Ayon-Beato1998 . In Ref. Bronnikov2001 , Bronnikov found a class of magnetically charged regular black holes in the framework of GR coupled with a specific NLED model (known as Bronnikov NLED model). Subsequently, Hayward proposed a concrete model which can describe both the collapse and evaporation of black holes Hayward2006 . One can see Refs. Dymnikova1992 ; Ayon-Beato2000 ; Elizalde2002 ; Nicolini2006 ; Ansoldi2007 ; Hossenfelder2010 ; Johannsen2013 ; Dymnikova2015 ; Rodrigues2016 ; Fan2016 ; Chinaglia2017 ; Nojiri2017 ; Yu2020 ; Pedro2020 for more regular black holes based on NLED models. In this paper, we mainly focus on two black hole models based on two particular above- mentioned NLED models, i.e., EH NLED model and Bronnikov NLED model, and investigate the weak deflection angle of light by these two black hole models. On the other hand, it is well known that light rays will be bent when traveling through a massive object, known as the gravitational lensing effect, which is one of the key predictions of GR. At present, the gravitational lensing is one of the most powerful tools in astronomy and cosmology, such as, measuring the mass of galaxies and clusters Hoekstra2013 ; Brouwer2018 ; Bellagamba2019 , detecting dark energy and dark matter Vanderveld2012 ; He2017 ; Cao2012 ; Huterer2018 ; Jung2019 ; Andrade2019 ; Turimov2019 . Since the first measurement of the gravitational bending of light by the sun, the gravitational lensing effects have been extensively investigated for black holes, wormholes, cosmic strings and other objects by the lens equation Keeton1998 ; Bhadra2003 ; Perlick2004 ; Whisker2005 ; Chen2009 ; Nandi2006 ; Eiroa2002 ; Mao1991 ; Bozza2002 ; Hoekstra2004 ; Virbhadra2002 ; Virbhadra2000 ; Gallo2011 ; Sharif2015 ; Gibbons1993 . In 2008, Gibbons and Werner introduced an alternative method to calculate the weak deflection angle of light in static asymptotically flat spacetimes by using the Gauss-Bonnet theorem and the optical geometry of the spacetime, where the light source and receiver are located at infinity Gibbons2008 . Later, this method was extended to stationary spacetimes by Werner Werner2012 . In Ref. Ishihara2016 , the authors investigated the weak deflection of light for the light source and receiver located at a finite distance. The weak deflection for the massive particles by this method was investigated in Refs. Crisnejo2018 ; Jusufi2018 ; Zonghai2020 . Besides, the weak deflection of light by a black hole immersed in a plasma medium was discussed in Ref. Crisnejo2018 . One can see Refs. Jusufi2016 ; Jusufi2017a ; Jusufi2017b ; Ono2017 ; Sakalli2017 ; Jusufi2018a ; Jusufi2018b ; Jusufi2018c ; Arakida2018 ; Ono2018 ; Gyulchev2019 ; Javed2019 ; Sakalli2019 ; Crisnejo2019 for more recent works. Although the black holes based on Einstein-Euler-Heisenberg (EEH) theory have been extensively studied in the literatures Yajima2001 ; Ruffini2013 ; Guerrero2020 ; Allahyari2020 ; Magos2020 , the weak deflection of light by these black holes have not been investigated yet. As a powerful tool to study the characteristics of black holes, it is interesting to investigate the weak deflection angle by the electrically charged EEH black hole and know what the effects are of the one-loop corrections to QED on the deflection angle. Besides, although there are many investigations on the NLED-based regular black holes, the weak deflection angle of light by such regular black holes are rarely investigated. In this paper, we take the Bronnikov NLED black hole with magnetic charge as an example and investigate the characteristics of this regular black hole by calculating its deflection angle. What’s more, most astrophysical objects including black holes are surrounded by a plasma medium. Thus, it is interesting to investigate the effects of the plasma medium on the deflection angle of light by these black holes. This paper is organized as follows. In Sec. II, we first give a brief review of the EEH black hole and then calculate the weak deflection angle of light by this black hole via two different methods, i.e., the method by using the Gauss-bonnet theorem and the traditional geodesic method. Then, the effects of the plasma on the weak deflection angle are studied. In Sec. III, we perform the same procedures for the Bronnikov NLED black hole and analyse the characteristics of the weak deflection angle of light by this regular magnetically charged black hole. Section IV comes with the conclusion. ## II Weak deflection angle of light by the Einstein-Euler-Heisenberg black holes In this section, we first give a brief review of the Einstein-Euler-Heisenberg theory and present the spherically symmetric and static solution to this theory. Then, we will use these results to calculate the weak deflection angle of light for this black hole by using the Gauss-Bonnet theorem. Besides, the weak deflection angle of light is also calculated with the null geodesic method as a verification to the former results. Finally, we will investigate the deflection angle of light for this black hole immersed in a cold non- magnetized plasma medium. ### II.1 Einstein-Euler-Heisenberg theory The action for the Einstein-Euler-Heisenberg theory is given by Allahyari2020 ; Magos2020 $\displaystyle S=\frac{1}{4\pi}\int d^{4}x\sqrt{-g}\left[\frac{1}{4}R-\mathcal{L}(F,G)\right],$ (1) where $\mathcal{L}(F,G)$ is the functional of the electromagnetic invariants, $F=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $G=\frac{1}{4}F^{\mu\nu}F^{}_{\mu\nu}$ with $F_{\mu\nu}$ the electromagnetic field strength and $F^{}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}F^{\sigma\rho}$ its dual. The Levi-Civita tensor satisfies $\epsilon_{\mu\nu\sigma\rho}\epsilon^{\mu\nu\sigma\rho}=-4!$. As one-loop corrections to quantum electrodynamics (QED), the Euler-Heisenberg Lagrangian is $\displaystyle\mathcal{L}(F,G)=-F+\frac{a}{2}F^{2}+\frac{7a}{8}G^{2},$ (2) where $a$ is the Euler-Heisenberg parameter. For $a=0$, the standard Maxwell electrodynamics is recovered. There are two frameworks in nonlinear electrodynamics. One is the $F$ framework constructed by the electromagnetic field tensor $F_{\mu\nu}$ and the other is the $P$ framework constructed by the tensor $P_{\mu\nu}$, defined by $\displaystyle P_{\mu\nu}=-(\mathcal{L}_{F}F_{\mu\nu}+F^{}_{\mu\nu}\mathcal{L}_{G}),$ (3) where $\mathcal{L}_{X}=\frac{\partial\mathcal{L}}{\partial X}$. Then, the $P_{\mu\nu}$ in the Euler-Heisenberg theory can be calculated as $\displaystyle P_{\mu\nu}=(1-aF)F_{\mu\nu}-\frac{7a}{4}F^{}_{\mu\nu}G.$ (4) In the $P$ framework, one can define two independent invariants $P$ and $O$, $\displaystyle P=-\frac{1}{4}P_{\mu\nu}P^{\mu\nu},\quad\quad O=-\frac{1}{4}P^{\mu\nu}P^{}_{\mu\nu},$ (5) where $P^{}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\rho}P^{\sigma\rho}$. The equations of motion can be derived as $\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ $\displaystyle=$ $\displaystyle 8\pi T_{\mu\nu},$ (6) $\displaystyle\nabla_{\mu}P^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$ (7) where the energy momentum tensor in the $P$ framework is given by $\displaystyle T_{\mu\nu}\\!\\!=\\!\\!\frac{1}{4\pi}\left[(1\\!-\\!aP)P_{\mu}^{\sigma}P_{\nu\sigma}\\!+\\!g_{\mu\nu}\left(P\\!-\\!\frac{3}{2}aP^{2}\\!-\\!\frac{7a}{8}O^{2}\right)\right].$ ### II.2 Spherically symmetric solution in the Einstein-Euler-Heisenberg theory The line element for a spherically symmetric and static black hole can be assumed as $\displaystyle~{}ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (9) where $\mu$ and $\nu$ run from $0$ to $3$, and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. According to the symmetry of the spacetime and restricting to the electric charge $Q$, the $P_{\mu\nu}$ can be calculated as $\displaystyle P_{\mu\nu}=\frac{Q}{r^{2}}\delta^{0}_{[\mu}\delta^{1}_{\nu]},$ (10) and the independent electromagnetic invariants are $\displaystyle P=\frac{Q^{2}}{2r^{4}},\quad\quad O=0.$ (11) Then the function in the metric can be solved as Allahyari2020 ; Magos2020 $\displaystyle f(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{aQ^{4}}{20r^{6}},$ (12) where $M$ is the mass of the black hole. ### II.3 Calculation of deflection angle with the Gauss-Bonnet theorem The null geodesics satisfies $ds^{2}=0$, which can be rearranged as $\displaystyle dt^{2}=\gamma_{ij}dx^{i}dx^{j}=\frac{1}{f^{2}}dr^{2}+\frac{r^{2}}{f}d\Omega^{2},~{}$ (13) where $i$ and $j$ run from $1$ to $3$, and $\gamma_{ij}$ is the so-called optical metric. After a coordinate transformation $dr^{}=\frac{1}{f}dr$, the above expression can be rewritten as $\displaystyle dt^{2}=dr^{2}+\tilde{f}^{2}(r^{})d\phi^{2},$ (14) where $\tilde{f}(r^{})\equiv\sqrt{\frac{r^{2}}{f}}$ and $\theta=\frac{\pi}{2}$. The Gaussian curvature of the optical spacetime can be calculated as $\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle\frac{R_{r\phi r\phi}}{\gamma}=\frac{1}{\sqrt{\gamma}}\left[\frac{\partial}{\partial\phi}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma^{\phi}_{rr}\right)-\frac{\partial}{\partial r}\left(\frac{\sqrt{\gamma}}{\gamma_{rr}}\Gamma^{\phi}_{r\phi}\right)\right]$ (15) $\displaystyle=$ $\displaystyle-\frac{2M}{r^{3}}\left(1-\frac{3M}{2r}\right)+\frac{Q^{2}}{r^{4}}\left(3-\frac{6M}{r}\right)$ $\displaystyle+$ $\displaystyle\frac{Q^{4}}{r^{6}}\left(2-\frac{21a}{20r^{2}}+\frac{19aM}{10r^{3}}\right)-\frac{9aQ^{6}}{10r^{10}}+\frac{3a^{2}Q^{8}}{100r^{14}},~{}$ where $\gamma\equiv\det(\gamma_{ij})$. Let the domain $D$ be a compact oriented nonsingular two-dimensional Riemannian surface with Euler characteristic $\chi(D)$ and Gaussian curvature $\mathcal{K}$, and bounded by a piecewise smooth curve with geodesic curvature $\kappa$. Then the Gauss-Bonnet theorem gives the relation between the deflection angle of light and the Gaussian curvature via $\displaystyle\int\int_{D}\mathcal{K}dS+\oint_{\partial D}\kappa dt+\sum_{i=1}\beta_{i}=2\pi\chi(D),~{}$ (16) where $dS$ is the surface element, $\kappa$ standards for the geodesic curvature of the boundary defined as $\kappa=\|\nabla_{\dot{C}}\dot{C}\|$, and $\beta_{i}$ denotes the $i^{\text{th}}$ exterior angles. For a specific $\tilde{D}$ bounded by a geodesic $C_{1}$ from the source $S$ to the observer $O$ and a circular curve $C_{R}$ intersecting $C_{1}$ in $S$ and $O$ at right angles, Eq. (16) reduces to $\displaystyle\int\int_{\tilde{D}}\mathcal{K}dS+\int_{C_{R}}\kappa(C_{R})dt=\pi,~{}$ (17) where we have used $\kappa(C_{1})=0$ and the Euler characteristic $\chi(\tilde{D})=1$. For the circular curve $C_{R}:=r(\phi)=R=\text{const}$, the non-zero part of the geodesic curvature can be calculated as $\displaystyle\kappa(C_{R})=\left(\nabla_{\dot{C}_{R}}\dot{C}_{R}\right)^{r}=\dot{C}^{\phi}_{R}(\partial_{\phi}\dot{C}^{r}_{R})+\Gamma^{r}_{\phi\phi}(\dot{C}^{\phi}_{R})^{2},$ (18) where $\dot{C}_{R}$ denotes the tangent vector of the circular curve $C_{R}$ and $\Gamma^{r}_{\phi\phi}$ is the Christoffel symbol related to the optical metric (13). In the last equation it is obvious that the first term vanishes, and $\Gamma^{r}_{\phi\phi}=-\tilde{f}(r^{})\tilde{f}^{\prime}(r^{})$, $(\dot{C}^{\phi}_{R})^{2}=\frac{1}{\tilde{f}^{2}(r^{})}$ in the second term. In the limit $R\rightarrow\infty$, one can obtain $\displaystyle\lim_{R\rightarrow\infty}\left[\kappa(C_{R})dt\right]$ (19) $\displaystyle=$ $\displaystyle\lim_{R\rightarrow\infty}[-\tilde{f}^{\prime}(r^{})]d\phi$ $\displaystyle=$ $\displaystyle\lim_{R\rightarrow\infty}\left(\frac{10R^{4}\left(R(R-3M)+2Q^{2}\right)-2aQ^{4}}{R^{3}\sqrt{100R^{4}\left(R(R-2M)+Q^{2}\right)-5aQ^{4}}}\right)d\phi$ $\displaystyle=$ $\displaystyle d\phi.~{}$ Inserting Eq. (19) into Eq. (17), one has $\displaystyle\int\int_{\tilde{D}_{R\rightarrow\infty}}\mathcal{K}dS+\int_{0}^{\pi+\alpha}d\phi=\pi.$ (20) Then the weak deflection angle of light can be calculated as $\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\int\int_{\tilde{D}}\mathcal{K}dS=-\int^{\pi}_{0}\int^{\infty}_{\frac{b}{\sin\phi}}\mathcal{K}dS$ (21) $\displaystyle\simeq$ $\displaystyle\frac{4M}{b}-\frac{3\pi Q^{2}}{4b^{2}}+\frac{7\pi aQ^{4}}{128b^{6}}+\mathcal{O}(M^{2},a^{2},Q^{4}),~{}$ where we have used the zero-order particle trajectory $r=b/\sin\phi$, $0\leq\phi\leq\pi$ at the weak deflection limit. It is obvious that the first two terms are the deflection angle of light by an electrically charged black hole based on the standard electrodynamics Jusufi2016 . The third term comes from the influences of the one-loop corrections to QED on the spacetime of the black hole. It is obvious that the deflection angle increases with the one- loop corrections while their effects are suppressed by the impact parameter. ### II.4 Calculation of deflection angle by the geodesic method The Lagrangian of the null geodesics of the Einstein-Euler-Heisenberg black hole is given by $\displaystyle 2\mathcal{L}_{}=-f(r)\dot{t}^{2}+f(r)^{-1}\dot{r}^{2}+r^{2}\big{(}\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\big{)},~{}$ (22) where $\dot{x}=\frac{dx}{d\tau}$, and $\tau$ is the affine parameter along the geodesics. Since the Lagrangian is independent on $t$ and $\phi$, one can obtain two conserved constants: $\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle\frac{\partial\mathcal{L}_{}}{\partial\dot{t}}=-f(r)\dot{t}=-E,$ (23) $\displaystyle p_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\partial\mathcal{L}_{}}{\partial\dot{\phi}}=r^{2}\dot{\phi}\sin^{2}\theta=L.$ (24) Then the null geodesic equation at the equatorial plane can be obtained as $\displaystyle\left(\frac{d\phi}{dr}\right)^{2}=\left(\frac{r^{4}}{b^{2}}-r^{2}f(r)\right)^{-1},~{}$ (25) where the impact parameter is defined as $b=r_{0}/\sqrt{f(r_{0})}$ with $r_{0}$ the radius of the circular orbit. The weak bending angle of the light coming from infinity and deflected by a black hole before arriving at infinity is given by $\displaystyle\alpha(r_{0})=\Delta\phi(r_{0})-\pi,$ (26) where $\Delta\phi(r_{0})$ can be solved from Eq. (25) as $\displaystyle\Delta\phi(r_{0})=2\int_{r_{0}}^{\infty}\left(\frac{r^{4}}{b^{2}}-r^{2}f(r)\right)^{-\frac{1}{2}}dr.~{}$ (27) It is convenient to define the dimensionless line element as $\displaystyle dS^{2}$ $\displaystyle=$ $\displaystyle(2M)^{-2}ds^{2}=-f(x)dT^{2}+f(x)^{-1}dx^{2}$ (28) $\displaystyle+$ $\displaystyle x^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$ where we have defined $\displaystyle x=\frac{r}{2M},\quad T=\frac{t}{2M},\quad q=\frac{Q}{2M},\quad\hat{\alpha}=\frac{a}{(2M)^{2}},$ (29) and the function $f(r)$ in the metric (9) can be reexpressed as $\displaystyle f(x)=1-\frac{1}{x}+\frac{q^{2}}{x^{2}}-\frac{\hat{\alpha}q^{4}}{20x^{6}}.$ (30) Then Eq. (27) can be rewritten as $\displaystyle\Delta\phi(x_{0})\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!2\int_{x_{0}}^{\infty}\\!\\!\sqrt{20}x^{2}x_{0}^{4}\Big{(}\hat{\alpha}q^{4}\left(x_{0}^{8}-x^{8}\right)$ (31) $\displaystyle+$ $\displaystyle\\!\\!20q^{2}x^{4}x_{0}^{4}\left(x^{4}-x_{0}^{4}\right)$ $\displaystyle+$ $\displaystyle\\!\\!20x^{5}x_{0}^{5}\left(x^{3}(x_{0}\\!-\\!1)\\!-\\!xx_{0}^{3}\\!+\\!x_{0}^{3}\right)\Big{)}^{-\frac{1}{2}}\\!\\!dx,$ and the impact parameter can be expressed as $\displaystyle\frac{b}{2M}=\frac{x_{0}}{\sqrt{f(x_{0})}}.~{}$ (32) After defining a new variable $z=\frac{x_{0}}{x}$, the above integral can be rewritten as $\displaystyle\Delta\phi(x_{0})$ $\displaystyle=$ $\displaystyle 2\int_{0}^{1}\sqrt{20}x_{0}^{3}\Big{(}\hat{\alpha}q^{4}\left(z^{8}-1\right)-20q^{2}x_{0}^{4}\left(z^{4}-1\right)$ (33) $\displaystyle-$ $\displaystyle 20x_{0}^{5}\left(x_{0}\left(z^{2}-1\right)-z^{3}+1\right)\Big{)}^{-\frac{1}{2}}dz.$ Considering the weak gravitational lensing limit $x_{0}\gg 1$ and expanding the above integrand about $\frac{1}{x_{0}}$, the above integral can be integrated out term by term as follows: $\displaystyle\alpha(x_{0})$ $\displaystyle=$ $\displaystyle\Delta\phi(x_{0})-\pi=\frac{2}{x_{0}}+\left(\frac{\pi}{4}\left(\frac{15}{4}-3q^{2}\right)-1\right)\frac{1}{x_{0}^{2}}+\left(\frac{3}{16}\pi\left(4q^{2}-5\right)-7q^{2}+\frac{61}{12}\right)\frac{1}{x_{0}^{3}}+\bigg{(}\frac{5}{8}\left(20q^{2}-13\right)$ (34) $\displaystyle+$ $\displaystyle\frac{3\pi\left(304q^{4}-2200q^{2}+1155\right)}{1024}\bigg{)}\frac{1}{x_{0}^{4}}+\left(\frac{1}{32}(632-105\pi)q^{4}+\frac{7}{64}(135\pi-536)q^{2}+\frac{7783}{320}-\frac{3465\pi}{512}\right)\frac{1}{x_{0}^{5}}$ $\displaystyle+$ $\displaystyle\bigg{(}\frac{7}{128}\pi\alpha q^{4}-\frac{1}{384}\left(28560q^{4}-59832q^{2}+21397\right)-\frac{105\pi}{16384}\left(192q^{6}-4816q^{4}+8676q^{2}-2959\right)\bigg{)}\frac{1}{x_{0}^{6}}$ $\displaystyle+$ $\displaystyle\mathcal{O}\left(\frac{1}{x_{0}^{7}}\right).~{}$ To obtain the deflection angle in terms of the impact parameter $b$, one needs the relation between $b$ and $x_{0}$ which can be solved from Eq. (32) in the weak deflection limit as $\displaystyle\frac{1}{x_{0}}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\frac{2M}{b}+\frac{1}{2}\left(\frac{2M}{b}\right)^{2}-\frac{1}{8}(4q^{2}-5)\left(\frac{2M}{b}\right)^{3}-\bigg{(}\frac{3q^{2}}{2}$ (35) $\displaystyle-$ $\displaystyle\\!\\!1\bigg{)}\left(\frac{2M}{b}\right)^{4}+\frac{7}{128}\Big{(}16q^{4}-72q^{2}+33\Big{)}\left(\frac{2M}{b}\right)^{5}$ $\displaystyle+$ $\displaystyle\\!\\!\left(5q^{4}\\!-\\!10q^{2}\\!+\\!\frac{7}{2}\right)\left(\frac{2M}{b}\right)^{6}\\!+\\!\mathcal{O}\left(\left(\frac{2M}{b}\right)^{7}\right).~{}$ Inserting Eq. (35) into Eq. (34), the weak deflection angle is found to be $\displaystyle\hat{\alpha}\simeq\frac{4M}{b}-\frac{3\pi Q}{4b^{2}}+\frac{7\pi aQ^{4}}{128b^{6}}+\mathcal{O}(M^{2},a^{2},Q^{4}).$ (36) It is obvious that the above result is in agreement with the result calculated by using the Gauss-Bonnet theorem. However, it should be noted that this agreement only holds for the first-order terms and breaks down for the higher- order corrections. ### II.5 Weak deflection angle in the presence of plasma In this subsection, we investigate the effects of a cold non-magnetized plasma on the deflection angle for the Einstein-Euler-Heisenberg black hole. The refractive index for this black hole is given by Perlick2015 , $\displaystyle n(r)=\sqrt{1-\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}f(r)},$ (37) where $\omega_{e}$ and $\omega_{\infty}$ denote the electron plasma frequency and the photon frequency measured by a static observer at infinity, respectively. The corresponding optical line element can be defined as $\displaystyle d\sigma^{2}$ $\displaystyle=$ $\displaystyle\gamma_{ij}dx^{i}dx^{j}=-\frac{n^{2}}{g_{00}}g_{ij}dx^{i}dx^{j}$ (38) $\displaystyle=$ $\displaystyle n^{2}\left(\frac{1}{f^{2}}dr^{2}+\frac{r^{2}}{f}d\phi^{2}\right),~{}$ which is conformally related to the induced metric on the spatial section with $\theta=\frac{\pi}{2}$. Then the Gaussian curvature can be calculated as $\displaystyle\tilde{\mathcal{K}}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!{40\Xi^{-3}r^{4}\left(3aQ^{4}+20r^{4}\left(Mr-Q^{2}\right)\right)^{2}}$ (39) $\displaystyle+$ $\displaystyle\\!\\!\Xi^{-1}\left(\frac{3aQ^{4}}{20r^{8}}+r^{-4}\left(Mr-Q^{2}\right)\right)\Big{(}aQ^{4}$ $\displaystyle-$ $\displaystyle\\!\\!20r^{4}\left(r(r-2M)+Q^{2}\right)\Big{)}-\Xi^{-2}\bigg{(}\frac{9a^{2}Q^{8}}{r^{2}}$ $\displaystyle+$ $\displaystyle\\!\\!20aQ^{4}r^{2}\left(r(18r-19M)+2Q^{2}\right)$ $\displaystyle+$ $\displaystyle\\!\\!400r^{6}\\!\left(-Q^{2}r(M\\!+\\!2r)\\!+\\!Mr^{2}(M\\!+\\!r)\\!+\\!Q^{4}\right)\\!\\!\bigg{)},$ where $\Xi=\left(a\delta Q^{4}+20r^{4}\left(r^{2}-\delta\left(r(r-2M)+Q^{2}\right)\right)\right)$ and the plasma parameter is defined by $\delta\equiv\frac{\omega_{e}^{2}}{\omega_{\infty}^{2}}$. For a photon can propagate in the plasma, one should require $\omega_{\infty}\geq\omega_{e}$, thus $0\leq\delta\leq 1$. For more details about the plasma, one can refer to Ref. Bisnovatyi-Kogan2010 . Besides, it follows from Eq. (38) that $\displaystyle\frac{d\sigma}{d\phi}\bigg{\|}_{\gamma_{R}}=n\sqrt{\frac{r^{2}}{f}},$ (40) which results in $\displaystyle\lim_{R\rightarrow\infty}\tilde{\kappa}(C_{R})\frac{d\sigma}{d\phi}\bigg{\|}_{\gamma_{R}}\approx 1.$ (41) By taking the zero-order particle trajectory $r=\frac{b}{\sin\phi}$ and for the limit $R\rightarrow\infty$, the Gauss-Bonnet theorem can be written as $\displaystyle\int^{\pi+\alpha}_{0}d\phi=\pi-\int^{\pi}_{0}\int^{\infty}_{\frac{b}{\sin\phi}}\tilde{\mathcal{K}}dS.$ (42) Then the deflection angle can be calculated as $\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\int^{\pi}_{0}\int^{\infty}_{\frac{b}{\sin\phi}}\tilde{\mathcal{K}}dS$ (43) $\displaystyle\simeq$ $\displaystyle\frac{2M}{b}\left(1+\frac{1}{1-\delta}\right)-\frac{\pi Q^{2}}{4b^{2}}\left(1+\frac{2}{1-\delta}\right)$ $\displaystyle+$ $\displaystyle\frac{a\pi Q^{4}}{128b^{6}}\left(1+\frac{6}{1-\delta}\right)+\mathcal{O}(M^{2},a^{2},Q^{4}).~{}$ It can be easily shown that Eq. (43) reduces to Eq. (21) when $\delta\rightarrow 0$, and the deflection angle increases with the plasma parameter $\delta$, which suggests that the lower the photon frequency measured by a static observer at infinity is, the larger the deflection angle of it will be for a fixed electron plasma frequency. ## III Weak deflection angle of light by Einstein-Bronnikov black holes In this section we will perform the same procedures of the previous section in the case of Einstein-Bronnikov theory, which is a particular NLED theory only consists of the relativistic invariant $F$, and wherein one can obtain regular black holes. ### III.1 The Einstein-Bronnikov theory The action for the Einstein-Bronnikov theory is given by Bronnikov2001 $\displaystyle S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}\left[R-\mathcal{L}(F)\right],$ (44) where $\displaystyle\mathcal{L}(F)=F\cosh^{-2}\left[\hat{a}\left(F/2\right)^{1/4}\right],$ (45) and the parameter $\hat{a}$ is related to the black hole mass $M$ and magnetic charge $Q_{m}$ via $\hat{a}=Q_{m}^{3/2}/(2M)$. The standard Einstein-Maxwell Lagrangian can be recovered with $\hat{a}\rightarrow 0$. The equations of motion can be derived as $\displaystyle R_{\mu\nu}\\!-\\!\frac{1}{2}g_{\mu\nu}R\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!8\pi T_{\mu\nu}\\!\\!=\\!\\!8\pi\\!\left(2\mathcal{L}_{F}F_{\rho\mu}F_{\nu}^{\rho}\\!-\\!\frac{1}{2}g_{\mu\nu}\mathcal{L}\right)\\!,$ (46) $\displaystyle\nabla_{\mu}(\mathcal{L}_{F}F^{\mu\nu})$ $\displaystyle=$ $\displaystyle 0.$ (47) Considering the spherically symmetric and static spacetime and restricting to the magnetic charge $Q_{m}$, the relevant function in the metric analogous to Eq. (9) can be obtained as Bronnikov2001 $\displaystyle~{}g(r)=1-\frac{2M}{r}\left(1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right),$ (48) and the gauge field is given by $A_{\mu}=-Q_{m}\cos\theta\delta^{\phi}_{\mu}$. It can be straightforwardly shown that the metric function (48) reduces to the Schwarzschild black hole solution with $Q_{m}\rightarrow 0$ and is regular as $r\rightarrow 0$, which suggests a regular black hole. ### III.2 Calculation of deflection angle by the Gauss-Bonnet theorem The null geodesics satisfying $ds^{2}=0$ can be rearranged as $\displaystyle dt^{2}=\gamma_{ij}dx^{i}dx^{j}=\frac{1}{g^{2}}dr^{2}+\frac{r^{2}}{g}d\Omega^{2}.$ (49) After a coordinate transformation $dr^{}=\frac{1}{g}dr$, the above line element can be rewritten as $\displaystyle dt^{2}=dr^{2}+\tilde{g}^{2}(r^{})d\phi^{2},$ (50) where $\tilde{g}(r^{})=\sqrt{\frac{r^{2}}{g}}$ and $\theta=\frac{\pi}{2}$. The Gaussian curvature of this optical spacetime can be calculated as $\displaystyle\mathcal{K}$ $\displaystyle=$ $\displaystyle-\frac{2M}{r^{3}}\left(1-\tanh\left(\frac{Q_{m}}{2Mr}\right)\right)+\frac{1}{r^{4}}\bigg{[}3M^{2}\left(1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right)^{2}+2Q_{m}^{2}\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)\bigg{]}$ (51) $\displaystyle-$ $\displaystyle\frac{Q_{m}^{2}}{2Mr^{5}}\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)\bigg{[}6M^{2}+(Q_{m}^{2}-6M^{2})\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\bigg{]}$ $\displaystyle-$ $\displaystyle\frac{Q_{m}^{4}}{4r^{6}}\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)\left(1-\tanh\left(\frac{Q_{m}}{2Mr}\right)\right)\left(1-3\tanh\left(\frac{Q_{m}}{2Mr}\right)\right).~{}$ Following the same procedures as the previous section, the weak deflection angle of light by this black hole can be obtained as $\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\int\int_{\tilde{D}}\mathcal{K}dS=-\int^{\pi}_{0}\int^{\infty}_{\frac{1}{u(\phi)}}\mathcal{K}dS$ (52) $\displaystyle\simeq$ $\displaystyle\frac{4M}{b}-\frac{3\pi Q_{m}^{2}}{4b^{2}}-\frac{16MQ_{m}^{2}}{b^{3}}+\mathcal{O}(M^{2},Q_{m}^{3}),~{}$ where $u(\phi)$ is given in Eq. (56). It is obvious that the first two terms are the same with the weak deflection angle of light by the Reissner- Nordstr$\ddot{\text{o}}$m black hole Jusufi2016 except the electric charge is replaced by the magnetic charge, and the minus sign in front of the third term indicates that the weak deflection angle of this regular magnetically charged black hole is smaller than the singular one. ### III.3 Calculation of deflection angle by the geodesic method The Lagrangian of the null geodesics of the Einstein-Bronnikov black hole is given by $\displaystyle 2\mathcal{L}_{}=-g(r)\dot{t}^{2}+g(r)^{-1}\dot{r}^{2}+r^{2}\big{(}\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}\big{)},~{}$ (53) where $\dot{x}=\frac{dx}{d\tau}$, and $\tau$ is the affine parameter along the geodesic. Then the null geodesic equation at the equatorial plane can be obtained as $\displaystyle\left(\frac{d\phi}{dr}\right)^{2}=\left(\frac{r^{4}}{b^{2}}-r^{2}g(r)\right)^{-1},~{}$ (54) where the impact parameter is defined as $b=\sqrt{\frac{r_{0}^{2}}{g(r_{0})}}$ with $r_{0}$ the radius of the circular orbit. After introducing a new variable $u(\phi)=\frac{1}{r}$, the above geodesic equation can be rewritten as $\displaystyle\left(\frac{du}{d\phi}\right)^{2}=\frac{1}{b^{2}}-u^{2}+2Mu^{3}\left[1-\tanh\left(\frac{Q_{m}^{2}u}{2M}\right)\right],~{}$ (55) which can be solved by iterative method as follows: $\displaystyle u(\phi)$ $\displaystyle=$ $\displaystyle\frac{\sin\phi}{b}+\frac{M\left(\cos^{2}\phi+1\right)}{b^{2}}-\frac{M^{2}\cos\phi}{8b^{3}}\Big{(}30\phi$ (56) $\displaystyle+$ $\displaystyle 3\sin(2\phi)-20\tan\phi\Big{)}-\frac{Q_{m}^{2}\cos\phi}{2b^{3}}\bigg{(}-\frac{3\phi}{2}$ $\displaystyle+$ $\displaystyle\frac{1}{4}\sin(2\phi)+\tan\phi\bigg{)}+\mathcal{O}(M^{3},Q_{m}^{3})~{}.$ Besides, the bending angle of light can be expressed as $\displaystyle\hat{\alpha}(r_{0})=\Delta\phi(r_{0})-\pi,$ (57) where $\Delta\phi(r_{0})$ can be obtained from Eq. (54) as $\displaystyle\Delta\phi(r_{0})$ $\displaystyle=$ $\displaystyle 2\int_{r_{0}}^{\infty}\bigg{(}\frac{r^{4}}{b^{2}}-r^{2}$ (58) $\displaystyle+$ $\displaystyle 2Mr\left[1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right]\bigg{)}^{-\frac{1}{2}}dr.~{}$ Defining the new dimensionless spacetime coordinates $x=\frac{r}{2M}$ and $T=\frac{t}{2M}$ and the dimensionless magnetic charge $q_{m}=\frac{Q_{m}}{2M}$, Eq. (58) can be reexpressed as $\displaystyle\Delta\phi(x_{0})\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!2\int_{x_{0}}^{\infty}\\!\\!x_{0}^{\frac{3}{2}}\Big{(}x^{4}(-1+x_{0})+xx_{0}^{3}(1-x)$ $\displaystyle-$ $\displaystyle\\!\\!xx_{0}^{3}\tanh\left(q_{m}^{2}/x\right)\\!+\\!x^{4}\tanh\left(q_{m}^{2}/x_{0}\right)\Big{)}^{-\frac{1}{2}}\\!\\!dx,$ and the impact parameter is given by $\displaystyle\frac{b}{2M}=\frac{x_{0}}{\sqrt{g(x_{0})}}.~{}$ (60) After defining a new variable $z=\frac{x_{0}}{x}$, the above integral becomes $\displaystyle\Delta\phi(x_{0})\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!2\int_{0}^{1}\sqrt{x_{0}}\Big{(}-1+(1-z^{2})x_{0}+z^{3}$ (61) $\displaystyle+$ $\displaystyle\\!\\!\tanh\left(q_{m}^{2}/x_{0}\right)\\!-\\!z^{3}\tanh\left(q_{m}^{2}z/x_{0}\right)\\!\Big{)}^{-\frac{1}{2}}\\!dz.$ Then the weak deflection angle can be integrated out term by term as follows: $\displaystyle\alpha(x_{0})$ $\displaystyle=$ $\displaystyle\Delta\phi(x_{0})-\pi=\frac{2}{x_{0}}+\left(\frac{\pi}{4}\left(\frac{15}{4}-3q_{m}^{2}\right)-1\right)\frac{1}{x_{0}^{2}}+\left(\frac{3}{16}\pi\left(4q_{m}^{2}-5\right)-7q_{m}^{2}+\frac{61}{12}\right)\frac{1}{x_{0}^{3}}+\bigg{(}\frac{5}{8}\left(20q_{m}^{2}-13\right)$ (62) $\displaystyle+$ $\displaystyle\frac{\pi\left(320q_{m}^{6}+912q_{m}^{4}-6600q_{m}^{2}+3465\right)}{1024}\bigg{)}\frac{1}{x_{0}^{4}}+\bigg{(}\frac{1}{120}(472-75\pi)q_{m}^{6}+\frac{1}{32}(632-105\pi)q_{m}^{4}$ $\displaystyle+$ $\displaystyle\frac{7}{64}(135\pi-536)q_{m}^{2}+\frac{7783}{320}-\frac{3465\pi}{512}\bigg{)}\frac{1}{x_{0}^{5}}+\bigg{(}\frac{7\pi q_{m}^{10}}{48}+\frac{49\pi q_{m}^{8}}{64}-\frac{5}{96}(208+63\pi)q_{m}^{6}-\frac{35}{1024}(2176+903\pi)q_{m}^{4}$ $\displaystyle+$ $\displaystyle\frac{9}{4096}(70912+25305\pi)q_{m}^{2}-\frac{21397}{384}-\frac{310695\pi}{16384}\bigg{)}\frac{1}{x_{0}^{6}}+\mathcal{O}\left(\frac{1}{x_{0}^{7}}\right).~{}$ Besides, the relation between $b$ and $x_{0}$ can be obtained from Eq. (60) as $\displaystyle\frac{1}{x_{0}}$ $\displaystyle=$ $\displaystyle\frac{2M}{b}+\frac{1}{2}\left(\frac{2M}{b}\right)^{2}-\frac{1}{8}(4q_{m}^{2}-5)\left(\frac{2M}{b}\right)^{3}$ (63) $\displaystyle-$ $\displaystyle\left(\frac{3q_{m}^{2}}{2}-1\right)\left(\frac{2M}{b}\right)^{4}+\bigg{(}\frac{q_{m}^{6}}{6}+\frac{7q_{m}^{4}}{8}-\frac{63q_{m}^{2}}{16}$ $\displaystyle+$ $\displaystyle\frac{231}{128}\bigg{)}\left(\frac{2M}{b}\right)^{5}+\bigg{(}\frac{2q_{m}^{6}}{3}+5q_{m}^{4}-10q_{m}^{2}$ $\displaystyle+$ $\displaystyle\frac{7}{2}\bigg{)}\left(\frac{2M}{b}\right)^{6}+\mathcal{O}\left(\left(\frac{2M}{b}\right)^{7}\right).~{}$ Inserting Eq. (63) into Eq.(62), the weak deflection angle is found to be $\displaystyle\hat{\alpha}\simeq\frac{4M}{b}-\frac{3\pi Q_{m}^{2}}{4b^{2}}-\frac{16MQ_{m}^{2}}{b^{3}}+\mathcal{O}(M^{2},Q_{m}^{3}),$ (64) which is in agreement with the result calculated by using the Gauss-Bonnet theorem. ### III.4 Weak deflection angle in the presence of plasma In this subsection, we investigate the effects of a cold non-magnetized plasma on the deflection angle of the black hole in Einstein-Bronnikov theory. The refractive index for this black hole is given by Perlick2015 , $\displaystyle n(r)=\sqrt{1-\delta^{2}g(r)}.$ (65) The corresponding optical metric is $\displaystyle d\sigma^{2}$ $\displaystyle=$ $\displaystyle\gamma_{ij}dx^{i}dx^{j}=-\frac{n^{2}}{g_{00}}g_{ij}dx^{i}dx^{j}$ (66) $\displaystyle=$ $\displaystyle n^{2}\left(\frac{1}{g^{2}}dr^{2}+\frac{r^{2}}{g}d\phi^{2}\right).~{}$ Then the Gaussian curvature is calculated as $\displaystyle\tilde{\mathcal{K}}$ $\displaystyle=$ $\displaystyle\frac{1}{4r^{4}\delta^{2}}\bigg{[}-3r^{2}+4M\delta r-2\delta\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)\left(Q_{m}^{2}+Mr\sinh\left(\frac{Q_{m}^{2}}{Mr}\right)\right)\bigg{]}$ (67) $\displaystyle+$ $\displaystyle\frac{6(\delta-1)Mr^{3}-2Q_{m}^{2}\delta\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)\left(Mr+Q_{m}^{2}\tanh\left(\frac{Q_{m}^{2}}{Mr}\right)\right)}{4Mr^{4}\delta^{2}\left(r(1-\delta)+2M\delta\left(1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right)\right)}$ $\displaystyle+$ $\displaystyle\frac{1}{4Mr^{4}\delta^{2}\left(r(1-\delta)+2M\delta\left(1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right)\right)^{2}}\bigg{[}Mr^{4}(-5+(8-3\delta)\delta)$ $\displaystyle+$ $\displaystyle 2MQ_{m}^{2}r^{2}\delta(3\delta-2)\text{sech}^{2}\left(\frac{Q_{m}^{2}}{2Mr}\right)-Q_{m}^{4}\delta\text{sech}^{4}\left(\frac{Q_{m}^{2}}{2Mr}\right)\left(3M\delta+r\sinh\left(\frac{Q_{m}^{2}}{Mr}\right)\right)\bigg{]}$ $\displaystyle+$ $\displaystyle\frac{2r\left(r^{2}(\delta-1)-Q_{m}^{2}\delta\text{sech}^{4}\left(\frac{Q_{m}^{2}}{2Mr}\right)\right)^{2}}{4Mr^{4}\delta^{2}\left(r(1-\delta)+2M\delta\left(1-\tanh\left(\frac{Q_{m}^{2}}{2Mr}\right)\right)\right)^{3}},$ and the deflection angle can be obtained as $\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\int^{\pi}_{0}\int^{\infty}_{\frac{1}{\sin\phi}}\tilde{\mathcal{K}}dS$ (68) $\displaystyle\simeq$ $\displaystyle\frac{2M}{b}\left(1+\frac{1}{1-\delta}\right)-\frac{\pi Q_{m}^{2}}{4b^{2}}\left(1+\frac{2}{1-\delta}\right)$ $\displaystyle+$ $\displaystyle\frac{2MQ_{m}^{2}}{b^{3}}\left(\frac{3\delta}{1-\delta}-\frac{8-10\delta}{(1-\delta)^{2}}\right)$ $\displaystyle+$ $\displaystyle\mathcal{O}(M^{2},Q_{m}^{3}),~{}$ It is obvious that Eq. (68) reduces to Eq. (52) when $\delta\rightarrow 0$, and the deflection angle increases with the plasma parameter $\delta$, which suggests that the lower the photon frequency measured by a static observer at infinity is, the larger the deflection angle of it will be for a fixed electron plasma frequency. ## IV Conclusion As two well-known nonlinear electrodynamic (NLED) theories, Euler-Heisenberg NLED model and Bronnikov NLED model are extensively studied in the literatures. In this paper, we considered the spherically symmetric and static black hole solutions based on these NLED models and calculated the weak deflection angle of light by these two black holes with the help of the Gauss- Bonnet theorem. To be specific, in the Einstein-Euler-Heisenberg black hole, we investigated the effects of the one-loop corrections to quantum electrodynamics on the deflection angle of light and found that the weak deflection angle increases with the one-loop corrections. In the Einstein- Bronnikov black hole, we calculated the weak deflection angle by this regular magnetically charged black hole and found that the deflection angle by this black hole is smaller than the singular one. Besides, the weak deflection angles of both black holes were also calculated via the geodesic method, which was confirmed in agreement with the method by using the Gauss-Bonnet theorem at least at low order. What’s more, the effects of a cold non-magnetized plasma on the weak deflection angle also were discussed and it was found that the deflection angle increases with the plasma parameter for both black holes, which indicates that the lower the photon frequency measured by a static observer at infinity is, the larger the deflection angle of it will be for a fixed electron plasma frequency. ###### Acknowledgements. 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# Zariski density of points with maximal arithmetic degree for surfaces Kaoru Sano, Takahiro Shibata Faculty of Science and Engineering, Doshisha University, Kyoto, 610-0394, Japan<EMAIL_ADDRESS>National University of Singapore, Singapore 119076, Republic of Singapore <EMAIL_ADDRESS> ###### Abstract. We prove that any surjective self-morphism $f$ with $\delta_{f}>1$ on a potentially dense smooth projective surface defined over a number field $K$ has densely many $L$-rational points for a finite extension $L/K$. ###### Key words and phrases: dynamical degree, arithmetic degree ###### 2010 Mathematics Subject Classification: Primary 37P55, Secondary 14G05 ###### Contents 1. 1 Introduction 2. 2 Definitions and Notation 3. 3 Lemmata 4. 4 Ptoof of Theorem 1.5 ## 1\. Introduction Let $X$ be a projective variety defined over a number field $K$ and $f\colon X\longrightarrow X$ a surjective self-morphism over $K$. Then we can define two dynamical quantities, the dynamical degree $\delta_{f}$ and the arithmetic degree $\alpha_{f}(x)$ at a point $x\in X(\overline{K})$, see Section 2 for the definitions. Relationships of those two quantities are studied in several papers. The following result is fundamental. ###### Theorem 1.1 ([KS16b, endomorphism case]). Let $X$ be a projective variety over a number field $K$, and $f\colon X\longrightarrow X$ a surjective self-morphism over $K$. Then the limit defining $\alpha_{f}(x)$ converges and the inequality $\alpha_{f}(x)\leq\delta_{f}$ holds for all $x\in X(\overline{K})$. See [KS16a, Theorem 26] and [Mat16, Theorem 1.4] for the case that $f$ is a dominant rational map. Matsuzawa, Meng, Zhang, and the second author gave the following conjecture in [MMSZ20]. ###### Conjecture 1.2. Let $X$ be a projective variety over a number field $K$, $f\colon X\longrightarrow X$ a surjective morphism, and $d>0$ a positive integer. Then the set $Z_{f}(d):=\\{x\in X(\overline{K})\ \|\ [K(x):K]\leq d,\alpha_{f}(x)<\delta_{f}\\}$ is not Zariski dense. Roughly speaking, Conjecture 1.2 says that the set of points with non-maximal arithmetic degree is small. On the other hand, the set of points with maximal arithmetic degree should be large. To give a precise statement, we prepare the notion of densely many rational points with maximal arithmetic degree. ###### Definition 1.3. Let $X$ be a projective variety over a number field $K$ and $f\colon X\longrightarrow X$ a surjective self-morphism over $K$. Fix an algebraic closure $\overline{K}$ of $K$ and let $L$ be an intermediate field extension $\overline{K}/L/K$. We say that $(X,f)$ has densely many $L$-rational points with maximal arithmetic degree if there is a subset $S\subset X(L)$ satisfying the following conditions: * (1) $S$ is Zariski dense, * (2) the equality $\alpha_{f}(x)=\delta_{f}$ holds for all $x\in S$, and * (3) $O_{f}(x)\cap O_{f}(y)=\emptyset$ for any distinct two points $x,y\in S$, where $O_{f}(x)$ is the forward $f$-orbit $\\{f^{n}(x)\ \|\ n\geq 0\\}$ of $x$. If $(X,f)$ has densely many $L$-rational points with maximal arithmetic degree, we also say that $(X,f)$ satisfies $(DR)_{L}$, for abbreviation. If there is a finite extension $L/K$ such that $(X,f)$ satisfies $(DR)_{L}$, we say that $(X,f)$ satisfies $(DR)$. In [SS20], the authors proved that for a surjective self-morphism $f\colon X\longrightarrow X$ on a projective variety $X$ both defined over a number field $K$, $(X,f)$ has densely many $\overline{K}$-rational points with maximal arithmetic degree. The authors also gave the following question: ###### Question 1.4. Let $X$ be a projective variety and $f\colon X\longrightarrow X$ a surjective self-morphism. If $X$ has potentially dense rational points i.e. there is a finite extension $L/K$ such that $X(L)$ is Zariski dense, then does $(X,f)$ satisfy $(DR)$? In fact, if $X$ is either a unirational variety, an abelian variety, or a $\mathbb{P}^{1}$-bundle over an elliptic curve, the authors gave affirmative answers for Question 1.4 in [SS20]. The main result of this paper is that Question 1.4 is affirmative if $X$ is a smooth projective surface and $f\colon X\longrightarrow X$ is a surjective self-morphism with $\delta_{f}>1$. ###### Theorem 1.5. Let $K$ be a number field, $X$ a smooth projective surface over $K$ having potentially dense rational points, and $f\colon X\longrightarrow X$ a surjective self-morphism with $\delta_{f}>1$. Then $(X,f)$ satisfies $(DR)$. The idea of the proof is as follows. Replacing the self-morphism by its iteration, a self-morphism on the minimal model is induced. So we may assume that the given surface is minimal. Since the case of abelian varieties and unirational varieties is proved in [SS20], the remaining case is automorphisms of K$3$ surfaces and non-isomorphic surjective self-morphisms of elliptic surfaces. These cases are treated in case-by-case analysis. The outline of this paper is as follows. In Section 2, we prepare some notation and definitions which we use in this paper. In Section 3, lemmata to be used in the proof of Theorem 1.5 are prepared. We prove Theorem 1.5 for automorphisms of K$3$ surfaces and non-isomorphic surjective self-morphisms of elliptic surfaces in Section 4.1 and Section 4.2, respectively. ###### Acknowledgments. The authors thank Professor Shu Kawaguchi for giving valuable comments and suggesting them writing this paper, and Professor Joseph Silverman for reading a draft and giving valuable comments. The first author is supported by JSPS KAKENHI Grant Number JP20K14300. The second author is supported by a Research Fellowship of NUS. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. ## 2\. Definitions and Notation ###### Notation and Conventions. * * • Throughout this article, we work over a fixed number field $K$. We fix an algebraic closure $\overline{K}$ of $K$. * • A variety means a geometrically integral separated scheme of finite type over $K$. * • The symbols $\sim$ (resp. $\sim_{\mathbb{Q}}$, $\sim_{\mathbb{R}}$) and $\equiv$ mean the linear equivalence (resp. $\mathbb{Q}$-linear equivalence, $\mathbb{R}$-linear equivalence) and the numerical equivalence on divisors. * • Let $\mathbb{K}=\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$. For a $\mathbb{K}$-linear endomorphism $\phi:V\to V$ on a $\mathbb{K}$-vector space $V$, $\rho(\phi)$ denotes the spectral radius of $f$, that is, the maximum of absolute values of eigenvalues (in $\mathbb{C}$) of $\phi$. * • Though the definition of the dynamical degree of dominant rational self-map is known, we need them only for surjective self-morphisms in this paper. Let $f\colon X\dashrightarrow X$ be a dominant rational map on a projective variety. We define the (first) dynamical degree $\delta_{f}$ of $f$ as $\delta_{f}=\lim_{n\to\infty}((f^{n})^{}H\cdot H^{\dim X-1})^{1/n}.$ Let $f^{}:\operatorname{NS}(X)_{\mathbb{R}}\to\operatorname{NS}(X)_{\mathbb{R}}$ be the pull-back action on the space of numerical classes of $\mathbb{R}$-Cartier divisors on $X$. If $f$ is a morphism, then $\delta_{f}=\rho(f^{})$, so $\delta_{f}$ is an algebraic integer. • Let $X$ be a projective variety. For an $\mathbb{R}$-Cartier divisor $D$ on $X$, a function $h_{D}:X(\overline{K})\to\mathbb{R}$ is determined up to the difference of a bounded function. $h_{D}$ is called the height function associated to $D$. For definition and properties of height functions, see e.g. [HS00, Part B] or [Lan83, Chapter 3]. * • Let $X$ be a projective variety and $f:X\longrightarrow X$ a surjective self- morphism. Fix an ample height function $h_{H}\geq 1$ on $X$. * • For $x\in X(\overline{K})$, we define $\alpha_{f}(x)=\limsup_{n\to\infty}h_{H}(f^{n}(x))^{1/n},$ which we call the arithmetic degree of $f$ at $x$. The convergence defining the arithmetic degree is known (cf. [KS16b]). Moreover, the arithmetic degree is independent of the choice of $H$ and $h_{H}$. ###### Remark 2.1. Dynamical degrees have the following invariance: if $\pi\colon X\dashrightarrow X^{\prime}$ is a dominant rational map between projective varieties of the same dimension, and $f\colon X\dashrightarrow X$ and $f^{\prime}\colon X^{\prime}\dashrightarrow X^{\prime}$ are dominant rational maps satisfying $\pi\circ f=f^{\prime}\circ\pi$, then the equality $\delta_{f}=\delta_{f^{\prime}}$ holds. For details on dynamical degrees, see [Tru20]. ## 3\. Lemmata In this section, we list lemmata used in the next section. ###### Lemma 3.1. Consider the following commutative diagram $\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{X}}$$\scriptstyle{\pi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{Y}}$$\textstyle{Y,}$ where $X,Y$ are smooth projective varieties and $f_{X}$, $f_{Y}$ are surjective self-morphisms. Suppose that there exists a non-empty open subset $U\subset Y$ such that $\pi\colon V:=\pi^{-1}(U)\longrightarrow U$ is finite. Let $x\in X(\overline{K})$, $y:=\pi(x)\in Y(\overline{K})$, $O_{f_{X}}(x)\subset V$, $O_{f_{Y}}(y)\subset U$. Then the equality $\alpha_{f_{X}}(x)=\alpha_{f_{Y}}(y)$ holds. ###### Proof. Since $f_{X}$ and $f_{Y}$ are morphism, the existence of the limit defining the arithmetic degrees are known. The equality $\alpha_{f_{X}}(x)=\alpha_{f_{Y}}(y)$ is a part of [MS20, Lemma 2.8]. ∎ ###### Lemma 3.2. Let $X,Y$ be projective varieties, $f_{X}\colon X\longrightarrow X$ and $f_{Y}\colon Y\longrightarrow$ surjective self-morphisms on $X$ and $Y$, respectively, and $\pi\colon X\longrightarrow Y$ a surjective morphism such that $\pi\circ f_{X}=f_{Y}\circ\pi$. * (a) If $\pi$ is birational and $(Y,f_{Y})$ satisfies $(DR)$, then $(X,f_{X})$ satisfies $(DR)$. * (b) If $\pi$ is finite and $(X,f_{X})$ satisfies the $(DR)$, then $(Y,f_{Y})$ satisfies $(DR)$. ###### Proof. * (a) Let $L$ be a finite extension of $K$ such that $(Y,f_{Y})$ satisfies $(DR)_{L}$. Let $S_{Y}=\\{y_{i}\\}_{i=1}^{\infty}\subset Y(L)$ be a sequence of $L$-rational points such that * – $S_{Y}$ is Zariski dense, * – $\alpha_{f_{Y}}(y_{i})=\delta_{f_{Y}}$ for $i\geq 1$, and * – $O_{f_{Y}}(y_{i})\cap O_{f_{Y}}(y_{j})=\emptyset$ for $i\neq j$. Let $\\{X_{j}\\}_{j=1}^{\infty}$ be the family of all the proper closed subsets of $X$. Let $U\subset X$ and $V\subset Y$ be open subsets such that $\pi\|_{U}\colon U\longrightarrow V$ is isomorphic. Then we can inductively take a subsequence $\\{y_{i_{j}}\\}_{j=1}^{\infty}\subset V$ of $S_{Y}$ such that $x_{j}:=(\pi\|_{U})^{-1}(y_{i_{j}})\not\in X_{j}$ for $j\geq 1$. Since $\alpha_{f_{X}}(x_{j})=\alpha_{f_{Y}}(y_{i_{j}})$ and $\delta_{f_{X}}=\delta_{f_{Y}}$ by Lemma 3.1, the sequence $S_{X}:=\\{x_{j}\\}_{j=1}^{\infty}$ satisfies * – $x_{j}\not\in X_{j}$ for $j\geq 1$, * – $\alpha_{f_{X}}(x_{j})=\delta_{f_{X}}$ for $j\geq 1$, and * – $O_{f_{X}}(x_{j})\cap O_{f_{X}}(x_{k})=\emptyset$ for $j\neq k$. * (b) Let $L$ be a finite extension of $K$ such that $(X,f_{X})$ satisfies $(DR)_{L}$. Let $S_{X}=\\{x_{i}\\}_{i=1}^{\infty}\subset X(L)$ be a sequence of $L$-rational points such that * – $S_{X}$ is Zariski dense, * – $\alpha_{f_{X}}(x_{i})=\delta_{f_{X}}$ for $i\geq 1$, and * – $O_{f_{X}}(x_{i})\cap O_{f_{X}}(x_{j})=\emptyset$ for $i\neq j$. Let $\\{Y_{j}\\}_{j=1}^{\infty}$ be the family of all the proper closed subsets of $Y$. Then since $\pi$ is finite, the number of $k\in\mathbb{Z}_{\geq 1}$ such that $O_{f_{Y}}(\pi(x_{k}))\cap O_{f_{Y}}(\pi(x_{i}))\neq\emptyset$ is finite for each $i\geq 1$. So we can inductively take a subset $\\{x_{i_{j}}\\}_{j=1}^{\infty}$ such that * – $y_{j}:=\pi(x_{i_{j}})\not\in Y_{j}$ for $j\geq 1$, * – $\alpha_{f_{Y}}(y_{j})=\alpha_{f_{X}}(x_{i_{j}})=\delta_{f_{X}}=\delta_{f_{Y}}$, and * – $O_{f_{Y}}(y_{j})\cap O_{f_{Y}}(y_{k})=\emptyset$ for $j\neq k$, where the second assertion follows by Lemma 3.1. ∎ ###### Lemma 3.3. Let $X$ be a projective variety and $f\colon X\longrightarrow X$ a surjective self-morphism. Then for any integer $t\geq 1$ and a finite extension $L/K$, $(X,f)$ satisfies $(DR)_{L}$ if and only if $(X,f^{t})$ satisfies $(DR)_{L}$. ###### Proof. Obviously $(X,f^{t})$ satisfies $(DR)_{L}$ if so does $(X,f)$. Conversely, assume that $(X,f^{t})$ satisfies $(DR)_{L}$. Let $S=\\{x_{i}\\}_{i=1}^{\infty}\subset X(L)$ be a subset such that * • $S$ is Zariski dense, * • $\alpha_{f^{t}}(x_{i})=\delta_{f^{t}}$ for $i=1,2,\ldots$, * • $O_{f^{t}}(x_{i})\cap O_{f^{t}}(x_{j})=\emptyset$ for $i\neq j$. Note that we have $\alpha_{f}(x)=\alpha_{f^{t}}(x)^{1/t}=\delta_{f^{t}}^{1/t}=\delta_{f}$ for $x\in S$. Thus, it is enough to prove that for any proper closed subset $Y\subset X$ and any points $x_{1}^{\prime},x_{2}^{\prime},\ldots x_{r}^{\prime}\in S$, there is a point $x\in S$ such that $x\not\in Y$ and $O_{f}(x)\cap O_{f}(x_{i}^{\prime})=\emptyset$ for $1\leq i\leq r$. Since $S$ is Zariski dense, there are infinitely many $i\geq 1$ such that $x_{i}\not\in Y$. Since the number of $i\geq 1$ such that $O_{f}(x_{i})\cap O_{f}(x_{j}^{\prime})\neq\emptyset$ is at most $t$ for each $j$. Hence we can get a point $x_{i}\in S$ which we wanted. ∎ ###### Theorem 3.4. Let $K$ be a number field. Then there is a constant $N$ such that, for any elliptic curve defined over $K$, the number of $K$-torsion points is at most $N$. ###### Proof. See [Mer96]. ∎ ###### Lemma 3.5. Let $\pi\colon X\longrightarrow B$ be an elliptic surface. Then the followings are equivalent. * (a) $X(K)$ is Zariski dense. * (b) There are infinitely many points $b\in B(K)$ such that $X_{b}=\pi^{-1}(b)$ has infinitely many $K$-rational points. ###### Proof. Clearly (b) implies (a). Assume that (b) does not hold. Then we can take an open subset $B^{\prime}\subset B$ such that $X^{\prime}=\pi^{-1}(B^{\prime})\overset{\pi}{\longrightarrow}B^{\prime}$ admits the structure of an abelian scheme and $X_{b}(K)$ is finite for any $b\in B^{\prime}(K)$. Let $N$ be an upper bound of the number of torsion points of elliptic curves defined over $K$, which we can take by Lemma 3.4. Let $[N!]\colon X^{\prime}\longrightarrow X^{\prime}$ be the morphism defined by the multiplication map by $N!$ on each fiber. Then we have $X(K)\subset\ker([N!])\ \cup\ (\pi^{-1}(B\setminus B^{\prime}))(K).$ Hence $X(K)$ is not dense. ∎ ###### Lemma 3.6. Let $f\colon X\longrightarrow X$ be a surjective self-morphism with $\delta_{f}>1$ on a normal projective variety. Assume that there is an ample $\mathbb{R}$-Cartier divisor $H$ such that $f^{\ast}H\equiv_{\mathbb{R}}\delta_{f}H$. * (a) The number of the preperiodic points are finite. * (b) $x\in X(\overline{K})$ is not preperiodic if and only if $\alpha_{f}(X)=\delta_{f}$. ###### Proof. By [MMSZZ20, Theorem 6.4 (1)], there is an ample $\mathbb{R}$-Cartier divisor $D^{\prime}$ such that $f^{\ast}D^{\prime}\sim_{\mathbb{R}}\delta_{f}D^{\prime}$. So the assertion follows from [CS93, Corollary 1.1.1]. ∎ ## 4\. Ptoof of Theorem 1.5 We divide the proof of Theorem 1.5 into two cases: the automorphism case and the non-isomorphic self-morphism case. ### 4.1. The automorphism case We start with listing known results on automorphisms of surfaces. ###### Lemma 4.1. Let $X$ be a smooth projective surface over $\mathbb{C}$, and $f\colon X\longrightarrow X$ be an automorphism with $\delta_{f}>1$. * (a) The set of eigenvalues of $f^{\ast}\|_{H^{2}(X,\mathbb{R})}$ with counted multiplicities is $\\{\delta_{f},\delta_{f}^{-1},\lambda_{1},\lambda_{2},\ldots\lambda_{\dim H^{2}(X,\mathbb{R})-2}\\},$ where $\|\lambda_{i}\|=1$ for all $i=1,2,\ldots,\dim H^{2}(X,\mathbb{R})-2$. * (b) The eigenvalues $\delta_{f}$ and $\delta_{f}^{-1}$ are irrational real numbers. Moreover, $\delta_{f}^{-1}$ is a Galois conjugate of $\delta_{f}$ over $\mathbb{Q}$. * (c) There are numerically non-zero nef $\mathbb{R}$-divisors $D^{+}$ and $D^{-}$ satisfying $f^{\ast}D^{+}\sim_{\mathbb{R}}\delta_{f}D^{+}$ and $f^{\ast}D^{-}\sim_{\mathbb{R}}\delta_{f}^{-1}D^{-}$, respectively. * (d) For a curve $C$ in $X$, $(C\cdot D^{+})=0$ holds if and only if $(C\cdot D^{-})=0$. * (e) Let $D:=D^{+}+D^{-}$. Then the set $\mathcal{C}_{f}$ of irreducible curves $C$ satisfying $(C\cdot D)=0$ is a finite set. * (f) The set $\mathcal{C}_{f}$ coincides with the set of $f$-periodic irreducible curves in $X$. * (g) For $\bullet\in\\{+,-\\}$, the set $\mathcal{C}_{f}$ coincides with the set of irreducible curves $C$ such that $(C\cdot D^{\bullet})=0$ holds. ###### Proof. * (a) See [McM02, Theorem 3.2] and [Kaw08, Theorem 2.1]. * (b) Since $f^{\ast}\|_{H^{2}(X,\mathbb{R})}$ is induced by the action of $f^{\ast}$ on $H^{2}(X,\mathbb{Z})$, an integer matrix represents $f^{\ast}\|_{H^{2}(X,\mathbb{R})}$. So $\delta_{f}$ and $\delta_{f}^{-1}$ are algebraic integers. If $\delta_{f}$ is a rational number, $\delta_{f}^{-1}$ is also a rational number, so $\delta_{f}$ and $\delta_{f}^{-1}$ are integers. Hence we get $\delta_{f}=\delta_{f}^{-1}=1$. This is contradiction. Since $f$ is an isomorphism, the constant term of the characteristic polynomial of $f^{\ast}\|_{H^{2}(X,\mathbb{R})}$ is $1$. Since the minimal polynomial of $\delta_{f}$ have integer coefficients and divides the characteristic polynomial of $f^{\ast}\|_{H^{2}(X,\mathbb{R})}$, the constant term of the minimal polynomial of $\delta_{f}$ is also $1$. Since $\|\lambda_{i}\|=1$ for $1\leq i\leq\dim H^{2}(X,\mathbb{R})-2$, the number $\delta_{f}^{-1}$ must be a Galois conjugate of $\delta_{f}$ over $\mathbb{Q}$. * (c) See [Kaw08, Proposition 2.5]. * (d) Let $F$ be the Galois closure of $\mathbb{Q}(\delta_{f})/\mathbb{Q}$. Let $\sigma\in\operatorname{Gal}(F/\mathbb{Q})$ be an automorphism sending $\delta_{f}$ to $\delta_{f}^{-1}$. Then since $D^{+}$ and $D^{-}$ lie in nef classes in $\operatorname{N}^{1}(X)\otimes_{\mathbb{Z}}\mathbb{Q}(\delta_{f})$, we have $(C\cdot D^{\bullet})\in\mathbb{Q}(\delta_{f})(\subset F)$ for any curve $C$ in $X$ and for $\bullet\in\\{+,-\\}$. Since $\delta_{f}$ and $\delta_{f}^{-1}$ are Galois conjugate over $\mathbb{Q}$, we have $\sigma D^{+}=\sigma D^{-}$, so $\sigma(C\cdot D^{+})=(C\cdot D^{-})$. * (e), (f) See [Kaw08, Proposition 3.1] * (g) If $(C\cdot D)=0$ holds, we have $(C\cdot D^{\bullet})=0$ for $\bullet\in\\{+,-\\}$ since $D^{+}$ and $D^{-}$ are nef. If $(C\cdot D^{+})=0$ holds, we get $(C\cdot D^{-})=0$ by (d), so $(C\cdot D)=(C\cdot D^{+})+(C\cdot D^{-})=0$. If $(C\cdot D^{-})=0$ holds, the similar is true. ∎ ###### Theorem 4.2. If $X$ is a smooth projective surface admitting an automorphism $f$ with $\delta_{f}>1$, then $X$ is either a non-minimal rational surface, or a surface birational to a K$3$ surface, an Enriques surface, or an abelian surface. ###### Proof. See [Can99, Proposition 1]. ∎ ###### Proposition 4.3. Let $X$ be a K$3$ surface over a number field $K$ with an infinite group of automorphisms. Then there is a rational curve $C\subset X$ such that $\\#\\{g(C)\ \|\ g\in\operatorname{Aut}(X)\\}=\infty$. ###### Proof. See the proof of [BT00, Theorem 4.10]. ∎ ###### Proposition 4.4. Let $X$ be a K$3$ surface defined over a number field $K$, and $f\colon X\longrightarrow X$ an automorphism with $\delta_{f}>1$. Then there exists a rational curve $C\subset X$ such that $\\#\\{f^{n}(C)\ \|\ n\geq 0\\}=\infty$ ###### Proof. Since $X$ contains only finitely many $f$-periodic curves by Lemma 4.1(e),(f), and there are infinitely many rational curves in $X$ by Proposition 4.3, there is a rational curve $C\subset X$ which is not $f$-periodic. ∎ ###### Proposition 4.5. Let $X$ be a projective variety and $f:X\longrightarrow X$ a surjective morphism with $\delta_{f}>1$. Assume the following condition: (†): There is a nef $\mathbb{R}$-Cartier divisor $D$ on $X$ such that $f^{}D\sim_{\mathbb{R}}\delta_{f}D$ and for any proper closed subset $Y\subset X_{\overline{K}}$, there exists a non-constant morphism $g:\mathbb{P}^{1}_{K}\longrightarrow X$ such that $g(\mathbb{P}^{1}_{K})\not\subset Y$ and $g^{}D$ is ample. Then $(X,f)$ satisfies the condition $(DR)_{K}$. ###### Proof. See [SS20, Theorem 4.1]. ∎ ###### Proof of Theorem 1.5 when $f$ is an automorphism. If $X$ is rational, our assertion is true by [SS20]. Assume that $X$ is irrational. Take a birational morphism $\mu\colon X\longrightarrow Y$ to the minimal model $Y$ of $X$ and let $g:Y\dashrightarrow Y$ be the birational automorphism on $Y$ defined as $g=\mu\circ f\circ\mu^{-1}$. Then $g$ is in fact an automorphism since, if $g$ has indeterminacy, $Y$ must have a $K_{Y}$-negative curve. By Theorem 4.2 and Lemma 3.2, we may assume that $X$ is either a K$3$ surface, an Enriques surface, or an abelian variety. If $X$ is an abelian variety, our assertion is true by [SS20]. If $X$ is an Enriques surface, take the universal covering $\pi\colon\tilde{X}\longrightarrow X$. Then an automorphism $\tilde{f}\colon\tilde{X}\longrightarrow\tilde{X}$ such that $\pi\circ\tilde{f}=f\circ\pi$ is induced and $\tilde{X}$ is a K$3$ surface. Hence by Lemma 3.2, we may assume that $X$ is a K$3$ surface. Now it is enough to prove that (†) in Proposition 4.5. Let $Y$ be a proper closed subset of $X$. Take a rational curve $\iota\colon C\hookrightarrow X$ suth that $\\#\\{f^{n}\circ\iota(C)\ \|\ n\geq 0\\}=\infty$ by Proposition 4.4. Then there is an integer $n_{Y}\geq 0$ such that $C_{Y}:=f^{n_{Y}}\circ\iota(C)\not\subset Y$. Let $g_{Y}:=f^{n_{Y}}\circ\iota$. Since $C_{Y}$ is not $f$-periodic, we have $(C_{Y}\cdot D^{+})>0$ by Lemma 4.1 (e), (f), so $g_{Y}^{\ast}D^{+}$ is ample. Hence (†) in Proposition 4.5 is proved. ∎ ### 4.2. The non-isomorphic surjective self-morphism case We prepare the following lemmata to reduce to a minimal surface. ###### Lemma 4.6. Let $f\colon X\longrightarrow X$ be a non-isomorphic surjective self-morphism on a smooth projective irrational surface $X$ with $\kappa(X)=-\infty$. Then there is a positive integer $t$, a birational morphism $\mu\colon X\longrightarrow X^{\prime}$ to a $\mathbb{P}^{1}$-bundle over a curve $B$ with $g(B)\geq 1$, and a surjective self-morphism $f^{\prime}\colon X^{\prime}\longrightarrow X^{\prime}$ such that the equality $\mu\circ f^{t}=f^{\prime}\circ\mu$ holds. ###### Proof. By [Nak02, Proposition 10], any $(-1)$-curve on $X$ is $f$-periodic. So the assertion follows. ∎ ###### Lemma 4.7. Let $f\colon X\longrightarrow X$ be a non-isomorphic surjective self-morphism on a smooth projective surface $X$ with $\kappa(X)\geq 0$. Then $X$ is minimal. ###### Proof. See [Fuj02, Lemma 2.3 and Proposition 3.1]. ∎ ###### Proof of Theorem 1.5 when $f$ is not an automorphism. We prove it by using the Enriques–Kodaira Classification and case-by-case analysis. * (I) $\kappa(X)=-\infty$. By Lemma 4.6, Lemma 3.2, and Lemma 3.3, we may assume that $X$ is either a rational surface or a $\mathbb{P}^{1}$-bundle over a curve $B$ with genus $g(B)\geq 1$. If $X$ is rational, the assertion follows by [SS20, Theorem 1.11]. If $g(B)\geq 2$, $X$ does not have potentially dense rational points. If $g(B)=1$, $X$ is a $\mathbb{P}^{1}$-bundle over an elliptic curve. This case is proved in [SS20, Theorem 6.1]. From now on, assume that $\kappa(X)\geq 0$. Then $X$ is minimal by Lemma 4.7. * (II) $\kappa(X)=0$. By [Fuj02, Theorem 3.2], $X$ is either a hyperelliptic surface or an abelian surface. * (II-i) The hyperelliptic surface case. In this case, there is an equivariant finite covering from an abelian variety. See e.g. the proof of [MSS18, Theorem 7.1] for details. By taking such an equivariant finite covering and applying Lemma 3.2, we can reduce to the abelian surface case. * (II-ii) The abelian surface case. More generally the abelian variety case is proved in [SS20, Theorem 1.12]. * (III) $\kappa(X)=1$. We treat this case below. * (IV) $\kappa(X)=2$. This case does not occur since any surjective self-morphisms on $X$ are automorphisms of finite order. Thus the remaining case is the $\kappa(X)=1$ case. Then $X$ admits an elliptic fibration $\pi\colon X\longrightarrow B$ and $f$ descends to an automorphism of finite order on $B$ (cf. [MZ19, Theorem A] or [MSS18, Theorem 8.1]). Replacing $f$ by some iteration $f^{t}$, we may assume that $f$ is a morphism over the base curve $B$. Let $\\{Y_{i}\\}_{i=1}^{\infty}$ be the set of all the proper closed subsets of $X$. It is enough to get $\\{x_{i}\\}_{i=1}^{\infty}\subset X(L)$ such that * • $x_{i}\not\in Y_{i}$, * • $\alpha_{f}(x_{i})=\delta_{f}$ for $i=1,2,\ldots$, and * • $O_{f}(x_{i})\cap O_{f}(x_{j})=\emptyset$ for $i\neq j$. Let $L$ be a finite extension field of $K$ such that $X(L)$ is Zariski dense. By Lemma 3.5, we can find an infinite subset $\\{b_{j}\\}_{j=1}^{\infty}\subset B(L)$ such that $X_{b_{j}}=\pi^{-1}(b_{j})$ contains infinitely many $L$-rational points for each $j$. Removing special fibers, we may assume that $X_{b_{j}}$ is an elliptic curve and $f\|_{X_{b_{j}}}$ has dynamical degree $\delta_{f}$. By Lemma 3.6, there are infinitely many $L$-rational points $x\in X_{b_{j}}(L)$ with $\alpha_{f}(x)=\delta_{f}$ for each $j$. Hence, letting $\\{b_{j_{i}}\\}_{i=1}^{\infty}$ be a subsequence of $\\{b_{j}\\}_{j=1}^{\infty}$ such that $X_{b_{j_{i}}}$ is not contained $Y_{i}$, we can find $x_{i}\in X_{b_{j_{i}}}(L)$ such that $x_{i}\not\in Y_{i}$. Since $f$ is a morphism over $B$, we have $O_{f}(x_{k})\cap O_{f}(x_{l})=\emptyset$ for $k\neq l$. The assertion is proved. ∎ ## References * [BT00] F. A. Bogomolov, Y. Tschinkel, Density of rational points on elliptic K3 surfaces, Asian J. Math. 4 (2000), no. 2, 351–368. * [CS93] G. S. Call, J. H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. * [Can99] S. Cantat, Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris S’er. I Math. 328 (1999), 901–906. * [Fuj02] Y. Fujimoto, Endomorphisms of smooth projective 3-folds with nonnegative Kodaira dimension, Publ. RIMS, Kyoto Univ. 38 (2002), 33–92. * [HS00] M. Hindry, J. H. Silverman, Diophantine Geometry: An Introduction, Springer-Verlag, New York, 2000. * [KS16a] S. Kawaguchi, J. H. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21–48. * [KS16b] S. Kawaguchi, J. H. Silverman, Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5009–5035. * [Kaw08] S. Kawaguchi, Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint, Amer. J. Math. 130 (2008), no. 1, 159–186. * [Lan83] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983. * [Mat16] Y. Matsuzawa, On upper bounds of arithmetic degrees, arXiv:1606.00598 * [McM02] C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002), 201–233. * [MS20] Y. Matsuzawa, K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655–1672. * [MSS18] Y. Matsuzawa, K. Sano, T. Shibata, Arithmetic degrees and dynamical degrees of endomorphisms on surfaces, Algebra Number Theory 12 (2018), no. 7, 1635–1657. * [MMSZ20] Y. 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4(0.5,1) To cite: Prerna Juneja and Tanushree Mitra. 2021. Auditing E-Commerce Platforms for Algorithmically Curated Vaccine Misinformation. In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems (CHI ’21). Association for Computing Machinery. DOI: https://doi.org/10.1145/3411764.3445250 # Auditing E-Commerce Platforms for Algorithmically Curated Vaccine Misinformation Prerna Juneja The Information School University of WashingtonSeattleWA, USA<EMAIL_ADDRESS>and Tanushree Mitra The Information School University of WashingtonSeattleUSA<EMAIL_ADDRESS> (2021) ###### Abstract. There is a growing concern that e-commerce platforms are amplifying vaccine- misinformation. To investigate, we conduct two-sets of algorithmic audits for vaccine misinformation on the search and recommendation algorithms of Amazon—world’s leading e-retailer. First, we systematically audit search- results belonging to vaccine-related search-queries without logging into the platform—unpersonalized audits. We find 10.47% of search-results promote misinformative health products. We also observe ranking-bias, with Amazon ranking misinformative search-results higher than debunking search-results. Next, we analyze the effects of personalization due to account-history, where history is built progressively by performing various real-world user-actions, such as clicking a product. We find evidence of filter-bubble effect in Amazon’s recommendations; accounts performing actions on misinformative products are presented with more misinformation compared to accounts performing actions on neutral and debunking products. Interestingly, once user clicks on a misinformative product, homepage recommendations become more contaminated compared to when user shows an intention to buy that product. search engines, health misinformation, vaccine misinformation, algorithmic bias, personalization, algorithmic audits, search results, recommendations, e-commerce platforms ††journalyear: 2021††copyright: acmlicensed††conference: CHI Conference on Human Factors in Computing Systems; May 8–13, 2021; Yokohama, Japan††booktitle: CHI Conference on Human Factors in Computing Systems (CHI ’21), May 8–13, 2021, Yokohama, Japan††price: 15.00††doi: 10.1145/3411764.3445250††isbn: 978-1-4503-8096-6/21/05††ccs: Information systems Personalization††ccs: Information systems Content ranking††ccs: Human- centered computing Human computer interaction (HCI)††ccs: Information systems Web crawling ## 1\. Introduction The recent onset of coronavirus pandemic has unleashed a barrage of online health misinformation (Ball and Maxmen, 2020; Financial, 2020) and renewed focus on the anti-vaccine movement, with anti-vax social media accounts witnessing a 19% increase in their follower base (Owen, 2020). As scientists work towards creating a vaccine for the disease, health experts worry that vaccine hesitancy could make it difficult to achieve herd immunity against the new virus (Ball, 2020). Battling health misinformation, especially anti- vaccine misinformation has never been more important. Statistics show that people increasingly rely on the internet (Rainie and Fox, 2000), and specifically online search engines (Center, 2006), for health information including information about medical treatments, immunizations, vaccinations and vaccine-related side effects (Fox, 2006; Bragazzi et al., 2017). Yet, the algorithms powering search engines are not traditionally designed to take into account the credibility and trustworthiness of such information. Search platforms being the primary gateway and reportedly the most trusted source (Edelman and Luca, 2014), persistent vaccine misinformation on them, can cause serious health ramifications (Kata, 2010). Thus, there has been a growing interest in empirically investigating search engine results for health misinformation. While multiple studies have performed audits on commercial search engines to investigate problematic behaviour (Hu et al., 2019; Robertson et al., 2018; Hussein et al., 2020), e-commerce platforms have received little to no attention ((Chen et al., 2016; Shin and Valente, 2020) are two exceptions), despite critics calling e-commerce platforms, like Amazon, a “dystopian” store for hosting anti- vaccine books (Diresta, 2019). Amazon specifically has faced criticism from several technology critics for not regulating health-related products on its platform (Reynolds, 2019; Belluz, 2016). Consider the most recent instance. Several medically unverified products for coronavirus treatment, like prayer healing, herbal treatments and antiviral vitamin supplements proliferated Amazon (Goldhill, 2020; Dreisbach, 2020), so much so that the company had to remove 1 million fake products after several instances of such treatments were reported by the media (Financial, 2020). The scale of the problematic content suggests that Amazon could be a great enabler of misinformation, especially health misinformation. It not only hosts problematic health-related content but its recommendation algorithms drive engagement by pushing potentially dubious health products to users of the system (Glaser, 2017; Shin and Valente, 2020). Thus, in this paper we investigate Amazon—world’s leading e-retailer—for most critical form of health misinformation—vaccine misinformation. What is the amount of misinformation present in Amazon’s search results and recommendations? How does personalization due to user history built progressively by performing real-world user actions, such as clicking or browsing certain products, impact the amount of misinformation returned in subsequent search results and recommendations? In this paper, we dabble into these questions. We conduct 2 sets of systematic audit experiments: _Unpersonalized audit_ and _Personalized audit_. In the _Unpersonalized audit_ , we adopt Information Retrieval metrics from prior work (Kulshrestha et al., 2017) to determine the amount of health misinformation users are exposed to when searching for vaccine-related queries. In particular, we examine search- results of 48 search queries belonging to 10 popular vaccine-related topics like ‘hpv vaccine’, ‘immunization’, ‘MMR vaccine and autism’, etc. We collect search results without logging in to Amazon to eliminate the influence of personalization. To gain in-depth insights about the platform’s searching and sorting algorithm, our _Unpersonalized audits_ ran for 15 consecutive days, sorting the search results across 5 different Amazon filters each day: “featured”, “price low to high”, “price high to low”, “average customer review” and “newest arrivals”. The first audit resulted in 36,000 search results and 16,815 product page recommendations which we later annotated for their stance on health misinformation—promoting, neutral or debunking. In our second set of audit— _Personalized audit_ , we determine the impact of personalization due to user history on the amount of health misinformation returned in search results, recommendations and auto-complete suggestions. User history is built progressively over 7 days by performing several real- world actions, such as “search” , “search + click” , “search + click + add to cart” , “search + click + mark top-rated all positive review as helpful” , “follow contributor” and “search on third party website” ( Google.com in our case) . We collect several Amazon components in our Personalized audit, like homepages, product pages, pre-purchase pages, search results, etc. Our audits reveal that Amazon hosts a plethora of health misinformative products belonging to several categories, including Books, Kindle eBooks, Amazon Fashion (e.g. apparel, t-shirt, etc.) and Health & Personal care items (e.g. dietary supplements). We also establish the presence of a filter-bubble effect in Amazon’s recommendations, where recommendations of misinformative health products contain more health misinformation. Below we present our formal research questions, key findings, contributions and implication of this study along with ethical considerations taken for conducting platform audits. ### 1.1. Research Questions and Findings In our first set of audits, we ask, RQ1 [_Unpersonalized audit_]: What is the amount of health misinformation returned in various Amazon components, given components are not affected by user personalization? * RQ1a: How much are the Amazon’s search results contaminated with misinformation? * RQ1b: How much are recommendations contaminated with misinformation? Is there a filter-bubble effect in recommendations? We find a higher percentage of products promoting health misinformation (10.47%) compared to products that debunk misinformation (8.99%) in the unpersonalized search results. We discover that Amazon returns high number of misinformative search results when users sort their searches by filter “featured” and high number of debunking results when they sort results by filter “newest arrivals”. We also find Amazon ranking misinformative results higher than debunking results especially when results are sorted by filters “average customer reviews” and “price low to high”. Overall, search results of topics “vaccination”, “andrew wakefield” and “hpv vaccine” contain the highest misinformation bias when sorted by default filter “featured”. Our analysis of product page recommendations suggests that recommendations of products promoting health misinformation contain more health misinformation when compared to recommendations of neutral and debunking products. RQ2 [_Personalized audit_]: What is the effect of personalization due to user history on the amount of health misinformation returned in various Amazon components, where user history is built progressively by performing certain actions? * RQ2a: How are _search results_ affected by various user actions? * RQ2b: How are _recommendations_ affected by various user actions? Is there a filter-bubble effect in the recommendations? * RQ2c: How are the _auto-complete suggestions_ affected by various user actions? Our _Personalized audit_ reveals that search results sorted by filters “average customer review”, “price low to high” and “newest arrivals” along with auto-complete suggestions are not personalized. Additionally, we find that user actions involving clicking a search product leads to personalized homepages. We find evidence of filter-bubble effect in various recommendations found in homepages, product and pre-purchase pages. Surprisingly, the amount of misinformation present in homepages of accounts building their history by performing actions “search + click” and “mark top-rated all positive review as helpful” on misinformative products was more than the amount of misinformation present in homepages of accounts that added the same misinformative products in cart. The finding suggests that Amazon nudges users more towards misinformation once a user shows interest in a misinformative product by clicking on it but hasn’t shown any intention of purchasing it. Overall, our audits suggest that Amazon has a severe vaccine/health misinformation problem exacerbated by its search and recommendation algorithms. Yet, the platform has not taken any steps to address this issue. ### 1.2. Contributions and Implications In the absence of an online regulatory body monitoring the quality of content created, sold and shared, vaccine misinformation is rampant on online platforms. Through our work, we specifically bring the focus on e-commerce platforms since they have the power to influence browsing as well as buying habits of millions of people. We believe our study is the first large-scale systematic audit of an e-commerce platform that investigates the role of its algorithms in surfacing and amplifying vaccine misinformation. Our work provides an elaborate understanding of how Amazon’s algorithm is introducing misinformation bias in product selection stage and ranking of search results across 5 Amazon filters for 10 impactful vaccine-related topics. We find that even use of different search filters on Amazon can dictate what kind of content a user can be exposed to. For example, use of default filter “featured” lead users to more health misinformation while sorting search results by filter “newest arrivals” lead users to products debunking health- related misinformation. Ours is also the first study to empirically establish how certain real-world actions on health misinformative products on Amazon could drive users into problematic echo chambers of health misinformation. Both our audit experiments resulted in a dataset of 4,997 unique Amazon products distributed across 48 search queries, 5 search filters, 15 recommendation types, and 6 user actions, conducted over 22 (15+7) days 111https://social-comp.github.io/AmazonAudit-data/. Our findings suggest that traditional recommendation algorithms should not be blindly applied to all topics equally. There is an urgent need for Amazon to treat vaccine related searches as searches of higher importance and ensure higher quality content for them. Finally, our findings also have several design implications that we discuss in detail in Section 7.4. ### 1.3. Ethical Considerations We took several steps to minimize the potential harm of our experiments to retailers. For example, buying and later returning an Amazon product for the purpose of our project can be deemed unethical and thus, we avoid performing this activity. Similarly, writing a fake positive review about an Amazon product containing misinformation could negatively influence the audience. Therefore, in our _Personalized audit_ we explored other alternatives that could mimic similar if not the same influence as the aforementioned activities. For example, instead of buying a product, we performed ”add to cart” action that shows users’ intent to purchase a product. Instead of writing positive reviews for products, we marked top rated positive review as helpful. Since, accounts did not have any purchase history, marking a review helpful did not increase the “Helpful” count for that review. Through this activity, the account shows positive reaction towards the product, at the same time avoids manipulation and thus, eliminates impacting potential buyers or users. Lastly, we refrained from performing the experiments on real-world users. Performing actions on misinformative products could contaminate users’ searches and recommendations. It could potentially have long-term consequences in terms of what types of products are pushed at participants. Thus, in our audit experiments, accounts were managed by bots that emulated the actions of actual users. ## 2\. Related work ### 2.1. Health misinformation in online systems The current research on online health misinformation including vaccine misinformation spans three broad themes: 1) quantifying the characteristics of anti-vaccine discourse (Mitra et al., 2016; Mønsted and Lehmann, 2019; Cossard et al., 2020), 2) building machine learning models to identify users engaging with health misinformation or instances of health misinformation itself (Ghenai and Mejova, 2018; Dai et al., 2020; Ghenai and Mejova, 2017) and 3) designing and evaluating effective interventions to ensure that users critically think when presented with health (mis)information (Kim et al., 2020; van der Meer and Jin, 2020). Most of these studies are post-hoc investigations of health misinformation, i.e the misinformation has already propagated. Moreover, existing scholarship rarely takes into account how the user encountered health misinformation or what role is played by the source of the misinformation. With the increasing reliance on online sources for health information, search engines have become the primary avenue of such information, with 55% of American adults relying on the web to get medical information (Rainie and Fox, 2000). A Pew survey reports that for 5.9M people, web search results influenced their decision to visit a doctor and 14.7M claimed that online information affected their decision on how to treat a disease (Rainie and Fox, 2000). Given how medical information can directly influence one’s health and well-being, it is essential that search engines return quality results in response to health related search queries. However, currently online health information has been contaminated by several outlets. These sources could be conspiracy groups or websites spreading misinformation due to vested interests or companies having commercial interests in selling herbal cures or fictitious medical treatments (Schwitzer, 2017). Moreover, online curation algorithms themselves are not built to take into account the credibility of information. Thus, it is of paramount importance that the role of search engines are investigated for harvesting health misinformation. How can we empirically and systematically probe search engines to investigate problematic behaviour like prevalence of health misinformation? In the next section, we briefly describe the emerging research field of “algorithmic auditing” that is focused on investigating search engines to reveal problematic biases. We discuss this field as well as our contribution to this growing research space in the next section. ### 2.2. Search engine audits Search engines are modern day gatekeepers and curators of information. Their black-box algorithm can shape user behaviour, alter beliefs and even affect voting behaviour either by impeding or facilitating the flow of certain kinds of information (Epstein and Robertson, 2015; Diakopoulos et al., 2018; Knobloch-Westerwick et al., 2015). Despite their importance and the power they exert, till date, search engine results and recommendations have mostly been unregulated. Information quality of search engine’s output is still measured in terms of relevance and it is up to the user to determine the credibility of information. Thus, researchers have advocated for making algorithms more accountable. One primary method to achieve this is to perform systematic audits to empirically establish the conditions under which problematic behavior surfaces. Raji et al provide the following definition of algorithmic audits. An algorithmic audit involves the collection and analysis of outcomes from a fixed algorithm or defined model within a system. Through the stimulation of a mock user population, these audits can uncover problematic patterns in models of interest (Raji and Buolamwini, 2019). (a) [Amazon homepage]Figure illustrates the Amazon homepage containing several books belonging to three different recommendation types specified in Table 1. (b) [Amazon pre-purchase page]Figure illustrates the Amazon pre-purchase page with several books belonging to different recommendation types. (c) [Amazon product page]Figure illustrates several books belonging to five different recommendation types present on the Amazon product page specified in Table 1. Figure 1. (a) Amazon homepage recommendations. (b) Pre-purchase recommendations displayed to users after adding a product to cart. (c) Product page recommendations. Previous audit studies have investigated the search engines for partisan bias (Robertson et al., 2018; Mustafaraj et al., 2020), gender bias (Chen et al., 2018; Kay et al., 2015), content diversity (Trielli and Diakopoulos, 2019; Steiner et al., 2020; Puschmann, 2019), and price discrimination (Hannak et al., 2014). However, only a few have systematically investigated search engines’ role in surfacing misinformation ((Hussein et al., 2020) is the only exception). Moreover, there is a dearth of systematic audits focusing specifically on health misinformation. The past literature, mostly consists of small-scale experiments that probe search engines with a handful of search queries. For example, an analysis of the first 30 pages of search results for query “vaccines autism” revealed that Google.com has 10% less anti-vaccine search results compared to the other search engines, like Qwant, Swisscows and Bing (Ghezzi et al., 2020). Whereas, search results present in the first 102 pages for the query “autism vaccine” on Google’s Turkey version returned 20% websites with incorrect information (Erden et al., 2019). One recently published work, closely related to this study, examined Amazon’s first 10 pages of search results in response to the query “vaccine”. They only collected and annotated books appearing in the searches for misinformation (Shin and Valente, 2020). The aforementioned studies probed the search engine for one single query and did the analysis on multiple search results pages. We, on the other hand, perform our _Unpersonalized audit_ on a curated list of 48 search queries belonging to 10 most searched vaccine-related topics, spanning various combinations of search filters and recommendation types, over multiple days—an aspect missing in prior work. Additionally, we are the first ones to experimentally quantify the prevalence of misinformation in various search queries, topics, and filters on an e-commerce platform. Furthermore, instead of just focusing on books, we analyze the platform for products belonging to different categories, resulting in an extensive all-category inclusive coding scheme for health misinformation. Another recent study on YouTube, audited the platform for various misinformative topics including vaccine controversies (Hussein et al., 2020). The work established the effect of personalization due to watching videos on the amount of misinformation present in search results and recommendations on YouTube. However, there are no studies investigating the impact of personalization on misinformation present in the product search engines of e-commerce platforms. Our work fills this gap by conducting a second set of audit— _Personalized audit_ where we shortlist several real-world user actions and investigate their role in amplifying misinformation in Amazon’s searches and recommendations. \| Recommend- --- ation page Recommendation types Homepage \| Related to items you’ve viewed Inspired by your shopping trends” \| Recommended items other customers often buy again --- Pre-purchase page \| \| Customers also bought these highly rated items --- Customers also shopped these items Related to items you’ve viewed Frequently bought together Related to items Sponsored products related Top picks for Product page \| Frequently bought together \| Customers who bought this item also bought --- \| Customers who viewed this item also viewed --- \| Sponsored products related to this item --- \| What other items customers buy after viewing this item --- Table 1. Table showing 15 recommendation types spread across 3 recommendation pages. (a) [Google Trends’ Related topics]Figure illustrates the Related topics section of the Google Trends page for topic vaccine. The section displays topics related to vaccine such as vaccination, influenza, HPV vaccine etc. (b) [Google Trends’ Related queries]Figure illustrates the Related queries section of the Google Trends page for topic vaccine. The section contains search queries related to topic vaccine such as vaccine, vaccines, vaccination, flu vaccine etc. (c) [Amazon’s auto-complete suggestions]Figure illustrates Amazon’s auto-complete suggestions for query anti vaccine. Some of the search query suggestions displayed are anti vaccine shirt, anti vaccine books,anti vaccine mask etc. Figure 2. (a) Google Trends’ Related Topics list for topic vaccine. People who searched for vaccine topic also searched for these topics. (b) Google Trends’ Related queries list for topic vaccine. These are the top search queries searched by people related to vaccine topic. (c) Amazon’s auto-complete suggestions displaying popular and trending search queries. ## 3\. Amazon components and terminology For the audits, we collected 3 major Amazon components and numerous sub- components. We list them below. 1. (1) Search results: These are products present on Amazon’s Search Engine Results Page (SERP) returned in response to a search query. SERP results can be sorted using five filters: “featured”, “price low to high,” “price high to low,” “average customer review” and “newest arrivals.” 2. (2) Auto-complete suggestions: These are the popular and trending search queries suggested by Amazon when a query is typed into the search box (see Figure 2(c)). 3. (3) Recommendations: Amazon presents several recommendations as users navigate through the platform. For the purpose of this project, we collect recommendations present on three different Amazon pages: homepage, pre- purchase page and product pages. Each page hosts several types of recommendations. Table 1 shows the 15 recommendation types collected across 3 recommendation pages. We describe all three recommendations below. 1. (a) Homepage recommendations: These recommendations are present on the homepage of a user’s Amazon account. They could be of three types namely, “Related to items you’ve viewed”, “Inspired by your shopping trends” and “Recommended items other customers often buy again” (see Figure 1(a)). Any of the three types together or separately could be present on the homepage depending on the actions performed by the user. For example, “Inspired by your shopping trends” recommendation type appears when a user performs one of two actions: either makes a purchase or adds a product to cart. 2. (b) Pre-purchase recommendations: These recommendations consist of product suggestions that are presented to users after they add product(s) to cart. These recommendations could be considered as a nudge to purchase other similar products. Figure 1(b) displays pre-purchase page. The page has several recommendations like “Frequently bought together”, “Customers also bought these highly rated items”, etc. We collectively call these recommendations as pre-purchase recommendations. 3. (c) Product recommendations: These are the recommendations present on the product page, also known as details page222https://sellercentral.amazon.com/gp/help/external/51. The page contains details of an Amazon product, like product title, category (e.g., Amazon Fashion, Books, Health & Personal care, etc.), description, price, star rating, number of reviews, and other metadata. The details page is home to several different types of recommendations. We extracted five: “Frequently bought together”, “What other items customers buy after viewing this item”, “Customers who viewed this item also viewed”, “Sponsored products related to this item” and “Customers who bought this item also bought”. Figure 1(c) presents an example of product page recommendations. Figure 3. Figure illustrating the breadth-wise topic discovery approach used to collect vaccine-related topics from Google Trends starting from two seed topics: vaccine and vaccine controversies. Each node in the tree denotes a vaccine-related topic. An edge A$\rightarrow$ B indicates that topic B was discovered from the Trends’ Related Topic list of topic A. For example, topics “vaccination” and “andrew wakefield” were obtained from the Trends’ Related Topic list of “vaccine controversies” topic. Then, topic “mmr vaccine and autism” was obtained from topic “andrew wakefield” and so on. indicates the topic was discarded during filtering. Similar colored square brackets indicate similar topics that were merged together. [Topic discovery approach]The figure contains two trees with roots as our seed topics namely, vaccine and vaccine controversies. The children of each node are the topics discovered from the Related Topic list of the parent. For example, topics vaccination and andrew wakefield are children of root node vaccine controversies. Furthermore, topics vaccination schedule, rabies vaccine and immunization are child nodes of topic vaccination. # \| Search topic \| Seed query \| \| Sample search --- queries \| # \| Search topic \| Seed query \| \| Sample search --- queries 1 \| vaccine controversies \| vaccine controversy/ anti vaccine \| anti vaccination \| \| 6 \| mmr vaccine and autism \| mmr autism/ vaccine autism \| autism anti vaccine shirt \| \| autism vaccine 2 \| vaccination \| vaccine/ vaccination \| vaccine \| \| 7 \| influenza vaccine \| varicella vaccine \| flu shot vaccine friendly me \| \| influenza vaccine 3 \| andrew wakefield \| andrew wakefield \| andrew wakefield \| \| 8 \| hepatitis vaccine \| hepatitis vaccine \| hepatitis b vaccine wakefield autism \| \| hepatitis a vaccine 4 \| hpv vaccine \| hpv vaccine \| vaccine hpv \| \| 9 \| varicella vaccine \| varicella vaccine \| chicken pox hpv vaccine on trial \| \| \| varicella vaccine 5 \| immunization \| immunization \| immunization \| \| 10 \| mmr vaccine \| mmr vaccine \| mmr vaccine immunization book \| \| \| measles vaccination Table 2. Sample search queries for each of the ten vaccine-related search topics. ## 4\. Methodology Here we present our audit methodology in detail. This section is organized as follows. We start by describing our approach to compile high impact vaccine related topics and associated search queries (section 4.1). Then, we present overview of each audit experiment followed by the details of numerous methodological decisions we took while designing our audits (section 4.2 and section 4.3). Next, we describe our qualitative coding scheme for annotating Amazon products for health misinformation (section 4.4). Finally, we discuss our approach to calculate misinformation bias in search results (section 4.5). ### 4.1. Compiling high impact vaccine-related topics and search queries Here, we present our methodology to curate high impact vaccine-related topics and search queries. #### 4.1.1. Selecting high impact search topics: The first step of any audit is to determine input—a viable set of topics and associated search queries that will be used to query the platform under investigation. We leveraged Google Trends (_Trends_ henceforth) to select and expand vaccine-related search topics. _Trends_ is an optimal choice since it shares past search trends and popular queries searched by people across the world. Since it is not practical to audit all topics present on _Trends_ , we designed a method to curate a reasonable number of high impact topics and associated search queries, i.e., topics that were searched by a large number of people for the longest period of time. We started with 2 seed topics and employed a breadth-wise search to expand our topic list. _Trends_ allows to search for any subject matter either as a topic or a term. Intuitively, topic can be considered as a collection of terms that share a common concept. Searching as a term returns results that include terms present in the search query while searching as a topic returns all search terms having same meaning as the topic333https://support.google.com/trends/answer/4359550?hl=en. We began our search with two seed words namely “vaccine” and “vaccine controversies” and decided to search them as topics. Starting our topic search by the aforementioned seed words ensured that the related topics will cover general vaccine-related topics and topics related to controversies surrounding the vaccines, offering us a holistic view of search interests. We set location to United States, date range to 2004-Present (this step was performed in Feb, 2020), categories to “All” and search service to “Web search”. The date range ensured that the topics are perennial, and have been popular for a long time (note that _Trends_ data is available from 1/1/2004 onwards). We selected the category setting as “All” so as to get a holistic view of the search trends encompassing all the categories together. Search service filter has options like ‘web search’, ‘YouTube search’, ‘Google Shopping’, etc. Although Google shopping is an e-commerce platform like Amazon, its selection returned handful to no results. Thus, we opted for ‘web search’ service filter. We employed _Trends’_ Related Topics feature for breadth-wise expansion of search topics (see Figure 2(a)). We viewed the Related Topics using “Top” filter which presents popular search topics in the selected time range that are related to the topic searched. We manually went through the top 15 Related Topics and retained relevant topics using the following guidelines. All generic topics like Infant, Travel, Side-Effects, Pregnancy CVS, etc. were discarded. Our focus was to only pick topics representing vaccine information. Thus, we discarded topics that were names of diseases but kept their corresponding vaccines. For example, we discarded topic Influenza but kept the topic Influenza vaccine. We kept track of duplicates and discarded them from the search. To further expand the topics list, we again went through the Related Topics list of the shortlisted topics and used the aforementioned filtering strategy to shortlist relevant topics. This step allowed us to expand our topic list to a reasonable number. After two levels of breadth-wise search, we obtained a list of 16 vaccine-related search topics (see Figure 3). Figure 4. Eight steps performed in Unpersonalized audit. The steps are described in detail in Section 4.2.4 [Experimental steps of Unpersonalized audit]Figure illustrates the eight steps performed in our Unpersonalized audit. These steps are explained in the method section in detail. Next, we combined multiple similar topics into a single topic. The idea is to collect search queries for both topics separately and then combine them under one single topic. For example, topics zoster vaccine and varicella vaccine were combined since both the vaccines are used to prevent chickenpox. Thus, later search queries of both topics were combined under topic varicella vaccine. All topics enclosed with similar colored boxes in Figure 3 were merged together. 11 topics remained after merging. #### 4.1.2. Selecting high impact search queries: After shortlisting a reasonable number of topics, next we determined the associated search queries per topic, to be later used for querying Amazon’s search engine. To compile search queries, we relied on both _Trends_ and Amazon’s auto-complete suggestions; _Trends_ , because it gives a list of popular queries that people searched on Google—the most popular search service, and Amazon, because it is the platform under investigation and it will provide popular trending queries specific to the platform. Searching for a topic on _Trends_ displays popular search queries related to the topic (see Figure 2(b)). We obtained top 3 queries per topic. Next, we collected Top 3 auto-complete suggestions obtained by typing seed query of each topic into Amazon’s search box (see Figure 2(c)). We removed all animal or pet related search queries (e.g “rabies vaccine for dogs”), overly specific queries (e.g. “callous disregard by andrew wakefield”) and replaced redundant and similar queries with a single search query selected at random. For example search queries “flu shots” and “flu shot” were replaced with a single search query “flu shot”. After these filtering steps, only one query remained in the query list of topic vaccination schedule, and thus, it was removed from the topic list. Finally, we had 48 search queries corresponding to 10 vaccine- related search topics. Table 2 presents sample search queries for all 10 search topics. ### 4.2. RQ1: Unpersonalized Audit #### 4.2.1. Overview The aim of the Unpersonalized audit is to determine the amount of misinformation present in Amazon’s search results and recommendations without the influence of personalization. We measure the amount of misinformation by determining the misinformation bias of the returned results. We explain the misinformation bias calculation in detail in Section 4.5. Intuitively, more the number of higher ranked misinformative results, higher the overall bias. We ran the Unpersonalized audit for 15 days, from 2 May, 2020 to 16 May, 2020. We took two important methodological decisions regarding which components to audit and what sources of noise to control for. We present these decisions as well as implementation details of the audit experiment below. #### 4.2.2. What components should we collect for our Unpersonalized audits? We collected SERPs sorted by all 5 Amazon filters: “featured”, “price low to high”, “price high to low”, “average customer review” and “newest arrivals”. For analysis, we extracted the top 10 search results from each SERP. Since 70% of Amazon users never click on search results beyond the first page (Baker, 2018), count 10 is a reasonable approximation of the number of search results users are likely to engage with. Recent statistics have also shown that the first three search results receive 75% of all clicks (Dean, 2019). Thus, we extracted the recommendations present on the product pages of the first three search results. We collected following 5 types of product page recommendations: “Frequently bought together”, “What other items customers buy after viewing this item”, “Customers who viewed this item also viewed”, “Sponsored products related to this item” and “Customers who bought this item also bought”. Refer Figure 1(c) for an example. We extracted the first product present in each recommendation type for analysis. Next, we annotated all collected components as promoting, neutral or debunking health misinformation. We describe our annotation scheme shortly in Section 4.4. #### 4.2.3. How can we control for noise? We controlled for potential confounding factors that may add noise to our audit measurements. To eliminate the effect of personalization, we ran the experiment on newly created virtual machines (VM) and freshly installed browser with empty browsing history, cookies and cache. Additionally, we ran search queries from the same version of Google Chrome in incognito mode to ensure that no history is built during our audit runs. To avoid cookie tracking, we erased cookies and cache before and after opening the incognito window and destroyed the window after each search. In sum, we performed searches on newly created incognito windows everyday. All VMs operated from same geolocation so that any effects due to location would affect all machines equally. To prevent machine speeds from affecting the experiment, all VMs had the same architecture and configuration. To control for temporal effect, we searched every single query at one particular time everyday for consecutive 15 days. Prior studies have established the presence of carry-over effect in search engines, where previously executed queries affect the results of the current query when both queries are issued subsequently within a small time interval (Hannak et al., 2013). Since, we destroyed browser windows and cleared session cookies and cache after every single search, carry over effect did not influence our experiment. # \| User action \| Type of history \| Tested values ---\|---\|---\|--- 1 \| Search product \| \| Product search history \| Product debunks vaccine or other health related misinformation (annotation value -1) & Neutral health information (annotation value 0) & Product promotes vaccine or other health related misinformation (annotation value 1) 2 \| Search + click product \| \| Product search and click history 3 \| Search + click + add to cart \| \| Intent to purchase history 4 \| Search + click + mark “Top rated, All positive review” helpful \| \| Searching, clicking and marking reviews helpful history 5 \| Following contributor by clicking follow button on contributor’s page \| \| Following history 6 \| Search product on Google (third party application) \| \| Third party search history Table 3. List of user actions employed to build account history. Every action and product type (misinformative, neutral or debunking) combination was performed on two accounts. One account sorted search results by filters “featured” and “average customer review”. The other account built history in the same way but sorted the search results by filters “price low to high” and “newest arrivals”. Overall, we created 40 Amazon accounts (6 actions X 3 tested values X 2 replicates for filters + 2 control accounts + 2 twin accounts). #### 4.2.4. Implementation details Figure 4 illustrates the eight steps for the _Unperonalized audit_. We used Amazon Web Services (AWS) infrastructure to create all the VMs. We created selenium bots to automate web browser actions. As a first step, each day at a particular time, the bot opened amazon.com in incognito window. Next, the bot searched for a single query, sorted the results by an Amazon filter and saved the SERPs. The bot then extracted the top 10 URLs of the products present in the results. The sixth step is an iterative step where the bot iteratively opened the product URLs and saved the product pages. In the last two steps, the bot cleared the browser cache and killed the browser window. We repeated steps 1 to 8 to collect search results sorted by all 5 Amazon filters. We added appropriate wait times after each step to prevent Amazon from detecting the account as a bot and blocking our experiment. We repeated these steps for 15 consecutive days for each of the 48 search queries. After completion of the experiment, we parsed the saved product pages to extract product metadata, like product category, contributors’ names (author, editor, etc.), star rating and number of ratings. We extracted product page recommendations for the top 3 search results only. ### 4.3. RQ2: Personalized Audit #### 4.3.1. Overview The goal of our Personalization Experiments is twofold. First, we assess whether user actions, such as clicking on a product, adding to cart would trigger personalization on Amazon. Second, and more importantly, we determine the impact of a user’s account history on the amount of misinformation presented to them in the search results page, recommendations, and auto- complete suggestions; account history is built progressively by performing a particular action for seven consecutive days. We ran our _Personalized audit_ from 12th to 18th August, 2020. We took several methodological decisions while designing this experimental setup. We discuss each of these decisions below. #### 4.3.2. What real-world user actions should we select to build account history? Users’ click history and purchase history trigger personalization and influence the price of commodities on e-commerce websites (Hannak et al., 2014). Account history also affects the amount of misinformation present in the personalized results (Hussein et al., 2020). Informed by the results of these studies, we selected six real-world user actions that could trigger personalization and thus, could potentially impact the amount of misinformation in search results and recommendations. The actions are (1) “search” (2) “search + click” (3) “search + click + add to cart” (4) “search + click + mark top-rated all positive review as helpful” (5) “follow contributor” and (6) “search on third party website” (Google.com in our case) . Table 3 provides an overview. First two actions involve searching for a product and/or clicking on it. Through the third and fourth action, a user shows positive reaction towards a product by adding it to cart and marking its top rated critical review as helpful respectively. Fifth action investigates the impact of following a contributor. For example, for a product in the Books category, the associated list of contributors include the author and editor of the book. The contributors have dedicated profile pages that a user can follow. Sixth action investigates the effect of searching for an Amazon product on Google.com. The user logs into Google using the email id used to register the Amazon account. The hypothesis is that Amazon search results could be affected by third party browsing history. After selecting the actions, we determined the products on which the actions needed to be performed. #### 4.3.3. What products and contributors should we select for building account history? To build user history, all user actions except “follow contributor” need to be performed on products. First, we annotated all products collected in the Unpersonalized audit run as debunking (-1), neutral (0) or promoting (1) health misinformation. We present the annotation details in Section 4.4. For each annotation value (-1, 0, 1), we selected top-rated products that had received maximum engagement and belonged to the most occurring category—‘Books’. We started by filtering Books belonging to each annotation value and eliminated the ones that did not have an “Add to cart” button on their product page at the time of product selection. Since users make navigation and engagement decisions based on information cues on the web (Pirolli, 2005), we considered cues present on Amazon such as customer ratings as a criteria to further shortlist Books. First, we sorted Books based on the accumulated engagement—number of customer ratings received. Next, we sorted the top 10 Books obtained from the previous sorting based on star ratings received by the Books to end up with highly rated, high-impact and high- engagement products. We selected top 7 books from the second sorting for the experiment (see Appendix, Table 9 for the shortlisted books). Action “follow contributor” is the only action that is performed on contributors’ Amazon profile pages 444The contributors could be authors, editors, people writing foreward of a book, publisher, etc.. We selected contributors who contributed to the most number of debunking (-1), neutral (0) and promoting (1) books. We retained only those who had a profile page on Amazon. Table 6 lists the selected contributors. # \| \| Contributors to debunking --- health products \| Contributors to neutral --- health products \| Contributors to misinformative --- health products name \| url code \| name \| url code \| name \| url code 1 \| Paul-A-Offit \| B001ILIGP6 \| Jason-Soft \| B078HP6TBD \| Andrew-J-Wakefield \| B003JS8YQC 2 \| Seth-Mnookin \| B001H6NG7A \| Joy-For-All-Art \| B07LDMJ1P4 \| Mary-Holland \| B004MZW7HS 3 \| Michael-Fitzpatrick \| B001H6L348 \| Peter-Pauper-Press \| B00P7QR4RO \| Kent-Heckenlively \| B00J08DNE8 4 \| Ziegler-Prize \| B00J8VZKBQ \| Geraldine-Dawson \| B00QIZY0MA \| Jenny-McCarthy \| B001IGJOUC 5 \| Ben-Goldacre \| B002C1VRBQ \| Tina-Payne-Bryson \| B005O0PL3W \| Forrest-Maready \| B0741C9TKH 6 \| Jennifer-A-Reich \| B001KDUUHY \| Vassil-St-Georgiev \| B001K8I8XC \| Wendy-Lydall \| B001K8LNVQ 7 \| Peter-J-Hotez \| B001HPIC48 \| Bryan-Anderson \| B087RL79G8 \| Neil-Z-Miller \| B001JP7UW6 Table 4. List of contributors who have contributed to the most number of books that are either debunking, neutral or promote health misinformation, selected for building account history for action “Follow contributors”. For example, Andrew J Wakefield, Mary Holland (both prominent vaccine deniers) have contributed to the most number of books that promote health misinformation.666The contributor’s Amazon web page can be accessed by forming the url “www.amazon.com/ + name + /e/ + url_code”. Figure 5. Steps performed by treatment and control accounts in Personalized audit corresponding to the 6 different features. [Experimental steps of Personalized audit]Figure illustrates how treatment accounts built histories by performing various actions and later collected homepage recommendations, SERPs for all 48 search queries followed by auto- complete suggestions. The control accounts did not build account history but collected SERPs and auto-complete suggestions for the 48 search queries at the same time as the treatment accounts. #### 4.3.4. How do we design the experimental setup? We performed all six actions explained in Section 4.3.2 and Table 3 on Books (or contributors of the books in case of action “follow contributor”) that are either all debunking, neutral or promoting health misinformation. Each action and product type combination was acted upon by two treatment accounts. One account built its search history by first performing searches on Amazon and then viewing search results sorted by filters “featured” and “average customer review” while the other did the same but sorted results by “price low to high” and “newest arrivals”777Every account created for this experiment was run by a bot. It was not possible for a bot to complete the following order of tasks in 24 hours because of wait times added after every action– building history using a particular action, searching for 48 search queries sorted by 4 filters and collecting auto-complete suggestions for those queries etc. Thus, every action-product type combination was performed on two accounts. First account, sorted the search results by two filters and second account sorted results using remaining two filters. We call these two accounts replicates since they built their history in the same way.. We did not use the filter “price high to low” since intuitively it is less likely to be used during searches. We also created 2 control accounts corresponding to 2 treatments that emulated the same actions as the treatments except that they did not build account histories by performing one of the 6 user actions. Like 2 treatment accounts, the first control account searched for 48 queries curated in Section 4.1.2 and sorted them by filters “Featured” and “Average customer Review” while the other control sorted them by the remaining two filters. Figure 5 outlines the experimental steps performed by treatment and control accounts. We also created twins for each of the control accounts. The twins performed the exact same tasks as the corresponding control. Any inconsistencies between a control account and its twin can be attributed to noise, and not personalization. Remember, Amazon’s algorithms are a black box. Even after controlling for all known possible sources of noise, there could be some sources that we are not aware of or the algorithm itself could be injecting some noise in the results. If the difference between search results of control and treatment is greater than the baseline noise, only then it can be attributed to personalization. Prior audit work have also adopted the strategy of creating a control and its twin to differentiate between the effect due to noise versus personalization (Hannak et al., 2014). Overall, we created 40 Amazon accounts (6 actions X 3 tested values X 2 replicates for filters + 2 control accounts + 2 twin accounts). Next, we discuss the components collected from each account. #### 4.3.5. What components should we collect for the personalized audit? We collected search results and auto-complete suggestions for treatment and control accounts to measure the extent of personalization. We collected recommendations only for the treatment accounts since they built history by clicking on product pages, pre-purchase pages, etc. Search results were sorted by filters ‘featured”, “average customer review”, “price low to high” and “newest arrivals”. Once users start building their account history, Amazon displays several recommendations to drive engagement on the platform. We collected various types of recommendations spread across three recommendation pages: homepage, product page and pre-purchase page. Pre-purchase pages were only collected for the accounts that performed “add to cart” action. Additionally, product pages were collected for accounts that clicked on search results while creating their respective account history. Each of the aforementioned pages consist of several recommendation types, such as “Customers who bought this item also bought”, etc. We collected the first product present in each of these recommendation types from both product pages and pre-purchase pages and two products from each type from the homepages for further analysis. Refer to Table 1 and Figures 1(a), 1(b) and 1(c) for examples of these recommendation types. #### 4.3.6. How do we control for noise? Just like our _Unpersonalized audit_ , we first controlled for VM configuration and geolocation. Next, we controlled for demographics by setting the same gender and age for newly creating Google accounts. Recall, that these Google accounts were used to sign-up for the Amazon accounts. Since, the VMs were newly created, the browser had no search history that could otherwise hint towards users’ demographics. All accounts created their histories at the same time. They also performed the searches at the same time each day, thus, controlling for temporal effects. Lastly, we did not account for carry over effects since it affected all the treatment and control accounts equally. #### 4.3.7. Implementation details Figure 5 illustrates the experimental steps. We ran 40 selenium bots on 40 VMs. Each selenium bot operated on a single Amazon account. On day 0, we manually logged in to each of the accounts by entering login credentials and performing account verification. Next day, experiment began at time t. All bots controlling treatment accounts started performing various actions to build history. Note, everyday bots built history by performing actions on a single Book/contributor. We gave bots sufficient time to build history (90 min) after which they collected and saved Amazon homepage. Later, all 40 accounts (control + treatment) searched for 48 queries with different search filters and saved the SERPs. Next, the bots collected and saved auto-complete suggestions for all 48 queries. We included appropriate wait times between every step to prevent accounts from being recognized as bots and getting banned in the process. We repeated these steps for a week. At the end of the week, for each treatment account we had collected personalized search results, recommendations and auto-complete suggestions. Next, we annotated the collected search results and recommendations to determine their stance on misinformation so that later we could analyze them to study the effect of user actions on the amount of misinformation presented to users in each component. A. Scale Value \| Annotation Description \| Annotation Heuristics \| Sample Amazon Products ---\|---\|---\|--- -1 \| debunks vaccine misinformation \| Product debunks, derides OR provides evidence against the myths/controversies surrounding vaccines OR helps understand anti-vaccination attitude OR promotes use of vaccination OR describes history of a disease and details how its vaccine was developed OR describes scientific facts about vaccines that help users to understand how they work OR debunks other health-related misinformation \| 0 \| neutral health related information \| All medicines and antibodies OR medical equipment (thermometer, syringes, record-books, etc.) OR dietary supplements that do not violate Amazon’s policy OR products about animal vaccination and diseases OR health-related products not promoting any conspiratorial views about health and vaccines \| 1 \| promotes vaccine and other health related misinformation \| Product promotes disuse of vaccines OR promotes anti-vaccine myths, controversies or conspiracy theories surrounding the vaccines OR advocates alternatives to vaccines and/or western medicine (diets, pseudoscience methods like homeopathy, hypnosis, etc.) OR product is a misleading dietary supplement that violates Amazon’s policy on dietary supplements- the supplement states that it can cure, mitigate, treat, or prevent a disease in humans, but the claim is not approved by the FDA OR it promotes other health-related misinformation \| 2 \| unknown \| Product’s description and metadata is not sufficient to annotate it as promoting, debunking or neutral information \| 3 \| removed \| Product’s URL is not accessible at the time of annotation \| - 4 \| Other language \| Product’s title and description is in language other than english \| 5 \| Unrelated \| Non-health related products \| Table 5. Description of annotation scale, heuristics along with sample products corresponding to each annotation value. ### 4.4. Annotating Amazon data for health misinformation Unlike partisan bias where bias could be determined by using features such as news source bias (Robertson et al., 2018), labelling a product for misinformation is hard and time-consuming. There are no pre-determined sources of misinformation such as list of sellers or authors of misinformative products on Amazon. Additionally, we found that the annotation process for some categories of products, like Books, Kindle ebooks, etc. required us to consider the product image, read the book’s preview, if available, and even perform external search about the authors. Therefore, we opted to manually annotate our data collection. We developed a qualitative coding scheme to label our Amazon data collection through an iterative process that required several rounds of discussions to reach an agreement on the annotation scale. In the first round, first author randomly sampled 200 Amazon products across different topics and categories. After multiple iterations of analyzing and interpreting each product, the author came up with an initial 7-point annotation scale. Then, six researchers with extensive work experience on online misinformation independently annotated 32 products, randomly selected from the 200 products. We discussed every product’s annotation value and the researchers’ annotation process. We refined the scale as well as the scheme based on the feedback. This process was repeated thrice after which all six annotators reached a consensus on the annotation scheme and process. In the fourth round, we gathered additional feedback from an external researcher from the Credibility Coalition group888https://credibilitycoalition.org/—an international organization of interdisciplinary researchers and practitioners dedicated to developing standards for news credibility and tackling the problem of online misinformation. The final result of the multi-stage iterative process (see Appendix, Figure 14) is a 7-point annotation scale comprising of values ranging from -1 to 5 (see Table 5). The scale measures the scientific quality of products that users are exposed to when they make vaccine-related searches on Amazon. #### 4.4.1. Annotation Guidelines In order to annotate an Amazon product, the annotators were required to go through several fields present on the product’s detail page in the following order: title, description, top critical and top positive reviews about the product, other metadata present on the detail page, such as editorial reviews, legal disclaimers, etc. If the product was a book, the annotators were also recommended to do the following three steps: (1) go through the first few pages in the book preview 999Amazon has introduced a Look Inside feature that allows users to preview few pages from the book., (2) see other books published by the authors, (3) perform a google search on the book and go through the first few links to discover more information about the book. Annotators were asked to see contextual information about the product from multiple sources to gain more context and perspective. This technique is grounded in lateral reading that has proven to be a good approach for credibility assessment (Spector, 2017). #### 4.4.2. Annotation scale and heuristics: Below we describe each value in our annotation scale. Table 5 presents examples. Debunking (-1): Annotation value ‘-1’ indicates that the product debunks vaccine misinformation or derides any vaccine-related myth or conspiracy theory or promotes the use of vaccination. As an example, consider the poster titled Immunization Poster 1979 Vintage Star Wars C-3PO R2-D2 Original (B00TFTS194)101010Every title of the Amazon product is followed by a URL id. This URL id can be converted into a url using the format: http://www.amazon.com/dp/url_id that encourages parents to vaccinate their children. Products helping users understand anti-vaccination attitude or those that describe the history about the development of vaccines or the science behind how vaccines work were also included in this category. Promoting (1): This category includes all products that support or substantiate any vaccine related myth or controversy and encourages parents to raise a vaccine-free child. For example, consider the following books that promote anti-vaccination agenda. In A Summary of the Proofs that Vaccination Does Not Prevent Small-pox but Really Increases It (B01G5QWIFM), the author talks about dangers of large scale vaccination and in Vaccine Epidemic: How Corporate Greed, Biased Science, and Coercive Government Threaten Our Human Rights, Our Health, and Our Children (B00CWSONCE), the authors question vaccine safety and present several narratives of vaccine injuries. We included several Amazon Fashion (B07R6PB2KP) and Amazon Home (B01HXAB7TM) merchandise in this category too since they contained anti-vaccine slogans like “Educate before you Vaccinate”, “Jesus wasn’t vaccinated”, etc. We also included all products advocating any alternatives to vaccines, products that promote other health-related misinformation, dietary supplements that claim to cure diseases in their description but are not approved by Food and Drug Administration (FDA) 111111Note that for dietary supplements category, Amazon asks sellers not to state that the products cure, mitigate, treat, or prevent a disease in humans in their details page, unless that statement is approved by the FDA (Central, 2020) in this category. Neutral (-0): We annotated all medical equipment and medicines as neutral (annotation value ‘0’). Note that it is beyond the scope of this project to determine the safety and veracity of the claims of each medicine sold on the Amazon platform. This means that the number of products that we have determined to be promoting (1) serve as the lower bound of the amount of misinformation present on the platform. This category also includes dietary supplements that do not violate Amazon’s policy, pet-related products and health-related products not advocating a conspiratorial view. Other annotations: We annotated a product as ‘2’ if the product’s description and metadata were not sufficient to determine its stance. We assigned values ‘3’ and ‘4’ to all products whose URL was not accessible at the time of the annotation and whose title and description was in a language other than English, respectively. We annotated all non-health related products (e.g. diary, carpet, electronic products, etc.) with value ‘5’. Both our audits resulted in a dataset of 4,997 Amazon products that were annotated by the first author and Amazon Mechanical Turk workers (MTurks). The first author being the expert annotated majority of products (3,367) to determine what would be a good task representation to obtain high quality annotations for the remaining 1,630 products from novice MTurks. We obtained three Turker ratings for each remaining product and used the majority response to assign the annotation value. Our task design worked. For 97.9% of the products, annotation values converged. Only 34 products had diverging responses. The first author then annotated these 34 products to obtain the final set of annotation values. We describe the AMT job in detail in Appendix A.1. \| Rank --- r Product \| \| Bias of each --- product \| Bias till --- rank r Bias value 1 \| $p_{1}$ \| $s_{1}$ \| B(1) \| $s_{1}$ 2 \| $p_{2}$ \| $s_{2}$ \| B(2) \| $\frac{1}{2}$ ($s_{1}$ \+ $s_{2}$) 3 \| $p_{3}$ \| $s_{3}$ \| B(3) \| $\frac{1}{3}$($s_{1}$ \+ $s_{2}$ \+ $s_{3}$) Input Bias (ib) \| $\frac{1}{3}$($s_{1}$ \+ $s_{2}$ \+ $s_{3}$) Output Bias (ob) \| $\frac{1}{3}$ [$s_{1}$(1 + $\frac{1}{2}$ \+ $\frac{1}{3}$) + $s_{2}$($\frac{1}{2}$ \+ $\frac{1}{3}$) + $s_{3}$($\frac{1}{3}$)] Rank Bias (rb) \| ob-ib Table 6. Example illustrating the bias calculations. For a given query, Amazon’s search engine presents users with the following products in the search results $p_{1}$, $p_{2}$ and $p_{3}$. The misinformation bias scores of the products are $s_{1}$, $s_{2}$ and $s_{3}$ respectively. The table has been adopted from previous work (Kulshrestha et al., 2017). A bias score larger than 0 indicates a lean towards misinformation. (a) Search results [Bar graph showing the percentage of search results belonging to different annotation values ] debunking (8.99%), neutral (40.81%), promoting (10.47%), unable to annotate (5.44%), URL not accessible (3.23%), other language (0.97%) and unrelated (30.06%). (b) Recommendations [Bar graph showing the percentage of recommendations belonging to different annotation values ]debunking (1.99%), neutral (37.56%), promoting (12.95%), unable to annotate (0.48%), URL not accessible (2.80%), other language (0.21%) and unrelated (43.98%). Figure 6. RQ1a: (a) Number (percentage) of search results belonging to each annotation value. While majority of products have a neutral stance (40.81%), products promoting health misinformation (10.47%) are greater than products debunking health misinformation (8.99%). (b) Number (percentage) of recommendations belonging to each annotation value. A high percentage of product recommendations promote misinformation (12.95%) while percentage of recommendations debunking health misinformation is very low (1.99%). Figure 7. RQ1a: Figure showing categories of promoting, neutral and debunking Amazon products (search results). All categories occurring less than 5% were combined and are presented as other category. Note that misinformation exists in various forms on Amazon. Products promoting health misinformation include books (Books, Kindle eBooks, Audible Audiobooks), apparel (Amazon Fashion) and dietary supplements (Health & Personal Care). Additionally, proportion of books promoting health misinformation is much greater than proportion of books debunking misinformation. [Categories of debunking, neutral and promoting Amazon products] Debunking products mostly belong to categories, Kindle eBooks, Books, Amazon fashion and Amazon home. Neutral products mostly belong to categories Books, Kindle eBooks, Health & Personal care and Amazon home. Promoting products belong to categories Books, Kindle eBooks, Health & Personal care and Amazon fashion. ### 4.5. Quantifying misinformation bias in SERPs: In this section, we describe our method to determine the amount of misinformation present in search results. How do we estimate the misinformation bias present in Amazon’s SERPs? First, we used our annotation scheme to assign misinformation bias scores ($s_{i}$) to individual products present in SERPs. We converted our 7 point (-1 to 5) scale to misinformation bias scores with values -1, 0 and 1. We mapped annotation values 2, 3, 4, and 5 to bias score 0. Merging “unknown” annotations to neutral will result in a conservative estimate of misinformation bias present in the search results. Now, a product can be assigned one of the three bias scores: -1 suggests that product debunks misinformation, 0 indicates a neutral stance and 1 implies that the product promotes misinformation. Next, to quantify misinformation bias in Amazon’s SERPs, we adopt the framework and metrics proposed in prior work to quantify partisan bias in Twitter search results (Kulshrestha et al., 2017). Below we discuss three kinds of bias proposed by the framework and delineate how we estimate each bias with respect to misinformation. Table 6 illustrates how we calculated the bias values. 1. (i) The input bias (ib) of a list of Amazon products is the mean of misinformation bias scores of the constituting products (Kulshrestha et al., 2017). Therefore, ib = ${\sum_{i=1}^{n}{s_{i}}}$, where n is the length of the list & ${s_{i}}$ is the misinformation bias score of ith product in the list. Input bias is an unweighted bias, i.e it is not affected by the rank/ordering of the items. 2. (ii) The output bias (ob) of a ranked list is the overall bias present in the SERPs and is the sum of biases introduced due to input and ranks of the input. We first calculate weighted bias score B(r) of every rank r, which is the average misinformation bias of products ranked from 1 to r. Thus, B(r) = $\frac{\sum_{i=1}^{r}{s_{i}}}{r}$, where ${s_{i}}$ is the misinformation bias score of ith product. Output bias (ob) is the average of weighted bias score B(r) for all ranks. Thus, by definition ob = $\frac{\sum_{i=1}^{r}{B(i)}}{r}$. 3. (iii) The ranking bias (rb) is introduced by the ranking algorithm of search engine (Kulshrestha et al., 2017). It is calculated by subtracting input bias from output bias. Thus, rb = ob-ib. In our case, high ranking bias indicates that search algorithm ranks misinformative products higher than neutral or debunking products. Why do we need three bias scores? Amazon’s search algorithm is not only selecting the products to be shown in the search results but it is also ranking them according to their internal algorithm. Therefore, the overall bias (ob) could be introduced either at the product selection stage (ib), or ranking stage (rb) or both. Studying all three biases gives us an elaborate understanding of how biases are introduced by the search algorithm. All three bias values (ib, ob and rb) lie between -1 and 1. A bias score larger than 0 indicates a lean towards misinformation. Conversely, a bias score less than 0 indicates a propensity towards debunking information. We only consider top 10 search results in each SERP. Thus, in the bias calculations, rank always varies from 1 to 10. ## 5\. RQ1 Results [Unpersonalized audit]: Quantify misinformation bias The aim of the Unpersonalized audit is to determine the amount of misinformation bias in search results. Below we present the input, rank, and output bias detected by our audit in search results of all 10 vaccine-related topics with respect to 5 search filters. Figure 8. RQ1a: Input, rank and output bias for all 10 vaccine-related topics across five search filters. The bias scores are average of scores obtained for each of the 15 days. Input and rank bias is positive (¿0) in the search results of majority of topics for filters “featured” and “average customer review”. A bias value greater than 0 indicates a lean towards misinformation. Topics “andrew wakefield” and “mmr vaccine & autism” have a positive input bias across all five filters indicating that search results of these topics contain large number of products promoting health misinformation irrespective of the filter used to sort the search results. Topic “vaccination” has the highest overall bias (output bias) of 0.63 followed by topic “andrew wakefield” that has output bias of 0.53 for filter “featured”. [Input, rank and output bias values]All topics except hepatitis have positive input and output bias for filter customer reviews. Additionally, all topics except mmr and influenza vaccine have positive input and output bias values for filter featured. Furthermore, all topics except andrew wakefield, mmr vaccine and autism, and vaccine controversies have a negative input and output bias for filter newest arrivals. On the other hand, ranking bias of all topics except vaccination and hepatitis for filter customer review and immunization and varicella vaccine for filter price low to high is positive. ### 5.1. RQ1a: Search results We collected 36,000 search results from our Unpersonalized audit run, out of which 3,180 were unique. Recall, we collected these products by searching for 48 search queries belonging to vaccine-related topics and sorting results by each of the 5 Amazon filters. We later extracted and annotated top 10 search results from all the collected SERPs resulting in 3,180 annotations. Figure 6(a) shows the number (and percentage) of products corresponding to each annotation value. Through our audits, we find a high percentage (10.47%) of misinformative products in the search results. Moreover, the number of misinformative products outnumbered the debunking products. Figure 7 illustrates the distribution of categories of Amazon products annotated as debunking (-1), neutral (0) and promoting (1). Note that the products promoting health misinformation primarily belong to categories Books (35.43%), Kindle eBooks (28.52%), Amazon Fashion (12.61%)—a category that includes t-shirts, apparel, etc. and Health & Personal Care (10.21%)—a category consisting of dietary supplements. Below we discuss the misinformation bias observed across all the vaccine-related topics, the Amazon search filters and search queries. #### 5.1.1. Misinformation bias in vaccine related topics We calculate the input, rank and output bias for each of the 10 search topics. All the bias scores presented are average of scores obtained across the 15 days of audit. The bias score for a topic is also the average across each of the constituting search queries. Figure 8 shows the bias scores for all the topics, search filters and bias combinations. Input bias: We observe a high input bias (¿0) for all topics except “hepatitis” for “average customer review” filter indicating presence of a large number of misinformative products in the SERPs when search results are sorted by this filter. Similarly, input biases for most topics is also positive for “featured” filter. Note, “featured” is the default Amazon filter. Thus, by default Amazon is presenting more misinformative search results to users searching for vaccine related queries. Topics “andrew wakefield”, “vaccination” and “vaccine controversies” have highest input biases for the both “featured” and “average customer review” filters. Another noteworthy trend is the negative input bias for 7 out of 10 topics with respect to filter “newest arrivals” indicating that there are more debunking products present in the SERP when users look for newly appearing products on Amazon. “Andrew wakefield” and “mmr vaccine & autism” are the only two topics that have positive input bias (¿0) across all the five filters. Interestingly, there is no topic that has negative input bias across all filters. Recall, a negative (¡0) bias indicates a debunking lean. Topics “mmr”, “influenza vaccine” and “hepatitis” have negative bias scores in four out of five filters. Figure 9. Input, rank and output bias for all filter types. [Bias values for all filter types]Input bias values: featured (0.21), avg customer reviews (0.3), price low to high (0.032), price high to low (0.056), newest arrival (-0.015). Rank bias: featured (0.018), avg customer reviews (0.034), price low to high (0.013), price high to low (0.011), newest arrival (-0.051). Output bias: featured (0.22), avg customer reviews (0.34), price low to high (0.045), price high to low (0.066), newest arrival (-0.067). Figure 10. Top 20 search query-filter combinations when sorted by output bias (ob). In other words, these query-filter combinations are the most problematic ones containing highest amount of misinformation (highest ob). [Search query - Amazon filter combinations containing highest amount of misinformation]Top five combinations: vaccination is not immunization - custReview (ob = 1), vaccination is not immunization - featured (ob = 1), vaccination is not immunization - priceHtoL (ob = 1), autism vaccine - custReview (ob = 0.99), vaccine - custReview (0.99) Rank bias: 8 out of 10 topics have positive (¿0) rank bias for filters “price low to high” and “average customer reviews” and 6 out of 10 topics have positive rank bias for filter “featured”. These results suggest that Amazon’s ranking algorithm favors misinformative products and ranks them higher when customers filter their search results by the aforementioned filters. Some topics have negative input bias but positive rank bias. Consider topic “mmr” with respect to filter “price low to high” whose input bias is -0.1 but the rank bias is 0.065. This observation suggests that although the SERPs obtained had more debunking products, a few misinformative products were still ranked higher. Rank bias for 8 out of 10 topics with respect to filter “newest arrivals” was negative, similar to what we observed for input bias. Output bias: Output bias is positive (¿0) for most topics with respect to filters “featured” and “average customer reviews”. Recall, a bias value greater than 0 indicates a lean towards misinformation. Topic “vaccination” has the highest output bias (0.63) for filter “featured”. On the other hand, topic “influenza vaccine” has least output bias (-0.24) for filter “price high to low”. #### 5.1.2. Misinformation bias in search filters Figure 9 shows the results for all 5 filters. Bias scores are averaged across all search queries. All filters except “newest arrivals” have positive input, rank, and output misinformation bias. Filter “average customer review” has the highest positive output bias indicating that misinformative products belonging to vaccine related topics receive higher ratings. We present the implications of these results in our discussion (Section 7). #### 5.1.3. Misinformation bias in search queries Figure 10 shows the top 20 search queries and filter combinations with highest output bias. Predictably, filter “newest arrivals” does not appear in any instance. Surprisingly, 9 search query-filter combinations have very high output biases (ob ¿ 0.9). Search query “vaccination is not immunization” has output bias of 1 for three filter types. Most of the search queries in Figure 10 have a negative connotation, i.e the queries themselves have a bias (e.g search queries anti vaccine books, vaccination is not immunization indicates an intent to search for misinformation). This observation reveals that if you search for anti vaccine stuff, you will get high amount of vaccine and health misinformation. This indicates how Information Retrieval systems currently work; they curate by relevance with no notion of veracity. The most troublesome observation is the presence of high output bias for generic and neutral search queries, “vaccine” (ob = 0.99) and “varicella vaccine” (ob = 0.79). These results indicate that, unlike companies like Pinterest, who have altered their search engines in response to vaccine related queries (Caron, 2019), Amazon has not made any modification to its search algorithm to push less anti vaccine products to users. (a) Customers who bought this item also bought (CBB) [CBB]There are several instances of red nodes connected to each other and green nodes connected to each other. Few of the green nodes are attached to red nodes too. (b) Customers who viewed this item also viewed (CVV) [CVV]There are several instances of red nodes connected to each other and green nodes connected to each other. Few of the green nodes are attached to red nodes too. (c) Frequently bought together (FBT) [FBT]There are large sized red nodes. Red nodes are attached to other red nodes and several green nodes are also attached together. (d) Sponsored products related to this item [Sponsored]The graph contains many large green nodes attached to other green nodes. Few large red nodes are also present that are attached to other red and green nodes. (e) What other items customers buy after viewing this item (CBV). Note that the recommendation graph for CBV recommendation type is indeed one figure. It consists of two disconnected components, indicating strong filter bubble effect. [CBV]The graph has two disconnected components. One component mostly consists of red nodes attached to one another. Other component comprises of green nodes and a few red nodes. Figure 11. Recommendation graphs for 5 different types of recommendations collected from the product pages of top three search-results obtained in response to 48 search queries, sorted by 5 filters over a duration of 15 days during Unpersonalized audit run. denotes products annotated as misinformative, as neutral and as debunking. Node size is proportional to the times the product was recommended in that recommendation type. Large sized red nodes coupled with several interconnections between red nodes indicate a strong filter-bubble effect where recommendations of misinformative products returned more misinformation. ### 5.2. RQ1b: Product page recommendations We extracted the product page recommendations of top 3 search results present in the SERPs. The product page constitutes of various types of recommendations. For analysis, we considered the first product present in 5 types of recommendations “Customers who bought this item also bought” (CBB), “Customers who viewed this item also viewed” (CVV), “Frequently bought together” (FBT), “Sponsored products related to this item” and “What other items customers buy after viewing this item” (CBV). The process resulted in 16,815 recommendations out of which 1,853 were unique. Figure 6(b) shows the number and percentage of recommendations belonging to different annotation values. The percentage of misinformative recommendations (12.95%) is much higher than the debunking recommendations (1.95%). The total input bias in all 16,815 recommendations is 0.417 while in all 1,853 unique recommendations is 0.109, indicating a lean towards misinformation. Does filter-bubble effect occur in product page recommendations? To answer, we compared the misinformation bias scores of all types of recommendations considered together (refer Table 7). Kruskal Wallis Anova test revealed the difference to be significant (KW H(2, N=16815) = 6,927.6, p=0.0). Post-hoc Tukey HSD test showed that the product page recommendations of misinformative products contain more misinformation when compared to recommendations of neutral and debunking products. Even more concerning is that the recommendations of debunking products have more misinformation than neutral products. To investigate further, we qualitatively studied the recommendation graphs of each of the five recommendation types (Figure 11). Each node in the graph represents an Amazon product. An edge A$\rightarrow$ B indicates that B was recommended in the product page of A. Node size is proportional to the number of times the product was recommended. Type of product page recommendations \| Kruskal Wallis Anova Test \| Post hoc Tukey HSD \| d \| n \| m ---\|---\|---\|---\|---\|--- All \| KW H(2, N=16815) = 6,927.6, p=0.0 \| M>D & M>N & D>N \| 37 \| 1576 \| 240 Cust. who bought this item also bought (CBB) \| KW H(2, N=3133) = 2136.03, p=0.0 \| M >D & M>N & N>D \| 11 \| 225 \| 66 Cust. who viewed this item also viewed (CVV) \| KW H(2, N=4485) = 2673.95, p=0.0 \| M>D & M>N & D>N \| 18 \| 331 \| 100 Frequently bought together (FBT) \| KW H(2, N=388) = 277.08, p=6.8e-61 \| M>D & M>N & D>N \| 1 \| 111 \| 16 Sponsored products related to this item \| KW H(2, N=6575) = 628.52, p=3.2e-137 \| M>D & M>N & D>N \| 7 \| 953 \| 98 \| What other items cust. buy after viewing --- this item (CBV) KW H(2, N=2234) = 1611.34, p=0.0 \| M>D & M>N & D>N \| 9 \| 230 \| 57 Table 7. RQ1b: Analyzing echo chamber effect in product page recommendations. M, N and D are the means of misinformation bias scores of products recommended in the product pages of misinformative, neutral and debunking Amazon products respectively. Higher means indicate that recommendations contain more misinformative products. For example, M>D indicates that recommendations of misinformative products have more misinformation than recommendations of debunking products. d, n and m are number of unique products annotated as debunking, neutral and promoting for each recommendation type. #### 5.2.1. Recommendation type- Customers who bought this item also bought (CBB) Misinformation bias scores of CBB are significantly different for debunking, neutral, and promoting products (KW H(2, N=3133) = 2136.03, p=0.0). Post hoc tests reveal that CBB recommendations of misinformative products have more misinformation when compared to CBB recommendations of neutral and debunking products. Additionally CBB recommendations of neutral products have more misinformation than CBB recommendations of debunking products. The findings are evident from Figure 11(a) too. For example, there are several instances of red nodes connected to each other. In other words, if you click on a misinformative search result, you will get misinformative products in CBB recommendations. Few of the green nodes are attached to red ones indicating that CBB recommendation of a neutral product sometimes contain a misinformative product. The most recommended product present in CBB is a misinformative Kindle book titled Miller’s Review of Critical Vaccine Studies: 400 Important Scientific Papers Summarized for Parents and Researchers (B07NQW27VD). #### 5.2.2. Recommendation type- Customers who viewed this item also viewed (CVV) Misinformation bias scores of CVV recommendations are significantly different for debunking, neutral and promoting products (KW H(2, N=4485) = 2673.95, p=0.0) . Post hoc test indicates that CVV recommendations of misinformative products have more misinformation than CVV recommendations of debunking and neutral products. Notably, CVV recommendations of debunking products contain more misinformation than CVV recommendations of neutral products. This is troubling since users who are clicking on products that present scientific information are pushed more misinformation in this recommendation type. In the recommendation graph (Figure 11(b) ), we see edges connecting multiple red nodes supporting our finding that CVV recommendations of misinformative products mostly contain other misinformative products. The most recommended product occurring in this recommendation type is a misinformative Kindle book titled Dissolving Illusions (B00E7FOA0U). #### 5.2.3. Recommendation type- Frequently bought together (FBT) Misinformation bias scores of FBT recommendations are significantly different for debunking, neutral and promoting products (KW H(2, N=388) = 277.08, p=6.8e-61). Post hoc tests reveal that amount of misinformation in FBB recommendations of misinformative products is significantly more than the FBB recommendations of neutral and debunking products. The finding is also evident from the graph (Figure 11(c)). There are large sized red nodes attached to other red nodes and several green nodes attached together indicating the presence of a strong filter-bubble effect. “Frequently bought together” can be considered an indicator of buying patterns on the platform. The post hoc tests indicate that people buy multiple misinformative products together. The most recommended product present in this recommendation type is a misinformative Paperback book titled Dissolving Illusions: Disease, Vaccines, and The Forgotten History (1480216895). #### 5.2.4. Recommendation type- Sponsored products related to this item Most of the sponsored recommendations are either neutral or promoting (Figure 11(d) and Table 7). Statistical test reveals that the misinformation bias score of sponsored recommendations are significantly different among debunking, neutral and promoting products (KW H(2, N=6575) = 628.52, p=3.2e-137). Post hoc tests reveal same results as for CVV recommendations. There are two most recommended sponsored books. First is a misinformative paperback book titled Vaccine Epidemic: How Corporate Greed, Biased Science, and Coercive Government Threaten Our Human Rights, Our Health, and Our Children (1620872129). Second is a neutral Kindle book titled SPANISH FLU 1918: Data and Reflections on the Consequences of the Deadliest Plague, What History Teaches, How Not to Repeat the Same Mistakes (B08774MCVP). #### 5.2.5. Recommendation type- What other items customers buy after viewing this item (CBV) Misinformation bias scores of CBV recommendations are significantly different for debunking, neutral and promoting products (KW H(2, N=2234) = 1611.34, p=0.0). Results of post hoc tests are same as that of CVV recommendations. The presence of an echo chamber is quite evident in the recommendation graph (see Figure 11(e)). The graph has two disconnected components, one comprising a mesh of misinformative products indicating a cluster of misinformative products that keep getting recommended. CBV is also indicative of buying patterns of Amazon users. The algorithm has learnt that people viewing misinformative products end up purchasing them. Thus, it pushes more misinformative items to users that click on them, creating a problematic feedback loop. The most recommended product in this recommendation type is a misinformative Kindle book titled Miller’s Review of Critical Vaccine Studies: 400 Important Scientific Papers Summarized for Parents and Researchers (B07NQW27VD). \| RQ2a \| RQ2b \| RQ2c ---\|---\|---\|--- \| Search results \| Recommendations \| \| Auto complete --- suggestions \| Featured \| \| Avg. --- customer reviews \| Price low --- to High \| Newest --- Arrivals Homepage \| Pre-purchase \| Product page \| \| \| Actions performed to build account history \| D \| N \| M \| D \| N \| M \| D \| N \| M \| D \| N \| M \| D \| N \| M \| D \| N \| M \| D \| N \| M \| D \| N \| M Search product \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| - \| - \| - \| X \| X \| X \| X \| X \| X \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| Search & click --- product IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| \| KW H(2, N=42) = 32.07, --- p = 1.08e-07 M>N>D X \| X \| X \| \| KW H(2, N=42) = 24.89, --- p = 3.94e-06 M>D & M>N NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| Search + click & --- add to cart product IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| \| KW H(2, N=42) = 33.48, --- p = 5.38e-08 M>N>D \| KW H(2, 42) = 32.63, --- p = 8.19e-08 M>N>D \| KW H(2, N=42) = 24.05, --- p = 5.98e-06 M>D & M>N NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| Search + click & --- mark “Top rated, All positive review” as helpful IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| \| KW H(2, N=42) = 32.33, --- p = 9.52e-08 M>N>D X \| X \| X \| \| KW H(2, 42) = 23.36, --- p = 8.44e-06 M>N & M>D NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| Following --- contributor IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| - \| - \| - \| X \| X \| X \| X \| X \| X \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| Search product --- on Google IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| IR [HTML]FFD9D7 \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| - \| - \| - \| X \| X \| X \| X \| X \| X \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC \| NP [HTML]E3DCDC Table 8. RQ2: Table summarizing RQ2 results. IR suggests noise and inconclusive results, i.e search results of control and its twin seldom matched. Thus, difference between treatment and control could either be attributed to noise or personalization, making it impossible to study the impact of personalization on misinformation. NP denotes little to no personalization. - indicates that the given activity had no impact on the component. X indicates that component was not collected for the activity. M, N and D indicate average per day bias in the component collected by accounts that built their history by performing actions on misinformative, neutral or debunking products. Higher mean value indicates more misinformation. For example, consider the cell corresponding to action “search + click & add to cart product” and “Homepage” recommendation. M¿N¿D indicates that accounts adding misinformative products to cart ends up with more misinformation in their homepage recommendations in comparison to accounts that add neutral or debunking products to cart. (a) [Average jaccard index of treatment and control accounts]Bar chart illustrating jaccard index between search results of control and its twin as well as control and the treatment accounts that performed “following contributors” action on misinformative, neutral and debunking products and later searched and sorted results using four Amazon filters. The jaccard index for control and its twin for filter “featured” is low (¡0.8). For other three search filters, “average customer review”,“price low to high” and “newest arrivals”, we see high (¿0.8) jaccard index and between and control and its twin and the metric values for treatment-control comparison are similar to that of control-twin comparison. (b) [Average kendall’s tau of treatment and control accounts]Bar chart illustrating kendall’s tau index between search results of control and its twin as well as control and the treatment accounts that performed “following contributors” action on misinformative, neutral and debunking products and later searched and sorted results using four Amazon filters. The kendall’s tau coefficient for control and its twin for filter “featured” is low (¡0.2). For other three search filters, “average customer review”,“price low to high” and “newest arrivals”, we see high (¿0.8) kendall’s tau coefficient between and control and its twin and the coefficient values for treatment-control comparison are similar to that of control-twin comparison. Figure 12. Investigating the presence and amount of personalization due to “following contributors” action by calculating (a) Jaccard index and (b) kendall’s tao metric between search results of treatment and control. M, N and D indicate results for accounts that follow contributors of misinformative, neutral and debunking products respectively. (a) [Input bias in homepages]Line graph showing input bias on the y axis and dates of experiment run (2020-08-12 to 2020-08-18) on x axis. Input bias for accounts performing actions on neutral products is 0 for all seven days. Input bias for accounts performing actions search+click and mark-review on misinformative products is greater than 0 for all seven days and becomes 1 (max value) from fourth day onwards. Input bias in homepages for accounts adding misinformative products to their cart is also greater than 0 for all seven days but the value is less than the bias in homepages for accounts performing actions search+click and mar review helpful from third day onwards. Input bias in homepages of accounts performing actions on debunking products becomes positive on the third day and after that drops below 0. (b) [Input bias in pre-purchase recommendations]Line graph showing input bias on the y axis and dates of experiment run (2020-08-12 to 2020-08-18) on x axis. Input bias for accounts that added neutral products to their cart remains 0 for all days except fifth and seventh when its 0.25. For accounts that added misinformative products to their cart, the bias value is greater than 0 for all seven days and for accounts adding debunking products to their cart, the bias value is less than 0 for all days except on fifth day when its 0 and sixth day when its 0.125 (c) [Input bias in product pages ]Line graph showing input bias on the y axis and dates of experiment run (2020-08-12 to 2020-08-18) on x axis. Input bias for accounts performing actions on neutral products is 0 while its greater than 0 for accounts performing actions on misinformative products for all seven days. Input bias in product pages of accounts performing actions on debunking products is less than zero on first, third, fourth and seventh day and unusually high (¿0) on sixth day. Figure 13. (a) Input bias in homepages of accounts performing actions ‘add to cart”, “search + click” and “mark top rated all positive review” for seven days of experiment run. (b) Input bias in pre-purchase recommendations of accounts for 7 days experiment run. These recommendations are only collected for accounts adding products to their carts. (c) Input bias in product pages of accounts performing actions “add to cart”, “search + click” and “mark top rated all positive review” for 7 days of experiment run. M, N and D indicate that the accounts performed actions on misinformative, neutral and debunking products respectively. ## 6\. RQ2 Results [Personalized audit]: Effect of personalization The aim of our Personalized audit was to determine the effect of personalization due to account history on the amount of misinformation returned in search results and various recommendations. Table 8 provides a summary. Below, we explain the effect of personalization on each component. ### 6.1. RQ2a: Search Results We measure personalization in search results for each Amazon filter using two metrics: Jaccard index and Kendall $\tau$ coefficient. Jaccard index determines similarity between two lists. A Jaccard index of 1 indicates that the two lists have same elements and zero indicates that the lists are completely different. On the other hand, Kendall $\tau$ coefficient, also known as Kendall rank correlation coefficient determines the ordinal correlation between two lists. It can take values between [-1,1] with -1 indicating that lists have inverse ordering, 0 signifying no correlation and 1 suggesting that items in the list have same ranks. First, we compare search results of control account and its twin. Recall we created twins for our 2 control accounts in the _Personalized audit_ to establish the baseline noise. Ideally, both should have Jaccard and Kendall rank correlation coefficient closer to 1 since the accounts do not build any history, are set up in a similar manner, perform searches at the same time and are in the same geolocation. Next, we compare search results of control account with treatment accounts that built account histories by performing different actions. If personalization is occurring, the difference between search results of treatment and control should be more than the baseline noise (or Jaccard index and Kendall $\tau$ should be less). Whereas, if the baseline noise itself is large, it indicates inconsistencies and randomness in the search results. Interestingly, we found significant noise in search results of control and its twin for “featured” filter with jaccard index ¡0.8 and Kendall’s rank correlation coefficient ¡0.2, that is, control and its twins seldom matched. Presence of noise suggests that Amazon is injecting some randomness in the “featured” search results. Unfortunately, this means that we would not be able to study the effect of personalization on the accounts for the “featured” search filter setting. For the other three search filters, “average customer review”, “price low to high” and “newest arrivals”, we see high (¿0.8) jaccard index and kendall $\tau$ metric values between and control and its twin. Additionally, we do not see any personalization for these filters since metrics values for treatment- control comparison are similar to that of control-twin comparison. Figure 12 shows the metrics calculation for control account and treatments that have built their search histories by following contributor’s of misinformative, neutral and debunking products. We see two minor inconsistencies for filter “average customer review” in accounts building their history on debunking products where treatment received more similar results to control than its twin account. In any case, the treatment does not see more inconsistency than the control and its twin indicating no personalization. Other user actions show similar results, hence, we have removed their results for brevity. ### 6.2. RQ2b: Recommendations We investigated the occurrence of personalization and its impact on the amount of misinformation in three different recommendation pages. We discuss them below. Homepage recommendations: We find that homepages are personalized only when a user performs click actions on the search results. Thus, actions “add to cart”, “search + click” and “mark top rated most positive review helpful” led to homepage personalization. On the other hand, homepages were not personalized for actions “follow contributor”, “search product” and “google search” actions. After identifying the actions leading to personalized homepages, we investigate the impact of personalization on the amount of misinformation. In other words, we investigate how misinformation bias in homepages is different for accounts building their history by performing actions on misinformative, neutral and debunking products. For each action, we had 6 accounts, two replicates for each action and product type (misinformation, neutral and debunking). For example, for action “add to cart” two accounts built their history by adding misinformative products to cart for 7 days, two added neutral products and two accounts added debunking products to their carts. We calculate per day input bias (ib) in homepages by averaging the misinformation bias scores of each recommended product present in the homepage. Therefore, for every account we have seven bias values. We consider only top two products in each recommendation type. Recall, homepages could contain three different types of recommendations ‘Inspired by your shopping trends”, “Recommended items other customers often buy again” and “Related to items you’ve viewed”. All the different types are considered together for analysis. Statistical tests reveal significant differences in the amount of misinformation present in homepages of accounts that built their histories by performing actions on misinformative, neutral and debunking products (see Table 8). This observation holds true for all three activities “add to cart”, “search + click” and “mark top rated most positive review helpful”. Post hoc test reveals an echo chamber effect. Amount of misinformation in recommendations of products performing actions on misinformative products is more than the amount of misinformation in homepages of accounts performing actions on neutral products which in turn is more than the misinformation present in homepages of accounts performing actions on debunking products. Figure 13(a) shows per day input bias of homepages of different accounts performing different actions. We take an average of the replicates for plotting the graph. Surprisingly, performing actions “mark top rated most positive review helpful” and “search + click” on a misinformative product leads to highest amount of misinformation in the homepages, even more than the homepages of accounts adding misinformative products to the cart. This means that amount of misinformation present in homepage is comparatively less once a user shows an intention to purchase a misinformative product but high if a user shows interest in the misinformative product but doesn’t show an indication to buy it. Figure 13(a) also shows that amount of misinformation present in homepages of accounts performing actions “mark top rated most positive review helpful” and “search + click” on misinformative products gradually increases and becomes 1 on day 4 (2020-08-15). Bias value 1 indicates that all analysed products in homepages were misinformative. Homepage recommendations of products performing actions on neutral objects show 0 bias constantly indicating all recommendations on all days were neutral. On the other hand, average bias in homepages of accounts building history on debunking accounts rose a little above 0 in the first three days but eventually fells below 0 indicating a debunking lean. Pre-purchase recommendations: These recommendations are only presented to users that add product(s) to their Amazon cart. Therefore, they were collected for 6 accounts, 2 of which added misinformative products to cart, 2 added neutral products and the other 2 added debunking products. These recommendations could be of several types. See Figure 1(b) for an example of pre-purchase page. For our analysis, we consider the first product present in each recommendation type. Statistical tests reveal significant difference in the amount of misinformation present in pre-purchase recommendations of accounts that added misinformative, neutral and debunking products to cart (KW H(2, 42) = 32.63, p = 8.19e-08). Those adding misinformative products to cart contain more misinformation than the accounts adding neutral or debunking products to their carts. Figure 13(b) shows the input bias in the pre-purchase recommendations for all the accounts. There is no coherent temporal trend, indicating that the input bias in this recommendation type depends on the particular product being added to cart. However, an echo chamber effect is evident. For example, bias in pre-purchase recommendations of accounts adding misinformative products to cart is above 0 for all 7 days. Product recommendations: We collect product recommendations for accounts performing “add to cart”, “search + click” and “mark top rated most positive review helpful” actions. We find significant difference in the amount of misinformation present in product page recommendations when accounts performed these actions on misinformative, neutral, and debunking products (refer Table 8). Post hoc analysis reveals that product page recommendations of misinformative products contain more misinformation than those of neutral and debunking products. Figure 13(c) shows the input bias present in product pages across accounts. The bias for neutral products is constantly 0 across the 7 days, but for misinformative products, it is constantly greater than 0 for all actions. We see an unusually high bias value on the 6th day (2020-08-17) of our experiment for accounts performing actions on debunking product titled Reasons to Vaccinate: Proof That Vaccines Save Lives (B086B8MM71). We checked the product page recommendations of this particular debunking book and found several misinformative recommendations on its product page. ### 6.3. RQ2c: Auto-complete suggestions We audited auto-complete suggestions to investigate how personalization affects the change in search query suggestions. Our initial hypothesis was that performing actions on misinformative products could increase the auto- complete suggestions of anti-vaccine search queries. However, we found little to no personalization in the auto-complete suggestions indicating that account history built by performing actions on vaccine-related misinformative, neutral or debunking products have little to no effect on how auto-complete suggestions of accounts change. In interest of brevity, we do not add the results and graphs for this component. ## 7\. Discussion There is a growing concern that e-commerce platforms are becoming hubs of dangerous medical misinformation. Unlike search engines where the motivation of the platform is to show relevant search results to sell advertisements, goal of e-commerce platforms is to sell products. The motivation to increase sales means that relevance in recommendations and search suggestions is driven by what people purchase after conducting a search or viewing an item, irrespective of whether the product serves credible information or not. As a result, due to lack of regulatory policies, websites like Amazon are providing a platform to people who are making money by selling misinformation—dangerous anti-vaccine ideas, pseudoscience treatments, or unproven dietary alternatives—some of which could have dangerous effects on people’s health and well-being. With a US market share of 49%, Amazon is the leading product search engine in the United States (Dayton, 2020). Therefore, any misinformation present in its search and recommendations could have a far reaching influence where they can negatively shape users’ viewing and purchasing patterns. Thus, in this paper we audited Amazon for the most dangerous form of health misinformation—vaccine misinformation. Our work resulted in several critical findings with far reaching implications. We discuss them below. ### 7.1. Amazon: a marketplace of multifaceted health misinformation Our analysis shows that Amazon hosts a variety of health misinformative products. Maximum number of such products belong to the category Books and Kindle eBooks (Figure 7). Despite the enormous amount of information available online, people still turn to books to gain information. A Pew Research survey revealed that 73% of Americans read atleast one book in a year (Perrin, 2016). Books are considered “intellectual heft”, have more presence than scientific journals and thus, leave “a wider long lasting wake” (Herr, 2017). Thus, anti- vaccine books could have a wider reach and can easily influence the audience negatively. Moreover, it does not help that a large number of anti-vaccine books are written by authors with medical degrees (Shin and Valente, 2020). Not just anti-vaccine books, there are abundant pseudoscience books on the platform, all suggesting unproven methods to cure diseases. We found diet books suggesting recipes with colloidal silver—an unsafe product, as an ingredient. Some of the books proposing cures for incurable diseases, like autism and auto immune diseases, can have a huge appeal for people suffering with such diseases (Reynolds, 2019). Thus, there is an urgent need to check the quality of health books presented to the users. The next most prominent category of health misinformative products is Amazon Fashion. Numerous apparels are sold on the platform with innovative anti- vaccine slogans, giving tools to the anti-vaccine propagandists to advocate their anti-vaccine agenda and gain visibility, not just in the online world, but in the offline world. During our annotation process, we also found many dietary supplements claiming to treat and cure diseases—a direct violation of Amazon’s policy on dietary supplements. Overall, we find that health misinformation exists on the platform in various forms—books, t-shirts and other merchandise. Additionally, it is very easy to sell problematic content because of lack of appropriate quality-control policies and their enforcement. ### 7.2. Amazon search results: a stockpile of health misinformation Analysis of our _Unpersonalized audit_ revealed that 10.47% of search results promote vaccine and other health-related misinformation. Notably, the higher percentage of products promoting misinformation compared to debunking suggests that anti-vaccine and problematic health-related content is churned out more and the attempts to debunk the existing misinformation is less. We also found that Amazon’s search algorithm puts more health misinformative products in search results than debunking products leading to high input bias for topics like “vaccination”, “vaccine controversies”, “hpv vaccine”, etc. This is specifically true for search filters “featured” and “average customer reviews”. Note, that “featured” is the default search filter indicating that by default users will see more misinformation when they search for the aforementioned topics. On the other hand, if users want to make a purchase decision based on product ratings, again users will be presented with more misinformation since our analysis indicates that sorting by filter ”average customer reviews” leads to highest misinformation bias in the search results. We also found a ranking bias in Amazon’s search algorithm with misinformative products getting ranked higher. Past research has shown that people trust higher ranked search results (Guan and Cutrell, 2007). Thus, more number of higher ranked misinformative products can make problematic ideas in these products appear mainstream. The only positive finding of our analysis was the presence of more debunking products in search results sorted by filter “newest arrivals”. This might indicate that more higher quality products are being sold on the platform in recent times. However, since there are no studies/surveys indicating which search filters are mostly used by people while making purchase decisions, it is difficult to conclude how beneficial this finding is. ### 7.3. Amazon recommendations: problematic echo chambers Many search engines and social media platforms employ personalization to enhance users’ experience on their platform by recommending them items that the algorithm think they will like based on their past browsing or purchasing history. But on the downside, if not checked, personalization can also lead users into a rabbit hole of problematic content. Our analysis of _Personalized audit_ revealed that an echo chamber exists on Amazon where users performing real-world actions on misinformative books are presented with more misinformation in various recommendations. Just a single click on an anti- vaccine book could fill your homepage with several other similar anti vaccine books. And if you proceed to add that book in your cart, Amazon again presents more anti-vaccine books, nudging you to purchase even more problematic content. The worst discovery is that your homepages get filled with more misinformation if you just show an interest in a misinformative product (by clicking on it) compared to when you show an intention to buy it by adding product to your cart. Additionally on the product page itself, you are presented 5 different kinds of recommendations each of which contains equally problematic content. In a nutshell, once you start engaging with misinformative products on the platform, you will be presented with more misinformative stuff at every point of your Amazon navigation route and at multiple places. These findings would not have been concerning if buying a milk chocolate would lead to recommendations of other chocolates of different brands. The problem is that Amazon is blindly applying its algorithms on all products including problematic content. Its algorithms do not differentiate or give special significance to vaccine-related topics. Amazon has learnt from users’ past viewing and purchasing behaviour and has categorized all the anti- vaccine and other problematic health cures together. It presents the problematic content to users performing actions on any of these products, creating a dangerous recommendation loop in the process. There is an urgent need for the platform to treat vaccine and other health related topics differently and ensure high quality searches and recommendations. In the next section, we present a few ways, based on our findings, that could assist the platform in combating health misinformation. ### 7.4. Combating health misinformation Tackling online health misinformation is a complex problem and there is no easy silver-bullet solution to curb its spread. However, the first step towards addressing is accepting that there is a problem. Many tech giants have acknowledged their social responsibility in ensuring high quality in health- related content and are actively taking many steps to ensure the same. For example, Google’s policy “Your Money Or Your Life” classifies medical and health-related search pages as pages of particular importance, whose content should come from reputable websites (McGee, 2013). Pinterest completely hobbled the search results of certain queries such as ‘anti-vax’ (Caron, 2019) and limited the search results for other vaccine-related queries to content from officially recognized health institutions (Hutchinsona, 2019). Even Facebook—a platform known to have questionable content moderation policies—banned anti-vaccine advertisements and demoted the anti-vaccine content in its search results to make its access difficult (Matsakis, 2019). Therefore, given the massive reach and user base of Amazon—206 million website visits every month (10under100, 2020)—it is disconcerting to see that Amazon has not yet joined the bandwagon. Till date, it has not taken any concrete steps towards addressing the problem of anti-vaccine content on its platform. Through our findings, we recommend several short-term and long-term strategies that the platform can adopt. #### 7.4.1. Short term strategies: design interventions. The simplest short term solution would be to introduce design interventions. Our Unpersonalized audit revealed high misinformation bias in search results. The platform can use interventions as an opportunity to communicate to users the quality of data presented to them by signalling misinformation bias. The platform could introduce a bias meter or scale that signals the amount of misinformation present in search results every time it detects a vaccine- related query in its search bar. The bias indicators could be coupled with informational interventions like showing Wikipedia and encyclopedia links, that have already been proven to be effective in reducing traffic to anti- vaccine content (Kim et al., 2020). The second intervention strategy could be to recognise and signal source bias. During our massive annotation process, we realized that several health misinformative books have been written by known anti-vaxxers like Andrew Wakefield, Jenny Mccarthy, Robert S. Mendelsohn, etc. We also present a list of authors who have contributed to most misinformative books in Table 6. Imagine a design where users are presented with a message “The author is a known anti-vaxxer and is known to write books that might contain health minformation” every time they click a book written by these authors. An another extreme short term solution could be to either enforce a platform-wide ban prohibiting sale of any anti-vaccine product or hobble search results for anti-vaccine search queries. #### 7.4.2. Long term strategies: algorithmic modifications and policy changes. Long term interventions would include modification of search, ranking and recommendation algorithm. Our investigations revealed that Amazon’s algorithm has learnt problematic patterns through consumer’s past viewing and buying patterns. It has categorized all products of similar stance together (see several edges connecting red nodes— products promoting misinformation in Figure 11). In some cases, it has also associated some misinformative products with neutral and debunking products (refer Figure 11) Amazon needs to “unlearn” this categorization. Additionally, the platform should incorporate misinformation bias in their search and recommendation algorithms to reduce the exposure to misinformative content. There is also an urgent need to introduce some policy changes. First and foremost, Amazon should stop promoting health misinformative books by sponsoring them. We found 98 misinformative products in the sponsored recommendations indicating that today, anti-vaccine outlets can easily promote their products by spending some money. Amazon should also introduce some minimum quality requirements that should be met before a product is allowed to be sponsored or sold on its platform. It can employ search quality raters to rate the quality of search results for various health-related search queries. Google has already set an example with its extensive Search Quality Rating process and guidelines (Google, 2020, 2019). In recent times Amazon introduced several policy and algorithmic changes including roll out of a new feature “verified purchase” to curb fake reviews problem on its platform (Roddy, 2019). Similar efforts are required to ensure product quality as well. Amazon can introduce a similar “verified quality” or “verified claims” tag with health-related products once they are evaluated by experts. Having a product base of millions of products can make any kind of review process tedious and challenging. Amazon can start by targeting specific health and vaccine related topics that are most likely to be searched. Our work itself presents a list of most popular vaccine- related topics that can be used as a starting point. Can we expect Amazon to make any changes to its current policies and algorithms without sustained pressure? We believe audit studies like ours are the way to reveal biases in the algorithms used by commercial platforms so that there is more awareness about the issues which in turn would create pressure on the organization to act. In the past, such audit studies have led platforms to make positive changes to their algorithms (Raji and Buolamwini, 2019). We hope our work acts as a call to action for Amazon and also inspires vaccine and health audits on other platforms. ## 8\. Limitations Our study is not without limitations. First, we only considered top products in each recommendation-type present on a page while determining bias of the entire page. Annotating and determining bias of all the recommendations occurring in a page would give a much more accurate logic of recommendation algorithms. However, past studies have shown that the top results receive the highest number of clicks, thus, are more likely to receive attention from users (Dean, 2019). Second, search queries themselves have inherent bias. For example query ‘anti vaccine t-shirt’ suggests that user is looking for anti- vax products. Higher bias in search results of neutral queries is much worse than that of biased queries. We did not segregate our analysis based on search query bias. Although, we did notice two neutral search queries namely ‘vaccine’ and ‘varicella vaccine’ appearing in the list of most problematic search-query and filter combinations. Third, while we audited various recommendations present on the platform, we did not analyse the email recommendations—product recommendations present outside the platform. A journalistic report pointed that email recommendations could be contaminated too if a user shows an interest in a misinformative product but leaves the platform without buying it (Diresta, 2019). We leave investigation of these recommendations to future work. Fourth, in our _Personalized audit_ , accounts only built history for a week. Moreover, experiments were only run on Amazon.com. We plan to continue to run our experiments and explore features such as geolocation for future audits. Fifth, our audit study only targeted results returned in response to vaccine-related queries. Since, Amazon is a vast platform that hosts variety of products and sellers, we cannot claim that our results are generalizable for other misinformative topics or conspiracy theories. However, our methodology is generic enough to be applied to other misinformative topics. Lastly, another major limitation of the study is that in the _Personalized audit_ account histories were built in a very conservative setting. Accounts performed actions on only one product each day. Additionally, the actions were only performed on products with the same stance. In real-world it will be tough to find users who only add misinformative products in their carts for seven days continuously. But in spite of this limitation, our study still provides a peek into the workings of Amazon’s algorithm and has paved way for future audits that could use our audit methodology and extensive qualitative coding scheme to perform experiments considering complex real world settings. ## 9\. Conclusion In this study, we conducted two sets of audit experiments on a popular e-commerce platform, Amazon to empirically determine the amount misinformation returned by its search and recommendation algorithm. We also investigated whether personalization due to user history plays any role in amplifying misinformation. Our audits resulted in a dataset of 4,997 Amazon products annotated for health misinformation. We found that search results returned for many vaccine-related queries contain large number of misinformative products leading to high misinformation bias. Moreover, misinformative products are also ranked higher than debunking products. Our study also suggests presence of a filter-bubble effect in recommendations, where users performing actions on misinformative products are presented with more misinformation in their homepages, product page recommendations and pre-purchase recommendations. We believe, our proposed methodology to audit vaccine misinformation can be applied to other platforms to investigate health misinformation bias. Overall, our study brings attention to the need for search engines to ensure high standards and quality of results for health related queries. ## References * (1) * 10under100 (2020) 10under100. 2020\. 20 Eye Opening Amazon Statistics & Facts For 2020. https://10under100.com/amazon-statistics-facts/ * Baker (2018) Loren Baker. 2018\. Amazon’s Search Engine Ranking Algorithm: What Marketers Need to Know. https://www.searchenginejournal.com/amazon-search-engine-ranking-algorithm-explained/265173/ * Ball (2020) P Ball. 2020. 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Additionally, we give details about our Amazon Mechanical Turk (AMT) task in Appendix, Section A.1. # \| Debunking products \| Neutral products \| Misinformative products ---\|---\|---\|--- title (url code) \| S \| R \| title (url code) \| S \| R \| title (url code) \| S \| R 1 \| Vaccinated: One Man’s Quest to Defeat the World’s Deadliest Diseases (006122796X) \| 4.7 \| 134 \| Baby’s Book: The First Five Years (Woodland Friends) 144131976X \| 4.9 \| 614 \| Dissolving Illusions: Disease, Vaccines, and The Forgotten History (1480216895) \| 4.9 \| 953 2 \| Epidemiology and Prevention of Vaccine-Preventable Diseases, 13th Edition (990449114) \| 4.5 \| 11 \| My Child’s Health Record Keeper (Log Book) (1441313842) \| 4.8 \| 983 \| The Vaccine Book: Making the Right Decision for Your Child (Sears Parenting Library) (0316180521) \| 4.8 \| 1013 3 \| The Panic Virus: The True Story Behind the Vaccine-Autism Controversy (1439158657) \| 4.4 \| 175 \| Ten Things Every Child with Autism Wishes You Knew, 3rd Edition: Revised and Updated paperback (1941765882) \| 4.8 \| 792 \| The Vaccine-Friendly Plan: Dr. Paul’s Safe and Effective Approach to Immunity and Health-from Pregnancy Through Your Child’s Teen Years (1101884231) \| 4.8 \| 877 4 \| Vaccines: Expert Consult - Online and Print (Vaccines (Plotkin)) (1455700908) \| 4.4 \| 18 \| Baby 411: Your Baby, Birth to Age 1! Everything you wanted to know but were afraid to ask about your newborn: breastfeeding, weaning, calming a fussy baby, milestones and more! Your baby bible! (1889392618)) \| 4.8 \| 580 \| How to End the Autism Epidemic (1603588248) \| 4.8 \| 717 5 \| Bad Science (865479186) \| 4.3 \| 967 \| Uniquely Human: A Different Way of Seeing Autism (1476776245) \| 4.8 \| 504 \| How to Raise a Healthy Child in Spite of Your Doctor: One of America’s Leading Pediatricians Puts Parents Back in Control of Their Children’s Health (0345342763) \| 4.8 \| 598 6 \| Reasons to Vaccinate: Proof That Vaccines Save Lives (B086B8MM71) \| 4.3 \| 232 \| The Whole-Brain Child: 12 Revolutionary Strategies to Nurture Your Child’s Developing Mind (0553386697) \| 4.7 \| 2347 \| Miller’s Review of Critical Vaccine Studies: 400 Important Scientific Papers Summarized for Parents and Researchers (188121740X) \| 4.8 \| 473 7 \| Deadly Choices: How the Anti-Vaccine Movement Threatens Us All (465057969) \| 4.2 \| 223 \| We’re Pregnant! The First Time Dad’s Pregnancy Handbook (1939754682) \| 4.7 \| 862 \| Herbal Antibiotics, 2nd Edition: Natural Alternatives for Treating Drug-resistant Bacteria (1603429875) \| 4.7 \| 644 Table 9. Books corresponding to each annotation value shortlisted to build account histories in our _Personalized audit_. S represents the star rating of the product and R denotes the number of ratings received by the book. Figure 14. Our multi-stage iterative qualitative coding process to obtain a coding scheme for annotating Amazon products for health misinformation. [Qualitative coding process]Three stages: (1) multiple iterations of qualitative codification by the first author, (2) multiple iterations of refining the codification scheme based on feedback obtained from 6 researchers followed by (3) feedback from an external researcher. ### A.1. Amazon Mechanical Turk Job #### A.1.1. Turk job description In this section, we describe how we obtained annotations for our study from Amazon Mechanical Turk workers (MTurks). Past research has shown that it is possible to get good data from crowd-sourcing platforms like Amazon Mechanical Turk (AMT) if the workers are screened and trained for the crowd-sourced task (Mitra et al., 2015). Below we describe the screening process and our annotation task briefly. #### A.1.2. Screening To get high quality annotations, we screened MTurks by adding 3 qualification requirements. First, we required MTurks to be Masters. Second, we required them to have atleast 90% approval rating. And lastly, we required them to get a full score of 100 in a Qualification Test. We introduced a test to ensure that MTurks attempting our annotation job had a good understanding of the annotation scheme. The test had one eligibility question asking them to confirm whether they are affiliated to authors’ University. Other three questions involved Mturks to annotate three Amazon products (see Figure 18 for a sample question). First author annotated these products and thus, their annotation values were known. To ensure MTurks understood the task and annotation scheme, we gave detailed instructions and described each annotation value in detail with various examples of Amazon products in the qualifying test (Figures 15, 16 and 17). Examples were added as visuals. In each example, we marked the meta data used used for the annotation and explained why a particular annotation value was assigned to the product (see Figure 17). We took two steps to ensure that instructions and test questions were easy to understand and attempt. First, we posted the test on subreddit r/mturk121212https://www.reddit.com/r/mturk/—a community of MTurks, to obtain feedback. Second, we did a pilot run by posting ten tasks along with the aforementioned screening requirements. After obtaining positive feedback from the community and successful pilot-run, we released our AMT job titled “Amazon product categorization task”. We paid the Turks according to the United States federal minimum wage ($7.25/hr). Additionally, we did not disapprove any worker’s responses. #### A.1.3. Amazon product categorization task We posted 1630 annotations (tasks) in batches of 50 at a time. The job was setup to get three responses for each annotation value. The majority response was selected to label the Amazon product. To avoid any MTurk bias, we did not explicitly reveal that the idea behind the task was to get misinformation annotations. We used the term ”Amazon product categorization” to describe our project and task throughout. For 34 products, all three MTurk responses differed. The first author then annotated these products to get annotation values. Figure 19 shows the interface of our AMT job. Figure 15. Figure illustrating Qualification Test instructions. Test included 4 questions including one eligibility question required to be added by authors’ University. A full score of 100 was required to qualify the test. [Qualitative test instructions:]The instructions were worded as “You will be graded on 4 questions in total including the eligibility question. You qualify if you fulfill our eligibility criteria and answer all three questions mentioned below correctly. Please read the instructions carefully before attempting the questions. In case you do not qualify, you can retake this test after 10 minutes.” Figure 16. Task description in the Qualification test. Same instructions were provided in the actual task. [Instructions and annotation description]The figure is a snapshot of the annotation instructions provided to MTurks including detailed description of each annotation value. Figure 17. Example explaining Turks how to determine the annotation value. [Figure illustrates how user can assign annotation value to a product by looking at its metadata]In the figure we highlight the description of an Amazon book The vaccine-friendly plan where the author addresses vaccines as aluminium shots and has come up with a vaccine schedule that is not approved by medical authorities. We also show the top critical review which suggests that book is anti-vaccine. Figure 18. Example of Qualification Test question. [Qualification Test question]Figure illustrates a qualification test question which has URL of the Amazon product along with radio buttons enlisting all the annotation values. MTurks had to select the annotation value that best suits the Amazon product. Figure 19. Interface of our Amazon product categorization task. [Interface of our AMT product categorization task]Each task had a URL of the Amazon product along with radio buttons enlisting all the annotation values.
# Some punctured codes of several families of binary linear codes Xiaoqiang Wang, Dabin Zheng, Cunsheng Ding Corresponding author. Xiaoqiang Wang and Dabin Zheng are with the Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China (E-mail<EMAIL_ADDRESS>[email protected]). Cunsheng Ding is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China (E-mail: [email protected]). Abstract. Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by $\mathcal{C}(f)=\left\\{{\rm Tr}(af(x)+bx)_{x\in\mathbb{F}_{q^{m}}^{}}:a,b\in\mathbb{F}_{q^{m}}\right\\}$, where $q$ is a prime power, ${\mathbb{F}}_{q^{m}}^{}={\mathbb{F}}_{q^{m}}\setminus\\{0\\}$, ${\mathrm{Tr}}$ is the trace function from ${\mathbb{F}}_{q^{m}}$ to ${\mathbb{F}}_{q}$, and $f(x)$ is a function from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{q^{m}}$ with $f(0)=0$. Almost bent functions, quadratic functions and some monomials on ${\mathbb{F}}_{2^{m}}$ were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper. Keywords. Boolean function, linear code, punctured code, distance-optimal code, weight distribution 2010 Mathematics Subject Classification. 94B05, 94B15 ## 1 Introduction of motivations, objectives, and methodology Let $q$ be a prime power and $n$ be a positive integer. An $[n,k,d]$ code $\mathcal{C}$ over the finite field $\mathbb{F}_{q}$ is a $k$-dimensional linear subspace of $\mathbb{F}_{q}^{n}$ with minimum Hamming distance $d$. The dual code, denoted by ${\mathcal{C}}^{\perp}$, of ${\mathcal{C}}$ is defined by ${\mathcal{C}}^{\perp}=\left\\{{\mathbf{x}}=(x_{0},\ldots,x_{n-1})\in{\mathbb{F}}_{q}^{n}:\sum_{i=0}^{n-1}x_{i}c_{i}=0\ \forall\ {\mathbf{c}}=(c_{0},\ldots,c_{n-1})\in{\mathcal{C}}\right\\}.$ The minimum distance of ${\mathcal{C}}^{\perp}$, denoted by $d^{\perp}$, is called the dual distance of ${\mathcal{C}}$. ${\mathcal{C}}$ is called a projective code if its dual distance is at least $3$. An $[n,k,d]$ code over ${\mathbb{F}}_{q}$ is said to be distance-optimal (respectively, dimension- optimal and length-optimal) if there is no $[n,k,d^{\prime}\geq d+1]$ (respectively, $[n,k^{\prime}\geq k+1,d]$ and $[n^{\prime}\leq n-1,k,d]$) linear code over ${\mathbb{F}}_{q}$. An optimal code is a code that is length- optimal, or dimension-optimal, or distance-optimal, or meets a bound for linear codes. A binary linear code $\mathcal{C}$ is called self-complementary if it contains the all-one vector. Let $A_{i}$ denote the number of codewords with Hamming weight $i$ in $\mathcal{C}$. The weight enumerator of $\mathcal{C}$ is defined by $1+A_{1}x+A_{2}x^{2}+\cdots+A_{n}x^{n}$. The weight distribution of $\mathcal{C}$ is defined by the sequence $(1,A_{1},\cdots,A_{n})$. If the number of nonzero $A_{i}$ in the sequence $(A_{1},\cdots,A_{n})$ is $t$, then the code $\mathcal{C}$ is said to be a $t$-weight code. By the parameters of a code, we mean its length, dimension and minimum distance. Coding theory has important applications in communications systems, data storage systems, consumer electronics, and cryptography. In addition, coding theory is closely related to many areas of mathematics, such as algebra, algebraic geometry, algebraic function fields, algebraic number theory, association schemes, combinatorics, finite fields, finite geometry, graph theory, and group theory. These are the major motivations of studying coding theory. Constructing linear codes with desired parameters and weight distributions has been an important task in the history of coding theory. Linear codes may be constructed directly with algebraic approaches, combinatorial approaches and other approaches. Alternatively, almost all linear codes over finite fields can be constructed from some known codes by the puncturing or shortening techniques. Let ${\mathcal{C}}$ be an $[n,k,d]$ code over ${\mathbb{F}}_{q}$, and let $T$ be a set of $t$ coordinate positions in ${\mathcal{C}}$. We puncture ${\mathcal{C}}$ by deleting all the coordinates in $T$ in each codeword of ${\mathcal{C}}$. The resulting code is still linear and has length $n-t$, where $t=\|T\|$. We denote the punctured code by ${\mathcal{C}}^{T}$. Let ${\mathcal{C}}(T)$ be the set of codewords which are $0$ on $T$. Then ${\mathcal{C}}(T)$ is a subcode of ${\mathcal{C}}$. We now puncture ${\mathcal{C}}(T)$ on $T$, and obtain a linear code over ${\mathbb{F}}_{q}$ with length $n-t$, which is called a _shortened code_ of ${\mathcal{C}}$, and is denoted by ${\mathcal{C}}_{T}$. The puncturing and shortening techniques are two very important tools for constructing new codes from old ones. It was shown that every projective linear code over ${\mathbb{F}}_{q}$ (i.e., the minimum distance of the dual code is at least 3) is a punctured code of a Simplex code over ${\mathbb{F}}_{q}$ and a shortened code of a Hamming code over ${\mathbb{F}}_{q}$ [37]. These facts justify the importance of the Simplex codes and the Hamming codes as well as the puncturing and shortening techniques. Note that the Simplex codes are optimal with respect to the Griesmer bound. Since every projective code is a punctured Simplex code, a punctured code of an optimal linear code may have good or bad parameters. To obtain a very good punctured code ${\mathcal{C}}^{T}$ from a good or optimal linear code ${\mathcal{C}}$, one has to choose a proper set $T$ of coordinate positions in ${\mathcal{C}}$. This is the difficulty of using the puncturing technique to construct new linear codes with good parameters from old ones [37, 56]. In this paper, we will use the puncturing technique to construct new codes with interesting and new parameters from some old linear codes. Linear codes with few weights have applications in secret sharing [1], strongly regular graphs [5], association schemes [4] and authentication codes [17]. In finite geometry, hyperovals in the projective geometry ${\mathrm{PG}}(2,2^{m})$ are the same as $[2^{m}+2,3,2^{m}]$ MDS codes with two weights [13, Chapter 12], maximal arcs in ${\mathrm{PG}}(2,2^{m})$ are the same as a special type of two-weight codes [13, Chapter 12], and ovoids in ${\mathrm{PG}}(3,q)$ are the same as a special type of two-weight codes [13, Chapter 13]. Many families of linear codes have been used to construct combinatorial $t$-designs [13, Chapters 5–13]. These are some of the motivations of studying linear codes with few weights in the literature. In the past two decades, a lot of progress on the construction of linear codes with few weights has been made. The reader is referred to [16, 11, 12, 18, 24, 34, 38, 43, 46, 50, 51, 52, 47, 54, 44, 60] and the references therein for information. One of the objectives of this paper is to construct binary linear codes with few weights. Functions and linear codes are closely connected. In the literature two general constructions of linear codes with functions over finite fields have been intensively investigated [12]. The first construction is given by $\displaystyle\mathcal{C}(f)=\left\\{{\rm Tr}(af(x)+bx)_{x\in\mathbb{F}_{q^{m}}^{}}\,:\,a,b\in\mathbb{F}_{q^{m}}\right\\},$ (1) where $q$ is a prime power, ${\mathbb{F}}_{q^{m}}^{}={\mathbb{F}}_{q^{m}}\setminus\\{0\\}$, ${\mathrm{Tr}}$ is the trace function from ${\mathbb{F}}_{q^{m}}$ to ${\mathbb{F}}_{q}$, and $f(x)$ is a function from $\mathbb{F}_{q^{m}}$ to $\mathbb{F}_{q^{m}}$ with $f(0)=0$. It is clear that $\mathcal{C}(f)$ is a linear code with length $q^{m}-1$ and dimension at most $2m$. If $f(x)$ is a monomial, then ${\mathcal{C}}(f)$ is permutation-equivalent to a cyclic code [7]. This general construction has a long history and its importance is supported by Delsarte’s Theorem [10]. The weight distribution of $\mathcal{C}(f)$ is closely related to the value distributions of certain exponential sums, and is difficult to settle in general. In order to determine the weight distribution of ${\mathcal{C}}(f)$, people usually choose $f(x)$ to be a special function such as a quadratic function, PN function, and APN function. Many good and optimal linear codes have been obtained with this construction. This is also a main method for constructing linear codes with few weights. The reader is referred to, for example, [7, 26, 20, 39, 33, 43, 51, 57] for information. The second general construction of linear codes is described as follows [16, 53]. Let $D=\\{d_{1},d_{2},\cdots,d_{n}\\}\subset{\mathbb{F}}_{q^{m}}^{}$ be a multiset. Define a linear code ${\mathcal{C}}_{D}=\left\\{\left({\rm Tr}(xd_{1}),{\rm Tr}(xd_{2}),\cdots,{\rm Tr}(xd_{n})\right):x\in{\mathbb{F}}_{q^{m}}\right\\},$ where $q$ is a prime power, ${\mathrm{Tr}}$ is the trace function from ${\mathbb{F}}_{q^{m}}$ to ${\mathbb{F}}_{q}$. The code ${\mathcal{C}}_{D}$ over ${\mathbb{F}}_{q}$ has length $n$ and dimension at most $m$, where $D$ is called the defining set of ${\mathcal{C}}_{D}$. This construction is fundamental in the sense that every linear code over ${\mathbb{F}}_{q}$ can be expressed as ${\mathcal{C}}_{D}$ for some positive integer $m$ and some subset $D$ of ${\mathbb{F}}_{q^{m}}$ [23, 55]. It is known that this construction is equivalent to the generator matrix construction of linear codes. The code $\mathcal{C}_{D}$ may have good parameters if the defining set is properly chosen. With the second general construction, many good linear codes with few weights have been constructed [11, 15, 19, 24, 36, 34, 25, 38, 43, 52, 50]. With some variants of the second construction, interesting linear codes were obtained in [48, 34, 32]. By the definition of the second construction above, ${\mathcal{C}}_{{\mathbb{F}}_{q^{m}}^{}}$ has parameters $[q^{m}-1,m,(q-1)q^{m-1}]$ and weight enumerator $1+(q^{m}-1)z^{(q-1)q^{m-1}}$. If $D\subset{\mathbb{F}}_{q^{m}}^{}$ does not contain repeated elements, let $\bar{D}={\mathbb{F}}_{q^{m}}^{}\setminus D$. In this case, we have ${\mathcal{C}}_{D}=({\mathcal{C}}_{{\mathbb{F}}_{q^{m}}^{}})^{\bar{D}}$, where the coordinate positions in ${\mathcal{C}}_{{\mathbb{F}}_{q^{m}}^{}}$ are indexed by the elements in ${\mathbb{F}}_{q^{m}}^{}$. This means that ${\mathcal{C}}_{D}$ is in fact a punctured code of the one-weight code ${\mathcal{C}}_{{\mathbb{F}}_{q^{m}}^{}}$, which is a concatenation of $(q-1)$ Simplex codes over ${\mathbb{F}}_{q}$ with the same parameters. Hence, the second construction above is in fact a puncture construction, and every projective linear code over ${\mathbb{F}}_{q}$ is a punctured code of the one- weight code ${\mathcal{C}}_{{\mathbb{F}}_{q^{m}}^{}}$. Motivated by the power of the puncture technique and the first construction, in this paper we study some punctured codes of several families of binary linear codes ${\mathcal{C}}(f)$ from special functions on ${\mathbb{F}}_{2^{m}}$. Specifically, we will study the following punctured codes. Let $f$ be a function on ${\mathbb{F}}_{2^{m}}$ with $f(0)=0$, and let $D=\\{d_{1},d_{2},\cdots,d_{n}\\}\subset{\mathbb{F}}_{2^{m}}^{}$ that does not contain any repeated elements. Define $\bar{D}={\mathbb{F}}_{2^{m}}^{}\setminus D$. In this paper, we will study the punctured code ${\mathcal{C}}(f)^{\bar{D}}=\left\\{{\mathbf{c}}(a,b)=\left({\rm Tr}(af(d_{1})+bd_{1}),\cdots,{\rm Tr}(af(d_{n})+bd_{n})\right):a,b\in{\mathbb{F}}_{2^{m}}\right\\},$ (2) where ${\mathrm{Tr}}$ is the trace function from ${\mathbb{F}}_{2^{m}}$ to ${\mathbb{F}}_{2}$ and the binary code ${\mathcal{C}}(f)$ was defined in (1). We call the set $D$ the position set of the code ${\mathcal{C}}(f)^{\bar{D}}$, as we index the coordinate positions of the code ${\mathcal{C}}(f)$ with the elements in ${\mathbb{F}}_{2^{m}}^{}$. The dimension of ${\mathcal{C}}(f)^{\bar{D}}$ is at most $2m$. The two objectives of this paper are to obtain binary linear codes ${\mathcal{C}}(f)^{\bar{D}}$ with new parameters and few weights and $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ with new and good parameters. To this end, we have to select $f$ and the position set $D$ carefully. Concretely, we first choose the position set to be $\begin{split}D=\left\\{x\in\mathbb{F}_{2^{m}}^{}\,:\,{\rm Tr}(\lambda f(x))=\nu\right\\}\end{split}$ (3) and determine the weight distributions of ${\mathcal{C}}(f)^{\bar{D}}$, where $\nu\in\left\\{0,1\right\\}$, $\lambda\in\mathbb{F}_{2^{m}}^{}$ and $f(x)$ is an almost bent function from ${\mathbb{F}}_{2^{m}}$ to itself. We show that ${\mathcal{C}}(f)^{\bar{D}}$ is a five-weight code if $\nu=0$ and a self- complementary six-weight code if $\nu=1$. Some of the codes ${\mathcal{C}}(f)^{\bar{D}}$ are optimal according to the tables of best codes known in [22]. The dual of ${\mathcal{C}}(f)^{\bar{D}}$ is distance-optimal with respect to the sphere packing bound if $\nu=1$. We then present several classes of four-weight or six-weight linear codes by choosing $f(x)$ to be some special quadratic functions, and the position set to be the support of ${\rm Tr}(x)$, i.e., $D=\left\\{x\in\mathbb{F}_{2^{m}}^{}\,:\,{\rm Tr}(x)=1\right\\}.$ (4) Several families of complementary binary linear codes are obtained. The parameters of the duals of ${\mathcal{C}}(f)^{\bar{D}}$ are also determined and almost all of them are distance-optimal with respect to the sphere packing bound. Finally, we present several classes of binary linear codes with three weights, or five weights or six weights by selecting the position sets to be some cyclotomic classes. Some of the codes and their duals are distance- optimal. The parameters of most of the codes presented in this paper are new. The rest of this paper is organized as follows. Section 2 introduces some preliminaries. Section 3 investigates the weight distribution of the linear code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual, where $f(x)$ is an almost bent function, $D=\left\\{x\in\mathbb{F}_{2^{m}}^{}:{\rm Tr}(\lambda f(x))=\nu\right\\}$, $\nu\in\left\\{0,1\right\\}$ and $\lambda\in\mathbb{F}_{2^{m}}^{}$. Section 4 determines the weight distribution of the linear code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual, where $f(x)$ is some special quadratic function and $D=\left\\{x\in\mathbb{F}_{2^{m}}^{}:{\rm Tr}(x)=1\right\\}$. Section 5 settles the weight distribution of the linear code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual, where $D$ is a cyclotomic class and $f$ is a monomial. Section 6 concludes this paper. ## 2 Preliminaries In this section, we introduce some special functions on ${\mathbb{F}}_{2^{m}}$, some exponential sums and some basic results in coding theory, which will be used later in this paper. ### 2.1 Notation used starting from now on Starting from now on, we assume $m\geq 4$ and adopt the following notation unless otherwise stated: $\bullet$ ${\mathbb{F}}_{2^{m}}$ is the finite field with $2^{m}$ elements and $\gamma$ is a primitive element of $\mathbb{F}_{2^{m}}$. $\bullet$ ${\mathbb{F}}_{2^{m}}^{}={\mathbb{F}}_{2^{m}}\setminus\\{0\\}$. $\bullet$ ${\rm Tr}(\cdot)$ is the absolute trace function from ${\mathbb{F}}_{2^{m}}$ to ${\mathbb{F}}_{2}$. $\bullet$ ${\rm Tr}_{u}^{v}(\cdot)$ is the trace function from ${\mathbb{F}}_{2^{v}}$ to ${\mathbb{F}}_{2^{u}}$, where $u,v$ are positive integers such that $u\,\|\,v$. $\bullet$ $v_{2}(\cdot)$ is the 2-adic order function with $v_{2}(0)=\infty$. $\bullet$ $\operatorname{wt_{H}}(\bf c)$ denotes the Hamming weight of a vector ${\mathbf{c}}$. $\bullet$ $d_{H}({\mathcal{C}})$ denotes the minimum distance of a linear code ${\mathcal{C}}$. ### 2.2 AB and APN functions Let $f(x)$ be a function from $\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2^{m}}$. The Walsh transform of $f(x)$ at $(a,b)\in\mathbb{F}_{2^{m}}^{2}$ is defined as $W_{f}(a,b)=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(af(x)+bx)}.$ (5) If $W_{f}(a,b)=0$ or $\pm 2^{\frac{m+1}{2}}$ for any pair $(a,b)\in{\mathbb{F}}_{2^{m}}^{2}$ with $a\neq 0$, then $f(x)$ is called an almost bent (AB) function. Almost bent functions exist only for odd $m$. Define $\delta_{f}(a,b)={\rm max}_{a\in\mathbb{F}_{2^{m}}^{},b\in\mathbb{F}_{2^{m}}}\|\\{x\in\mathbb{F}_{2^{m}}\,:\,f(x+a)+f(x)=b\\}\|,$ then $f(x)$ is called an almost perfect nonlinear (APN) function if $\delta_{f}(a,b)=2$. APN and AB functions have applications in coding theory, combinatorics, cryptography, finite geometry and sequence design. Many good linear codes over finite fields have been constructed with APN and AB functions [6, 11, 12, 33, 43]. AB functions and APN functions have the following relationship. ###### Lemma 2.1 [3] Let $\mathbb{F}_{2^{m}}$ be a finite field with $2^{m}$ elements. If $f(x)$ is an almost bent function over $\mathbb{F}_{2^{m}}$, then $f(x)$ is an almost perfect nonlinear function over $\mathbb{F}_{2^{m}}$. The converse is not true for Lemma 2.1, as almost bent functions exist only for $m$ being odd while almost perfect nonlinear functions exist for $m$ being even too. ### 2.3 Quadratic functions By identifying the finite field ${\mathbb{F}}_{2^{m}}$ with the $m$-dimensional vector space ${\mathbb{F}}_{2}^{m}$ over ${\mathbb{F}}_{2}$, a function $f$ from ${\mathbb{F}}_{2^{m}}$ to ${\mathbb{F}}_{2}$ can be viewed as an $m$-variable polynomial over ${\mathbb{F}}_{2}$. In the sequel, we fix a basis of ${\mathbb{F}}_{2^{m}}$ over ${\mathbb{F}}_{2}$ and identify $x\in{\mathbb{F}}_{2^{m}}$ with a vector $(x_{1},x_{2},\cdots,x_{m})\in{\mathbb{F}}_{2}^{m}$, a quadratic function over ${\mathbb{F}}_{2}$ is of the form: $Q(x_{1},x_{2},\cdots,x_{m})=(x_{1},x_{2},\cdots,x_{m})A(x_{1},x_{2},\cdots,x_{m})^{T},$ where $A=(a_{ij})_{m\times m},\,a_{ij}\in{\mathbb{F}}_{2}$, is an upper triangular matrix. The matrix $A+A^{T}$ is called an alternate matrix and its rank must be even [49]. By the theory of linear equations, the rank $r$ of the matrix $A+A^{T}$ is equal to the codimension of the ${\mathbb{F}}_{2}$-linear subspace $V=\\{x\in{\mathbb{F}}_{2^{m}}:Q(x+z)+Q(x)+Q(z)=0\mbox{ for all }z\in{\mathbb{F}}_{2^{m}}\\},$ (6) i.e. $r=m-\dim_{{\mathbb{F}}_{2}}V$. Let $G(x)$ be a linear polynomial over $\mathbb{F}_{2^{m}}$, then $\begin{split}\left(\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(x)+G(x))}\right)^{2}&=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(x)+G(x))}\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(y)+G(y))}\\\ &=\sum_{x,y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(x+y)+G(x+y)+Q(x)+G(x))}\\\ &=\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(y)+G(y))}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(x+y)+Q(x)+Q(y))}\\\ &=2^{m}\cdot\sum_{y\in V}(-1)^{{\rm Tr}(Q(y)+G(y))},\end{split}$ where $V$ was defined in $(\ref{eq:quadraticform1})$. It is easy to check that ${{\rm Tr}\left(Q(x+y)+G(x+y)\right)}={{\rm Tr}\left(Q(x)+G(x)\right)}+{{\rm Tr}\left(Q(y)+G(y)\right)}$ for any $x,y\in V$. Then $\begin{split}\left(\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(Q(x)+G(x))}\right)^{2}=\begin{cases}2^{m+r},&\text{if ${{\rm Tr}\left(Q(y)+G(y)\right)}=0$ for all $y\in V$,}\\\ 0,&\text{otherwise},\end{cases}\end{split}$ (7) where $r$ is the rank of $Q(x)$ and $r=m-\dim_{{\mathbb{F}}_{2}}V$. The following are some well known results about quadratic forms, which will be needed in this paper. ###### Lemma 2.2 [8, 9] Let $m$ and $k$ be non-negative integers with $v_{2}(m)\leq v_{2}(k)$ and $a,b\in\mathbb{F}_{2^{m}}$ with $a\neq 0$. Let $\begin{split}S(a,b)=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}\left(ax^{2^{k}+1}+bx\right)},\end{split}$ (8) then the possible values of $S(a,b)$ are in the set $\\{0,\pm 2^{\frac{m+\ell}{2}}\\}$, where $\ell=\gcd(m,k)$. ###### Lemma 2.3 [8, 9] Let $m$ and $k$ be non-negative integers with $v_{2}(m)>v_{2}(k)$ and $a,b\in\mathbb{F}_{2^{m}}$ with $a\neq 0$. Let $S(a,b)$ be defined in (8). Then $S(a,b)=0$ unless the equation $a^{2^{k}}x^{2^{2k}}+ax+b^{2^{k}}=0$ is solvable. Let $\gamma$ be a primitive element of $\mathbb{F}_{2^{m}}$. Let $\ell=\gcd(m,k)$. Assume $a^{2^{k}}x^{2^{2k}}+ax+b^{2^{k}}=0$ is solvable. Then there are two possibilities as follows. (i) If $a\neq\gamma^{s\left(2^{\ell}+1\right)}$ for any integer $s$, then the equation has a unique solution $x_{b}$ for any $b\in\mathbb{F}_{2^{m}}$, and $S(a,b)=(-1)^{\frac{m}{2\ell}-{\rm Tr}\left(ax_{b}^{2^{k}+1}\right)}2^{\frac{m}{2}}.$ (ii) If $a=\gamma^{s\left(2^{\ell}+1\right)}$ for some integer $s$, then the equation is solvable if and only if ${\rm Tr}_{2\ell}^{m}\left(b\beta^{-s}\right)=0$, where $\beta\in\mathbb{F}_{2^{m}}^{}$ is the unique element satisfying $\beta^{\frac{2^{k}+1}{2^{\ell}+1}}=\gamma$. In such case, $S(a,b)=-(-1)^{\frac{m}{2\ell}-{\rm Tr}\left(ax_{b}^{2^{k}+1}\right)}2^{\frac{m}{2}+\ell},$ where $x_{b}$ is a solution to $a^{2^{k}}x^{2^{2k}}+ax+b^{2^{k}}=0$. ###### Lemma 2.4 [42] Let $\gamma$ be a primitive element of $\mathbb{F}_{2^{m}}$. Assume that $m=2sh$ and $\ell\,\|\,(2^{h}+1)$. Then $\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(\gamma^{i}x^{\ell})}=\left\\{\begin{array}[]{lcl}(-1)^{s}2^{\frac{m}{2}},&{\rm if}\,\,\,i\not\equiv 0\pmod{\ell},\\\ (-1)^{s-1}(\ell-1)2^{\frac{m}{2}},&{\rm if}\,\,\,i\equiv 0\pmod{\ell}.\end{array}\right.$ ###### Lemma 2.5 [40] Let $\ell=\gcd(\frac{m}{2},k)$ and $\ell^{\prime}=\gcd(\frac{m}{2}+k,2k)$. Let $S_{1}(a,b)=(-1)^{{\rm Tr}\left(ax^{2^{k}+1}+bx^{2^{\frac{m}{2}}+1}\right)}.$ If $\ell^{\prime}=2\ell$ and $(a,b)$ runs over $\mathbb{F}_{2^{m}}\times\mathbb{F}_{2^{\frac{m}{2}}}$, then $\begin{split}S_{1}(a,b)=\begin{cases}2^{m},&{\rm occuring}\,\,\,1\,\,{\rm time},\\\ -2^{\frac{m}{2}},&{\rm occuring}\,\,\,\frac{2^{3k}(2^{\frac{m}{2}}-1)(2^{m}-2^{m-2k}-2^{m-3k}+2^{\frac{m}{2}}-2^{\frac{m}{2}-k}+1}{(2^{k}+1)(2^{2k}-1)}\,\,{\rm times},\\\ 2^{\frac{m}{2}+k},&{\rm occuring}\,\,\,\frac{2^{k}(2^{m}-1)(2^{m}-2^{m-\ell}+2^{m-2\ell}+1)}{(2^{k}+1)^{2}}\,\,{\rm times},\\\ -2^{\frac{m}{2}+2k},&{\rm occuring}\,\,\,\frac{(2^{\frac{m}{2}-\ell}-1)(2^{m}-1)}{(2^{k}+1)(2^{2k}-1)}\,\,{\rm times}.\end{cases}\end{split}$ ### 2.4 Pless power moments and the sphere packing bound To study the parameters of the duals of the punctured binary codes ${\mathcal{C}}(f)^{\bar{D}}$, we need the Pless power moments of linear codes. Let $\mathcal{C}$ be a binary $[n,k]$ code, and denote its dual by $\mathcal{C}^{\perp}$. Let $A_{i}$ and $A^{\perp}_{i}$ be the number of codewords of weight $i$ in $\mathcal{C}$ and $\mathcal{C}^{\perp}$, respectively. The first five Pless power moments are the following [41, p. 131]: $\begin{split}&\sum_{i=0}^{n}A_{i}=2^{k};\\\ &\sum_{i=0}^{n}iA_{i}=2^{k-1}(n-A_{1}^{\perp});\\\ &\sum_{i=0}^{n}i^{2}A_{i}=2^{k-2}[n(n+1)-2nA_{1}^{\perp}+2A_{2}^{\perp}];\\\ &\sum_{i=0}^{n}i^{3}A_{i}=2^{k-3}[n^{2}(n+3)-(3n^{2}+3n-2)A_{1}^{\perp}+6nA_{2}^{\perp}-6A_{3}^{\perp}];\\\ &\sum_{i=0}^{n}i^{4}A_{i}=2^{k-4}[n(n+1)(n^{2}+5n-2)-4n(n^{2}+3n-2)A_{1}^{\perp}+4(3n^{2}+3n-4)A_{2}^{\perp}-24nA_{3}^{\perp}+24A_{4}^{\perp}].\\\ \end{split}$ If $A_{1}^{\perp}=A_{2}^{\perp}=A_{3}^{\perp}=A_{4}^{\perp}=0$, then the sixth Pless power moment becomes the following: $\sum_{i=0}^{n}i^{5}A_{i}=2^{k-5}\cdot n^{5}+5\cdot 2^{k-4}\cdot n^{4}+15\cdot 2^{k-5}\cdot n^{3}-5\cdot 2^{k-4}\cdot n^{2}-A_{5}^{\perp}\cdot 2^{k-5}\cdot 120.$ We will need the following bound for binary linear codes later. ###### Lemma 2.6 (The sphere packing bound) Let $\mathcal{C}$ be an $[n,k,d]$ binary code. Then $2^{n}\geq 2^{k}\sum_{i=0}^{\lfloor\frac{d-1}{2}\rfloor}\left(\begin{array}[]{cccc}n\\\ i\\\ \end{array}\right).$ ## 3 Some punctured codes of the binary codes from almost bent functions Recall the code ${\mathcal{C}}(f)$ defined in (1). When $q=2$ and $f(x)=x^{2^{h}+1}$ with $\gcd(h,m)=1$ and $m$ being odd, the parameters and weight distribution of the binary code ${\mathcal{C}}(f)$ were settled in [29, 30]. When $q=2$, $m$ is odd and $f(x)$ is an almost bent function on ${\mathbb{F}}_{2^{m}}$, the parameters and weight distribution of the binary code ${\mathcal{C}}(f)$ were settled in [6]. The binary code ${\mathcal{C}}(f)$ has parameters $[2^{m}-1,2m,2^{m-1}-2^{(m-1)/2}]$ and three nonzero weights [6]. Let ${\mathcal{C}}(f)^{\bar{D}}$ be the binary punctured code defined in (2) with position set $D$ in (3), where $f(x)$ is an almost bent function from ${\mathbb{F}}_{2^{m}}$ to itself. In this section, we investigate the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual. We first give the length of the linear code ${\mathcal{C}}(f)^{\bar{D}}$ in the following lemma. ###### Lemma 3.1 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (3), where $f(x)$ is an almost bent function from $\mathbb{F}_{2^{m}}$ to itself. Then the length $n$ of ${\mathcal{C}}(f)^{\bar{D}}$ is $\begin{split}n=\|D\|=\begin{cases}2^{m-1}-(-1)^{\nu}2^{\frac{m-1}{2}}-1+\nu,&{\rm if}\,\,\,W_{f}(\lambda,0)=-2^{\frac{m+1}{2}},\\\ 2^{m-1}+(-1)^{\nu}2^{\frac{m-1}{2}}-1+\nu,&{\rm if}\,\,\,W_{f}(\lambda,0)=2^{\frac{m+1}{2}},\\\ 2^{m-1}-1+\nu,&{\rm if}\,\,\,W_{f}(\lambda,0)=0,\end{cases}\end{split}$ where $W_{f}(\lambda,0)$ was defined in (5) and $\nu\in\\{0,1\\}$. In order to apply the Pless power moments to determine the multiplicity of each Hamming weight of ${\mathcal{C}}(f)^{\bar{D}}$, we need to investigate the minimum Hamming distance of its dual. ###### Lemma 3.2 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (3), where $f(x)$ is an almost bent function from $\mathbb{F}_{2^{m}}$ to itself. Then the dual distance is lower bounded by $\begin{split}d_{H}\left(\left({\mathcal{C}}(f)^{\bar{D}}\right)^{\perp}\right)\geq\begin{cases}5,&{\rm if}\,\,\,\nu=0,\\\ 6,&{\rm if}\,\,\,\nu=1.\\\ \end{cases}\end{split}$ Proof. It is easy to see $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\geq 3$ from the definition of ${\mathcal{C}}(f)^{\bar{D}}$. Next, we show that $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\neq 4$. The case of $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\neq 3$ can be shown similarly, and we omit the details of the proof. If $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)=4$, then there are four pairwise-distinct elements $x_{1}$, $x_{2}$, $x_{3}$ and $x_{4}$ in ${\mathbb{F}}_{2^{m}}^{}$ such that $\begin{split}\begin{cases}{\rm Tr}(\lambda f(x_{1}))={\rm Tr}(\lambda f(x_{2}))={\rm Tr}(\lambda f(x_{3}))={\rm Tr}(\lambda f(x_{4}))=\nu,\\\ a(x_{1}+x_{2}+x_{3}+x_{4})+b(f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4}))=0\end{cases}\end{split}$ for any $a,b\in\mathbb{F}_{2^{m}}$. Then, $\begin{split}\begin{cases}{\rm Tr}(\lambda f(x_{1}))={\rm Tr}(\lambda f(x_{2}))={\rm Tr}(\lambda f(x_{3}))={\rm Tr}(\lambda f(x_{4}))=\nu,\\\ x_{1}+x_{2}+x_{3}+x_{4}=0,\\\ f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})=0.\end{cases}\end{split}$ (9) The second and third equations in (9) can be rewritten as $\begin{split}\begin{cases}x_{1}+x_{2}=\alpha\,\,\text{and}\,\,x_{3}+x_{4}=\alpha,\\\ f(x_{1})+f(x_{2})=\beta\,\,\text{and}\,\,f(x_{3})+f(x_{4})=\beta,\end{cases}\end{split}$ where $\alpha,\beta\in\mathbb{F}_{2^{m}}$ with $\alpha\neq 0$. Hence, there are four different elements $x_{1}$, $x_{1}+\alpha$, $x_{3}$ and $x_{3}+\alpha$ satisfying the equation $f(x)+f(x+\alpha)=\beta$. This contradicts Lemma 2.1, as $f(x)$ is an almost perfect nonlinear function. Therefore, $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\geq 5$. If $\nu=1$ and $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)=5$, there are five pairwise-distinct elements $x_{1},x_{2},x_{3},x_{4},x_{5}$ in ${\mathbb{F}}_{2^{m}}^{}$ such that $f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})+f(x_{5})=0$ by the definition of ${\mathcal{C}}(f)^{\bar{D}}$, then ${\rm Tr}(\lambda(f(x_{1})+f(x_{2})+f(x_{3})+f(x_{4})+f(x_{5})))=0$, which is contradictory to ${\rm Tr}(\lambda f(x_{1}))={\rm Tr}(\lambda f(x_{2}))={\rm Tr}(\lambda f(x_{3}))={\rm Tr}(\lambda f(x_{4}))={\rm Tr}(\lambda f(x_{5}))=1.$ Hence, $\begin{split}d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\geq\begin{cases}5,&\text{if $\nu=0$,}\\\ 6,&\text{if $\nu=1$.}\end{cases}\end{split}$ This completes the proof of this lemma. $\square$ We now give the weight distribution of the binary code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual as follows. ###### Theorem 3.3 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (3), where $f(x)$ is an almost bent function from $\mathbb{F}_{2^{m}}$ to itself. Then the following statements hold. (1) If $\nu=0$, then ${\mathcal{C}}(f)^{\bar{D}}$ is an $[n,2m-1,\frac{n+1}{2}-2^{\frac{m-3}{2}}]$ code with the weight distribution in Table 1, where $n$ was given in Lemma 3.1. Its dual has parameters $[n,n-2m+1,5]$. Table 1: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $\nu=0$ in Theorem 3.3 Weight \| Multiplicity ---\|--- $0$ \| $1$ $\frac{n+1}{2}$ \| $2^{2m-1}-(n+1)^{4}2^{-2m}+5(n+1)^{2}2^{-m-1}-5(n+1)2^{m-2}+\frac{3}{2}n^{2}+2n-\frac{1}{2}$ $\frac{n+1}{2}\pm 2^{\frac{m-1}{2}}$ \| $\begin{array}[]{c}\pm\frac{1}{6}\big{(}(n+1)^{3}2^{\frac{1-3m}{2}}-(3n+1)2^{\frac{m-1}{2}}-(n+1)2^{-\frac{m+1}{2}}+2^{\frac{3m-3}{2}}\big{)}-\\\ \frac{1}{6}(n+1)^{4}2^{-2m}+\frac{1}{6}(n+1)^{2}2^{-m-1}-\frac{1}{6}(n+1)2^{m-2}+\frac{1}{4}n^{2}+\frac{1}{3}n+\frac{1}{12}\end{array}$ $\frac{n+1}{2}\pm 2^{\frac{m-3}{2}}$ \| $\begin{array}[]{c}\pm\frac{1}{6}\big{(}-(n+1)^{3}2^{\frac{3-3m}{2}}+(n+1)2^{\frac{5-m}{2}}+2^{\frac{m+1}{2}}-2^{\frac{3+3m}{2}}+6n\cdot 2^{\frac{m-1}{2}}\big{)}+2^{2-2m}\cdot n^{2}+\\\ \frac{1}{3}(n^{4}+4n^{3}+4n+1)2^{1-2m}-\frac{1}{3}(n+1)^{2}2^{2-m}+\frac{1}{3}(n+1)2^{1+m}-n^{2}-\frac{4}{3}n-\frac{1}{3}\end{array}$ (2) If $\nu=1$, then ${\mathcal{C}}(f)^{\bar{D}}$ is an $[n,2m,\frac{n}{2}-2^{\frac{m-3}{2}}]$ code with the weight distribution in Table 2, where $n$ was given in Lemma 3.1. Its dual has parameters $[n,n-2m,6]$, and is distance-optimal with respect to the sphere packing bound. Table 2: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $\nu=1$ in Theorem 3.3 Weight \| Multiplicity ---\|--- $0$ \| $1$ $\frac{n}{2}$ \| $2^{2m}-5n\cdot 2^{m-1}+5n^{2}\cdot 2^{-m}-2^{1-2m}n^{4}+3n^{2}-2n-2$ $\frac{n}{2}\pm 2^{\frac{m-1}{2}}$ \| $-\frac{1}{3}n^{4}2^{-2m}+\frac{1}{6}(2^{-m}n^{2}+3n^{2}-2^{m-1}n-2n)$ $\frac{n}{2}\pm 2^{\frac{m-3}{2}}$ \| $\frac{4n}{3}(2^{-2m}n^{3}-2^{1-m}n-\frac{3n}{2}+2^{m}+1)$ $n$ \| $1$ Proof. It follows from (2) that the Hamming weight of the codeword ${\mathbf{c}}(a,b)$ in ${\mathcal{C}}(f)^{\bar{D}}$ is given by $\begin{split}{\rm wt_{H}}(\mathbf{c}(a,b))&=\|D\|-\left\|\left\\{x\in D:\,\,{\rm Tr}\left(af(x)+bx\right)=0\right\\}\right\|\\\ &=\frac{\|D\|}{2}-\frac{1}{2}\sum_{x\in D}(-1)^{{\rm Tr}(af(x)+bx)}\\\ &=\frac{\|D\|}{2}-\frac{1}{2}\sum_{x\in\mathbb{F}_{2^{m}}\setminus\\{0\\}}\left(\frac{1}{2}\sum_{y\in\mathbb{F}_{2}}(-1)^{y{({\rm Tr}(\lambda f(x))-\nu)}}\right)(-1)^{{\rm Tr}\left(af(x)+bx\right)}\\\ &=\frac{\|D\|}{2}-\frac{1}{4}\sum_{x\in\mathbb{F}_{2^{m}}}\left(\sum_{y\in\mathbb{F}_{2}}(-1)^{y{({\rm Tr}(\lambda f(x))-\nu)}}\right)(-1)^{{\rm Tr}(af(x)+bx)}+\frac{1}{4}\sum_{y\in\mathbb{F}_{2}}(-1)^{y\nu}\\\ &=\frac{\|D\|}{2}-\frac{1}{4}\sum_{x\in\mathbb{F}_{2^{m}}}\left(1+(-1)^{({\rm Tr}(\lambda f(x))-\nu)}\right)(-1)^{{\rm Tr}(af(x)+bx)}+\frac{1}{4}\sum_{y\in\mathbb{F}_{2}}(-1)^{y\nu}\\\ &=\frac{\|D\|}{2}-\frac{1}{4}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(af(x)+bx)}-(-1)^{\nu}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}((\lambda+a)f(x)+bx)}+\frac{1}{4}\sum_{y\in\mathbb{F}_{2}}(-1)^{zy\nu}\\\ &=\frac{\|D\|}{2}-\frac{1}{4}W_{f}(a,b)-\frac{(-1)^{\nu}}{4}W_{f}(a+\lambda,b)+\frac{1}{4}\sum_{y\in\mathbb{F}_{2}}(-1)^{z_{0}\nu},\end{split}$ (10) where $W_{f}(a,b)$ was defined in (5). By the definition of almost bent functions, for any $(a,b)\in\mathbb{F}_{2^{m}}^{2}\setminus\\{(0,0)\\}$, we know that $W_{f}(a,b)\in\\{0,\pm 2^{\frac{m+1}{2}}\\}$. So, $\frac{1}{4}\left(W_{f}(a,b)\pm W_{f}(a+\lambda,b)\right)\in\left\\{0,\pm 2^{\frac{m-1}{2}},\pm 2^{\frac{m-3}{2}}\right\\}$ (11) for any $(a,b)\in\mathbb{F}_{2^{m}}^{2}\setminus\\{(0,0),(\lambda,0)\\}$. In the following, we prove this theorem case by case. Case 1: $\nu=0$, i.e., $D=\\{x\in\mathbb{F}_{2^{m}}^{}:{\rm Tr}(\lambda f(x))=0\\}$. By (10) and (11), when $(a,b)$ runs over $\mathbb{F}_{2^{m}}^{2}\setminus\\{(0,0),(\lambda,0)\\}$, the possible values of $\operatorname{wt_{H}}(\mathbf{c}(a,b))$ are $\frac{n+1}{2},\,\,\frac{n+1}{2}\pm 2^{\frac{m-1}{2}},\,\,\text{and}\,\,\frac{n+1}{2}\pm 2^{\frac{m-3}{2}},$ where $n$ was given in Lemma 3.1. It is easy to see that $\operatorname{wt_{H}}(\mathbf{c}(a,b))=0$ if and only if $(a,b)=(0,0)$ or $(a,b)=(\lambda,0)$. So, the dimension of ${\mathcal{C}}(f)^{\bar{D}}$ is $2m-1$. Denote $w_{1}=\frac{n+1}{2}$, $w_{2}=\frac{n+1}{2}+2^{\frac{m-1}{2}}$, $w_{3}=\frac{n+1}{2}-2^{\frac{m-1}{2}}$, $w_{4}=\frac{n+1}{2}+2^{\frac{m-3}{2}}$ and $w_{5}=\frac{n+1}{2}-2^{\frac{m-3}{2}}$. Let $A_{w_{i}}$ be the number of the codewords with weight $w_{i}$ in ${\mathcal{C}}(f)^{\bar{D}}$. By Lemma 3.2, we know that $A_{1}^{\perp}=A_{2}^{\perp}=A_{3}^{\perp}=A_{4}^{\perp}=0$. From the first five Pless power moments, we have the following system of equations: $\begin{split}\begin{cases}\sum_{i=1}^{5}A_{w_{i}}=2^{2m-1}-1;\\\ \sum_{i=1}^{5}w_{i}A_{w_{i}}=2^{2m-2}n;\\\ \sum_{i=1}^{5}w_{i}^{2}A_{w_{i}}=2^{2m-3}n(n+1);\\\ \sum_{i=1}^{5}w_{i}^{3}A_{w_{i}}=2^{2m-4}n^{2}(n+3);\\\ \sum_{i=1}^{5}w_{i}^{4}A_{w_{i}}=2^{2m-5}n(n+1)(n^{2}+5n-2).\\\ \end{cases}\end{split}$ Solving this system of equations, we obtain the desired values of $A_{w_{1}}$, $A_{w_{2}}$, $A_{w_{3}}$, $A_{w_{4}}$ and $A_{w_{5}}$ in Table 1. We now determine the parameters of the dual of ${\mathcal{C}}(f)^{\bar{D}}$. We consider only the case $n=2^{m-1}-1$, i.e., the value of $W_{f}(\lambda,0)$ is zero. The other two cases can be shown similarly. Substituting the value of $n=2^{m-1}-1$ in Table 1, we obtain that $A_{w_{1}}=3\cdot 2^{2m-4}+2^{m-3}-1$, $A_{w_{2}}=2^{2m-5}-2^{\frac{3m-7}{2}}+2^{\frac{m-5}{2}}-2^{m-4}$, $A_{w_{3}}=2^{2m-5}+2^{\frac{3m-7}{2}}-2^{\frac{m-5}{2}}-2^{m-4}$, $A_{w_{4}}=2^{2m-3}-2^{\frac{3m-5}{2}}$ and $A_{w_{5}}=2^{2m-3}+2^{\frac{3m-5}{2}}$. By Lemma 3.2, $A_{1}^{\perp}=A_{2}^{\perp}=A_{3}^{\perp}=A_{4}^{\perp}=0$. Then from the sixth Pless power moment, we have $\begin{split}\sum_{i=1}^{5}w_{i}^{5}A_{w_{i}}&=2^{2m-6}\cdot(2^{m-1}-1)^{5}+5\cdot 2^{2m-5}\cdot(2^{m-1}-1)^{4}\\\ &+15\cdot 2^{2m-6}\cdot(2^{m-1}-1)^{3}-5\cdot 2^{2m-5}\cdot(2^{m-1}-1)^{2}-A_{5}^{\perp}\cdot 2^{2m-6}\cdot 120.\\\ \end{split}$ Solving this equation, we obtain $A_{5}^{\perp}=(11\cdot 2^{m}+2^{3m-4}-13\cdot 2^{2m-3}-2^{4})/120\neq 0$. Hence, $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ has parameters $[2^{m-1}-1,2^{m-1}-2m,5]$. Case 2: $\nu=1$, i.e., $D=\\{x\in\mathbb{F}_{2^{m}}^{}:{\rm Tr}(\lambda f(x))=1\\}$. By (10) and (11), when $(a,b)$ runs over $\mathbb{F}_{2^{m}}^{2}\setminus\\{(0,0),(\lambda,0)\\}$, the possible values of $\operatorname{wt_{H}}(\mathbf{c}(a,b))$ are $\frac{n}{2},\,\,\frac{n}{2}\pm 2^{\frac{m-1}{2}}\,\,\text{and}\,\,\frac{n}{2}\pm 2^{\frac{m-3}{2}},$ where $n$ was given in Lemma 3.1. Moreover, $\operatorname{wt_{H}}(\mathbf{c}(a,b))=0$ if and only if $(a,b)=(0,0)$ and $\operatorname{wt_{H}}(\mathbf{c}(a,b))=n$ if $(a,b)=(\lambda,0)$. So, the dimension of ${\mathcal{C}}(f)^{\bar{D}}$ is $2m$. Denote $w_{1}=2^{m-2}$, $w_{2}=2^{m-2}+2^{\frac{m-1}{2}}$, $w_{3}=2^{m-2}-2^{\frac{m-1}{2}}$, $w_{4}=2^{m-2}+2^{\frac{m-3}{2}}$ and $w_{5}=2^{m-2}-2^{\frac{m-3}{2}}$. Let $A_{w_{i}}$ be the number of the codewords with weight $w_{i}$ in ${\mathcal{C}}(f)^{\bar{D}}$. From Lemma 3.2 we know that $A_{1}^{\perp}=A_{2}^{\perp}=A_{3}^{\perp}=A_{4}^{\perp}=0$. Then the first five Pless power moments lead to the following system of equations: $\begin{split}\begin{cases}\sum_{i=1}^{5}A_{w_{i}}=2^{2m}-2;\\\ \sum_{i=1}^{5}w_{i}A_{w_{i}}=2^{2m-1}n-n;\\\ \sum_{i=1}^{5}w_{i}^{2}A_{w_{i}}=2^{2m-2}n(n+1)-n^{2};\\\ \sum_{i=1}^{5}w_{i}^{3}A_{w_{i}}=2^{2m-3}n^{2}(n+3)-n^{3};\\\ \sum_{i=1}^{5}w_{i}^{4}A_{w_{i}}=2^{2m-4}n(n+1)(n^{2}+5n-2)-n^{4}.\end{cases}\end{split}$ Solving this system of equations, we obtain the desired values of $A_{w_{1}}$, $A_{w_{2}}$, $A_{w_{3}}$, $A_{w_{4}}$ and $A_{w_{5}}$ in Table 2. We now determine the parameters of the dual of ${\mathcal{C}}(f)^{\bar{D}}$. We treat only the case $n=2^{m-1}$ and the other two cases can be treated similarly. Substituting the value of $n=2^{m-1}$ in Table 2, we obtain that $A_{w_{1}}=3\cdot 2^{2m-3}+2^{m-2}-2$, $A_{w_{2}}=A_{w_{3}}=2^{2m-4}-2^{m-3}$ and $A_{w_{4}}=A_{w_{5}}=2^{2m-2}$. If $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)>6$, then $\begin{split}\sum_{i=0}^{3}\left(\begin{array}[]{cccc}2^{m-1}\\\ i\\\ \end{array}\right)=1+2^{m-1}+2^{m-2}\cdot(2^{m-1}-1)+\frac{2^{m-2}\cdot(2^{m-1}-1)\cdot(2^{m-1}-2)}{3}>2^{2m},\end{split}$ which contradicts the sphere packing bound. From Lemma 3.2, we then deduce that $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)=6$, and $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is distance-optimal with respect to the sphere packing bound. $\square$ ###### Example 3.4 Let $m=7$ and $f(x)$ be an almost bent function from $\mathbb{F}_{2^{7}}$ to $\mathbb{F}_{2^{7}}$ with $W_{f}(1,0)=2^{\frac{7+1}{2}}$. Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 3.3. (1) If $\nu=0$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[71,13,28]$ and its dual has parameters $[71,58,5]$. (2) If $\nu=1$ then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[56,14,20]$ and its dual has parameters $[56,42,6]$. The four codes are optimal according to the tables of best codes known in [22]. ###### Remark 3.5 In [36], the authors proposed the following open problem (Problem 4.4): Let $\lambda\in\mathbb{F}_{2^{s}}^{}$, $F$ be a function from $\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2^{s}}$ and $D$ be the support of ${\rm Tr}_{1}^{s}(\lambda F(x))$. Define a linear code ${\mathcal{C}}^{\prime}(F)^{\bar{D}}$ over $\mathbb{F}_{2}$ by ${\mathcal{C}}^{\prime}(F)^{\bar{D}}=\\{({\rm Tr}_{1}^{m}(xh)+{\rm Tr}_{1}^{s}(yF(h)))_{h\in D}\,:\,x\in\mathbb{F}_{2^{m}},y\in\mathbb{F}_{2^{s}}\\}.$ Determining the weight distributions of the linear codes if $F$ is a vectorial bent function with $m\neq 2s$ or an almost bent function but not the Gold type. Clearly, if $F$ is an almost bent function, then $s=m$. Table 2 in Theorem 3.3 has given the weight distribution of ${\mathcal{C}}^{\prime}(F)^{\bar{D}}$ for $F$ being an almost bent function. The following is a list of known almost bent monomials $f(x)=x^{d}$ on $\mathbb{F}_{2^{m}}$ for an odd $m$: * • $d=2^{h}+1$, where $\gcd(m,h)=1$ is odd [21]; * • $d=2^{2h}-2^{h}+1$, where $h\geq 2$ and $\gcd(m,h)=1$ is odd [31]; * • $d=2^{\frac{m-1}{2}}+3$, where $m$ is odd[31]; * • $d=2^{\frac{m-1}{2}}+2^{\frac{m-1}{4}}-1$, where $m\equiv 1\pmod{4}$ [26, 27]; * • $d=2^{\frac{m-1}{2}}+2^{\frac{3m-1}{4}}-1$, where $m\equiv 3\pmod{4}$[26, 27]. All almost bent monomials $f(x)=x^{d}$ for $d$ in the list above are permutation polynomials on $\mathbb{F}_{2^{m}}$. Hence, the length of ${\mathcal{C}}(f)^{\bar{D}}$ is $n=2^{m-1}-1$ if $\nu=0$ and $n=2^{m-1}$ if $\nu=1$, respectively. Substituting the value of $n$ into Theorem 3.3, we obtain the following results. ###### Corollary 3.6 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (3). If $f(x)=x^{d}$ for some integer $d$ in the list above, then the following statements hold. (1) If $\nu=0$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1}-1,2m-1,2^{m-2}-2^{\frac{m-3}{2}}]$ code with the weight distribution in Table 3. Its dual has parameters $[2^{m-1}-1,2^{m-1}-2m,5]$. Table 3: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $\nu=0$ in Corollary 3.6 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $3\cdot 2^{2m-4}+2^{m-3}-1$ $2^{m-2}\pm 2^{\frac{m-1}{2}}$ \| $2^{2m-5}\mp 2^{\frac{3m-7}{2}}\pm 2^{\frac{m-5}{2}}-2^{m-4}$ $2^{m-2}\pm 2^{\frac{m-3}{2}}$ \| $2^{2m-3}\mp 2^{\frac{3m-5}{2}}$ (2) If $\nu=1$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},2m,2^{m-2}-2^{\frac{m-3}{2}}]$ code with the weight distribution in Table 4. Its dual has parameters $[2^{m-1},2^{m-1}-2m,6]$, and is distance- optimal with respect to the sphere packing bound. Table 4: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $\nu=1$ in Corollary 3.6 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $3\cdot 2^{2m-3}+2^{m-2}-2$ $2^{m-2}\pm 2^{\frac{m-1}{2}}$ \| $2^{2m-4}-2^{m-3}$ $2^{m-2}\pm 2^{\frac{m-3}{2}}$ \| $2^{2m-2}$ $2^{m-1}$ \| $1$ ###### Example 3.7 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Corollary 3.6. (1) If $m=7$, $\nu=0$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[63,13,24]$ and its dual has parameters $[63,50,5]$. (2) If $m=7$, $\nu=1$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[64,14,24]$ and its dual has parameters $[64,50,6]$. The four codes are optimal according to the tables of best codes known in [22]. ## 4 Some punctured codes of binary linear codes from quadratic functions Let ${\mathcal{C}}(f)^{\bar{D}}$ be the binary punctured code defined in (2) with the position set $D$ in (4). It is clear that the length of ${\mathcal{C}}(f)^{\bar{D}}$ is equal to $2^{m-1}$, as $\|D\|=\|\\{x\in{\mathbb{F}}_{2^{m}}^{}:{\rm Tr}_{1}^{m}(x)=1\\}\|=2^{m-1}$. As shown in (10), the Hamming weight of each codeword in this case can be expressed as $\begin{split}{\rm wt_{H}}(\mathbf{c}(a,b))&=2^{m-2}-\frac{1}{4}\left(W_{f}(a,b)-W_{f}(a,b+1)\right),\end{split}$ (12) where $W_{f}(a,b)$ was given in (5). In this section, we investigate the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ with the position set $D$ in (4), where $f$ is a quadratic function in the list below and the parameters of its dual. • $f(x)=x^{2^{k}+1}$, where $k$ is an integer with $1\leq k\leq m-1$; * • $f(x)=x^{t_{1}}+x^{t_{2}}$, where $3\,\|\,m$, $m\geq 9$ and $t_{1},t_{2}\in\\{2^{\frac{m}{3}}+1,2^{\frac{2m}{3}}+1,2^{\frac{2m}{3}}+2^{\frac{m}{3}}\\}$ with $t_{1}\neq t_{2}$; * • $f(x)={\rm Tr}_{k}^{m}(x^{2^{k}+1})$, where $m,k$ are positive integers such that $k\,\|\,m$. When $f(x)=x^{2^{k}+1}$, the parameters and weight distribution of the binary code ${\mathcal{C}}(f)$ were settled in [29, 30]. In this section we will investigate the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ with a different position set $D=\\{x\in{\mathbb{F}}_{2^{m}}^{}:{\rm Tr}_{1}^{m}(x)=1\\}$. It is open if the binary code ${\mathcal{C}}(f)$ was studied in the literature or not when $f$ is one of the other two quadratic functions in the list above. ### 4.1 The case that $f(x)=x^{2^{k}+1}$ In this subsection, we study the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ in (2) and determine its weight distribution, where $f(x)=x^{2^{k}+1}$ and $D=\\{x\in{\mathbb{F}}_{2^{m}}^{}:{\rm Tr}_{1}^{m}(x)=1\\}$. When $k=0$, $f(x)=x^{2}$. In this case, it can be proved that the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ is permutation-equivalent to the first-order Reed-Muller code. In the following, we investigate the linear code ${\mathcal{C}}(f)^{\bar{D}}$ for $f(x)=x^{2^{k}+1}$ with $1\leq k<m$. We start with the following two lemmas. ###### Lemma 4.1 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (4). Let $A_{i}^{\perp}$ denote the number of codewords with weight $i$ in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$. If $f(x)=x^{2^{k}+1}$ with $1\leq k<m$, then $A_{1}^{\perp}=A_{2}^{\perp}=A_{3}^{\perp}=A_{5}^{\perp}=0\,\,\text{and}\,\,A_{4}^{\perp}=\frac{2^{m-1}\cdot(2^{m-2}-1)\cdot(2^{\ell}-2)}{4!},$ where $\ell=\gcd(k,m)$. Proof. From the definition of the linear code ${\mathcal{C}}(f)^{\bar{D}}$, we know that $A_{i}^{\perp}$ is equal to the number of sets $\\{x_{1},x_{2},\cdots,x_{i}\\}$ with $i$ pairwise-distinct nonzero elements in ${\mathbb{F}}_{2^{m}}$ such that $\begin{split}\begin{cases}{\rm Tr}(x_{1})={\rm Tr}(x_{2})=\cdots={\rm Tr}(x_{i})=1,\\\ x_{1}+x_{2}+\cdots+x_{i}=0,\\\ x_{1}^{2^{k}+1}+x_{2}^{2^{k}+1}+\cdots+x_{i}^{2^{k}+1}=0.\\\ \end{cases}\end{split}$ It is clear that $A_{1}^{\perp}=A_{2}^{\perp}=0$. From the first and second equations, we see that $A_{i}^{\perp}=0$ if $i$ is odd. Hence, $A_{3}^{\perp}=A_{5}^{\perp}=0$. In the following, we determine the value of $A_{4}^{\perp}$, which is equal to the number of sets $\\{x_{1},x_{2},x_{3},x_{4}\\}$ with $4$ pairwise-distinct nonzero elements in ${\mathbb{F}}_{2^{m}}$ such that $\begin{split}\begin{cases}{\rm Tr}(x_{1})={\rm Tr}(x_{2})={\rm Tr}(x_{3})={\rm Tr}(x_{4})=1,\\\ x_{1}+x_{2}+x_{3}+x_{4}=0,\\\ x_{1}^{2^{k}+1}+x_{2}^{2^{k}+1}+x_{3}^{2^{k}+1}+x_{4}^{2^{k}+1}=0.\\\ \end{cases}\end{split}$ (13) Assume that $x_{1}=\mu$, $x_{2}=\mu+\beta$, $x_{3}=\gamma$ and $x_{4}=\gamma+\beta$, where $\mu\neq 0,\beta,\gamma,\gamma+\beta$, and $\gamma\neq 0,\beta$, and $\beta\neq 0$. From (13) we know that $A_{4}^{\perp}$ is equal to the number of the sets of the form $\\{\mu,\mu+\beta,\gamma,\gamma+\beta\\}$ such that $\mu^{2^{k}+1}+(\mu+\beta)^{2^{k}+1}=\gamma^{2^{k}+1}+(\gamma+\beta)^{2^{k}+1},\,\,{\rm Tr}(\mu)={\rm Tr}(\gamma)=1\,\,\text{and}\,\,{\rm Tr}(\beta)=0,$ i.e., $(\mu+\gamma)^{2^{k}-1}=\beta^{2^{k}-1},\,\,{\rm Tr}(\mu)={\rm Tr}(\gamma)=1\,\,\text{and}\,\,{\rm Tr}(\beta)=0.$ It is clear that $(\mu+\gamma)^{2^{k}-1}=\beta^{2^{k}-1}$ if and only if there is a $\delta\in\mathbb{F}_{2^{\ell}}$ such that $\mu+\gamma=\delta\beta$ as $\gcd(2^{m}-1,2^{k}-1)=2^{\ell}-1$. Then $A_{4}^{\perp}$ is equal to the number of the sets of the form $\\{\mu,\mu+\beta,\mu+\delta\beta,\mu+\beta(\delta+1)\\}$ such that ${\rm Tr}(\mu)=1$ and ${\rm Tr}(\beta)={\rm Tr}(\delta\beta)=0$, where $\delta\in\mathbb{F}_{2^{\ell}}\backslash\\{0,1\\}$, $\mu\neq 0,\beta,\delta\beta,\beta(\delta+1)$ and $\beta\neq 0$. Hence, $\begin{split}A_{4}^{\perp}&=\frac{1}{8\cdot 4!}\sum_{z_{0}\in\mathbb{F}_{2}}\sum_{\mu\in\mathbb{F}_{2^{m}}^{}\backslash\\{\beta,\delta\beta,\beta(\delta+1)\\}}(-1)^{z_{0}({\rm Tr}(\mu)-1)}\sum_{z_{1}\in\mathbb{F}_{2}}\sum_{\beta\in\mathbb{F}_{2^{m}}^{}}(-1)^{z_{1}{\rm Tr}(\beta)}\sum_{z_{2}\in\mathbb{F}_{2}}\sum_{\delta\in\mathbb{F}_{2^{\ell}}^{}\backslash\\{1\\}}(-1)^{z_{2}{\rm Tr}(\delta\beta)}\\\ &=\frac{2^{m-3}}{4!}\sum_{z_{1}\in\mathbb{F}_{2}}\sum_{\beta\in\mathbb{F}_{2^{m}}^{}}(-1)^{z_{1}{\rm Tr}(\beta)}\sum_{z_{2}\in\mathbb{F}_{2}}\sum_{\delta\in\mathbb{F}_{2^{\ell}}^{}\backslash\\{1\\}}(-1)^{z_{2}{\rm Tr}(\delta\beta)}\\\ &=\frac{2^{m-3}}{4!}\sum_{z_{1}\in\mathbb{F}_{2}}\sum_{z_{2}\in\mathbb{F}_{2}}\sum_{\beta\in\mathbb{F}_{2^{m}}^{}}\sum_{\gamma\in\mathbb{F}_{2^{\ell}}^{}\backslash\\{1\\}}(-1)^{{\rm Tr}((z_{1}+z_{2}\gamma)\beta)}\\\ &=\frac{2^{m-3}}{4!}\left(\sum_{z_{1}\in\mathbb{F}_{2}}\sum_{z_{2}\in\mathbb{F}_{2}}\sum_{\beta\in\mathbb{F}_{2^{m}}}\sum_{\gamma\in\mathbb{F}_{2^{\ell}}^{}\backslash\\{1\\}}(-1)^{{\rm Tr}((z_{1}+z_{2}\gamma)\beta)}-2^{2}\cdot(2^{\ell}-2)\right)\\\ &=\frac{2^{m-3}}{4!}\left(2^{m}\cdot(2^{\ell}-2)-2^{2}\cdot(2^{\ell}-2)\right)=\frac{2^{m-1}\cdot(2^{m-2}-1)\cdot(2^{\ell}-2)}{4!}.\end{split}$ The desired conclusion then follows. $\square$ ###### Theorem 4.2 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (3). Let $k$ be a positive integer with $k<m$ and $\ell=\gcd(k,m)$. If $f(x)=x^{2^{k}+1}$, then the following statements hold. (1) If $v_{2}(m)\leq v_{2}(k)$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},2m,2^{m-2}-2^{\frac{m+\ell-4}{2}}]$ code with the weight distribution in Table 5. If $\ell\geq 2$, then its dual has parameters $[2^{m-1},2^{m-1}-2m,4]$. If $\ell=1$, then its dual has parameters $[2^{m-1},2^{m-1}-2m,6]$, and is distance-optimal with respect to the sphere packing bound. Table 5: The weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.2 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $2^{2m}-2^{2m-\ell+1}+3\cdot 2^{2m-2\ell-1}+2^{m-\ell-1}-2$ $2^{m-2}\pm 2^{\frac{m+\ell-2}{2}}$ \| $2^{2m-2\ell-2}-2^{m-\ell-2}$ $2^{m-2}\pm 2^{\frac{m+\ell-4}{2}}$ \| $2^{2m-\ell}-2^{2m-2\ell}$ $2^{m-1}$ \| $1$ (2) If $v_{2}(m)>v_{2}(k)$ and $\gcd(m,k)=1$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},2m,2^{m-2}-2^{\frac{m}{2}}]$ code with the weight distribution in Table 6. Its dual has parameters $[2^{m-1},2^{m-1}-2m,6]$, and is distance-optimal with respect to the sphere packing bound. Table 6: The weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.2 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $17\cdot 2^{2m-5}+3\cdot 2^{m-3}-2$ $2^{m-2}\pm 2^{\frac{m}{2}}$ \| $\frac{1}{3}\left(2^{2m-6}-2^{m-4}\right)$ $2^{m-2}\pm 2^{\frac{m-2}{2}}$ \| $\frac{1}{6}\left(11\cdot 2^{2m-3}-2^{m}\right)$ $2^{m-1}$ \| $1$ (3) If $k=\frac{m}{2}$ and $m\geq 4$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},\frac{3m}{2},2^{m-2}-2^{\frac{m-2}{2}}]$ code with the weight distribution in Table 7. Its dual has parameters $[2^{m-1},2^{m-1}-\frac{3m}{2},4]$, and is distance-optimal with respect to the sphere packing bound. Table 7: The weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.2 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $2^{\frac{3m}{2}-1}+2^{m-1}-2$ $2^{m-2}\pm 2^{\frac{m-2}{2}}$ \| $2^{\frac{3m}{2}-2}-2^{m-2}$ $2^{m-1}$ \| $1$ Proof. We prove the desired conclusions for Cases (1) and (3) only. The conclusions in Case (2) can be proved in a similar way. If $a=0$, it is easy to see that ${\rm wt_{H}}\left({\bf c}(a,b)\right)=2^{m-2}-\frac{1}{4}\left(W_{f}(0,b)-W_{f}(0,b+1)\right)=\begin{cases}0,&{\rm if}\,\,\,b=0,\\\ 2^{m-1},&{\rm if}\,\,\,b=1,\\\ 2^{m-2},&{\rm if}\,\,\,b\neq 0,\,\,1.\end{cases}$ If $a\neq 0$ and $v_{2}(m)\leq v_{2}(k)$, then Lemma 2.2 shows that $W_{f}(a,b)\in\\{0,\pm 2^{\frac{m+\ell}{2}}\\}$. Consequently, in this case we have $W_{f}(a,b)-W_{f}(a,b+1)\in\\{0,\pm 2^{\frac{m+\ell}{2}},\pm 2^{\frac{m+\ell+2}{2}}\\}$. From (12) we see that the set of possible nonzero weights of ${\mathcal{C}}(f)^{\bar{D}}$ is $\\{2^{m-1},2^{m-2},2^{m-2}\pm 2^{\frac{m+\ell-2}{2}},2^{m-2}\pm 2^{\frac{m+\ell-4}{2}}\\}$ and ${\mathcal{C}}(f)^{\bar{D}}$ has dimension $2m$. Set $w_{1}=2^{m-1},w_{2}=2^{m-2},w_{3}=2^{m-2}+2^{\frac{m+\ell-2}{2}}$, $w_{4}=2^{m-2}-2^{\frac{m+\ell-2}{2}}$, $w_{5}=2^{m-2}+2^{\frac{m+\ell-4}{2}}$ and $w_{6}=2^{m-2}-2^{\frac{m+\ell-4}{2}}$. It is known that $A_{w_{1}}=1$. From Lemma 4.1 and the first five Pless power moments we have $\left\\{\begin{array}[]{lll}\sum_{i=2}^{6}A_{w_{i}}=2^{2m}-2;\\\ \sum_{i=2}^{6}w_{i}A_{w_{i}}=2^{m-1}(2^{2m-1}-1);\\\ \sum_{i=2}^{6}w_{i}^{2}A_{w_{i}}=2^{2m-2}(2^{2m-2}+2^{m-1}-1);\\\ \sum_{i=2}^{6}w_{i}^{3}A_{w_{i}}=2^{3m-3}(2^{2m-2}+3\cdot 2^{m-1}-1);\\\ \sum_{i=2}^{6}w_{i}^{4}A_{w_{i}}=2^{3m-5}\left((2^{m-1}+1)(2^{2m-2}+5\cdot 2^{m-1}-2)-(2^{m-2}-1)(2^{\ell}-1)\right)-2^{4(m-1)}.\end{array}\right.$ (14) Solving the linear equations in (14), we get the desired values of $A_{w_{i}}$ in Table 5. If $\ell>1$, by Lemma 4.1, $A_{4}^{\perp}>0$. Consequently, the dual distance of the code equals $4$. If $\ell=1$, by Lemma 4.1, $A_{4}^{\perp}=0$. Since all weights in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ are even, the minimum distance of $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is at least $6$. By the sphere packing bound, the minimum distance of $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ cannot be $8$ or more. Consequently, the minimum distance of $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is equal to $6$. This completes the proof of the conclusions in Case (1). Next, we prove the conclusions for Case (3). Assume that $k=\frac{m}{2}$ and $f(x)=x^{2^{m/2}+1}$, then $\begin{split}W_{f}^{2}(a,b)&=\sum_{x_{0}\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(ax_{0}^{2^{m/2}+1}+bx_{0})}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(ax^{2^{m/2}+1}+bx)}\\\ &=\sum_{x,y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a(x+y)^{2^{m/2}+1}+b(x+y)+ax^{2^{m/2}+1}+bx)}\\\ &=\sum_{x,y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a(y^{2^{m/2}+1}+xy^{2^{m/2}}+x^{2^{m/2}}y)+by)}\\\ &=\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(ay^{2^{m/2}+1}+by)}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a(xy^{2^{m/2}}+x^{2^{m/2}}y))}\\\ &=2^{m}\sum_{\small\begin{array}[]{c}y\in\mathbb{F}_{2^{m}}\\\ (a+a^{2^{m/2}})y=0\end{array}}(-1)^{{\rm Tr}(ay^{2^{m/2}+1}+by)}\\\ &=\begin{cases}2^{m}W_{f}(a,b),&\text{if $a\in\mathbb{F}_{2^{\frac{m}{2}}}$,}\\\ 2^{m},&\text{otherwise}.\end{cases}\end{split}$ (15) If $a\in\mathbb{F}_{2^{\frac{m}{2}}}$, then ${\rm Tr}(ay^{2^{m/2}+1})=0$ and the possible values of $W_{f}(a,b)$ are as follows: $W_{f}(a,b)=\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(ay^{2^{m/2}+1}+by)}=\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(by)}=\begin{cases}2^{m},&\text{if $b=0$,}\\\ 0,&\text{otherwise}.\end{cases}$ Hence, $\begin{split}W_{f}(a,b)=\begin{cases}2^{m},&\text{if $a\in\mathbb{F}_{2^{\frac{m}{2}}}$ \text{and} $b=0$,}\\\ 0,&\text{if $a\in\mathbb{F}_{2^{\frac{m}{2}}}$ \text{and} $b\neq 0$,}\\\ \pm 2^{\frac{m}{2}},&\text{otherwise.}\end{cases}\end{split}$ When $(a,b)$ runs through $\mathbb{F}_{2^{m}}^{2}$, we know that $W_{f}(a,b)-W_{f}(a,b+1)\in\\{0,\pm 2^{\frac{m+2}{2}},\pm 2^{m}\\}$ and the value $2^{m}$ occurs $2^{\frac{m}{2}}$ times. Then $\operatorname{wt_{H}}({\mathbf{c}}(a,b))\in\\{0,2^{m-2}\pm 2^{\frac{m-2}{2}},2^{m-1}\\}$ and $\operatorname{wt_{H}}({\mathbf{c}}(a,b))=0$ occurs $2^{\frac{m}{2}}$ times by (15). So, ${\mathcal{C}}(f)^{\bar{D}}$ has dimension $\frac{3m}{2}$ and we obtain the weight distribution in Table 7 from the first three Pless power moments. From the sphere packing bound and Lemma 4.1, the desired conclusions on ${\mathcal{C}}(f)^{\bar{D}}$ then follow. In this case, $\ell=m/2>1$. It then follows from Lemma 4.1, $A_{4}^{\perp}>0$. Consequently, the dual distance of the code equals $4$. $\square$ ###### Example 4.3 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 4.2. (1) Let $m=5$, $k=1$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[16,10,4]$ and its dual has parameters $[16,6,6]$. (2) Let $m=8$, $k=4$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[128,12,56]$ and its dual has parameters $[128,116,4]$. All the four codes are optimal according to the tables of best codes known in [22]. ### 4.2 The case that $f(x)=x^{t_{1}}+x^{t_{2}}$ In this subsection, we investigate the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual for $f(x)=x^{t_{1}}+x^{t_{2}}$, where $3\,\|\,m$, $m\geq 9$ and $t_{1},t_{2}\in\\{2^{\frac{m}{3}}+1,2^{\frac{2m}{3}}+1,2^{\frac{2m}{3}}+2^{\frac{m}{3}}\\}$ with $t_{1}\neq t_{2}$. We first determine all possible Hamming weights in ${\mathcal{C}}(f)^{\bar{D}}$. ###### Lemma 4.4 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (2) with the position set $D$ in (4). Let $3\,\|\,m$, $m\geq 9$ and $f(x)=x^{t_{1}}+x^{t_{2}}$, where $t_{1},t_{2}\in\\{2^{\frac{m}{3}}+1,2^{\frac{2m}{3}}+1,2^{\frac{2m}{3}}+2^{\frac{m}{3}}\\}$ with $t_{1}\neq t_{2}$. Then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},\frac{5m}{3}]$ code with nonzero weights in the set $\\{2^{m-2},\,2^{m-1},\,2^{m-2}\pm 2^{\frac{2m}{3}-1}\\}$. Proof. We prove the conclusions only for the case $t_{1}=2^{\frac{2m}{3}}+1$ and $t_{2}=2^{\frac{2m}{3}}+2^{\frac{m}{3}}$. The conclusions in the other two cases can be similarly proved. In this case, we have $W_{f}(a,b)=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a(x^{2^{2m/3}+1}+x^{2^{2m/3}+2^{m/3}})+bx)}.$ If $a\in\mathbb{F}_{2^{\frac{m}{3}}}$, then $a+a^{2^{{m/3}}}=0$ and $a+a^{2^{{2m/3}}}=0$. In this case, $W_{f}(a,b)=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(bx)}=\left\\{\begin{array}[]{lll}2^{m},&\text{if $b=0$},\\\ 0,&\text{if $b\neq 0$.}\end{array}\right.$ Hence, when $(a,b)$ runs over $\mathbb{F}_{2^{\frac{m}{3}}}\times\mathbb{F}_{2^{m}}$, we obtain $W_{f}(a,b)-W_{f}(a,b+1)=\left\\{\begin{array}[]{lll}0,&\text{occuring $2^{2m}-2^{\frac{m}{3}+1}$ times},\\\ 2^{m},&\text{occuring $2^{\frac{m}{3}}$ times},\\\ -2^{m},&\text{occuring $2^{\frac{m}{3}}$ times}.\\\ \end{array}\right.$ (16) If $a\in\mathbb{F}_{2^{m}}\setminus\mathbb{F}_{2^{\frac{m}{3}}}$, similar to the calculations in (15), we have $\begin{split}W_{f}^{2}(a,b)&=\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a(y^{2^{2m/3}+1}+y^{2^{2m/3}+2^{m/3}})+by)}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}((ay^{2^{m/3}}+a^{2^{m/3}}y+a^{2^{2m/3}}y^{2^{m/3}}+ay)x^{2^{2m/3}})}\\\ &=2^{m}\sum_{\small\begin{array}[]{c}y\in\mathbb{F}_{2^{m}}\\\ (a+a^{2^{{2m/3}}})y^{2^{m/3}}+(a^{2^{m/3}}+a)y=0\end{array}}(-1)^{{\rm Tr}(a(y^{2^{2m/3}+1}+y^{2^{2m/3}+2^{m/3}})+by)}.\end{split}$ Let $L_{a}(y)=(a+a^{2^{2m/3}})y^{2^{m/3}}+(a^{2^{m/3}}+a)y$, then $\textrm{Ker}(L_{a}(y))=\\{\,y\in\mathbb{F}_{2^{m}}\,\,\|\,\,L_{a}(y)=0\,\\}=\left\\{(a^{2^{2m/3}}+a)z:\,\,z\in\mathbb{F}_{2^{\frac{m}{3}}}\right\\}.$ From (7) we get $W_{f}^{2}(a,b)=\begin{cases}2^{\frac{4m}{3}},&\text{if ${{\rm Tr}\left(a(y^{2^{2m/3}+1}+y^{2^{2m/3}+2^{m/3}})+by\right)}=0$ for all $y\in$Ker$(L_{a}(y))$,}\\\ 0,&\text{otherwise}.\end{cases}$ If ${{\rm Tr}\left(a(y^{2^{2m/3}+1}+y^{2^{2m/3}+2^{m/3}})+by\right)}=0$ for all $y\in$Ker$(L_{a}(y))$, then ${{\rm Tr}\left(a(y^{2^{2m/3}+1}+y^{2^{2m/3}+2^{m/3}})+(b+1)y\right)}={\rm Tr}(y)={\rm Tr}_{1}^{\frac{m}{3}}\left({\rm Tr}_{\frac{m}{3}}^{m}((a^{2^{2m/3}}+a)t)\right)=0$ because $t\in\mathbb{F}_{2^{\frac{m}{3}}}$. Hence, $W_{f}(a,b)-W_{f}(a,b+1)\in\left\\{0,\pm 2^{\frac{2m}{3}+1}\right\\}$ for $a\in\mathbb{F}_{2^{m}}\backslash\mathbb{F}_{2^{\frac{m}{3}}}$. Combining this with (16), when $(a,b)$ runs through $\mathbb{F}_{2^{m}}\times\mathbb{F}_{2^{m}}$, we have $W_{f}(a,b)-W_{f}(a,b+1)\in\left\\{0,\pm 2^{m},\pm 2^{\frac{2m}{3}+1}\right\\}$ and each of the values $\pm 2^{m}$ occurs $2^{\frac{m}{3}}$ times. Then from (12) we know that ${\rm wt_{H}}({\mathbf{c}}(a,b))=0$ and ${\rm wt_{H}}({\mathbf{c}}(a,b))=2^{m-1}$ both occur $2^{\frac{m}{3}}$ times and the nonzero weights in ${\mathcal{C}}(f)^{\bar{D}}$ belong to the set $\\{2^{m-2},\,2^{m-1},\,2^{m-2}\pm 2^{\frac{2m}{3}-1}\\}$. It then follows that ${\mathcal{C}}(f)^{\bar{D}}$ is degenerate and has dimension $\frac{5m}{3}$. This completes the proof. $\square$ ###### Theorem 4.5 Follow the notation and conditions introduced in Lemma 4.4. Then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},\frac{5m}{3},2^{m-2}-2^{\frac{2m}{3}-1}]$ code with the weight distribution in Table 8. Its dual has parameters $[2^{m-1},2^{m-1}-\frac{5m}{3},4]$, and is distance-optimal with respect to the sphere packing bound. Table 8: The weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.5 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $2^{\frac{5m}{3}}-2^{\frac{4m}{3}-1}+2^{\frac{2m}{3}-1}-2$ $2^{m-2}\pm 2^{\frac{2m}{3}-1}$ \| $2^{\frac{4m}{3}-2}-2^{\frac{2m}{3}-2}$ $2^{m-1}$ \| $1$ Proof. From Lemma 4.4, we conclude that the dimension of ${\mathcal{C}}(f)^{\bar{D}}$ is $\frac{5m}{3}$, the possible weights in ${\mathcal{C}}(f)^{\bar{D}}$ are given in the set $\\{0,2^{m-1},2^{m-2},$ $2^{m-2}\pm 2^{\frac{2m}{3}-1}\\}$ and the weight $2^{m-1}$ occurs $1$ time. Denote $w_{1}=2^{m-2}$, $w_{2}=2^{m-2}-2^{\frac{2m}{3}-1}$ and $w_{3}=2^{m-2}+2^{\frac{2m}{3}-1}$. Let $A_{w_{i}}$ be the number of the codewords with weight $w_{i}$ in ${\mathcal{C}}(f)^{\bar{D}}$. Note that the all-one vector is a codeword of ${\mathcal{C}}(f)^{\bar{D}}$. It then follows that all codewords in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ have even weights. It is easily seen that the minimum weight in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ cannot be $2$. Consequently, the minimum weight in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is at least $4$. From the first three Pless power moments, we have $\begin{split}\begin{cases}\sum_{i=1}^{3}A_{w_{i}}=2^{\frac{5m}{3}}-2;\\\ \sum_{i=1}^{3}w_{i}A_{w_{i}}=2^{\frac{8m}{3}-2}-2^{m-1};\\\ \sum_{i=1}^{3}w_{i}^{2}A_{w_{i}}=2^{\frac{8m}{3}-2}(2^{m-1}+1)-2^{2m-2}.\end{cases}\end{split}$ Solving this system of equations, we obtain $A_{w_{1}}=2^{\frac{5m}{3}}-2^{\frac{4m}{3}-1}+2^{\frac{2m}{3}-1}-2$, $A_{w_{2}}=A_{w_{3}}=2^{\frac{4m}{3}-2}-2^{\frac{2m}{3}-2}$. We now consider the minimum distance of $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$. We have already proved that $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)\geq 4.$ If there exists a $[2^{m-1},2^{m-1}-\frac{5m}{3}]$ binary code with Hamming distance at least $5$, then $\begin{split}\sum_{i=0}^{2}\left(\begin{array}[]{cccc}2^{m-1}\\\ i\\\ \end{array}\right)=1+2^{m-1}+2^{m-2}\cdot(2^{m-1}-1)>2^{\frac{5m}{3}},\end{split}$ which contradicts the sphere packing bound. Hence, $d_{H}\left(({\mathcal{C}}(f)^{\bar{D}})^{\perp}\right)=4$ and $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is distance-optimal to the sphere packing bound. $\square$ ###### Example 4.6 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 4.5. Let $m=9$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[256,15,96]$ and its dual has parameters $[256,241,4]$. We settled the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.5, but do not know if the corresponding code ${\mathcal{C}}(f)$ was studied in the literature or not. ### 4.3 The case that $f(x)={\rm Tr}_{k}^{m}(x^{2^{k}+1})$ In this subsection, we study the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual for $f(x)={\rm Tr}_{k}^{m}(x^{2^{k}+1})$, where $k$ divides $m$. It is easy to see that $f(x)=0$ if $k=\frac{m}{2}$. In the following, we just consider the case that $k\not\in\\{m,\frac{m}{2}\\}$. We begin with the following lemma. ###### Lemma 4.7 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the punctured code defined in (2) with the position set $D$ in (4). Let $f(x)={\rm Tr}_{k}^{m}(x^{2^{k}+1})$, where $k$ divides $m$ and $k\not\in\\{m,\frac{m}{2}\\}$. Let $t=2^{\frac{m+2k-2}{2}}$ if $v_{2}(m)>v_{2}(k)+1$, and $t=2^{\frac{m+2k-4}{2}}$ if $v_{2}(m)=v_{2}(k)+1$, and $t=2^{\frac{m+k-4}{2}}$ if $v_{2}(m)=v_{2}(k)$. Then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},k+m]$ code whose nonzero weights are in the set $\left\\{2^{m-2},\,\,2^{m-1},\,\,2^{m-2}\pm t\right\\}$. Proof. We prove the conclusions only for the case $v_{2}(m)>v_{2}(k)+1$. The conclusions for the other two cases can be similarly proved. We first determine the possible values of $W_{f}(a,b)$ for $(a,b)\in\mathbb{F}_{2^{m}}^{2}$, where $W_{f}(a,b)$ was defined in (5). Note that ${\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1}))={\rm Tr}({\rm Tr}_{k}^{m}(a)x^{2^{k}+1})$. If ${\rm Tr}_{k}^{m}(a)=0$, then $W_{f}(a,b)=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(bx)}=\left\\{\begin{array}[]{lll}2^{m},&\text{if $b=0$},\\\ 0,&\text{if $b\neq 0$.}\end{array}\right.$ Let $L=\\{a\in\mathbb{F}_{2^{m}}:{\rm Tr}_{k}^{m}(a)=0\\}$, then $\|L\|=2^{m-k}$. Hence, when $(a,b)$ runs over $L\times\mathbb{F}_{2^{m}}$, we have $W_{f}(a,b)-W_{f}(a,b+1)=\left\\{\begin{array}[]{lll}0,&\text{occuring $2^{m+k}-2^{m-k+1}$ times},\\\ 2^{m},&\text{occuring $2^{m-k}$ times},\\\ -2^{m},&\text{occuring $2^{m-k}$ times}.\\\ \end{array}\right.$ (17) If ${\rm Tr}_{k}^{m}(a)\neq 0$, similar to the discussions in (15), we have $\begin{split}W_{f}^{2}(a,b)&=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+bx)}\sum_{y\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a{\rm Tr}_{k}^{m}(xy^{2^{k}}+x^{2^{k}}y))}\\\ &=2^{m}\sum_{\small\begin{array}[]{c}x\in\mathbb{F}_{2^{m}}\\\ x+x^{2k}=0\end{array}}(-1)^{{\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+bx)}\\\ &=2^{m}\cdot\sum_{x\in\mathbb{F}_{2^{2k}}}(-1)^{{\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+bx)},\end{split}$ as $v_{2}(m)>v_{2}(k)+1$. Then by (7) we obtain $W_{f}^{2}(a,b)=\begin{cases}2^{m+2k},&\text{if ${{\rm Tr}\left(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+bx\right)}=0$ for all $x\in\mathbb{F}_{2^{2k}}$,}\\\ 0,&\text{otherwise}.\end{cases}$ Clearly, if ${{\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+bx)}=0$ for all $x\in\mathbb{F}_{2^{2k}}$, then ${{\rm Tr}(a{\rm Tr}_{k}^{m}(x^{2^{k}+1})+(b+1)x)}={{\rm Tr}\left(x\right)}=\frac{m}{2k}{{\rm Tr}_{1}^{2k}\left(x\right)}$. Hence, $W_{f}(a,b)-W_{f}(a,b+1)\in\\{0,\pm 2^{\frac{m+2k+2}{2}}\\}$ for ${\rm Tr}_{k}^{m}(a)\neq 0$. Combining this with (17), when $(a,b)$ runs through $\mathbb{F}_{2^{m}}^{2}$, we have $W_{f}(a,b)-W_{f}(a,b+1)\in\left\\{0,2^{m},-2^{m},\pm 2^{\frac{m+2k+2}{2}}\right\\}$ and each of the values $\pm 2^{m}$ occurs $2^{m-k}$ times. Then ${\rm wt_{H}}({\mathbf{c}}(a,b))=0$ and ${\rm wt_{H}}({\mathbf{c}}(a,b))=2^{m-1}$ both occur $2^{m-k}$ times and every nonzero weight in ${\mathcal{C}}(f)^{\bar{D}}$ belongs to the set $\\{2^{m-2},\,2^{m-1},\,2^{m-2}\pm 2^{\frac{m+2k-2}{2}}\\}$ by (12) . Hence, ${\mathcal{C}}(f)^{\bar{D}}$ is degenerate and has dimension $m+k$. This completes the proof. $\square$ Using Lemma 4.7 and similar discussions in the proof of Theorem 4.2, one can prove the following theorem. ###### Theorem 4.8 Follow the notation and conditions introduced in Lemma 4.7. Then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{m-1},m+k,2^{m-2}-t]$ code with the weight distribution in Table 9. Its dual has parameters $\left[2^{m-1},2^{m-1}-m-k,4\right]$, and is distance-optimal with respect to the sphere packing bound. Table 9: The weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.8 Weight \| Multiplicity ---\|--- $0$ \| $1$ $2^{m-2}$ \| $2^{m+k}-2+2^{2m-3}\cdot(1-2^{k})/t^{2}$ $2^{m-2}\pm t$ \| $2^{2m-4}\cdot(2^{k}-1)/t^{2}$ $2^{m-1}$ \| $1$ ###### Example 4.9 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 4.8. Let $m=5$ and $k=1$. Then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[16,6,6]$ and its dual has parameters $[16,10,4]$. Both codes are optimal according to the tables of best codes known in [22]. We settled the parameters and weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 4.8, but do not know if the corresponding code ${\mathcal{C}}(f)$ was studied in the literature or not. ## 5 Some punctured codes of binary linear codes from cyclotomic classes In this section, we settle the weight distribution of the punctured code ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual, where the position set $D$ is a cyclotomic class and $f(x)=x^{d}$ for some integer $d$. Let $\gamma$ be a primitive element of $\mathbb{F}_{2^{m}}$ and let $t$ be a positive integer dividing $2^{m}-1$. Let $D=\langle\gamma^{t}\rangle$, which is the subgroup of $\mathbb{F}_{2^{m}}^{}$ generated by $\gamma^{t}$. The multiplicative cosets of $D$ are called the cyclotomic classes of order $t$ in $\mathbb{F}_{2^{m}}^{}$. Recall that the binary punctured code is ${\mathcal{C}}(f)^{\bar{D}}=\\{\mathbf{c}(a,b)=({\rm Tr}(ax^{d}+bx))_{x\in D}:a,b\in{\mathbb{F}}_{2^{m}}\\}$ (18) if $f(x)=x^{d}$. Since $\|D\|=\frac{2^{m}-1}{t}$, the length $n$ of ${\mathcal{C}}(f)^{\bar{D}}$ is $\frac{2^{m}-1}{t}$. It is easily seen that the Hamming weight of the codeword ${\mathbf{c}}(a,b)$ is given by $\begin{split}{\rm wt_{H}}({\bf c}(a,b))&=n-\left\|\left\\{x\in D:\,\,{\rm Tr}\left(ax^{d}+bx\right)=0\right\\}\right\|=\frac{1}{2}\left(n-\sum_{x\in D}(-1)^{{\rm Tr}(ax^{d}+bx)}\right).\end{split}$ (19) To determine the weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$, we need to determine the value distribution of $T(a,b)=\sum_{x\in D}(-1)^{{\rm Tr}(ax^{d}+bx)}$ (20) for $(a,b)$ running through $\mathbb{F}_{2^{m}}^{2}$. In the following, we propose several classes of linear codes with few weights by choosing proper $d$ and $t$. ### 5.1 The case that $d=\frac{2^{m}-1}{3}$ and lcm$(3,t)\,\|\,(2^{\frac{m}{2}}+1)$ In this subsection, we always assume that $v_{2}(m)=1$, $d=\frac{2^{m}-1}{3}$ and $t$ is a positive integer satisfying lcm$(3,t)\,\|\,(2^{\frac{m}{2}}+1)$. If $3\,\|\,t$, then $x^{\frac{2^{m}-1}{3}}=1$ for any $x\in D$. From (20) we have $\begin{split}T(a,b)=\sum_{x\in D}(-1)^{{\rm Tr}(a+bx)}.\end{split}$ (21) If $3\,\nmid\,t$, then $\begin{split}T(a,b)&=\sum_{x\in\langle\gamma^{3t}\rangle}(-1)^{{\rm Tr}(a+bx)}+\sum_{x\in\langle\gamma^{3t}\rangle}(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}}+b\gamma^{t}x)}+\sum_{x\in\langle\gamma^{3t}\rangle}(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}}+b\gamma^{2t}x)}\\\ &=\frac{1}{3t}\left(\sum_{x\in\mathbb{F}_{2^{m}}^{}}(-1)^{{\rm Tr}(a+bx^{3t})}+\sum_{x\in\mathbb{F}_{2^{m}}^{}}(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}}+b\gamma^{t}x^{3t})}+\sum_{x\in\mathbb{F}_{2^{m}}^{}}(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}}+b\gamma^{2t}x^{3t})}\right)\\\ &=\frac{1}{3t}\left(\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a+bx^{3t})}+\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}}+b\gamma^{t}x^{3t})}+\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}}+b\gamma^{2t}x^{3t})}\right)\\\ &-\frac{1}{3t}\left((-1)^{{\rm Tr}(a)}+(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})}+(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})}\right).\end{split}$ (22) In order to obtain the possible values of $T(a,b)$ for $3\,\nmid\,t$, we need the following lemma. ###### Lemma 5.1 Let $N$ be the number of zeros in the sequence $\left({\rm Tr}(a),\,\,{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}}),\,\,{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})\right)$. When $a$ runs over $\mathbb{F}_{2^{m}}$, we have $N=\left\\{\begin{array}[]{lll}3,&{\rm occuring}\,\,\,2^{m-2}\,\,\,{\rm times},\\\ 1,&{\rm occuring}\,\,\,3\cdot 2^{m-2}\,\,\,{\rm times}.\end{array}\right.$ Proof. Obviously, the possible values of $N$ are 0, 1, 2 or 3. Let $N_{i}$ denote the number of times that $N=i$ when $a$ runs over $\mathbb{F}_{2^{m}}$, where $i\in\\{0,1,2,3\\}$. Then $\begin{split}N_{3}&=\frac{1}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{0}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(y_{1}a)}\sum_{y_{1}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(y_{1}a\gamma^{\frac{2^{m}-1}{3}})}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(y_{2}a\gamma^{\frac{2(2^{m}-1)}{3}})}\\\ &=\frac{1}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{0}\in\mathbb{F}_{2}}\sum_{y_{1}\in\mathbb{F}_{2}}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}\left(a(y_{0}+y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}})\right)}.\end{split}$ Note that $y_{0}+y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}}=0$ if and only if $y_{0}=y_{1}=y_{2}=0$ or $y_{0}=y_{1}=y_{2}=1$. Then $N_{3}=\frac{1}{2^{3}}\left(2^{m}+2^{m}\right)=2^{m-2}.$ Due to symmetry, we have $\displaystyle N_{2}$ $\displaystyle=$ $\displaystyle\frac{3}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{0}\in\mathbb{F}_{2}}(-1)^{y_{0}({\rm Tr}(a)-1)}\sum_{y_{1}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(y_{1}a\gamma^{\frac{2^{m}-1}{3}})}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(y_{2}a\gamma^{\frac{2(2^{m}-1)}{3}})}$ $\displaystyle=$ $\displaystyle\frac{3}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{1}\in\mathbb{F}_{2}}\sum_{y_{0}\in\mathbb{F}_{2}}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}(a(y_{0}+y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}})-y_{0})}$ $\displaystyle=$ $\displaystyle\frac{3}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{1}\in\mathbb{F}_{2}}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}\left(a(y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}})\right)}-$ $\displaystyle\frac{3}{2^{3}}\sum_{a\in\mathbb{F}_{2^{m}}}\sum_{y_{1}\in\mathbb{F}_{2}}\sum_{y_{2}\in\mathbb{F}_{2}}(-1)^{{\rm Tr}\left(a(1+y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}})\right)}.$ Note that $y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}}=0$ if and only if $y_{1}=y_{2}=0$, and $1+y_{1}\gamma^{\frac{2^{m}-1}{3}}+y_{2}\gamma^{\frac{2(2^{m}-1)}{3}}=0$ if and only if $y_{1}=y_{2}=1$. Then $N_{2}=\frac{1}{2^{3}}\left(2^{m}-2^{m}\right)=0.$ Similarly, we can prove that $N_{1}=3\cdot 2^{m-2}$ and $N_{0}=0$. $\square$ ###### Theorem 5.2 Let $v_{2}(m)=1$, $d=\frac{2^{m}-1}{3}$ and $t$ be a positive integer satisfying $\textrm{lcm}(3,t)\,\|\,(2^{\frac{m}{2}}+1)$. Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (18) and $D=\langle\gamma^{t}\rangle$. If $t\neq 2^{\frac{m}{2}}+1$, then the following statements hold. (1) If $3\,\|\,t$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[\frac{2^{m}-1}{t},m+1]$ code with the weight distribution in Table 10. Its dual has parameters $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-m-1,4]$, and is distance-optimal with respect to the sphere packing bound. Table 10: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $3\,\|\,t$ in Theorem 5.2 Weight \| Multiplicity ---\|--- $0$ \| $1$ $\frac{1}{2t}\left(2^{m}-2-2^{\frac{m}{2}}\right)$ \| $\frac{(t-1)(2^{m}-1)}{t}$ $\frac{1}{2t}\left(2^{m}+2^{\frac{m}{2}}\right)$ \| $\frac{(t-1)(2^{m}-1)}{t}$ $\frac{1}{2t}\left(2^{m}-2+(t-1)2^{\frac{m}{2}}\right)$ \| $\frac{(2^{m}-1)}{t}$ $\frac{1}{2t}\left(2^{m}-(t-1)2^{\frac{m}{2}}\right)$ \| $\frac{(2^{m}-1)}{t}$ $\frac{2^{m}-1}{t}$ \| $1$ (2) If $3\,\nmid\,t$, then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[\frac{2^{m}-1}{t},m+2]$ code with the weight distribution in Table 11. Its dual has parameters $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-m-2,3]$. Table 11: Weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ for $3\,\nmid\,t$ in Theorem 5.2 Weight \| Multiplicity ---\|--- $0$ \| $1$ $\frac{2^{m}-1}{2t}-\frac{1}{2t}\left((t-1)2^{\frac{m}{2}}-1\right)$ \| $\frac{2^{m}-1}{t}$ $\frac{2^{m}-1}{2t}+\frac{1}{6t}\left((3t-1)2^{\frac{m}{2}}-1\right)$ \| $\frac{2^{m+1}-2}{t}$ $\frac{1}{2t}\left(2^{m}+2^{\frac{m}{2}}\right)$ \| $\frac{(t-1)(2^{m}-1)}{t}$ $\frac{2^{m}-1}{2t}-\frac{1}{6t}\left(2^{\frac{m}{2}}+1\right)$ \| $\frac{3(t-1)(2^{m}-1)}{t}$ $\frac{2^{m}-1}{2t}-\frac{1}{6t}\left((3t+1)2^{\frac{m}{2}}+1\right)$ \| $\frac{2^{m}-1}{t}$ $\frac{2(2^{m}-1)}{3t}$ \| $3$ Proof. We prove this theorem case by case as follows. Case 1: $3\,\|\,t$. From (21) we have $\begin{split}T(a,b)&=(-1)^{{\rm Tr}(a)}\sum_{x\in E}(-1)^{{\rm Tr}(bx)}=\frac{1}{t}(-1)^{{\rm Tr}(a)}\sum_{x\in\mathbb{F}_{2^{m}}^{}}(-1)^{{\rm Tr}(bx^{t})}\\\ &=\frac{1}{t}(-1)^{{\rm Tr}(a)}-\frac{1}{t}(-1)^{{\rm Tr}(a)}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(bx^{t})}.\end{split}$ If $b=0$, it is clear that $\begin{split}T(a,0)&=\left\\{\begin{array}[]{lcl}\frac{(1-2^{m})}{t},&{\rm if}\,\,\,{\rm Tr}(a)=0,\\\ \frac{(2^{m}-1)}{t},&{\rm if}\,\,\,{\rm Tr}(a)=1.\end{array}\right.\end{split}$ (23) If $b\neq 0$, then $b$ can be written as $b=\gamma^{i}$, where $\gamma$ is a primitive element of $\mathbb{F}_{2^{m}}$ and $1\leq i\leq 2^{m}-1$. From Lemma 2.4 we have $\begin{split}T(a,\gamma^{i})&=\left\\{\begin{array}[]{lcl}\frac{1}{t}(-1)^{{\rm Tr}(a)}-\frac{1}{t}(-1)^{{\rm Tr}(a)}(-1)^{s}2^{\frac{m}{2}},&{\rm if}\,\,\,i\not\equiv 0\pmod{t},\\\ \frac{1}{t}(-1)^{{\rm Tr}(a)}-\frac{1}{t}(-1)^{{\rm Tr}(a)}(-1)^{s-1}(t-1)2^{\frac{m}{2}},&{\rm if}\,\,\,i\equiv 0\pmod{t},\end{array}\right.\end{split}$ (24) as $t$ is a positive integer such that lcm$(3,t)\,\|\,(2^{m/2}+1)$. Hence, when $(a,b)$ runs over $\mathbb{F}_{2^{m}}^{2}$, by (23) and (24), the value distribution of $T(a,b)$ is given as follows: $\begin{split}T(a,b)&=\left\\{\begin{array}[]{lll}\frac{(1-2^{m})}{t},&{\rm occuring}\,\,2^{m-1}\,\,{\rm times},\\\ \frac{(2^{m}-1)}{t},&{\rm occuring}\,\,2^{m-1}\,\,{\rm times},\\\ \frac{1}{t}\left(2^{\frac{m}{2}}+1\right),&{\rm occuring}\,\,\frac{2^{m-1}(t-1)(2^{m}-1)}{t}\,\,{\rm times},\\\ -\frac{1}{t}\left(2^{\frac{m}{2}}+1\right),&{\rm occuring}\,\,\frac{2^{m-1}(t-1)(2^{m}-1)}{t}\,\,{\rm times},\\\ \frac{1}{t}-\frac{1}{t}(t-1)2^{\frac{m}{2}},&{\rm occuring}\,\,\frac{2^{m-1}(2^{m}-1)}{t}\,\,{\rm times},\\\ -\frac{1}{t}+\frac{1}{t}(t-1)2^{\frac{m}{2}},&{\rm occuring}\,\,\frac{2^{m-1}(2^{m}-1)}{t}\,\,{\rm times}.\end{array}\right.\end{split}$ (25) From (19) and (25), we know that the Hamming weight $0$ occurs $2^{m-1}$ times when $(a,b)$ runs through $\mathbb{F}_{2^{m}}^{2}$. Hence, in this case, ${\mathcal{C}}(f)^{\bar{D}}$ is degenerate and has dimension $m+1$. Dividing each frequency by $2^{m-1}$ in (25), we get the weight distribution in Table 10 from (19). From the first five Pless power moments and the weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$, we deduce that the dual of ${\mathcal{C}}(f)^{\bar{D}}$ is a $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-m-1,4]$ code. If there exists a $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-m-1]$ binary code with Hamming distance at least $5$, then we have $\begin{split}\sum_{i=0}^{2}\left(\begin{array}[]{cccc}\frac{2^{m}-1}{t}\\\ i\\\ \end{array}\right)=1+\frac{2^{m}-1}{t}+\frac{2^{m}-1}{2t}\cdot(\frac{2^{m}-1}{t}-1)>2^{m+1}\end{split}$ as $t\neq 2^{\frac{m}{2}}+1$, which is contrary to the sphere packing bound. Hence, the dual code $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is distance- optimal with respect to the sphere packing bound. Case 2: $3\,\nmid\,t$. From (22) we have $\begin{split}T(a,b)&=\frac{1}{3t}\Bigg{(}(-1)^{{\rm Tr}(a)}\big{(}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(bx^{3t})}-1\big{)}+(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})}\big{(}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(b\gamma^{t}x^{3t})}-1\big{)}\\\ &+(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})}\big{(}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{(b\gamma^{2t}x^{3t})}-1\big{)}\Bigg{)}.\end{split}$ If $b=0$, it is clear that $\begin{split}T(a,0)&=\frac{2^{m}-1}{3t}\left((-1)^{{\rm Tr}(a)}+(-1)^{{\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})}+(-1)^{{\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})}\right).\end{split}$ From Lemma 5.1 we have $\begin{split}T(a,0)&=\left\\{\begin{array}[]{lll}\frac{(2^{m}-1)}{t},&{\rm occuring}\,\,\,2^{m-2}\,\,\,{\rm times},\\\ -\frac{(2^{m}-1)}{3t},&{\rm occuring}\,\,\,3\cdot 2^{m-2}\,\,\,{\rm times}.\end{array}\right.\end{split}$ (26) If $b\neq 0$, then $b$ can be written as $b=\gamma^{i}$, where $\gamma$ is a primitive element of $\mathbb{F}_{2^{m}}$ and $1\leq i\leq 2^{m}-1$. From Lemma 2.4 we have $\begin{split}T(a,\gamma^{i})=\begin{cases}\frac{1}{3t}+\frac{1}{3t}\sum_{x\in\mathbb{F}_{2^{m}}}\left((-1)^{{\rm Tr}(\gamma^{i}x^{3t})}-(-1)^{{\rm Tr}(\gamma^{i+t}x^{3t})}-(-1)^{{\rm Tr}(\gamma^{i+2t}x^{3t})}\right),\\\ &\hskip-199.16928pt\text{ if \ ${\rm Tr}(a)=0$, ${\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})=1$ and ${\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})=1$,}\\\ \frac{1}{3t}+\frac{1}{3t}\sum_{x\in\mathbb{F}_{2^{m}}}\left(-(-1)^{{\rm Tr}(\gamma^{i}x^{3t})}+(-1)^{{\rm Tr}(\gamma^{i+t}x^{3t})}-(-1)^{{\rm Tr}(\gamma^{i+2t}x^{3t})}\right),\\\ &\hskip-199.16928pt\text{ if \ ${\rm Tr}(a)=1$, ${\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})=0$ and ${\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})=1$,}\\\ \frac{1}{3t}+\frac{1}{3t}\sum_{x\in\mathbb{F}_{2^{m}}}\left(-(-1)^{-{\rm Tr}(\gamma^{i}x^{3t})}-(-1)^{{\rm Tr}(\gamma^{i+t}x^{3t})}+(-1)^{{\rm Tr}(\gamma^{i+2t}x^{3t})}\right),\\\ &\hskip-199.16928pt\text{ if \ ${\rm Tr}(a)=1$, ${\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})=1$ and ${\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})=0$,}\\\ -\frac{1}{t}+\frac{1}{3t}\sum_{x\in\mathbb{F}_{2^{m}}}\left((-1)^{{\rm Tr}(\gamma^{i}x^{3t})}+(-1)^{{\rm Tr}(\gamma^{i+t}x^{3t})}+(-1)^{{\rm Tr}(\gamma^{i+2t}x^{3t})}\right),\\\ &\hskip-199.16928pt\text{ if \ ${\rm Tr}(a)={\rm Tr}(a\gamma^{\frac{t(2^{m}-1)}{3}})={\rm Tr}(a\gamma^{\frac{2t(2^{m}-1)}{3}})=0$.}\\\ \end{cases}\end{split}$ (27) Clearly, one of $3t\,\|\,i$, $3t\,\|\,(i+t)$ and $3t\,\|\,(i+2t)$ holds if and only if $t\,\|\,i$ for any positive integer $t$ and $1\leq i\leq 2^{m}-1$. Otherwise, $3t\,\nmid\,i$, $3t\,\nmid\,(i+t)$ and $3t\,\nmid\,(i+2t)$. Then combining Lemma 2.4, (26) and (27), it is not hard to see that when $(a,b)$ runs over $\mathbb{F}_{2^{m}}^{2}$, the value distribution of $T(a,b)$ is given as follows: $\begin{split}T(a,b)&=\left\\{\begin{array}[]{lll}\frac{(2^{m}-1)}{t},&{\rm occuring}\,\,2^{m-2}\,\,{\rm times},\\\ -\frac{(2^{m}-1)}{3t},&{\rm occuring}\,\,3\cdot 2^{m-2}\,\,{\rm times},\\\ -\frac{1}{t}+\frac{1}{t}\left((t-1)2^{\frac{m}{2}}\right),&{\rm occuring}\,\,\frac{(2^{m}-1)2^{m-2}}{t}\,\,{\rm times},\\\ \frac{1}{3t}\left(1-3\cdot 2^{\frac{m}{2}}\right),&{\rm occuring}\,\,\frac{(t-1)(2^{m}-1)2^{m-2}}{t}\,\,{\rm times},\\\ \frac{1}{3t}\left(1-(3t-1)2^{\frac{m}{2}}\right),&{\rm occuring}\,\,\frac{(2^{m}-1)2^{m-1}}{t}\,\,{\rm times},\\\ \frac{1}{3t}\left(2^{\frac{m}{2}}+1\right),&{\rm occuring}\,\,\frac{3(t-1)(2^{m}-1)2^{m-2}}{t}\,\,{\rm times},\\\ \frac{1}{3t}\left((3t+1)2^{\frac{m}{2}}+1\right),&{\rm occuring}\,\,\frac{(2^{m}-1)2^{m-2}}{t}\,\,{\rm times}.\\\ \end{array}\right.\end{split}$ (28) By a similar analysis to Case 1, we obtain the weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of its dual. This completes the proof. $\square$ ###### Example 5.3 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 5.2. Let $m=6$ and $t=3$, then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[21,7,8]$ and its dual has parameters $[21,14,4]$. The two codes are optimal according to the tables of best codes known in [22]. ###### Remark 5.4 If $t=2^{\frac{m}{2}}+1$, it is easy to check that ${\mathcal{C}}(f)^{\bar{D}}$ is a $[2^{\frac{m}{2}}-1,\frac{m}{2}+1,2^{\frac{m}{2}-1}-1]$ code with the weight enumerator $1+(2^{\frac{m}{2}}-1)(x^{2^{\frac{m}{2}-1}-1}+x^{2^{\frac{m}{2}-1}})+x^{2^{\frac{m}{2}}-1},$ which is optimal with respect to the Griesmer bound. Its dual has parameters $[2^{\frac{m}{2}}-1,\frac{m}{2}+1,4]$, which is distance-optimal with respect to the sphere packing bound. By the Assmus-Mattson theorem [2], the code ${\mathcal{C}}(f)^{\bar{D}}$ and its dual support $2$-designs [13, Chapter 4]. The reader is informed that in the special case $t=2^{\frac{m}{2}}+1$, the code ${\mathcal{C}}(f)^{\bar{D}}$ is permutation-equivalent to a singly punctured code of the first-order Reed-Muller code [37]. ### 5.2 The case that $d(2^{k}+1)\equiv 2^{\frac{m}{2}}+1\pmod{2^{m}-1}$ and $t=2^{k}+1$ In this subsection, we always assume that $m$ is even, $d(2^{k}+1)\equiv 2^{\frac{m}{2}}+1\pmod{2^{m}-1}$ and $t=2^{k}+1$. From (20) it follows that $\begin{split}T(a,b)&=\sum_{x\in D}(-1)^{{\rm Tr}(ax^{d}+bx)}=\frac{1}{2^{k}+1}\sum_{x\in\mathbb{F}_{2^{m}}^{}}(-1)^{{\rm Tr}(ax^{2^{m/2}+1}+bx^{2^{k}+1})}\\\ &=\frac{1}{2^{k}+1}\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{{\rm Tr}(ax^{2^{m/2}+1}+bx^{2^{k}+1})}-\frac{1}{2^{k}+1}.\end{split}$ (29) If $k=\frac{m}{2}$, by Lemma 2.4, (19) and (29), ${\mathcal{C}}(f)^{\bar{D}}$ is a one-weight code with parameters $[2^{\frac{m}{2}}-1,\frac{m}{2},2^{\frac{m}{2}-1}]$, and is permutation- equivalent to a Simplex code. In the following, we determine the parameters and the weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ and the parameters of the dual code $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ for $k\,\neq\,\frac{m}{2}$. ###### Theorem 5.5 Let $d$ satisfy the condition $d(2^{k}+1)\equiv 2^{\frac{m}{2}}+1\pmod{2^{m}-1}$. Let $t=2^{k}+1$ and $k\,\neq\frac{m}{2}$. Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code defined in (18). Then ${\mathcal{C}}(f)^{\bar{D}}$ is a $[\frac{2^{m}-1}{t},\frac{3m}{2},\frac{2^{m-1}-2^{\frac{m}{2}+k-1}}{t}]$ code with the weight distribution in Table 12. If $k>1$, its dual has parameters $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-\frac{3m}{2},3]$. If $k=1$ and $m\neq 6$, its dual has parameters $[\frac{2^{m}-1}{t},\frac{2^{m}-1}{t}-\frac{3m}{2},4]$, and is distance- optimal with respect to the sphere packing bound. If $k=1$ and $m=6$, its dual has parameters $[21,12,5]$, and is optimal according to the tables of best codes known in [22]. Table 12: Weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 5.5 Weight \| Multiplicity ---\|--- $0$ \| $1$ $\frac{2^{m-1}+2^{\frac{m}{2}-1}}{t}$ \| $\frac{2^{3k}(2^{\frac{m}{2}}-1)(2^{m}-2^{m-2k}-2^{m-3k}+2^{\frac{m}{2}}-2^{\frac{m}{2}-k}+1)}{t^{2}(2^{k}-1)}$ $\frac{2^{m-1}-2^{\frac{m}{2}+k-1}}{t}$ \| $\frac{2^{k}(2^{m}-1)(2^{\frac{m}{2}}+2^{\frac{m}{2}-k}+2^{\frac{m}{2}-2k}+1)}{t^{2}}$ $\frac{2^{m-1}+2^{\frac{m}{2}+2k-1}}{t}$ \| $\frac{(2^{\frac{m}{2}-k}-1)(2^{m}-1)}{t^{2}(2^{k}-1)}$ Proof. It is clear that ${\rm Tr}\left(ax^{2^{m/2}+1}\right)={\rm Tr}_{1}^{\frac{m}{2}}\left((a+a^{2^{m/2}})x^{2^{m/2}+1}\right)$ and $a+a^{2^{m/2}}\in\mathbb{F}_{2^{\frac{m}{2}}}$ for any $a\in\mathbb{F}_{2^{m}}$. Obviously, $a+a^{2^{m/2}}$ runs through $\mathbb{F}_{2^{\frac{m}{2}}}$ with multiplicity $2^{\frac{m}{2}}$ when $a$ runs through $\mathbb{F}_{2^{m}}$. Let $K=\left\\{x\in\mathbb{F}_{2^{m}}:x+x^{2^{m/2}}\right\\}.$ Then $\mathbf{c}(a,b)=\mathbf{c}(a,b+\delta)$ for any $\delta\in K$. Since $t\,\|\,(2^{m}-1)$ and $t=2^{k}+1$, it is easy to prove that there exists a positive integer $\ell$ such that $\ell(2^{k}+1)\equiv 2^{\frac{m}{2}}+1\pmod{2^{m}-1}$ if and only if $v_{2}(k)=v_{2}(\frac{m}{2})$ and $k\,\|\,\frac{m}{2}$. From Lemma 2.5, (19) and (29), the desired weight distribution of ${\mathcal{C}}(f)^{\bar{D}}$ follows. Let $A^{\perp}_{i}$ be the number of the codewords with weight $i$ in $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$. By the first four Pless power moments, we get that $A_{1}^{\perp}=A_{2}^{\perp}=0$ and $\begin{split}A_{3}^{\perp}&=\frac{1}{48(2^{k}+1)^{5}(2^{2k}-1)}\big{(}(2^{7k}+2^{6k+1}+6\cdot 2^{2k}+4\cdot 2^{3k}+7\cdot 2^{k}+2-3\cdot 2^{5k}-10\cdot 2^{4k}-9\cdot 2^{3k})\cdot 2^{\frac{3m}{2}}\\\ &+(3\cdot 2^{5k}+10\cdot 2^{4k}+5\cdot 2^{3k}-6\cdot 2^{2k}-7\cdot 2^{k}-2)\cdot 2^{\frac{5m}{2}}\big{)}.\end{split}$ We can check that $A_{3}=0$ if $k=1$ and $A_{3}\neq 0$ if $k>1$. If $k=1$, by the fifth Pless power moment, we obtain that $\begin{split}A_{4}^{\perp}=\frac{1}{6^{4}}(2^{4m}+70\cdot 2^{\frac{5m}{2}}-6\cdot 2^{\frac{7m}{2}}-25\cdot 2^{3m})+\frac{2^{2m}}{54}-\frac{2^{\frac{3m}{2}+2}}{81}.\end{split}$ It is easy to check that $A_{4}^{\perp}=0$ if and only if $m=6$. Similarly to the proof of Theorem 5.2, we can show that $({\mathcal{C}}(f)^{\bar{D}})^{\perp}$ is distance-optimal with respect to the sphere packing bound if $k=1$ and $m\neq 6$. By the sixth Pless power moment, we obtain $A_{5}^{\perp}\neq 0$. This completes the proof. $\square$ ###### Example 5.6 Let ${\mathcal{C}}(f)^{\bar{D}}$ be the linear code in Theorem 5.5. Let $m=10$ and $k=1$. Then ${\mathcal{C}}(f)^{\bar{D}}$ has parameters $[341,15,160]$. Its dual has parameters $[341,326,4]$ and is distance-optimal with respect to the sphere packing bound. We settled the parameters and weight distribution of the code ${\mathcal{C}}(f)^{\bar{D}}$ in Theorem 5.5, but do not know if the corresponding code ${\mathcal{C}}(f)$ was studied in the literature or not. ## 6 Summary and concluding remarks The main contributions of this paper are the following: 1. 1. We obtained several classes of binary punctured codes with three weights, or four weights, or five weights, or six weights, and determined their weight distributions (see Theorem 3.3, Corollary 3.6, Theorem 4.2, Theorem 4.5, Theorem 4.8, Theorem 5.2 and Theorem 5.5). 2. 2. We presented several classes of self-complementary linear codes. Almost all of their duals are distance-optimal with respect to the sphere packing bound (see Theorem 3.3, Corollary 3.6, Theorem 4.2, Theorem 4.5, Theorem 4.8, Theorem 5.2 and Theorem 5.5). 3. 3. We got some distance-optimal codes with specific parameters (see Example 3.4, Example 3.7, Example 4.3, Example 4.9 and Example 5.3). A constructed binary linear code ${\mathcal{C}}$ is new if one of the following happens: • No binary linear code with the same parameters was documented in the literature. * • Some binary linear codes with the same parameters as ${\mathcal{C}}$ were documented in the literature, but their weight distributions are different from the weight distribution of ${\mathcal{C}}$. * • Some binary linear codes with the same parameters and weight distribution as those of ${\mathcal{C}}$ were documented in the literature, but they are not permutation-equivalent to ${\mathcal{C}}$. Except the class of codes in Remark 5.4, every other class of binary codes presented in this paper would be new, as we have not found a class of binary codes with the same parameters and weight distributions in the literature as those codes documented in this paper. Starting from Section 2, we restricted our discussions on finite fields with characteristic 2. The position sets were constructed from some trace functions and cyclotomic classes. It would be interesting to extend some of the results in this paper to the case that $q\geq 3$. Finally, we make some comments on the puncturing and shortening techniques. As mentioned in the introductory section, every projective linear code over ${\mathbb{F}}_{q}$ is a punctured Simplex code over ${\mathbb{F}}_{q}$ and a shortened code of a Hamming code over ${\mathbb{F}}_{q}$. However, it is in general very hard to determine the parameters of punctured codes of Simplex codes and shortened codes of Hamming codes [37, 56]. 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# Malware Detection and Analysis: Challenges and Research Opportunities Zahid Akhtar Department of Network and Computer Security, State University of New York Polytechnic Institute, USA. Email<EMAIL_ADDRESS> ###### Abstract Malwares are continuously growing in sophistication and numbers. Over the last decade, remarkable progress has been achieved in anti-malware mechanisms. However, several pressing issues (e.g., unknown malware samples detection) still need to be addressed adequately. This article first presents a concise overview of malware along with anti-malware and then summarizes various research challenges. This is a theoretical and perspective article that is hoped to complement earlier articles and works. ## I Introduction Use of personal computers and mobile devices coupled with internet has now become integral part of everyday life. This ubiquity of high interconnectivity has prompted many serious privacy and security menaces as well as different other malicious activities. For instance, 117 million LinkedIn user’s email and password were made publicly available by hackers in 2016. In 2017, Uber revealed that its server was attacked and 57 million drivers and riders data were stolen. While, in 2018 almost 50 million Facebook accounts were compromised due to security breach. Similarly, cyberattacks on Norway’s ‘Health South East RHF’ healthcare organization in 2018 exposed health record of more than half of country’s population. Moreover, it is estimated that on an average every 10 second a new malicious code specimen is released to attack mobile devices [1]. A surge of cyberattacks with increasing number and sophistication can be seen with each passing year, which is impacting governments, enterprises and individual alike and causing severe reputation, financial and social damages. For example, malicious cyber activities in 2016 cost U.S. economy alone up to 109 billion USD [2]. Different types of cyberattacks are presently being performed by cybercriminals, e.g., man-in-the-middle, malware or birthday attack. In particular, malware attacks have advanced as one of the main formidable issues in cybersecurity domain as well as primary tool utilized by cybercriminals [3]. Malware is a short form of _mal_ icious _soft_ ware. In French language, ‘mal’ means ‘bad’. Malware is a catch-all term widely employed to denote various kinds of unwanted harmful software programs with malicious motives [4]. When malware is executed on a system or computing device it attempts to breach the system/device’s security policies regarding integrity, confidentiality and availability of data. Other names for malware are badware, malicious code, malicious executable and malicious program. Malwares are developed or used by hobbyists and cyber-offenders trying to show their ability by causing havoc and to steal information potentially for monetary gains, respectively. They are popularly known as hackers, black hats and crackers, and could be external/internal menace, industrial spies or foreign governments. Malwares can be used to change or erase data from victim computers, to collect confidential information, or to hijack systems in order to attack other devices, send spams, host and share illicit contents, bring down servers, penetrate networks, and cripple critical infrastructures. Consequently, a broad range of tools and schemes have been devised to detect and mitigate malware attacks [1]. Anti-malware systems thwart malwares by determining whether given program has malign intent or not [4]. Despite great advancement of malware defense techniques and their incessant evolution, badwares still can bypass the anti-malware solutions owing to mainly sophisticated packers and weakest link, i.e., humans. Namely, most anti- malware methods do not exhibit low enough error rates. Additionally, their performance particularly drops when they face unknown malwares. While, daily 360,000 novel malware samples hit the scene [4]. As anti-malware becomes more avant-garde so as malwares in the wild, thereby escalating the arms race between malware guardians and writers. The quests for scalable and robust automated malware detection frameworks still have to go a long way. This article presents an overview of malwares and their defenses formulated in recent years, and highlights challenges, open issues and research opportunities for researchers and engineers. It is a perspective and academic article, which is aimed at complementing existing studies and prompt interdisciplinary research. ## II Malware Categories Malwares, as depicted in Fig. 1, can be divided into various types depending on how they infect, propagate or exploit the target system as follows [3]. Please note that some of the malware types/tools/names fall in gray area of features intended for begin purposes as well, e.g., cookie, Wireshark, etc. ### II-A Virus A piece of code that duplicates, reproduces or propagates itself across files, programs and machines if they are network-connected. Viruses cannot execute independently, therefore they are mainly appended to ‘host’ programs/files (e.g., executable files, master boot record). When executed by ‘host’ can corrupt or destroy files, programs, computer’s functioning and shared network that may result in denial of service and system’s performance degradation. Examples of viruses are Melissa virus and Creeper virus. Figure 1: General taxonomy of malware. ### II-B Worm Unlike virus, worm does not need ‘host’ but can run independently. Worms are self-replicating and self-propagating via a network or storage devices. Worms exploit operation system vulnerabilities, but do not corrupt user or system files. They consume computing and network resources by residing in main memory while replicating and spreading themselves causing DoS and SPD. Examples are MyDoom and SQL Slammer. ### II-C Trojan Trojan surfaces as benign program but performs malevolent activities in the backend without user’s knowledge. Trojans usually do not infect files or replicate themselves, rather create backdoors for unauthorized system access to delete files, install programs or extricate private data (e.g., passwords). Examples are Zeus and Zitmo. ### II-D Spyware Spyware spies on users without their knowledge or consent. It is generally used to surveil user activities, gather keystrokes or harvest sensitive information (e.g., login credentials). Examples of spyware are LogKext and GPSSpy. Following are popular spyware sub-categories: #### II-D1 Adware Adware dispenses either spiteful code/content or ads to infected users via web browser, mobile app or PC’s desktop to generate attacker’s revenue. Another name of this malware is malvertising, as it may use reputed companies/banners to distribute malicious codes. It can be considered as a subcategory of spyware, but unlikely leading to a big harm until coupled with other spywares. Examples are AllSearchApp and Plankton. Pornware is also seen as a subclass of adware, when installed maliciously without user’s knowledge to display pornographic materials. #### II-D2 Keylogger This malware is also called keystroke logger, password grabbers, sniffer or information-stealing malware, which is employed by attackers to record each keystroke to steal sensitive data (e.g., passwords, credit card numbers). Keylogger is generally transferred to a system when spiteful-software is installed or -site is visited. Examples are SpyEye and Formgrabber. #### II-D3 Trackware Trackware is unwanted software that tracks and collects user activities and habits then share data with a third party. Though trackwares harm user’s privacy, they do not harvest confidential or personally identifiable information. Examples are Trackware. Rewardnet and Win32/WebHancer.A. #### II-D4 Cookie Cookies are plain text files with information of user’s web browsing sessions. They are stored on user’s computer/device for future use. Although cookies seemingly are not detrimental, they can be menace if exploited by some spyware. Likewise, tracking cookies can be utilized by hackers to gain user’s personal details. #### II-D5 Riskware Riskware (aka grayware) is a genuine program when utilized by the attacker can cause damage to system security or data by deletion, duplication, or modification. Authentic programs for riskware could be IRC client, file downloaders, etc. #### II-D6 Sniffer It is a malicious program that observes and captures the network traffics. It analyzes various fields of packets and gather data for preparation of malware attacks. Sniffers can be ‘Ethereal’ (i.e., legitimate used for troubleshooting) and ‘BUTTSniffer’ (i.e., illegitimate for malign purposes). Examples of sniffers are Wireshark and Aircrack-ng. ### II-E Ransomware Ransomwares are covertly installed on a victim computer to execute a cryptovirology attack. This malware type encrypts the data or locks down the system, thereby restricting user access till ransom is paid. Specifically, ransomwares can be classified in two main groups, viz. locker ransomwares that decline access to the system/device functionalities, and crypto ransomware that avert access to files/data. Ransomwares examples are FakeDefender and TorrentLocker. ### II-F Scareware Scareware deludes people into buying/downloading inessential and potentially perilous security software, opening attachments or visiting a malevolent website. It mostly attempts to frighten users (e.g., by displaying false warning messages), and when installed it collects stored information from victim system, which maybe sold to cybercriminals. Examples are Mac Defender and Registry Cleaner XP. ### II-G Diallerware Diallerware send premium-rate SMS/multimedia messages without mobile user’s knowledge, thereby causing monetary sums to user. The premium-rate SMS/multimedia messages/calls provide value-added services, which can be abused by attackers. Attackers lure mobile owners to sign up for the premium services managed by themselves, e.g., HippoSMS. Diallerware blocks the messages from service providers to users to avoid user’s awareness of unwanted additional charges. ### II-H Bot A bot (abbreviated from robot) is a malicious program, which enables attacker (aka botmaster or bot herder) to remotely control infected machine without user’s knowledge via a command and control (C&C) channel from a system called C&C server. A cluster of bots controlled by a sole server is known as botnet. Botnets can be employed to organize DDoS attacks, phishing fraud, sending spams, etc. Well-known examples are Sdbot and Agobot. #### II-H1 Spamware Spamware (aka spam sending malware or spambot) is malicious software designed to search and compile list of email addresses as well as sending large number of spam emails. It is an element of a botnet functioning as a distributed spam-sending network. Spamware can use infected user’s email ID or IP address to send emails, which may consume great amount of bandwidth and slow down the system. Examples are Trik Spam and Necurs botnet. #### II-H2 Reverse Shell A reverse shell is an unauthorized program (malware) that provides access of undermined computer to the attacker. Reverse shell enables attacker to run and type command on host as the attacker is local. Examples are Netcat and JSP web shell. ### II-I Rootkit A rootkit is a stealthy software that is devised to conceal specific programs/processes and enabling privileged access to computer/data. Rootkit allows the attacker accessing and controlling the system remotely without being detected, as it normally runs with root privileges and subverts system logs and security software. Examples are NTRootkit and Stuxnet. #### II-I1 Bootkit Bootkit is an advanced form of rootkits that infects master boot record or volume boot record. Since it resides in boot sector, it is difficult to be detected by security software, and also stays active after system reboot. Well-known examples are BOOTRASH and FinFisher. ### II-J Backdoor Backdoor is a malware that installs by itself and creates secret entrance for attackers to bypass system’s authentication procedures and to access and perform illegitimate activities. Backdoors are never utilized alone but as foregoing malware attacks of other kinds, as they do not harm but furnish wider attack surfaces. A notable backdoor tool is Remote Access Terminal/Trojan (RAT). Other examples are Basebridge and Olyx. ### II-K Browser Hijackers It is an undesired software that alters settings of web browser without user’s consent either to inject ads in the browser or replace home/error page and search engine. Some of them may access sensitive data with spyware. Examples are CoolWebSearch and RocketTab. ### II-L Downloader It is a malicious program that downloads and installs/runs new versions of malwares from internet on compromised computers. Downloader is usually embedded in websites and software. Examples are Trojan-Downloader:W32/JQCN and Trojan-Downloader:OSX/Jahlev.A. Figure 2: A generic malware detection and analysis system. First, input sample is provided to feature extraction module that yields feature representation vector. A feature reduction/selection process is carried out on feature representation vector to obtain fixed dimensionality regardless of length of input sample for enhanced performance. A classification/clustering technique is trained on available set of malware and benign samples. During detection/analysis, unseen sample is reported by the classification/clustering techniques as malware or not. Further analysis is also sometimes performed, e.g., describing suspicious (or benign) characteristics present in the sample. ## III Malware Concealment Techniques To evade anti-malwares, malware writers have applied following different malware camouflage approaches [3]: ### III-A Encryption Encrypted malware by this method consists of encryption and decryption algorithms, keys and malicious codes. Each time attacker employs new encryption algorithm and key to generate novel malware version. Since decryption algorithm remains same, there is a higher probability to be detected. The main target of this procedure is to avoid static analysis and delaying investigation process. CASCADE was reported as the first encrypted malware in 1987. ### III-B Packing Packing mechanism is utilized to compress/encrypt malware executable file. To detect malwares with packing technique, reverse engineering methods or correct unpacking algorithm is needed, which sometime is hard as it requires knowledge of true packing/compression algorithm. UPX and Upack are examples of packing. ### III-C Obfuscation This technique obscures program’s principal logic to stop others gaining associated knowledge of the code. Malwares with obfuscation and their deleterious functionality stay unintelligible till activated. Quintessential obfuscation strategies are inessential jumps and including garbage commands. ### III-D Polymorphism Polymorphic malware is designed to alter its appearance every time it is executed while retaining original code entirely. Compared to encryption technique, boundless number of encryption algorithms can be utilized by a polymorphic malware such that in each implementation a decryption code’s portion is mutated. Transformation engine is generally kept in encrypted malware. When any mutation occurs, a random encryption algorithm is produced to re-encrypt the engine and malware with new decryption key. Different inimical actions can be embedded under encryption operations. Since original code remains intact, polymorphic malwares comparatively become easy to be detected. First known polymorphic virus developed in 1990 is 1260. ### III-E Metamorphism Metamorphism malware (aka body-polymorphic malware) mutates its malevolent codes in each execution to create novel instance that has no similitude with native codes, but functionality yet remains the same. There are two categories of metamorphic malwares. _Open-world malware_ that mutates by communicating with other sites over net, e.g., Conficker worm. _Open-world malware_ that reprograms itself without external communication by mutating binary code (i.e., binary transformer) or employing pseudocode representation, e.g., Win32/Apparition virus. ## IV Malware Detection and Analysis System As Fig. 2 shows, a generic malware detection system consists of four main modules: feature extraction, feature selection, classification/clustering, and decision. The raw data sample is input to feature extraction module, which extricates salient attributes as a feature set. Next, feature selection is performed to tackle the curse of dimensionality, to reduce the computational complexity, and to increase performance of the system by quantifying feature correlations. The resultant feature vector is given to a classifier/clustering scheme. Finally, decision module is employed either to acquire the final binary decision: malware or benign (cleanware), or/and for additional malware analysis such as malware variants detection (i.e., recognizing variant and families), malware category detection (i.e., categorizing based on malwares’ prominent behaviors and objectives), malware similarity and novelty detection (i.e., acquiring knowledge about unknown sample by specific similarities and differences against known ones), malware development detection (i.e., finding out if the malware writer has previously submitted it to online defense tools), and malware attribution (i.e., identifying its programming language, from where its launched and actor/group involved). ### IV-A Malware Analysis In general, malware analysis is deployed both for detecting/classification and other investigations (e.g., understanding the working to devise novel identification schemes) of malware. Different features such as strings (i.e., frequency of code fragments, names, etc.), byte sequences (i.e., characterization of byte-level contents), opcodes (i.e., identification of machine/assembly-level operations), system/APIs calls (i.e., analyses of execution traces/disassembly code or characterization of APIs’ executed actions), call graphs and data dependent (i.e., analyses of data being exchanged between process calls), control flow graphs (i.e., behavior relationships of data flow between system resources), multilayer dependency chains (i.e., characterization of sub-behaviors to capture interactions among samples and system levels), causal dependency graphs (i.e., tracking persistent state changes on target system), influence graphs (i.e., encoding of downloads by malware), memory accesses (i.e., analyses of memory during malware executions), file system (i.e., frequency of created/deleted/modified files), system registry (i.e., count of queried/deleted/modified registry keys), CPU registers (i.e., frequency of registers usages/changes), function length (i.e., number of bytes in a function), exceptions (i.e., exceptions prompted during malware execution) and network traffic (i.e., analyses of incoming and outgoing packets, visited addresses, etc.) are being used for malware analysis. Malware analysis can be conducted in following three ways: #### IV-A1 Static Analysis It is also called signature-based, code analysis, white-box or misuse detection approach. Methods in this category generally review statically the code-structure for traits of infections using a pre-defined list of known assails’ signatures without executing the sample [4]. However, advanced static analysis techniques may run the sample by deploying reverse engineering, i.e., obtaining binary and assembly codes using decompiler, disassembler and debugger. Hellal _et al._ [5] presented a call code graph mining based static analysis scheme, called minimal contrast frequent subgraph miner algorithm, to distinguish variants of malware in Windows environment. Schultz _et al._ [6] used features like list of DLLs functions, system calls and hex-dump to detect novel and unseen malicious executables. Martin _et al._ [7] designed a malware detection method that uses third-party API calls in Java files and multi- objective optimization classification. While, Narayanan _et al._ [8] developed a mutli-view (MKLDROID) framework utilizing a graph kernel with multiple kernel learning to determine sets of semantic structures and contextual information from Android apps for malware/malicious code localization. Yerima and Sezer [9] proposed Android malware detection that analyzes permissions and intents from the apps via multilevel classifier rank fusion architecture. Recenlty, Cakir _et al._ [10] designed a shallow deep learning based method that employed word2vec features via opcodes and a Gradient boosting classifier. Though static analysis techniques are capable of fast recognizing malwares in versatile applications and pose no risk of infection while analyzing malwares, they need huge number of pre-defined signature dataset. Moreover, they suffer from runtime overhead and cannot discriminate variations of known- or obscure- malwares and zero-day intrusions. #### IV-A2 Dynamic Analysis It is also called behavior-based, behavioral analysis, anomaly-based, specification-based or black-box approach. Methods in this category assess samples via their activities by executing them in a confined/simulated environment, e.g., sandboxed, simulator, debugger, virtual machine or emulator. Miao _et al._ [11] proposed a bi-layer behavior abstraction technique via semantic examination of dynamic API sequences in Windows environment. Lower- and higher-layer behaviors were captured using data dependence of APIs and complex good interpretability of lower abstractions, respectively. In [12], authors developed a graph-based model harnessing relations (i.e., dependency graphs) among system-calls’ groups for smartphone malicious software detection, but the model requires high time consumption. Authors in [13] presented a compression-based feature mining on system/API calls’ quantitative information flow graphs to detect Windows malware. Mao _et al._ [14] designed a security dependence network from access behaviors to evaluate importance of system resources (e.g., files, registry, and processes) and malware detection. While, Egele _et al._ [15] presented a dynamic blanket execution function that employs high-level API-relevant semantic features. Enck _et al._ [16] presented _TaintDroid_ for dynamic taint examination to trace leakage of sensitive data (e.g., microphone, GPS and camera) in third-party apps. Ye _et al._ [17] presented a deep learning strategy comprised of AutoEncoder, multilayer restricted Boltzmann machines and associative memory. The framework detects malware in embedded systems via Windows API calls extricated from portable executable files. Though dynamic analysis techniques are independent of malware source-code and can detect unknown and zero-day malware instances, they require more resources (e.g., memory, CPU time and disk space) and have high computational cost and false positive rates. #### IV-A3 Hybrid Analysis It is also called gray-box approach. Neither static- nor dynamic-analysis methods are unable to provide perfect anti-malware solutions. Thus, hybrid- analysis approaches, which combine benefits of both static and dynamic analyses, is more desirable. For instance, Santos _et al._ [18] designed a hybrid method that integrates static (i.e., opcodes frequency) and dynamic (i.e., executable’s execution trace data) features with multitude of classifiers. Authors in [19] proposed a hybrid technique that collects system calling runtime data and then utilizes a static scheme for mobile malware detection. While, Dali _et al._ [20] developed a method that uses FlowDroid static analysis tool and sensitive sources data flows with deep learning-based classifier. ### IV-B Feature selection The performance of malware detection depends on choice of feature representation and length. The feature selection/dimensionality reduction is conducted to attain a set of more discriminative features for enhanced performance. Various anti-malwares have been presented using filter, wrapper and embedding based feature selection algorithms such as distributed-, hierarchical-, correlation-, low-rank matrix approximation-, forward-, backward-, local sensitive hashing-, max relevance, adaptive feature scaling-, spectral graph theory-, F1-score, F2-score, mean decrease impurity-, document frequency-, information gain-, information gain ratio-, principal component analysis- and latent dirichlet allocation [3]. ### IV-C Classification/Clustering To identify if a given sample is malicious or/and to determine malware family, various binary and multiclass classification techniques such as Multilayer Perceptron, Support Vector Machines, Naïve Bayes, Decision Tree, Rule-based, Random Forests, Multiple Kernel Learning, $K$-Nearest Neighbors, Logistic Regression, Ensemble, Multi-Objective Evolutionary by Genetic Algorithm, Deep Belief Networks have been employed [4]. Hierarchical-, $K$-means-, meanShift-, $K$-medoid partitional-, density-based spatial-, prototype-, self-organizing maps-, single-linkage- and locality sensitive hashing-based clustering techniques have been utilized to categorize malware samples exhibiting identical behaviors into different groups or to generate signatures for detection [3]. ### IV-D Evaluation Metrics Performance of malware detection methods is generally evaluated by False Positive Rate = FP/(FP + TN), True Positive Rate = TP/(TP + FN), specificity = TN/(TN + FP), precision = TP/(TP + FP), accuracy = (TP + TN)/(TP + TN + FP + FN), where TP, FP, TN and FN are true positives, false positives, true negatives and false negatives, respectively. Malware samples are commonly considered as positive instances. Moreover, Matthews correlation coefficient, F-score, Kappa statistic, confusion matrix, receiver operating characteristic and under the curve measures have been used. While, for clustering-based algorithms Macro-F1 and Micro-F1 metrics, respectively, for accentuating the performance on rare and common categories [3, 4]. ## V Research challenges and opportunities The ever-growing demand of minimized failure rates of anti-malware solutions have opened up exigent research opportunities and challenges to be resolved yet. ### V-A Issues in existing anti-malware methods Malwares are still exponentially evolving in sophistication, and more difficult plights lie ahead. Most prior static and dynamic or hybrid methods do not work for novel/unknown/zero-day signatures and require virtual environment plus are time consuming, respectively. Nonetheless, virtual environments are becoming less effective as malware writers are usually one step ahead by implementing high-level new techniques to conceal malicious features. Though efforts are afoot to design multi-level and parallel processing system, existing anti-malware methods/tools all in all are not adequate or potent for higher levels of concealments. Current anti-malware systems also face challenges like scalability, lack of truly real-world representative datasets, irreproducibility of published results, low generalization and detection disagreement among them for the same samples. There is a need of improved and comprehensive malware countermeasures, which could be developed by utilizing recent advanced-machine/deep learning, -data mining and -adaptive schemes. Also, approaches embodying anomaly analysis with behavioral data should be designed to investigate what the malware is doing rather than how it is doing. This may result in minimized error and false alarm rates. ### V-B Advanced machine learning (AML) techniques for anti-malware Quintessential anti-malwares often depend on non-linear adversary explicit models and expert domain knowledge, thereby making them prone to overfitting and lower overall reliability. Conversely, AML techniques attempt to imitate attackers with various content, contexts, etc. rather than explicit models/systems/attacks. Few preliminary studies on shallow AML usage for anti- malware has been conducted, but still a lot of efforts to be done regarding AML anti-malware. For improved accuracy, flexibility and scalability on wide range and unknown samples, AML paradigms like open set recognition, more complex and residual deep learning, dictionary learning and data mining should be explored for feature segmentation/representation learning/selection/classification and determining temporal relationships within and between malware sections. ### V-C Mobile device malwares Smart-devices connected to internet is growing exponentially, so as malwares (especially via third party apps) against them. Insubstantial studies have been conducted on mobile device malwares. Moreover, most existing anti-malware techniques are not real-time and unsuited for mobile devices because of high computational cost and/or features complexity used for analysis. Thus, real- time lightweight mobile anti-malwares via Bayesian classification is an interesting research direction to be explored. Multiple information from in- built sensors (e.g., accelerometer) may enhance mobile anti-malware performance. Mobile hardware malware detection and removal is another issue that needs serious exploration. Sooner mobile anti-malware-inspired techniques will substantially impact smart-devices design. Anyway, smart-device malwares should be tackled both by preventive and effective countermeasures. App developers should assure that their apps are abiding security and privacy policies. App stores administrators should vet and remove dubious apps. Users should use superior anti-malwares and install trusted apps. On the whole, wearable and mobile devices malware and anti-malware are a new research field in cybersecurity with pressing problems worth researching like malware affecting device’s master boot record or stealthily exploiting device to mine cryptocurrency, and how a malware performing well on benchmark data will be better under real-world environments. ### V-D Large-scale benchmark databases Advancement in malware research deeply depends on the public availability of comprehensive benchmark datasets incorporating accurate labels and contexts. Most existing databases suffers from limitations like small size, missing information/features, imbalanced-classes, and not publicly available. Lack of adequate large-scale public datasets has stymied research on malware. Benchmark public datasets will assist to compare independent anti-malware schemes, determine inter and intra relationships between security infringement phenomena and unify malware findings to draw determined conclusions with reference to statistical significance. Nevertheless, collecting large-scale heterogenous annotated databases is challenging and time- and resource- consuming due to malware attributes, forms and behaviors diversity. Crowdsourcing may help accumulating different annotated large-scale databsets. ### V-E Graph-based malware analysis Malwares with concealments are dominant nowadays and effectual in evading conventional anti-malwares that largely disregard learning and identifying the underlying relationships between samples and variants, and contextual information. Graph-based relationship representations and features (e.g., data- and control-flow graphs, call graphs, data-, program-, and control- dependency graphs) offer interesting possibility even when malware code is altered as it helps in tracking malware genealogy in different settings. Devising graph-based anti-malwares yet have issues from data heterogeneity, noisy and incomplete labels, and computational cost during real-time detection. Up to some extend such challenges may be addressed in decentralized fashion. Furthermore, use of multiple directed and undirected graphs, multi- view spectral clustering, heterogeneous networks, multiple graph kernel learning, dynamic graph mining and deep graph convolution kernels to capture contextual and structural information could be fruitful area of research. ### V-F Bio-inspired anti-malware Several limitations of traditional anti-malwares could be suppressed by bio- inspired (e.g., biological immune system, biological evolution, genetic algorithms and swarm intelligence) techniques. Comparatively these techniques are lightweight, highly scalable and less resource-constrained. Adaptive bio- inspired techniques that is used both for intelligent concealment-invariant feature extraction and classification can dramatically enhance accuracy in the wild. Bio-inspired methods that define particular objective functions to discriminate a system under attack from a malfunctioning or failing may also help strengthening the security. Combination of bio-inspired algorithms with deep neural networks is one of the most promising direction, however has been explored less in anti-malwares. ### V-G Defense-in-depth anti-malware Anti-malware strategy that is composed of multiple defense levels/lines rather than single is called defense-in-depth. Such strong defensive mechanism is expected to be more robust as it doesn’t depend on one defense technique and if one is breached the others aren’t. Each machine/cyber-system architecture can be divided in various levels of depth, e.g., in a power grid system, the meters, communication frameworks, and smaller components, respectively, could be envisaged as lowest, intermediate and highest level. Another solution is active or adaptive malware defense. Active defense has received little attention due to inherent complexity, where developer anticipates attack scenarios at different level and accordingly devises malware countermeasures. In adaptive defense, the system is persistently updated by retraining/appending novel features or dynamically adjusted corresponding to reshaping environments. Adaptive defenses would require fast, automated and computationally effective and could use unsupervised learning and concept drift. ### V-H Internet of things (IoT) attacks IoT are progressively being used in different domains ranging from smart- cities to smart- and military-grids. Despite finest security endeavors, IoT devices/systems can also be compromised by innovative cyber-attacks. Security of IoT technology is more crucial as it is always connected to a network. IoT cyber-security is relatively new research realm and quite challenging owing to heterogeneous networks with multisource data and several categories of nodes. To this end, different routes (e.g., predictive and blockchain) could be effective. Predictive security is attaining cyber resiliency by devising models that predict future attacks and prevent in advance. As there is a strong correlation between security infractions and human blunders, predictive models should consider computer networking, social sciences, descriptive theory, uncertain behavior theory and psychology from attackers, users and administrators’ perspectives at different granularity levels. Blockchain can be utilized for self-healing of compromised devices/systems. Models could be devised that exploit e.g., redundancy to heal corrupted codes/software by good codes replacements, since in blockchain one can trace and roll back the firmware versions. However, such models should also be capable to handle resource, energy and communication constraints, which may be achieved by lightweight machine/transfer/reinforcement learning based access control protocols. ### V-I Deception and moving target anti-malware techniques Deception techniques (e.g., honeypot) are being used to detect and prevent malwares, which lures adversaries to strike in order to mislead them with false information. There are two kinds of honeypots, i.e., client and server. Honeypot helps to reduce false positives and prevent DDoS attacks. Complex attacks/tools (e.g., polymorphic malware) is increasing to identify honeypots or to alter their behaviors to deceive honeypots themselves. Also, honeypot can be exploited by attackers to undermine other sensitive parts of frameworks. More complicated honeypot and honey nets (i.e., bunch of honeypots) schemes (e.g., shadow honeypots) should be devised as compromised honeypot will put security of whole organization in danger. Moving target techniques (aka dynamic platform methods-DPMs) dynamically randomizes system components to suppress successful attacks’ likelihood and shorten attack lifetime. Though adversary must undermine all platforms not one to evade DPMs, DPMs require complicated application state synchronization among varying platforms, and expand the system’s attack surface. Much less efforts have been dedicated to developing well-articulated attack models and how to upgrade deception elements and strategy to confront dynamic changes in attack behaviors. Future research should concentrate on devising unorthodox methodologies, performing real-world analyses to compute and compare effectiveness of deception and DPMs techniques, and studying if DPMs conflict or can co-exist with other anti-malwares. ### V-J Decentralized anti-malware Data sharing and trust management hinder current anti-malwares advancement, which can be resolved by decentralized malware detectors using blockchain technology. But it has received little attention till now. For intersection of anti-malware and blockchain technology, future directions will include exploring overhead traffic handling, quality and sparse malware signatures, building accurate dynamic normal nature of traffics, reducing massive false alerts, energy and cost, blockchain latency, case-by-case scenario investigation, and more proof-of-concept implementations. ### V-K Botnet countermeasures Thwarting botnets has become key area. Several botnet detection and defense architectures have been proposed. Various issues surround botnet countermeasure study, e.g., difficulties in testing devised botnet defenses in real scenarios/data. Besides, lack of widely acknowledged benchmark or standard methodology to quantitative evaluate or compare bot defenses presumably due to privacy and data sharing concerns. Botnets, including IoT bot and socialbot, will continue to rise until effective means both technical and non-technical are taken. Technical factors include passive internet service providers and unassertive software. Non-technical factors include establishing distributed global environment, local and multinational legal issues and poor user awareness. ### V-L Privacy preservation Malwares that steal sensitive information has received much attention. However, preserving user privacy in malware analysis (especially at the cloud or third party server) and malware data sharing is yet an open and seldom touched concern. Establishing privacy and regaining trust in commercial anti- malwares would become difficult if user’s privacy/data is compromised once. Majority of prior anti-malwares overlook the privacy and security of user, data and network. Thus, reasonably little has been worked on privacy protection frameworks to respect public and law opinions. Privacy preservation mechanisms that do not influence the detection performance is practically worthy of contemplation. Formulating lightweight detection and privacy protection systems usable on mobile devices to balance security, efficacy, privacy and power consumption demands special considerations. More innovative privacy preservation approaches (e.g., allowing user to stabilize privacy, convenience and security levels) in malware analysis has been highlighted by many experts as an essential future research to be carried out. ### V-M Big data malware analysis The demand for big data malware analysis frameworks is steadily expanding. Practitioners are working to resolve big data malware challenges such as volume (e.g., collecting, cleaning and compressing data/labels), velocity (e.g., real-time online training, learning, processing or streaming big data), variety (e.g., heterogeneous multi-view data learning/embedding), veracity (e.g., leaning with contradicting and unreliable data), and value (explainable ML based malware analysis). Another promising future research direction is devising large-scale feature selection techniques, which are less-dependent on feature engineering, via distributed feature selection, low-rank matrix approximation, adaptive feature scaling, spectral graph theory, and fuzzy and neuro-fuzzy clustering. Rigorous efforts need to be made to investigate use of synchronous parallel processing (e.g., Spark) and to develop body of knowledge on pros and cons of big data anti-malware tools to assist practitioners. ### V-N Malware analysis visualization systems Existing methods to analyze malwares are time-consuming for malware analysts. Highly interactive visual analysis would aid researchers and practitioners to forensically investigate, summarize, classify and compare malwares more easily. Most prior techniques are very limited with regard to interactivity, mapping temporal dimensions, scalability and representation space (e.g., they are superficially 2D rather than 3D). The field of developing malware visualization systems covering consequential rang of malware types and environments is vital and emerging. Encyclopedic visualization systems will lead analysts/researchers to ascertain novel research domains in the years to come. ### V-O Multimodal anti-malwares Multimodal anti-malwares, which consolidate evidences from different kinds of features/sources (e.g., string, permission, elements converted to image matrices) can overcome numerous constraints in frameworks that consider only one/fewer features. Multimodal frameworks are more flexible and can significantly enhance the accuracy of unimodal ones in the wild. Multimodal may include multiple sensors, algorithms and instances, and information can be fused at feature, score or decision level. There is ample room to develop novel fusion architectures. Moreover, multimodal frameworks are expected to be intrinsically more robust to concealments, but no study investigated how robust are they to concealments. ### V-P Clustering for malware analysis Previous works have shown that clustering could be a useful tool to effectively classify unknown malwares for improved generalization, to underline unseen family’s behaviors for thorough analysis that may help more robust anti-malware schemes, and to label huge volumes of malwares in fast and automatic fashion that has become major challenge. Future goal should be further improving accuracy of clustering-based malware analysis using cluster quality measurements, contextual/metadata information, and boosted genetic algorithms, etc. Attentions should also be given to rectify security issues, e.g., poisoning and obfuscation attacks against targeted clusters. ### V-Q Hardware-based Solutions Hardware-based detectors are recently getting momentum against proliferation of malware. Such detection mechanisms utilize low-level architectural features, which are obtained by redesigning the micro-architecture of computer processors, e.g., CPUs with special registers providing hardware and software anomaly events. Nevertheless, research in this domain and trustworthy systems (i.e., inherently secure and reliable against human errors and hostile parties) is yet in its initial genesis and has to go a long way. Furthermore, there is dearth of studies on efficacy of anti-malwares combining hardware- and software-based techniques that have exceptional potential to uncover extra elaborate malwares. Likewise, smart devices’ sensors (e.g., GPS and ambient light sensors) data could also be used as additional feature vector to profile malware. ### V-R Malware adversarial learning Machine-learning (ML) recently has been used to achieve effective malware defenses, however they are not designed for situations where an adversary is actively trying to impact outcomes. Specially, deep learning-based countermeasures lack robustness against adversarial examples (i.e., sample crafted from genuine samples with careful minor perturbations). Attackers can also inject poisoning samples in (online/adaptive) training database with the aim to remarkably decrease ML-malware countermeasure’s accuracy at testing phase. A comprehensive analysis for each malware considering attacker’s capability and what features to what extend should be modified to avoid detection has not been done yet. It is still difficult task to design ML anti- malwares that are robust in adversarial setting. Researchers should explore malware adversarial ML in identifying probable countermeasures’ vulnerabilities both at train and test stages, devising homologous assails and their impacts and developing techniques to enhance robustness of ML-based anti-malwares. ### V-S Performance evaluation framework Malware analysis accuracy/performance, which is used to evaluate, compare, or configure anti-malwares, in general lacks standardization. A unified and comprehensive evaluation framework should be developed to rank present and future methods, that incorporates static and dynamic techniques, adversary’s goal, knowledge and capability, attack strategies at train and test phase, evaluation metrics (i.e., security- and privacy-relevant error rates as most current methods do not cover all aspects), and common parlance to elucidate anti-malware performances. Any such framework with common criteria and open online platform for evaluating resilient, malware sophistication, decision making, policies, experimental setups, big databases, and open source codes will surely help both in reporting baseline performances without giving a false sense of progress and encouraging reproducible research on scalability and challenges in real-world scenarios. ### V-T Malware education Most malwares succeed contemplating humans as weakest link. Additionally, there is growing demand for cybersecurity workforces, therefore it is imperative to educate people about malware safety. In academic institutions, malware analysis and related courses should be taught both at undergraduate and graduate levels. Nonetheless, relatively very limited colleges/universities offer malware courses, which maybe because of the shortage of agreement on fundamental topics among institutions, book and training providers, and ethical sensitivity of educating/creating white-hats. Moreover, most academic courses being offered are practitioner-oriented but not science-/research-oriented and heavily rely on text books that are not current. Some training camps/workshops are being held by companies/organizations also for general public, but they are exceptionally expensive. More on-line free-to-access training courses will surely diminish malware damages. ### V-U Interdisciplinary research To advance state-of-the-art malware analysis, the research and industrial communities need to support and promote interdisciplinary fundamental science research and development (including contributions from machine learning, human psychology, computer engineering, etc.) to accomplish dependable, natural, and generalized anti-malware techniques. ## VI Conclusion Malwares, including in mobile and smart devices, have become more sophisticated and greater in frequency during recent years. Although there exist many defense tools and mechanisms, malware detection and analysis are still challenging tasks, since malware developers continuously conceal the information in attacks or evolve cyber-attacks to circumvent newer security techniques, plus some prior methods face low generalization to unknown malwares and scalability issues. It is hoped that this academic and perspective article will stimulate focused interdisciplinary research and development in anti-malware towards aggrandizing its full potential in different cyberspace applications. ## References * [1] P. Faruki, A. Bharmal, V. Laxmi, V. Ganmoor, M. S. Gaur, M. Conti, M. 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# Learning Based Signal Detection for MIMO Systems with Unknown Noise Statistics Ke He, Le He, Lisheng Fan, Yansha Deng, George K. Karagiannidis, _Fellow, IEEE_ , and Arumugam Nallanathan, _Fellow, IEEE_ K. He, L. He and L. Fan are all with the School of Computer Science and Cyber Engineering, Guangzhou University, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>[email protected]).Y. Deng is with the Department of Informatics, King’s College London, London WC2R 2LS, UK (e-mail: [email protected]).G. K. Karagiannidis is with the Wireless Communications Systems Group (WCSG), Aristotle University of Thessaloniki, Thessaloniki 54 124, Greece (e-mail: [email protected]).A. Nallanathan is with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London, U.K (e-mail: [email protected]). ###### Abstract This paper aims to devise a generalized maximum likelihood (ML) estimator to robustly detect signals with unknown noise statistics in multiple-input multiple-output (MIMO) systems. In practice, there is little or even no statistical knowledge on the system noise, which in many cases is non- Gaussian, impulsive and not analyzable. Existing detection methods have mainly focused on specific noise models, which are not robust enough with unknown noise statistics. To tackle this issue, we propose a novel ML detection framework to effectively recover the desired signal. Our framework is a fully probabilistic one that can efficiently approximate the unknown noise distribution through a normalizing flow. Importantly, this framework is driven by an unsupervised learning approach, where only the noise samples are required. To reduce the computational complexity, we further present a low- complexity version of the framework, by utilizing an initial estimation to reduce the search space. Simulation results show that our framework outperforms other existing algorithms in terms of bit error rate (BER) in non- analytical noise environments, while it can reach the ML performance bound in analytical noise environments. The code of this paper is available at https://github.com/skypitcher/manfe. ###### Index Terms: Signal detection, MIMO, impulsive noise, unknown noise statistics, unsupervised learning, generative models. ## I Introduction Consider the linear inverse problem encountered in signal processing, where the aim is to recover a signal vector $\bm{x}\in\mathbb{C}^{N\times 1}$ given the noisy observation $\bm{y}\in\mathbb{C}^{M\times 1}$, and the channel response matrix $\bm{H}\in\mathbb{C}^{M\times N}$. Formally, the observation vector can be expressed as $\displaystyle\bm{y}=\bm{H}\bm{x}+\bm{w},$ (1) where $\bm{w}\in\mathbb{C}^{M\times 1}$ is an additive measurement noise, that is independent and identically distributed (i.i.d) with an unknown distribution $p_{\bm{w}}(\bm{w})$. From a Bayesian perspective, the optimal solution to the above problem is the maximum a posteriori (MAP) estimation $\displaystyle\hat{\bm{x}}_{MAP}=$ $\displaystyle\arg\max_{\bm{x}\in\mathcal{X}}p(\bm{x}\|\bm{y}),$ (2) $\displaystyle=$ $\displaystyle\arg\max_{\bm{x}\in\mathcal{X}}p(\bm{y}\|\bm{x})p(\bm{x}),$ (3) where $\mathcal{X}$ denotes the set of all possible signal vectors. When there is no prior knowledge on the transmitted symbols, the MAP estimate is equivalent to the maximum likelihood estimation (MLE), which can be expressed as $\displaystyle\hat{\bm{x}}_{MAP}=\hat{\bm{x}}_{MLE}=$ $\displaystyle\arg\max_{\bm{x}\in\mathcal{X}}p(\bm{y}\|\bm{x}),$ (4) $\displaystyle=$ $\displaystyle\arg\max_{\bm{x}\in\mathcal{X}}p_{\bm{w}}(\bm{y}-\bm{H}\bm{x}).$ (5) In most of the existing works in the literature, the noise $\bm{w}$ is assumed to be additive white Gaussian noise (AWGN), whose probability density function (PDF) is analytical and the associated likelihood of each possible signal vector is tractable. In this case, the MLE in (5) becomes $\displaystyle\hat{\bm{x}}_{\text{E-MLE}}=\arg\min_{\bm{x}\in\mathcal{X}}\\|\bm{y}-\bm{H}\bm{x}\\|^{2},$ (6) which aims to minimize the Euclidean distance, referred to as E-MLE. However, in practical communication scenarios, we may have little or even no statistical knowledge on the noise. In particular, the noise may present some impulsive characteristics and may be not analyzable. For example, the noise distribution becomes unknown and mainly impulsive for scenarios like long- wave, underwater communications, and multiple access systems [1, 2, 3, 4]. In these cases, the performance of E-MLE will deteriorate severely [5]. In contrast to the Gaussian case, the exact PDF of impulsive noise is usually unknown and not analytical [3, 4, 6, 7], which means that the exact likelihood $p(\bm{y}\|\bm{x})$ is computationally intractable. ### I-A Related Research In general, there are two major approaches to solve the problem of signal detection in MIMO systems: model-driven and data-driven. Next, we briefly present both of them. #### I-A1 Model-Driven Methods Model-driven approaches have been extensively studied in the literature for MIMO signal detection, by assuming that the noise is Gaussian. Among them, the approximate message passing (AMP) algorithm is an attractive method, which assumes Gaussian noise and well-posed channel matrix [8]. The AMP can detect the desired signal by iteratively predicting and minimizing the mean squared error (MSE) with a state evolution process [8, 9]. Combined with deep learning methods, AMP is unfolded into a number of neural layers to improve the performance with ill-posed channel matrix [10, 11]. Moreover, an efficient iterative MIMO detector has been proposed in [12] to leverage the channel- aware local search (CA-LS) technology to significantly improve the signal detection under Gaussian noise environment. When the noise has an arbitrary density function, a generalized AMP (GAMP) algorithm can be designed for the generalized linear mixing model [9], where the sum-product version of the GAMP algorithm can be treated as a hybrid of the iterative soft-threshold algorithm (ISTA) and alternating direction method of multipliers (ADMM) [13]. The Gaussian GAMP (G-GAMP) is equivalent to the AMP, and the GAMP with MMSE denoiser can be rigorously characterized with a scalar state evolution whose fixed points, when unique, are Bayes-optimal [14, 15]. Since the GAMP algorithm extends the AMP algorithm to adapt to arbitrary noise whose PDF is analytical, it still requires numerical or approximate methods to compute the marginal posterior, when the noise statistics is unknown [9]. In addition, for some specific non-Gaussian noises, A. Mathur et al. have investigated the system performance and found the ML detectors in [16, 17, 18], which is critical for the development of the model-driven methods. When there is no prior knowledge on the noise statistics, researchers proposed other model-driven approaches to approximate the unknown noise distribution $p_{\bm{w}}(\bm{w})$ and compute the approximate likelihood $p(\bm{y}\|\bm{x})$ accordingly. However, this requires a huge amount of computation or sampling loops. As a result, these methods can not be applied efficiently in practical scenarios. For example, the expectation-maximization (EM) algorithm will be extremely slow in this case, since the dimension of data can be very high and the data set can be very large [19]. Moreover, in order to select an appropriate approximate model, the EM algorithm requires to have some knowledge on the noise statistics, otherwise it may performs worst[20]. Besides, for the variational inference based approximations in [21, 22, 23], the noise is assumed to depend on a hidden variable $\bm{\theta}$, so that we can approximate the associated a posteriori probability $p(\bm{\theta}\|\bm{w})$ with a simpler distribution $q(\bm{w})$, by maximizing a lower bound of the reverse KL-divergence $D\left(q(\bm{w})\\|p(\bm{w}\|\bm{\theta})\right)$. This indicates that the exact marginal probability of the noise $p(\bm{w})=\int p(\bm{w},\bm{\theta})\mathrm{d}\bm{\theta}$ as well as the likelihood of signal vectors remains computationally intractable. #### I-A2 Data-Driven Methods In recent years, thanks to the tremendous success of deep learning, researchers have developed some data-driven methods to solve the problems encountered in various communication areas [24, 25, 26, 27]. For example, Y.-S. Jeon et al. have proposed a supervised-learning based novel communication framework to construct a robust nonlinear MIMO system, which consisted of the concatenation of a wireless channel and a quantization function used at the ADCs for data detection [28]. For widely connected internet of things (IoT) devices, a novel deep learning-constructed joint transmission-recognition scheme was introduced in [29] to tackle the crucial and challenging transmission and recognition problems. It effectively improves the data transmission and recognition by jointly considering the transmission bandwidth, transmission reliability, complexity, and recognition accuracy. For the aspect of signal detection, the authors in [30] proposed a projection gradient descent (PGD) based signal detection neural network (DetNet), by unfolding the iterative PGD process into a series of neural layers. However, its performance is not guaranteed when the noise statistics is unknown, for which the gradient is computed based on the maximum likelihood criterion of Gaussian noise in DetNet. Moreover, when the noise is dynamically correlated in time or frequency domain, the authors proposed a deep learning based detection framework to improve the performance of MIMO detectors in [31, 32]. In addition, some generative models based on deep learning have been proposed to learn the unknown distribution of random variables. In particular, the generative models are probabilistic, driven by unsupervised learning approaches. Currently, there are three major types of generative models [33, 34], which are variational auto encoders (VAEs) [35, 36], generative adversarial networks (GANs) [37, 38] and normalizing flows (NFlows) [39, 40, 41]. Recently, the deep generative models are adopted in literature to solve the linear inverse problems [42]. For example, the images can be restored with high quality from the noisy observations by approximating the natural distribution of images with a generative model [43], which inspires us to try to solve the linear inverse problem with the aid of data-driven generative models rather than noise statistics. ### I-B Contributions In this paper, we propose an effective MLE method to detect the signal, when the noise statistics is unknown. Specifically, we propose a novel signal detection framework, named _maximum a normalizing flow estimate_ (MANFE). This is a fully probabilistic model, which can efficiently perform MLE by approximating the unknown noise distribution through a normalizing flow. To reduce the computational complexity of MLE, we further devise a low-complexity version of the MANFE, namely G-GAMP-MANFE, by jointly integrating the G-GAMP algorithm and MANFE. The main contributions of this work can be summarized as follow: * • We propose a novel and effective MLE method, when only noise samples are available rather than statistical knowledge. Experiments show that this method achieves much better performance than other relevant algorithms under impulsive noise environments. Also, it can still reach the performance bound of MLE in Gaussian noise environments. * • The proposed detection framework is very flexible, since it does not require any statistical knowledge on the noise. In addition, it is driven by an unsupervised learning approach, which does not need any labels for training. * • The proposed detection framework is robust to impulsive environments, since it performs better a more effective MLE with comparison compared to E-MLE, when the noise statistics is unknown. * • We extend the MANFE by presenting a low-complexity version in order to reduce the computational complexity of MLE. The complexity of this version is very low, so that it can be easily implemented in practical applications. Further experiments show that its performance can even outperform the E-MLE under highly impulsive noise environments. ### I-C Organization In section II, we first overview the prototype of normalizing flows and discuss the reasons why we choose normalizing flows to solve the problem under investigation. The proposed detection framework and the implementation details are presented in Section III. We present various simulation results and discussions in Section IV to show the effectiveness of the proposed methods. Finally, we conclude the contribution of this paper in Section V. ## II Unknown Distribution Approximation In this section, we firstly present the maximum likelihood approach for the distribution approximation, and then provide the concept of normalizing flows. Furthermore, we compare the normalizing flows with other generative models, and explain the reason why we choose the former method to approximate an unknown noise distribution. ### II-A Maximum Likelihood Approximation Let $\bm{w}$ be a random vector with an unknown distribution $p_{\bm{w}}(\bm{w})$ and $\mathcal{D}_{\bm{w}}=\\{\bm{w}^{(1)},\bm{w}^{(2)},\cdots,\bm{w}^{(L)}\\}$ is a collected data set consisting of $L$ i.i.d data samples. Using $\mathcal{D}_{\bm{w}}$, the distribution $p_{\bm{w}}(\bm{w})$ can be approximated by maximizing the total likelihood of the data set on the selected model $q(\bm{w};\bm{\theta})$ parameterized by $\bm{\theta}$, such as the mixture Gaussian models. In this case, the loss function is the sum of the negative log-likelihoods of the collected data set, which can be expressed as $\displaystyle\mathcal{L}(\bm{\theta})=-\frac{1}{L}\sum_{l=1}^{L}\log q\left(\bm{w}^{(l)};\bm{\theta}\right).$ (7) Clearly, (7) measures how well the model $q(\bm{w};\bm{\theta})$ fits the data set drawn from the distribution $p_{\bm{w}}(\bm{w})$. Since $q(\bm{w};\bm{\theta})$ is a valid PDF, it is always nonnegative. In particular, it reaches its minimum if the selected model $q(\bm{w};\bm{\theta})$ perfectly fits the data, i.e. $q(\bm{w};\bm{\theta})\equiv p_{\bm{w}}(\bm{w})$. Otherwise, it enlarges if $q(\bm{w};\bm{\theta})$ deviates from $p_{\bm{w}}(\bm{w})$. Hence, the training objective is to minimize the loss and find the optimal parameters as $\displaystyle\bm{\theta}^{}=\arg\min_{\bm{\theta}}\mathcal{L}(\bm{\theta}),$ (8) where $\bm{\theta}$ can be optimized by some methods, such as the stochastic gradient descent (SGD) with mini-batches of data [44]. This is an unsupervised learning approach, since the objective does not require any labeled data. However, it is not flexible enough if the optimization is performed directly on the selected model $q(\bm{w};\bm{\theta})$, since the knowledge of the true distribution is needed in order to choose an appropriate model for approximation. ### II-B Normalizing Flow As a generative model, _normalizing flow_ allows to perform efficient inference on the latent variables [36]. More importantly, the computation of log-likelihood on the data set is accomplished by using the change of variable formula rather than computing on the model directly. For the observation $\bm{w}\in\mathcal{D}_{\bm{w}}$, it depends on a latent variable $\bm{z}$ whose density function $p(\bm{z};\bm{\theta})$ is simple and computationally tractable (e.g. spherical multivariate Gaussian distribution), and we can describe the generative process as $\displaystyle\bm{z}$ $\displaystyle\sim p(\bm{z};\bm{\theta}),$ (9) $\displaystyle\bm{w}$ $\displaystyle=g(\bm{z}),$ (10) where $g(\cdot)$ is an invertible function (aka bijection), so that we can infer the latent variables efficiently by applying the inversion $\bm{z}=f(\bm{w})=g^{-1}(\bm{w})$. By using (10), we can model the approximate distribution as $\displaystyle\log q(\bm{w};\bm{\theta})=\log p(\bm{z};\bm{\theta})+\log\bigg{\|}\det\bigg{(}\frac{\mathrm{d}\bm{z}}{\mathrm{d}{\bm{w}}}\bigg{)}\bigg{\|},$ (11) where the so called _log-determinant_ term $\log\bigg{\|}\det\bigg{(}\frac{\mathrm{d}\bm{z}}{\mathrm{d}{\bm{w}}}\bigg{)}\bigg{\|}$ denotes the logarithm of the absolute value of the determinant on the Jacobian matrix $\left(\frac{\mathrm{d}\bm{z}}{\mathrm{d}\bm{w}}\right)$. To improve the model flexibility, it is recognized that the invertible function $f(\cdot)$ is composed of $K$ invertible subfunctions $f(\cdot)=f_{1}(\cdot)\otimes f_{2}(\cdot)\otimes\cdots f_{k}(\cdot)\cdots\otimes f_{K}(\cdot).$ (12) From the above equation, we can infer the latent variables $\bm{z}$ by $\displaystyle\bm{w}\stackrel{{\scriptstyle f_{1}}}{{\longrightarrow}}\bm{h_{1}}\stackrel{{\scriptstyle f_{2}}}{{\longrightarrow}}\bm{h_{2}}\cdots\stackrel{{\scriptstyle f_{k}}}{{\longrightarrow}}\bm{h}_{k}\cdots\stackrel{{\scriptstyle f_{K}}}{{\longrightarrow}}\bm{z}.$ (13) By using the definitions of $\bm{h_{0}}\triangleq\bm{w}$ and $\bm{h_{K}}\triangleq\bm{z}$, we can rewrite the loss function in (7) as $\displaystyle\mathcal{L}(\bm{\theta})$ $\displaystyle=-\frac{1}{L}\sum_{l=1}^{L}\log q(\bm{w}^{(l)};\bm{\theta})$ (14) $\displaystyle=-\frac{1}{L}\sum_{l=1}^{L}\left(\log p(\bm{z}^{(l)};\bm{\theta})+\sum_{k=1}^{K}\log\bigg{\|}\det\bigg{(}\frac{\mathrm{d}\bm{h}^{(l)}_{k}}{\mathrm{d}\bm{h}^{(l)}_{k-1}}\bigg{)}\bigg{\|}\right).$ (15) Using the above, we can treat each subfunction as a step of the flow, parameterized by trainable parameters. By putting all the $K$ flow steps together, a normalizing flow is constructed to enable us to perform approximate inference and efficient computation of the log-probability. In general, the normalizing flow is inspired by the change of variable technique. It assumes that the observed random variable comes from the invertible change of a latent variable which follows a specific distribution. Hence, the normalizing flow is actually an approximate representation of the invertible change. From this perspective, one simple example is the general Gaussian distribution. Let us treat the normalizing flow as a black box, and simply use $f(\cdot)$ to represent the revertible function parameterized by the normalizing flow. Since any Gaussian variable $X\sim\mathcal{N}(\mu,\sigma)$ can be derived via the change of the standard Gaussian variable $Y\sim\mathcal{N}(0,1)$, the normalizing flow actually represents the approximation of the perfect inversion, saying that $Y=\frac{X-\mu}{\sigma}\approx f(X)$. In this case, one can easily compute the approximation of the corresponding latent variable as $Y\approx f(X)$. Therefore, when the observed variable follows different unknown distributions, the only difference is that the network parameters are fine tuned to different values for different distributions, which makes the principle of computing the associated latent variables become rather simple. To summarize the normalizing flow, its name has the following interpretations: • “Normalizing” indicates that the density is normalized by the reversible function and the change of variables, and * • “Flow” means that the reversible functions can be more complex by incorporating other invertible functions. ### II-C Why to use the Normalizing Flow? Similar to normalizing flow, the other generative models like VAE and GAN map the observation into a latent space. However, the exact computation of log- likelihood is totally different in these models. Specifically, in the VAE model the latent variable is inferred by approximating the posterior distribution $p(\bm{z}\|\bm{w})$. Hence, the exact log-likelihood can be computed through the marginal probability $p(\bm{w})=\int p(\bm{w},\bm{z})\mathrm{d}\bm{z}$, with the help of numerical methods like Monte Carlo. Therefore, in this case the computational complexity of the log- likelihood is very high. On the other hand, the GAN model do not maximize the log-likelihood. Instead of the training via the maximum likelihood, it will train a generator and a discriminator. The generator maps the observation into a latent space and draws samples from the latent space, while the discriminator decides whether the samples drawn from the generator fit the collected data set. Hence, both generator and discriminator play a min-max game. In this case, drawing samples from GANs is easy, while the exact computation of the log-likelihood is computationally intractable. For the reversible generative models like normalizing flows, the inference of latent variables can be exactly computed. The benefit of this approach is that we are able to compute the corresponding log-likelihood efficiently. In consequence, the normalizing flow is a good choice among these three generative models to perform exact Bayesian inference, especially when the noise statistics is unknown. ## III Proposed Signal Detection Framework In this section, we propose an unsupervised learning driven and normalizing flow based signal detection framework, which enables the effective and fast evaluation of MLE without knowledge of the noise statistics. As shown in Fig. 1, the proposed detection framework is built upon a normalizing flow, which includes three kinds of components: squeeze layer, $K$ flow steps, and an unsqueeze layer. To perform the MLE, given $\bm{y}$ and $\bm{H}$, we firstly compute the associated noise vector $\bm{w}_{i}=\bm{y}-\bm{H}\bm{x}_{i}$ for each possible signal vector $\bm{x}_{i}\in\mathcal{X}$. Then, by using the normalizing flow, we can map $\bm{w}_{i}$ into the latent space, and infer the associated latent variable $\bm{z}_{i}$ as well as the log-determinant. Therefore we can compute the corresponding log-likelihood $p(\bm{y}\|\bm{x}_{i})$ from (11) and determine the final estimation of the maximum log-likelihood. Since most neural networks focus on real-valued input, we can use a well-known real-valued representation to express the complex-valued input equivalently as $\displaystyle\underbrace{\begin{bmatrix}\mathop{R}(\bm{y})\\\ \mathop{I}(\bm{y})\\\ \end{bmatrix}}_{\bar{\bm{y}}}=\underbrace{\begin{bmatrix}\mathop{R}(\bm{H})&-\mathop{I}(\bm{H})\\\ \mathop{I}(\bm{H})&\mathop{R}(\bm{H})\\\ \end{bmatrix}}_{\bar{\bm{H}}}\underbrace{\begin{bmatrix}\mathop{R}(\bm{x})\\\ \mathop{I}(\bm{x})\\\ \end{bmatrix}}_{\bar{\bm{x}}}+\underbrace{\begin{bmatrix}\mathop{R}(\bm{w})\\\ \mathop{I}(\bm{w})\end{bmatrix}}_{\bar{\bm{w}}},$ (16) where $\mathop{R}(\cdot)$ and $\mathop{I}(\cdot)$ denote the real and imaginary part of the input, respectively. Then, we can rewrite (16) into a form like (1) as $\displaystyle\bar{\bm{y}}=\bar{\bm{H}}\bar{\bm{x}}+\bar{\bm{w}}.$ (17) Based on the above real-valued representation, the squeeze layer will process the complex-valued input by separating the real and imaginary part from the input, while the unsqueeze layer performs inverselly. In general, an $M$-dimensional complex-valued input $\bm{h}=\begin{bmatrix}h_{1},h_{2},\cdots,h_{M}\end{bmatrix}$ for a flow step can be represented by a 2-D tensor with a shape of $M\times 2$ and the following structure $\displaystyle\begin{bmatrix}\mathop{R}(h_{1})&\mathop{I}(h_{1})\\\ \mathop{R}(h_{2})&\mathop{I}(h_{2})\\\ \vdots&\vdots\\\ \mathop{R}(h_{M})&\mathop{I}(h_{M})\\\ \end{bmatrix},$ (18) where the first column/channel includes the real and the second column/channel the imaginary part, respectively. Figure 1: Architecture of the detection framework with normalizing flow. From Section II we can conclude that a critical point when implementing a normalizing flow is to carefully design the architecture, in order to ensure that the subfunctions, represented by neural blocks, are invertible and flexible, and the corresponding log-determinants are computationally tractable. Accordingly, as shown in Fig. 2, each flow step consists of three hidden layers: an activation normalization, an invertible $1\times 1$ convolution layer, and an alternating affine coupling layer. In particular, all these hidden layers except squeeze and unsequeeze, would not change the shape of input tensor. This means that its output tensor will have the same structure as the input one. In the rest of this section, we introduce these hidden layers one by one and then we present how the detection framework works. Figure 2: Structure of one step of the normalizing flow. ### III-A Activation Normalization To accelerate the training of the deep neural network, we adopt a batch normalization technique [45] and employ an activation normalization layer [41] to perform inversely affine transformation of activations. The activation normalization at the $k$-th layer utilizes trainable scale and bias parameters for each channel, and the corresponding initial value depends on the initial mini-batch of input data $\bm{h}_{k-1}$ from the prior layer $\displaystyle\bm{s}_{k}$ $\displaystyle=\frac{1}{\sqrt{\mathbb{V}\left[\bm{h}_{k-1}\right]}},$ (19) $\displaystyle\bm{b}_{k}$ $\displaystyle=-\mathbb{E}\left[\bm{h}_{k-1}\right],$ (20) where $\mathbb{E}[\cdot]$ and $\mathbb{V}[\cdot]$ denote the expectation and variance, respectively. Once the initialization is performed, the scale and bias are treated as trainable parameters and then the activation normalization layer output is $\displaystyle\bm{h}_{k}=\bm{h}_{k-1}\odot\bm{s}_{k}+\bm{b}_{k},$ (21) with $\odot$ being the element-wise multiplication on the channel axis. As explained in [40] and [41], the Jacobian of the transformation of coupling layers depends on the associated diagonal matrix. Due to that the determinant of a triangular matrix is equal to the product of the diagonal elements, the corresponding log-determinant can be easily computed. Specifically, the log- determinant of (21) is given by $\displaystyle\log\left\|\det\left(\frac{\mathrm{d}\bm{h}_{k}}{\mathrm{d}\bm{h}_{k-1}}\right)\right\|=M\text{sum}\left(\log\lvert\sqrt{\bm{s}_{k}}\rvert\right),$ (22) where the operator $\text{sum}(\cdot)$ means sum over all the elements of a tensor. ### III-B Invertible $1\times 1$ Convolution In order to improve the model flexibility, we employ an invertible $1\times 1$ convolutional layer. This can incorporate the permutation into the deep neural network, while it does not change the channel size and it can be treated as a generalized permutation operation [41]. For an invertible $1\times 1$ convolution with a $2\times 2$ learnable weight matrix $\bm{W}_{k}$, the log- determinant is computed directly as [41] $\displaystyle\log\left\|\det\left(\frac{\mathrm{d}\bm{h}_{k}}{\mathrm{d}\bm{h}_{k-1}}\right)\right\|=M\log\lvert\det(\bm{W}_{k})\rvert.$ (23) In order to construct an identity transformation at the beginning, we set the weight $\bm{W}_{k}$ to be a random rotation matrix with a zero log-determinant at initialization. ### III-C Alternating Affine Coupling Layer Affine coupling layer is a powerful and learnable reversible function, since its log-determinant is computationally tractable[39, 40]. For an $M$-dimensional input $\bm{h}_{k-1}$, the affine coupling layer will change some part of the input based on another part of the input. To do this, we first separate the input into two parts, which is given by $\displaystyle\bm{q}_{k}$ $\displaystyle=\bm{h}_{k-1}(1\colon m),$ (24a) $\displaystyle\bm{u}_{k}$ $\displaystyle=\bm{h}_{k-1}(m+1\colon M),$ (24b) where $\bm{h}_{k-1}(1\colon m)$ represents the first part of the input, and $\bm{h}_{k-1}(m+1\colon M)$ denotes the rest. After that, we couple the two parts together in some order $\displaystyle\bm{s}_{k}$ $\displaystyle=\mathcal{G}(\bm{q_{k}})\quad\mbox{or}\quad\mathcal{G}(\bm{u_{k}}),$ (24c) $\displaystyle\bm{b}_{k}$ $\displaystyle=\mathcal{H}(\bm{q_{k}})\quad\mbox{or}\quad\mathcal{H}(\bm{u_{k}}),$ (24d) $\displaystyle\bm{h}_{k}$ $\displaystyle=\bm{h}_{k-1},$ (24e) $\displaystyle\bm{h}_{k}({m+1\colon M})$ $\displaystyle=\bm{u}_{k}\odot\bm{s}_{k}+\bm{b}_{k},$ (24f) $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ represent two learnable neural networks, which dynamically and nonlinearly compute the corresponding scale and bias. Similarly, the determinant of a triangular matrix is again equal to the product of the diagonal elements [40], which indicates that we can readily compute the corresponding log-determinant as $\displaystyle\log\left\|\det\left(\frac{\mathrm{d}\bm{h}_{k}}{\mathrm{d}\bm{h}_{k-1}}\right)\right\|=\text{sum}\left(\log\lvert\bm{s}_{k}\rvert\right).$ (25) Basically, the architecture of our flow steps is developed based on the implementations suggested in [39, 40, 41]. However, there are two key differences between our architecture and them. One key difference is that our implementation can handle the complex-valued input by using the squeeze layer and the unsqueeze layer, in order to separate and recover the real parts and imaginary parts. Another key difference is that we just need to ensure the existence of inversions rather than investigating their exact analytical forms, since there is no need to draw samples from the latent space in signal detection. This difference leads to more generalities and flexibilities enabled in our implementation. Specifically, as shown in Fig. 2, we alternatively combine two affine coupling layers together in a flow step, and they can change any part of the input without considering whether it has been changed or not at the prior coupling layer. As a contrast, the existing implementations suggested in [39, 40, 41] must change the part that has not been changed yet by the prior layer. Therefore, we incorporate a powerful alternating pattern into the network to help enhance the generality and flexibility, and eventually improve the network’s ability to approach unknown distributions. ### III-D Signal Detection Input: Received signal $\bm{y}$, channel matrix $\bm{H}$, Output: Recovered signal $\bm{x}^{}$ 1 foreach _$\bm{x}_{i}\in\mathcal{X}$_ do 2 $\bm{w}_{i}\leftarrow\bm{y}-\bm{H}\bm{x}_{i}$ 3 $\bm{z}_{i}\leftarrow f(\bm{w}_{i})$ 4 $\mathcal{L}_{\bm{x}_{i}}\leftarrow\log p(\bm{z}_{i};\bm{\theta})+\log\left\|\det\left(\frac{\mathrm{d}\bm{z}_{i}}{\mathrm{d}\bm{w}_{i}}\right)\right\|$ 5 end foreach 6$\bm{x}^{}\leftarrow\arg\underset{\bm{x}_{i}}{\max}(\mathcal{L}_{\bm{x}_{i}})$ return $\bm{x}^{}$ Algorithm 1 MANFE As discussed above, by jointly using the change of latent variables and the nonlinearity of neural networks, the approximate distribution parameterized by a normalizing flow can be highly flexible to approach the unknown true distribution. In this case, we are able to perform MLE in (5) by evaluating the log-likelihood through the normalizing flow. Accordingly, we devise a signal detection algorithm for the proposed detection framework. In contrast to E-MLE, the proposed detection framework estimates the signal by finding the maximum likelihood computed from a normalizing flow, so that we call it _maximum a normalizing flow estimate_ (MANFE). Specifically, in the MANFE algorithm, we first evaluate the corresponding noise vector $\bm{w}_{i}$ given received signal $\bm{y}$ and channel maxtrix $\bm{H}$ for each possible signal vector $\bm{x}_{i}\in\mathcal{X}$. Then, the algorithm maps the noise vector $\bm{w}_{i}$ into the latent space to infer the corresponding latent variable $\bm{z}_{i}$. After that, we compute the corresponding log-likelihood by evaluating $\displaystyle\mathcal{L}_{\bm{x}_{i}}$ $\displaystyle=\log p(\bm{y}\|\bm{x}_{i})$ (26) $\displaystyle\approx\log q(\bm{w}_{i};\bm{\theta})$ (27) $\displaystyle=\log p(\bm{z}_{i};\bm{\theta})+\log\left\|\det\left(\frac{\mathrm{d}\bm{z}_{i}}{\mathrm{d}\bm{w}_{i}}\right)\right\|.$ (28) Finally, by finding the most possible signal vector which has the maximum log- likelihood, we get the MLE of desired signal as $\displaystyle\bm{x}^{}=\arg\underset{\bm{x}_{i}\in\mathcal{X}}{\max}(\mathcal{L}_{\bm{x}_{i}}).$ (29) The whole procedures of the MANFE algorithm is summarized in Algorithm 1. Intuitively, the major difference between MANFE and E-MLE is that MANFE is a generalized ML estimator which approximates the unknown noise distributions and thereby compute the log-likelihoods accordingly in different noise environments. On the contrary, E-MLE is only designed for Gaussian noises so that it can be treated as a perfectly trained MANFE under specific noise environment, which results that it loses flexibilities with comparison to MANFE. ### III-E Low-Complexity Version of MANFE As the MLE is an NP-hard problem, we have to exhaust all possible candidates to check out which one has the maximum probability. In particular, the computational complexity of MLE is $\mathcal{O}(P^{N})$, which indicates that the complexity increases exponentially with the constellation size $P$ and the antenna number $N$. Hence, it is difficult to implement a perfect MLE in practice. To solve this problem, some empirical approaches have been proposed to reduce the complexity of MLE, by utilizing an initial guess to reduce the searching space [46]. The initial guess can be estimated by some kind of low-complexity detectors, such as ZF detector, MMSE detector, and GAMP detector. Though the selection range of low-complexity detectors is broadly wide, we should choose a detector which has a lower complexity and fine bit error rate (BER) performance. From this viewpoint and in order to reduce the complexity of MLE, we propose to jointly use the low-complexity G-GAMP detection algorithm and the MAFNE, which is named as G-GAMP-MANFE. Specifically, in the G-GAMP-MANFE algorithm, we first get an initial estimate came from G-GAMP algorithm, where the details about the GAMP algorithm can be found in the existing works such as [8, 9, 13]. Since the initial guess is approximate, we can assume that there exist at most $E$ ($0\leq E\leq N$) error symbols at the initial guess. Accordingly, we will only require to compare $\sum_{i=0}^{E}C_{N}^{i}(P-1)^{i}$ signal candidates instead of $P^{N}$ ones. The number of error symbols can be sufficiently small so that the total complexity can be reduced significantly, especially when the channel is in good condition. For example, we only need to compare $1+N(P-1)$ candidates when $E=1$. Hence, the searching space is reduced as well as the total complexity of MANFE. The whole procedures of the G-GAMP-MANFE algorithm is summarized in Algorithm 2, as shown at the top of the next page. Basically, the low-complexity version of MANFE is a generalized framework that helps improve the detection performance of other low-complexity detectors under unknown noise environments. In this paper, we use G-GAMP detector for the initial estimation as its performance and complexity are both acceptable under most common scenarios [11]. Obviously, the BER performance of the low- complexity MANFE depends on two factors. One is that the choice of initial detector significantly affects the BER performance of the low-complexity MANFE, while the other is the choice of $E$. If the problem scale is not too large and we can tolerate a high complexity, we can increase $E$ to improve the BER performance. On the other hand, if the systems are sensitive to the computational complexity, it would be better to maintain $E$ at a lower level. In other worlds, the choice of $E$ is quite flexible and users would set the value of $E$ based on their specific needs. In particular, as an iterative detection algorithm, G-GAMP-MANFE’s convergence mainly depends on the G-GAMP, whose convergence can be guaranteed when the noise is Gaussian and the channel matrix is a large i.i.d. sub-Gaussian matrix, where the details can be found in the literature such as [14, 13, 15]. Input: Received signal $\bm{y}$, channel matrix $\bm{H}$ Output: Recovered signal $\bm{x}^{}$ 1 Get an initial estimate $\bm{x}_{0}$ from G-GAMP algorithm 2 Get a subset $\mathcal{X}_{E}$ from $\mathcal{X}$ where there exist at most $E$ different symbols between a possible signal candidate and the initial estimate $\bm{x}_{0}$ 3 foreach _$\bm{x}_{i}\in\mathcal{X}_{E}$_ do 4 $\bm{w}_{i}\leftarrow\bm{y}-\bm{H}\bm{x}_{i}$ 5 $\bm{z}_{i}\leftarrow f(\bm{w}_{i})$ 6 $\mathcal{L}_{\bm{x}_{i}}\leftarrow\log p(\bm{z}_{i};\bm{\theta})+\log\left\|\det\left(\frac{\mathrm{d}\bm{z}_{i}}{\mathrm{d}\bm{w}_{i}}\right)\right\|$ 7 end foreach 8$\bm{x}^{}\leftarrow\arg\underset{\bm{x}_{i}}{\max}(\mathcal{L}_{\bm{x}_{i}})$ return $\bm{x}^{}$ Algorithm 2 Combine G-GAMP With MANFE (G-GAMP-MANFE) ### III-F Complexity Analysis In this part, we provide some analysis on the computational complexity for the MANFE and G-GAMP-MANFE algorithms. There are three kinds of subfunction in the MANFE detection framework and all these subfunctions operate in an element- wise manner. Accordingly, the computational complexity of a flow step depends on the element-wise summation of log-determinant, which is about $\mathcal{O}(M)$ for a single flow step. Therefore, the computational cost to compute the log-likelihood of a possible signal vector $\bm{x}_{i}$ depends on the matrix multiplication $\bm{H}\bm{x}_{i}$ and the cost of $K$ flow steps, which is about $\mathcal{O}(KM+MN)$. Hence, the total computational complexity for MANFE can be expressed as $\mathcal{O}((KM+MN)P^{N})$, for which we have to exhaust all $P^{N}$ possible signal candidates. As to the G-GAMP-MANFE algorithm, since the G-GAMP has a computational complexity of $\mathcal{O}(T(M+NP))$ for $T$ iterations and we have to compare $\sum_{i=0}^{E}C_{N}^{i}(P-1)^{i}$ candidates, the computational complexity for the G-GAMP-MANFE is $\mathcal{O}\big{(}T(M+NP)+(KM+MN)\sum_{i=0}^{E}C_{N}^{i}(P-1)^{i}\big{)}$. More specifically, when $E=1$, the complexity of the G-GAMP-MANFE is only of $\mathcal{O}\big{(}T(M+NP)+(KM+MN)(1+N(P-1))\big{)}$, which indicates that it can be easily implemented in practical MIMO systems. ## IV Simulations and Discussions In this section, we perform simulations to verify the effectiveness of the proposed detection framework. In particular, we first introduce the environment setup of these simulations as well as the implementation details of the deep neural network, and then, we present some simulation results and give the related discussions. ### IV-A Environment Setup The simulations are performed in a MIMO communication system in the presence of several typical additive non-Gaussian noises, such as Gaussian mixture noise, Nakagami-$m$ noise, and impulsive noise. The numbers of antennas are $N$ and $M$ at the transmitter and the receiver, respectively. The modulation scheme is quadrature phase shift keying (QPSK) with $P=4$, and the channel experiences Rayleigh flat fading. The receiver has perfect knowledge on the channel state information (CSI). To model the impulsive noise, we employ a typical impulsive model, named symmetric $\alpha$-stable (S$\alpha$S) noise[7, 6, 3, 4]. In particular, an S$\alpha$S random variable $w$ has the following characteristic function $\displaystyle\psi_{w}(\theta)=\mathbb{E}[e^{jw\theta}]=e^{-\sigma^{\alpha}\lvert\theta\rvert^{\alpha}},$ (30) where $\mathbb{E}[\cdot]$ represents the statistical expectation, $\sigma>0$ is the scale exponent, and $\alpha\in(0,2]$ is the characteristic exponent. When $\alpha$ decreases, the noise becomes heavy-tailed and impulsive. For practical scenarios, $\alpha$ usually falls into $[1,2]$. Especially, the S$\alpha$S distribution turns into a Cauchy distribution when $\alpha=1$, while it is a Gaussian distribution when $\alpha=2$. The density function $f_{w}(w)$ can be expressed by [7] $\displaystyle f_{w}(w)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{-\lvert\theta\rvert\sigma^{\alpha}-j\theta w}\mathrm{d}\theta.$ (31) Unfortunately, we can only compute the approximate density through numerical methods since $f_{w}(w)$ does not have a closed-form expression when $\alpha\in(1,2)$. In other words, the exact MLE under S$\alpha$S noise is computationally intractable and the performance of E-MLE will severely deviate from the situation of Gaussian noise. In particular, we mix two Gaussian distributed noises subject to $\mathcal{CN}(-\mathbf{I},2\mathbf{I})$ and $\mathcal{CN}(\mathbf{I},\mathbf{I})$ equably as the instance of the Gaussian mixture noise. Notice that any statistical knowledge on these noises such as the value of $\alpha$ which indicates the impulse level is not utilized during the training and testing processes, which can simulate the situation that the noise statistics is unknown. ### IV-B Training Details In the proposed framework, the hyper-parameters that need to be chosen carefully are concluded as follows: • The total number of flow steps $K$, * • The specified partition parameter $m$ in alternative affine coupling layers, * • The specified structure of the two neural networks $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ in alternative affine coupling layers. Obviously, $K$ significantly affects the complexity and the effectiveness of our methods, and we find that $K=4$ is a good choice based on the experiences. As $\bm{q}_{k}$ is often the half part of the input $\bm{h}_{k-1}$, we can set the partition parameter $m$ to $\frac{M}{2}$. Additionally, the scale and bias functions $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ are both implemented by a neural network with three fully-connected layers, where the activation functions are rectified linear unit (ReLU) functions and the hidden layer sizes are all constantly set to $8$ for both $4\times 4$ and $8\times 8$ MIMO systems. In practice, we consider that the latent variables follow a multivariate Gaussian distribution with trainable mean $\bm{\mu}$ and variance $\bm{\Sigma}$. Therefore, the trainable parameters of the proposed framework can be summarized below: * • The scale vectors $\bm{s}_{k}$ and bias vectors $\bm{b}_{k}$ for each activation normalization layers introduced in (19), * • The learnable weight matrix $\bm{W}_{k}$ for each $1\times 1$ convolution layer introduced in (23), * • The network parameters in $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ for each affine coupling layer introduced in (24), * • The mean $\bm{\mu}$ and variance $\bm{\Sigma}$ of the latent variable’s multivariate Gaussian distribution. Hence, we can find that there are only a handful of trainable parameters inside a single flow step and the total number of flow steps is small too. This indicates that the training overhead of our framework is affordable. Specifically, we use the TensorFlow framework [47] to train and test the model, and there are $10$ millions training noise samples and $2$ millions test noise samples generated from the aforementioned noise models in the training phase. Since our model is driven by unsupervised learning, the data sets only consist of noise samples. To refine the trainable parameters, (7) is adopted as the loss function, and the Adam optimizer [48] with learning rate set to $0.001$ is adopted to update the trainable parameters by using the gradient descent method. Furthermore, engineering details and reproducible implementations can be found in our open-source codes located at https://github.com/skypitcher/manfe for interested readers. TABLE I: Computational Complexity of Competing Algorithms Abbreviations \| Complexity ---\|--- G-GAMP($T$) \| $\mathcal{O}\left(T(N+M)\right)$ DetNet($T$) \| $\mathcal{O}\left(T(N^{2}+(3N+2L_{z})L_{h})\right)$ 111$L_{z}$ and $L_{h}$ are the size of a latent parameter and the size of hidden layers, which are both suggested to be $2N$ in the paper of DetNet [30], respectively. E-MLE \| $\mathcal{O}\left(MNP^{N}\right)$ MANFE \| $\mathcal{O}\left((KM+MN)P^{N}\right)$ G-GAMP($T$)-MANFE($E$) \| $\mathcal{O}\left(T(M+NP)+(KM+MN)\sum_{i=0}^{E}C_{N}^{i}(P-1)^{i}\right)$ TABLE II: Computational Complexity of Competing Algorithms in $4\times 4$ QPSK Modulated MIMO Systems Algorithm \| Complexity ---\|--- G-GAMP($30$) \| $\mathcal{O}\left(480\right)$ DetNet($30$) \| $\mathcal{O}(28800)$ E-MLE \| $\mathcal{O}(16384)$ MANFE \| $\mathcal{O}(24576)$ G-GAMP($30$)-MANFE($2$) \| $\mathcal{O}(7152)$ ### IV-C Comparison with Relevant Algorithms In order to verify the effectiveness of the proposed detection framework, we compare the proposed methods with various competitive algorithms. For convenience, we use the following abbreviations, * • G-GAMP($T$): Gaussian GAMP algorithm with $T$ iterations. See [8, 9, 13] for more details. * • DetNet($T$): Deep learning driven and projected gradient descent (PGD) based detection network introduced in [30] with $T$ layers (iterations). * • E-MLE: Euclidean distance based maximum likelihood estimate introduced in (6). * • MANFE: Maximum a normalizing flow estimate introduced in Section III-D. * • G-GAMP($T$)-MANFE($E$): A low-complexity version of MANFE combined by the G-GAMP algorithm with $T$ iterations and the assumption that there exist at most $E$ error symbols at the initial guess came from the G-GAMP algorithm. See Section III-D and III-E for more information. * • G-GAMP($T$)-E-MLE($E$): As similar to the G-GAMP-MANFE, the G-GAMP-E-MLE is a low-complexity version of E-MLE combined by the G-GAMP algorithm with $T$ iterations and the assumption that there are at most $E$ error symbols at the initial guess came from the G-GAMP algorithm. For the purpose of complexity comparison, we provide two tables to present the complexity comparison among the competing algorithms. Specifically, Table I lists the theoretical computational complexity of the competing algorithms, while Table II presents the corresponding complexities in $4\times 4$ QPSK modulated MIMO systems. From these two tables, we can find that although the computational complexity of the proposed MANFE is slightly higher than that of the conventional E-MLE, the complexity of the proposed G-GAMP-MANFE is affordable with a fine detection performance. ### IV-D Simulation Results and Discussions Fig. 3 demonstrates the detection BER performances of the aforementioned detection methods, where the QPSK modulated $4\times 4$ MIMO system is used with SNR$=25$ dB and $\alpha$ varies from $1$ to $2$. In particular, $\alpha=1$ and $\alpha=2$ correspond to the typical Cauchy and Gaussian distributed noise, respectively. We can find from Fig. 3 that the proposed MANFE outerperforms the other several detectors in the impulsive noise environments, in the terms of BER performance. Specifically, when $\alpha=1.9$ where the noise is slightly impulsive and deviates a little from the Gaussian distribution, the MANFE can significantly reduce the detection error of E-MLE to about $1.1$%. This indicates that the MANFE can compute an effective log- likelihood with contrast to E-MLE, especially when the noise is non-Gaussian with unknown statistics. More importantly, when $\alpha$ changes from $2.0$ to $1.9$ associated with a little impulsiveness, the E-MLE meets a severe performance degradation, whereas the MANFE has a relatively slight performance dagradation. This indicates that the MANFE is robust to the impulsive noise compared with E-MLE. In further, when the impulsive noise approaches to the Gaussian distribution with $\alpha=2$, the MANFE achieves the performance bound of MLE, indicating that the MANFE is very flexible to approach unknown noise distribution even when the noise statistics is unknown. Figure 3: BER performance comparison versus $\alpha$ with SNR=25 dB for $4\times 4$ MIMO systems. In addition, we can observe from Fig. 3 that as a low-complexity version of MANFE, the G-GAMP-MANFE, still outperforms the other low-complexity detectors in impulsive noise environments. In particular, the G-GAMP(30)-MANFE(1) can even achieve the same performance as the E-MLE when $\alpha\leq 1.7$. In these cases, the G-GAMP(30)-MANFE(1) has a much lower computational complexity with comparison to the E-MLE, which is only about $5.08\%$222When $E=1$, the G-GAMP(30)-MANFE(1) can reduce the computational complexity of the E-MLE to about $\frac{1+N(P-1)}{P^{N}}$, which is equal to $\frac{13}{256}\approx 5.08\%$. of the E-MLE. When $\alpha\leq 1.9$, we can find that if we can tolerate a moderate computational complexity by increasing $E$ from $1$ to $2$, the G-GAMP(30)-MANFE(2) will have a better BER performance than the E-MLE, which reduces the detection error of the E-MLE to about $47.5\%$ and meanwhile decreases the computational complexity to about $26.17\%$333Similarly, when $E=2$, the G-GAMP(30)-MANFE(2) can reduce the computational complexity of the E-MLE to about $\frac{\sum_{i=0}^{2}C_{N}^{i}(P-1)^{i}}{P^{N}}$, which is equal to $\frac{67}{256}\approx 26.17\%$. with respect to the E-MLE. By comparing with the other low-complexity detectors, the G-GAMP(30)-MANFE(1) can reduce the detection error of the DetNet(30) and the G-GAMP(30) to about $39.61\%$ and $57.55\%$, respectively, when $\alpha=1.9$. These results further verify the effectiveness of the proposed detection framework. Figure 4: BER performance comparison versus SNR with $\alpha=1.9$ for $4\times 4$ MIMO systems. To verify the effectiveness of the proposed detection framework under different channel conditions and different levels of impulsive noise, Figs. 4-6 illustrate the BER performance comparison among several detectors for $4\times 4$ MIMO systems in impulsive noise environments, where SNR varies from $10$ dB to $30$ dB. Specifically, Fig. 4, Fig. 5 and Fig. 6 are associated with $\alpha=1.9$, $\alpha=1.5$ and $\alpha=1.1$, respectively. From Figs. 4-6, we can find that the BER performance gap between the MANFE and the E-MLE enlarges when the corresponding SNR increases. For example, when $\alpha=1.9$ and the values of SNR are set to $20$ dB, $25$ dB and $30$ dB, the MANFE can reduce the detection error of E-MLE to about $16.18$%, $1.06$% and $0.1$%, respectively. In particular, for $\alpha=1.9$ when the noise is slightly impulsive, the SNR gain of the MANFE over the E-MLE is $10$ dB at the BER level of $10^{-4}$. Similarly, when the noises have moderate and strong level of impulsiveness where the corresponding values of $\alpha$ are equal to $1.5$ and $1.1$, the SNR gains of the MANFE over the E-MLE are about $10.5$ dB and $12$ dB at the BER levels of $10^{-3}$ and $10^{-2}$, respectively. This further verifies that the MANFE can perform the efficient MLE even under highly impulsive situations. Figure 5: BER performance comparison versus SNR with $\alpha=1.5$ for $4\times 4$ MIMO systems. Figure 6: BER performance comparison versus SNR with $\alpha=1.1$ for $4\times 4$ MIMO systems. Moreover, for the low-complexity detectors in Figs. 4-6, we can observe that the BER performance gap of the G-GAMP(30)-MANFE over the G-GAMP(30) and DetNet(30) enlarges with the imcreasing SNR. Specifically, when $\alpha=1.9$ where the noise is nearly Gaussian, the G-GAMP(30)-MANFE(1) can reduce the detection error of G-GMAP(30) to about $89.7$%, $63.9$% and $51.9$% at the SNR levels of $10$ dB, $15$ dB and $20$ dB, respectively. For the same situation with regards to the DetNet(30), the above results are about $97$%, $78$% and $61$%, respectively. In general, the SNR gains of the G-GAMP(30)-MANFE(1) over the G-GAMP(30) are $6$ dB, $4$ dB and $3$ dB at the BER level of $10^{-2}$ for $\alpha=1.9$, $\alpha=1.5$ and $\alpha=1.1$, respectively. In addition, the SNR gains of the G-GAMP(30)-MANFE(1) over the DetNet(30) are $9$ dB, $4$ dB and $3$ dB at the BER level of $10^{-2}$ for $\alpha=1.9$, $\alpha=1.5$ and $\alpha=1.1$, respectively. More importantly, when $\alpha=1.1$ and $\alpha=1.5$, the G-GAMP(30)-MANFE(1) has almost the same BER performance as the E-MLE in a wide range of SNR. This indicates that the G-GAMP(30)-MANFE(1) can obtain the same BER performance as the E-MLE under highly impulsive environments. In further, we can see from Fig. 4-6 that the G-GAMP(30)-MANFE(2) with a moderate computational complexity achieves the BER performance at least not worse than the E-MLE, for MIMO systems in impulsive noise environments. For example, the SNR gains of the G-GAMP(30)-MAFNet(2) over the E-MLE are about $5$ dB and $6.5$ dB at the BER level of $10^{-2}$ for $\alpha=1.5$ and $\alpha=1.1$, respectively. In particular, when the MIMO system is significantly affected by the impulsive noise, the BER performance of the G-GAMP(30)-MANFE(2) will be much better than that of the E-MLE. Figure 7: BER performance comparison versus $\alpha$ with SNR=25 dB for $8\times 8$ MIMO systems. To further verify the effectiveness of the low-complexity version of the MANFE, we use Figs. 7 \- 9 to demonstrate the BER performance of $8\times 8$ MIMO systems in impulsive noise environments. Specifically, Fig. 7 shows the BER performance versus $\alpha$ with SNR $=25$ dB, while Fig. 8 and Fig. 9 correspond to the BER performance versus SNR with $\alpha=1.9$ and $\alpha=1.7$, respectively. In these figures, we did not plot the BER performances of E-MLE and MANFE, due to that the computational complexities of these two detectors are too high to implement in practice for $8\times 8$ MIMO systems. Instead, we plot the BER performance of the G-GAMP-E-MLE for comparison. From Fig. 7, we can observe that the BER performance of G-GAMP(30)-MANFE(2) is much better than that of the other detectors when $\alpha$ varies from $1$ to $2$. Specifically, when $\alpha=1.9$ and SNR$=25$ dB, the G-GAMP(30)-MANFE(2) can sufficiently reduce the detection error of the G-GAMP(30)-E-MLE(2), G-GAMP(30) and DetNet(30) to about 70%, 30% and 21.2%, respectively. Moreover, the performance gain of G-GAMP(30)-E-MLE(2) over the G-GAMP(30) algorithm vanishes with the raise of the impulse strength, while the G-GAMP(30)-MANFE(2) still outperforms the G-GAMP(30) algorithm under significantly impulsive noise environments. In further, the G-GAMP(30)-MANFE(2) achieves the performance bound of the G-GAMP(30)-E-MLE(2) when the noise falls into Gaussian distribution. This further indicates that the proposed detection framework has the ability to effectively approximate the unknown distribution. Figure 8: BER performance comparison versus SNR with $\alpha=1.9$ for $8\times 8$ MIMO systems. Figure 9: BER performance comparison versus SNR with $\alpha=1.7$ for $8\times 8$ MIMO systems. In addition, from Figs. 8-9, we can find that the G-GAMP(30)-MANFE(2) still outperforms the other low-complexity detectors in a wide range of SNR. More specifically, when $\alpha=1.9$ where the noise is nearly Gaussian distributed and slightly impulsive, the SNR gains of the G-GAMP(30)-MANFE(2) over the G-GAMP(30)-E-MLE(2), G-GAMP(30) and DetNet(30) are $1.8$ dB, $4.5$ dB and $7$ dB at the BER level of $10^{-4}$, respectively. In the case that $\alpha=1.7$ and the noise becomes more impulsive, the SNR gains of G-GAMP(30)-MANFE(2) with respect to the G-GAMP(30)-E-MLE(2), G-GAMP(30) and DetNet(30) are $2$ dB, $4.5$ dB and $6$ dB at the BER level of $10^{-3}$, respectively. These results verify the effectiveness of the proposed framework furthermore. Figure 10: BER performance comparison versus SNR for $4\times 4$ MIMO systems with Nakagami-$m$ noises. Figure 11: BER performance comparison versus SNR for $4\times 4$ MIMO systems with Gaussian mixture noises. In order to further exam the effectiveness of the proposed detection framework under different non-Gaussian noise environments, Figs. 10-11 illustrate the BER performance comparisons among several detectors for the $4\times 4$ QPSK modulated MIMO system, where the noises are subject to the Nakagami-$m$ ($m=2$) distribution and Gaussian mixture distribution Fig. 10 and Fig. 11, respectively. In particular, the ML estimates are available and provided in the two figures, since the two distributions are both analytical. In Fig. 10, we can observe that MANFE can still achieve the optimal ML performance, while E-MLE fails to work in nakagami-$m$ noise environments. As to the low- complexity detectors, G-GAMP-MANFE can still outperform the conventional detectors in nakagami-$m$ noise environment. Similar to the results in Fig. 10, the simulation results in Fig. 11 show that MANFE can still achieve almost the optimal ML performance under the Gaussian mixture noise environment, and G-GAMP-MANFE can still outperform the conventional sub-optimal detectors. From the simulation results of Gaussian, impulsive and nakagami-$m$ distributed noises in Figs. 3-11, we can find that the proposed method can almost achieve the optimal ML performance under various analytical noises, and it also outperforms the conventional detectors under non-analytical noises, which further verified the generality, flexibility, and effectiveness of the proposed method. ## V Conclusions In this paper, we have investigated the signal detection problem in the presence of noise whose statistics is unknown. We have devised a novel detection framework to recover the signal by approximating the unknown noise distribution with a flexible normalizing flow. The proposed detection framework does not require any statistical knowledge about the noise since it is a fully probabilistic model and driven by the unsupervised learning approach. Moreover, we have developed a low-complexity version of the proposed detection framework with the purpose of reducing the computational complexity of MLE. Since the practical MIMO systems may suffer from various additive noise with some impulsiveness of nature and unknown statistics, we believe that the proposed detection framework can effectively improve the robustness of the MIMO systems in practical scenarios. Indeed, to the best of our knowledge, the main contribution of this work is that our methods are the first attempt in the literature to address the maximum likelihood based signal detection problem without any statistical knowledge on the noise suffered in MIMO systems. Nevertheless, there are still some interesting issues for future researches. 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# Kubo formulae for first-order spin hydrodynamics Jin Hu Department of Physics, Tsinghua University, Beijing 100084, China ###### Abstract We derive Kubo formulae for first-order spin hydrodynamics based on non- equilibrium statistical operators method. In first-order spin hydrodynamics, there are two new transport coefficients besides the ordinary ones appearing in first-order viscous hydrodynamics. They emerge due to the incorporation of the spin degree of freedom into fluids and the spin-orbital coupling. Zubarev’s non-equilibrium statistical operator method can be well applied to investigate these quantum effects in fluids. The Kubo formulae, based on the method of non-equilibrium statistical operators, are related to equilibrium (imaginary-time) infrared Green’s functions, and all the transport coefficients can be determined when the microscopic theory is specified. ## I Introduction Recent developments in relativistic heavy-ion collisions have seen great progress in studying observables with spin dependence. The measurements of spin polarization of $\Lambda$ hyperons show that a fraction of the spin of quarks within the hyperons takes one particular direction Adamczyk et al. (2017); Alpatov (2020), which implies the media, quark-gluon plasma (QGP), should carry a large magnitude of angular momentum. Such a significant magnitude of vorticity leads to the phenomenon of spin alignments as a result of the well-known spin-orbital coupling. Theoretical researches on global polarization of $\Lambda$ hyperons can be found in Wei et al. (2019); Karpenko and Becattini (2017); Csernai et al. (2019); Li et al. (2017); Bzdak (2017); Shi et al. (2019); Sun and Ko (2017); Ivanov et al. (2020); Xie et al. (2017). The results of theoretical calculations fit the data well. Later, the STAR Collaboration published the measurements of differential spin polarization, namely, the dependence of $\Lambda$ polarization on the azimuthal angle and transverse momentum Adam et al. (2019, 2018). However, theoretical calculation can not provide satisfying explanation to experimental data, which is usually called “spin sign problem” Becattini and Karpenko (2018); Xia et al. (2018) , also see Liu and Huang (2020) and Gao et al. (2020) for a review. To resolve this problem, new theoretical frameworks are necessary. One promising framework is hydrodynamics with the spin degree of freedom included. In other words, these direct experimental measurements of quantum effects in relativistic heavy-ion collisions motivate the incorporation of the quantum spin degree of freedom into the evolution of fluids. To well describe the macroscopic dynamics of spin, it is intuitive to generalize ordinary hydrodynamics, making it a spinful one. There are many efforts following this direction. “Ideal” relativistic hydrodynamics with spin freedom was proposed in the context of the QGP Florkowski et al. (2018). Some relevant discussions can also be seen in Becattini and Tinti (2010); Montenegro et al. (2017); Florkowski et al. (2019). Recently, viscous spin hydrodynamics has also been put into consideration Hattori et al. (2019); Fukushima and Pu (2020a). In these works, two new transport coefficients arise reflecting new physical effects with spin freedom, which will be the main focus of our interest herein. Transport coefficients are important quantities manifesting the transport property of the medium in the field of heavy-ion collisions. In the case of spin hydrodynamics, new transport coefficients well capture the property of slow dynamics in the spinful fluid system Hattori et al. (2019). The usual dissipative hydrodynamics have been widely used to describe the collective behavior of the QGP in the last decade. Once the spin degree of freedom was considered, as mentioned in Liu and Huang (2020) , a numerical implementation of causal spin hydrodynamics is a promising one, particularly, it is hopeful to get insight into the “spin sign problem”. For the purpose of quantitatively describing the evolution of the fluid system, transport coefficients are indispensable inputs. There are many methods for calculating the transport coefficients. The kinetic theory based on transport equation offers us an effective tool for investigating transport properties Xu et al. (2008); Xu and Greiner (2008). Noting that transport methods, for example, Boltzmann equation, rely on the picture of quasi-particle, so we are supposed to apply them with caution. An alternative approach is based on the Kubo method, in which the correlation functions represent the response of an equilibrated system to a perturbation. Based on this method transport coefficients are directly linked to real-time retarded Green’s functions, which can be evaluated by analytic continuation of equilibrium Green’s functions formulated in imaginary time. In this paper, we utilize the non-equilibrium statistical operator method developed by Zubarev Zubarev (1974); Hosoya et al. (1984); Horsley and Schoenmaker (1987) to derive Kubo formulae for transport coefficients of relativistic spinful fluids. The non-equilibrium statistical operator is a generalization of the equilibrium Gibbs statistical operator to non- equilibrium states. Using this approach, we are able to separate naturally the equilibrium part and the non-equilibrium part which takes the form of gradients of the thermodynamical parameters. From the viewpoint of linear response, transport coefficients can be obtained from linear perturbations of the non-equilibrium statistical operator around its equilibrium expectation value. This paper is organized as follows. In Sec. II we present a brief review of relativistic spin hydrodynamics in dissipative cases based on Hattori et al. (2019); Fukushima and Pu (2020a). In Sec. III we adopt the non-equilibrium statistical operator method to derive Kubo formulae in first-order spin hydrodynamics, which relates transport coefficients to retarded correlation functions defined in terms of the underlying elementary fields. Discussion and outlook are given in Sec. IV. Natural units $\hbar=k_{B}=c=1$ are used. The metric tensor here is given by $g^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$, while $\Delta^{\mu\nu}\equiv g^{\mu\nu}-u^{\mu}u^{\nu}$ is the projection tensor orthogonal to the four-vector fluid velocity $u^{\mu}$. In addition, we employ the symmetric/antisymmetric shorthand notations: $\displaystyle X^{(\mu\nu)}$ $\displaystyle\equiv$ $\displaystyle(X^{\mu\nu}+X^{\nu\mu})/2,$ (1) $\displaystyle X^{[\mu\nu]}$ $\displaystyle\equiv$ $\displaystyle(X^{\mu\nu}-X^{\nu\mu})/2,$ (2) $\displaystyle X^{\langle\mu\nu\rangle}$ $\displaystyle\equiv$ $\displaystyle\bigg{(}\frac{\Delta^{\mu}_{\alpha}\Delta^{\nu}_{\beta}+\Delta^{\nu}_{\alpha}\Delta^{\mu}_{\beta}}{2}-\frac{\Delta^{\mu\nu}\Delta_{\alpha\beta}}{3}\bigg{)}X^{\alpha\beta}.$ (3) ## II Review of first order spin hydrodynamics Hydrodynamics is based on basic conservation laws Landau and Lifshitz (1987), which are conservation of the energy-momentum $T^{\mu\nu}$ and conserved current $N^{\mu}$ for the spinless case, $\displaystyle\partial_{\mu}T^{\mu\nu}=0\,,$ (4) $\displaystyle\partial_{\mu}N^{\mu}=0\,.$ (5) Problem comes when we need to take into account the spin degree of freedom. Spin angular momentum plays a big role in the evolution of spinful fluids, and one needs to refer to another conserve law: the conservation of total angular momentum, which is expressed as: $\displaystyle\partial_{\lambda}\Sigma^{\lambda\mu\nu}=0\,.$ (6) Microscopically, the rank three tensor $\Sigma^{\lambda\mu\nu}$ can be decomposed into two distinct components by calculating Noether current for the Lorentz symmetry which reads: $\Sigma^{\mu\alpha\beta}=(x^{\alpha}T^{\mu\beta}-x^{\beta}T^{\mu\alpha})+S^{\mu\alpha\beta}$, where $S^{\mu\alpha\beta}=-S^{\mu\beta\alpha}$ Fukushima and Pu (2020b). $S^{\mu\alpha\beta}$ arises from the invariance with respect to the representation of the Lorentz group acting on a field under consideration, and is naturally identified with spin angular momentum. On the other hand, the orbital angular momentum part comes from the coordinate transformation of the argument of the field. Here $T^{\mu\nu}$ is the canonical energy-momentum tensor, featured as having both symmetric and antisymmetric components: $T^{\mu\nu}\equiv T^{\mu\nu}_{(s)}+T^{\mu\nu}_{(a)}$. In this work, we will keep using canonical form of energy-momentum tensor. Discussions about details of pseudo gauge transformed form, for instance, Belinfante form can be seen in Fukushima and Pu (2020a). Explicitly (6) can be rewritten as: $\displaystyle\partial_{\lambda}S^{\lambda\mu\nu}=T^{\nu\mu}-T^{\mu\nu}\,.$ (7) First, recalls that the thermodynamic relation in equilibrium as well as the first law of thermodynamics, which read as: $\displaystyle Ts+\mu n=e+p-\omega_{\mu\nu}S^{\mu\nu},$ (8) $\displaystyle Tds+\mu dn=de-\omega_{\mu\nu}dS^{\mu\nu},$ (9) $\displaystyle sdT+nd\mu=dp-S^{\mu\nu}d\omega_{\mu\nu},$ (10) where $T$, $s$, $\mu$, $n$, $e$, and $p$ denote the local temperature, entropy density, chemical potential, conserved charge density, energy density, and pressure, respectively. In this paper, we consider only one conserved charge. Here, analogous to the relation of chemical potential and charge density, a “spin potential” $\omega_{\mu\nu}$ is introduced conjugate to the spin density $S^{\mu\nu}$. And one thing that needs to be paid attention to is $\omega_{\mu\nu}=O(\partial^{1})$ in derivative expansion, and we will show the reason for this counting later. On the basis of a derivative expansion, we obtain the constitutive relations: $\displaystyle T^{\mu\nu}=eu^{\mu}u^{\nu}+p\Delta^{\mu\nu}+T^{\mu\nu}_{(1)}\,,$ (11) $\displaystyle N^{\mu}=nu^{\mu}+j^{\mu}_{(1)}\,,$ (12) $\displaystyle\Sigma^{\mu\alpha\beta}=u^{\mu}S^{\alpha\beta}+\Sigma^{\mu\alpha\beta}_{(1)}\,.$ (13) The normalization of the fluid velocity reads $u^{\mu}u_{\mu}=1$, and we also use $\nabla^{\mu}\equiv\Delta^{\mu\nu}\partial_{\nu}$, $D\equiv u^{\mu}\partial_{\mu}$ as the spatial and temporal component of derivative, respectively. It is not hard to notice that the spin density $S^{\mu\nu}$ satisfies the antisymmetric property $S^{\mu\nu}=-S^{\nu\mu}$. Accordingly, we have $\omega_{\mu\nu}=-\omega_{\nu\mu}$. The thermodynamic second law puts additional limits onto the entropy production. Following the prescription of Israel and Stewart (1979), we make assumptions about $s^{\mu}$ in the presence of spin freedom: $\displaystyle s^{\mu}$ $\displaystyle=\frac{u_{\nu}}{T}T^{\mu\nu}+\frac{p}{T}u^{\mu}s-\frac{\mu}{T}j^{\mu}-\frac{1}{T}\omega_{\alpha\beta}S^{\alpha\beta}u^{\mu}+O(\partial^{2})$ $\displaystyle=su^{\mu}+\frac{u_{\nu}}{T}T^{\mu\nu}_{(1)}-\frac{\mu}{T}j^{\mu}_{(1)}+O(\partial^{2}).$ (14) Combined with thermodynamic relation (8) and the constituent equation of energy, $u_{\nu}\partial_{\mu}T^{\mu\nu}=0$ , the entropy production is simplified as: $\partial_{\mu}s^{\mu}=-j^{\mu}_{(1)}\partial_{\mu}\frac{\mu}{T}+T^{\mu\nu}_{(1)}\partial_{\mu}\frac{u_{\nu}}{T}+\frac{1}{T}\omega_{\alpha\beta}\partial_{\mu}(S^{\alpha\beta}u^{\mu})\,.$ (15) Taking into account that $T^{\mu\nu}_{(1)}$ has the symmetric and antisymmetric part, we then have: $\displaystyle T^{\mu\nu}_{(1s)}=2h^{(\mu}u^{\nu)}+\pi^{\mu\nu}+\Pi\Delta^{\mu\nu},$ (16) $\displaystyle T^{\mu\nu}_{(1a)}=2q^{[\mu}u^{\nu]}+\tau^{\mu\nu},$ (17) where $\pi^{\mu\nu}$ and $\Pi$ represent shear stress tensor and bulk viscous pressure, and $h^{\mu}$ is heat flow. Meanwhile, $\tau^{\mu\nu}$ and $q^{\mu}$ are antisymmetric counterparts of $\pi^{\mu\nu}$ and $h^{\mu}$, respectively. These five quantities are all of the first order in gradient expansion. One can further find $\pi^{\mu\nu}=\pi^{\nu\mu}$, $\tau^{\mu\nu}=-\tau^{\nu\mu}$, and $h^{\mu}u_{\mu}=q^{\mu}u_{\mu}=\tau^{\mu\nu}u_{\nu}=\pi^{\mu\nu}u_{\nu}=0$. Putting $T^{\mu\nu}_{(1)}$ into $\partial_{\mu}s^{\mu}$, and neglecting the terms of $O(\partial^{3})$, we obtain full form of the entropy production in first-order spin hydrodynamics: $\displaystyle\partial_{\mu}s^{\mu}$ $\displaystyle=\Big{(}h^{\mu}-\frac{e+p}{n}j^{\mu}_{(1)}\Big{)}\frac{n}{e+p}\nabla_{\mu}\frac{\mu}{T}+\frac{\pi^{\mu\nu}}{T}\partial_{\langle\mu}u_{\nu\rangle}$ $\displaystyle\ -\frac{\Pi}{T}\theta+q^{\mu}\Big{(}-\frac{u\cdot\partial}{T}u_{\mu}+\partial_{\mu}\frac{1}{T}+\frac{4\omega_{\mu\nu}u^{\nu}}{T}\Big{)}$ $\displaystyle\ +\tau^{\mu\nu}\Big{[}\Delta_{\mu\rho}\Delta_{\nu\sigma}\Big{(}\partial^{\rho}\frac{u^{\sigma}}{T}-\partial^{\sigma}\frac{u^{\rho}}{T}\Big{)}+2\frac{\omega_{\mu\nu}}{T}\Big{]},$ (18) where the notation $\theta=\partial_{\mu}u^{\mu}$ is used. Noting that, when the system is in equilibrium, the entropy production must cease and we obtain $\omega_{\mu\nu}=-\frac{T}{2}\omega^{th}_{\mu\nu}$ with the thermal vorticity $\omega^{th}_{\mu\nu}=\Delta_{\mu\rho}\Delta_{\nu\sigma}(\partial^{\rho}\frac{u^{\sigma}}{T}-\partial^{\sigma}\frac{u^{\rho}}{T})$ Becattini (2012); Becattini et al. (2019). According to this argument, the counting of $\omega_{\mu\nu}$ can be estimated as $O(\partial^{1})$ assuming the system is not far away from global equilibrium. Following the routine of first-order hydrodynamics, we impose the sufficient conditions of semipositive entropy production $\partial_{\mu}s^{\mu}\geq 0$, this is, cast every term into positive semidefinite quadratic form so that the entropy production can be seen as a sum of squares. Therefore we have: $\displaystyle\pi^{\mu\nu}=2\eta\nabla^{\langle\mu}u^{\nu\rangle},$ (19) $\displaystyle\Pi=-\zeta\theta,$ (20) $\displaystyle h^{\mu}-\frac{e+p}{n}j^{\mu}_{(1)}=-\kappa\frac{nT}{e+p}\nabla^{\mu}\frac{\mu}{T},$ (21) $\displaystyle\tau^{\mu\nu}=2\gamma\big{(}\omega_{th}^{\mu\nu}+2\Delta^{\mu\rho}\Delta^{\nu\sigma}\omega_{\rho\sigma}\big{)},$ (22) $\displaystyle q^{\mu}=\lambda\big{(}Du^{\mu}+\frac{\nabla^{\mu}T}{T}-4\omega^{\mu\nu}u_{\nu}\big{)},$ (23) $\eta$, $\zeta$ and $\kappa$ represent shear viscosity, bulk viscosity and heat conductivity respectively, $\gamma$ and $\lambda$ are new transport coefficients of spin hydrodynamics, which are identified as “rotational viscosity” in de Groot and Mazur (2011) and “boost heat conductivity” in Hattori et al. (2019). In the next section, we will derive Kubo formulae for these transport coefficients. ## III Non-equilibrium Statistical Operators and Kubo formulae The method of non-equilibrium statistical operators (NESO) developed by Zubarev starts from constructing a statistical ensemble encoding thermodynamic information of the macroscopic state of the system in non-equilibriumn state. In present case we consider the system in the hydrodynamic regime which is near local equilibrium and thermodynamic parameters such as temperature and chemical potentials can be well defined locally. See Hosoya et al. (1984) and Huang et al. (2011) for reference about NESO. Following Zubarev’s practice, the form of NESO in the textbook is Zubarev (1974); Hosoya et al. (1984); Huang et al. (2011) $\displaystyle\hat{\rho}(t)=Q^{-1}\exp\left[-\int d^{3}\bm{x}\,\hat{Z}(\bm{x},t)\right],$ (24) $\displaystyle Q=\mathop{\mathrm{Tr}}\exp\left[-\int d^{3}\bm{x}\,\hat{Z}(\bm{x},t)\right],$ (25) where the operator $\hat{Z}$ is defined as $\displaystyle\hat{Z}(\bm{x},t)=\epsilon\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{[}\beta^{\nu}(\bm{x},t^{\prime})\hat{T}_{0\nu}(\bm{x},t^{\prime})$ $\displaystyle\ \ \qquad-\alpha(\bm{x},t^{\prime})\hat{N}^{0}(\bm{x},t^{\prime})-\frac{1}{2}\omega_{\rho\sigma}(\bm{x},t^{\prime})\hat{S}^{0\rho\sigma}(\bm{x},t^{\prime})\Big{]},$ (26) with $\epsilon\rightarrow+0$ after taking thermodynamic limit. Here we have introduced new Lagrange multiplier $\omega^{\rho\sigma}(\bm{x},t)$ and the operator $\hat{S}^{0\rho\sigma}$ coupled to it, which can be understood as the incorporation of total angular momentum. From the form of total angular momentum $\Sigma^{\mu\alpha\beta}$, we can deduce that the conservation condition of total angular momentum brings only new information in the part of spin angular momentum (the information of orbital part can be reproduced by energy-momentum tensor). More details can be found in Becattini et al. (2019). Other Lagrange multipliers are written explicitly as: $\displaystyle\beta^{\nu}(\bm{x},t)=\beta(\bm{x},t)u^{\nu}(\bm{x},t),$ (27) $\displaystyle\alpha(\bm{x},t)=\beta(\bm{x},t)\mu(\bm{x},t).$ (28) The parameters $\beta$ stand for the inverse local equilibrium temperature. We need to identify these parameters in the language of statistical operators, which will be deferred below. We here express three local conservation laws with statistical operators: $\displaystyle\partial_{\mu}\hat{T}^{\mu\nu}=0\;,\;\;\partial_{\mu}\hat{N}^{\mu}=0\;,\;\;\partial_{\mu}\hat{S}^{\mu\rho\sigma}=\hat{T}^{\sigma\rho}-\hat{T}^{\rho\sigma}.$ (29) Integrating Eq.(III) by parts and utilizing Eq.(29), we can get: $\displaystyle\int d^{3}\bm{x}\,\hat{Z}(\bm{x},t)=\hat{A}-\hat{B},$ (30) $\displaystyle\hat{A}=\int d^{3}\bm{x}\Big{[}\beta^{\nu}(\bm{x},t)\hat{T}_{0\nu}(\bm{x},t)-\alpha(\bm{x},t)\hat{N}^{0}(\bm{x},t)$ $\displaystyle\quad-\frac{1}{2}\omega_{\rho\sigma}(\bm{x},t)\hat{S}^{0\rho\sigma}(\bm{x},t)\Big{]},$ (31) $\displaystyle\hat{B}=\int d^{3}\bm{x}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{[}\hat{T}_{\mu\nu}(\bm{x},t^{\prime})\partial^{\mu}\beta^{\nu}(\bm{x},t^{\prime})$ $\displaystyle\quad-\hat{N}^{\mu}(\bm{x},t^{\prime})\partial_{\mu}\alpha(\bm{x},t^{\prime})+\omega_{\mu\nu}(\bm{x},t^{\prime})\hat{T}^{[\mu\nu]}(\bm{x},t^{\prime})\Big{]}.$ (32) In deriving Eq.(30), we notice that integrating by parts will bring in three- dimensional surface integrals that are often discarded, but for temporal dimension, it is different because of definite integration upper limit. If we take $t$ to infinity, the derivative term would all vanish showing that the system should go to equilibrium given long enough evolution time. Due to the counting of $\omega_{\mu\nu}$, the term of $\partial_{\mu}\omega^{\mu\nu}$ is the order of $O(\partial^{2})$, so we neglect this term in Eq.(30). Following the spirit of non-equilibrium statistical mechanics, we treat the derivative terms as thermodynamic forces that lead to dissipation. By doing so, we are able to decompose statistical operators into the local equilibrium part and non-equilibrium part. We define the local equilibrium statistical operator as: $\displaystyle\hat{\rho}_{\rm{leq}}\equiv Q_{\rm{leq}}^{-1}\exp\Big{(}-\hat{A}\,\Big{)},$ (33) $\displaystyle Q_{\rm{leq}}=\mathop{\mathrm{Tr}}\exp\Big{(}-\hat{A}\,\Big{)},$ (34) and the complete statistical operator as: $\displaystyle\hat{\rho}\equiv Q^{-1}\exp\Big{(}-\hat{A}+\hat{B}\,\Big{)},$ (35) $\displaystyle Q=\mathop{\mathrm{Tr}}\exp\Big{(}-\hat{A}+\hat{B}\,\Big{)}.$ (36) Now comes the question of how to handle the complete statistical operator taking in the form of the exponential function of the sum of two operators. Noting $[\hat{A},\hat{B}]\neq 0$ , $[\hat{A},[\hat{A},\hat{B}]]\neq 0$ and $[\hat{B},[\hat{A},\hat{B}]]\neq 0$, the form of exponential expansion is complex. Here we adopt an approach of operators expansion proposed in Zubarev (1974). We focus on small perturbation around the equilibrium system, that is, the thermodynamic forces can be treated as perturbations. In this case, it is safe to say that the relation of these forces and irreversible dissipative currents is linear so that we can expand Eq.(35), keep only linear term, and approximate the complete statistical operator as: $\displaystyle\hat{\rho}=\left[1+\int_{0}^{1}d\tau\left(e^{-\hat{A}\tau}\hat{B}e^{\hat{A}\tau}-\langle\hat{B}\rangle_{\rm{leq}}\right)\right]\hat{\rho}_{\rm{leq}},$ (37) where $\langle\hat{B}\rangle_{\rm{leq}}=\mathop{\mathrm{Tr}}[\hat{\rho}_{\rm{leq}}\hat{B}]$ is the expectation over the local equilibrium operator. We consider energy- momentum tensor in non-equilibrium state. First, we evaluate the energy- momentum tensor averaged over the non-equilibrium distribution: $\displaystyle\langle{\hat{T}^{\mu\nu}(\bm{x},t)}\rangle=\langle{\hat{T}^{\mu\nu}(\bm{x},t)}\rangle_{\rm{leq}}+\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}$ $\displaystyle\qquad\quad\quad\ \ \Big{[}\Big{(}\hat{T}^{\mu\nu}(\bm{x},t)\,,\,\hat{T}^{\rho\sigma}(\bm{x}^{\prime},t^{\prime})\Big{)}\partial_{\rho}\beta_{\sigma}(\bm{x}^{\prime},t^{\prime})$ $\displaystyle\qquad\quad\quad\ \ -\Big{(}\hat{T}^{\mu\nu}(\bm{x},t)\,,\,\hat{N}^{\rho}(\bm{x}^{\prime},t^{\prime})\Big{)}\partial_{\rho}\alpha(\bm{x}^{\prime},t^{\prime})$ $\displaystyle\qquad\quad\quad\ \ +\Big{(}\hat{T}^{\mu\nu}(\bm{x},t)\,,\,\hat{T}^{[\rho\sigma]}(\bm{x}^{\prime},t^{\prime})\Big{)}\omega_{\rho\sigma}(\bm{x}^{\prime},t^{\prime})\Big{]},$ (38) where $\langle\hat{B}\rangle=\mathop{\mathrm{Tr}}[\hat{\rho}\hat{B}]$ is the expectation over the complete operator, with the binary operator $\displaystyle\left(\hat{X}(\bm{x},t),\hat{Y}(\bm{x}^{\prime},t^{\prime})\right)$ $\displaystyle\equiv\int_{0}^{1}d\tau\Big{\langle}\hat{X}(\bm{x},t)\big{(}e^{-\hat{A}\tau}\hat{Y}(\bm{x}^{\prime},t^{\prime})e^{\hat{A}\tau}-\langle\hat{Y}(\bm{x}^{\prime},t^{\prime})\rangle_{\rm{leq}}\big{)}\Big{\rangle}_{\rm{leq}}$ (39) being the Kubo correlation function. We precede to match the relevant tensors $T^{\mu\nu}$ and $N^{\mu}$ in hydrodynamics with the corresponding operators. A straightforward and natural tensor decomposition reads as: $\displaystyle\hat{T}^{\mu\nu}=\hat{e}u^{\mu}u^{\nu}-\hat{p}\Delta^{\mu\nu}+\hat{T}^{\mu\nu}_{(1s)}+\hat{T}^{\mu\nu}_{(1a)}\,,$ (40) $\displaystyle\hat{T}^{\mu\nu}_{(1s)}=2\hat{h}^{(\mu}u^{\nu)}+\hat{\pi}^{\mu\nu}+\hat{\Pi}\Delta^{\mu\nu}\,,$ (41) $\displaystyle\hat{T}^{\mu\nu}_{(1a)}=2\hat{q}^{[\mu}u^{\nu]}+\hat{\tau}^{\mu\nu},\,$ (42) $\displaystyle\hat{N}^{\mu}=\hat{n}u^{\mu}+\hat{j}^{\mu}_{(1)},$ (43) which consistently matches the form of ${T}^{\mu\nu}$ and $N^{\mu}$ in hydrodynamics. We also need to specify the parameters coupled to $\hat{T}^{\mu\nu}$, $\hat{N}^{\mu}$ and $\hat{S}^{\lambda\mu\nu}$. By imposing Landau matching conditions $u^{\mu}\delta\langle\hat{T}_{\mu\nu}\rangle u^{\nu}=0$ , $u^{\mu}\delta\langle\hat{N}_{\mu}\rangle=0$ Landau and Lifshitz (1987) with the notation $\delta\langle\hat{X}\rangle=\langle\hat{X}\rangle-\langle\hat{X}\rangle_{\rm{leq}}$, the parameters $\beta$ and $\alpha$ are identified as the inverse of temperature and the ratio of chemical potential to temperature. As for the identification of $\omega_{\mu\nu}$ with spin potential, details can be found in Becattini et al. (2019) and we will keep using this identification hereafter. Then the expectation of $\hat{T}_{\mu\nu}$ is evaluated over non- equilibrium operator and compared with the result with that of first-order spin hydrodynamics. To that end, we first rewrite the terms $\hat{T}^{\rho\sigma}\partial_{\rho}\beta_{\sigma}$, $N^{\mu}\partial_{\mu}\alpha$ and $\hat{T}^{\rho\sigma}\omega_{\rho\sigma}$ with the identification of these Lagrange multipliers: $\displaystyle\hat{T}^{\mu\nu}\partial_{\mu}\beta_{\nu}=\beta\hat{\pi}^{\mu\nu}\partial_{\mu}u_{\nu}+\beta\hat{h}^{\mu}\big{(}\beta^{-1}\partial_{\mu}\beta+Du_{\mu}\big{)}$ $\displaystyle\qquad\qquad+\beta\hat{\tau}^{\mu\nu}\partial_{\mu}u_{\nu}+\beta\hat{q}^{\mu}\big{(}-\beta^{-1}\partial_{\mu}\beta+Du_{\mu}\big{)}$ $\displaystyle\qquad\qquad+\hat{e}D\beta-\beta\big{(}\hat{p}-\hat{\Pi}\big{)}\theta,$ (44) $\displaystyle\hat{N}^{\mu}\partial_{\mu}\alpha=\hat{n}D\alpha+\hat{j}^{\mu}_{(1)}\nabla_{\mu}\alpha,$ (45) $\displaystyle\hat{T}^{[\mu\nu]}\omega_{\mu\nu}=2\hat{q}^{[\mu}u^{\nu]}\omega_{\mu\nu}+\hat{\tau}^{\mu\nu}\omega_{\mu\nu}.$ (46) In order to match thermodynamic forces in first-order hydrodynamics, we substitute $D\beta$ and $D\alpha$ with $\theta$ by using the equations of zero-order hydrodynamics, $\displaystyle\partial_{\mu}\langle\hat{T}^{\mu\nu}\rangle_{\rm{leq}}=0,$ (47) $\displaystyle\partial_{\mu}\langle\hat{N}^{\mu}\rangle_{\rm{leq}}=0.$ (48) Taking the scalar product of these equations with the four velocity $u^{\nu}$, we get: $\displaystyle D\langle\hat{e}\rangle_{\rm{leq}}=-(\langle\hat{e}\rangle_{\rm{leq}}+\langle\hat{p}\rangle_{\rm{leq}})\theta,$ (49) $\displaystyle D\langle\hat{n}\rangle_{\rm{leq}}=-\langle\hat{n}\rangle_{\rm{leq}}\theta.$ (50) Noting that the matching conditions ensures that $e=\langle\hat{e}\rangle_{\rm{leq}}$ and $n=\langle\hat{n}\rangle_{\rm{leq}}$, we shall using these notations in the following paragraphs. Straightforward calculation leads us to: $\displaystyle D\beta=\frac{\theta}{J}\Big{(}-\big{(}e+p\big{)}\frac{\partial n}{\partial\alpha}+n\frac{\partial e}{\partial\alpha}\,\Big{)},$ (51) $\displaystyle D\alpha=\frac{\theta}{J^{\prime}}\Big{(}-\big{(}e+p\big{)}\frac{\partial n}{\partial\beta}+n\frac{\partial e}{\partial\beta}\,\Big{)},$ (52) $\displaystyle J=$ $\displaystyle\frac{\partial e}{\partial\beta}\frac{\partial n}{\partial\alpha}-\frac{\partial n}{\partial\beta}\frac{\partial e}{\partial\alpha},\quad J^{\prime}=\frac{\partial e}{\partial\alpha}\frac{\partial n}{\partial\beta}-\frac{\partial n}{\partial\alpha}\frac{\partial e}{\partial\beta},$ (53) with the derivative of the thermodynamic functions with respect to $\alpha$ calculated holding $\beta$ fixed and vice versa. Now Eq.(III) and (45) can be cast into: $\displaystyle\hat{T}^{\mu\nu}\partial_{\mu}\beta_{\nu}=\beta\hat{\pi}^{\mu\nu}\partial_{\mu}u_{\nu}+\beta\hat{h}^{\mu}\left(\beta^{-1}\partial_{\mu}\beta+Du_{\mu}\right)$ $\displaystyle\qquad\qquad+\beta\hat{\tau}^{\mu\nu}\partial_{\mu}u_{\nu}+\beta\hat{q}^{\mu}\left(-\beta^{-1}\partial_{\mu}\beta+Du_{\mu}\right)$ $\displaystyle\qquad\qquad-\beta\hat{p}^{\prime}\theta,$ (54) $\displaystyle\hat{N}^{\mu}\partial_{\mu}\alpha=\hat{j}^{\mu}_{(1)}\nabla_{\mu}\alpha+\frac{\hat{n}}{J^{\prime}}\Big{[}-\big{(}e+p\big{)}\frac{\partial n}{\partial\beta}+n\frac{\partial e}{\partial\beta}\,\Big{]}\theta,$ (55) $\displaystyle\hat{p}^{\prime}=\hat{p}-\hat{\Pi}-\frac{1}{J\beta}\Big{[}-\big{(}e+p\big{)}\frac{\partial n}{\partial\alpha}+\,n\frac{\partial e}{\partial\alpha}\,\Big{]}\,\hat{e}.$ (56) From now on we will handle Kubo correlations so as to get final results. In an isotropic medium, one can turn to Curie’s principle for help and that has been used to simplify Eq.(III). Curie’s principle shows that the correlation function between operators of different ranks and spatial parity vanishes. The remaining Kubo correlations can be expressed in the comoving frame as: $\displaystyle\left(\hat{h}_{k}^{\prime},\hat{h}_{l}^{\prime}\right)=L_{h^{\prime}}\delta_{kl},$ $\displaystyle\left(\hat{\pi}^{kl},\hat{\pi}^{mn}\right)=L_{\pi}\frac{1}{2}\left(\delta^{km}\delta^{ln}+\delta^{kn}\delta^{lm}-\frac{2}{3}\delta^{kl}\delta^{mn}\right),$ $\displaystyle\left(\hat{q}^{k},\hat{q}^{l}\right)=L_{q}\delta^{kl},$ $\displaystyle\left(\hat{\tau}^{kl},\hat{\tau}^{mn}\right)=L_{\tau}\frac{1}{2}\left(\delta^{km}\delta^{ln}-\delta^{kn}\delta^{lm}\right),$ (57) with $\hat{h}^{\prime}_{\mu}=\hat{h}_{\mu}-\frac{e+p}{n}\hat{j}_{\mu}$, $\delta^{ij}$ being the Kronecker symbol, and $L_{i}$ are scalar functions that can be determined by taking trace in both sides of Eq.(III). And we conclude that the correlation between symmetric tensor and antisymmetric tensor is zero and we needn’t consider the contribution of these cross parts. A simple explanation is put following. First, we assume that: $\displaystyle\left(\hat{\pi}^{kl},\hat{\tau}^{mn}\right)\partial_{m}u_{n}=L_{1}\partial^{k}u^{l}+L_{2}\partial^{l}u^{k}+L_{3}\delta^{kl}\theta,$ (58) with $L_{i}(i=1,2,3)$ being scalar functions. The reason why we can make such an ansatz is that we have constrained the form of current-force relation as Eq.(19). It is this simple linear current-force relation that leads to this ansatz. Moving on, for the symmetry of exchanging $\mu$ and $\nu$, $L_{1}$ must equal $L_{2}$. Equivalently, we have: $\displaystyle\left(\hat{\pi}^{kl},\hat{\tau}^{mn}\right)=L_{1}\left(\delta^{km}\delta^{ln}+\delta^{kn}\delta^{lm}\right)+L_{3}\delta^{kl}\delta^{mn}.$ (59) Because the right-hand side of Eq.(59) is not antisymmetric when exchanging $m$ and $n$, which is in conflict with the left-hand side, the correlation between symmetric tensor and antisymmetric tensor is exactly zero. Then we boost to general reference frame, substitute the Kronecker symbol $\delta^{\mu\nu}$ with $-\Delta^{\mu\nu}$, and take trace to get all $L$ functions Zubarev (1974); Hosoya et al. (1984). In order to extract transport coefficients, we suppose the changes of thermodynamic forces within correlation length are sufficiently small so that we can factorize them out of Eq.(III). Thus we obtain the linear thermodynamic current-force relation combining Eqs.(40), (41), (42), (III), (56), and (III) together: $\displaystyle\langle\hat{\pi}^{\mu\nu}\rangle=2\eta\nabla^{\langle\mu}u^{\nu\rangle},$ (60) $\displaystyle\langle\hat{\Pi}\rangle=-\zeta\theta,$ (61) $\displaystyle\langle\hat{h}^{\mu}\rangle-\frac{e+p}{n}\langle\hat{j}_{(1)}^{\mu}\rangle=-\kappa\frac{nT}{e+p}\nabla^{\mu}\frac{\mu}{T},$ (62) $\displaystyle\langle\hat{\tau}^{\mu\nu}\rangle=2\gamma\Big{(}\omega_{th}^{\mu\nu}+2\Delta^{\mu\rho}\Delta^{\nu\sigma}\omega_{\rho\sigma}\Big{)},$ (63) $\displaystyle\langle\hat{q}^{\mu}\rangle=\lambda\bigg{(}Du^{\mu}+\frac{\nabla^{\mu}T}{T}-4\omega^{\mu\nu}u_{\nu}\bigg{)},$ (64) where the Gibbs–Duhem relation has been employed. By comparing these equations with Eqs.(19), (20), (21), (22) and (23), we conclude that we have reproduced the linear law of first-order spin hydrodynamics with the method of statistical operators. Throughout the derivation, the four-vector fluid velocity is not specified, which means the results we obtain is frame independent up to first order in gradients. After finishing the above manipulations, all the transport coefficients can be given in terms of Kubo correlation functions: $\displaystyle\eta=$ $\displaystyle\frac{\beta}{10}\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{(}\hat{\pi}^{\mu\nu}(\bm{x},t),\hat{\pi}_{\mu\nu}(\bm{x}^{\prime},t^{\prime})\Big{)},$ (65) $\displaystyle\kappa=$ $\displaystyle\frac{-\beta}{3}\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{(}\hat{h}^{\prime\mu}(\bm{x},t),\hat{h}_{\mu}^{\prime}(\bm{x}^{\prime},t^{\prime})\Big{)},$ (66) $\displaystyle\zeta=$ $\displaystyle\beta\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{(}\hat{p}^{\star}(\bm{x},t),\hat{p}^{\star}(\bm{x}^{\prime},t^{\prime})\Big{)},$ (67) $\displaystyle\gamma=$ $\displaystyle\frac{-\beta}{6}\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{(}\hat{\tau}^{\mu\nu}(\bm{x},t),\hat{\tau}_{\mu\nu}(\bm{x}^{\prime},t^{\prime})\Big{)},$ (68) $\displaystyle\lambda=$ $\displaystyle\frac{-\beta}{3}\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\Big{(}\hat{q}^{\mu}(\bm{x},t),\hat{q}_{\mu}(\bm{x}^{\prime},t^{\prime})\Big{)},$ (69) with $\hat{p}^{}=\hat{p}^{\prime}+\frac{\hat{n}}{\beta\theta}D\alpha$. In the following paragraphs, we will build up the connection between Kubo correlation functions and retarded Green functions. The discussion of the following paragraphs is similar to that in Huang et al. (2011). Direct evaluation of Eq.(III) leads to: $\displaystyle\left(\hat{X}(\bm{x},t),\hat{Y}(\bm{x}^{\prime},t^{\prime})\right)$ $\displaystyle\equiv\int_{0}^{1}d\tau\Big{\langle}\hat{X}(\bm{x},t)\left[e^{-\hat{A}\tau}\hat{Y}(\bm{x}^{\prime},t^{\prime})e^{\hat{A}\tau}-\langle{\hat{Y}}(\bm{x}^{\prime},t^{\prime})\rangle_{\rm{leq}}\right]\Big{\rangle}_{\rm{leq}}$ $\displaystyle=\frac{i}{\beta}\int_{-\infty}^{t^{\prime}}ds\Big{\langle}\Big{[}{\hat{X}}(\bm{x},t),{\hat{Y}}(\bm{x}^{\prime},s)\Big{]}\Big{\rangle}_{\rm{leq}},$ (70) where we supposed in the last step that the correlation of two operators vanishes when time goes to distant past. In deriving Eq.(III), we have utilized the conclusion that $\hat{A}$ can be treated as Hamiltonian operator. To see that, we are informed of three points. First, when we choose local rest frame or comoving frame, the first term within the integrand of Eq.(III) is exactly $\beta\hat{H}$. Second, taking the second term means we are doing the calculation of the grand canonical ensemble. In finite temperature field theory, this term can always be added to Hamiltonian to construct the grand canonical Hamiltonian operator. Third, the third term related to the coupling of spin and vorticity, which is the covariant form of the scalar product of angular velocity and angular momentum $\bm{\omega}\cdot\bm{J}$ Becattini (2012), can also be included in Hamiltonian in a rotating system. Combining these three considerations, we conclude that $e^{\hat{A}\tau}$ is a quantum mechanical evolution operator. Keep following this procedure: $\displaystyle\rm{I}$ $\displaystyle=\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\frac{i}{\beta}\int_{-\infty}^{t^{\prime}}ds\Big{\langle}\Big{[}{\hat{X}}(\bm{x},t),{\hat{Y}}(\bm{x}^{\prime},s)\Big{]}\Big{\rangle}_{\rm{leq}}$ $\displaystyle=-\int d^{3}\bm{x}^{\prime}\int_{-\infty}^{t}dt^{\prime}e^{\epsilon(t^{\prime}-t)}\frac{1}{\beta}\int_{-\infty}^{t^{\prime}}dsG_{R}(\bm{x}-\bm{x}^{\prime},t-s)$ $\displaystyle=\frac{i}{\beta}\lim_{\omega\rightarrow 0}\lim_{\bm{k}\rightarrow 0}\frac{\partial}{\partial\omega}G_{R}(\bm{k},\omega).$ (71) In obtaining this equation, the definition of retarded Green function is required: $\displaystyle G_{\hat{A}\hat{B}}^{R}(\bm{x},t)\equiv-i\theta(t)\left[\hat{A}(\bm{x},t),\hat{B}({\bf 0},0)\right].$ (72) So far we have proved Kubo correlation is exactly related to retarded Green function. Because formulae for transport coefficients are all relevant to self-correlation, we keep our focus on this case. Suppose $A,B$ represent the same operator, the imaginary(real) part of retarded Green is even(odd) function of $\omega$ according to the Onsager’s reciprocal principle Zubarev (1974) such that: $\displaystyle\rm{I}=-\frac{1}{\beta}\lim_{\omega\rightarrow 0}\lim_{\bm{k}\rightarrow 0}\frac{\partial}{\partial\omega}ImG^{R}_{\hat{A}\hat{A}}(\bm{k},\omega).$ (73) Collect all the results we obtained: $\displaystyle\eta=-\frac{1}{10}\lim_{\omega\rightarrow 0}\frac{\partial}{\partial\omega}\rm{Im}G^{R}_{\hat{\pi}\hat{\pi}}(\b{0},\omega),$ (74) $\displaystyle\zeta=-\lim_{\omega\rightarrow 0}\frac{\partial}{\partial\omega}\rm{Im}G^{R}_{\hat{p}\hat{p}}(\b{0},\omega),$ (75) $\displaystyle\kappa=\frac{1}{3}\lim_{\omega\rightarrow 0}\frac{\partial}{\partial\omega}\rm{Im}G^{R}_{\hat{h}^{\prime}\hat{h}^{\prime}}(\b{0},\omega),$ (76) $\displaystyle\gamma=\frac{1}{6}\lim_{\omega\rightarrow 0}\frac{\partial}{\partial\omega}\rm{Im}G^{R}_{\hat{\tau}\hat{\tau}}(\b{0},\omega),$ (77) $\displaystyle\lambda=\frac{1}{3}\lim_{\omega\rightarrow 0}\frac{\partial}{\partial\omega}\rm{Im}G^{R}_{\hat{q}\hat{q}}(\b{0},\omega).$ (78) The operators arising in the subscripts are all defined in the previous paragraphs. And the first three transport coefficients are consistent with the results of Huang et al. (2011) and Hosoya et al. (1984). We note that there is a factor $2$ difference compared to the result of $\eta$ in Hosoya et al. (1984), which is due to the different definition of shear viscosity, [see Eq.(19)]. The last two transport coefficients are exactly what we want, which can give a description of new transport properties of spinful fluids. ## IV Summary and Outlook We have evaluated Kubo formulae for transport coefficients arising in first- order spin hydrodynamics based on the approach of the non-equilibrium statistical operator. We apply Zubarev’s statistical operator method to linearize the non-equilibrium corrections, and study how a thermal system respond to such linear perturbations. The Kubo formulae we obtained are related to equilibrium (imaginary-time) infrared Green’s functions. Given specific microscopic theory, the imaginary-time Green’s functions in finite temperature field can be formulated. Thus, by analytical continuation, the real-time retarded Green’s functions can be calculated to obtain final results of these transport coefficients, which in turn are the basis of numerical simulation of the evolution of spinful fluids. According for the spin degree of freedom, one would need to perform the calculation based on the theory of spinor or vector field, which would be a non-trivial extension of that based on a scalar field theory Jeon (1993). Then it would be interesting to see to what extent the results obtained are the same compared to the calculation of Montenegro and Torrieri (2020), which is also based on linear response theory. On the other hand, it is an efficient way to use transport methods to determine these coefficients if the Green’s functions show good behavior during some period of the evolution of the system Arnold and Yaffe (1998). 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# Spin liquid behavior of a three-dimensional magnetic system Ba3NiIr2O9 with $S$ = 1 Siddharth Kumar Department of Physics, Indian Institute of Science, Bengaluru 560012, India S. K. Panda Department of Physics, Bennett University, Greater Noida 201310, Uttar Pradesh, India Manju Mishra Patidar UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452017, India Shashank Kumar Ojha Department of Physics, Indian Institute of Science, Bengaluru 560012, India Prithwijit Mandal Department of Physics, Indian Institute of Science, Bengaluru 560012, India Gangadhar Das Chemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru, 560064 India J. W. Freeland Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA V. Ganesan UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452017, India Peter J. Baker ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot OX11 0QX, United Kingdom S. Middey<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science, Bengaluru 560012, India ###### Abstract The quantum spin liquid (QSL) is an exotic phase of magnetic materials where the spins continue to fluctuate without any symmetry breaking down to zero temperature. Among the handful reports of QSL with spin $S\geq$1, examples with magnetic ions on a three-dimensional magnetic lattice are extremely rare since both larger spin and higher dimension tend to suppress quantum fluctuations. In this work, we offer a new strategy to achieve 3-D QSL with high spin by utilizing two types of transition metal ions, both are magnetically active but located at crystallographically inequivalent positions. We design a 3-D magnetic system Ba3NiIr2O9 consisting of interconnected corner shared NiO6 octahedra and face shared Ir2O9 dimer, both having triangular arrangements in a-b plane. X-ray absorption spectroscopy measurements confirm the presence of Ni2+ ($S$=1). Our detailed thermodynamic and magnetic measurements reveal that this compound is a realization of gapless QSL state down to at least 100 mK. Ab-initio calculations find a strong magnetic exchange between Ir and Ni sublattices and in-plane antiferromagnetic coupling between the dimers, resulting in dynamically fluctuating magnetic moments on both the Ir and Ni sublattice. Experimental realization and theoretical description of the highly entangled quantum spin liquid phase remain challenging topics of quantum many-body physics Anderson (1973). Over the last fifteen years, several spin-1/2 systems with two-dimensional frustrated lattice have been reported as probable candidates with QSL behavior Balents (2010); Savary and Balents (2016); Zhou, Kanoda, and Ng (2017); Knolle and Moessner (2019); Broholm _et al._ (2020). Since experimental reports of QSL with either spin $(S)\geq$ 1 Cheng _et al._ (2011); Chamorro _et al._ (2018) or three dimensional arrangement of spins Okamoto _et al._ (2007a); Koteswararao _et al._ (2014); Balz _et al._ (2016); Gao _et al._ (2019); Chillal _et al._ (2020) are very few, it can be easily anticipated that chances of having a QSL with both higher spin and 3-D geometry is extremely low Plumb _et al._ (2019). Many of the attempts to obtain spin-1 QSL have been focused on stabilizing Ni within various structural network Nakatsuji _et al._ (2005); Cheng _et al._ (2011); Lu _et al._ (2018); Chamorro _et al._ (2018); Plumb _et al._ (2019); Medrano _et al._ (2018). However, the magnetic behavior of these $S$ = 1 systems at low temperature differs widely from each other, even in the compounds with similar structural geometry. For example, unlike to the well known 120∘ spin structure Starykh (2015), as observed in $A_{3}$NiNb2O9 Lu _et al._ (2018), Ba3NiSb2O9 shows characteristic spin liquid behavior Cheng _et al._ (2011); Fåk _et al._ (2017); Quilliam _et al._ (2016), whereas NiGa2S4 hosts a spin nematic phase Bhattacharjee, Shenoy, and Senthil (2006). The interaction of such Ni- based $S$ = 1 triangular lattice with another magnetically active sublattice might result in an exotic magnetic phase in three-dimensional materials. However, only very few 3-D compounds with such feasibility exist Treiber, Kemmler-Sack, and Ehmann (1982); Lightfoot and Battle (1990); Ferreira _et al._ (2018). Six-layered hexagonal (6$H$) perovskite $A_{3}MM^{\prime}_{2}$O9 (Fig. 1(a)) with magnetic $M$ and $M^{\prime}$ ions constitutes a 3-D spin system. Both, $M$O6 octahedral units and face-shared $M^{\prime}_{2}$O9 dimers form a triangular lattice in $a$-$b$ plane (Fig. 1(b)-(c)) and would become geometrically frustrated in the presence of antiferromagnetic interaction. Moreover, the $M$-O-$M^{\prime}$ connectivity constitutes a buckled honeycomb lattice (Fig. 1(d)), which could host a Kitaev spin liquid phase in case of spin-1/2 ions with strong spin-orbit coupling (SOC) Kitaev (2006); Takagi _et al._ (2019). In search of SOC driven elusive nonmagnetic $J$ = 0 state and excitonic magnetism in $d^{4}$ system, several Ba${}_{3}M$Ir2O9 compounds with nonmagnetic $M^{2+}$ have been investigated recently Khaliullin (2013); Nag _et al._ (2016, 2018a); Khan _et al._ (2019). However, the comparable strength of SOC, non-cubic crystal field, Hund’s coupling, and superexchange interaction gives rise to a small but finite moment on the Ir sublattice. Moreover, interdimer hopping results in a spin-orbital liquid phase in Ba3ZnIr2O9 Nag _et al._ (2016). Replacing the nonmagnetic Zn2+ by an isovalent magnetic ion such as Ni2+ should provide a unique opportunity to investigate the magnetic response of a triangular lattice with $S$ = 1 in presence of the interconnected Ir sublattice with fluctuating magnetic moments. If both Ni and Ir moments of Ba3NiIr2O9 fluctuate dynamically, then it would offer a new route to realize 3-D QSL by involving two different magnetic ions. Also, it would be a 3-D QSL with a new type of structural network as compared to all existing examples with pyrochlore Gao _et al._ (2019); Plumb _et al._ (2019), hyperkagome Okamoto _et al._ (2007a), and hyper hyperkagome structure Koteswararao _et al._ (2014); Chillal _et al._ (2020). Figure 1: (a) Unit cell of 6$H$ $A_{3}MM^{\prime}$O9 without any disorder. (b), & (c) shows the triangular lattice arrangement of $M^{\prime}_{2}$O9 dimers, $M$O6 octahedra, respectively, in a-b plane. (d) $M$-O-$M\prime$ connectivity forms buckled honeycomb lattice. $A$, $M$ and $M^{\prime}$ corresponds to Ba, Ni, and Ir, respectively, for BNIO. Different magnetic exchange pathways ($J_{i}$) are also shown in (a)-(d). (e) Observed and refined powder XRD pattern of BNIO. (f) XAS spectrum of Ni $L_{3,2}$-edge of BNIO along with NiO and NdNiO3 for comparison. The XAS data for NdNiO3 has been adapted from Ref. Freeland, Van Veenendaal, and Chakhalian, 2016. In this paper, we report on the electronic and magnetic behavior of Ba3NiIr2O9 (BNIO). The phase purity and absence of any cationic site disorder have been demonstrated by powder X-ray diffraction (XRD) measurement. X-ray absorption spectroscopy (XAS) experiments have confirmed the desired +2 oxidation state of Ni. Persisting spin fluctuations down to 100 mK have been revealed by magnetization, specific heat and muon spin-rotation ($\mu$SR) measurements. We have also investigated BNIO by density functional theory calculations including Hubbard $U$ and SOC within the framework of LSDA+$U$ (local spin density approximation + $U$) approach. We have found not only appreciable magnetic exchange between Ni and Ir-sublattice but also antiferromagnetic coupling in the triangular sublattice of Ir. This geometrical frustration prohibits any long-range magnetic ordering and makes BNIO a rare example of three-dimensional QSL involving $S$ = 1. ## Results Polycrystalline BNIO was synthesized by solid state synthesis route. Members of the $A_{3}MM^{\prime}_{2}$O9 series can have site disorder between face shared, and corner shared octahedra Middey _et al._ (2011). It is also well known that structural disorder often jeopardizes QSL behavior, resulting in magnetic order or spin glass freezing Zhong _et al._ (2019). All peaks of the powder XRD pattern of BNIO (Fig. 1(e)) can be indexed and refined with 6$H$ structure having space group $P6_{3}/mmc$. The refinement also confirms that all corner-shared (face-shared) octahderal units are occupied by Ni (Ir) without any Ni-Ir site disorder. The structural parameters obtained from the refinement have been listed in SM sup . The temperature dependent XRD measurements down to 15 K also rules out any structural transition. Having confirmed that both Ni and Ir ions form triangular lattices in a-b plane without any disorder, we want to verify whether Ni has indeed $S$ = 1 state. For this purpose, XAS measurements were carried out Freeland, Van Veenendaal, and Chakhalian (2016). The comparison of Ni $L_{3,2}$ XAS line shape and energy position of BNIO, Ni2+O, and NdNi3+O3 (Fig. 1(e)) testifies the desired +2 oxidation of Ni in present case. The octahedral crystal field of Ni2+ ($d^{8}$: $t_{2g}^{6}$, $e_{g}^{2}$) ensures $S$ = 1 on Ni sublattice. Electrical measurement demonstrates insulating nature of the sample (inset of Fig.2 (a)), which can be fitted using Mott’s variable range hopping (VRH) Hill (1976) model in three-dimensions ($\rho=\rho_{o}\exp(T_{o}/T)^{1/4}$) as shown in Fig.2 (a). We also note that the insulating behavior of Ba3ZnIr2O9 is well described by VRH in two-dimensions. This difference between two compounds is a manifestation of electron hopping paths along the Ni-O-Ir bonds. Our electronic structure calculations (see SM sup ) further demonstrate that the insulating state can be obtained only by considering both electron correlation and SOC, implying that BNIO is a SOC driven Mott insulator Kim _et al._ (2008). Figure 2: (a) Fitting of $\rho$ by 3-D variable range hopping model for BNIO. Inset shows $\rho$ vs. $T$. (b) $\chi$ vs T on left axis and $1/(\chi-\chi_{0})$ along with fitting on right axis. (c) M-H at 2 K. The temperature dependent field cooled and zero-field cooled magnetic susceptibility does not differ from each other and also do not show any anomaly (Fig. 2(b)). This strongly implies absence of any long range magnetic ordering and spin glass behavior. We have fitted the data by a modified Cuire- Weiss law ($\chi$ = $\chi_{0}$ \+ $\frac{C_{W}}{T-\theta_{CW}}$) where $\chi_{0}$, $C_{W}$ and $\theta_{CW}$ represents temperature independent susceptibility contribution, Curie constant and Curie temperature, respectively. The fitting, shown as plot of 1/($\chi$-$\chi_{0}$) vs. $T$ in right axis of Fig. 2(b), results a $\theta_{CW}\sim$ -15 K. Negative values of Curie-Weiss temperature signify net antiferromagnetic interaction among the spins of BNIO. The relatively smaller value of $\theta_{CW}$ is related to the presence of multiple exchange pathways, which will be discussed in later part of this paper. The effective magnetic moment $\mu_{eff}$ (= $\sqrt{8C_{W}}$) is found to be $\sim$ 3.65 $\mu_{B}$ from the fitting. Interestingly, the effective magnetic moment of a similar compound Ba3NiSb2O9, with nonmagnetic Sb5+ was reported to be around 3.54 $\mu_{B}$ Cheng _et al._ (2011). This gives an estimate of $g$-factor $\sim$ 2.5, similar to other Ni2+ based systems carlin_paramagnetism_1986 . If we assume similar value for the present compound, the effective Ir moment turns out to be 0.9 $\mu_{B}$ per dimer (= $\sqrt{\mu_{eff}^{2}-\mu_{Ni}^{2}}$) i.e. 0.64 $\mu_{B}$/Ir. This is very similar to the Ir moment (0.5 - 0.6 $\mu_{B}$) reported for Ba3MgIr2O9 Nag _et al._ (2018a), though a nonmagnetic $J$ = 0 state is expected for Ir5+ from a pure ionic picture. Thus, our analysis highlights both Ni2+ and Ir5+ participate in magnetism of BNIO compound. The estimated magnetic moments from our LSDA+U+SOC calculations (shown in SM sup ) are also in good agreement with our experimental results. Fig. 2 (c) shows $M$-$H$ done at 2 K between $\pm$ 9 T. The absence of any hysteresis again confirms absence of ferromagnetism and spin glass freezing at 2 K. The presence of antiferromagnetic interaction without any long range magnetic ordering or spin glass transition strongly indicates that BNIO is a favorable candidate of QSL. Figure 3: (a) $C_{p}$ vs T curves at various fields. (b) Magnetic specific heat ($C_{m}$) extracted by subtracting lattice contribution. (c) Low temperature part of $C_{m}$ plotted in log-log plot. (d) Magnetic entropy for zero field. In order to further investigate the magnetic nature of the sample, we have measured specific heat ($C_{p}$) from 100 mK to 200 K. $C_{p}$ not only probes the presence/absence of any long-range magnetic ordering but also provides very crucial information about the nature of low energy excitation. The absence of any $\lambda$-anomaly (Fig. 3(a)) again confirms absence of long- range order and/or any structural transition down to 100 mK, consistent with the magnetic measurement and XRD results, respectively. For an insulating compound with magnetic spins, $C_{p}$ consists of lattice specific heat ($C_{lat}$) and magnetic specific heat ($C_{m}$). In absence of any analogous non-magnetic compound, the contribution of $C_{lat}$ has been evaluated by fitting $C_{p}$ in 30 K - 200 K range by a Debye-Einstein equation with one Debye term and two Einstein terms (details are in SM sup ) and extrapolating the fitted curve down to 100 mK. A broad peak is observed around 7 K in $C_{m}$ vs. $T$ plot (Fig. 3(a)). We can not capture this feature by Schottky anomaly, arising due to energy level splitting (see SM sup ). On the other hand, such feature is commonly observed in spin liquid materials and thus, could be considered as a signature of crossover from thermally disordered paramagnet to quantum disordered spin liquid state Okamoto _et al._ (2007b); Balents (2010); Li _et al._ (2015); Dey _et al._ (2017). The position of this broad peak shows negligible shifts with the application of magnetic field (shifts $\sim$ 1 K for applied field of 12 T). At low temperature, $C_{m}$ follows power-law behavior $C_{m}$ = $\gamma T^{\alpha}$ (Fig. 3(c)). For zero field, the magnitude of coefficient $\gamma$ is 45 mJ/mol K2 and the exponent $\alpha$ is 1.0$\pm$0.05 within 0.1 K - 0.6 K range. The value of $\gamma$ is very similar to other gapless spin liquid candidates: like Ba3CuSb2O9 (43.4 mJ/mol K2) Zhou _et al._ (2011), Ba2YIrO6 (44 mJ/mol K2) Nag _et al._ (2018b), Ba3ZnIr2O9 (25.9 mJ/mol K2) Nag _et al._ (2016), Sr2Cu(Te0.5W0.5)O6 (54.2 mJ/mol K2) Mustonen _et al._ (2018). Also, the linear $T$ behavior with nonzero $\gamma$ in an insulator comes due to gapless spinon excitations with a Fermi surface and has been reported in several organic and inorganic spin liquid candidates Yamashita _et al._ (2008, 2011); Zhou _et al._ (2011); Cheng _et al._ (2011); Mustonen _et al._ (2018); Clark _et al._ (2014); Uematsu and Kawamura (2019). Otherwise, any gapped excitation would result in an exponential dependence of $C_{m}$ on $T$. $\alpha$ becomes 2.6$\pm$0.05 within 0.8 K - 2.1 K. Interestingly, the application of an external field destroys the linear $T$ behavior of $C_{m}$. For $\mu_{0}H$ = 4 T, $\alpha$ becomes 2 for 0.15 K $\leq T\leq$ 0.50 K and 2.9 for 0.5 K $\leq T\leq$ 2.4 K. We note that $\alpha$ is found to be between 2 to 3 for several spin nematic phase Nakatsuji _et al._ (2005, 2007); Povarov _et al._ (2019); Kohama _et al._ (2019). Further studies are necessary to investigate the possibility of transition from spin liquid to spin nematic phase by the application of a magnetic field. The amount of released magnetic entropy ($S_{m}$) is evaluated by integrating $C_{m}/T$ w.r.t. $T$ and is shown in Fig. 3(d). For BNIO, the entropy saturates at 6.9 J/mol K for zero field measurement, which is only 75% of the total entropy expected for even a $S=1$ system [$R$ln(2$S$+1), where $R$ is universal gas constant]. The retention of a large amount of entropy at low temperature is another signature of the spin-liquid nature of BNIO, which has been reported as well for many other QSL Zhou _et al._ (2011); Cheng _et al._ (2011); Mustonen _et al._ (2018); Yamashita _et al._ (2008). To have a further understanding of the magnetic behavior at low temperature, we have performed $\mu$SR measurements, which is a very sensitive local probe to detect a weak magnetic field, even of the order of 0.1 Oe Blundell (1999). Fig. 4(a) shows asymmetry vs time curves for zero-field (ZF) measurements at selected temperatures. Neither any oscillation nor the characteristic 1/3rd recovery of the asymmetry is observed down to 60 mK, strongly implying the absence of long-range magnetic ordering or spin freezing. For a system with two interacting spin networks, the local magnetic field, felt by a muon at a stopping site is contributed by both magnetic sublattices. In such cases, the muon relaxation function is generally described by a product of two response functions, representing local fields from two spin networks Uemura _et al._ (1985). However, our such attempts considering different possible combinations of relaxation functions, including a spin glass like relaxation Morenzoni _et al._ (2008); Uemura _et al._ (1985), did not provide a satisfactory fitting of the experimental observed data (see SM sup ). This further supports the absence of spin glass freezing in present case. Figure 4: (a) Asymmetry vs time curves at various temperatures taken at zero magnetic field and fitted curve (solid line) using equation 1. (b) Variation of $\nu$ to temperature (shaded area is a guide to the eye). Inset shows variation of $\nu$ with applied LF at 100 mK Interestingly, similar to the other hexagonal Ba${}_{3}M$Ir2O9 Nag _et al._ (2016, 2018a), these asymmetry curves consist of one fast relaxing, one moderately relaxing, and one hardly relaxing components. We have fitted these curves using a combination of two dynamical relaxation functions with a common fluctuation rate $\nu$ and a Kubo-Toyabe function (KT) Hayano _et al._ (1979), $A(t)=A_{1}G(t,\Delta H_{1},\nu)+A_{2}G(t,\Delta H_{2},\nu)+A_{3}KT(t,\delta)$ (1) where $A_{1}$, $A_{2}$, $A_{3}$ are amplitudes. The static KT function, corresponding to the hardly relaxing component, accounts for the muons stopping at the silver sample holder as we find that the relaxation curve from the bare sample holder can also be described by a KT function with similar $\delta$. The dynamical relaxation, arising due to the presence of a field distribution ($\Delta H$) with a fluctuation rate ($\nu$) is represented by the Keren function $G(t,\Delta H,\nu)$ Keren (1994). The presence of two dynamical relaxations implies two inequivalent muon stopping sites, which are likely to be related with the two types of crystallographically inequivalent oxygen in hexagonal Ba${}_{3}M$Ir2O9 Nag _et al._ (2016, 2018a). The asymmetry data over a large temperature range (60 mK - 150 K) have been fitted by allowing $\nu$ to vary with $T$ and, the extracted values of $\nu$ has been shown as a function of $T$ in Fig. 4(b). The background contribution is different between measurements in dilution refrigerator and helium cryostat and, has been kept fixed for our analysis within the corresponding temperature range. The inequality $\nu>\gamma\Delta H$ ( $\gamma$ = muon gyromagnetic ratio = 2$\pi\times$135.5 Mrad s-1 T-1) holds for both relaxing components as $\gamma\Delta H_{1}\sim$0.425 MHz and $\gamma\Delta H_{2}\sim$ 0.09 MHz for the lowest temperature of our measurement (60 mK). This justifies the use of dynamical relaxation functions and establishes spin liquid nature of BNIO. We note that the value of $\nu$ ($\sim$ 4 MHz) at low temperature for BNIO is one order of magnitude smaller than Ba3ZnIr2O9 Nag _et al._ (2016) and is likely to be related with the involvement of large spins on the Ni sublattice. $\mu$-SR spectra, recorded at 100 mK in presence of an applied longitudinal field (LF) have further corroborated QSL nature of BNIO. In case of relaxation arising from a static internal field distribution with width $\Delta H_{i}$, an applied LF $\sim$ 5-10$\Delta H_{i}$ would completely suppress the relaxation. From the analysis of ZF $\mu$SR data, we found $\Delta H_{1}\sim$ 5 Gauss and $\Delta H_{2}\sim$ 1 Gauss. No such decoupling is observed in the present case in measurement up to 200 Gauss (see inset of Fig. 4(b) and SM sup ), establishing the dynamic nature of spins in BNIO down to atleast 100 mK. ## Discussion To understand the underlying mechanism of the observed QSL state, we estimated the inter-atomic magnetic exchange interactions from the converged LSDA+$U$+SOC calculations using the formalism of Ref. Kvashnin _et al._ , 2015a (see Method section and SM sup for details). As shown in Table-I, the strongest interaction is antiferromagnetic, which is between the Ir ions of the structural dimer. The strong Ir-Ni interaction further testifies three- dimensional nature of BNIO. Most importantly, Ir-Ir exchange in the a-b plane ($J_{4}$) is found to be antiferromagnetic, resulting in-plane magnetic frustration and explains the origin of the QSL behavior of the present system. However, the presence of strong Ir-Ni and Ni-Ni ferromagnetic exchange reduces the net antiferromagnetic exchange of this system, resulting a relatively lower value of negative $\theta_{CW}$. We further note that the ferromagnetic Ni-Ni exchange has been observed also in antiferromagnetic phase of analogous compound Ba3NiRu2O9 Lightfoot and Battle (1990). Table 1: Exchange couplings obtained from $ab$-initio calculations. Exchange pathways have been shown in Fig. 1. AFM and FM refers to antiferromagnetic and ferromagnetic interaction, respectively. Exchange \| Interacting \| Number \| Magnitude \| Type \| $\|$z${}_{i}J_{i}$/$J_{1}\|$ ---\|---\|---\|---\|---\|--- ($J_{i}$) \| pair \| of neighbor ($z_{i}$) \| (meV) \| \| $J_{1}$ \| Ir-Ir \| 1 \| -8.91 \| AFM \| 1 $J_{2}$ \| Ir-Ni \| 3 \| 0.96 \| FM \| 0.32 $J_{3}$ \| Ir-Ir \| 3 \| 0.10 \| FM \| 0.03 $J_{4}$ \| Ir-Ir \| 6 \| -0.17 \| AFM \| 0.11 $J_{5}$ \| Ni-Ni \| 6 \| 0.09 \| FM \| 0.06 To summarize, our detailed measurements reveal that 6$H$ BNIO containing $S$=1 hosts a gapless spin liquid phase below 2 K. The involvement of Ir5+ and Ni2+ in the magnetic properties of BNIO is revealed by dc magnetic measurements, $\mu$SR experiments, and electronic structure calculation. The antiferromagnetic interaction between Ir2O9 dimers in a-b plane facilitates geometrical frustration driven QSL phase of BNIO. Since many Ba3$MM_{2}^{\prime}$O9 compounds can be stabilized in 6$H$ structure, we believe this work will lead to the realization of many 3-D QSLs with large spin by judicial choice of $M$ and $M^{\prime}$. ### Method Stoichiometric amount of BaCO3, NiO and Ir metal power were used as starting materials for the solid state synthesis of BNIO. The mixture was heated multiple times at 1175∘ C with intermediate grindings till the desired phase is formed. Powder XRD was carried out using a lab based Rigaku Smartlab diffractometer and also in the Indian beamline (BL-18B) at the Photon Factory, KEK, Japan. The diffraction pattern of the final phase was refined by Reitveild method using FULLPROF Rodríguez-Carvajal (1993). XAS spectra of Ni $L_{3,2}$-edges were recorded in bulk sensitive total fluorescence yield mode in 4-ID-C beam line of Advanced Photon Source, USA. DC magnetic measurements were carried using a Quantum Design (QD) SQUID magnetometer. Heat capacity measurements ($C_{p}$) were done in a dilution refrigerator insert coupled with a 16T QD-PPMS system using relaxation calorimetry. $\mu$SR experiments down to 60 mK were performed using pulsed muon beam at MuSR spectrometer of ISIS Neutron and Muon Source, UK. A dilution fridge was used to record $\mu$SR data from 60 mK to 4 K and a cryostat was used for temperatures above 1.5 K. The density functional theory (DFT) calculations have been performed in the local spin-density approximation + Hubbard $U$ (LSDA+U) approach with and without including spin-orbit coupling (SOC) by means of a full potential linearized muffin-tin orbital method (FP-LMTO) Andersen (1975); Wills and Cooper (1987) as implemented in the RSPt code Wills _et al._ (2000). The Brillouin-zone (BZ) integration is carried out by using the thermal smearing method with 10 $\times$ 10 $\times$ 4 k-mesh. For the charge density and potential angular decomposition inside the muffin-tin (MT) spheres, the value of maximum angular momentum was taken equal to $l_{max}=8$. To describe the electron-electron correlation within LSDA+U approach, we have taken $U$ = 6 eV, $J$ =0.8 eV for Ni-$d$ states and $U$ = 2 eV, $J$ =0.6 eV for the Ir-$d$ states. The set of the correlated orbitals located on Ni and Ir sites were obtained by projecting the electron density onto the corresponding MT sphere with a certain angular character (so-called “MT-heads” projectionGrechnev _et al._ (2007)). After obtaining the self-consistent fully converged LSDA+$U$+SOC calculations, the magnetic force theorem Liechtenstein _et al._ (1987); Katsnelson and Lichtenstein (2000) was used to extract the effective inter-site magnetic- interaction parameters ($J_{ij}$). In this approach the magnetic system is mapped onto the Heisenberg Hamiltonian: $\hat{H}=-\sum_{i\neq j}J_{ij}\vec{S_{i}}\cdot\vec{S_{j}}.$ (2) Further, $J_{ij}$ are extracted in a linear-response manner via Green’s function technique. A detailed discussion of the implementation of the magnetic force theorem in RSPt is provided in Ref. Kvashnin _et al._ , 2015b. This method is considered to be one of the most accurate techniques for the estimation of exchange interactions and also been successfully employed for many transition metal compounds Panda _et al._ (2016). ### Acknowledgement S.M. acknowledges financial support from ISRO-IISc Space Technology Cell and Infosys Foundation, Bangalore. S.M. and S.K. acknowledge insightful discussions with Dr. Subhro Bhattacharjee, Dr. Yogesh Singh, Dr. Pabitra Biswas. 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# All-Day Object Tracking for Unmanned Aerial Vehicle Bowen Li, Changhong Fu, Fangqiang Ding, Junjie Ye, and Fuling Lin, Corresponding author The aurhors are with the School of Mechanical Engineering, Tongji University, 201804 Shanghai, China. (Email<EMAIL_ADDRESS> ###### Abstract Visual object tracking, which is representing a major interest in image processing field, has facilitated numerous real-world applications. Among them, equipping unmanned aerial vehicle (UAV) with real-time robust visual trackers for all-day aerial maneuver, is currently attracting incremental attention and has remarkably broadened the scope of applications of object tracking. However, prior tracking methods have merely focused on robust tracking in the well-illuminated scenes, while ignoring trackers’ capabilities to be deployed in the dark. In darkness, the conditions can be more complex and harsh, easily posing inferior robust tracking or even tracking failure. To this end, this work proposed a novel discriminative correlation filter-based tracker with illumination adaptive and anti-dark capability, namely ADTrack. ADTrack firstly exploits image illuminance information to enable adaptability of the model to the given light condition. Then, by virtue of an efficient and effective image enhancer, ADTrack carries out image pretreatment, where a target-aware mask is generated. Benefiting from the mask, ADTrack aims to solve a dual regression problem where dual filters, i.e., the context filter and target-focused filter, are trained with mutual constraint. Thus ADTrack is able to maintain continuously favorable performance in all-day conditions. Besides, this work also constructed one UAV nighttime tracking benchmark UAVDark135, comprising of more than 125k manually annotated frames, which is also very first UAV dark tracking benchmark. Exhaustive experiments are extended on authoritative daytime benchmarks, i.e., UAV123@10fps, DTB70, and the newly built dark benchmark UAVDark135. Our results have validated the superiority of ADTrack in both bright and dark conditions compared with other arts. Meanwhile, ADTrack realizes a real-time speed of over 30 frames/s on a single CPU, remarkably ensuring real-world UAV object tracking under all-day scenes. ###### Index Terms: Unmanned aerial vehicle, visual object tracking, discriminative correlation filter, dark tracking benchmark, image illumination based mask, dual regression model. ## I Introduction Standing as one of the hotspots in image processing field, visual object tracking aims at estimating the location and scale of an initial given object in the subsequent frames of an image sequence. Applying such flourishing approach onboard unmanned aerial vehicle (UAV) has enabled extensive applications in practice, e.g., path optimization and planning [1], disaster response [2], target following [3], and autonomous landing [4]. Specifically, transmission-line inspection [5], collision avoidance [6], and aircraft tracking [7], often need around-the-clock operation. Figure 1: Performance comparison of SOTA trackers on the newly constructed UAVDark135, where tracking scenes are generally much darker and more onerous. Clearly, ADTrack outperforms the other trackers in both distance precision (DP), and area under curve (AUC), maintaining favorable robustness even in harsh dark conditions. Figure 2: The first frames of representative scenes in newly constructed UAVDark135. Here, target ground-truths are marked out by green boxes and sequence names are located at the top left corner of the images. Dark special challenges like objects’ unreliable color feature and objects’ merging into the dark can be seen clearly. Unfortunately, state-of-the-art (SOTA) trackers [8, 9, 10, 11, 12, 13, 14, 15, 16] only focus on tracking in bright environment, where external illumination condition is favorable and the inline texture as well as outline of the object is representatvie. When the night falls, despite that the content of the scene is discernible, the visual quality of images captured is barely satisfactory, hurting the performance of methods that are primarily designed for object tracking with high-visibility inputs. Compared with common tracking scenes, tracking in the dark onboard UAV confronts special undesirable hardships such as: * • Color distortion: Since the objects in the dark are not well-illuminated, very little light is reflected on their surfaces, making them nearly all-dark. In this case, the objects’ color is distorted. Without representative color features, the discriminative ability of the tracker can decrease in a notable margin. * • Low visibility: When the object enters dimmer region in the dark, like the shadow, it can merge into the dark background, ending up in low visibility. Such scenes set barrier to trackers’ robust localization and precise scale estimation of the object. * • High-level noise: Images captured by UAV at night inevitably contains random noise, which may distract trackers, resulting in inferior robust tracking. * • Limited computation: For cost issue and scarce battery power, UAV generally carries a single CPU as its computation platform. While in order to handle tough darkness, more modules are needed to maintain tracking robustness, making real-time processing even more arduous. Though the outstanding trackers [17, 8, 9, 10, 11, 18, 19, 14, 15, 16] have achieved promising performance in well-illuminated scenes. The deep trackers [18, 19, 14, 15, 16] on the one hand introduce too much computation burden to be deployed on a single CPU for real-time tracking. On the other, the discriminative ability of deep semantic features were proved to drop significantly in our experiment since the off-the-shelf network is trained by bright images. While the brilliant discriminative correlation filter (CF)-based trackers [17, 8, 9, 10, 11], which utilize online learned handcrafted features, easily lose accuracy and robustness under low-light nighttime scenes due to the challenges mentioned above. In Fig. 1, both deep and handcrafted CF-based methods fails to meet our expectation. Prior work contributed very few to robust tracking in the dark, while there is an urgent need to develop to broaden the service life and usage scenarios of UAV. In this regard, this work proposes a novel and pioneer tracker with illuminance adaptive and anti-dark function (ADTrack) to achieve robust all- day real-time UAV tracking. Fig. 1 exhibits the superiority of proposed ADTrack against other arts in nighttime UAV tracking scene. To be specific, ADTrack explores image illumination processing methods [20, 21] and proposes more innovative module to be embedded into efficient robust CF-based tracker [10]. To achieve robust tracking under 2 distinct light condition, i.e., day and night, ADTrack firstly exploits image illumination [20] to detect and adapt to the condition, which is inspired by human visual system and proves to be effective and efficient. Then, a fast enhancer [21] generates appropriate image samples according to the detection result for training. Surprisingly, we found the image illumination map in [21] can be utilized to obtain a target-aware mask. By virtue of the mask, ADTrack solves a dual regression model to train target-focused and context filters with mutual constraint. During detection phase, dual responses, i.e., target-focused and context response maps, are fused using weight sum to achieve more precise object localization. With the proposed dual filters, ADTrack proves to be more robust in around-the-clock UAV tracking scenes. Besides, to the best of our knowledge, there exists no dark UAV tracking benchmark for large-scale evaluation in literature. Thus, this work also builds the very first UAV dark tracking benchmark, i.e., UAVDark135. UAVDark135 consists of totally 135 sequences, most of which were newly shot by a standard UAV at night, including more than 125k manually annotated frames. The benchmark covers a wide range of scenes, e.g., road, ocean, street, highway, and lakeside, including a large number of objects, such as person, car, building, athlete, truck, and bike. Fig. 2 displays representative scenes in UAVDark135, where dark special challenges are distinct. Contributions111The source code of the proposed ADTrack and newly constructed benchmark UAVDark135 are located at https://github.com/vision4robotics/ADTrack_v2. of this work can be summarized as: * • This work proposes a novel tracker with illumination adaptive and anti-dark function (ADTrack). * • The proposed ADTrack exploits image illumination to acquire target-aware mask, which can be creatively utilized to solve a dual regression. * • This work constructed the very pioneer UAV dark tracking benchmark UAVDark135 to perform large-scale evaluation. * • Exhaustive experiments have been conducted on two authoritative daytime benchmark UAV123@10fps, DTB70 and the newly built nighttime benchmark UAVDark135 to validate the surprising ability of proposed ADTrack in around- the-clock tracking performance. The remainder of this work is organized as follows. Section II summarizes related work about image enhancing approaches, CF-based tracking methods, and target-aware tracking approaches. Section III elaborately interprets the proposed ADTrack, which can be epitomized as 4 modules, respectively, illumination adaptation, pretreatment, filter training, and object detection. Section IV gives a thorough introduction of the newly built UAVDark135, including its statistics, platform, annotation, and attributes. Section V exhibits comprehensive experimental evaluation on various benchmarks, i.e., UAV123@10fps [22], DTB70 [23], and UAVDark135, where the superiority of ADTrack is apparent. Finally, Section VI gives an integrated conclusion of the full article. ## II Related Work ### II-A Low-Light Image Enhancers Existing SOTA enhancers can be generally divided into 2 categories, i.e., learning-based and model-based. learning-based methods aim at training an end- to-end network specialized for domain transformation with paired images [24, 25, 26]. To name a few, C. Chen et al. [25] carefully designed an encoder- decoder structured network and trained it with paired short-exposure low-light images and corresponding long-exposure images, which can generate high-quality enhanced images. W. Ren et al. [26] proposed a deep hybrid network made up of distinct content stream and edge stream, which can handle the degraded images captured at night. Such methods are usually carried out on GPU due to their huge amount of calculation brought by convolution layers, which cannot realize real-time processing on a single CPU for tracking onboard UAV. Model-based methods [27, 28, 21, 29] are in view of the famous retinex theory [30], which need no off-line training. For instance, M. Li et al. [27] creatively proposed to estimate a noise map based on tradition retinex model, which is able to obtain de-noised enhanced images. Z. Rahman et al. [29] replaced the original logarithm function in multi-scale retinex model with a sigmoid function which can suppress noise speckles in extreme low light areas. Specially, the enhancer [21] proves to be both efficient and effective. This work introduces it into the UAV tracking structure and constructs an illumination adaptive anti-dark tracker. Different from roughly applying the enhancer to preprocess the frames, ADTrack achieves light condition awareness and dual regression with the in-depth integration of image enhancement and visual tracking. ### II-B CF-Based Tracking Methods The key idea of CF-based tracking methods is to train a discriminative correlation filter using current image samples and utilize the trained filter to search for the object in the coming frame [31, 32, 10, 33, 11, 9, 8, 34]. J. F. Henriques et al. introduced circular sample matrix, ridge regression and kernel correlation in the KCF tracker [31], considered as the foundation of all CF-based trackers. M. Danelljan et al. proposed scale filter in the DSST tracker [11], settling scale estimation efficiently. H. K. Galoogahi et al. put forward cropping matrix in the training regression equation [10], not only immensely expanding the real negative samples but also alleviating boundary effect. Generally speaking, the key enablers of the superiority of CF type methods involve: $1$) its simplicity of calculation by discrete Fourier transformation as well as a myriad of implicitly circular shift samples generated in this duration, $2$) its online learning schemes which make the filter maintain satisfying robustness in various scenarios. UAV tracking, owing to its limited computation resource and wide application, is right the field where CF-based trackers can shine brilliantly [35, 17, 36, 9, 8]. To be specific, Z. Huang et al. exploited response map aberrance and proposed the ARCF tracker [9], which ensures high robustness under volatile tracking distractors. The AutoTrack tracker [8] proposed by Y. Li et al. aimed at the onerous mode adjustment procedure and adapted to various given condition automatically. Though the trackers mentioned above can strike a balance between accuracy and speed in common bright environment, they lose robustness and accuracy in the dark, when the light condition becomes abominable. To realize all-day real- time UAV tracking, this work proposed ADTrack, which can adapt to the given light condition and maintain predominant robustness even in the terrible darkness. ### II-C Target-Aware CF-Based Tracking Target-aware mask, which is a foreground matrix, concentrating the filter’s attention on pixels more possible included within the object outline. Such strategy can raise the importance of features indeed represent the object characteristics. M. Danelljan et al. proposed a spatial punishment term in the SRDCF tracker [32], which can be considered as a foreground mask, making the filter learn center region more to alleviate boundary effect. A. Lukezic et al. improved the fixed spatial regularization term in the SRDCF tracker [32] into alterable reliability mask in the CSR-DCF tracker [37], which is based on Bayes principle. Recently, saliency-aware methods are also introduced [35, 38]. Specifically, C. Fu et al. [35] creatively utilized an efficient saliency detection method to generate effective mask, which raised the robustness tracker onboard UAV tremendously. Figure 3: Pipeline of ADTrack. The proposed tracker contains four stages, i.e., illumination adaptation, pretreatment, training, and detection stages, which are marked out by boxes in different colors. In well-illuminated daytime and dim nighttime, ADTrack is able to adjust its tracking modules automatically, according to light condition judgment in illumination adaptation stage. In training and detection stage, ADTrack adopts dual filter training and dual response fusion, respectively, target-focused level and context level, as is shown in different routes. Despite that the methods mentioned can improve tracking performance, they have two obvious drawbacks. Firstly, it is hard to obtain valid alterable masks like [37], [38], and [35] in the dark, since nighttime images lack sufficient information. Secondly, the aforementioned trackers directly embed the masks into the CF in training process as a regularization term, assigning higher weights to the predicted target region in the filter. When an invalid mask generates, wrong pixels that odd with actual target region of the CF will possess higher importance, easily leading to tracking failure. Totally different from the above, ADTrack exploits image illumination information to obtain effective masks in arbitrary light condition. By virtue of the target-aware mask, ADTrack proposes a dual regression, where target- focused and context filters are trained with mutual constriction. In this way, both background and target information are learned and utilized, greatly improving tracking robustness. ## III Illumination Adaptive and Anti-Dark Object Tracking The pipeline of the proposed ADTrack can be divided into four stages, i.e., illumination adaptive stage, pretreatment stage, training stage, and detection stage. As is clearly exhibited in Fig. 3, for a given dark tracking scene, ADTrack firstly implements an illumination adaptation decider in the first frame captured by UAV to judge whether it is at daytime or nighttime. The determined outcome can enable mode switching automatically. In the subsequent frame $f$, ADTrack carries out pretreament stage with the illumination judgment result, where appropriate samples (from original image or enhanced image) and target-aware mask are obtained simultaneously. Then in training stage, dual filters are jointly trained by focusing on both context and target appearance. As next frame $f+1$ comes, the trained filters generate dual response maps which are fused to obtain the final response for target localization as detection stage. ### III-A Illumination Adaptation To realize all-day adaptation, we consider an effective illumination expression algorithm, which transforms complex image illuminance information into a simple constant. For a given RGB image $\mathcal{I}\in\mathbb{R}^{w\times h\times 3}$, the pixel-level world illumination value $\mathcal{L}^{\mathrm{W}}(\mathcal{I})$ is firstly computed as: $\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})=\sum_{\mathrm{m}}\alpha_{\mathrm{m}}\Psi_{\mathrm{m}}(\mathcal{I}(x,y)),~{}\mathrm{m}\in\\{\mathrm{R,G,B}\\}~{},$ (1) where $\Psi_{\mathrm{m}}(\mathcal{I}(x,y))$ denotes the intensity value of image $\mathcal{I}$ at coordinate $(x,y)$ in color channel $\mathrm{m}$, e.g., $\Psi_{\mathrm{G}}(\mathcal{I}(x,y))$ denotes the value in green channel. The channel weight parameters $\alpha_{\mathrm{R}},\alpha_{\mathrm{G}},\alpha_{\mathrm{B}}$ meet $\alpha_{\mathrm{R}}+\alpha_{\mathrm{G}}+\alpha_{\mathrm{B}}=1$. Then, the log-average luminance $\tilde{\mathcal{L}}^{W}(\mathcal{I})$ is given as in [20]: $\tilde{\mathcal{L}}^{\mathrm{W}}(\mathcal{I})={\mathrm{exp}}\Big{(}\frac{1}{wh}\sum_{x,y}\mathrm{log}(\delta+\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I}))\Big{)}~{},$ (2) where $\delta$ is a small value, to avoid zero value. Remark 1: Our experiment has proved that the log-average luminance $\tilde{\mathcal{L}}^{W}(\mathcal{I})$ can express the light condition of image $\mathcal{I}$ effectively. Different light condition, e.g., in the dark or in daytime, the log-average luminance varies largely. Thus, ADTrack introduced a threshold $\tau$ for illumination judgment as: $S(\mathcal{I})=\left\\{\begin{array}[]{rcl}1&&{\tilde{\mathcal{L}}^{W}(\mathcal{I})<\tau}\\\ 0&&{\tilde{\mathcal{L}}^{W}(\mathcal{I})\geq\tau}\end{array}\right.~{},$ (3) where $S(\mathcal{L})$ can be seen as the night identifier, $S(\mathcal{I})=1$ indicates that image $\mathcal{I}$ is a night image. Remark 2: To test the validity of the above night decider and determine a proper $\tau$, this work selected first frames in benchmark UAV123@10fps [22] as daytime test samples and newly constructed benchmark UAVDark135 as nighttime test samples. The deciding success rate of different thresholds $\tau$ are shown in TABLE I, where the decider can achieve a surprising result of over 99%. Remark 3: During UAV tracking, ADTrack implements illumination adaptation decider in the first frame of a given sequence, then adjusts its mode and following pretreatment stage automatically according to the judgment outcome $S(\mathcal{I})$ in Eq. (3). TABLE I: Success rates of proposed illuminance decider with different thresholds $\tau$. Clearly, the results can surprisingly achieve over 99%, enabling effective night judgment. Thresholds $\tau$ \| 0.12 \| 0.13 \| 0.14 \| 0.15 \| 0.16 \| 0.17 \| 0.18 \| 0.19 ---\|---\|---\|---\|---\|---\|---\|---\|--- Success rate \| 0.979 \| 0.983 \| 0.987 \| 0.991 \| 0.987 \| 0.987 \| 0.991 \| 0.983 ### III-B Pretreatment The pretreatment stage can achieve two purposes. Firstly, for determined night sequences, ADTrack adopts an efficient and effective image enhancer [21] to obtain enhanced images for the subsequent training and detection stages. Secondly, the target-aware mask is acquired by virtue of image illuminance information, so that dual filters learning in the training process and dual response maps generation in the detection process can be realized. Remark 4: To our excitement, the two purposes can be both based on world illumination value $\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})$ in Eq. (1) and log-average luminance $\tilde{\mathcal{L}}^{W}(\mathcal{I})$ in Eq. (2). To realize image enhancing, the global adaptation factor $\mathcal{L}_{\mathrm{g}}(x,y,\mathcal{I})$, which is based on the original world illumination map, is firstly calculated as in [21]: $\mathcal{L}_{\mathrm{g}}(x,y,\mathcal{I})=\frac{\mathrm{log}(\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})/\tilde{\mathcal{L}}^{\mathrm{W}}(\mathcal{I})+1)}{\mathrm{log}(\mathcal{L}^{\mathrm{W}}_{\mathrm{max}}(\mathcal{I})/\tilde{\mathcal{L}}^{\mathrm{W}}(\mathcal{I})+1)}~{},$ (4) where $\mathcal{L}^{\mathrm{W}}_{\mathrm{max}}(\mathcal{I})=\mathrm{\mathrm{max}}(\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I}))$. The calculated factor can be referred to change the pixel value in three intensity channels of each pixel to realize image enhancement as: $\Psi_{\mathrm{m}}(\mathcal{I}_{\mathrm{e}}(x,y))=\Psi_{\mathrm{m}}(\mathcal{I}(x,y))\cdot\frac{\mathcal{L}_{\mathrm{g}}(x,y,\mathcal{I})}{\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})}~{},$ (5) where $\mathcal{I}_{\mathrm{e}}$ denotes the enhanced image based on original $\mathcal{I}$. Since $\mathcal{L}_{\mathrm{g}}(x,y,\mathcal{I})$ varies in different regions that process different illumination, Eq. (5) can readjust the brightness of the whole image while keeping the proportion of the three color channels a constant, i.e., the image color unchanged. Remark 5: The aforementioned strategy is merely the fast first stage of [21], Eq. (5) shows its efficacy for image enhancing. For target-aware mask generation, ADTrack considers the illuminance change $\bm{\Theta}_{\mathcal{L}}(\mathcal{I})$ after enhancement, which can be written as: $\begin{split}\bm{\Theta}_{\mathcal{L}}(\mathcal{I})&=\mathcal{L}^{\mathrm{W}}(\mathcal{I})-\mathcal{L}^{\mathrm{W}}(\mathcal{I}_{\mathrm{e}})\\\ &=\frac{\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})-\mathrm{log}\Big{(}\frac{\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})}{\tilde{\mathcal{L}^{\mathrm{W}}}(\mathcal{I})+1}\Big{)}}{\mathrm{log}(\mathcal{L}^{\mathrm{W}}_{\mathrm{max}}(\mathcal{I})/\tilde{\mathcal{L}}^{\mathrm{w}}(\mathcal{I})+1)}~{}.\end{split}$ (6) Remark 6: The illumination change $\bm{\Theta}_{\mathcal{L}}(\mathcal{I})(x,y)$ of pixel $(x,y)$ in a given image depends only on the first 2 terms in the numerator of Eq. (6), i.e., $\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})-\rm{log}(\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I}))$. Since $\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})\in[0,1]$, the value of $\bm{\Theta}_{\mathcal{L}}(\mathcal{I})$ apparently varies according to the original illumination $\mathcal{L}^{\mathrm{W}}(x,y,\mathcal{I})$. Assume that different objects’ illumination are different under similar light condition in an image due to their various reflectivity. By virtue of Eq. (6), the class of pixels can be indicated as the target or the context. To be specific, consider that the target is located at the center part, the average value $\mu$ and standard deviation $\sigma$ of the center region of $\mathbf{\Theta}_{\mathcal{L}}$ are computed. Following a three-sigma criterion in statistics, which reflects the probability distribution characteristics of samples, pixels in the range $\mu\pm 3\sigma$ are considered targets while others are the context. Then, a binary mask $\mathbf{m}_{r}$ is generated as: $\mathbf{m}_{r}(x,y)=\left\\{\begin{array}[]{rcl}1&&{\mu-3\sigma\leq\mathbf{\Theta}_{\mathcal{L}}(x,y)\leq\mu+3\sigma}\\\ 0&&{\mathrm{else}}\end{array}\right.~{}.$ (7) Ultimately, the expected mask is obtained by $\mathbf{m}=\mathbf{m}_{r}\odot\mathbf{P}$, where $\odot$ denotes element-wise product. $\mathbf{P}\in\mathbb{R}^{w\times h}$ is the cropping matrix, which extracts the value of the target-size area in the middle of the raw mask $\mathbf{m}_{r}$, and set the value of the remaining area to 0 to shield the interference of similar brightness objects in the background. Fig. 4 displays some representative examples of mask generation in all-day conditions. Figure 4: Visualization of mask generation in both nighttime and daytime. From top to bottom, the images denote original patch, illumination map, and generated mask. The sequences person10_1, pedestrian5_2 are from newly constructed UAVDark135, and boat2, person20 are from UAV123@10fps [22]. It’s clear that in both conditions, the proposed method can obtain valid mask with vivid object contour. Thus, according to the outcome in Eq. (3), for $S(\mathcal{I})=1$, ADTrack uses Eq. (5) to obtain enhanced image $\mathcal{I}_{e}$, and for $S(\mathcal{I})=0$, original image $\mathcal{I}$ is utilized. Both daytime sequences and nighttime sequences can adopt Eq. (6) for mask generation. ### III-C Filter Training #### III-C1 Review of BACF Due to both its robustness and efficiency, this work adopts background-aware correlation filter (BACF) [10] as baseline. The BACF tracker achieves its satisfying performance mainly by virtue of the introduction of the cropping matrix $\mathbf{P}$, which expands the training samples hugely without inletting much boundary effect. The training regression equation of the BACF tracker can be expressed as: $\mathcal{E}(\mathbf{w})=\frac{1}{2}\sum_{j=1}^{T}\left\\|\sum_{c=1}^{D}\mathbf{w}^{c\top}\mathbf{P}\mathbf{C}^{j}\mathbf{x}^{c}-\mathbf{y}(j)\right\\|_{2}^{2}+\frac{\lambda}{2}\sum_{c=1}^{D}\left\\|\mathbf{w}^{c}\right\\|_{2}^{2}~{},$ (8) where $\mathbf{w}^{c}\in\mathbb{R}^{N}(c=1,2,\cdots,D)$ is the filter in the $c$-th channel obtained in current frame and $\mathbf{w}=[\mathbf{w}^{1},\mathbf{w}^{2},\cdots,\mathbf{w}^{D}]$ denotes the whole filter. $\mathbf{x}^{c}\in\mathbb{R}^{T}$ is the $c$-th channel of extracted feature map and $\mathbf{y}(j)$ denotes the $j$-th element in the expected Gaussian-shape regression label $\mathbf{y}\in\mathbb{R}^{T}$. Cropping matrix $\mathbf{P}\in\mathbb{R}^{N\times T}$ aims at cropping the center region of samples $\mathbf{x}^{c}$ for training and cyclic shift matrix $\mathbf{C}^{j}\in\mathbb{R}^{T\times T}$ is the same in [31], which is employed to obtain cyclic samples. $\lambda$ is the regularization term parameter. Remark 7: Since in Eq. (8), $T$ and $N$ meet $T>>N$, the filter $\mathbf{w}$ learns far more samples, the negative samples in particular, than in other CF- based trackers. Such strategy makes the filter aware of the background information, resulting in its better discriminative ability. #### III-C2 Proposed ADTrack Apart from BACF [10], which trains single filter $\mathbf{w}$ with both negative and positive target-size samples, ADTrack trains dual filters $\mathbf{w}_{g}$ and $\mathbf{w}_{o}$, which learns context and target information separately. Besides, a constraint term is added into the overall objective to promise more robust tracking on-the-fly. The proposed regression objective can be written as: $\begin{split}\mathcal{E}(\mathbf{w}_{g},\mathbf{w}_{o})=&\sum_{k}(\frac{1}{2}\left\\|\sum_{c=1}^{D}\mathbf{P}^{\top}\mathbf{w}_{k}^{c}\star\mathbf{x}_{k}^{c}-\mathbf{y}\right\\|_{2}^{2}\\!\\!+\frac{\lambda}{2}\sum_{c=1}^{D}\left\\|\mathbf{w}_{k}^{c}\right\\|_{2}^{2})\\\ &+\frac{\mu}{2}\sum_{c=1}^{D}\left\\|\mathbf{w}_{g}^{c}-\mathbf{w}_{o}^{c}\right\\|_{2}^{2}~{},k\in\\{g,o\\}~{},\end{split}$ (9) where $\star$ denotes circular correlation operator, which implicitly executes sample augmentation by circular shift. Differently, $\mathbf{x}_{g}$ denotes the context feature map, while $\mathbf{x}_{o}$ indicates the target region feature map, which is generated using the mask $\mathbf{m}$, i.e., $\mathbf{x}_{o}=\mathbf{m}\odot\mathbf{x}_{g}$. The second and fourth term in Eq. (9) serve as the regularization term to prevent overfitting of the filters. The last term can be considered as the constraint term, where $\mathbf{w}_{g}$ and $\mathbf{w}_{o}$ bind each other during training. In this case, the discriminative ability of both filters will be more robust. $\mu$ is a parameter used to control the impact of the constraint term. Remark 8: In order to maintain historic appearance information of object, this work follows a conventional fashion in [10] for adaptive feature updates using linear interpolation strategy with a fixed learning rate $\eta$ as: $\begin{split}\mathbf{x}^{f}_{k,\mathrm{model}}=\mathbf{x}^{f-1}_{k,\mathrm{model}}+\eta\mathbf{x}^{f}_{k}~{},k\in\\{g,o\\}~{},\end{split}$ (10) where $\mathbf{x}^{f}_{k,\mathrm{model}}$ denotes the training sample in the $f$-th frame, which is utilized to train dual filters in Eq. (9). #### III-C3 Optimization Assume that $\mathbf{w}_{o}$ is given, ADTrack firstly finds the optimal solution of $\mathbf{w}_{g}$. Defining an auxiliary variable $\mathbf{v}$, i.e., $\mathbf{v}=\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}_{g}\in\mathbb{R}^{TD}$, where $\otimes$ denotes Kronecker product, $\mathbf{I}_{N}$ an $N$-order identical matrix. Here, $\mathbf{w}_{g}=[\mathbf{w}^{1\top}_{g},\mathbf{w}^{2\top}_{g},\cdots,\mathbf{w}^{D\top}_{g}]^{\top}\in\mathbb{R}^{ND}$. Then, the augmented Lagrangian form of Eq. (9) is formulated as: $\begin{split}\mathcal{E}(\mathbf{w}_{g},\mathbf{v},\bm{\theta})&=\frac{1}{2}\left\\|\mathbf{v}\star\mathbf{x}-\mathbf{y}\right\\|^{2}_{2}+\frac{\lambda}{2}\left\\|\mathbf{w}_{g}\right\\|_{2}^{2}\\\ &+\frac{\mu}{2}\left\\|\mathbf{w}_{g}-\mathbf{w}_{o}\right\\|_{2}^{2}+(\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}_{g}-\mathbf{v})^{\top}\bm{\theta}\\\ &+\frac{\gamma}{2}\left\\|\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}_{g}-\mathbf{v}\right\\|^{2}_{2}~{},\end{split}$ (11) where $\bm{\theta}=[\bm{\theta}^{1\top},\bm{\theta}^{2\top},\cdots,\bm{\theta}^{D\top}]^{\top}\in\mathbb{R}^{TD}$ is the Lagrangian vector and $\gamma$ denotes a penalty factor. Adopting ADMM [39], Eq. (11) can be dissected and solved by iteratively solving the following three subproblems: $\left\\{\begin{aligned} \mathbf{w}^{e+1}_{g}&=\rm{arg}\min_{\mathbf{w}}\Big{\\{}\frac{\lambda}{2}\left\\|\mathbf{w}^{e}_{g}\right\\|_{2}^{2}+\frac{\mu}{2}\left\\|\mathbf{w}^{e}_{g}-\mathbf{w}_{o}\right\\|_{2}^{2}\\\ &+(\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}^{e}_{g}-\mathbf{v})^{\top}\bm{\theta}+\frac{\gamma}{2}\left\\|\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}^{e}_{g}-\mathbf{v}\right\\|^{2}_{2}\Big{\\}}\\\ \mathbf{v}^{e+1}&=\rm{arg}\min_{\mathbf{v}}\Big{\\{}\frac{1}{2}\left\\|\mathbf{v}^{e}\star\mathbf{x}-\mathbf{y}\right\\|^{2}_{2}+\\\ &(\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}_{g}-\mathbf{v}^{e})^{\top}\bm{\theta}+\frac{\gamma}{2}\left\\|\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{w}_{g}-\mathbf{v}^{e}\right\\|^{2}_{2}\Big{\\}}\\\ \bm{\theta}^{e+1}&=\bm{\theta}^{e}+\gamma(\mathbf{v}^{e+1}-(\mathbf{FP}^{\top}\otimes\mathbf{I}_{D})\mathbf{w}^{e+1}_{g})~{},\\\ \end{aligned}\right.$ (12) where the superscript $\cdot^{e}$ indicates $e$-th iteration. Following superscript ′ represents the optimization objectives. Subproblem $\mathbf{w}^{\prime}_{g}$: By setting the partial derivative of the first subproblem in Eq. (12) with respect to $\mathbf{w}_{o}$ as zero, we can find the closed-form solution of $\mathbf{w}^{\prime}_{g}$, which is expressed as: $\mathbf{w}^{\prime}_{g}=\frac{\mu\mathbf{w}_{o}+T\bm{\theta}+\gamma T\mathbf{v}}{\lambda+\mu+\gamma T}~{}.$ (13) Subproblem $\mathbf{v}^{\prime}$: To effectively obtain the closed-form of $\mathbf{v}$, this work firstly turn the second subproblem in Eq. (12) into Fourier domain using discrete Fourier transform (DFT) as: $\begin{split}\mathbf{v}^{\prime}=\rm{arg}\min_{\hat{\mathbf{v}}}&\Big{\\{}\frac{1}{2T}\left\\|\hat{\mathbf{v}}^{}\odot\hat{\mathbf{x}}-\hat{\mathbf{y}}\right\\|^{2}_{2}+\hat{\bm{\theta}}^{\top}(\sqrt{T}\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{F}_{N}\mathbf{w}_{g}\\\ &-\hat{\mathbf{v}})+\frac{\gamma}{2T}\left\\|\sqrt{T}\mathbf{I}_{N}\otimes\mathbf{P}^{\top}\mathbf{F}_{N}\mathbf{w}_{g}-\hat{\mathbf{v}}\right\\|^{2}_{2}\Big{\\}}~{},\\\ \end{split}$ (14) where $\hat{\cdot}$ denotes the Fourier form of a variable, i.e., $\hat{\mathbf{x}}=\sqrt{T}\mathbf{F}_{T}\mathbf{x}$. $\mathbf{F}_{T}\in\mathbb{C}^{T\times T}$ is the Fourier matrix. Superscript $\cdot^{}$ indicates the complex conjugate. Remark 9: Since circular correlation in time domain is turned into element- wise product in Fourier domain, separating sample in Eq. (14) across pixels, e.g., $\mathbf{x}(t)=[\mathbf{x}^{1}(t),\mathbf{x}^{2}(t),\cdots,\mathbf{x}^{D}(t)]\in\mathbb{R}^{T\times D},(t=1,2,\cdots,T)$, each $\hat{\mathbf{v}}^{\prime}(t)$ can be solved as: $\begin{split}\hat{\mathbf{v}}^{\prime}(t)=&\Big{(}\hat{\mathbf{x}}(t)\hat{\mathbf{x}}(t)^{\top}+T\gamma\mathbf{I}_{D}\Big{)}^{-1}\\\ &\times\Big{(}\hat{\mathbf{y}}(t)\hat{\mathbf{x}}(t)-T\hat{\bm{\theta}}(t)+T\gamma\hat{\mathbf{w}}_{g}(t)\Big{)}~{}.\end{split}$ (15) Sherman-Morrison formula [40] is applied to avoid the time-consuming matrix inversion operation and Eq. (15) is turned into: $\begin{split}\hat{\mathbf{v}}^{\prime}(t)=\frac{1}{\gamma T}\Big{(}\hat{\mathbf{y}}(t)\hat{\mathbf{x}}(t)-T\hat{\bm{\theta}}(t)+\gamma T\hat{\mathbf{w}}_{g}(t)\Big{)}-\\\ \frac{\hat{\mathbf{x}}(t)}{\gamma Tb}\Big{(}\hat{\mathbf{y}}(t)\hat{\mathbf{s}}_{\mathbf{x}}(t)-T\hat{\mathbf{s}}_{\bm{\theta}}(t)+\gamma T\hat{\mathbf{s}}_{\bm{w}_{g}}(t)\Big{)}~{},\end{split}$ (16) where $\hat{\mathbf{s}}_{\mathbf{x}}(t)=\hat{\mathbf{x}}(t)^{\top}\hat{\mathbf{x}},\hat{\mathbf{s}}_{\bm{\theta}}=\hat{\mathbf{x}}(t)^{\top}\hat{\mathbf{\theta}},\hat{\mathbf{s}}_{\bm{w}_{g}}=\hat{\mathbf{x}}(t)^{\top}(t)\hat{\mathbf{w}}_{g}$, and $b=\hat{\mathbf{s}}_{\mathbf{x}}(t)+T\gamma$ are scalar. The positions of $\mathbf{w}_{g}$ and $\mathbf{w}_{o}$ in Eq. (9) are equivalent. When an solving iteration of $\mathbf{w}_{g}$ is completed, then the same ADMM iteration operation is performed to obtain the optimized solution of $\mathbf{w}_{o}$. ### III-D Target Detection Given the expected filter $\mathbf{w}^{f}_{g}$ and $\mathbf{w}^{f}_{o}$ in the $f$-th frame, the response map $\mathbf{R}$ regarding the detection samples $\mathbf{z}^{f+1}$ in the $(f+1)$-th frame can be obtained by: $\begin{split}\mathbf{R}=\mathcal{F}^{-1}\sum_{c=1}^{D}\big{(}\hat{\mathbf{w}}^{f,c}_{g}\odot\hat{\mathbf{z}}^{f+1,c}_{g}+\psi\hat{\mathbf{w}}^{f,c}_{o}\odot\hat{\mathbf{z}}^{f+1,c}_{o}\big{)}~{},\end{split}$ (17) where $\mathcal{F}^{-1}$ means inverse discrete Fourier transform. $\mathbf{z}_{g}^{f+1,c}$ denotes the $c$-th channel of resized search region samples extracted in the $(f+1)$-th frame, and $\mathbf{z}_{o}^{f+1,c}$ is the $c$-th channel of the masked samples similar to $\mathbf{x}_{o}$. $\psi$ is a weight parameter that controls the impact response map generated by context filter and object filter. Finally, the object location in the $(f+1)$-th frame can be estimated at the peak of response map $\mathbf{R}$. The holonomic pipeline pseudo code of ADTrack is summarized in Algorithm 1. 1 Input: A video sequence of $F$ frames. Position ($\mathbf{p}^{1}$) and size ($\mathbf{s}^{1}$) of the tracked object in the first frame $\mathcal{I}^{1}$. Output: Estimated position ($\mathbf{p}^{f}$) and size ($\mathbf{s}^{f}$) of the object in all upcoming frames. 2 Construct the Gaussian label function $\mathbf{y}$. 3 for _frame number $f=1$ to end_ do 4 if _$f=1$_ then 5 Calclate log-average illuminance $\tilde{\mathcal{L}}^{\mathrm{W}}(\mathcal{I}^{1})$ and dark identifier $S(\mathcal{I}^{1})$ and adjust the mode of the tracker (Sec. III-A). 6 Crop the training patch from $\mathcal{I}^{1}$ with $\mathbf{p}^{1}$ and $\mathbf{sc}\times\mathbf{s}^{1}$, where $\mathbf{sc}$ is a predefined scale factor. 7 if _$S(\mathcal{I}^{1})==1$_ then 8 Do image enhancing to obtain enhanced patch (Sec. III-B). 9 end if 10 Obtain target-aware mask $\mathbf{m}$ (Sec. III-B). 11 Extract context features $\mathbf{x}^{1}_{g}$ and target features $\mathbf{x}^{1}_{o}$ of the obtained patch. 12 Update the appearance model $\mathbf{x}^{1}_{k,\mathrm{model}}=\mathbf{x}^{1}_{k}~{},k\in\\{g,o\\}$ for filter training. 13 Learn context and target filter $\mathbf{w}_{k}~{},k\in\\{g,o\\}$ (Sec. III-C). 14 15 else 16 Crop the search patch from $\mathcal{I}^{f}$ with $\mathbf{p}^{f-1}$ and $\mathbf{sc}\times\mathbf{s}^{f-1}$. 17 if _$S(\mathcal{I}^{1})==1$_ then 18 Do image enhancing to obtain enhanced patch (Sec. III-B). 19 end if 20 Obtain target-aware mask $\mathbf{m}$ (Sec. III-B). 21 Extract context and target search features $\mathbf{z}^{f}_{k}~{},k\in\\{g,o\\}$ of the obtained patch. 22 Generate the fused response map $\mathbf{R}$ (Sec. III-D). 23 Estimate $\mathbf{p}^{f}$ and $\mathbf{s}^{f}$. 24 Crop the training patch from $\mathcal{I}^{f}$ with $\mathbf{p}^{f}$ and $\mathbf{sc}\times\mathbf{s}^{f}$. 25 if _$S(\mathcal{I}^{1})==1$_ then 26 Do image enhancing to obtain enhanced patch (Sec. III-B). 27 end if 28 Obtain target-aware mask $\mathbf{m}$ (Sec. III-B). 29 Extract context features $\mathbf{x}^{f}_{g}$ and target features $\mathbf{x}^{f}_{o}$ of the obtained patch. 30 Update the appearance model using Eq. (10) for dual filter training. 31 Learn context and target filter $\mathbf{w}_{k}~{},k\in\\{g,o\\}$ (Sec. III-C). 32 end if 33 34 end for Algorithm 1 ADTrack tracker ## IV UAVDark135 Tracking Benchmark TABLE II: Numbers of sequences, minimum, maximum, mean frames in each sequence, and total frames in 6 benchmarks, i.e., newly constructed UAVDark135, UAV123, UAV123@10fps [22], DTB70 [23], UAVDT [41], and VisDrone2019-SOT [42]. Red, green, and blue denotes the first, second and third place respectively. Benchmark \| UAVDark135 \| UAV123 \| UAV123@10fps \| DTB70 \| UAVDT \| VisDrone2019-SOT ---\|---\|---\|---\|---\|---\|--- Sequences \| 135 \| 123 \| 123 \| 70 \| 50 \| 132 Each sequence \| 216 \| 4571 \| 929 \| 109 \| 3085 \| 915 \| 37 \| 1029 \| 306 \| 68 \| 699 \| 225 \| 82 \| 2969 \| 742 \| 90 \| 2970 \| 833 Min \| Max \| Mean Total frames \| 125466 \| 112578 \| 37607 \| 15777 \| 37084 \| 109909 TABLE III: Detailed explanation of the attributes in newly built UAVDark135, which are commonly confronted in UAV tracking. Attributes \| Explanation ---\|--- VC \| \| Viewpoint Change: In the sequence, different aspects, e.g., --- front, side, and top aspect, of the tracking object are captured and involved. FM \| \| Fast Motion: There exist two continuous frames, where the --- center locations of the tracking object change more than 20 pixels. LR \| \| Low Resolution: There exist frames, where the --- tracking object is small, whose total resolution is fewer than 20 pixels. OCC \| \| Occlusion: There exist frames, where the --- tracking object is partially or fully occluded by obstacles. IV \| \| Illumination Variation: In the sequence, the --- tracking object undergoes various light conditions. ### IV-A Platform and Statistics Standing as the first UAV dark tracking benchmark, the UAVDark135 contains totally 135 sequences captured by a standard UAV222This work utilized Parrot Bebop 2 drone as shooting platform. More detailed information can be found at https://support.parrot.com/us/support/products/parrot-bebop-2. at night. The benchmark includes various tracking scenes, e.g., crossings, t-junctions, road, highway, and consists of different kinds of tracked objects like people, boat, bus, car, truck, athletes, house, etc. To extent the covered scenes, the benchmark also contains some sequences from YouTube, which were shot on the sea. The total frames, mean frames, maximum frames, and minimum frames of the benchmark are 125466, 929, 4571, and 216 respectively, making it suitable for large-scale evaluation. TABLE LABEL:tab:benchmarks exhibits main statistics of UAVDark135 against existing UAV tracking benchmarks, i.e., UAV123@10fps, UAV123 [22], DTB70 [23], UAVDT [41], and VisDrone2019-SOT [42] (VisDrone2019-SOT testset-challenge is not included). The videos are captured at a frame-rate of 30 frames/s (FPS), with the resolution of 1920$\times$1080. Remark 10: Despite the fact that there exist some sequences captured at night in benchmark UAVDT [41] and VisDrone2019-SOT [42], it is far from an exhaustive dark tracking evaluation. Besides, the night sequences are actually well-illuminated in [41, 42], which can not represent more common dark tracking scenes in UAVDark135, where the light conditions are much more hash. ### IV-B Annotation The frames in UAVDark135 are all manually annotated, where a sequence is completely processed by the same annotator to ensure consistency. Since in some dark scenes the object is nearly invisible, annotation process is much more strenuous. After the first round, 5 professional annotators carefully checked the results and made revision for several rounds to reduce errors as much as possible in nearly 2 months. Since the boundary contour of the object is not obvious in the dark, the result boxes of the first annotation fluctuates in continuous image frames. However, the actual motion process of the object should be smooth. In these considerations, we record the original annotation every 5 frames for the sequence with extremely severe vibration, and the results of the remaining frames are obtained by linear interpolation, which is closer to the position and scale variation of the real object. Figure 5: Sequences distribution comparison of 6 UAV tracking benchmarks. The abscissa are 5 attributes, and the ordinate is the numbers of the sequences. The sequences in different benchmarks are marked by different colors, which are explained in the legend. Note that not all benchmarks made contribution to all 5 attributes. (a) (b) (c) Figure 6: Overall performance of SOTA handcrafted CF-based trackers on UAV123@10fps [22], DTB70 [23], and newly built UAVDark135. The evaluation metric in precision plot is the distance precision (DP) under center location error (CLE) = 20 pixels, and the metric in success rate plot is the area under curve (AUC). Clearly ADTrack maintains its robustness in all 3 benchmarks by virtue of its dual regression. ### IV-C Attributes To better evaluate the trackers’ abilities under special challenges, UAVDark135 also provides 5 commonly encountered challenge attributes in UAV tracking, following our prior work [43], i.e., viewpoint change (VC), fast motion (FM), low resolution (LR), occlusion (OCC), and illumination variation (IV). TABLE III elaborately explains the criterion for each attribute. Additionally, Fig. 5 displays sequences distribute comparison of 6 UAV tracking benchmarks. Clearly, UAVDark135 distributes both evenly and considerably in the five attributes. ## V Experiment and Evaluation TABLE IV: Average results by all 328 sequences of handcrafted trackers on benchmarks UAV123@10fps [22], DTB70 [23], and newly constructed UAVDark135. Red, green, and blue denotes the first, second and third place respectively. Here, the abilities of the trackers under all-day conditions are evaluated. Tracker \| ADTrack \| AutoTrack \| ARCF-HC \| ARCF-H \| MCCT-H \| STRCF \| KCC \| fDSST \| DSST \| BACF \| CSR-DCF \| ECO-HC \| Staple_CA \| Staple \| KCF \| SRDCF \| SAMF ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Venue \| Ours \| ’20CVPR \| ’19ICCV \| ’19ICCV \| ’18CVPR \| ’18CVPR \| ’18AAAI \| ’17TPAMI \| ’17TPAMI \| ’17ICCV \| ’17CVPR \| ’17CVPR \| ’17CVPR \| ’16CVPR \| ’15TPAMI \| ’15ICCV \| ’14ECCV DP \| 0.659 \| 0.651 \| 0.638 \| 0.591 \| 0.514 \| 0.611 \| 0.459 \| 0.465 \| 0.420 \| 0.581 \| 0.576 \| 0.601 \| 0.495 \| 0.510 \| 0.391 \| 0.566 \| 0.434 AUC \| 0.480 \| 0.468 \| 0.462 \| 0.433 \| 0.380 \| 0.451 \| 0.330 \| 0.354 \| 0.321 \| 0.429 \| 0.415 \| 0.449 \| 0.367 \| 0.377 \| 0.266 \| 0.427 \| 0.312 FPS \| 31.621 \| 45.485 \| 22.585 \| 32.320 \| 44.858 \| 20.438 \| 29.393 \| 122.976 \| 58.113 \| 33.911 \| 9.274 \| 53.571 \| 46.829 \| 81.216 \| 374.912 \| 8.583 \| 7.518 This section displays exhaustive experimental evaluations, involving night benchmark UAVDark135, daytime benchmark UAV123@10fps [22], and DTB70 [23]. In subsection V-A, implementation details including experiment platform, parameter settings, features, and metrics are introduced. Subsection V-B gives a comprehensive comparison of the handcrafted CF-based trackers on the benchmarks, where the superiority of proposed ADTrack for all-day UAV tracking is demonstrated. Subsection V-C presents attribute-based evaluation of the handcrafted CF-based trackers to test their abilities under UAV special challenges. In subsection V-D, we also invite SOTA deep trackers that utilize convolution neural network (CNN) for dark tracking comparison. Lastly, in subsection V-E, we extended ablation study and parameter analysis to further demonstrate the validity of different modules in proposed ADTrack. ### V-A Implementation Details #### V-A1 Platform The experiments extended in this work were mainly performed on MATLAB R2019a. The main hardware adopted consists of an Intel Core I7-8700K CPU and 32GB RAM. #### V-A2 Parameters To guarantee the fairness and objectivity of the evaluation, the tested trackers from other works have maintained their official initial parameters. The different parameters of two conditions in ADTrack are as follows: In daytime mode, $\mu$ is set as 280. During detection, weight $\psi$ is set as $0.02$. Translation filter takes learning rate $\eta_{t}=0.032$ for model update and for scale filter, $\eta_{s}$ is set as 0.016. In nighttime mode, $\mu$ is set as 200. During detection, weight $\psi$ is set as $0.01$. Translation filter takes learning rate $\eta_{t}=0.024$ for model update and for scale filter, $\eta_{s}$ is set as 0.023. Remark 11: ADTrack can adapt to the given light condition in its first stage, where the tracking mode mentioned is switched automatically without manually adjusting. #### V-A3 Features and Scale Estimation ADTrack uses handcrafted features for appearance representations, i.e., gray- scale, a fast version of histogram of oriented gradient (fHOG) [44], and color names (CN) [45]. Note that gray-scale and CN features can be valid in ADTrack thanks to low-light enhancement. The cell size for feature extraction is set as $4\times 4$. ADTrack adopts the scale filter proposed by [11] to perform accurate scale estimation. #### V-A4 Metrics In the experiment evaluation, we mainly use two metrics: distance precision (DP) and area under curve (AUC). DP is based on the distance between the center points of the predicted box and the target ground-truth, and AUC is based on the intersection ratio of the predicted box and the target ground- truth box. ### V-B Overall Evaluation Using merely handcrafted features, most handcrafted CF-based trackers can achieve satisfying running speed, by virtue of their light calculation, while ensuring their robustness under various tracking scenes onboard UAV. This work employs proposed ADTrack and 16 SOTA handcrafted CF-based trackers, i.e., AutoTrack [8], KCF [31], SAMF [46], SRDCF [32], STRCF [33], BACF [10], DSST & fDSST [11], ECO-HC [47], ARCF-HC & ARCF-H [9], KCC [48], MCCT-H [49], CSR-DCF [37], Staple [34], and Staple_CA [50], for evaluation on tracking benchmarks, UAV123@10fps [22], DTB70 [23], and UAVDark135 to demonstrate the robustness of the proposed ADTrack in all-day UAV tracking comprehensively. Figure 7: Visualization of some typical tracking scenes in both daytime and nighttime. Sequences Car2, RcCar6, and uav3 are from the daytime benchmarks DTB70 and UAV123@10fps. Sequences car13, pedestrian_l, and running_man are from the new nighttime benchmark UAVDark135. Clearly, ADTrack favorably maintains its robustness in all-day UAV tracking challenges. The typical sequences were made into video, which can be found at https://youtu.be/cJMUKF4J38A. #### V-B1 Daytime Performance In Fig. 6, DP and AUC comparison on benchmarks, respectively, UAV123@10fps [22], DTB70 [23] is exhibited in the first 2 columns, where ADTrack ranks first in both metrics. Specifically, in Fig. 6, ADTrack surpasses the second- best AutoTrack tracker (0.671) [8] by 1.6% (0.682). In terms of AUC, ADTrack surpasses its baseline BACF tracker (0.413) [10] by over 19% (0.482). Fig. 6 shows that ADTrack brings its baseline (0.581) up by 24% (0.722) in DP, and exceeds the brilliant AutoTrack tracker (0.478) by nearly 4% (0.497) in AUC. The outstanding results achieved by ADTrack in daytime indicates its strong robustness in real-world UAV tracking scenes by virtue of its dual filter learning. #### V-B2 Nighttime Performance Fig. 6 displays the excellent handcrafted CF-based trackers’ DPs under newly constructed benchmark UAVDark135, where clearly ADTrack exceeds all other trackers, surpassing the second best tracker (0.599) by over 1% (0.605). Additionally, ADTrack enjoys its satisfying advantages in success rate as well, exceeding the baseline tracker (0.460) by over 1.9% (0.469). In dark scenes like UAVDark135, ADTrack maintains its robustness, providing a favorable choice for UAV all-day tracking. TABLE V: Average performance of the handcrafted trackers by UAV special attributes. Obviously, ADTrack maintains its superiority in most challenges. Tracker Metric \| DP \| AUC ---\|---\|--- VC \| FM \| LR \| OCC \| IV \| VC \| FM \| LR \| OCC \| IV ADTrack \| 0.637 \| 0.63 \| 0.668 \| 0.622 \| 0.605 \| 0.464 \| 0.464 \| 0.471 \| 0.434 \| 0.437 AutoTrack \| 0.622 \| 0.588 \| 0.651 \| 0.598 \| 0.599 \| 0.448 \| 0.433 \| 0.455 \| 0.412 \| 0.431 ARCF-HC \| 0.61 \| 0.61 \| 0.649 \| 0.595 \| 0.597 \| 0.438 \| 0.448 \| 0.458 \| 0.417 \| 0.433 ARCF-H \| 0.565 \| 0.551 \| 0.606 \| 0.537 \| 0.56 \| 0.413 \| 0.4 \| 0.427 \| 0.373 \| 0.411 MCCT-H \| 0.504 \| 0.471 \| 0.504 \| 0.503 \| 0.476 \| 0.366 \| 0.361 \| 0.367 \| 0.353 \| 0.353 STRCF \| 0.584 \| 0.568 \| 0.611 \| 0.584 \| 0.59 \| 0.424 \| 0.43 \| 0.442 \| 0.406 \| 0.437 KCC \| 0.425 \| 0.451 \| 0.493 \| 0.427 \| 0.459 \| 0.309 \| 0.329 \| 0.348 \| 0.297 \| 0.326 fDSST \| 0.462 \| 0.406 \| 0.481 \| 0.436 \| 0.424 \| 0.343 \| 0.327 \| 0.363 \| 0.317 \| 0.329 DSST \| 0.425 \| 0.342 \| 0.413 \| 0.385 \| 0.391 \| 0.316 \| 0.275 \| 0.303 \| 0.274 \| 0.298 BACF \| 0.55 \| 0.554 \| 0.582 \| 0.517 \| 0.537 \| 0.41 \| 0.411 \| 0.414 \| 0.371 \| 0.402 CSR-DCF \| 0.57 \| 0.536 \| 0.576 \| 0.561 \| 0.54 \| 0.399 \| 0.383 \| 0.405 \| 0.387 \| 0.381 ECO-HC \| 0.584 \| 0.524 \| 0.572 \| 0.599 \| 0.56 \| 0.434 \| 0.409 \| 0.426 \| 0.423 \| 0.421 Staple_CA \| 0.476 \| 0.465 \| 0.534 \| 0.484 \| 0.486 \| 0.353 \| 0.353 \| 0.387 \| 0.346 \| 0.36 Staple \| 0.463 \| 0.498 \| 0.567 \| 0.491 \| 0.512 \| 0.343 \| 0.379 \| 0.407 \| 0.349 \| 0.377 KCF \| 0.376 \| 0.31 \| 0.38 \| 0.363 \| 0.353 \| 0.251 \| 0.227 \| 0.262 \| 0.242 \| 0.24 SRDCF \| 0.526 \| 0.549 \| 0.587 \| 0.509 \| 0.55 \| 0.403 \| 0.418 \| 0.43 \| 0.365 \| 0.418 SAMF \| 0.414 \| 0.36 \| 0.427 \| 0.418 \| 0.391 \| 0.293 \| 0.267 \| 0.303 \| 0.288 \| 0.281 #### V-B3 All-Day Performance To evaluate the abilities of the trackers in all-day tracking scenes, this part takes the average results on 3 benchmarks by sequences, i.e. daytime benchmark UAV123@10fps [22], DTB70 [23], and nighttime benchmark UAVDark135, together 328 sequences. TABLE IV exhibits the average results of SOTA handcrafted CF-based trackers. Obviously, the proposed ADTrack possesses great advantages over all the other trackers in both DP and AUC. In specific, ADTrack (0.659) improves the precision of its baseline (0.581) by more than 13%, surpassing second place (0.651) by over 1%. In terms of AUC, ADTrack (0.480) is far ahead of the second place (0.468), up nearly 2.6%. Fig. 7 displays some representative tracking scenes in all-day condition, where ADTrack exhibits competitive in robustness against other arts. In addition to the satisfying tracking performance, ADTrack achieves an average speed of over 31 FPS on a single CPU, meeting real-time requirement of onboard UAV tracking. Evidently, ADTrack achieves promising tracking performance in all-day condition, through day and night, thus greatly expanding the tracking based application onboard UAV around-the-clock. ### V-C Attribute-Based Evaluation To clearly evaluate the abilities of the tracker under UAV specific challenge, this part displays their performance in the aforementioned 5 UAV tracking attributes, i.e., VC, FM, LR, OCC, and IV. TABLE V gives the average results on all 328 sequences on 3 benchmarks. For the authoritative daytime benchmarks, this subsection follows our precious work [43] to rewrite their official attributes. Clearly, ADTrack outperforms all other trackers in most attributes under 2 evaluation metrics. Specially, in FM, ADTrack surpasses second-best tracker ARCF-HC [9] by over 3% in both DP and AUC. The results demonstrate the satisfying comprehensive tracking performance and favorable robustness of ADTrack in common challenges. ### V-D Against Deep Trackers This subsection focuses on comparison between proposed ADTrack and deep trackers which utilize off-line trained deep network for feature extraction or template matching. This work invites totally 11 SOTA deep trackers, i.e., SiamRPN++ [18], DaSiamRPN [51], SiamFC++ [15], ASRCF [52], ECO [47], UDT+ [14], HCFT [53], CoKCF [54], CFWCR [55], DeepSTRCF [33], and MCCT [49], to evaluate their performance in UAVDark135. From Fig. 8, ADTrack outperforms all deep trackers in terms of DP and AUC under benchmark UAVDark135. Using merely single CPU, ADTrack still achieves a real-time speed at over 30 FPS, while many deep trackers are far from real-time even on GPU, demonstrating the excellence of ADTrack for real-time UAV tracking against the deep trackers. Remark 12: The results illustrate that the top-ranked deep trackers in recent years, especially the trackers without online update, e.g., SiamRPN++ [18], DaSiamRPN [51], SiamFC++ [15], fail to maintain their robustness in real-world common dark scenes, since the off-the-shelf CNNs they utilize are trained by daytime images, ending up in their huge inferiority compared with online- learned ADTrack in the dark. Since there lacks sufficient dark images for training, off-line trained deep trackers fall short onboard UAV tracking at night. Figure 8: Comparison of proposed ADTrack and the deep trackers on self- constructed UAVDark135. Evidently, most deep trackers fall short in hash dark scenes, while ADTrack maintains its robustness. ### V-E Ablation Study and Parameter Analysis #### V-E1 Component Validity Analysis To demonstrate the effectiveness of the proposed components, ablation study is conducted on all three benchmarks. The average AUC results by sequences on 3 benchmarks are displayed in Fig. 9, where from bottom to top, proposed components, i.e., weight sum, dual filter constraint, and illumination adaptation, were disabled one by one. ADTrack_aed denotes ADTrack without the weight sum in detection phase. In ADTrack_ae, dual filter training constraint is disabled on the basis of ADTrack_aed. ADTrack_a denotes ADTrack_ae without the image enhancer. And ADTrack_b_day, ADTrack_b_dark respectively denotes ADTrack with daytime parameters and nighttime parameters without any proposed module. The first 3 bars in Fig. 9 has demonstrated the validity of illumination adaptation module, and the last 3 bars illustrates how dual filter constraint and weight sum boosts the trackers’ performance. #### V-E2 Impacts of Key Parameters The key parameters in ADTrack are the constraint parameter $\mu$ in the training regression equation and weight parameter $\psi$ in detection stage. We investigate the impact of two parameters on tracking results on nighttime benchmark UAVDark135, i.e., DP and AUC, which is shown in Fig. 10. Note that when $\mu$ varies, $\psi$ remains 0.01 unchanged, and when $\psi$ changes, $\mu$ is settled as 200. Figure 9: Ablation study of the modules in ADTrack. ADTrack, ADTrack_aed, ADTrack_ae, ADTrack_a, ADTrack_b_day, and ADTrack_b_dark respectively denote ADTrack with different component activated. Note that ADTrack_ae performs slightly inferior than ADTrack_a since the enhancer needs to cooperate with the dual filter learning and dual response fusion modules to work effectively. Figure 10: Parameter analysis of ADTrack on newly built benchmark UAVDark135. With other parameters remaining fixed, the tracking performance with different $\mu$ (Top) and with different $\psi$ (Bottom) are displayed. The chosen parameters along with their results are marked out by dotted lines. ## VI Conclusion This work puts forward a novel real-time tracker with illumination adaptation and anti-dark function, i.e., ADTrack. ADTrack first implements illumination adaptation to decide day-night condition and switches its tracking mode. Then, pretreatment is carried out where proper training patch and target-aware mask are generated based on an image enhancer. With the mask, ADTrack proposes innovative dual filter regression model, in which the dual filters restrict each other in training and compensate each other in detection. In addition, the first large-scale dark tracking benchmark, UAVDark135, is also built in this work for visual tracking community. 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# Allocating Opportunities in a Dynamic Model of Intergenerational Mobility Hoda Heidari Carnegie Mellon University <EMAIL_ADDRESS> Jon Kleinberg Cornell University <EMAIL_ADDRESS> ###### Abstract Opportunities such as higher education can promote intergenerational mobility, leading individuals to achieve levels of socioeconomic status above that of their parents. We develop a dynamic model for allocating such opportunities in a society that exhibits bottlenecks in mobility; the problem of optimal allocation reflects a trade-off between the benefits conferred by the opportunities in the current generation and the potential to elevate the socioeconomic status of recipients, shaping the composition of future generations in ways that can benefit further from the opportunities. We show how optimal allocations in our model arise as solutions to continuous optimization problems over multiple generations, and we find in general that these optimal solutions can favor recipients of low socioeconomic status over slightly higher-performing individuals of high socioeconomic status — a form of socioeconomic affirmative action that the society in our model discovers in the pursuit of purely payoff-maximizing goals. We characterize how the structure of the model can lead to either temporary or persistent affirmative action, and we consider extensions of the model with more complex processes modulating the movement between different levels of socioeconomic status. ## 1 Introduction Intergenerational mobility — the extent to which an individual’s socioeconomic status differs from the status of their prior generations of family members — has emerged as a central notion in our understanding of inequality. A large amount of empirical work has gone into estimating the extent of mobility for different subsets of society; while many of the effects are complex and challenging to measure, two broad and fairly robust principles emerge from this work. First, socioeconomic status is persistent across generations: an individual’s socioeconomic status is strongly dependent on parental status. As Lee and Solon (2009) write in the opening to their survey of this topic, “Over the past two decades, a large body of research has documented that the intergenerational transmission of economic status in the United States is much stronger than earlier sociological and economic analyses had suggested”. Second, certain types of opportunities can serve as strong catalysts for socioeconomic mobility; a canonical example is higher education, which has the potential to raise an individual’s socioeconomic status (and, by the previous principle, that of their current or future children as well). As Chetty et al. (2014) write, “The fact that the college attendance is a good proxy for income mobility is intuitive given the strong association between higher education and subsequent earnings”. An important question from a social planning perspective is thus the choice of policy for allocating opportunities to people of different levels of socioeconomic status. (Again, we can think of access to higher education as a running example in this discussion.) Many goals can motivate the choice of policy, including the reduction of socioeconomic inequality and the prioritization of opportunities to those most in need. Such goals are often viewed as operating in tension with the aim of maximizing the achievable payoff from the available opportunities, which would seem to suggest targeting the opportunities based only on the anticipated performance of the recipient, not their socioeconomic status. In this view, society is implicitly being asked to choose between these goals; this consideration forms a central ingredient in the informal discourse and debate around the allocation of opportunity. But through all of this, a challenging question remains: to what extent is the tension between these goals genuine, and to what extent can they be viewed as at least partially in alignment? A large body of work in economics compares various allocation policies in terms of the above seemingly-competing criteria — typically in simplified settings in which only two generations are considered. The literature includes seminal work by Nobel Laureate Garry Becker with Nigel Tomes (Becker and Tomes, 1986) and by Glenn Loury (Loury, 1981). In multigenerational settings, however, deriving the optimal policy becomes exceedingly challenging, and it has been highlighted as a class of open questions in this literature. For example, in his work on models of college admissions and intergenerational mobility, Durlauf (2008) notes: “ A college admissions rule has intergenerational effects because it not only influences the human capital of the next generation of adults, but also affects the initial human capital of the generation after next. […] Efficiency in student allocation [in this case] is far more complicated than before. I am unaware of any simple way of describing efficiency conditions for college assignment rules analogous to [the above setting].” In this work, we address this challenge and the associated open questions concerning the behavior of multigenerational models. A key ingredient in our progress on these questions is the development of methods for working with a class of Markov Decision Processes that operate over continuous states and continuous actions. Our analysis of multigenerational models enables us to investigate the apparent tension between efficiency and fairness considerations in allocating opportunities. #### Allocating Opportunities in a Payoff-Maximizing Society. We work with a simple mathematical model representing a purely payoff- maximizing society, operating over multiple generations. As we discuss briefly in Section 1 and at more length in Appendix A, our model is grounded in the types of models proposed in economic theory work on these problems. The society must decide how to allocate opportunities in each generation across a population heterogeneous in its socioeconomic status. The payoff to the society is the total performance of everyone who receives the opportunities, summed (with discounting) over all generations. Although the set-up of the model is highly streamlined, the analysis of the model becomes quite subtle since society must solve a continuous-valued dynamic programming problem over multiple generations. What we find from the model is that the optimal solution will in general tend to offer opportunities to individuals of lower socioeconomic status over comparable individuals of higher socioeconomic status, even when these competing individuals are predicted to have a slightly better performance from receiving the opportunity. This is not arising because the optimal solution has any a priori interest in reducing socioeconomic inequality (although such goals are important in their own right (Forde-Mazrui, 2004)); rather it is strictly trying to maximize payoff over multiple generations. But given two individuals of equal predicted performance, the one with lower socioeconomic status confers an added benefit to the payoff function: their success would grow the size of the socioeconomically advantaged class, resulting in higher payoffs in future generations. Because the difference in payoff contributions between these two individuals is strictly positive, the same decision would be optimal even if the individual of lower socioeconomic status had a slightly lower predicted performance from receiving the opportunity. The optimal solution should still favor the candidate with lower status in this case. In other words, the society in this model discovers a form of socioeconomic affirmative action in allocating opportunities, based purely on payoff- maximizing motives. The model thus offers a view of a system in which reducing inequality is compatible with direct payoff maximization. In this sense, our results belong to a genre of analyses (popularized by Page (Page, 2008) and others) asserting that policies and interventions that we think of as motivated by equity concerns, can also be motivated by purely performance- maximizing considerations: even if society only cares about performance, not equity, it should still (at least in the underlying models) undertake these policies. In addition to providing a purely utilitarian motivation for socioeconomic affirmative action, our model provides novel insights regarding the shape and extent of effective affirmative action policies by specifying the way in which criteria for receiving the opportunity should be adjusted based on socioeconomic status to maximize society’s performance across multiple generations. We now give a rough overview of the model and results; a complete description of the model is provided in the following section. #### A Model for Allocating Opportunities. We consider a population that is partitioned into two groups of different socioeconomic status: $D$ (disadvantaged), consisting of a $\phi_{0}$ fraction of the population, and $A$ (advantaged), consisting of a $\phi_{1}=1-\phi_{0}$ fraction of the population. Each agent $i$ (from either group) has an ability $a_{i}$ drawn uniformly at random from the interval $[0,1]$, Society has the ability to offer an opportunity to an $\alpha$ fraction of the population. Note that the parameter $\alpha$ specifies the inherent limitation on the amount of opportunities available. Since opportunities are limited, the society has to wrestle with the question of how to allocate them. An individual $i$ in group $D$ who is offered the opportunity has a probability $\sigma a_{i}$ of succeeding at it, for a parameter $0<\sigma<1$. An individual $i$ in group $A$ who is offered the opportunity has a probability $\sigma a_{i}+\tau$ of succeeding at it, for the same $\sigma$ and an additional parameter $0<\tau\leq 1-\sigma$ reflecting the advantage. We will refer to the above quantities as the success probabilities of the agents. Success probabilities reflect various levels of performance when agents are offered the opportunity. Anyone in group $D$ who is offered the opportunity and succeeds at it moves up to group $A$. Each individual is then replaced by one offspring of the same socioeconomic status and the process continues to the next generation. In the general form of the model, there is also some probability that an individual’s offspring does not perfectly inherit their socioeconomic status. The payoff to society is the number of individuals who succeed at the opportunity summed over all generations, with the generation $t$ steps into future multiplied by $\gamma^{t}$ for a discount factor $0<\gamma<1$. #### Summary of Results. In any given generation, society’s policy will consist of a threshold for group $D$ and a (possibly different) threshold for group $A$: the opportunity is given to every individual whose success probability is above the threshold for their group. The optimal policy is given by a dynamic program over the continuous set of all possible choices for the population composition $(\phi_{0},\phi_{1})$ as state variables. We solve the most basic version of the model analytically. We computationally solve more complex versions of the model by discretizing the state space, then applying standard dynamic programming solutions for finite decision processes. If the problem of allocating the opportunity only spanned a single generation, then the payoff-maximizing policy would use the same threshold for both groups. But given the discounting sum over multiple generations, we find that society’s optimal policy can, in general, use a lower threshold for group $D$ than for group $A$. The difference in thresholds is a form of socioeconomic affirmative action, and it arises due to the intuition discussed above: boosting the number of individuals from group $D$ who receive the opportunity will increase the number of available candidates from group $A$ in future generations, each of whom provides a (discounted) payoff in future generations via their enhanced performance. Finding the correct trade-off in allocating opportunity thus involves a delicate balance between immediate and future utility. (a) (b) (c) Figure 1: Visualizing the triples $(\alpha,\tau,\gamma)$ for which the optimal policy uses persistent affirmative actions. (We set $\sigma=1-\tau$ for these plots.) Points that lie above the surfaces in panels (a) and (c), and below the surface in panel (b), correspond to parameter values yielding persistent affirmative action. (1(a)) When $1-\frac{\alpha\sigma}{\tau}<0$, any $\gamma>0$ suffices for persistent affirmative action and eventually moving the entire population to group $A$. When $1-\frac{\alpha\sigma}{\tau}>0$, there exists some $\gamma<1$ (and hence a finite level of patience $\left(\frac{\gamma}{1-\gamma}\right))$ that suffices for persistent affirmative action. (1(b)) When $\tau$ is sufficiently large, the optimal policy does not use persistent affirmative action; this is because for a large $\tau$, the extent of affirmative action required to pick up the best performing members of $D$ is large—which in turn significantly reduces the immediate payoff. For any given value of $\alpha$, there exists a sufficiently small $\tau$ that guarantees persistent affirmative action. (1(c)) When $\alpha$ is small relative to $\tau$, the optimal policy does not use persistent affirmative action; this is because the cost of picking the best performing members of $D$ is very high and a small $A$ group suffices for filling the available opportunities. Note that for some values of $\tau$, no matter how large $\alpha$ is, the optimal policy never employs persistent affirmative action. Whether socioeconomic affirmative action is employed by the optimal solution — and the extent to which it is employed — depends on the fraction $\phi_{0}$ of individuals from group $D$; in the most basic model, the amount of affirmative action decreases monotonically as $\phi_{0}$ is reduced. The extent of affirmative action is also determined by the amount of opportunity available ($\alpha$), the dependence of success on ability and socioeconomic status ($\sigma$ and $\tau$), and society’s patience in trading off immediate payoff in return for payoff from future generations ($\gamma$). We characterize the optimal solution in this respect as a function of these parameters, finding that for some regions of the parameter space, the society employs temporary affirmative action, reducing the size of group $D$ to a given level before equalizing thresholds in subsequent generations; in other parts of the parameter space, the society employs persistent affirmative action, in which the threshold for group $D$ is strictly lower in every generation and the size of group $D$ converges to 0 over time. Figure 1 provides some ways of describing the regions of parameter space in which the optimal solution uses persistent affirmative action. As the partitions of the space there make apparent, the interactions among the key parameters is fairly subtle. First, persistent affirmative action is promoted by large values of $\alpha$ and small values of $\tau$, since these make it easier to include high-performing members of group $D$ without a large difference in thresholds; and it is promoted by larger values of $\gamma$, indicating greater concern for the payoffs in future generations. One might have suspected that persistent affirmative action would only be realized in the optimal solution in the limit as society’s patience (essentially $\gamma/(1-\gamma)$) goes to infinity; but in fact, a sufficiently large finite amount of patience is sufficient for the optimal policy to use persistent affirmative action. In our model, we include a probabilistic background process by which individuals can also move between groups $A$ and $D$; this reflects the idea that there are many mechanisms operating simultaneously for socioeconomic mobility, and we are studying only one of these mechanisms via the opportunity under consideration. The most basic version posits a single probability $p$ that each individual independently loses their group membership and re-samples it from the current distribution of group sizes. We also consider a version of the model in which this probability of loss of group membership is different for groups $A$ and $D$; in this case, we are only able to solve the model computationally, and these computational results reveal interesting non- monotonicities in the amount of affirmative action employed as a function of the relative size of group $D$ ($\phi_{0}$). #### Utilitarianism, Prioritarianism, and the Desert Principle. Our simple mathematical model allows us to represent and distinguish among several distinct worldviews toward allocation policies (see, e.g., (Arneson, 2013) for further discussion of these views): (1) a _utilitarian_ view, which generally favors slightly lower-ability members of $A$ to comparable, but slightly higher ability members of $D$ in pursuit of maximizing social utility and productivity (recall that membership in $A$ confers a boost in success probability); (2) a _prioritarian_ view, which evaluates a policy according to its impact on the well-being of the worse-off members of society. Our model can capture the priority view through large discount factors (recall that as the society’s patience increases, it effectively increases the priority assigned to the disadvantaged group members), or by adjusting the welfare function; (3) a _desert-principle_ view, which advocates for allocating opportunities based on some notion of deservingness. Deservingness in this view is often defined in terms of the contributions people make to the social utility. Hence success probability in our model is arguably the closest match to individual desert. With that definition for desert, desert-based principles would allocate opportunities myopically in each generation. As our analysis illustrates, such policies often fail to maximize the social utility in the long-run. #### Limitations and Interpretations. Our model is designed to incorporate the basic points we just mentioned in as simplified a fashion as possible; as such, it is important to note some of its key limitations. First, it is intended to model the effect of a single opportunity, and it treats other forms of mobility probabilistically in the background. It also assumes that the fundamental parameters ($\alpha,\sigma,\tau,\gamma$) are constant over all generations as well as over individuals within one generation. It treats an individual’s group membership ($A$ and $D$) and ability as a complete description of their performance, rather than including any dependence on the group membership of the individual’s parent. (That is, an individual in group $A$ performs the same in the model regardless of whether their parent belonged to group $A$ or $D$.) All of these would be interesting restrictions to relax in an extension of the model. Second, much of the past theoretical work on intergenerational mobility focuses on an issue that we do not consider here: the strategic considerations faced by parents as they decide how much to consume in the present generation and how much to pass on to their children. Our interest instead has been in the optimization problem faced by a social planner in allocating opportunities, treating the behavior of the agents as fixed and simple. Here too, it would be interesting to explore models that address these issues in combination. Finally, because our focus is on intergenerational mobility in a socioeconomic sense, we do not model discrimination based on race, ethnicity, or gender, and the role of race- or gender-based affirmative action in combatting these effects. The model is instead concerned with _socio-economic_ or _class-based_ (Malamud, 1995; Kahlenberg, 1996) affirmative action. That said, the ingredients here could be combined with models of statistical or taste-based discrimination on these attributes to better understand their interaction (as outlined in Section 5). The simplicity of our model, however, does allow us to make a correspondingly fundamental point: that even a purely payoff-maximizing society can discover affirmative action policies from first principles as it seeks to optimize the allocation of opportunities over multiple generations. Moreover, the optimal allocation policy is deeply connected to dynamic programming over the generations; the society is essentially attempting to “steer” the balance of group $A$ and group $D$ over time, making sure not to turn things too abruptly (giving up present benefit) or too gradually (giving up future benefit). This idea that society is searching for a way to turn optimally toward a better outcome is not specific to our model; it is an image that has arisen in qualitative discourse over several centuries. It can be seen in a quote popularized by Martin Luther King, that “the arc of the moral universe is long, but it bends toward justice” (Cohen, 2006). Interestingly, the original form of this quote, by the American minister Theodore Parker in 1853, has an even more abstractly mathematical flavor: “I do not pretend to understand the moral universe; the arc is a long one, my eye reaches but little ways. I cannot calculate the curve and complete the figure by the experience of sight; I can divine it by conscience. And from what I see I am sure it bends towards justice” (Parker, 1853). It is a curiously apt image for the way in which our optimal solutions gradually turn through the state space to reshape the distribution of socioeconomic groups, and it can be seen as added motivation for the issues at the heart of the model. ### Related Work Here, we briefly mention several lines of scholarship that are closely related to our work. See Appendix A for a more in-depth discussion. #### Long-term Implications of Fair ML. Several recent articles study the long-term impact of ML-based decision-making and fairness interventions on society, including the enforcement of statistical parity in hiring (Hu and Chen, 2018), and responses by individuals and populations to an ML-based decision rule (Liu et al., 2018; Mouzannar et al., 2019; Kannan et al., 2019). Liu et al. (2018), for example, study the conditions under which the choices of a myopic profit-maximizing institution (e.g., a bank lending money to individuals) work in the interest of the disadvantaged group. Dong et al. (2018); Hu et al. (2019); Milli et al. (2019) address _strategic classification_ where the goal is to design classifiers robust to strategic manipulation. This body of research focuses on strategic responses by agents being evaluated; in contrast, we assume idealized prediction so as to focus on the perspective of a social planner who seeks optimal allocation of opportunities over time. #### Intergenerational Income Mobility. A substantial literature in economics studies how higher inequality results in lower income mobility across generations (see, e.g., (Becker and Tomes, 1979; Maoz and Moav, 1999; Corak, 2013)). The precise measurements of inequality and mobility significantly influence the strength of this effect (see, e.g., (Solon, 1992; Piketty, 2000)). Theoretical models of income mobility have studied utility-maximizing parents deciding how much of their capital to consume and how much of it to invest in their offspring (Becker and Tomes, 1986; Becker and Tomes, 1979; Loury, 1981; Solon, 1999). We deliberately set aside parental strategic considerations and focus instead on deriving the optimal policy that maximizes the discounted payoff over generations. #### Affirmative Action Policies. A rich body of work in economics investigates statistical discrimination (Arrow, 1973) and the role of affirmative action in redressing it (Fang and Moro, 2011). Outcome-based policies including affirmative action targets have long been proposed and implemented as temporary remedies to eliminate group- level inequalities. Race-based affirmative action and socioeconomic affirmative action can be viewed as distinct categories of intervenions (Kahlenberg, 1996; Carnevale and Rose, 2013), with the former addressing long- term effects of racial bias and the latter facilitating access for economically disadvantaged individuals (Reardon et al., 2017). While the relationship between them is complex and contested in the literature (Kane, 1998; Reardon et al., 2006; Gaertner and Hart, 2013), they can co-exist without necessarily competing. #### Affirmative Action in College Admissions and Comparison with (Durlauf, 2008). Durlauf (2008) provides a model to compare the equality and efficiency of affirmative action policies with those of meritocratic policies in the context of admission rules to public universities. In his concluding remarks, Durlauf poses the key question our work sets out to answer: how do multigenerational considerations impact the optimal allocation policy? As we address this open question that he poses, we follow his model in many respects, although we depart from it in a few key areas. 111For a more detailed comparison, see Section A.4. Similar to our work, Durlauf provides a condition under which efficiency and equality considerations are aligned, but he focuses on settings where a diverse body of students on college campuses improves the human capital development for all of them. Durlauf assumes the policymaker aims to maximize the level of human capital among _the next generation_ of adult citizens. He restricts attention to two generations only, and instead of solving for the optimal policy, he compares the meritocratic policy with affirmative action in terms of the average human capital of the next generation. In contrast, we model the dynamics of human-capital development across _multiple generations_ and derive the _optimal allocation policy_. Moreover, groups in model corresponds to _socio-economic tiers_ , whereas Durlauf defines them in terms of _race_. In both models, generations are represented as follows: in each time step, a new member is born into each family/dynasty, and he/she replaces the current member of the family in the next generation. The _initial human capital_ in Durlauf’s model corresponds to our notion of _success probability_. _Adult human capital_ in his model is determined by college attendance, and it roughly maps to our notion of _success_ (i.e., whether the individual succeeds if given the opportunity.) In both models, an admission rule maps a student’s human capital and group membership into a binary outcome indicating whether the student is given the opportunity. In both models, the admission rule may vary across time and generations; the level of state expenditures on education is assumed constant across generations ($\alpha$ is fixed in our model); the only output of universities is human capital (our objective function is made up of the percentage of the population who succeed in each generation); and finally, the initial human capital is accurately measured for every student (we assume ability and success probability are perfectly observable.) ## 2 A Dynamic Model ### Agents, Circumstances, Abilities, & Generations We consider a model of a society that consists of a continuum of agents in two different sets of socioeconomic circumstances — a disadvantaged circumstance $D$ and an advantaged circumstance $A$. These circumstances are (probabilistically) inherited from one generation to the next, but we can try to increase the number of agents in the advantaged circumstance in future generations by offering opportunities to disadvantaged agents in the current generation. This comes with a trade-off, however, since a competing option is to offer these opportunities to advantaged agents in the current generation. Our goal is to model this trade-off. We say that an agent $i$ has circumstance $c_{i}=0$ if they are disadvantaged ($i\in D$), and circumstance $c_{i}=1$ if they are advantaged ($i\in A$). Each agent $i$ also has an ability $a_{i}$, which is a real number in $[0,1]$. Time advances in discrete periods, beginning with period $t=0$. We think of these as generations. Consider an agent $i$ who has circumstance $c^{\text{init}}_{i}$ at the beginning of time $t$. Depending on whether $i$ receives the opportunity, his/her circumstance may change to $c_{i}^{\text{post}}$. At the end of time step $t$, $i$ produces a new agent $i^{\prime}$ in generation $t+1$. This new agent $i^{\prime}$ has an ability $a_{i^{\prime}}$ drawn uniformly at random from $[0,1]$. With some fixed probability (specified below) $i^{\prime}$ inherits the ex-post circumstance of $i$ (so $c_{i^{\prime}}=c^{\text{post}}_{i}$), otherwise, it takes on a circumstance randomly selected from the background distribution of circumstances within the population in generation $t$. More specifically, in a given period $t$, let $\phi_{j}(t)$ denote the fraction of agents who have circumstance $j$, for $j=0,1$. If $c^{\text{post}}_{i}=0$, then with a fixed probability $1-p_{D}$, $i^{\prime}$ inherits circumstance $D$, and with fixed probability $p_{D}$, it receives a circumstance randomly selected from the background distribution $(\phi_{0}(t),\phi_{1}(t))$. Similarly, If $c^{\text{post}}_{i}=1$, then with a fixed probability $1-p_{A}$ , $i^{\prime}$ inherits circumstance $A$, and with probability $p_{A}$, it receives a circumstance randomly selected from the background distribution $(\phi_{0}(t),\phi_{1}(t))$. The movement probabilities, $p_{A}$ and $p_{D}$, capture all processes—other than the opportunity we seek to allocate optimally—through which individuals can change their circumstance from their parental inheritance. For example, in the college admissions example, while our model focuses on how admission decisions can reshape circumstances over generations, there are many other forces and processes that impact the evolution of circumstances within society (e.g., number of jobs in the economy, training opportunities outside college, or pure luck). The movement probabilities summarize and capture all these alternative upward or downward movement possibilities. ### Opportunities and Payoffs We consider the problem of performing an intervention in this society, which consists of offering an opportunity to a subset of the population. We only have the resources to offer the opportunity to an $\alpha$ fraction of the population. An agent who is offered the opportunity has some probability of succeeding at it, as a function of their ability and circumstances that we specify below. Succeeding at the opportunity confers two benefits on society: 1. (i) it produces an immediate payoff/reward to the society in the form of productivity; 2. (ii) if the agent is disadvantaged, it moves them (and subsequently their future generations) into the advantaged group. The central problem to be solved in the model, as we will see below, is how to balance the immediate gains from (i) against the long-term gains from (ii) over multiple generations. In particular, if an agent of ability $a_{i}\in[0,1]$ and circumstance $c_{i}\in\\{0,1\\}$ is offered the opportunity, their probability of succeeding at it is $a_{i}\sigma+c_{i}\tau,$ where $\sigma,\tau>0$ and $\sigma+\tau\leq 1$. Note that since $c_{i}\in\\{0,1\\}$, this simply means that $\tau$ gets added to the success probability of all agents whose circumstance is equal to $1$. Our payoff (or reward) $r(t)$ in period $t$ is simply the fraction of the population that both receives the opportunity and succeeds at it. Our total payoff is a discounted sum of payoffs over all periods, with discount factor $0<\gamma<1$; that is, the total payoff $r$ is equal to $\sum_{t=0}^{\infty}\gamma^{t}r(t)$. As noted earlier, agents with circumstance $0$ who receive the opportunity and succeed at it will produce offspring who (are more likely to) have circumstance $1$; this matters for the payoff because the total payoff $r=\sum_{t=0}^{\infty}\gamma^{t}r(t)$ depends on the fraction of agents with each type of circumstance in all time periods. ### Thresholds and Interventions The way we allocate the opportunity at time $t$ is to set a threshold $\theta_{j}(t)$ for agents with circumstance $j$, for $j\in\\{0,1\\}$, and to offer the opportunity to all agents $i$ with circumstance $j$ whose success probability, $a_{i}\sigma+c_{i}\tau$, is at least $\theta_{j}(t)$. That is, $a_{i}\sigma+c_{i}\tau\geq\theta_{j}.$ We will sometimes write the threshold $\theta_{j}$ and the population fraction with each circumstance $\phi_{j}$ without the explicit dependence “$(t)$” when we are considering a single fixed time period. Agents of circumstance $0$ make up a $\phi_{0}$ fraction of the population, and a $1-\frac{\theta_{0}}{\sigma}$ fraction of them receive the opportunity, for a total fraction of the population equal to $\phi_{0}\times(1-\frac{\theta_{0}}{\sigma})$. Similarly, agents of circumstance $1$ make up a $\phi_{1}$ fraction of the population, and a $1-\frac{\theta_{1}-\tau}{\sigma}$ fraction of them receive the opportunity, for a total fraction of the population equal to $\phi_{1}\times(1-\frac{\theta_{1}-\tau}{\sigma})$. The sum of these two fractions must add up to $\alpha$ — the portion of the population to whom we can offer the opportunity: $\displaystyle\forall 0\leq\theta_{0}\leq\sigma\text{ and }\tau\leq\theta_{1}\leq\sigma+\tau:\text{ }\phi_{0}\times\left(1-\frac{\theta_{0}}{\sigma}\right)+\phi_{1}\times\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)=\alpha$ (1) $\displaystyle\Leftrightarrow$ $\displaystyle\forall 0\leq\theta_{0}\leq\sigma\text{ and }\tau\leq\theta_{1}\leq\sigma+\tau:\text{ }\phi_{0}\theta_{0}+\phi_{1}\theta_{1}=\sigma(1-\alpha)+\phi_{1}\tau.$ This also shows how our choice of thresholds is a one-dimensional problem in the single variable $\theta_{0}$ (or equivalent in the single variable $\theta_{1}$), since after setting one of the two thresholds, the other is determined by this equation. More precisely, we have that: $\begin{cases}\theta_{1}=\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\quad\forall\phi_{0}<1\\\ \theta_{1}\in[\tau,\sigma+\tau]\quad\text{ for }\phi_{0}=1.\end{cases}$ (2) Note that if $\phi_{0}=1$, $\theta_{1}$ can take on any value in $[\tau,\sigma+\tau]$, the threshold will not affect how opportunities are allocated. The same holds for $\phi_{0}=0$ and any $\phi_{0}\in[0,\sigma]$. ### Dynamics in a Single Period Recall that our payoff $r(t)$ in period $t$ is simply the fraction of the population that both receives the opportunity and succeeds at it. We can decompose this as follows. * • Agents of circumstance $0$ make up a $\phi_{0}$ fraction of the population, and a $1-\frac{\theta_{0}}{\sigma}$ fraction of them receive the opportunity, for a total fraction of the population equal to $\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)$. Not all of these agents succeed at the opportunity; the average success probability in this group is $\frac{1}{2}(\sigma+\theta_{0})$, so the expected quantity that succeeds is $\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)\frac{(\sigma+\theta_{0})}{2}=\frac{\phi_{0}}{2\sigma}\left(\sigma^{2}-\theta_{0}^{2}\right).$ * • Agents of circumstance $1$ make up a $\phi_{1}$ fraction of the population, and a $1-\frac{\theta_{1}-\tau}{\sigma}$ fraction of them receive the opportunity, for a total fraction of the population equal to $\phi_{1}\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)$. Again, not all of these agents succeed at the opportunity; the average success probability in this group is $\frac{1}{2}(\sigma+\tau+\theta_{1})$, so the expected quantity that succeeds is $\phi_{1}\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)\frac{(\sigma+\tau+\theta_{1})}{2}=\frac{\phi_{1}}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}^{2}\right).$ The total payoff in period $t$ is the sum of these two terms: $r(t)=\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)+\frac{\phi_{1}(t)}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}(t)^{2}\right).$ (3) ### Dynamics over Multiple Periods If we were just optimizing the payoff in this single time period, then we’d have a single-variable optimization problem in the variable $\theta_{0}$ (or equivalently in $\theta_{1}$), with the objective function given by (3). But since there is also the set of discounted payoffs in future time periods, we also need to look at the effect of our decisions on the quantities $\phi_{0}(.)$ and $\phi_{1}(.)$ in future periods. If $p_{A}=p_{D}=0$, $\phi_{1}(t+1)$ grows relative to $\phi_{i}(t)$ depending on the fraction of the population that transitions from circumstance $0$ to circumstance $1$ by succeeding at the opportunity. Thus we have $\phi_{0}(t+1)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right).$ (4) More generally when $p_{A}$ or $p_{D}$ are non-zero, let’s define $\phi^{\text{post}}_{0}(t)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$. (For simplicity, we drop “$(t)$” and simply use $\phi^{\text{post}}_{0}$ in the remainder of this section). We have: $\phi_{0}(t+1)=\phi^{\text{post}}_{0}(1-p_{D})+\phi^{\text{post}}_{0}p_{D}\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)p_{A}\phi^{\text{post}}_{0},$ (5) It is easy to see that: ###### Proposition 1 If $p_{A}=p_{D}$, then $\phi_{0}(t+1)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$. Proof Suppose $p_{A}=p_{D}=p$. Then we can re-write (5) as follows: $\displaystyle\phi_{0}(t+1)$ $\displaystyle=$ $\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)p\phi^{\text{post}}_{0}$ $\displaystyle=$ $\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p\left(\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)\right)$ $\displaystyle=$ $\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p$ $\displaystyle=$ $\displaystyle\phi^{\text{post}}_{0}=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$ where in the last line, we replace $\phi^{\text{post}}_{0}$ with its definition. The above proposition shows that with respect to dynamics and optimal policy, settings in which $p_{A}=p_{D}$, are essentially equivalent to settings in which $p_{A}=p_{D}=0$. In summary, the full problem is to choose thresholds $\theta_{0}(t),\theta_{1}(t)$ for each time period $t$ so as to maximize the infinite sum $r=\sum_{t=0}^{\infty}\gamma^{t}r(t)$. Each term $r(t)$ depends not just on the chosen thresholds but also on the fractions of agents with each type of circumstance $\phi_{0}(t),\phi_{1}(t)$, which evolve according to the recurrence in Equation (5). Note that the intuitive trade-off between $\theta_{0}$ and $\theta_{1}$ shows up in the formulation of (3) and (4): lowering $\theta_{0}$ and lowering $\theta_{1}$ have different effects, both in period $t$ and in future time periods. ## 3 Theoretical Analysis for $p_{D}\geq p_{A}$ In this section, we focus on settings in which $p_{D}\geq p_{A}$ and characterize the optimal policy to maximize the discounted payoff over generations. (We only provide the analysis for settings of $p_{D}=p_{A}$ but the extension to $p_{D}>p_{A}$ is straightforward). We cast the problem as deriving the infinite-time-horizon optimal policy in a continuous state- and action-space (Markov) decision process. We characterize the optimal threshold and value function for every state $\phi_{0}\in[0,1]$. Importantly, we show that there exists a tipping point $\phi^{}_{0}$ that splits the state space into two distinct regions: states at which the optimal threshold uses strict affirmative action, and states at which the optimal policy consists of imposing equally high thresholds on both $A$ and $D$ groups. ### The Decision Process Given $\alpha,\sigma,\tau$, and $\gamma$, we define a decision process $\mathcal{D}_{\alpha,\sigma,\tau,\gamma}=(\Phi,\Theta,S,R)$ (or $\mathcal{D}$ for short) with a continuous state space $\Phi=[0,1]$, action space $\Theta=[0,\sigma]$, state transition $S:\Phi\times\Theta\rightarrow\Phi$, and reward function $R:\Phi\times\Theta\rightarrow[0,1]$. Each state $\phi_{0}\in\Phi$ corresponds to a particular fraction of disadvantaged individuals within the population. For instance, the states $0$ (or $1$) represents a society in which no one (or everyone) belongs to $D$. The set of thresholds admissible in each state $\phi_{0}$ is denoted by $\Theta_{\phi_{0}}$. $\Theta_{\phi_{0}}$ consist of all thresholds $0\leq\theta_{0}\leq\sigma$ that can satisfy the capacity constraint, $\alpha$. In other words, for any $\in\theta_{0}\Theta_{\phi_{0}}$ if we impose the threshold $\theta_{0}\in[0,\sigma]$ on group $D$, we can find a threshold $\theta_{1}\in[0,\sigma+\tau]$ for group $A$ such that exactly $\alpha$ fraction of the overall population receives the opportunity. This capacity constraint translates into two conditions on $\Theta_{\phi_{0}}$: 1. 1. A threshold $\theta_{0}\in\Theta_{\phi_{0}}$ should not give the opportunity to _more than_ $\alpha$ fraction of the population. Formally, $\forall\theta_{0}\in\Theta_{\phi_{0}}:\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)\leq\alpha$, which is equivalent to: $\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad\theta_{0}\geq\sigma\left(1-\frac{\alpha}{\phi_{0}}\right).$ (6) 2. 2. A threshold $\theta_{0}\in\Theta_{\phi_{0}}$ should not waste opportunities, that is, it should give the opportunity to _at least_ $\alpha$ fraction of the overall population. Formally, $\forall\theta_{0}\in\Theta_{\phi_{0}}:\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)+\phi_{1}\geq\alpha.$ Replacing $\phi_{1}$ with $1-\phi_{0}$ and rearranging terms, the above is equivalent to $\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad\theta_{0}\leq\frac{\sigma(1-\alpha)}{\phi_{0}}.$ (7) Figure 2 illustrates the actions satisfying conditions (6) and (7) for every state $\phi_{0}\in[0,1]$. Figure 2: The admissible thresholds for every state of the decision process, $\mathcal{D}$. The x-axis specifies the state $\phi_{0}$, and the y-axis highlights (in light red) the admissible thresholds, $\Theta_{\phi_{0}}$, at every state, $\phi_{0}$. The state transition function, $S:\Phi\times\Theta\rightarrow\Phi$, specifies the state transitioned to for every (state, admissible threshold) pair. Formally, $\forall\phi_{0}\in\Phi,\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad S(\phi_{0},\theta_{0})=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2}).$ (8) The reward function, $R:\Phi\times\Theta\rightarrow[0,1]$, is defined as follows: $R(\phi_{0},\theta_{0})$ denotes the immediate reward/payoff of imposing threshold $\theta_{0}$ at state $\phi_{0}$. Formally, $\forall\phi_{0}\in\Phi,\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad R(\phi_{0},\theta_{0})=\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{\phi_{1}}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}^{2}\right).$ Replacing $\phi_{1}$ with $1-\phi_{0}$ and $\theta_{1}$ with the right hand side of (2), we obtain the following equivalent expression for $R$: $R(\phi_{0},\theta_{0})=\begin{cases}\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})\quad\text{ for }\phi_{0}=1\text{, otherwise: }\\\ \frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)\end{cases}$ (9) ### Characterization of the Optimal Policy Next, we illustrate and characterize the optimal policy for the decision process $\mathcal{D}$ defined above. (For further information and references on continuous-state decision processes, see Appendix A.5.) A deterministic policy $\pi$ for a decision process $\mathcal{D}$ is a mapping $\pi:\Phi\rightarrow\Theta$ such that $\pi(\phi_{0})$ prescribes the threshold at state $\phi_{0}$. The value $V_{\pi}(\phi_{0})$ of a state $\phi_{0}$ under policy $\pi$ is the discounted reward of executing policy $\pi$ on $\mathcal{D}$ starting with initial state $\phi_{0}$. A policy $\pi$ is optimal if its value function $V_{\pi}$ satisfies Bellman Optimality—defined recursively as follows: $V_{\pi}(\phi_{0})=\max_{\theta_{0}}R(\phi_{0},\theta_{0})+\gamma V_{\pi}(S(\phi_{0},\theta_{0})).$ (10) We establish in Appendix C that for our decision process $\mathcal{D}$, the value function satisfying the above functional equation is _unique_ , _continuous_ , and _differentiable_. (We prove these facts utilizing tools from recursive analysis and dynamic programming (Stokey, 1989; Cotter and Park, 2006)). For simplicity, from this point on we refer to this unique optimal value function as $V(.)$ and drop the subscript $\pi$. Let the correspondence $\Pi^{}_{0}:\Phi\rightarrow 2^{\Theta}$ denote all optimal policies for $\mathcal{D}$. More precisely, for any $0\leq\phi_{0}\leq 1$, the set $\Pi^{}_{0}(\phi_{0})$ contains all optimal threshold values at $\phi_{0}$. Figure 3(a) illustrates $\Pi^{}_{0}$ for a sample setting of the parameters $\alpha,\sigma,\tau,\delta$. (See Figure 6 in the Appendix for more instances.) Figure 7 in the Appendix illustrates the value function $V$. Note that the value function is consistently decreasing and concave. (a) The optimal $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$. The optimal threshold decreases with $\phi_{0}$. The dashed lines indicate the tipping points, $\phi^{}_{0}$, below which $\sigma$ is the only optimal threshold, and above it $\sigma$ is not optimal. (b) The difference between $\theta_{0}$ and $\theta_{1}$ at every state $0\leq\phi_{0}\leq 1$. The dashed lines specify the tipping points. Strict affirmative action is employed beyond $\phi^{}_{0}$ only and the extent of affirmative action is increasing in $\phi_{0}$. (c) The state to which the optimal policy converges given the initial state $\phi_{0}$. The dashed lines specify the tipping point, $\phi^{}_{0}$. Note that the optimal policy never shrinks the size of group $D$ to a value less than $\phi^{}_{0}$. Figure 3: Illustration of the (a) optimal policy, (b) extent of affirmative action, and (c) absorbing state. We say that the optimal policy, $\Pi^{}_{0}$, uses affirmative action at a state $\phi_{0}$ if at $\phi_{0}$, it imposes a lower threshold on group $D$ compared to $A$. ###### Definition 1 ((Strict) Affirmative Action) The optimal policy uses (strict) affirmative action at state $\phi_{0}$ if for all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, $\theta_{0}<\theta_{1}$. If the inequality above is not strict (i.e., $\theta_{0}\leq\theta_{1}$), we say the optimal policy uses _weak_ affirmative action. Figure 3(b) shows the extent of affirmative action for a sample setting of the parameters $\alpha,\sigma,\tau,\delta$. (See Figure 8 in the Appendix for more instances). Our main result (Theorem 1) characterizes the states at which the optimal policy employs affirmative action. In particular, it establishes the existence of a tipping point $0\leq\phi^{}_{0}\leq 1-\alpha$ that designate the region of affirmative action: At any state $\phi_{0}\leq\phi^{}_{0}$, the optimal policy assigns similar high threshold to both the advantaged and disadvantaged groups. In contract, at any state $\phi_{0}\leq\phi^{}_{0}$ the optimal policy imposes a strictly lower threshold on group $D$ compared to $A$. ###### Theorem 1 Given $\alpha,\sigma,\tau,\gamma$ and the decision process $\mathcal{D}$, let $\phi^{}_{0}=\max\left\\{0,\min\left\\{1-\alpha,1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\right\\}\right\\}.$ (11) • For any state $\phi_{0}\leq\phi^{}_{0}$, there exists $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$ such that $\theta_{0}=\theta_{1}$. • For any state $\phi_{0}>\phi^{}_{0}$, for all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$ such that $\theta_{0}<\theta_{1}$. We prove Theorem 1 by establishing a series of Lemmas. Lemma 1 determines the largest state $\phi^{}_{0}\in[0,1]$ below which the optimal policy consists of applying the high threshold of $\sigma$ to group $D$. Clearly, below such point, the optimal policy does not use affirmative action. Next, we investigate states larger than $\phi^{}_{0}$. For every state $\phi_{0}>\phi^{}_{0}$, Lemma 2 identifies the set of thresholds that exhibit affirmative action. Lemma 3 establishes that the optimal policy uses weak affirmative action beyond $\phi^{}_{0}$. That is, it shows that for any state $\phi_{0}>\phi^{}_{0}$, it is never optimal to impose a strictly higher threshold on D compared to A. Proposition 2 shows that beyond $\phi^{}_{0}$, the optimal policy in fact uses _strict_ affirmative action. That is, at every state $\phi_{0}>\phi^{}_{0}$, the optimal policy imposes a strictly lower threshold on D compared to A. Lemma 1 determines the state $\phi^{}_{0}\in[0,1]$ up to which $\sigma$ is an optimal threshold. Note that if $\sigma$ is an optimal threshold at a state $\phi_{0}$, the optimal policy does not use affirmative action at $\phi_{0}$. To see this, note that when $\sigma$ is optimal, any action $\theta_{0}>\sigma$ is also optimal (they all pick a 0-fraction of group $D$ and are, therefore, effectively equivalent). (All omitted proof can be found in Appendix C.1.) ###### Lemma 1 (The Tipping Point) For any state $\phi_{0}<\phi^{}_{0}$, $\sigma\in\Pi^{}_{0}(\phi_{0})$. The following Lemma characterizes the region of affirmative action in the state-action space. ###### Lemma 2 (Region of Affirmative Action) For a state $\phi_{0}^{}<\phi_{0}<1$, the threshold $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ uses affirmative action if and only if $\theta_{0}\leq\sigma(1-\alpha)+\tau(1-\phi_{0})$. Proof Let $\theta$ be the threshold that if applied to both D and A at $\phi_{0}$, allocates opportunities exactly. We have that $\phi_{0}\left(1-\frac{\theta}{\sigma}\right)+(1-\phi_{0})\left(1-\frac{\theta-\tau}{\sigma}\right)=\alpha$ or equivalently, $\theta=\sigma(1-\alpha)+\tau(1-\phi_{0}).$ Note that $\theta_{0}<\theta$ if and only if $\theta_{1}>\theta$—otherwise the capacity constraints would not be maintained. Therefore, for any $\theta_{0}<\theta$, $\theta_{1}>\theta_{0}$, which implies affirmative action. The following Lemma shows that beyond $\phi^{}_{0}$, the optimal threshold for the disadvantaged is never higher than that for the advantaged. ###### Lemma 3 (Weak Affirmative Action) Consider a state $\phi_{0}>\phi^{}_{0}$. For all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, $\theta_{0}\leq\theta_{1}$. Proof Suppose not and there exists $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$ such that $\theta_{0}>\theta_{1}$. If we lower $\theta_{0}$ down to $\sigma(1-\alpha)+\tau(1-\phi_{0})$ and increase $\theta_{1}$ up to $\sigma(1-\alpha)+\tau(1-\phi_{0})$, we maintain the capacity constraints and at the same time achieve the following: (a) We improve the immediate reward of the current time step— because we replace advantaged agents with low success probabilities in the range of $[\theta_{1},\sigma(1-\alpha)+\tau(1-\phi_{0})]$ with disadvantaged agents with higher success probabilities in $[\sigma(1-\alpha)+\tau(1-\phi_{0}),\theta_{0}]$. * (b) We move to a state with a relatively smaller size of group $D$ (simply because we gave the opportunity to more disadvantaged agents). The value function is strictly decreasing in $\phi_{0}$. Therefore, this new next state has a higher value compared to the previous one. The fact that we can improve the value contradicts the optimality of $\theta_{0},\theta_{1}$. Therefore, $\theta_{0}>\theta_{1}$ cannot hold The following Proposition establishes that beyond $\phi^{}_{0}$, the optimal policy uses strict affirmative action. (The proofs establishing the proposition can be found in Appendix C.1.) ###### Proposition 2 (Strict Affirmative Action) Consider a state $\phi_{0}>\phi^{}_{0}$, and let $V^{\prime}(\phi_{0})$ denote the derivative of the value function $V$ evaluated at $\phi_{0}$. If $V^{\prime}(\phi_{0})<0$, then for all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, $\theta_{0}<\theta_{1}$. #### Insights from the Analysis. We end this section by making several observations about the optimal policy: First, note that the optimal policy never shrinks the size of $D$ to a value less than $\phi^{}_{0}$. For every initial state $\phi_{0}\in[0,1]$, Figure 9 shows the state one converges to when the optimal policy is simulated for $1000$ steps on $\mathcal{D}$. In other words, affirmative action is optimal from a utilitarian point of view as long as group $D$ is sufficiently large. Second, the precise derivation of $\phi_{0}^{}$, as specified in (11), allows us to gain new insights into how the interaction between the parameters of our model can give rise to or avert affirmative action. Figure 1 depicts the status of persistent affirmative action (i.e., $\phi_{0}^{}\leq 0$) for $\alpha,\tau,\gamma$ in settings where $\sigma=1-\tau$. (The derivation behind the plots can be found in Appendix C.2.) Notice that persistent affirmative action is promoted by large values of $\alpha$ and small values of $\tau$, since these make it easier to include high-performing members of group $D$ without a large difference in thresholds. Persistent affirmative action is also promoted by larger values of $\gamma$, indicating greater concern for the rewards in future generations. Note, however, that a finite level of patience often suffices for persistent affirmative action to be optimal. Finally, a frequent objection to affirmative action polices is their potential for reverse discrimination (see, e.g., Hopwood v. Texas, 78 F.3d 932 (5th Cir. 1996)). Translating these concerns into the terminology of our model, one may object that “if the highest-ability member of $D$ has ability below the lowest-ability member of $A$, then _any_ affirmative action in this scenario will violate the desert principle”. Note, however, that in our model, abilities for both $A$ and $D$ group members are uniform in the $[0,1]$ interval. So the optimal policy will never favor a lower-ability member of $D$ to a higher ability member of $A$. ## 4 Computational Analysis for $p_{D}<p_{A}$ The focus of our theoretical analysis was on the settings in which $p_{A}\leq p_{D}$. Next, we computationally investigate the optimal policy for cases where $p_{A}>p_{D}$. Recall that when $p_{A}$ and $p_{D}$ are non-zero, the circumstance of an offspring is not deterministically specified by that of their parent. Instead, $p_{A}$ and $p_{D}$ specify the offspring’s probability of spontaneous movement to the background distribution of circumstances in society, that is, $(\phi_{0},\phi_{1})$. When $p_{A}\leq p_{D}$, both the spontaneous movement dynamics and the allocation of opportunities work toward reducing the relative size of group $D$ over time. This fact allowed us to utilize backward induction to characterize the optimal policy theoretically. When $p_{A}>p_{D}$, however, the spontaneous movement work in the opposite direction of opportunity allocation: with no opportunity allocated, the spontaneous movement dynamics gradually shift the entire population to group $D$. In such settings, the role of the allocation policy consists of combatting the natural flow of the population to $D$. Although in settings with $p_{A}>p_{D}$, it is significantly more challenging to characterize the optimal allocation policy, we can still approximately compute this policy via discretization followed by solution methods for finite decision problems. More specifically, we discretize the state- and action- space of the decision process, $\mathcal{D}$, then compute the optimal policy of the resulting finite decision process using methods such as policy iteration. In what follows, we report the result of our simulations when the state- and action-space are approximated by 1000 equidistant states and actions (i.e., $\tilde{\Phi}=(0,0.001,0.002,\cdots,1)$ and $\tilde{\Theta}=\sigma(0,0.001,0.002,\cdots,1)$) and the reward and transition functions are approximated accordingly. Figure 4: The optimal policy $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$ for various $p_{A}$ values. Figure 5: The absorbing state for every initial state $0\leq\phi_{0}\leq 1$. The optimal policy is non-monotone and exhibits sharp drops when better absorbing states become cost-effective. Figure 4 shows how the optimal policy varies with $p_{A}$ for a fixed value of $p_{D}$. Compared to the optimal policy for settings of $p_{A}\leq p_{D}$, we observe two significant differences here: • The optimal policy can be non-monotone in $\phi_{0}$. * • There exists a sharp drop in the optimal threshold at some intermediate state $\phi_{0}\in[0,1]$. Next, we provide some justifications for both the intuitive and potentially counter-intuitive aspects of these phenomena. For every initial state $\phi_{0}\in[0,1]$, Figure 5 illustrates the state one converges to (i.e., the absorbing state) if the optimal policy is simulated for 1000 steps. Note that in all cases, the optimal policy successfully reduces the absorbing state to a more desirable value less than 1 (recall that with no intervention, we always converge to $\phi_{0}=1$ through the spontaneous movements). As illustrated in Figure 5, the sharp drop in the optimal threshold coincides with an abrupt change in the absorbing state. When the initial state is close to 1, the allocation policy must employ extensive affirmative action to reduce and maintain a better absorbing state. This feat, however, comes at the cost of immediate reward. The optimal policy, therefore, settles for reaching and obtaining a slightly more desirable absorbing state (a value between 0.6 and 0.8 in Figure 5). For smaller initial states, the cost of employing extensive affirmative action goes down, and at some point, it becomes viable to reach and maintain a much better absorbing state (a value between 0 to 0.3 in Figure 5). The sharp drop in the optimal threshold happens precisely at the state where the benefit of extensive affirmative action outweighs its cost. In summary, when $p_{A}>p_{D}$, we need persistent (albeit non-monotone) affirmative action to reach and maintain more desirable absorbing states. The optimal extent of affirmative action crucially depends on the initial state of the population: If the disadvantaged group is large to begin with, the optimal policy has to forgo extensive affirmative action to maintain sufficient short- term return. If the advantaged group initially has a sufficient mass, extensive affirmative action becomes optimal, and in the long-run, manages to significantly increases the fraction of advantaged individuals in the population. These findings are in sharp contrast with settings of $p_{A}\leq p_{D}$. In those settings, affirmative action is optimal under the much more straightforward condition that the disadvantaged group exceeds a $\phi_{0}^{}$ fraction of the population, for a constant $\phi_{0}^{}$ that we can specify precisely, and it ceases to be optimal as soon as society gains at least $1-\phi_{0}^{}$ mass in the advantaged group. ## 5 Discussion and Future Directions In this paper, we developed and analyzed a model for allocating opportunities in a society with limited intergenerational mobility. These opportunities produce benefits in the present generation, but they can also raise the socioeconomic status of the recipients and their descendants. This creates a trade-off: whether to maximize the current value achievable from the opportunities or to increase the value achievable in future generations. We have shown how to resolve this trade-off by solving a continuous optimization problem over multiple generations, and we have seen how optimal solutions to this problem can exhibit a form of socioeconomic affirmative action, favoring individuals of low socioeconomic status over slightly higher-performing individuals of high socioeconomic status. Characterizing the conditions under which this type of affirmative action occurs in the model provides insights into the interaction between the amount of opportunity available, the magnitude of the gap between different socioeconomic classes, and the “patience”’ of the society in evaluating the benefits achievable from future generations. #### Insights and Implications. Our work provides a purely utilitarian account of a society–with no intrinsic interest in reducing inequality– which nonetheless chooses to employ affirmative action along socioeconomic lines. In that sense, our work responds to concerns around affirmative action hurting social utility. In addition, our analysis present new insights on the shape and extent of effective socioeconomic affirmative action policies across multiple generations. Our findings offer several important insights to the Fair-ML literature: (a) If temporal dynamics are taken into account, there are important cases where fairness interventions can be fully aligned with performance-maximizing goals. (b) Effective fairness interventions should often adapt to population changes over time. (c) For a comprehensive assessment of fairness for an allocation policy/algorithm, we may need to look beyond its immediate impact on its direct subjects. Our work, for instance, characterizes optimal allocations when decisions made for individuals today impact their future generations. #### Toward a Broader Class of Models Leading to Affirmative Action. Our work’s main focus was on _intergenerational dynamics_ and their impact on the effectiveness of _socioeconomic affirmative action policies_. But we hope that our work also serves as a stepping stone toward a broader class of models characterizing conditions under which affirmative action comes out of optimal policy derivations. In particular, a generalized version of our decision process can capture _race-dependent_ dynamics of movement between various socioeconomic levels. Race-dependent dynamics are motivated by the observation that advantaged Black people are more likely to be downwardly mobile than their advantaged white counterparts (Chetty et al., 2020). A more general version of our decision process would consist of race-dependent $\sigma$’s and $\tau$’s. Analyzing the resulting dynamics can shed light on the tradeoffs between race/ethnicity-based and socioeconomic affirmative action policies. We leave this analysis as a crucial direction for future work. There are a number of interesting further research directions suggested by the results in the paper, and several of them address the limits of the model noted in the introduction. In particular, it would be interesting to consider extensions of the model in which basic parameters of the society varied over time rather than being fixed and could only be estimated with delayed feedback (e.g., after several years); the ability distributions were non-uniform; the dynamics would amplify the interaction between ability and socioeconomic advantage; or there were multiple opportunities being allocated concurrently. It would be interesting to incorporate strategic considerations as in other work on intergenerational mobility. And finally, there is a natural opportunity to try integrating the models here with work on the nature of statistical discrimination and the design of interventions to alleviate it. #### On the Scope and Limitations of Mathematical Models. Our work addresses one potential role that affirmative action policies can have on future generations and mitigating socioeconomic inequality. Our work follows a long tradition in the mathematical social sciences, where a stylized model is proposed to capture certain aspects of a complex societal issue; the model is then rigorously analyzed with the hope that the formal analysis provides new insights about alternative policy choices and their counterfactual effects. These insights can in turn inform policymakers at a qualitative level. We conclude this article by acknowledging that all mathematical models are by definition highly simplified representations of the phenomena at hand, and as such it is important to understand and interpret them keeping their limitations and scope of applicability in mind. We have grounded our work in a broad literature from economics so as to draw on the insights that earlier modelers have brought to this setting. But we emphasize that theoretical models and their implications should never be taken as exact representations of the way complex societal processes operate and evolve. As such, it is important to not draw policy interpretations on the basis of such models alone. ## Acknowledgments This work was supported in part by a Simons Investigator Award, a Vannevar Bush Faculty Fellowship, a MURI grant, AFOSR grant FA9550-19-1-0183, and grants from the ARO and the MacArthur Foundation. We are grateful to Lawrence E. Blume, Alexandra Chouldechova, Zachary C. 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Is a college degree still the great equalizer? intergenerational mobility across levels of schooling in the united states. American Journal of Sociology, 117(3):763–807, 2011. ## Appendix A Expanded Related Work ### A.1 Long-term Implications of Fair-ML Several recent papers study the long-term impact of decision-making models and fairness interventions on society and individuals. Liu et al. [2018] and Kannan et al. [2019] study how a utility-maximizing _decision-maker_ may respond to the predictions made by the model. For instance, the decision-maker may interpret and use the predictions in a certain way, or update the model entirely. Dong et al. [2018]; Hu et al. [2019]; Milli et al. [2019] address _strategic classification_ —a setting in which decision subjects are assumed to respond _strategically_ and potentially _untruthfully_ to the choice of the classification model, and the goal is to design classifiers that are robust to strategic manipulation. [Hu and Chen, 2018] and [Mouzannar et al., 2019] take inspiration from existing models of statistical discrimination and affirmative action. Hu and Chen [2018] study the impact of enforcing statistical parity on hiring decisions made in a temporary labor market that precedes a permanent labor market. They show that under certain conditions, statistical parity can result in an equilibrium that Pareto-dominates the one that would emerge in an unconstrained labor market. Mouzannar et al. [2019] model the dynamics of how a population reacts to a selection rule by changing their qualifications. The qualification of an individual is assumed to be estimated via some classification function and this estimate is assumed to be the same as their true qualification/label of the individual. They model a selection rule by its selection rates across the qualified and unqualified members of two socially salient groups. In other words, the rule is modeled by four non-negative numbers. They assume that there exist continuously differentiable functions $f_{1}$ and $f_{2}$ that map the selection rates in each group to the percentage of qualified individuals in that group in the next round. These functions model how qualifications change over time in response to the selection rule. Within this model, Mouzannar et al. [2019] study two types of myopic utility-maximizing policies: an affirmative action policy which forces the selection rates to be equal across the two groups, and an unconstrained policy that simply picks the qualified in each group. They then provide several conditions under which each of these policies lead to social equality (i.e., groups that are indistinguishable in terms of their qualifications). As for affirmative action policies, under-acceptance of qualified individuals in one group (to guarantee equal selection rates) is shown to be inefficient (both in terms of the decision-maker’s utility and average qualifications). Over-acceptance of the unqualified, on the other hand, can lead to a more qualified population (although it may fail to guarantee social equality). ### A.2 Intergenerational Income Mobility A large economic literature addresses the relationship between income inequality and intergenerational mobility. At a high-level, the literature suggests that higher inequality results in lower income mobility across generations (see e.g., [Maoz and Moav, 1999; Corak, 2013]). The precise measurements of inequality and mobility significantly influence the strength of this effect (see, e.g., [Solon, 1992; Piketty, 2000]). The literature recognizes two main mechanisms through which a parent may affect its offspring income: (1) by transmitting endowments (e.g., innate abilities, connections, etc.) to the next generation; (2) by investing in the human capital of the next generation [Becker and Tomes, 1979; Solon, 1999]. In the existing theoretical models of income mobility, utility-maximizing parents decide how much of their income/human capital to consume and how much of it to invest in their offspring (depending on their level of altruism) [Becker and Tomes, 1986; Becker and Tomes, 1979; Loury, 1981; Solon, 1999]. This decision, combined with the social institutions and the technology that converts parental investment to the next generation’s human capital, determine the fate of the next generation. For example, Loury [1981] provides a dynamic model of how the earnings distributions of the next generation depends on the parent’s earnings and his/her investment decision for his/her offspring. The earnings of the offspring is determined by their innate ability/endowment $\alpha$222Similar to ours, Loury’s model the ability of each individual is drawn randomly from some fixed distribution. The ability of the offspring is only observed after the parent makes their investment decision. and how much the parent invests in their training $e$ through a function $h(\alpha,e)$. Unlike our model, Loury assumes the the parents are expected-utility maximizing when making their investment decisions for their offspring. In particular, he assumes the utility $u(.,.)$ of a parent with earnings $y$ is a function of his/her own consumption $c$ and the expected utility of the offspring as the result of the investment he/she makes in their training, which is $y-c$. This recursive definition of utility specifies how the utility of one generation depends on that of all their subsequent generations (not just the immediate offspring). Loury defines the indirect utility associated with an earning $y$ to be the function the parent uses to estimate their offspring’s utility. If this function is the same as the one obtained by solving the parent’s utility maximization problem, the indirect utility function is called _consistent_. He goes on to show that under certain (weak) assumptions, there is a unique consistent indirect utility function. He also defines an _equilibrium_ earnings distribution as one that if it characterizes the earnings distribution for one generation, it continues to do so for all subsequent generations. He then provides several comparative statics results for various policies (e.g., “education specific tax policies”). For example, he shows that under certain conditions, egalitarian policies that redistribute earnings of the next generation have insurance effects that make every member of society today better off. Other factors that have been shown to causally affect income mobility are neighborhood [Chetty and Hendren, 2018, 2018], parental education [Torche, 2011], and family background characteristics (e.g., welfare receipt, health, attitudes and social behavior [Black and Devereux, 2010]). ### A.3 Affirmative Action Policies A rich body of work in economics investigates sources of statistical discrimination333According to [Fang and Moro, 2011] “Statistical discrimination generally refers to the phenomenon of a decision-maker using observable characteristics of individuals as a proxy for unobservable, but outcome-relevant, characteristics.” and the role of affirmative action policies in redressing it. For an excellent survey of this literature, see [Fang and Moro, 2011]. In contrast to taste-based theories of discrimination—which attribute group inequality to racial or gender preference against members of certain groups—statistical discrimination theories cast group inequality as a consequence of interaction between two _rational_ parties: 1. 1. A utility maximizing decision-maker (e.g., an employer) who has imperfect information about a decision subject’s characteristics and uses his/her group membership as a signal for his/her outcome-relevant unobservable characteristics (e.g., employee’s productivity); 2. 2. Several groups of individuals (e.g., racial or gender groups) that are ex-ante identical in terms of the distribution of qualifications. Individuals are utility maximizing and best-respond to the decision maker’s strategy by adjusting their investment in qualifications. In a seminal work, Arrow [1973] argues that differences between groups can be explained as a form of _coordination failure_ : In equilibrium, the decision maker holds asymmetric beliefs about group qualifications, and this serves as a _self-fulfilling stereotype_. Because of this belief, members of the disadvantaged group don’t have enough incentive to invest in skills and qualifications—precisely because they know that the decision maker will treat them unfavorably). This in turn rationalizes the decision maker’s belief—members of the disadvantaged group indeed end up being less qualified (on average) than the advantage group. Outcome-based policies, such as affirmative action quotas or the application of disparate impact tests, have long been proposed and implemented as a _temporary_ remedy to eliminate group-level inequalities. Such policies may seem particularly effective when self-fulfilling stereotypes are the primary cause of group inequalities. Imposing quota constraints can lead the players to coordinate on a symmetric outcome, that is, and equilibrium in which the decision maker holds symmetric beliefs about different groups. This, however, is not the only possible consequence of imposing quota constraints. Coate and Loury [1993] have shown that quota constraints may reduce the disadvantaged group’s incentives to invest in skills, and subsequently, result in an even worse asymmetric the equilibrium. Coate and Loury call this phenomenon _patronization_. (This is a potential consequence of algorithmic fairness- enhancing interventions.) Advocates of affirmative action have often argued that the larger representation of minorities in higher social positions can generate _role models_ that can positively influence future generations of minorities in their investment decisions. Chung [2000] formalizes these arguments, by allowing for groups to differ in their costs of investment. While the existing literature on affirmative action largely focuses on race- based interventions and policies, some scholars have advocated for _socio- economic_ or _class-based_ affirmative action as a race-neutral alternative (see e.g., [Kahlenberg, 1996; Carnevale and Rose, 2013]). Race-neutral alternatives become particularly important to understand when race-sensitive interventions are constitutionally challenged (for example, see the case of Fisher v. University of Texas at Austin (2013)). The extent to which substituting socio-economic status for race in college admissions is effective in improving minority enrollment is contested (see, e.g., [Kane, 1998; Reardon et al., 2006; Gaertner and Hart, 2013; Reardon et al., 2017]). However, socio- economic affirmative action can certainly facilitate access to college for economically disadvantaged students. The resulting socioeconomic diversity is a desirable outcome in and of itself [Reardon et al., 2017] and in essence, it does not have to compete with or replace race-sensitive admission policies. ### A.4 Comparison with [Durlauf, 2008] [Durlauf, 2008] is one of the closest papers to our work—in terms of the motivation and modeling choices. In his concluding remarks, Durlauf poses the key question our work sets out to answer: how do multigenerational considerations impact the optimal allocation policy? Next, we provide a brief summary of [Durlauf, 2008] and compare our work with it in terms of modeling choices and results. Durlauf focuses on a condition under which efficiency and equality considerations are aligned, and that is when a diverse body of students on college campuses improves the human capital development for all of them. Durlauf [2008] provides a model to compare the equality and efficiency of affirmative action policies with those of meritocratic policies in the context of admission rules to public universities. An admission rule maps the initial human capital of students (and possibly their demographic information, such as race) to their admission decisions. If a student is admitted, their human capital is then improved by the quality of the college they attend and the average human capital of their classmates in that college. The state wishes to employ an admission policy that maximizes the aggregate human capital. He argues that while the meritocratic policy may be efficient under certain conditions, the same holds for affirmative action policies (e.g., when “diversity improves education for all students.”). In many respects, our model is similar to [Durlauf, 2008], although we depart from his model and analysis in a few key areas. Most importantly, we model the dynamics of human-capital development across _multiple generations_ and derive the _optimal allocation policy_. Similar to our work, Durlauf studies a multi-generational model of a society, wherein each generation, a new member is born into every family/dynasty, and he/she replaces the current adult member of the family in the next generation. Even though our model is not expressed in terms of _overlapping generations_ , one can equivalently cast it in those terms. The _initial human capital_ in Durlauf’s model corresponds to our notion of _success probability_. _Adult human capital_ in his model is determined by college attendance, and it roughly maps to our notion of _success_ (i.e., whether the individual succeeds if given the opportunity.) In Durlauf’s model, students always prefer classmates with higher human capital and colleges with higher fundamental quality. Unlike his model, we do not capture the _spillover effects_ of students attending the same college on each other. We also don’t allow for various levels of college quality. (It may be worth noting that while in his general model, Durlauf allows for various college qualities, in his analysis he focuses on at most two quality levels—one college of high quality, the other of low quality.) In both models, an admission rule maps a student’s human capital and group membership into a binary outcome indicating whether the student is admitted. In our work, groups corresponds to _socio-economic tiers_ , whereas in Durlauf’s, they correspond to _racial groups_. In both models, the admission rule may vary across time and generations. Other simplifying assumptions in Durlaufs model are: * • The level of state expenditures on education is fixed across generations. Similarly, alpha is fixed in our model. * • The only output of universities is human capital. Similarly, we only care about the percentage of the population who succeed. * • The initial human capital is accurately measured for every student. Similarly, we assume ability and success probability are perfectly observable. As for the objective function, Durlauf assumes the policymaker wants to maximize the level of human capital among _the next generation_ of adult citizens. He restricts attention to two generations only, but he notes that “When one moves to this dynamic perspective, one also needs to consider an _inter-temporal_ objective function; in parallel to my simplest specification, which is based on the level of human capital at one point in time, it is appropriate to consider a _weighted average_ of the levels across time. A standard way to make this move involves working with […] a _discounted sum_ of adult human capital across two generations.” The latter is precisely the objective function we aim to optimize. Unlike our work, Durlauf does not solve for the optimal policy. He merely compares the meritocratic policy with affirmative action in terms of the average human capital of the next generation. (Similar to our work, he defines a meritocratic admissions policy to mean that each student is admitted to college exclusively on the basis of their initial human capital. An affirmative action policy in his view is one that takes group membership into account as well.) Both models take the perspective of a policymaker who needs to choose between affirmative action and meritocratic rules and aim to understand how efficiency and equity interact. Similar to our findings, he finds that depending on the parameters of the model, affirmative action can be more efficient than myopic meritocratic admission rules. In our model, this happens because the human capital of a student is directly impacted by that of his/her parent. In Durlauf’s work, in contrast, this can happen because of the spillover effects of students on each other and the impact of college quality on developing adult human capital. In more concrete terms, Durlauf’s model can capture the competing claims that ”diversity improves education for all students on campus”, or ”stronger students going to the same college leads to higher social capital for all of them”. Our model does not capture such _spillover_ effects. We instead focus on the _intertemporal_ or intergenerational effects of admission rules. As mentioned earlier, Durlauf emphasizes the importance of intergenerational factors in Section 6.2 of ”Affirmative action, meritocracy, and efficiency”. ### A.5 MDPs with Continuous State and Action Spaces Finding the optimal policy for a continuous state- and action-space MDP is often a difficult task [Marecki et al., 2006]. Two main categories of algorithms have been proposed to (approximately) calculate the optimal policy: (1) A typical approach is to discretize the state and action space, then solve the resulting MDP using standard methods, such policy or value iteration. This approach suffers from the well-known “curse of dimensionality”. (2) Another approach approximates the optimal value function with a parametric form, then sets out to fit those parameters such that they satisfy the Bellman equation (see, e.g., [Li and Littman, 2005; Hauskrecht and Kveton, 2004]. ## Appendix B Limitations and Interpretations Our model is designed to incorporate the basic points we mentioned in the introduction in as simplified a fashion as possible; as such, it is important to note some of its key limitations. First, it is intended to model the effect of a single opportunity, and it treats other forms of mobility probabilistically in the background. It also assumes that its fundamental parameters ($\alpha,\sigma,\tau,\gamma$) as constant over all generations. It treats an individual’s group membership ($A$ and $D$) and ability as a complete description of their performance, rather than including any dependence on the group membership of the individual’s parent. (That is, an individual in group $A$ performs the same in the model regardless of whether their parent belonged to group $A$ or $D$.) All of these would be interesting restrictions to relax in an extension of the model. Much of the past theoretical work on intergenerational mobility focuses on an issue that we do not consider here: the strategic considerations faced by parents as they decide how much to consume in the present generation and how much to pass on to their children. Our interest instead has been in the optimization problem faced by a social planner in allocating opportunities, treating the behavior of the agents as fixed and simple. Here too, it would be interesting to explore models that address these issues in combination. Finally, because our focus is on intergenerational mobility in a socioeconomic sense, we do not model discrimination based on race, ethnicity, or gender, and the role of race-based and gender-based affirmative action in combatting these effects. The model is instead concerned with _socio-economic_ or _class-based_ [Malamud, 1995; Kahlenberg, 1996] affirmative action. That said, the ingredients here could be combined with models of statistical or taste-based discrimination on these attributes to better understand their interaction. The simplicity of our model, however, does allow us to make a correspondingly fundamental point: that even a purely payoff-maximizing society can discover affirmative action policies from first principles as it seeks to optimize the allocation of opportunities over multiple generations. Moreover, the optimal allocation policy is deeply connected to dynamic programming over the generations; the society is essentially attempting to “steer” the balance of group $A$ and group $D$ over time, making sure not to turn things too abruptly (giving up present benefit) or too gradually (giving up future benefit). ## Appendix C Properties of the Value Function In this section, we show that the value function $V(.)$ for the decision process $\mathcal{D}$ is unique, continuous, differentiable, and monotonically decreasing. We begin by offering an alternative formulation of the decision process that has the exact same optimal policy and value function, but is more conducive to recursive analysis. Next, we prove several properties of the state space $\Phi$ and the reward function $R^{\prime}$ for this alternative decision process. Then we apply previously-established theorems from dynamic programming and recursive analysis [Stokey, 1989] to establish the aforementioned properties of $V(.)$. #### A Decision Process Equivalent to $\mathcal{D}$ Given the decision process $\mathcal{D}=(\Phi,\Theta,S,R)$ (as defined in Section 3), we first provide an alternative formulation, called $\mathcal{D}^{\prime}$, that fits the standard representation of decision processes in the Dynamic Programming literature. This standard formulation allows us to import tools and theorems from Dynamic Programming with minor modifications. We construct $\mathcal{D}^{\prime}$ from $\mathcal{D}$ by re-defining the action space—not in terms of thresholds, but in terms of the states reachable through an admissible threshold from a given state. The state transitions in this formulation become trivial, but we need to re-write the reward function in terms of the new parameters (i.e., current and the next state). Formally, given $\mathcal{D}$, we define $\mathcal{D}^{\prime}=(\Phi,\Gamma,I,R^{\prime})$ as follows: * • The correspondence $\Gamma:\Phi\rightarrow\Phi$ specifies the states reachable by one admissible $\theta_{0}$ from any given state $\phi_{0}\in\Phi$. More precisely $\Gamma(\phi_{0})=\\{\omega\in\Phi\|\exists\theta_{0}\in\Theta(\phi_{0})\text{ s.t. }\omega=S(\phi_{0},\theta_{0})\\}.$ * • $I:\Phi_{0}\times\Phi_{0}\rightarrow\Phi_{0}$ simply returns its second argument, that is, for all $\phi_{0}\in[0,1]$, $I(\phi_{0},\omega)=\omega$. * • To recast the reward function in terms of $(\phi_{0},\omega)$, we first write $\theta_{0}$ as a function of $\phi_{0},\omega$: $\displaystyle\omega=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta^{2}_{0})$ $\displaystyle\Leftrightarrow$ $\displaystyle 2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma=\theta^{2}_{0}$ $\displaystyle\Leftrightarrow$ $\displaystyle\theta_{0}=\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}$ Given the above change of variables, we can write: $R^{\prime}(\phi_{0},\omega_{0})=\begin{cases}R\left(\phi_{0},\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}\right)\quad\text{ if }\phi_{0}>0\\\ R(0,\sigma)\quad\text{ if }\phi_{0}=0\end{cases}$ ###### Property 1 $\Phi$ is a convex subset of $\mathbb{R}$, and the correspondence $\Gamma$ is nonempty, compact-valued, and continuous. Proof $\Phi=[0,1]$, which is clearly a convex subset of $\mathbb{R}$. The correspondence $\Gamma$ can be characterized as follows: $\displaystyle\Gamma(\phi_{0})$ $\displaystyle=$ $\displaystyle\\{\omega\in\Phi\|\exists\theta_{0}\in\Theta(\phi_{0})\text{ s.t. }\omega=S(\phi_{0},\theta_{0})\\}$ $\displaystyle=$ $\displaystyle\left\\{\omega\in\Phi\|\omega=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})\text{ where }\sigma\left(1-\frac{\alpha}{\phi_{0}}\right)\leq\theta_{0}\leq\frac{\sigma(1-\alpha)}{\phi_{0}}\text{ and }0\leq\theta_{0}\leq 1\right\\}$ $\displaystyle=$ $\displaystyle\begin{cases}\omega\in\left[\phi_{0}\left(1-\frac{1}{2\sigma}\right),\phi_{0}\right]\quad\text{ if }\phi_{0}\leq\alpha\\\ \omega\in\left[\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right),\phi_{0}\right]\quad\text{ if }\alpha<\phi_{0}\leq 1-\alpha\\\ \omega\in\left[\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right),\phi_{0}-\frac{\phi_{0}}{2\sigma}\left(1-\frac{1-\alpha}{\phi_{0}}\right)\left(1+\frac{1-\alpha}{\phi_{0}}\right)\right]\quad\text{ if }\phi_{0}\geq 1-\alpha\end{cases}$ $\displaystyle=$ $\displaystyle\left[\max\left\\{\phi_{0}\left(1-\frac{1}{2\sigma}\right),\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right)\right\\},\min\left\\{\phi_{0},\phi_{0}-\frac{\phi_{0}}{2\sigma}\left(1-\frac{1-\alpha}{\phi_{0}}\right)\left(1+\frac{1-\alpha}{\phi_{0}}\right)\right\\}\right]$ Given the above definition, it is trivial to verify that $\Gamma$ is indeed nonempty, compact-valued, and continuous. ###### Property 2 The reward function $R^{\prime}$ is bounded and continuous. Proof The reward function $R$ specifies the fraction of the population who succeed if given the opportunity. So clearly $0\leq R(.,.)\leq 1$ is bounded. As a result $R^{\prime}$ is also bounded. To establish continuity of $R^{\prime}$, we first establish the continuity of $R$. It is trivial to see that $R(.,.)$ (defined in (9)) is continuous at any $\phi_{0}<1$. At $\phi_{0}=1$, we have $\displaystyle\lim_{\phi_{0}\rightarrow 1}\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow 1}\frac{1-\phi_{0}}{2\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow 1}\frac{1}{2\sigma}\frac{\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}\right)^{2}}{1-\phi_{0}}$ ($\Gamma(1)=\\{\sigma(1-\alpha)\\}$) $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow 1}\frac{1}{2\sigma}\frac{\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma(1-\alpha)\right)^{2}}{1-\phi_{0}}$ (L’Hospital’s rule) $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow 1}\frac{1}{\sigma}\frac{(\tau+\sigma(1-\alpha))\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma(1-\alpha)\right)}{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-0$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})$ $\displaystyle=$ $\displaystyle R(1,\theta_{0}).$ So $R$ is continuous at $\phi_{0}=1$, as well. Finally, note that $\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}$ is continuous function of $(\phi_{0},\omega)\in\Gamma(\phi_{0})$, so $R^{\prime}$ is also continuous. Let $C(\Phi)$ be the space of bounded continuous functions $f:\Phi_{0}\rightarrow\mathbb{R}$ with the sup norm $\\|f\\|=\max_{\phi_{0}\in\Phi}\|f(\phi_{0})\|$. We define the operator $T$ on the space $C(\Phi)$ as follows: $(Tf)(\phi_{0})=\max_{\omega\in\Gamma(\phi_{0})}R^{\prime}(\phi_{0},\omega)+\gamma f(\omega).$ ###### Theorem 2 (Adapted version of Theorem 4.6 in [Stokey, 1989]) Let $\Phi$, $\Gamma$, and $R^{\prime}$ satisfy Properties 1 and 2. Then the operator $T$ maps $C(\Phi)$ into itself, $T:C(\Phi)\rightarrow C(\Phi)$, and $T$ has a unique fixed point $V\in C(\Phi)$. Moreover, given $V$, the optimal policy correspondence $\Pi^{}:\Phi\rightarrow\Phi$ is compact-valued and upper hemi continuous (u.h.c). According to Cotter and Park [2006], if the reward function $R^{\prime}$ is differentiable, the value function $V$ is differentiable on any interior point that is an optimal “next state” for some current state. More precisely, ###### Theorem 3 (Adapted version of Theorem 2 in [Cotter and Park, 2006]) Suppose $\omega\in\Pi^{}(\phi_{0})\cap(0,1)$ for some $\phi_{0}\in[0,1]$. If $R^{\prime}$ is continuously differentiable, then $V$ is differentiable at $\omega$, with $\frac{\partial}{\partial\phi_{0}}V\|_{\omega}=\frac{\partial}{\partial\phi_{0}}R^{\prime}\|_{(\omega,\omega^{\prime})}$ for any $\omega^{\prime}\in\Pi^{}(\omega)$. It only remains to show that: ###### Property 3 $R^{\prime}$ is continuously differentiable on the interior of its domain. Proof To establish continuous differentiablity of $R^{\prime}$ with respect to $\omega$, note that: $\frac{\partial}{\partial\omega}R^{\prime}(\phi_{0},\omega)=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})\frac{\partial}{\partial\omega}\theta_{0}.$ and both terms on the right hand side of the above equation are continuous. $\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})=\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)$ $\frac{\partial}{\partial\omega}\theta_{0}=\frac{\sigma}{\sqrt{2\sigma\omega\phi_{0}+(\sigma^{2}-2\sigma)\phi_{0}^{2}}}$ Therefore, $\frac{\partial}{\partial\omega}R^{\prime}(\phi_{0},\omega)$ is trivially continuous at any $(\phi_{0},\omega)\in(0,1)^{2}$. To establish continuous differentiability of $R^{\prime}$ with respect to $\phi_{0}$, note that: $\frac{\partial}{\partial\phi_{0}}R^{\prime}(\phi_{0},\omega)=\frac{\partial}{\partial\phi_{0}}R(\phi_{0},\theta_{0})+\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}\theta_{0}.$ It is easy to see that all three terms in the right hand side of the above are continuous: $\displaystyle\frac{\partial}{\partial\phi_{0}}R(\phi_{0},\theta_{0})$ $\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\frac{1}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)$ $\displaystyle-\frac{1}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)\left(\frac{\sigma(1-\alpha)-\theta_{0}}{1-\phi_{0}}\right)$ $\frac{\partial}{\partial\phi_{0}}\theta_{0}=\frac{\sigma\omega}{\phi_{0}^{2}\sqrt{2\sigma\omega\phi_{0}+(\sigma^{2}-2\sigma)\phi_{0}^{2}}}$ Therefore, $\frac{\partial}{\partial\phi_{0}}R^{\prime}(\phi_{0},\omega)$ is continuous at any $(\phi_{0},\omega)$ in the interior of $R^{\prime}$s domain. ###### Proposition 3 $V(\phi_{0})$ is monotonically decreasing at all $\phi_{0}\in(0,1)$. Proof For any $\phi_{0}\leq\phi^{}_{0}$, the statement is easy to verify given the closed form expression for the value function in (12). For any $\phi_{0}>\phi^{}_{0}$ we show that $V$ is decreasing in an open neighborhood on the left side of $\phi_{0}$. Since $V$ is differentiable at $\phi_{0}$, this implies $V^{\prime}(\phi_{0})\leq 0$. Let $\phi^{\prime}_{0}$ be the state we get to if we apply the optimal threshold $\theta_{0}$ at $\phi_{0}$. We have that $\phi^{\prime}_{0}<\phi_{0}$ (note that at $\phi_{0}>\phi^{}_{0}$, $\theta_{0}<\sigma$, which implies $\phi^{\prime}_{0}<\phi_{0}$). Now for any state $\phi^{\prime}_{0}<\phi^{\prime\prime}_{0}<\phi_{0}$, we can show that $V(\phi^{\prime\prime}_{0})>V(\phi_{0})$. This is simply because we can reach $\phi^{\prime}_{0}$ from $\phi^{\prime\prime}_{0}$ using a threshold $\theta^{\prime\prime}_{0}>\theta_{0}$. To see this, note that: $\displaystyle S(\phi_{0},\theta_{0})=\phi^{\prime}_{0}=S(\phi^{\prime\prime}_{0},\theta^{\prime\prime}_{0})$ $\displaystyle\Leftrightarrow$ $\displaystyle\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})=\phi^{\prime\prime}_{0}-\frac{\phi^{\prime\prime}_{0}}{2\sigma}(\sigma^{2}-{\theta^{\prime\prime}_{0}}^{2})$ $\displaystyle\Leftrightarrow$ $\displaystyle\phi_{0}(1-\frac{\sigma}{2}+\frac{\theta_{0}^{2}}{2\sigma})=\phi^{\prime\prime}_{0}(1-\frac{\sigma}{2}+\frac{{\theta^{\prime\prime}_{0}}^{2}}{2\sigma})$ Since $\phi_{0}>\phi^{\prime\prime}_{0}$, it must be the case that ${\theta^{\prime\prime}_{0}}>\theta_{0}$ for the above equation to hold. Next, observe that compared to applying $\theta_{0}$ at $\phi_{0}$, using the higher threshold of $\theta^{\prime\prime}_{0}$ at $\phi^{\prime\prime}_{0}$ leads to a higher immediate reward and the same next state value at $\phi^{\prime\prime}_{0}$. From this observation, we can conclude that the value of $\phi^{\prime\prime}_{0}$ is higher than that of $\phi_{0}$, because: $\displaystyle V(\phi^{\prime\prime}_{0})$ $\displaystyle\geq$ $\displaystyle r(\phi^{\prime\prime}_{0},\theta^{\prime\prime}_{0})+\gamma V(\phi^{\prime}_{0})$ $\displaystyle\geq$ $\displaystyle r(\phi_{0},\theta_{0})+\gamma V(\phi^{\prime}_{0})$ $\displaystyle=$ $\displaystyle V(\phi_{0}).$ Figure 6: The optimal policy $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$ for various settings of $\alpha,\sigma,\tau$, and $\gamma$. Note that in all cases, the optimal threshold is monotonically decreasing with $\phi_{0}$. Moreover, there exists a point below which $\sigma$ is the only optimal threshold, and above which $\sigma$ is no longer optimal. Observe that this point coincides with the tipping point, $\phi^{}_{0}$, established in Theorem 1 (depicted by dashed lines). The dotted red line illustrates the line of affirmative action, derived in Lemma 2. Notice that beyond $\phi^{}_{0}$, the optimal policy is always below the line of affirmative action. Figure 7: The scaled value function at every state $0\leq\phi_{0}\leq 1$ for various settings of $\alpha,\sigma,\tau$, and $\gamma$. The dashed lines specify the tipping point, $\phi^{}_{0}$. Note that in all cases, the value function is continuous, concave, and decreasing. Figure 8: The difference between $\theta_{0}$ and $\theta_{1}$ at every state $0\leq\phi_{0}\leq 1$ for various settings of $\alpha,\sigma,\tau$, and $\gamma$. The dashed lines specify the tipping point, $\phi^{}_{0}$. Note that strict affirmative action is only employed beyond $\phi^{}_{0}$. Also note that the extent of affirmative action is monotonically increasing in $\phi_{0}$. Figure 9: The state the optimal policy converges to with the initial state $\phi_{0}$. The dashed lines specify the tipping point, $\phi^{}_{0}$. Note that the optimal policy never shrinks the size of group $D$ to a value less than $\phi^{}_{0}$. ### C.1 Omitted Proofs #### Proof of Lemma 1. Proof We utilize _backward induction_ to pinpoint the largest state at which $\sigma$ is an optimal threshold. First, observe that if $\phi_{0}$ if sufficiently small, $\sigma\in\Pi^{}_{0}(\phi_{0})$. To see this, note that at least for $\phi_{0}=0$, $\sigma\in\Pi^{}_{0}(\phi_{0})$. Second, note that if $\sigma\in\Pi^{}_{0}(\phi_{0})$), Bellman optimality (10) must hold: $\displaystyle V(\phi_{0})$ $\displaystyle=$ $\displaystyle R(\phi_{0},\sigma)+\gamma V(S(\phi_{0},\sigma))$ $\displaystyle=$ $\displaystyle R(\phi_{0},\sigma)+\gamma V(\phi_{0}),$ where in the second line we utilized the fact that $S(\phi_{0},\sigma)=\phi_{0}$ for all $\phi_{0}\in[0,1]$. This fact can be readily verified through (8). Rearranging the above equation, we obtain: $V(\phi_{0})=\frac{1}{1-\gamma}R(\phi_{0},\sigma).$ Replacing $R$ above with its definition (9) and plugging $\sigma$ in place of $\theta_{0}$, we obtain: $\displaystyle V(\phi_{0})$ $\displaystyle=$ $\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma}{1-\phi_{0}}\right)^{2}\right)$ (12) $\displaystyle=$ $\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{-\sigma\alpha+(1-\phi_{0})\tau+(1-\phi_{0})\sigma}{1-\phi_{0}}\right)^{2}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\sigma+\tau-\frac{\sigma\alpha}{1-\phi_{0}}\right)^{2}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\frac{\sigma\alpha}{1-\phi_{0}}\left(2(\sigma+\tau)-\frac{\sigma\alpha}{1-\phi_{0}}\right)$ $\displaystyle=$ $\displaystyle\frac{\alpha}{(1-\gamma)}\left((\sigma+\tau)-\frac{\sigma\alpha}{2(1-\phi_{0})}\right)$ A corollary of the above is that $V(0)=\frac{\alpha}{(1-\gamma)}\left((\sigma+\tau)-\frac{\sigma\alpha}{2}\right).$ Next, we derive the largest $\phi_{0}$ at which $\sigma$ remains an optimal threshold. Let’s denote this point by $\tilde{\phi}_{0}$. From Bellman optimality, we know that an action is optimal if and only if it is optimal with respect to the (optimal) value function. So taking (12) as the value function $V$ up to $\tilde{\phi}_{0}$, any optimal threshold at $\phi_{0}\leq\tilde{\phi}_{0}$ must maximize $R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))$. Since $R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))$ is differentiable and $\tilde{\phi}_{0}$ is the largest state at which $\sigma$ is an optimal policy, $\sigma$ must satisfy the first order condition at $\tilde{\phi}_{0}$—that is, the partial derivative of $R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))$ (w.r.t. $\theta_{0}$) must be 0 at $(\tilde{\phi}_{0},\sigma)$. We can derive the derivative of $R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))$ as follows: $\frac{\partial}{\partial\theta_{0}}\left\\{R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))\right\\}=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})+\gamma\frac{\partial}{\partial\theta_{0}}\ S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0})).$ (13) We can calculate each term in (13) as follows: $\displaystyle\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$ $\displaystyle=$ $\displaystyle-\frac{\phi_{0}\theta_{0}}{\sigma}+\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)$ (14) $\displaystyle=$ $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}-\theta_{0}\right)$ $\displaystyle=$ $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right).$ $\frac{\partial}{\partial\theta_{0}}S(\phi_{0},\theta_{0})=\frac{\phi_{0}\theta_{0}}{\sigma}.$ (15) $\frac{\partial}{\partial\phi_{0}}V(\phi_{0})=-\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0})^{2}}.$ (16) Plugging (8), (16), (14), and (15) into (13), we obtain that: $\displaystyle\frac{\partial}{\partial\theta_{0}}\left\\{R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))\right\\}=$ (17) $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)-\gamma\frac{\phi_{0}\theta_{0}}{\sigma}\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0}+\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2}))^{2}}.$ As mentioned earlier, at $(\tilde{\phi}_{0},\sigma)$, the derivative (17) must amount to 0. Therefore, to find $\tilde{\phi}_{0}$, we must solve the following equation (obtained by replacing $\theta_{0}$ in (17) with $\sigma$): $\displaystyle 0=\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\sigma}{1-\phi_{0}}\right)-\gamma\phi_{0}\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0})^{2}}$ $\displaystyle\Rightarrow$ $\displaystyle 0=(-\alpha\sigma+(1-\phi_{0})\tau)(1-\phi_{0})-\gamma\frac{\sigma^{2}\alpha^{2}}{2(1-\gamma)}$ $\displaystyle\Rightarrow$ $\displaystyle 0=\tau(1-\phi_{0})^{2}-\alpha\sigma(1-\phi_{0})-\frac{\gamma\sigma^{2}\alpha^{2}}{2(1-\gamma)}$ (Note that in the second line of the derivation above, we multiplied both sides by $\sigma(1-\phi_{0})^{2}/\phi_{0})$.) Solving the above quadratic equation for $(1-\phi_{0})$, we have: $\tilde{\phi}_{0}\in 1-\frac{\alpha\sigma}{2\tau}\left(1\pm\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ To obtain the tightest bound, we pick the smaller value among the above two possibilities: $\tilde{\phi}_{0}=1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ Note that $\tilde{\phi}_{0}$ must always be between $0$ and $(1-\alpha)$ (due to the budget constraints illustrated in Figure 2). So the above derivation only goes through if $\tilde{\phi}_{0}\leq(1-\alpha)$. Therefore, we have: $\phi^{}_{0}=\max\left\\{0,\min\left\\{1-\alpha,\tilde{\phi}_{0}\right\\}\right\\}.$ #### Proof of Proposition 2. Proof Note that according to Lemma 3, for all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, $\theta_{0}\leq\theta_{1}$. It only remains to show that the inequality is strict. That is, for all $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and all $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, $\theta_{0}\neq\theta_{1}$. Suppose not and there exists $\theta_{0}\in\Pi^{}_{0}(\phi_{0})$ and $\theta_{1}\in\Pi^{}_{1}(\phi_{0})$, such that $\theta_{0}=\theta_{1}$. According to Lemma 2, this implies that $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$. Next we show that $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ cannot be an optimal threshold at $\phi_{0}$. Recall that $\Pi^{}_{0}(\phi_{0})=\arg\max_{\theta_{0}}R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0})).$ So if $\sigma(1-\alpha)+\tau(1-\phi_{0})\in\Pi^{}_{0}(\phi_{0})$, it must satisfy the following first-order condition: $\displaystyle\frac{\partial}{\partial\theta_{0}}\left(R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))\right)=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})+\gamma\frac{\partial}{\partial\theta_{0}}\ S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))=0$ But note that $\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$ at $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ is 0: $\displaystyle\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$ $\displaystyle=$ $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)$ $\displaystyle=$ $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\left(\sigma(1-\alpha)+\tau(1-\phi_{0})\right)}{1-\phi_{0}}\right)$ $\displaystyle=$ $\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{0}{1-\phi_{0}}\right)=0$ So $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ can only be optimal if $\frac{\partial}{\partial\theta_{0}}\ S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))=0.$ But this equality cannot hold because $\frac{\partial}{\partial\theta_{0}}S(\phi_{0},\theta_{0})=\frac{\phi_{0}\theta_{0}}{\sigma}>0,$ and $\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))<0$. So $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ cannot be the optimal threshold at $\phi_{0}$. ### C.2 Derivation of $\gamma^{}$, $\tau^{}$, and $\alpha^{}$ The precise derivation of $\phi_{0}^{}$ in Equation 11 allows us to gain insight into how the interaction between the primitive parameters of our model can promote or avert affirmative action. In Figure 1, we focus on the interactions among $\alpha,\tau,\gamma$ (for simplicity assuming that $\sigma=1-\tau$) and illustrate the regimes of persistent affirmative action (i.e., $\phi_{0}^{}\leq 0$). We define and investigate the following quantities: * • Given $\tau$ and $\alpha$, $\gamma^{}$ specifies the minimum discount factor required for $\phi_{0}^{}\leq 0$. * • Given $\gamma$ and $\alpha$, $\tau^{}$ specifies the maximum level of $\tau$ that can maintain $\phi_{0}^{}\leq 0$. * • Given $\tau$ and $\gamma$, $\alpha^{}$ specifies the minimum level opportunities required for $\phi_{0}^{}\leq 0$. Next, we derive the above quantities using Equation 11. In what follows, we assume $\tau>\alpha\sigma$, because we have $\displaystyle 1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ $\displaystyle\leq$ $\displaystyle 1-\frac{\alpha\sigma}{2\tau}\left(1+1\right)$ $\displaystyle\leq$ $\displaystyle 1-\frac{\alpha\sigma}{\tau}$ and if $\tau<\alpha\sigma$, then $1-\frac{\alpha\sigma}{\tau}<1-1=0$ and $\phi^{}_{0}=0$. #### Derivation of $\tau^{}$ Assuming that $\alpha>0$, $\tau,\gamma\in(0,1)$, $\sigma=1-\tau$ and $2\tau>\sigma\alpha$, $\displaystyle 1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle 1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ (assuming $\alpha>0$ and $0<\tau<1$) $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha\sigma}\leq 1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha\sigma}-1\leq\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ (assuming $2\tau>\alpha(1-\tau)$) $\displaystyle\Leftrightarrow$ $\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}\leq 1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{4\tau^{2}}{\alpha^{2}\sigma^{2}}-\frac{4\tau}{\alpha\sigma}+1\leq 1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{4\tau^{2}}{\alpha^{2}\sigma^{2}}-\frac{4\tau}{\alpha\sigma}\leq\frac{2\tau\gamma}{1-\gamma}$ (divide by $\tau$ assuming $\tau>0$) $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha^{2}\sigma^{2}}-\frac{2}{\alpha\sigma}\leq\frac{\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle 2\tau-2\alpha\sigma\leq\frac{\gamma\alpha^{2}\sigma^{2}}{1-\gamma}$ (18) $\displaystyle\Leftrightarrow$ $\displaystyle 2\tau-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$ $\displaystyle\Leftrightarrow$ $\displaystyle 2-2+2\tau-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$ $\displaystyle\Leftrightarrow$ $\displaystyle 2-2(1-\tau)-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$ $\displaystyle\Leftrightarrow$ $\displaystyle 0\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}+2(1+\alpha)(1-\tau)-2$ $\displaystyle\Leftrightarrow$ $\displaystyle 0\leq\frac{\gamma\alpha^{2}}{2(1-\gamma)}(1-\tau)^{2}+(1+\alpha)(1-\tau)-1$ $\displaystyle\Leftrightarrow$ $\displaystyle\tau\leq 1-\frac{-(1+\alpha)+\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}$ where the last line is derived by obtaining the roots of the quadratic function $q(x)=\frac{\gamma\alpha^{2}}{2(1-\gamma)}x^{2}+(1+\alpha)x-1$, as follows: $x^{}_{1}=\frac{-(1+\alpha)-\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}\text{ , }x^{}_{2}=\frac{-(1+\alpha)+\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}.$ Note that $\frac{\gamma\alpha^{2}}{2(1-\gamma)}>0$, so $q(x)\geq 0$ if and only if $x<x^{}_{1}$ or $x>x^{}_{2}$. Note that $x^{}_{1}<0$ so if we know $x$ is positive, then $q(x)\geq 0$ if and only if $x>x^{}_{2}$. Replacing $x$ with $(1-\tau)$, gives us equation (18). #### Derivation of $\alpha^{}$ Assuming that $\tau\in(0,1)$ and $\gamma<1$, $\displaystyle 1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle 1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\sigma}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)^{-1}\leq\alpha$ #### Derivation of $\gamma^{}$ Assuming that $\alpha,\tau,\sigma,\gamma\in(0,1)$ and $\tau>\sigma\alpha$, $\displaystyle 1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq 0$ $\displaystyle\Leftrightarrow$ $\displaystyle 1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha\sigma}\leq 1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha\sigma}-1\leq\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ $\displaystyle\Leftrightarrow$ $\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}\leq 1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-1\leq\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle\frac{1}{2\tau}\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-\frac{1}{2\tau}\leq\frac{\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$ $\displaystyle 1-\left(\frac{1}{2\tau}\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-\frac{1}{2\tau}+1\right)^{-1}\leq\gamma$
Simple finite elements and multigrid for efficient mass-consistent wind downscaling in a coupled fire-atmosphere model J. Mandel 1, A. Farguell 2, A. K. Kochanski 2, D. V. Mallia 3, K. Hilburn 4 ${}^{1}\,$University of Colorado Denver, Denver, CO 2San José State University, San José, CA 3University of Utah, Salt Lake City, UT 4Colorado State University, Fort Collins, CO ## 1 Introduction In the coupled atmosphere-fire model WRF-SFIRE [6, 7], the Weather Research Forecasting (WRF) model [12] runs at 300m–1km horizontal resolution, while the fire model runs at the resolution of 30m or finer. The wind has a fundamental effect on fire behavior and the topography details have a strong effect on the wind, but WRF does not see the topography on the fire grid scale. We want to downscale the wind from WRF to account for the fine-scale terrain. For this purpose, we fit the wind from WRF with a divergence-free flow over the detailed terrain. Such methods, called mass-consistent approximations, were originally proposed on regular grids [10, 11] for urban and complex terrain modeling, with terrain and surface features modeled by excluding entire grid cells from the domain. For fire applications, WindNinja [13] uses finite elements on a terrain-following grid. The resulting equations are generally solved by iterative methods such as SOR, which converge slowly, so use of GPUs is of interest [2]. A multigrid method with a terrain-following grid by a change of coordinates was proposed in [15]. The method proposed here is to be used in every time step of WRF-SFIRE in the place of interpolation to the fire model grid. Therefore, it needs to have the potential to (1) scale to hundreds or thousands of processors using WRF parallel infrastructure [14]; (2) scale to domains size at least 100km by 100km horizontally, with $3000\times 3000\times 15$ grid cells and more; (3) have reasonable memory requirements per grid point; (4) not add to the cost of the time step significantly when started from the solution in the previous time step; and, (5) adapt to the problem automatically, with minimum or no parameters to be set by the user. ## 2 Finite element formulation Given vector field $\boldsymbol{u}_{0}$ on domain $\Omega\subset\mathbb{R}^{d}$, subset $\Gamma\subset\partial\Omega$, and $d\times d$ symmetric positive definite coefficient matrix $\boldsymbol{A}=\boldsymbol{A}\left(\boldsymbol{x}\right)$, we want to find the closest divergence-free vector field $\boldsymbol{u}$ by solving the problem $\min_{\boldsymbol{u}}\frac{1}{2}\int\limits_{\Omega}\left(\boldsymbol{u}-\boldsymbol{u}_{0}\right)\cdot\boldsymbol{A}\left(\boldsymbol{u}-\boldsymbol{u}_{0}\right)d\boldsymbol{x}\text{\quad subject to }\operatorname{div}\boldsymbol{u}=0\text{ in }\Omega\text{ and }\boldsymbol{u}\cdot\boldsymbol{n}=0\text{ on }\Gamma,$ (1) where $\Gamma$ is the bottom of the domain (the surface), and $\boldsymbol{A}\left(\boldsymbol{x}\right)$ is a $3\times 3$ diagonal matrix with penalty constants $a_{1}^{2},a_{2}^{2},a_{3}^{2}$ on the diagonal. Enforcing the constraints in (1) by a Lagrange multiplier $\lambda$, we obtain the solution $\left(\boldsymbol{u},\lambda\right)$ as a stationary point of the Lagrangean $\mathcal{L}\left(\boldsymbol{u},\lambda\right)=\frac{1}{2}\int\limits_{\Omega}\boldsymbol{A}\left(\boldsymbol{u}-\boldsymbol{u}_{0}\right)\cdot\left(\boldsymbol{u}-\boldsymbol{u}_{0}\right)d\boldsymbol{x}+\int\limits_{\Omega}\lambda\operatorname{div}\boldsymbol{u}d\boldsymbol{x}-\int\limits_{\Gamma}\lambda\boldsymbol{n}\cdot\boldsymbol{u}d\boldsymbol{s}.$ (2) Eliminating $\boldsymbol{u}$ from the stationarity conditions $\partial\mathcal{L}(\boldsymbol{u},\lambda)/\partial\lambda=0$ and $\partial\mathcal{L}(\boldsymbol{u},\lambda)/\partial\boldsymbol{u}=0$ by $\boldsymbol{u}=\boldsymbol{u}_{0}+\boldsymbol{A}^{-1}\operatorname{grad}\lambda$ (3) leads to the generalized Poisson equation for Lagrange multiplier $\lambda$, $-\operatorname{div}\boldsymbol{A}^{-1}\operatorname{grad}\lambda=\operatorname{div}\boldsymbol{u}_{0}\text{ on }\Omega,\quad\lambda=0\text{ on }\partial\Omega\setminus\Gamma,\text{ \quad}\boldsymbol{n\cdot A}^{-1}\operatorname{grad}\lambda=-\boldsymbol{n\cdot u}_{0}\text{ on }\Gamma.$ (4) Multiplication of (4) by a test function $\mu$, $\mu=0$ on $\partial\Omega\setminus\Gamma$, and integration by parts yields the variational form to find $\lambda$ such that $\lambda=0$ on $\partial\Omega\setminus\Gamma$ and $\int_{\Omega}\boldsymbol{A}^{-1}\operatorname{grad}\lambda\cdot\operatorname{grad}\mu\,d\boldsymbol{x}=-\int_{\Omega}\operatorname{grad}\mu\cdot\boldsymbol{u}_{0}d\boldsymbol{x}$ (5) for all $\mu$ such that $\mu=0$ on $\partial\Omega\setminus\Gamma$. The solution is then recovered from (3). We proceed formally here; see [5] for a different derivation of (5) in a functional spaces setting. The variational problem (5) is discretized by standard isoparametric 8-node hexahedral finite elements, e.g., [4]. The integral on the left-hand side of (5) is evaluated by tensor-product Gauss quadrature with two nodes in each dimension, while for the right-hand side, one-node quadrature at the center of the element is sufficient. The same code for the derivatives of a finite element function is used to evaluate $\operatorname{grad}$ $\lambda$ in (3) at the center of each element. The unknown $\lambda$ is represented by its values at element vertices, and the wind vector is represented naturally by its values at element centers. No numerical differentiation of $\lambda$ from its nodal values, computation of the divergence of the initial wind field $\boldsymbol{u}_{0}$, or explicit implementation of the boundary condition on $\operatorname{grad}\lambda$ in (4) is needed. These are all taken care of by the finite elements naturally. ## 3 Multigrid iterations The finite element method for (5) results in a system of linear equations $Ku=f$. The values of the solution are defined on a grid, which we will call a _fine grid_. One cycle of the multigrid method consists of several iterations of a basic iterative method, such as Gauss-Seidel, called a _smoother_ , followed by a _coarse-grid correction_. A prolongation matrix $P$ is constructed to interpolate values from a coarse grid, in the simplest case consisting of every other node, to the fine grid. For a given approximate solution $u$ after the smoothing, we seek an improved solution in the form $u+Pu_{c}$ variationally, by solving $P^{\top}K\left(u+Pu_{c}\right)=P^{\top}f$ (6) for $u_{c}$, and obtain the coarse-grid correction procedure as $\displaystyle f_{c}=P^{\top}\left(f-Ku\right)\qquad$ form the coarse right- hand side $\displaystyle K_{c}=P^{\top}KP\qquad$ form the coarse stiffness matrix $\displaystyle K_{c}u_{c}=f_{c}\qquad$ solve the coarse-grid problem (7) $\displaystyle u\leftarrow u+Pu_{c}\qquad$ insert the coarse-grid correction The coarse grid correction is followed by several more smoothing steps, which completes the multigrid cycle. In the simplest case, $P$ is a linear interpolation and the coarse stiffness matrix $K_{c}$ is the stiffness matrix for a coarse finite element discretization on a grid with each coarse-grid element taking the place of a $2\times 2\times 2$ agglomeration of fine-grid elements. That makes it possible to apply the same method to the coarse-grid problem (7) recursively. This process creates a hierarchy of coarser grids. Eventually, the coarsest grid problem is solved by a direct method, or one can just do some more iterations on it. Multigrid methods gain their efficiency from the fact that simple iterative methods like Gauss-Seidel change the values of the solution at a node from differences of the values between this and neighboring nodes. When the error values at neighboring nodes become close, the error can be well approximated in the range of the prolongation $P$ and the coarse-grid correction can find $u_{c}$ such that $u+Pu_{c}$ is a much better approximation of the solution. For analysis of variational multigrid methods and further references, see [1, 8]. Multigrid methods are very efficient. For simple elliptic problems, such as the Poisson equation on a regular grid, convergence rates of about $0.1$ (reduction of the error by a factor of $10$) at the cost of $4$ to $5$ Gauss- Seidel sweeps on the finest grid are expected [3]. However, the convergence rates get worse on more realistic grids, and adaptations are needed. We choose as the smoother vertical sweeps of Gauss-Seidel from the bottom up to the top, with red-black ordering horizontally into $4$ groups. For the base method, we use $2\times 2\times 2$ coarsening and construct $P$ so that the vertices of every $2\times 2\times 2$ agglomeration of elements interpolate to the fine- grid nodes in the agglomeration, with the same weights as the trilinear interpolation on a regular grid. The interpolation is still trilinear on a stretched grid, but only approximately trilinear on a deformed terrain- following grid. The base method works as expected as long as some grid layers are not tightly coupled. If they are, we mitigate the slower convergence by semicoarsening [9]: After smoothing, the error is smoother in the tightly coupled direction(s), which indicates that we should not coarsen the other direction(s). When the grid is stretched vertically away from the ground, the nodes are relatively closer and thus tightly coupled in the horizontal direction. Similarly, when the penalty coefficient $a_{3}$ in the vertical direction is larger than $a_{1}$ and $a_{2}$ in the horizontal directions, the neighboring nodes in the vertical direction are tightly coupled numerically. The algorithm to decide on coarsening we use is: Suppose that the penalty coefficients are $a_{1}=a_{2}=1$ and $a_{3}\geq 1$, and at the bottom of the grid, the grid spacing is $h_{1}=h_{2}$ (horizontal) and $h_{3}$ (vertical). If $h_{3}/(h_{1}a_{3})>1/3$, coarsen in the horizontal directions by $2$, otherwise do not coarsen. Then, replace $h_{1}$ and $h_{2}$ by their new values, corsened (multiplied by 2) or not, and for every horizontal layer from the ground up, if $h_{3}/(h_{1}a_{3})]<3$, coarsen about that layer vertically, otherwise do not coarsen. This algorithm retains the coarse grids as logically cartesian, which is important for computational efficiency and keeping the code simple, and it controls the convergence rate to remain up to about $0.28$ with four smoothing steps per cycle. ## 4 Conclusion We have presented a simple and efficient finite element formulation of mass- consistent approximation, and a multigrid iterative method with adaptive semicoarsening, which maintains the convergence of iteration over a range of grids and penalty coefficients. A prototype code is available at https://github.com/openwfm/wrf-fire-matlab/tree/femwind/femwind. Acknowledgement: This work has been supported by NSF grant ICER-1664175 and NASA grant 80NSSC19K1091. ## References * [1] R. E. Bank and T. 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Inhalation drug delivery has seen a swift rise in the use of dry powder inhalers (DPIs) to treat chronic respiratory conditions. However, universal adoption of DPIs has been restrained due to their low efficiencies and significant drug losses in the mouth-throat region. Aerosol efficiency of DPIs is closely related to the fluid-dynamics characteristics of the inhalation flow generated from the devices, which in turn are influenced by the device design. In-vitro and particle image velocimetry (PIV) have been used in this study to assess the aerosol performance of a model carrier formulation delivered by DPI devices and to investigate their flow characteristics. Four DPI device models, with modification to their tangential inlets and addition of a grid, have been explored. Similar aerosol performances were observed for all four device models, with FPF larger than 50%, indicating desirable lung deposition. A high swirling and recirculating jet-flow emerging from the mouthpiece of the DPI models without the grid was observed, which contributed to particle deposition in the throat. DPI models where the grid was present showed a straightened outflow without undesired lateral spreading, that reduced particle deposition in the throat and mass retention in the device. These findings demonstrate that PIV measurements strengthen evaluation and can be jointly used to develop high-performance DPIs. § INTRODUCTION The last few decades have seen dry powder inhalers (DPIs) evolve as a clinically-appropriate and preferred device to treat chronic respiratory conditions via aerosol drug delivery. This rapid development is due to the advantages that DPIs offer, which include delivery of larger doses, greater drug stability, and ease of use. In addition, the patient's inspiratory flow is the primary energy source for drug detachment and dispersion, thereby removing the requirement for the patient's coordination during inhalation. The need for a forceful and deep inhalation, however, creates disadvantages, such as large variability in the required inhalation effort Azouz2015, especially for patients with severe airflow limitation, low dose-emission uniformity Hindle1995, and high mouth-throat losses DeHaan2004, restricting the widespread use of DPIs in different patient populations. Moreover, despite the advances in the last decades, DPIs still suffer from poor efficiency, with conventional devices delivering approximately 20 to 30% of the nominal dose to the lungs at a normal inhalation flow rate Buttini2016. DPI's performance is evaluated by the particle size distribution delivered to the lungs. Efficiency is assessed based on metrics of mean mass aerodynamic diameter, emitted dose, fine particle dose and fine particle fraction using impaction studies. Various factors affect this performance, such as particle entrainment and de-agglomeration, the device resistance at a given inhalation flow rate, and the formulation and properties of the drug Frijlink2004, Atkins2005. The first two factors are critical as they are mainly controlled by the device design, which in turn significantly affects not only the generation and properties of delivered aerosol, but also drug losses due to particle deposition in the mouth-throat region DeBoer2017. The characteristics of the resulting particle aerosol flow that emerges as a particle-laden jet from the DPI mouthpiece is closely correlated with the fluid-dynamics characteristics of that jet. When coupled with fluid motion in the human respiratory tract, these characteristics strongly control fine particle deposition in the lungs. Characterisation of DPI jet-flow in this study has been performed using the experimental technique of particle image velocimetry (PIV). PIV is a non-intrusive, laser-based, optical imaging technique that enables measurement of the instantaneous 2-component - 2-dimensional (2C-2D) velocity field with high spatial and temporal resolutions, and has been used previously to investigate the fluid-dynamics characteristics of flows in DPIs. The velocity fields at different planes normal to the longitudinal axis of the mouthpiece of a Spiros® model DPI for three different flow rates were measured using PIV by Han et al. Han2002. It was found that tangential inlet jets produced a cross-flow resulting in large re-circulation flow zones close to the mouthpiece wall. PIV measurements have also been performed in the mouthpiece of an idealized DPI model with an upstream grid, which showed an increase in turbulence intensities with grid voidage Ngoc2013. Furthermore, powder dispersion measurements in a Rotahaler® model DPI using PIV have revealed that particle-grid collisions and drag force were responsible for powder de-agglomeration, whereas particle-particle and particle-grid collisions assisted in dispersion of the de-agglomerated powder Kou2016. Pasquali et al. Pasquali2015 measured the axial and radial velocity components across a plane perpendicular to the mouthpiece exit of a Nexthaler® DPI. The device was tested at a transient inhalation airflow that had a peak flow rate of 60 and a rise time of 0.3 s. They found a slight asymmetry in the velocity magnitude field across the jet center-line indicating the presence of high swirl levels in the internal flow. Although the mean velocities represented moving average over only 10 vector fields, it showed that the increase in mean velocity correlated with the decrease in measured pressure difference across the inhaler. Voss and Finlay Voss2002 used laser doppler velocimetry to measure turbulent flow velocities in an entrainment tube rig and a Diskhaler® DPI. They found that although higher turbulence velocities caused greater particle deagglomeration, turbulence might not be the only or most effective de-agglomeration mechanism in DPIs. Wang et al. Wang2004 examined experimentally the effect of an impinging air jet on powder dispersion in a Ventodisk® DPI for various jet velocities, nozzle-to-surface separation distances, and dosing cup shapes. Optimum dispersion was found to occur at higher jet velocities and nozzle-to-surface separation distance of 5 jet diameters. A recent experimental study in a channel flow with a grid placed upstream of a pocket bed of lactose carrier powder Elserfy2020 has shown that powder de-agglomeration at air flow rates of 60 and above depends more on the action of aerodynamic shear forces on the agglomerates generated by higher mean flow and grid turbulence, than on the powder properties. An extensive examination of the flow emerging from a DPI is important to fully understand aerosol dispersion from the device. The jet-flow from a DPI has not been extensively quantified in any of the previous experimental studies, including the distribution of mean and turbulent flow statistics, as well as quantification of changes in flow upon modifications due to the DPI design. The present study addresses these issues by carrying out an experimental investigation of the fluid-dynamics characteristics of flows originating from DPIs having different inlet configurations and grid positions. These results are then used to corroborate the findings of studies performed on the same DPIs. § MATERIALS AND METHODS §.§ Fluid Mechanics Scaling An important dimensionless quantity that characterises fluid flows is the Reynolds number, which for a DPI can be defined as \begin{equation} Re_a = \frac{U_aD_a}{\nu_a} \end{equation} where ${U_a}$ is the characteristic velocity, taken as the average flow velocity at the DPI mouthpiece exit, ${D_a}$ is the characteristic length, taken as the mouthpiece exit inner-diameter, and ${\nu_a}$ is the kinematic viscosity of the fluid. The subscript $'a'$ here refers to air as the fluid. For an actual DPI with ${D_a}$ = 10 mm and an inspiratory air flow-rate of ${Q_a}$ = 60, as recommended for medium airflow resistance DPIs Byron1994, Ari2020, DeBoer2003, this yields ${U_a}$ = 12.74 m/s and ${Re_a \approx}$ 8400. PIV experiments can then be performed at dynamically similar conditions if the experimental flow is at the same Reynolds number. At this point, let us consider that these experiments are to be performed using water as the working fluid, such that \begin{equation} Re_w = \frac{U_wD_w}{\nu_w}={Re_a} \end{equation} where the subscript, $w$, refers to water in this case. This results in the following relationship between the geometric and dynamic flow conditions required between the water-based experiment and the dynamically equivalent air-based DPI flows as \begin{equation} \frac{U_w}{U_a} \frac{D_w}{D_a} = \frac{\nu_w}{\nu_a} \end{equation} The value of ${\nu_w}/{\nu_a}$ at a normal room temperature of $20 \degree$C is 0.066, which means that the geometric scaling factor, defined by ${S_f}={D_w}/{D_a}$, can be chosen to be greater than 1 such that ${U_w}<{U_a}$. So, for a scale factor of ${S_f} = 3$, ${D_w}$ = 30 mm with ${U_w}$ = 0.281 m/s, which is two orders of magnitude lower than ${U_a}$. Thus, this permits PIV measurements in the water-based model to be performed with higher spatial and temporal resolution than in the dynamically equivalent smaller air-based model. §.§ DPI Device Models Four DPI models have been used in this study. The DPI models used with air for study and with water for PIV experiments are geometrically similar, with the models for the latter being scaled-up by a factor of three, as explained in the previous section. These models have a fixed mouthpiece exit inner-diameter of ${D_a}$ = 10 mm and ${D_w}$ = 30 mm respectively, with a uniform circular inner cross-section for the mouthpiece. The models differ in their configurations of tangential inlets and grid positions as shown in Fig. <ref>. The four models shown in the top row were used for the study, while those in the bottom row were used for PIV experiments. The model in Fig. <ref>(a) had 2 tangential inlets spaced $180\degree$ apart, while the model in Fig. <ref>(b) had 6 tangential inlets spaced $60\degree$ apart, with the summed area of the tangential inlets in the two models being the same. The model in Fig. <ref>(c) had a grid positioned just above the tangential inlets, whereas the model in Fig. <ref>(d) has the same grid positioned at the mouthpiece exit. The grid for the models presented square holes of side 1 mm and spaced 0.5 mm apart, while for the experimental models it was geometrically scaled-up by a factor of three. Each model had the same dosing cup design, which is a hollow hemisphere. For the experimental DPI models the dosing cup was integrated into a base fixture with a bottom flange to facilitate mounting the model in the experimental rig. DPI device models examined in this study The models were 3D printed in FormLabs Clear Resin v4 Methylacrylic Oligomer 75-90%, Methylacrylic monomer 25-50%, Diphenyl (2,4,6-trimethylbenzoyl) phosphine oxide < 1%, in a FormLab 3 3D Printer (FormLabs, Summervile, USA). The support material was removed after printing and the models were washed in the Form Wash (FormLabs, Sumervile, USA) using fresh isopropyl alcohol for 15 minutes, followed by curing in the Form Cure (FormLabs, Sumervile, USA) for 30 minutes at 65$^{\circ}$C. The experimental models were 3D printed in ABSplus thermoplastic material with a layer thickness of 0.254 mm, and the model outer surfaces were coated with urethane to prevent any structural porosity while immersed in water. §.§ In-vitro Measurements §.§.§ Materials The formulation used, comprising of micronised beclamethasone dipropionate (BDP), the pre-blend composed of micronised magnesium stearate and micronised lactose, coarse granular α-lactose monohydrate (212 - 350) was prepared as described by Yeung et al. Yeung2019. High-performance liquid chromatography (HPLC) grade methanol was purchased from Honeywell (North Carolina, USA), ethanol and isopropanol were obtained from ChemSupply (Sydney, Australia). Deionised water, used in this study, was purified by reverse osmosis (MilliQ, MilliPore, Australia). Brij35 and glycerol were purchased from Sigma (Sigma Aldrich, USA). §.§.§ Formulation preparation Due to the direct influence of formulation on aerosol performance, and to study the effect of different device parameters on aerosol performance, a model formulation of 1% BDP (w/w) with high aerosol performance was used Yeung2018. This formulation includes a pre-blend of magnesium stereate as adjuvant to reduce the cohesive forces between the API (BDP) and the carrier (lactose), facilitating API dettachment. A BDP-carrier based formulation containing 1% BDP (w/w), coarse lactose at 89.1% (w/w) and pre-blend magnesium stearate at 9.9% (w/w) was used in the study. Formulation was prepared as described by Yeung et al. Yeung2019. Briefly, a pre-blend of magnesium stearate and coarse granular α-lactose were mixed at a 1:9 ratio (w/w) for 4 h at 32 rpm using a low shear 3-dimensional shaker-mixer (Alphie-03, Hexagon Product Development PVT. LTD., Vadodara, India). Micronised BDP was added to the carrier 24 h after carrier:pre-blend preparation to minimize the effect of electrostatic charges, and mixed for 90 minutes at 32 rpm. To remove any potential agglomerates the formulation was sieved through a 400 sieve, and mixed for further 30 min at 32 rpm using low shear 3-dimensional shaker-mixer. A resting period of 24 h in a desiccator was used prior to any further analysis to minimize electrostatic charges effect. §.§.§ Particle size distribution The particle size distribution of micronised BDP was assessed using laser diffraction with a Mastersizer 3000 (Malvern, Worcestershire, UK) equipped with a Mastersizer® Aero S™ dry powder dispersion unit (Malvern, Worcestershire, UK), tray and hopper. Samples were dispersed at 4 mbar shear pressure. The refractive index used was 1.56. Five measurements were performed and analysed in Mastersizer 3000 Software (Version 3.81). Results are depicted as D_10 , D_50 and D_90 . §.§.§ Particle morphology The morphology of the micronised BDP and BDP-loaded formulation was visualized under a scanning electron microscope (SEM, JCM-6000 Neoscope Scanning Electron Microscope, Joel Ltd., Akishima Tokyo, Japan). The samples were placed on a circular carbon tape and coated with 15 thickness of gold using a sputter gold-coater (Smart Coater, Joel Ltd., Akishima Tokyo, Japan). The samples were imaged at an accelerating voltage of 10 kV. §.§.§ Drug content uniformity Drug content uniformity of the formulation was assessed to ensure that a homogeneous blend was obtained. The assay was performed based on British Pharmacopoeia Office2017. Briefly, after dispersion on wax paper, ten random powder samples of 10 mg were collected, and dissolved in methanol:water (80:20, v/v) solution to dissolve the drug. Samples were vortexed for 30 s to ensure drug dissolution, and filtered in 0.45 PTFE filter (Aireka Scientific, Zhejiang, China). Concentration of BDP was determined using a validated HPLC as described in the following section. The content uniformity of the drug in the formulation is expressed as a mean percentage of the theoretical loaded dose ± standard deviation. The acceptance value (AV) was also calculated based on the British Pharmacopoeia Office2017. §.§.§ Drug quantification via HPLC The concentration of BDP was quantified in a Shimadzu HPLC system consisting of a LC20AT pump, the SIL20AHT autosampler and an SPD-20A UV-VIS detector (Shimadzu, Sydney, NSW, Australia) using a previously validated method Yeung2019. Chromatographic separation of BDP was achieved using a Luna C-18 column (150 $\times$ 4.6 mm, 3, Phenomenex, Torrance, USA). Samples were run in methanol:water (80:20) mobile phase, in isocratic flow of 0.8. BDP was detected at 243. Injection volume was kept at 100ł. A calibration curve between 0.1 - 100 was used to extrapolate the concentration of BDP in the samples. §.§.§ In-vitro aerosol performance using cascade impactor Aerosol deposition profile was conducted using British Pharmacopoeia Apparatus E – Next Generation Impactor (NGI, Copley, UK) connected to a critical flow controller (TPK 2100-R, Copley, UK) and a rotary pump (Copley, UK). Flow rate was set for ${Q_a}$ = 60 using a calibrated flow meter (Model 4040, TSI Precision Measurement Instruments, Aachen, Germany). To minimize particle bouncing, 50ł of Brij 35:glycerol:ethanol (10:50:40) solution was used per stage, to coat all the stages of the NGI (S1-S7 and micro orifice collector, MOC). The USP induction port was coated with 2 ml of BriJ35:glycerol:ethanol solution and spread onto its internal surface using a brush to prevent particle bouncing and assess throat deposition. Excess coating solution was removed by verting the induction port for 1 minute prior to deposition. For this assay, the cascade impactor was tilted at 45$^{\circ}$ angle to the bench to minimise any potential loss of the formulation loaded to the device from the air inlets. The NGI pre-separator was used to collect the remaining lactose carrier particles. Due to the tilted position of the NGI, a glass microfibre disc (Sartorius Stedin, Goettingen, Germany) was cut to fit into the central cup of the pre-separator and wetted with 2 ml of mobile phase, as a replacement of the 15 ml of mobile phase recommended by the Pharmacopoeia to collect the samples. For each actuation, 10 mg of the BDP-loaded formulation was weighted on to the device dosing cup. Aerosol deposition was conducted for 4 s, based on the cutoff diameters of the NGI at 60 (Cutoffs: s1, 8.06; s2, 4.46; s3, 2.82; s4, 1.66; s5, 0.94; s6, 0.55; s7, 0.34; and micro-orifice collector, MOC, 0.00). The drug deposited in each stage was recovered using the mobile phase methanol:water (80:20) with the following volumes: device, 5 ml; adapter, 5 ml; induction port, 10 ml; pre-separator, 35 ml; S1 and MOC, 10 ml; S2-S7, 5 ml. All solutions were filtered using 0.45 PTFE filters prior to HPLC detection. The devices were weighted before and after actuation to determine shot weight. As required by the British Pharmacopoeia Office2017, total mass recovery was set within 85 - 115% of the nominal dose. Each device was tested in triplicate, with one actuation performed per run. Data were analysed in Copley Inhaler Testing Data Analysis Software (CITDAS) (Version 3.10 Wibu, Copley, Nottingham, UK) based on the derived parameters of delivered dose (DD, total dose in that was recovered per experiment), fine particle dose (FPD, mass in of particles below 5), fine particle fraction (FPF% emitted dose, percentage of particles below 5), mass median aerodynamic diameter (MMAD, calculated as the 50^th percentile of the particle size distribution), geometric standard deviation (GSD, calculate as the square root of the 84.13^th/15.87^th percentile). As the throat was coated with Brij solution to assess the effect of the device design on throat deposition, the results of throat and pre-separator depositions have been combined. §.§.§ Statistical analysis of in-vitro analysis Data are presented as mean ± standard deviation of three independent experiments (n=3). Statistical analysis was performed using GraphPad Prism Software version 8.0 (GraphPad, San Diego, USA). and device models were compared by two-tailed t-test assuming gaussian distribution at 95% CI. The effect of the grid position was compared amongst the , and device models using one-way analysis of variance (ANOVA) followed by Tukey post hoc test. Differences were considered statistically different at 95% CI (* P0.05, P0.01, * P0.001 and **** P0.0001). §.§.§ Pressure drop and device intrinsic resistance The intrinsic resistance and pressure drop of each device was measured by connecting the induction port measurement adapter (Copley, UK) between the device and the induction port of the NGI cascade impactor. The system was connected to the critical flow controller to measure the pressure drop ($\Delta$P) over the inhaler under test (4 s), when the flow rate was set for ${Q_a} = \SI{60}{\litre\per\minute}$ using a calibrated flow meter (Model 4040, TSI Precision Measurement Instruments, Aachen, Germany). Pressure drop is expressed as the mean of three independent measurements using different devices. The intrinsic resistance of the device was calculated from $\sqrt{{\Delta}P}/{Q_a}$, and is expressed in units of ^0.5.^-1. §.§ PIV Experiments §.§.§ Experimental apparatus A schematic of the PIV experimental setup is shown in Fig. <ref>. The experimental rig comprises the DPI model placed in a Perspex tank with a closed-loop water flow system (represented by blue lines). The inflow is at the tank base, while the outflow is from the tank top. The tank has a circular channel milled in its base from the midpoint of one of its edges to the tank centre. Water flows in through this channel with an axial outflow, impinging on a circular plate which is cut out from the base to form an annular region as shown in Fig. <ref>. Flow enters the tank from this annular region at an average velocity that is an order of magnitude lower than ${U_w}$, and then flows into the DPI model through the tangential inlets. A confining plate with sides equal to the inner cross-section of the tank (300 mm $\times$ 300 mm) is placed flush with the DPI model mouthpiece end to ensure that flow exits from the mouthpiece to avoid leakage from the interfaces between the plate, DPI model, and inner surfaces of the tank. The distance from the mouthpiece exit to the top of the tank is approximately 9${D_w}$. A magnetically coupled centrifugal pump provides the required pressure difference to drive the flow through the rig. The water flow rate is controlled via a globe valve and measured using a variable area flow meter placed downstream of the pump. A steady water flow rate ${Q_w}$ of approximately 12 is maintained resulting in ${Re_w \approx}$ 8400. Experimental setup for PIV §.§.§ PIV system and parameters The PIV system is shown at the right of Fig. <ref>. Water in the tank is seeded with hollow glass spheres that have a mean particle diameter of 11. These particles, which have been used in numerous previous fluid mechanics studies Kostas2005, Buchner2012, Gonzalez-Espinosa2014, have a density of 1.1 g/cc and a relaxation time of 7.38 in water such that they faithfully follow the flow with high fidelity. A double-cavity pulsed Nd:YAG laser (New Wave Research) emitting at a wavelength of 532 nm is used to illuminate these particles. This laser can generate 120 mJ double-pulses of duration 3 - 5 ns at a repetition frequency of 15 Hz. The output laser beam is directed towards the experimental rig using an articulated mirror arm and then shaped into a thin light sheet using a telescope and plano-concave lens arrangement. The light sheet is approximately 1 mm thick and is aligned coincident with a vertical plane passing through the center of mouthpiece exit as shown in Fig. <ref>. Single-exposed double-frame PIV images are acquired using the array sensor (4008 px $\times$ 2672 px, 9) of a CCD camera (PCO AG pco.4000) at a frame rate of 2 Hz, with a time delay of 820 between two laser pulses. A 105 mm Micro-Nikkor lens set at an aperture of f/4 is used for these experiments with an image magnification of 0.26, resulting in a spatial resolution of 35.2/px. The synchronous timing signals to control the laser and the PIV image acquisition using the CCD camera are generated from a fully programmable in-house developed BBB control computer Fedrizzi2015. The coordinate system used in this study is shown in the bottom left of Fig. <ref>, where $x$ represents the axial direction and $y$ the radial direction, with $u$ and $v$ being their respective velocity components. The particle images occupy an area of approximately 3.5${D_w}$ $\times$ 3${D_w}$ ($x$ $\times$ $y$) outwards from the mouthpiece exit. A total of 8000 PIV images were acquired for each experimental model. §.§.§ PIV processing algorithm Analysis of the single-exposed double-frame images is performed using multi-grid/multi-pass cross-correlation digital particle image velocimetry (MCCDPIV). The algorithm was developed by Soria Soria1994 and is described in Soria Soria1996 and Soria et al. Soria1999. It employs an iterative and adaptive cross-correlation algorithm to increase the dynamic range of the measurements. This is done by adapting the sample window size to the local flow conditions and offsetting the discrete sampling window in the second frame by an amount approximately equal to the estimated particle displacement in the sampling window. The final sample window size is (48 px $\times$ 32 px) with a grid spacing of (24 px $\times$ 16 px). The algorithm also employs a correlation based correction by comparing correlation data from adjacent sampling windows to improve sub-pixel accuracy and eliminate spurious vectors Hart2000. A two-dimensional Gaussian function is least-square fitted around the correlation peak region to locate the maximum spatial cross-correlation function value to sub-pixel accuracy. A median value test and a dynamic mean value operator test are then performed to validate the resulting displacement vectors Westerweel1994. The PIV velocity vector components are then computed by dividing the measured pixel displacements in each sampling window by the time between the image double-frames (time delay between two laser pulses) and the optical magnification. The spacing between the vectors is 0.8424 mm $\times$ 0.5616 mm ($x$ $\times$ $y$). The performance and accuracy of this algorithm in the analysis of single-exposed double-frame PIV images is reported in Soria Soria1998, wherein the uncertainty of a single-sample measurement is $\pm$ 0.06 px at 95% confidence interval. §.§.§ Measurement uncertainties An uncertainty analysis based on the methodology reported by Moffat Moffat1988 was performed. The uncertainty in measurement of the mouthpiece exit inner-diameter is $\pm$ 0.029 mm. The variable area flow meter has an accuracy of $\pm$ 6.66% at a flow rate of ${Q_w}$ = 12. The pulse generator has a timing uncertainty of $\pm$ 5 at 2 Hz. The uncertainty in measuring the optical magnification is $\pm$ 0.61/px, based on an uncertainty of $\pm$ 1 px in the image space of the calibration reference points and an uncertainty of $\pm$ 0.1 mm in the physical distance between these reference points. This gives an uncertainty of ${\epsilon_u}/{U_w}$ = 1.78% in the PIV velocity measurement. The uncertainty in ensemble average velocity components is ${\epsilon_U}/{U_w}$ = ${\epsilon_V}/{U_w}$ = 0.02%. § RESULTS §.§ In-vitro Measurements The particle size distribution of the micronised BDP, measured via laser diffraction (Fig. <ref>(a)), showed that 90% of the particles were below 4.14 ± 0.38, indicating that these particles are suitable for deep lung deposition. Mean diameter of micronized BDP was 1.24 ± 0.06, smaller than those previously reported by Yeung et al. Yeung2018 who used the same formulation to assess aerosol performance of lactose-loaded BDP. Scanning electron micrographs confirmed the particle size of BDP (refer Fig. <ref>(b)), and showed that the drug was successfully loaded onto the surface of the lactose carrier by the blending procedure (Fig. <ref>(c)). The use of fine and cohesive particles, like the micronised BDP in this study, may lead to issues regarding drug content, as the dose of API loaded to the carrier is low, and the mixing process can vary. To ensure that a homogeneous blend was produced a content uniformity assay was carried out and showed a mean content of 94.55 ± 2.23%, with acceptance value of 9.31, complying with the British Pharmacopoeia requirements. (a) Particle Size distribution of micronized BDP, scanning electron micrographs of (b) micronised BDP and (c) and pre-blend carrier system loaded with BDP (1% w/w) Aerosol performance of all devices was assessed at ${Q_a}$ = 60 using a standard 1% BDP-loaded lactose formulation (Fig. <ref>). The pressure drop and intrinsic resistance of all device models tested here are shown in Table 2. The addition of air-inlets to the device models significantly decreased the mass of BDP remaining in the device from 25.17 $\pm$ 1.40 to 16.26 $\pm$ 1.95 (P0.0001). As the total dose observed for the model compared with the model was significantly lower (P0.05), the aerosol deposition was compared based on the percentage of the total dose delivered (%TD) using a two-tailed t-test. Throat deposition was similar between the devices (P0.05), with a significant increase in mass of BDP deposited on S4 and S5 (particles of aerodynamic size between 2.82 - 1.66 and 1.66 - 0.94, respectively) (P0.05). Similar mass depositions were observed in the remaining stages of the NGI (P0.05). When comparing the aerosol performance parameters (Table 1), although a significant increase in the ED (%TD) was observed for the model compared with the model, the change in FPF (%ED) from 52.83 $\pm$ 3.45 to 61.41 $\pm$ 7.58 was not significant (P0.05). No significant differences were also observed for FPD and MMAD values (P0.05), despite the lower intrinsic resistance observed for the device model. Aerosol Parameters of all 4 DPI models using a 1% (w/w) BPD – lactose formulation 2c 2c 2c 2c Mean SD Mean SD Mean SD Mean SD Total Dose () 104.96 2.74 90.91 b 4.10 100.90 8.47 97.53 8.94 Emitted Dose (%TD) 76.01 1.47 82.03 a 2.97 79.27 1.74 73.78 3.73 Fine Particle Dose () 42.13 2.62 45.76 5.89 42.69 8.73 40.42 4.53 Fine Particle Fraction (%ED) 52.83 3.45 61.41 7.58 53.05 7.17 56.25 4.54 MMAD () 1.59 0.09 1.72 0.04 1.86 a 0.07 1.78 a 0.07 GSD 1.91 0.09 1.84 0.15 1.99 0.04 1.94 0.09 Means were compared with model using a two-tailed t-test a: P0.05; b: P0.01 The effect of the flow straightener (grid) in the aerosol performance was investigated using the model. Compared with the $\twin$ model without the grid, the addition of the grid increased the intrinsic device resistance to 0.0339 and 0.0374 for the $\teng$ and $\texg$, respectively, without affecting the percentage of BDP that remained in the device. A comparison between the location of the grid in the entry or exit positions showed that a greater mass of BDP remained in the device for the model (P0.01), the device with higher intrinsic resistance. The addition of the grid at the exit position also led to a decrease in BDP deposited in the throat + pre-separator (P0.05). No significant differences were observed amongst the actual , and models for the remaining stages of the NGI (P0.05). Similarly, no changes in the aerodynamic performance parameters of ED (%TD), FPD, FPF (%ED) were observed, although a significantly greater MMAD () was observed for the models with a grid (Table 1.). Aerosol performance of (a,c) and models, (b,d) , , and models, expressed as (a,b) mass of BDP deposited () and (c,d) percentage of TD (% TD) Pressure drop and intrinsic device resistance measured at ${Q_a}$ = 60 1cPressure drop (kPa) 1cIntrinsic resistance (^0.5.^-1) $\twin$ 1.79 0.0223 $\sxin$ 1.37 0.0195 $\teng$ 4.14 0.0339 $\texg$ 5.04 0.0374 §.§ PIV Experiments The mean velocities $U$, $V$ and root-mean-square (RMS) velocity fluctuations $u_{rms}$, $v_{rms}$ in the region outside of the DPI model mouthpiece are presented here. The mean velocities are ensemble average velocities calculated over the measured 8000 PIV velocity fields, whereas the RMS fluctuations are standard deviations of those velocity components. The mean velocities and RMS fluctuations are non-dimensionalized using ${U_w}$, while the spatial coordinates are non-dimensionalized with ${D_w}$. The results are shown up to a location of $x$/${D_w}$ = 3 from the mouthpiece exit, wherein the location $x$/${D_w}$ = 0 is approximately 4 mm ($x$/${D_w}$ = 0.13) above of the mouthpiece exit plane. The mean axial velocity $U/{U_w}$ distributions over the radial coordinate $y/{D_w}$ across the jet cross-section for all four models are shown in Fig. <ref>. The profiles for and models in Fig. <ref>(a) and <ref>(b) are similar to each other with negative velocities in the jet central region (core) and maximum positive velocities in the jet edge regions (shear layers). Negative mean axial velocities signify a reverse flow region which begins at the mouthpiece exit at and persists up to , with the most negative values observed at . These characteristics of mean axial velocities are exemplary of a high swirling jet flow that produces axial recirculation in the form a central toroidal recirculation zone due to strong radial and axial pressure gradients near the nozzle exit gupta1984swirl, Chigier1967, Giannadakis2008. The profiles for the model in Fig. <ref>(c) are representative of an axisymmetric jet without any swirl (Flow B in Liang and Maxworthy Liang2005), which is a result of the flow-straightening effect produced by the grid placed after the tangential inlets. In contrast, the placement of the grid at the mouthpiece exit in the model does not entirely eliminate the flow swirl close to the mouthpiece exit, as shown by the negative mean axial velocities at in Fig. <ref>(d), which occur due to the grid square holes having a side and an axial length of 3 mm that allows the flow to exit while retaining most of its swirl level. However, the profiles from to show that the flow straightening effect begins to manifest, only further outside of the mouthpiece exit. Mean axial velocities for: (a) ; (b) ; (c) ; (d) ; at $x$/${D_w}$ = 0, $x$/${D_w}$ = 1, $x$/${D_w}$ = 2, and $x$/${D_w}$ = 3 For the and models, as we move downstream from the mouthpiece exit, the maximum values of mean axial velocities decrease and the radial location at which they occur shifts away from the longitudinal jet axis $y/{D_w}$ = 0. On the other hand, for the model, maximum mean axial velocities occur close to longitudinal jet axis and do not have large variations in their values, whereas for the model, the radial locations of the maximum mean axial velocities move closer to the jet axis with increasing $y/{D_w}$. These observations show that the jet emerging from a DPI mouthpiece spreads and decays faster if there is a high level of swirl present. Mean radial velocities for: (a) ; (b) ; (c) ; (d) ; at $x$/${D_w}$ = 0, $x$/${D_w}$ = 1, $x$/${D_w}$ = 2, and $x$/${D_w}$ = 3 The mean radial velocity $V/{U_w}$ distributions along the radial direction are shown in Fig. <ref>. The distributions for the and models in Fig. <ref>(a) and Fig. <ref>(b), almost mirror each other, with maximum positive and negative values at the mouthpiece exit attained at $y/{D_w}$ = $\pm$ 0.5, respectively. The negative mean radial velocities occurring at locations $y/D_w > 0.6$, and the positive mean radial velocities occurring at locations $y/D_w < -0.6$, signify entertainment of the ambient fluid into the jet core. This entrainment appears to be large for the and models, as the maximum negative and positive values of $V/{U_w}$ at $y/D_w = \pm 0.6$, respectively, reach at least 30% of $U_w$ due to high swirl levels. The mean radial velocities for the model in Fig. <ref>(c) are very small when compared with those for the and models, which reinforces the flow-straightening effect of the grid. However, for the model at in Fig. <ref>(d), the mean radial velocities are larger than those for the model but lower than those for the model, with maximum negative and positive values occurring at $y/{D_w}$ = $\pm$ 0.5, respectively. The asymmetry in the mean velocity profiles across the jet centerline $y/{D_w}$ = 0, observed in Figs. <ref> and <ref>, arise from non-uniform inner cross-sections of the mouthpiece resulting from the 3D printing process and also as a consequence of swirling jet flow, which has been reported in previous studies Toh2010,Vanierschot2014. RMS axial velocity fluctuations for: (a) ; (b) ; (c) ; (d) ; at $x$/${D_w}$ = 0, $x$/${D_w}$ = 1, $x$/${D_w}$ = 2, and $x$/${D_w}$ = 3 RMS axial fluctuations $u_{rms}$/${U_w}$ at in Fig. <ref> attain maximum values of the order of ${U_w}$ for all models except the model. These occur in the jet shear layers where there are large velocity gradients due to sudden expansion and mixing of the jet with quiescent surrounding fluid. The fluctuations decrease while spreading out radially as the jet decays outside of the mouthpiece exit due to further mixing with the ambient fluid. A similar observation for the RMS radial fluctuations $v_{rms}$/${U_w}$ can be found in Fig. <ref>. Maximum radial fluctuations for the and models occur at the jet central axis which is the boundary of the recirculation zone where the mean radial velocities are zero gupta1984swirl. The RMS axial and radial fluctuations for the model are relatively very small when compared with the other models at all axial distances reported, which is due to the elimination of flow swirl that in turn reduces the amount of jet mixing with the ambient fluid. RMS radial velocity fluctuations for: (a) ; (b) ; (c) ; (d) ; at $x$/${D_w}$ = 0, $x$/${D_w}$ = 1, $x$/${D_w}$ = 2, and $x$/${D_w}$ = 3 Figure <ref> shows streamlines in the jet flow emerging from all DPI models, overlaid on the velocity magnitude contours. Large recirculation zones in the central region can be seen in Fig. <ref>(a) and Fig. <ref>(b), for the jet flows from the and models, which occur due to high levels of swirl. Swirl induces a radial pressure gradient to balance the centrifugal forces in the flow thereby creating a toroidal vortex (vortex ring) with a low pressure region in the jet central region Percin2017. This sub-ambient pressure region leads to reverse axial flow directed towards the jet nozzle. For a strong swirl, such as that present in flows from these two DPI models, this reverse flow occurs close to the nozzle exit, in this case the DPI mouthpiece exit. The two regions of high velocity magnitude on either sides of the central jet axis are a result of high swirling flow emerging from inside the device, which was previously observed in the Nexthaler® DPI Pasquali2015. The flow straightening effect of the grid in the model is clearly visible in Fig. <ref>(c), which shows the jet radial spread to be much lower than that for the and models. For the model in Fig. <ref>(d), there is a small recirculation and stagnation zone in the jet core that extends up to about $x$/${D_w}$ = 0.3. The size of this recirculation zone is smaller than that observed for the model, because of the impact of the upstream swirling flow in the mouthpiece with the grid which reduces the flow swirl-level. Velocity magnitude contours and streamlines for: (a) ; (b) ; (c) ; (d) § DISCUSSION DPI devices currently available in the market have a wide range of efficiencies, from as low as 19.3% and 22.9% of FPF for salmeterol xinafoate and fluticasone propionate, respectively, in Seretide® Diskus®, to as high as 62.3% and 62.6% for budesonide and formoterol fumarate (FF), respectively, in the Symbicort® Turbuhaler®, and 66.3% and 64.4% for BDP and FF, respectively, in the Foster® Nexthaler® Buttini2016. This large variation in aerosol performance confirms that despite various studies investigating DPI performance, the interaction and inter-dependence of the powder formulation used, the device design, and the energy supplied by the inspiratory flow affects DPI efficiency in a complex way which is not fully understood. In order to generate inhaled particle aerosol from a DPI, both fluidization and de-agglomeration processes must occur. These processes are strongly affected by the characteristics of flow generated from the device and the powder formulation used in the device. The adhesive and cohesive forces between the particles must be overcome by aerodynamic and inertial forces in the inhaled air flow in order to achieve particle entrainment (pick-up) and detachment Frijlink2004. Micronised particles (APIs) have high cohesive forces that form strong agglomerates and require generation of higher forces in the inspiratory flow of a patient for their de-agglomeration and dispersion into the air stream. Thus, carriers, such as lactose, are blended with APIs do not only prevent agglomeration of cohesive APIs but also to bulk the formulation thereby facilitating powder flowability. In addition, particle engineers have also used ternary force control agents (FCA) like MgSt and lactose fines, which modify the surface of the carrier, altering interparticle interaction, decreasing API's adhesion to the carrier and thereby facilitating its detachment during aerosolization Jetzer2018. Therefore, the present study has used a model carrier-based formulation containing MgSt as a FCA agent to enhance API detachment from the lactose carrier. The in-vitro results show that the model has lower drug mass retained in the device compared with the model. This is a consequence of the more uniform swirl in the flow cross-section, and hence, more uniform mixing being produced from 6 tangential inlets than that produced from only 2 tangential inlets in the model. The large BDP deposition of around 30% TD, observed in the throat + pre-separator stages for the and models, occurs due to the axially recirculating and radially spreading jet flow emerging from the mouthpieces of these DPIs, as shown in Fig. <ref>(a) and (b), respectively. Since micronised API particles detached from the carrier have very low Stokes number because of their mean size of 1.24, their trajectories closely follow the jet flow, causing them to spread outwards and impact with the mouth cavity surface and on the throat. The increased BDP deposition on stages S4 and S5 for the model indicates a greater detachment/dispersion of API from the carrier than for the model. As similar jet-flows are observed emerging from these models, the aforesaid differences can be due to different flow characteristics within the device, particularly the swirl levels and distribution. Nevertheless, high FPF (larger than 50% ED) observed for both models demonstrates that high swirling flow produced inside the device is able to generate sufficient forces that detach the API from the carrier and entrain particles in the airflow. The use of a grid in the model reduces mass retention in the device when compared with the model. In this case, the grid acts as a flow straightener as well as an `additional structure' for particle impaction/detachment. The flow straightening effect in eliminating flow swirl is evident in Figs. <ref>(c), <ref>(c), and <ref>(c), and has also been shown to exist in both carrier-based Zhou2013 and carrier-free formulation Wong2011 DPIs. API detachment from the carrier, as in the present carrier-free formulation, mainly occurs when most of the particles are trapped upstream of the grid where they are subjected to particle-particle and particle-obstacle (grid) collisions. This was also observed in the powder dispersion study by Kou et al. Kou2016. Placement of the grid at the mouthpiece exit in the model shows an increase in the % of BDP (%TD) in the device, which indicates lower particle de-agglomeration due to fewer particle collisions in the absence of the grid closer to the tangential inlets. The reduction in throat deposition for the model is due to the flow-straightening effect of the grid, illustrated in Fig. <ref>(d), as opposed to the larger throat deposition for the model, because of the emerging high swirling and radially spreading jet-flow, Fig. <ref>(a). However, this flow straightening effect increases particle deposition on stage S1 (cut off 8.06), instead on the lower stages of the NGI, which indicates that the API are still adhered to the lactose carrier or agglomerated after exiting the device, as the particle size distribution shows that 90% of the API was smaller than 4.14. Although a flow straightening effect is observed by addition of the grid, there is no significant change in FPD or FPF for the and models. This similarity in aerosol performance can be attributed to powder de-agglomeration due to larger number of inter particle and particle-grid collisions when the grid is placed after the tangential inlets and a similar de-agglomeration potential achieved when high velocity particles collide with the grid placed at the mouthpiece exit. Such high velocity particle-grid collisions occur when a high swirling flow in the DPI mouthpiece impacts an obstacle (grid) placed at the exit, resulting in a jet-flow that has a higher velocity magnitude from that when the grid is placed at the entry, as shown in Figs. <ref>(c) and <ref>(d). To summarize, the preceding discussion synthesizes the results between the performance and the fluid-dynamic characteristics of flows emerging from the four DPI models examined in this study. The increase in the number of tangential inlets leads to a lower drug retention within the device without changing the flow characteristics emerging from the DPIs. The addition of a grid either close to the tangential inlets or at the mouthpiece exit, leads to a flow straightening effect that removes swirl in the DPI flow. Although similar aerosol performances (FPD and FDF) are observed for all device models used in this study, the aerosol efficiency of these devices are high, with FPF larger than 50% ED and FPD larger than 40. The present work highlights the conjoint application of PIV and techniques to extensively characterise flows emerging from DPIs and quantify their aerosol performance, respectively, in order to improve DPI designs for effective pulmonary delivery. § CONCLUSION Particle image velocimetry was used in this study to experimentally characterise the jet-flows emerging from four different DPI models, while studies on the same DPIs have been performed to corroborate their aerosol performance with their fluid-dynamics characteristics. The DPI models for which these performances and characteristics have been ascertained include modifications to their tangential inlets and the addition of a grid. DPI models without the grid have a highly swirling and recirculating jet-flow that emerges from the mouthpiece, whereas those with the grid have a straightened-flow without the undesirable radial spreading, which yields a reduction in the particle deposition in the mouth cavity and the throat. Similar aerosol performances were observed for all four device models, with FPF larger than 50%, indicating a desirable lung deposition. § ACKNOWLEDGMENTS The research was supported by the Australian Research Council. The research was also benefited from computational resources provided by the Pawsey Supercomputing Centre and through the NCMAS, supported by the Australian Government. The computational facilities supporting this project included the NCI Facility, the partner share of the NCI facility provided by Monash University through a ARC LIEF grant and the Multi-modal Australian ScienceS Imaging and Visualisation Environment (MASSIVE). Missing 'biblatex' package The bibliography requires the 'biblatex' package. Francis Ltd abstractIntroduction: Aerosolized medications are commonly prescribed for the treatment of patients with pulmonary diseases, and there has been an increased interest in the development of aerosol delivery devices over the years. Technical innovations have advanced device design, novel features such as breath actuation, dose tracking, portability, and feedback mechanism during treatment that improved the performance of aerosol devices, and effectiveness of inhalation therapy. Areas covered: The purpose of this paper is to review recent advances in aerosol devices for delivery of inhaled medications. Expert opinion: Drug formulations and device designs are rapidly evolving to make more consistent dosing across a broad range of inspiratory efforts, to maximize dose and target specific areas of the diseased lung. journaltitleExpert Opinion on Drug Delivery titleRecent advances in aerosol devices for the delivery of inhaled medications Aerosols,dry powder inhalers,inhaled drugs,metered-dose inhalers,nebulizers Respiratory Care journaltitleRespiratory Care titleDry powder inhalers: an overview Mary Ann Liebert Inc. abstractBackground: The characteristics of each inhalation maneuver when patients use dry powder inhalers (DPIs) are important, because they control the quality of the emitted dose. Methods: We have measured the inhalation profiles of asthmatic children [CHILD; n=16, mean forced expiratory volume in 1 sec (FEV<inf>1</inf>) 79% predicted], asthmatic adults (ADULT; n=53, mean predicted FEV<inf>1</inf> 72%), and chronic obstructive pulmonary disease (COPD; n=29, mean predicted FEV<inf>1</inf> 42%) patients when they inhaled through an Aerolizer, Diskus, Turbuhaler, and Easyhaler using their "real-life" DPI inhalation technique. These are low-, medium-, medium/high-, and high-resistance DPIs, respectively. The inhalation flow against time was recorded to provide the peak inhalation flow (PIF; in L/min), the maximum pressure change ($\Delta$P; in kPa), acceleration rates (ACCEL; in kPa/sec), time to maximum inhalation, the length of each inhalation (in sec), and the inhalation volume (IV; in liters) of each inhalation maneuver. Results: PIF, $\Delta$P, and ACCEL values were consistent with the order of the inhaler's resistance. For each device, the inhalation characteristics were in the order ADULT>COPD>CHILD for PIF, $\Delta$P, and ACCEL (p<0.001). The results showed a large variability in inhalation characteristics and demonstrate the advantages of $\Delta$P and ACCEL rather than PIFs. Overall inhaled volumes were low, and only one patient achieved an IV >4 L and $\Delta$P >4 kPa. Conclusion: The large variability of these inhalation characteristics and their range highlights that if inhalation profiles were used with compendial in vitro dose emission measurements, then the results would provide useful information about the dose patients inhale during routine use. The inhalation characteristics highlight that adults with asthma have greater inspiratory capacity than patients with COPD, whereas children with asthma have the lowest. The significance of the inhaled volume to empty doses from each device requires investigation. journaltitleJournal of Aerosol Medicine and Pulmonary Drug Delivery titleThe Inhalation characteristics of patients when they use different dry powder inhalers COPD,asthma,dry powder inhalers,inhalation profiles,inhaled therapy abstractThis paper applies particle image velocimetry (PIV) to a simplified, canonical, pitch-hold-return problem of a pitching plate in order to gain some understanding of how three dimensionality develops in such flows. Data from a progression of PIV studies, from stereoscopic PIV yielding three-component, two-dimensional (3C-2D) data to tomographic PIV yielding three-component, three-dimensional (3C-3D) data are presented thus providing progressively more detailed information. A comparison of results is made between the two techniques. The PIV study is performed in a water tunnel facility with cross-sectional area 500 × 500 mm, and involves a full-span (nominally two-dimensional) plate, suspended between a wall end boundary condition and a free surface, pitching at a dimensionless pitch rate of K c = 0.93 in flow at Re = 7,500. Results demonstrate the existence of spanwise flows in both the leading edge and trailing edge vortices, but with strong directionality in the leading edge vortex towards the wall end boundary condition. Observations of instantaneous flow patterns suggest also the existence of three-dimensional coherent vortex filament structures in the outer regions of the leading edge vortex. © 2011 Springer-Verlag. journaltitleExperiments in Fluids titleStereoscopic and tomographic PIV of a pitching plate ermodynamics,Fluid,Heat and Mass Transfer,and Aerodynamics Mary Ann Liebert Inc. abstractBackground: European and United States Pharmacopoeia compendial procedures for assessing the in vitro emitted dose and aerodynamic size distribution of a dry powder inhaler require that 4.0 L of air at a pressure drop of 4 kPa be drawn through the inhaler. However, the product performance should be investigated using conditions more representative of what is achievable by the patient population. This work compares the delivered dose and the drug deposition profile at different flow rates (30, 40, 60, and 90 L/min) of Foster NEXThaler® (beclomethasone dipropionate/formoterol fumarate), Seretide® Diskus® (fluticasone propionate/salmeterol xinafoate), and Symbicort® Turbohaler® (budesonide/formoterol fumarate). Methods: The delivered dose uniformity was tested using a dose unit sampling apparatus (DUSA) at inhalation volumes either 2.0 or 4.0 L and flow rates 30, 40, 60, or 90 L/min. The aerodynamic assessment was carried out using a Next Generation Impactor by discharging each inhaler at 30, 40, 60, or 90 L/min for a time sufficient to obtain an air volume of 4 L. Results: Foster® NEXThaler® and Seretide® Diskus® showed a consistent dose delivery for both the drugs included in the formulation, independently of the applied flow rate. Contrary, Symbicort® Turbohaler® showed a high decrease of the emitted dose for both budesonide and formoterol fumarate when the device was operated at airflow rate lower that 60 L/min. The aerosolizing performance of NEXThaler® and Diskus® was unaffected by the flow rate applied. Turbohaler® proved to be the inhaler most sensitive to changes in flow rate in terms of fine particle fraction (FPF) for both components. Among the combinations tested, Foster NEXThaler® was the only one capable to deliver around 50% of extra-fine particles relative to delivered dose. Conclusions: NEXThaler® and Diskus® were substantially unaffected by flow rate through the inhaler in terms of both delivered dose and fine particle mass. journaltitleJournal of Aerosol Medicine and Pulmonary Drug Delivery titleEffect of flow rate on in vitro aerodynamic performance of NEXThaler® in comparison with Diskus® and Turbohaler® dry powder inhalers Diskus®,Extra-fine particlemass,NEXThaler®,NGI flow rate,Turbohaler®,aerodynamic assessment US Pharmacopeial Convention 12601 Twinbrook Pkwy, Rockville, MD 20852 booktitlePharmacopeial Forum titleRecommendations of the USP advisory panel on aerosols on the USP general chapters on aerosols (601) and uniformity of dosage units (905) American Society of Mechanical Engineers Digital Collection abstractExperiments have been carried out in a series of axisymmetric free turbulent jets with degrees of swirl covering the weak, moderate, and strong ranges, including the case of the onset of reversed flow in the central region of the jet. Measurements are reported of mean axial and swirl velocities, static pressure, and jet width at axial stations up to 15 orifice diameters. Mean velocity and pressure profiles are shown to be effectively similar from an axial distance of four diameters for weak and moderate siuirl. For the case of strong swirl, a vortex is generated in the region close to the orifice resulting in a displacement of the axial velocity maximum from the jet axis. After a distance of 10 diameters, the influence of the vortex motion becomes small, and similarity of the profiles is obtained farther doivnstream. Experimentally determined profiles are described in terms of Gaussian error curves and third-order polynomials. Jet width and mass flow rates of entrained fluid are shown to increase according to the degree of swirl so that, for strong svvirl, jet width and rate of entrainment are almost twice ihose for the nonswirling jet. Results of the decay of velocity and pressure along the axis are compared with values predicted by an approximate theory based on the integration of the Reynolds' equations of motion. Good agreement is found between results and predictions, and a set of semi-empirical equations is provided from which a complete description of the mean velocity and pressure fields can be obtained for swirling jets. © 1967 by ASME. journaltitleJournal of Applied Mechanics, Transactions ASME titleExperimental investigation of swirling vortex motion in jets Displacement,Flow (Dynamics),Jets,Pressure,Swirling flow,Turbulence,Vortex motion family=De Boer, abstractAir classifier technology (ACT) is introduced as part of formulation integrated dry powder inhaler development (FIDPI) to optimise the de-agglomeration of inhalation powders. Carrier retention and de-agglomeration results obtained with a basic classifier concept are discussed. The theoretical cut-off diameter for lactose of the classifier used, is between 35 and 15$\mu$m for flow rates ranging from 20 to 70l/min. Carrier retention of narrow size fractions is higher than 80% for flow rates between 30 and 60l/min, inhalation times up to 6s and classifier payloads between 0 and 30mg. The de-agglomeration efficiency for adhesive mixtures, derived from carrier residue (CR) measurement, increases both with increasing flow rate and inhalation time. At 30l/min, 60% fine particle detachment can be obtained within 3s circulation time, whereas at 60l/min only 0.5s is necessary to release more than 70%. More detailed information of the change of detachment rate within the first 0.5s of inhalation is obtained from laser diffraction analysis (LDA) of the aerosol cloud. The experimental results can be explained with a novel force distribution concept (FDC) which is introduced to better understand the complex effects of mixing and inhalation parameters on the size distributions of adhesion and removal forces and their relevance to the de-agglomeration in the classifier. © 2003 Elsevier B.V. All rights reserved. journaltitleInternational Journal of Pharmaceutics titleAir classifier technology (ACT) in dry powder inhalation: Part 1. Introduction of a novel force distribution concept (FDC) explaining the performance of a basic air classifier on adhesive mixtures Adhesive mixtures,Air classifier technology,Carrier retention,Dry powder inhalation,Force distribution concept,Powder dispersion family=De Boer, Francis Ltd abstractIntroduction: Early dry powder inhalers (DPIs) were designed for low drug doses in asthma and COPD therapy. Nearly all concepts contained carrier-based formulations and lacked efficient dispersion principles. Therefore, particle engineering and powder processing are increasingly applied to achieve acceptable lung deposition with these poorly designed inhalers. Areas covered: The consequences of the choices made for early DPI development with respect of efficacy, production costs and safety and the tremendous amount of energy put into understanding and controlling the dispersion performance of adhesive mixtures are discussed. Also newly developed particle manufacturing and powder formulation processes are presented as well as the challenges, objectives, and new tools available for future DPI design. Expert opinion: Improved inhaler design is desired to make DPIs for future applications cost-effective and safe. With an increasing interest in high dose drug delivery, vaccination and systemic delivery via the lungs, innovative formulation technologies alone may not be sufficient. Safety is served by increasing patient adherence to the therapy, minimizing the use of unnecessary excipients and designing simple and self-intuitive inhalers, which give good feedback to the patient about the inhalation maneuver. For some applications, like vaccination and delivery of hygroscopic formulations, disposable inhalers may be preferred. journaltitleExpert Opinion on Drug Delivery titleDry powder inhalation: past, present and future Adhesive mixtures,drug deposition,drug formulation,dry powder inhaler design,inhalation,particle engineering Elsevier Ltd abstractThe deposition of monodisperse aerosols entering an idealized oral cavity geometry through a variety of inlets was experimentally measured. Aerosol particles with diameters of 2.5, 3.8 and 5.0 $\mu$m were investigated at flow rates ranging from 15 to 90 L/min. The tested inlets ranged in diameter from 3 to 17 mm and included contraction nozzles, straight tubes, a turbulence generator and six commercially available dry powder inhalers (DPIs). A model for predicting the oral cavity deposition was derived from the data based on the particle Stokes number near the primary impaction location modified to incorporate the turbulent kinetic energy at the inlet. The model predicted similar (but slightly underestimated) deposition for monodisperse aerosols entering through DPIs, with increasing deposition for decreasing inlet diameter. The model was then extended to predict extrathoracic deposition for polydisperse aerosol formulations in vivo. Improved agreement was found between the in vitro predictions and the in vivo measurements compared to previous attempts. © 2003 Elsevier Ltd. All rights reserved. journaltitleJournal of Aerosol Science titlePredicting extrathoracic deposition from dry powder inhalers Dry powder inhaler,Impaction,In vitro,Mouth,Oral cavity,Stokes number Elsevier B.V. abstractThe influence of grid generated mixing on the fluidization of pharmaceutical carrier powders is studied in a channel-flow experiment using direct high-speed imaging and particle image velocimetry (PIV). Four different lactose powders with mass median diameters that range between 61 µm and 121 µm are used. The degree of powder mixing in the flow as a function of grid position relative to the powder bed and grid area blockage ratios (ranging from 25% to 40%) is studied for a range of flow-rates. The study presents comprehensive mappings of how pharmaceutical powders are fluidised under the influence of mixing, by examining powder bed morphology, powder emptying rate, and the local flow-field surrounding the pocket. The use of a grid results in higher evacuation percentages (void fraction) and a faster evacuation rate but is associated with randomized evacuation behaviour as observed from the powder bed morphology. Use of a grid can enable evacuation of powder at lower overall flow-rates, which may have important implications on respiratory drug delivery. PIV results show the trend of mean velocities with the mass median powder diameter and demonstrates how a grid with lower blockage ratio can increase the degree of mixing of the evacuating powder and make the evacuation process more rapid. This study contributes towards a better understanding of fluidization processes as relevant to dry powder inhaler devices and sheds light on how simple design alterations, such as adding an upstream grid, can be incorporated to optimise device effectiveness. journaltitleInternational Journal of Pharmaceutics titleEffect of an upstream grid on the fluidization of pharmaceutical carrier powders Dry powder inhalers,Fluidization,Pharmaceutical powders Institute of Physics Publishing abstractA BeagleBone Black (BBB) single-board open-source computer was implemented as a low-cost fully programmable pulse generator. The pulse generator makes use of the BBB Programmable Real-Time Unit (PRU) subsystem to achieve a deterministic temporal resolution of 5 ns, an RMS jitter of 290 ps and a timebase stability on the order of 10 ppm. A Python-based software framework has also been developed to simplify the usage of the pulse generator. journaltitleMeasurement Science and Technology titleApplication of a single-board computer as a low-cost pulse generator BeagleBone Black,experiment timing control,pulse generator family=De Boer, Expert Opin Drug Deliv abstractThe pulmonary route is an interesting route for drug administration, both for effective local therapy (asthma, chronic obstructive pulmonary disease or cystic fibrosis) and for the systemic administration of drugs (e.g., peptides and proteins). Well-designed dry powder inhalers are highly efficient systems for pulmonary drug delivery. However, they are also complicated systems, the the performance of which relies on many aspects, including the design of the inhaler (e.g., resistance to air flow and the used de-agglomeration principle to generate the inhalation aerosol), the powder formulation and the air flow generated by the patient. The technical background of these aspects, and how they may be tuned in order to obtain desired performance profiles, is reviewed. In light of the technical background, new developments and possibilities for further improvements are discussed. © 2004 Ashley Publications Ltd. journaltitleExpert Opinion on Drug Delivery titleDry powder inhalers for pulmonary drug delivery Adhesion,Aerosol,Asthma,COPD,Cohesion,Cystic fibrosis,De-agglomeration,Devices,Dry powder inhaler,Flow-rate dependency,Lung deposition,Particle engineering,Peptides,Powder formulation,Proteins,Pulmonary drug administration,Size distribution,Systemic drug administration,Vaccines abstractThe recirculating flow field generated by a swirling jet and a coaxial annular stream entering a pipe is investigated with the use of 2D-DPIV. Parametric change of inlet flow rates (constant tangential injection with change of annular flow and vice versa) is being considered in order to study the mean and turbulent flow field. A recirculation bubble stabilized close to the swirler exit is the dominating feature of the interaction between the inner swirling jet and the annular stream. Results are discussed in terms of bubble topology and dynamics on the basis of a modified Rossby number that appears to describe the trends of the complex flow field. © 2008 Elsevier Inc. All rights reserved. journaltitleExperimental Thermal and Fluid Science titleA swirling jet under the influence of a coaxial flow Coaxial flow,DPIV,Swirling jets,Vortex breakdown Elsevier Inc. abstractThis paper presents time-averaged and temporally evolving mean flow and turbulence statistics for a turbulent zero-net-mass-flux (ZNMF) jet at high Reynolds and Strouhal numbers in the far-field in a plane perpendicular to the jet axis. The measurements have been obtained using time-resolved stereo particle image velocimetry (TR-SPIV). The jet is generated by the oscillation of a piston, which discharges filtered water to the quiescent fluid in a tank through a round orifice. A multigrid cross correlation digital particle image velocimetry algorithm (MCCDPIV) has been used to compute the images from each camera that subsequently have been combined to obtain the three components of the velocity. Velocity and turbulence statistics are in good agreement with results obtained in previous work. A comparative study has been performed to determine the contribution to the spreading rate due to the displacement of the center of the jet as rigid body. Different criteria to define the instantaneous jet center have been considered. For the different cases carried out for this study it has been observed that subtracting the movement as a rigid body the width of the jet becomes similar to that of a continuous jet. Velocity fluctuation statistics are also influenced by this restriction. The image series have also been analyzed to determine the temporal evolution of the velocity field and its fluctuations. The fast Fourier transforms (FFT) have been used to calculate the power spectra of these variables. © 2014 Elsevier Inc. journaltitleExperimental Thermal and Fluid Science titleTime-resolved stereo PIV measurements in the far-field of a turbulent zero-net-mass-flux jet SPIV,Spreading rate,ZNMF Tunbridge Wells, Kent : Abacus Press seriesEnergy and engineering science series titleSwirl Flows journaltitleAerosol Science & Technology titleFlow field measurement inside the mouthpiece of the Spiros inhaler using particle image velocimetry journaltitleExperiments in fluids titlePIV error correction abstractThe dose emission characteristics of eight marketed dry powder inhalers (DPIs: Intal Spinhaler®, Ventolin and Becotide Diskhalers®, Ventolin and Becotide Rotahalers®, Bricanyl and Pulmicort Turbohalers®, Berotec Inhalator® have been investigated using the proposed USP dosage unit sampling apparatus for DPIs. Intra- and inter-device variation in emitted doses was determined at air flow rates of 60 and 100 1/min using a 4 1 air throughput in each case except Inhalator®, which was tested at 30 l/min only. The sampling apparatus was found to be suitable for quantifying single emitted doses from all of these devices which comprise examples of low, medium and high airflow resistance DPIs (Table 1 footnote). Dose emissions from the DPIs are presented as percentages of the manufacturers' label claims. Under all test flow conditions variability was high, when compared to the uniformity of content standards usually applied to pharmaceutical products; in some cases relative standard deviations (RSD) were greater than 15%, both within and between devices. However, under the proposed USP test flow rate conditions, the total RSD (n = 25) was < 15% around the average emitted dose in all cases except Pulmicort Turbohaler®; such variance (RSD< 15%) is proposed to be acceptable for DPIs delivering current medications. Only the Intal Spinhaler® emitted an average dose similar to its label claim. Testing at 100 1/min vs 60 1/min significantly increased DPI drug emission and reduced the device retention of both the Ventolin® and Becotide® versions of the low resistance devices, Rotahaler® and Diskhaler®. Using these same flow rates for testing the dose emissions from the medium resistance Bricanyl and Pulmicort Turbohalers®, there was no significant difference in drug output between the two flow rates. © 1995. journaltitleInternational Journal of Pharmaceutics titleDose emissions from marketed dry powder inhalers Aerosol,Dose emission,Dry powder inhaler,Flow rate,Pharmacopeial test,Resistance Elsevier B.V. abstractThe potential of the force control agent magnesium stearate (MgSt) to enhance the aerosol performance of lactose-based dry powder inhaled (DPI) formulations was investigated in this study. The excipient-blends were investigated with analytical techniques including time-of-flight secondary ion mass spectrometry and single particle aerosol mass spectrometry (SPAMS), and particle size, morphology, and surface properties were evaluated. Excipient-blends were manufactured either by high-shear or low-shear blending lactose carrier with different amounts of MgSt in the range from 0% to 10% (w/w). Fluticasone propionate (FP) and salmeterol xinafoate (SX) used as model active pharmaceutical ingredients were added by low-shear mixing. The in vitro aerosol performance in terms of aerodynamic particle size distribution and fine particle fraction (FPF) of the FP and SX DPI formulations was evaluated with the Next Generation Impactor and also with SPAMS using a Breezhaler® inhalation device. The distribution of MgSt on the lactose carrier in the blends was visualized and found to depend strongly on the blending method. This affected drug particle detachment from the carrier and thus impacted aerosol performance for FP and SX. Compared with blends without force control agent, low-shear blending of MgSt increases the FPF of the model drug SX, whereas high-shear blending significantly increased FPF of both SX and FP. The interactions between drug and carrier particles were substantially affected by the choice of blending technique of MgSt with lactose. This allows detailed control of aerosol performance of a DPI by an adequate choice of the blending technique. SPAMS successfully demonstrated that it is capable to distinguish changes in DPI formulations blended with different amounts of MgSt, and additional information in terms of dispersibility of fine particles could be generated. journaltitleJournal of Pharmaceutical Sciences titleInvestigations on the mechanism of magnesium stearate to modify aerosol performance in dry powder inhaled formulations blending method,dispersibility,dry powder inhaler (DPI),force control agent (FCA),next generation impactor (NGI),single particle aerosol mass spectrometry (SPAMS),time-of-flight secondary ion mass spectrometry (To abstractProper orthogonal decomposition (POD) was performed on both the fluctuating velocity and vorticity fields of a backward-facing step (BFS) flow at Reynolds numbers of 580 and 4,660. The data was obtained from particle image velocimetry (PIV) measurements. The vorticity decomposition captured the fluctuating enstrophy more efficiently than the equivalent velocity field decomposition for a given number of modes. Coherent structures in the flow are also more easily identifiable using vorticity-based POD. A common structure of the low-order vorticity POD modes suggests that a large-scale similarity, independent of the Reynolds number, may be present for the BFS flow. The POD modes obtained from a vorticity-based decomposition would help in determining a basis for constructing simplified vortex skeletons and low-order flow descriptions based on the vorticity of turbulent flows. © Springer-Verlag 2005. journaltitleExperiments in Fluids titleA comparison between snapshot POD analysis of PIV velocity and vorticity data ermodynamics,Fluid,Heat and Mass Transfer,and Aerodynamics Elsevier B.V. abstractThe goal of this work was to evaluate the ability of Particle Image Velocimetry (PIV) to visually assess dry powder dispersion within an inhaler. Herein, the study reports particle movement characterization of entrained low-micron particles within an inhaler to further scheme of potential mechanisms. Carrier based DPI formulations were prepared and placed in a transparent model Rotahaler® chamber for the aerosolization experiments. Then using the PIV, a high-speed camera, the dried powder dispersion was directly observed and analyzed for all, neat, binary and ternary systems. Powder dispersion mechanisms proposed include drag force, impact with obstacle and particle-particle collision; these different mechanisms depended on the powder flow properties. A revised ratio of aerodynamic response time ($\tau$A) to the mean time between collisions ($\tau$C) was found to be 6.8 indicating that particle collisions were of strong influence to particle dispersion. With image analysis techniques, visualization of particle flow pattern and collision regions was possible; suggesting that the various mechanisms proposed did govern the powder dispersion. journaltitleInternational Journal of Pharmaceutics titlePowder dispersion mechanisms within a dry powder inhaler using microscale particle image velocimetry Collision,DPI,Dispersion mechanism,Image analysis,Impact,PIV Cambridge University Press abstractThe ‘plug' flow emerging from a long rotating tube into a large stationary reservoir has been used in an experimental investigation of centrifugally unstable swirling jets. A moderate Reynolds number, . journaltitleJournal of Fluid Mechanics titleAn experimental investigation of swirling jets abstractIt is no longer acceptable, in most circles, to present experimental results without describing the uncertainties involved. Besides its obvious role in publishing, uncertainty analysis provides the experimenter a rational way of evaluating the significance of the scatter on repeated trials. This can be a powerful tool in locating the source of trouble in a misbehaving experiment. To the user of the data, a statement (by the experimenter) of the range within which the results of the present experiment might have fallen by chance alone is of great help in deciding whether the present data agree with past results or differ from them. These benefits can be realized only if both the experimenter and the reader understand what an uncertainty analysis is, what it can do (and cannot do), and how to interpret its results. This paper begins with a general description of the sources of errors in engineering measurements and the relationship between error and uncertainty. Then the path of an uncertainty analysis is traced from its first step, identifying the intended true value of a measurement, through the quantitative estimation of the individual errors, to the end objective-the interpretation and reporting of the results. The basic mathematics of both single-sample and multiple-sample analysis are presented, as well as a technique for numerically executing uncertainty analyses when computerized data interpretation is involved. The material presented in this paper covers the method of describing the uncertainties in an engineering experiment and the necessary background material. © 1988. journaltitleExperimental Thermal and Fluid Science titleDescribing the uncertainties in experimental results analysis,error analysis,experimental uncertainty,multiple-sample analysis,single-sample,system errors abstractThe study aims to investigate the impact of various design parameters of a dry powder inhaler on the turbulence intensities generated and the performance of the dry powder inhaler. The flow fields and turbulence intensities in the dry powder inhaler are measured using particle image velocimetry (PIV) techniques. In vitro aerosolization and deposition a blend of budesonide and lactose are measured using an Andersen Cascade Impactor. Design parameters such as inhaler grid hole diameter, grid voidage and chamber length are considered. The experimental results reveal that the hole diameter on the grid has negligible impact on the turbulence intensity generated in the chamber. On the other hand, hole diameters smaller than a critical size can lead to performance degradation due to excessive particle-grid collisions. An increase in grid voidage can improve the inhaler performance but the effect diminishes at high grid voidage. An increase in the chamber length can enhance the turbulence intensity generated but also increases the powder adhesion on the inhaler wall. © 2013 Elsevier B.V. All rights reserved. journaltitleInternational Journal of Pharmaceutics titleExperimental investigation of design parameters on dry powder inhaler performance Dry powder inhaler,Emitted dose,Fine particle fraction,Flow field,In vitro deposition,Turbulence intensity The Stationery Office seriesBritish Pharmacopoeia titleBritish Pharmacopoeia Elsevier B.V. abstractEffective drug delivery to the lungs by a DPI device requires the air-stream through the device to have sufficient power to aerosolise the powder. Furthermore, sufficient turbulence must be induced, along with particle-wall and particle-particle collisions, in order to de-aggregate small drug particles from large carrier particles. As a result, the emitted and the fine particle doses produced by many commercially available DPI devices tend to be strongly affected by the natural inter-patient variability of the inhaled air flow. The Nexthaler® is a multi-dose breath-actuated dry-powder inhaler with minimum drug delivery-flow rate dependency and incorporating a dose protector. The actuation mechanism of the dose-protector ensures that the dose is only exposed to the inhaled air flow if the flow has sufficient power to cause complete aerosolisation. For this study, a proprietary lactose placebo powder blend was filled into "transparent" Nexthaler® to allow application of high-speed imaging and particle image velocimetry (PIV) techniques to successfully interrogate and reveal details of the powder entrainment and emission processes coupled with characterisation of the flow environment in the vicinity of the mouthpiece exit. The study showed that fluidisation of the bulk of the powder occurs very quickly (∼20 ms) after withdrawal of the dose protector followed by powder emission from the device within ∼50 ms thereafter. The bulk of the metered placebo dose was emitted within 100-200 ms. The visualisation study also revealed that a very small fraction of powder fines is emitted whilst the dose protector still covers the dosing cup as the flow rate through the device accelerates. The PIV results show that the flow exiting the device is highly turbulent with a rotating flow structure, which forces the particles to follow internal paths having a high probability of wall impacts, suggesting that the flow environment inside the Nexthaler® DPI will be very beneficial for carrier-drug de-aggregation. journaltitleInternational Journal of Pharmaceutics titleOptical diagnostics study of air flow and powder fluidisation in Nexthaler® - Part I: Studies with lactose placebo formulation Breath-actuated,Dry powder inhaler,High-speed imaging,Optical diagnostics,Particle image velocimetry,Powder emission,Powder fluidization Springer Verlag abstractIn this paper, we investigate the flow structures and pressure fields of a free annular swirling jet flow undergoing vortex breakdown. The flow field is analyzed by means of time-resolved tomographic particle image velocimetry measurements, which enable the reconstruction of the three-dimensional time-resolved pressure fields using the governing flow equations. Both time-averaged and instantaneous flow structures are discussed, including a characterization of the first- and second-order statistical moments. A Reynolds decomposition of the flow field shows that the time-averaged flow is axisymmetric with regions of high anisotropic Reynolds stresses. Two recirculation zones exist that are surrounded by regions of very intense mixing. Notwithstanding the axisymmetric nature of the time-averaged flow, a non-axisymmetric structure of the instantaneous flow is revealed, comprising a central vortex core which breaks up into a precessing vortex core. The winding sense of this helical structure is opposite to the swirl direction and it is wrapped around the vortex breakdown bubble. It precesses around the central axis of the flow at a frequency corresponding to a Strouhal number of 0.27. The precessing vortex core is associated with a low-pressure region along the central axis of the jet and the maximum pressure fluctuations occur upstream of the vortex breakdown location, where the azimuthal velocity component also reaches peak values as a result of the inward motion of the fluid and the conservation of angular momentum. The POD analysis of the pressure fields suggests that the precessing helical vortex formation is the dominant coherent structure in the instantaneous flow. journaltitleExperiments in Fluids titleAnalysis of the pressure fields in a swirling annular jet flow ic particle image velocimetry,Vortex breakdown CSIRO Australia booktitleInternational Colloquium on Jets, Wakes and Shear Layers titleDigital cross-correlation particle image velocimetry measurements in the near wake of a circular cylinder Elsevier Inc. abstractA cross-correlation particle image velocimetry (PIV) technique has been developed to measure the spatiotemporal in-plane velocity vector field evolution of time-dependent flows. A novel iterative two-stage cross-correlation scheme of two sequential images of flow tracers has been incorporated in the image analysis. The implementation in hardware and software of this complete recording and analysis system are described. The expected accuracy of the velocity measurements was investigated and is discussed. The technique has been applied to study the near wake behind a circular cylinder at low Reynolds numbers (Red). The measurements presented pertain to cylinders with d = 12.5 and 25 mm (l/d = 19.5 and 9.8, respectively). The respective Reynolds numbers Red are 875 and 769. Two planes of this flow are considered in this study: (1) plane normal to the cylinder axis (xy plane) and (2) a plane containing the cylinder axis and the stream direction (xz plane). Instantaneous in-plane velocity vector fields and out-of-plane vorticity fields are presented for both planes. The effect of spatial resolution on peak vorticity is discussed using velocity vector field measurements in the near wake of the cylinder that were conducted using different spatial resolutions. The three-dimensional nature of the near wake of circular cylinders at low Red is demonstrated using quantitative in-plane velocity vector field and out-of-plane vorticity measurements. An upstream influx of relatively high velocity fluid into the stagnant near-wake region in the xy plane and the subsequent deflection of the fluid normal to this plane as it approaches the stagnation region at the cylinder are shown to be responsible for the generation of three-dimensional flow in the near wake of a circular cylinder. journaltitleExperimental Thermal and Fluid Science titleAn investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique Cross-correlation PIV,Cylinder wake,Vorticity measurement Monash University booktitle13th Australasian Fluid Mechanics Conference titleMultigrid approach to cross-correlation digital PIV and HPIV analysis Elsevier Science Ltd abstractThis paper describes a novel approach of acquiring two single-exposed, non-overlapping images using half frame image shift (HFIS) recording on photographic film. This technique permits the recording of two single-exposed, non-overlapping images of seed particles in a flow plane on high spatial resolution film with any arbitrary time delay between exposures. A new multigrid CCDPIV (MCCDPIV) analysis method is used to analyze the single-exposed, non-overlapping sequential images resulting in PIV measurements with a larger velocity dynamic range, lower random error and better spatial resolution than standard CCDPIV analysis. HFIS recording followed by MCCDPIV analysis was employed to measure the spatio-temporal evolution of the in-plane velocity vector and the out-of-plane vorticity fields of a turbulent starting jet at Reynolds numbers based on the orifice diameter and piston velocity of 10,780 and 13,860. journaltitleOptics and Laser Technology titleHigh resolution multigrid cross-correlation digital PIV measurements of a turbulent starting jet using half frame image shift film recording abstractThis paper presents an experimental investigation on swirling jets with well-defined initial conditions. The axial, radial, and azimuthal velocity components, with their respective fluctuations were measured using high spatial-resolution particle image velocimetry. These detailed measurements allow the initial conditions of the swirling jets to be established and the jets to be characterized using various swirl number definitions. The significance of each term in the swirl number calculations are quantified, and the effect of the common assumptions and simplifications are examined. The characteristics of the jets in relation to the initial conditions are then investigated and compared with the previous studies using similar characterization parameters. Jets with Reynolds number of approximately 5700 and swirl conditions ranging from a non-swirling reference case to high swirl are studied. General properties of swirling jets such as higher spreading rate, higher centerline velocity decay, and higher turbulence level are observed. When the degree of swirl is sufficiently high, vortex breakdown occurs. A swirl number of 0.94 is recorded for a high swirl case prior to vortex breakdown, much higher than the critical swirl number reported in the literature. This behavior is attributed to the effect of the initial conditions on the swirl number calculation. © 2009 Springer-Verlag. journaltitleExperiments in Fluids titleAxial plus tangential entry swirling jet ermodynamics,Fluid,Heat and Mass Transfer,and Aerodynamics family=Van Dyck, family=Van den Bulck, American Institute of Physics Inc. abstractIn this paper, the flow dynamics in the wake of a turbulent annular jet is studied using Time-Resolved Stereoscopic Particle Image Velocimetry and Proper Orthogonal Decomposition (POD). In this wake, a central recirculation zone is present which, under certain conditions, shows a low-frequency precessing motion. POD analysis of the measured velocity data shows that at zero swirl, an asymmetry is present in the wake, which motion is random in time. This asymmetry originates from a bifurcation of the flow once a threshold Reynolds number is exceeded. For low-swirl numbers, ranging from 0 < S < 0.12, the asymmetry is still present and its motion becomes structured into a well defined precession. For S > 0.12, the precession is gone and the motion of the asymmetric wake is again random in time, similar like the non-swirling jet. In this paper, a model is developed to describe the influence of swirl on the wake dynamics. The model assumes that perturbations in the inner shear layer near the bluff body wall are convected towards the stagnation point. These perturbations cause a shift in the stagnation points position. This shift is convected back to the inner shear layer through convection in the recirculating flow. The dynamics of this feedback mechanism can be modeled by the nonlinear delayed saturation model. In this paper, the model is adapted for swirling flow and simulations show that good agreement is found with the experiments. journaltitlePhysics of Fluids titleSymmetry breaking and vortex precession in low-swirling annular jets abstractThe effect of turbulence and mechanical impaction on dry powder aerosol deaggregation was tested using a novel powder deagglomeration rig, with fine particle fraction (FPFED<5.6 $\mu$m), defined here as particles sized smaller than 5.6 $\mu$m, measured using an Anderson inertial impactor. Powder from GlaxoSmithKline Ventodisks™ was deaggregated either using turbulence generated with a ring of impinging jets, or by impacting the powder on bars of a wire mesh. This deaggregation was compared with deaggregation achieved with the GlaxoSmithKline Diskhaler. The turbulence levels in the test rig and at the exit of the Diskhaler were quantified using laser Doppler velocimetry (LDV). In addition, the Ventodisk powder's auto-adhesion properties were altered by introducing the powder into a high humidity environment (25°C and 25% R.H.) and then deagglomerated by both the rig (using turbulence as the primary deagglomeration mechanism) and the Diskhaler. Fine particle fractions were found to increase from 13 to 24% as the level of turbulence in the rig was increased. However, fine particle fractions found with the Diskhaler were 35%. Turbulence levels found in the rig at the highest jet flow rate were significantly higher than that at the outlet of the Diskhaler, leading to the conclusion that turbulence is not the only method of deaggregation in this inhaler. The humidified powders were significantly more difficult to deaggregate, giving a FPFED<5.6 $\mu$m of 9% when using the rig and 15% when using the Diskhaler. Fine particle fractions produced when deagglomerating the powder with the wire meshes were similar to those produced without a mesh, showing that mechanical impaction had little effect. The results underline the utility of having a rig that can explore the ability of a powder to deagglomerate with controlled variations in the deaggregation forces. © 2002 Elsevier Science B.V. All rights reserved. journaltitleInternational Journal of Pharmaceutics titleDeagglomeration of dry powder pharmaceutical aerosols Aerosol,Anderson impactor,Diskhaler,Dry powder inhaler,Powder deagglomeration,Powder deaggregation,Powder inhalation,Turbulence Int J Pharm abstractThe dispersion of Ventodisk® (salbutamol sulphate with lactose) from different drug reservoirs by an air jet at normal impingement is examined experimentally. The effect on dispersion efficiency of jet velocity, nozzle location, reservoir size and shape, and the loaded dose is investigated for possible design of new dosing methods or inhalers. Results show that higher jet velocity (as high as feasible), lower drug loading (2mg or smaller), a cylindrical hole reservoir (6mm in diameter and 3mm in depth) and a medium distance (approximately 5 jet diameters) from the nozzle to the reservoir yield optimum dispersion. The dispersed fine particle dose improves by a factor of 2-3 times between optimized conditions and poor conditions. © 2004 Elsevier B.V. All rights reserved. journaltitleInternational Journal of Pharmaceutics titleUse of an impinging jet for dispersion of dry powder inhalation aerosols Aerosol,Cascade impactor,Dry powder inhaler,Impinging jet,Powder deagglomeration,Powder dispersion abstractA statistical model is introduced that describes the occurence of spurious vectors in PIV data. This model is used to investigate the performance of three different post-interrogation procedures: the global-mean, the local-mean and the local-median test. The model is also used to optimize the performance of these procedures. Predicted performances agree very well with those obtained from an artificially generated PIV record. It is demonstrated that the "detectability" as the conventional measure for the reliability of a measured displacement vector is very inefficient, compared to the three tests described here. The local-median test has the highest efficiency. © 1994 Springer-Verlag. journaltitleExperiments in Fluids titleEfficient detection of spurious vectors in particle image velocimetry data ermodynamics,Fluid,Heat and Mass Transfer,and Aerodynamics John Wiley Sons Inc. abstractThis study aimed to investigate the influence of grid structures on the break-up and aerosol performance of a model inhalation formulation through the use of standardised entrainment tubes in combination with computational fluid dynamics (CFD). A series of entrainment tubes with grid structures of different aperture size and wire diameters were designed in silico and constructed using three-dimensional printing. The flow characteristics were simulated using CFD, and the deposition and aerosol performance of a model agglomerate system (496.3-789.2 $\mu$m agglomerates containing 3.91 $\mu$m median diameter mannitol particles) were evaluated by chemical analysis and laser diffraction, respectively. Analysis of the mannitol recovery from the assembly indicated that mass deposition was primarily on the grid structure with little before or after the grid. Mass deposition was minimal down to 532 $\mu$m; however, for smaller grid apertures, significant blockage was observed at all airflow rates (60-140 L·min-1). Analysis of the particle size distribution exiting the impactor assembly suggested that mannitol aerosolisation was dependent on the void percentage of the grid structure. It is proposed that initial particle-grid impaction results in a shearing force causing agglomerate fragmentation followed by immediate re-entrainment into the turbulent airstream within the grid apertures which causes further dispersion of the fine particles. Such observations have significant implications in the design of dry powder inhaler devices. © 2011 Wiley-Liss, Inc. journaltitleJournal of Pharmaceutical Sciences titleParticle aerosolisation and break-up in dry powder inhalers: Evaluation and modelling of the influence of grid structures for agglomerated systems Aerosols,Agglomerate,CFD,Deagglomeration,Dry powder inhaler,Impaction,In silico modelling,Particle size,Pulmonary drug delivery,Simulations Elsevier B.V. abstractPurpose: This study was performed to investigate how increasing the active pharmaceutical ingredient (API) content within a formulation affects the dispersion of particles and the aerosol performance efficiency of a carrier based dry powder inhalable (DPI) formulation, using a custom dry powder inhaler (DPI) development rig. Methods: Five formulations with varying concentrations of API beclomethasone dipropionate (BDP) between 1% and 30% (w/w) were formulated as a multi-component carrier system containing coarse lactose and fine lactose with magnesium stearate. The morphology of the formulation and each component were investigated using scanning electron micrographs while the particle size was measured by laser diffraction. The aerosol performance, in terms of aerodynamic diameter, was assessed using the British pharmacopeia Apparatus E cascade impactor (Next generation impactor). Chemical analysis of the API was observed by high performance liquid chromatography (HPLC). Results: Increasing the concentration of BDP in the blend resulted in increasing numbers and size of individual agglomerates and densely packed BDP multi-layers on the surface of the lactose carrier. BDP present within the multi-layer did not disperse as individual primary particles but as dense agglomerates, which led to a decrease in aerosol performance and increased percentage of BDP deposition within the Apparatus E induction port and pre-separator. Conclusion: As the BDP concentration in the blends increases, aerosol performance of the formulation decreases, in an inversely proportional manner. Concurrently, the percentage of API deposition in the induction port and pre-separator could also be linked to the amount of micronized particles (BDP and Micronized composite carrier) present in the formulation. The effect of such dose increase on the behaviour of aerosol dispersion was investigated to gain greater insight in the development and optimisation of higher dosed carrier-based formulations. journaltitleInternational Journal of Pharmaceutics titleLimitations of high dose carrier based formulations Agglomerates,Carrier,Dry powder inhaler,High dose formulation,Multi-layer Springer New York LLC abstractThis study aims to investigate the implications of loaded formulation mass on aerosol performance using a reservoir novel dry powder inhaler containing a custom dosing cup to deliver carrier-based formulation to the lungs. A 3D printed dosing cup with volume size of 133.04 mm 3 was manufactured to allow for the progressive loading of different carrier formulation masses of 1% beclomethasone dipropionate BDP (w/w) formulation (10 to 60 mg, with increments of 10 mg), in a novel customizable DPI device. Scanning electron micrographs were used to investigate BDP detachment from carrier particles post-aerosolisation and particle deposition on the USP induction port. The subsequent aerosol performance analysis was performed using the next generation impactor (NGI). Incrementally increasing the loading mass to 60 mg led to decreases in BDP detachment from carrier particles, resulting in significant decreases in aerosol performance. Increases in loading dose mass led to progressively decreased detachment of BDP from the carrier and the overall aerosol performance in comparison to the initial mass of 10 mg. These results are likely to be due to a decrease in void volume within the dosing cup with increased loading mass leading to altered airflow, decreased impaction forces and the possibility of a significant quantity of large carrier particles introducing a ‘sweeping' effect on the inhaler inner surface. This study has shown that despite the decreased BDP detachment from the carrier and decreased aerosol performance, the dose delivered to the lung still increased due to the higher loaded dose. journaltitleAAPS PharmSciTech titleAssessing aerosol performance of a dry powder carrier formulation with increasing doses using a novel inhaler aerosol performance,carrier formulation,dispersion forces,loading dose,novel dry powder inhaler AAPS J abstractThe objective of this study is to investigate the effect of device design of the Aerolizer® on the aerosolization of a carrier-based dry powder inhaler formulation (Foradile®). The Aerolizer was modified by reducing the air inlet size and mouthpiece length to 1/3 of the original dimensions, or by increasing the grid voidage. Aerosolization of the powder formulation was assessed on a multi-stage liquid impinger at air flow rates of 30, 60, and 100 L/min. Coupled CFD-DEM simulations were performed to investigate the air flow pattern and particle impaction. There was no significant difference in the aerosolization behavior between the original and 1/3 mouthpiece length devices. Significant increases in FPF total and FPF emitted were demonstrated when the inlet size was reduced, and the results were explained by the increases in air velocity and turbulence from the CFD analysis. No significant differences were shown in FPF total and FPF emitted when the grid voidage was increased, but more drugs were found to deposit in induction port and to a lesser extent, the mouthpiece. This was supported by the CFD-DEM analysis which showed the particle-device collisions mainly occurred in the inhaler chamber, and the cross-grid design increased the particle-device collisions on both mouthpiece and induction port. The air inlet size and grid structure of the Aerolizer® were found to impact significantly on the aerosolization of the carrier-based powder. © 2013 American Association of Pharmaceutical Scientists. journaltitleAAPS Journal titleEffect of device design on the aerosolization of a carrier-based dry powder inhaler - A case study on Aerolizer® Foradile® aerosolization,computational fluid dynamics,device design,discrete element method,dry powder inhalers
# UNIT: Unifying Tensorized Instruction Compilation Jian Weng12, Animesh Jain2, Jie Wang12, Leyuan Wang2, Yida Wang2, Tony Nowatzki1 1University of California, Los Angeles, USA 2Amazon Web Services, USA<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Because of the increasing demand for intensive computation in deep neural networks, researchers have developed both hardware and software mechanisms to reduce the compute and memory burden. A widely adopted approach is to use mixed precision data types. However, it is hard to benefit from mixed precision without hardware specialization because of the overhead of data casting. Recently, hardware vendors offer _tensorized_ instructions specialized for mixed-precision tensor operations, such as Intel VNNI, Nvidia Tensor Core, and ARM DOT. These instructions involve a new computing idiom, which reduces multiple low precision elements into one high precision element. The lack of compilation techniques for this emerging idiom makes it hard to utilize these instructions. In practice, one approach is to use vendor- provided libraries for computationally-intensive kernels, but this is inflexible and prevents further optimizations. Another approach is to manually write hardware intrinsics, which is error-prone and difficult for programmers. Some prior works tried to address this problem by creating compilers for each instruction. This requires excessive efforts when it comes to many tensorized instructions. In this work, we develop a compiler framework, UNIT, to unify the compilation for tensorized instructions. The key to this approach is a unified semantics abstraction which makes the integration of new instructions easy, and the reuse of the analysis and transformations possible. Tensorized instructions from different platforms can be compiled via UNIT with moderate effort for favorable performance. Given a tensorized instruction and a tensor operation, UNIT automatically detects the applicability of the instruction, transforms the loop organization of the operation, and rewrites the loop body to take advantage of the tensorized instruction. According to our evaluation, UNIT is able to target various mainstream hardware platforms. The generated end-to-end inference model achieves 1.3$\times$ speedup over Intel oneDNN on an x86 CPU, 1.75$\times$ speedup over Nvidia cuDNN on an Nvidia GPU, and 1.13$\times$ speedup over a carefully tuned TVM solution for ARM DOT on an ARM CPU. 1212footnotetext: Work done during Jian and Jie’s internship at AWS. ## I Introduction Dense tensor operations like matrix multiplication (Matmul) and convolution (Conv) have long been the workhorses in many domains, including deep learning workloads [14]. The popularity of deep learning means that aggressively optimizing these operations has a high payoff. Essentially, Matmul and Conv are a series of multiply-accumulate (MAC) operations, which perform accumulation over a number of elementwise multiplications. To capture the reduction behavior and perform it more efficiently, recent general-purpose processors offer native tensor operation specialized instructions (hereinafter referred to as _tensorized instructions_), like Intel VNNI [2], Nvidia Tensor Core [5], and ARM DOT [1]. Unlike the conventional SIMD instructions, after performing elementwise arithmetic operations, these instructions introduce a “horizontal computation” to accumulate elementwise results. Further, tensorized instructions are often mixed-precision, meaning that elementwise operations use less precise and lower bitwidth operands (e.g., fp16 and int8), while accumulation occurs with higher bitwidth, where it is needed. This offers a good balance between data width and precision that is generally sufficient for deep learning workloads [24, 18], and enables the use of quantized data types. Mixed-precision is difficult to express in a single SIMD instruction, because the output vector width is different than the input vector width. In most ISAs this paradigm requires multiple SIMD instructions to express. In a tensorized instruction, by definition there are fewer outputs, so allocating more bitwidth to them for the output vector is natural. In addition, tensorized instructions sometimes reuse the same inputs multiple times, which reduces the required register file bandwidth. Overall, tensorized instructions offer significant advantages over SIMD for executing MACs. While promising, the absence of appropriate compilation techniques limit c the applicability of these tensorized instructions. Conventional SIMD instructions are vector instructions, so industry standard compilers only try parallelizing the innermost loops. In addition, it is difficult for the high-level language programmer to express the compute flow in a tensorization-friendly way and hint the compiler to try tensorization upon a loop nest, because the dependency of reduction is more complicated and error-prone. In practice, there are normally two options to leverage tensorized instructions. One way is to call the vendor-provided libraries such as Intel oneDNN [6], Nvidia cuBLAS and cuDNN [4], which provides highly optimized performance in some pre-defined single kernels using tensorized instructions [17, 44]. However, it also brings inflexibility when it comes to new workloads or when further performance exploitation is desired. The other option is to manually write assembly intrinsics, which sets a high bar to ordinary developers and hence lacks productivity. Some prior works tried to solve this problem by developing a compiler [35, 36] for each instruction. This requires too much effort when there are many tensorized instructions, both within and across hardware platforms. Our Goal: Although different processors may provide different tensorized instructions, in the context of deep learning workloads, we observe that these instructions essentially handle a similar compute pattern, i.e., elementwise multiplication and then horizontal accumulation. They primarily differ in the number of elementwise computation lanes and the accepting data types. Therefore, we aim to develop a unified approach to compile these tensorized instructions on multiple platforms to optimize the tensor operations in deep learning workloads. Our techniques are extensible to the tensorized instructions with other data types and operations as well. Challenges: There are several challenges to attain a unified compilation pipeline: * • _Instructions Integration:_ Instead of building a new specialized compiler for each new instruction, it is desirable to create a unified and extensible compilation flow; * • _Detecting the applicability:_ Given a tensorized instruction, a first question is whether and how this instruction can be applied to the target tensor operation, which may require loop reorganization to make it applicable; * • _Code rewriting:_ When applicable, the compiler must determine how the loops involved should be rewritten by the tensorized instruction, and how the loops should be rearranged to achieve high performance. Our Insight: We envision that the key to addressing these three challenges is to have a unified semantics abstraction for tensorized instructions so that the analysis and transformation can also be unified. This paper presents UNIT, an end-to-end compilation pipeline to surmount the above three challenges. UNIT takes the tensorized instructions (e.g., Intel VNNI instructions on CPUs, or Nvidia Tensor Core instructions on GPUs) and a deep learning model as input, lowers the tensor operations of the model into loop-based IRs to identify the tensorizable components, and inserts the tensorized instructions by transforming and rewriting the loop. It achieves high performance for tensor operations, and consequently, model inference. To the best of our knowledge, this is the first work to tackle tensorized instruction compilation and optimization with a unified solution. UNIT not only achieves high performance for single tensor operations, but also provides desirable model inference latency in practice. Key Results: According to our evaluation, UNIT is expressive enough to target many tensorized instructions on multiple hardware platforms, including Intel VNNI, Nvidia Tensor Core, and ARM DOT. The generated programs for end-to-end model inference are 1.3$\times$ and 1.75$\times$ faster than the solutions backed up by Intel oneDNN and Nvidia cuDNN on CPU and GPU, respectively. In addition, UNIT can be extended to new tensorized instructions with moderate effort. Although we designed UNIT to target Intel CPUs and Nvidia GPUs, on an ARM Cortex A-72 CPU with DOT instructions, UNIT achieves up to 1.13$\times$ speedup against a carefully manual tuned solution. To sum up, our contribution is an end-to-end compilation pipeline of tensorized instructions for deep learning workloads, which includes: * • A unified abstraction for tensorized instructions. * • An algorithm that detects the applicability of these tensorized instructions. * • A rewriting and tuning mechanism that looks for favorable loop transformations of the tensor operations to plug in the tensorized instructions for high performance. Paper Organization: We first introduce the background and challenges of tensorized compilation in Section II. The design of UNIT is presented in Section III. We explain the implementation details in Section IV. We clarify our experiment methodology in Section V, and evaluate our work in Section VI. Finally, we discuss the related work in Section VII. ## II Background UNIT is an end-to-end compilation pipeline capable of automatically mapping tensorized instructions to the deep learning tensor operations. It defines the tensorized instruction’s semantics using a suitable intermediate representation (IR) and inserts them in proper places of the program of tensor operations. In this section, we give an overview of popular mixed precision tensorized instructions, followed by the limitations of existing solutions in automatic mapping of these tensorized instructions. Finally, we discuss the background of tensor domain specific language and the multi-level intermediate representation. ### II-A Mixed Precision Tensorized Instructions Deep learning is computationally expensive, requiring substantial compute and memory resources. As deep learning becomes more pervasive, researchers are designing both software and hardware techniques to reduce the compute and memory burden. A widely adopted approach in this context is using mixed precision for expensive operations, e.g., convolution or dense operations [24, 18]. In practice, this means representing 32-bit floating point (fp32) operands with a lower bitwidth datatype - 16-bit floating point numbers (fp16) or 8/16-bit integer numbers (int8, int16). To keep the accuracy in check, it is helpful to accumulate the results in higher precision (fp32 or int32). This type of mixed precision computation is often called _quantization_ for integer values [18]. In this paper, we will always use _mixed precision_ for brevity. Figure 1: Performance comparison on Nvidia V100-SXM2 between fp32 and fp16 without mixed precision instruction support. While using mixed precision data types reduces memory footprint, it might not necessarily lead to performance improvement. To investigate this, we conducted an experiment to compare the performance of Nvidia cuDNN performance for fp16 and fp32 in the absence of Nvidia mixed precision tensorized instructions (Tensor Core). As shown in Figure 1, we observe that blindly using mixed precision leads to substantial slowdown because of the overhead of casting between two data types. Therefore, mainstream hardware vendors (Intel, ARM and Nvidia) have introduced mixed precision tensorized instructions to achieve better performance. These instructions add mixed precision arithmetic support where operands are of lower precision while the accumulation happens in higher precision, potentially leading to 2$\times$ \- 4$\times$ speedup. The most popular examples of these tensorized instructions are Intel VNNI, ARM DOT and Nvidia Tensor Core. We will discuss the semantics of these operations in Section III. Figure 2: The semantics of Intel VNNI and Nvidia Tensor Core. The text beside is the name of the corresponding LLVM intrinsic. Hardware vendors have a long history of adding new instructions to accelerate important applications. However, the mixed precision tensorized instructions introduce a unique idiom - horizontal accumulation. These tensorized instructions typically conduct a sequence of elementwise multiplications governed by a memory access pattern, followed by a horizontal accumulation. The accumulation is termed horizontal because all values to be accumulated are present in the same vector register. For example, as it is shown in Figure 2(a), Intel VNNI executes a dot product of two vectors, each having 4 int8 elements, while performing the accumulation in int32. We observe a similar pattern, though with different numbers of entries and data types, for Nvidia Tensor Core (in Figure 2(b)) and ARM DOT instructions (this is omitted, because it is similar to VNNI). ### II-B Limitations of Existing Solutions Though tensorized instructions seem promising, their adoption pace is limited because of the absence of an automatic technique that can detect and use these instructions seamlessly. Currently, their usage in the deep learning domain is limited to hardware vendor libraries like Intel oneDNN and Nvidia cuDNN, which may provide high performance for the pre-defined operations but are inflexible as discussed in Section I. Similarly, conventional loop vectorizers find it hard to exploit the profitability of these tensorized instructions, as they are not designed to work with the horizontal reduction idiom. Conventional loop vectorizers in general-purpose compilers like GCC and LLVM mainly focus on either analyzing the innermost loop body or combining instructions in the unrolled loop bodies. When it comes to the horizontal reduction idiom, these compilers often reorder the computation and generate epilogue reduction, preventing us from using the tensorized instructions. There have been some recent works in compiling programs to leverage tensorized instructions. PolyDL [36] generates CPU programs for convolution kernels in neural networks that call a GEMM micro-kernel using Intel VNNI instructions. Bhaskaracharya et al. [35] generate CUDA programs for matrix computation leveraging Nvidia Tensor Core. However, these works are limited to one platform and its specific instruction, which lacks generalizability. A generic solution to handle tensorized instructions from multiple platforms together is still missing. ### II-C Multi-Level Intermediate Representation Compilers often have multiple levels of intermediate representation (IR) to express the program; each level is designed to enable different analyses and transformations. In this section, we describe the background of a tensor domain specific language (DSL) and the multi-level IR. #### II-C1 Graph-Level IR Deep learning compilers like TVM [10], Glow [34], and XLA [43] adopt a graph- level IR to represent a deep learning model as a directed acyclic graph (DAG) of operations. This graph-level IR is useful for inter-tensor-operation optimization, like tensor shape padding, operation fusion, and choosing the proper data layout [23]. Our tensorized analysis relies on tensor padding so that loops can be tiled by the number of lanes of the instruction perfectly. However, this IR has little knowledge about the implementation of each tensor operation. When compiling a graph-level IR, each node of the DAG will be dispatched to its implementation in tensor DSL as explained next. #### II-C2 Tensor DSL Tensor domain-specific languages, like Halide [31], TVM [10], and Tensor Comprehension [37], have been developed to productively and portably express tensor programs while enabling efficient performance tuning. As shown in Figure 4 and Figure 5, programs written in tensor DSLs follow this paradigm: Users first declare the tensors and the loop variables, and then the computation is described by expressions involving the declared tensors and loop variables. These DSLs also provide interfaces to split, reorder, and annotate loops without affecting the computation semantics for performance tuning. All the information gathered from the tensor DSL frontend will be stored in a _tensor Op_ data structure, including the declared tensors, loop variables, expressions, and loop manipulation. #### II-C3 Tensor IR Each _tensor Op_ is then lowered to _Tensor IR_ , which is an imperative program IR with additional constraints: All the loops are canonical (starting from 0, and increased by 1 each time), and all the array operations are restricted (i.e., an element cannot be accessed by two different pointers). These two properties enable making strong assumptions for analysis and transformation. Our work conducts analysis on the _tensor Op_ data structure level and then performs transformation on the tensor IR. Although the tensor IR provides essentially identical information for analysis, as discussed above, it is easier to reorganize the loops via the _tensor Op_ data structure. #### II-C4 Low-Level IR The tensor IR is lowered to a general-purposed low-level IR like LLVM, after all the specialized analysis and transformations on the tensor IR are done, to get ready for assembly code generation. Figure 3: The overview of our framework, UNIT. ## III Unified Tensorization Our goal is to automatically _tensorize_ 111We coin the word to mean rewrite and optimize a given code by the tensorized instruction. mixed-precision deep learning tensor operations across a variety of hardware platforms. We resolve the challenges discussed in Section I by presenting UNIT with the following techniques: 1. 1. _Tensorized Instruction in Tensor DSL:_ To abstract the diverse tensorized instructions on different hardware platforms, we leverage the existing tensor DSL to represent their semantics. 2. 2. _Applicability Inspection:_ To determine if and how a tensorized instruction can be applied to a tensor operation, we developed an analysis pass in the _Inspector_ component of UNIT, which analyzes the _tensor Op_ data structure of both the instruction and the operation. The result of analysis will guide the loop reorganization and instruction injection. 3. 3. _Code Rewriter:_ Once the tensorized instruction is determined applicable, the Rewriter reorganizes the loop nests in accordance with the Inspector so that the innermost loop nests resemble the tensorized instruction and are ready to be replaced. Finally, it sets up the tuning space for the remaining loop nests to exploit high performance. These components of UNIT together enable a unified compilation flow to simplify the mapping of tensorized instructions across a variety of hardware platforms. In the rest of this section, the details of each of the above steps will be discussed. Figure 4: Tensorized instructions as abstracted in the tensor DSL. ### III-A Semantics Abstraction - Tensor DSL In order to unify the compilation of tensorized instructions from different platforms and keep the system open to integrate new instructions, the first question to answer is how to have a unified description of the semantics of tensorized instructions. As explained in Section II, we employ ubiquitous tensor DSL and tensor IR to solve the abstraction problem. All mixed precision tensorized instructions perform some elementwise operations for vectors, followed by a horizontal reduction. Each tensorized instruction, therefore, can be regarded as a small tensor operation program written in the tensor DSL. Figure 4(a) shows how an Intel VNNI instruction is described in the tensor DSL. Three source operands of Intel VNNI are 512-bit registers. Two of them are 64 lanes of unsigned 8-bit integers (uint8) and signed 8-bit integers (int8), and the other one is 16 lanes of signed 32-bit integers (int32), which correspond to the tensors a, b, c we defined. The arithmetic behavior is defined by the loop variables and the expression of d[i]. Here we annotate that loop i is data parallel, since these 16 elements are independent from each other; loop j is reduction since for every independent element it sums up 4 elements along with this loop. A similar loop pattern appears in the other tensor operations shown in Figure 5. The description of ARM DOT, shown in Figure 4(b), is similar to Intel VNNI, with a different number of lanes and data types. Nvidia Tensor Core, on the other hand, performs a $16^{3}$ square matrix multiplication as shown in Figure 4(c). Comparing with (a) and (b), a key difference is that it requires the accumulator register to be the same as the addition register (note the +=). This is due to the data type opaqueness of the Tensor Core instruction, which prevents us from giving arbitrary initial values for the accumulators. We describe the semantics of each tensorized instruction in tensor DSL. The deep learning compiler pipeline parses the operation into _tensor Op_ , which preserves tensor information like the expression tree, the loop trip count, and the array buffers. This information is essential for the analysis and transformation passes in Inspector and Rewriter. ### III-B Applicability Detection - Inspector To determine if a tensorized instruction can be applied to a tensor operation, the Inspector pass uses a two-step approach. It first determines if (part of) the tensor operation program and the instruction can be arithmetically equivalent by checking a form of isomorphism between their associated expression trees. After that, it inspects the data access pattern to confirm the assembly operands can be prepared so as to guide the Rewriter transformation. Figure 5: An example of applying Intel VNNI to Conv using UNIT. Algorithm 1: Determine the isomorphism between expression trees. _a_ is for the instruction, and _b_ is for the operation. function Inspect(a,b) if a.type=b.type then if isleaf(a)$\land$isleaf(b) then if a is not bound then bind[a]:=b else if bind[a]$\neq$b then return False end if return True else if isarith(a), isarith(b) then cond:=a.opcode=b.opcode cond:=cond$\land$Inspect(a.lhs, b.lhs) cond:=cond$\land$Inspect(a.rhs, b.rhs) return cond end if end if return False end function #### III-B1 Compute Isomorphism Algorithm 1 shows the algorithm we adopt to determine the isomorphism of two expression trees. It recursively traverses both trees and matches the data type and opcode of each pair of nodes. Figure 5(b).1 shows that the two trees of convolution and pbpdusd (an Intel VNNI instruction) are in exactly the same topology and data type, so these two programs are arithmetically isomorphic. This analysis also finds a mapping from the operands in the tensor program to the operands in the tensorized instruction. As we explained, tensor operands in the tensorized instruction are the abstraction for registers. Therefore, a register cannot correspond to multiple data sources. This property still requires further checks, which will be explained in the next section. #### III-B2 Array Access Isomorphism Once compute isomorphism is determined, the next concern is how the data are fed to this instruction. The enforcement explained in the last subsection already determines each register operand only corresponds to one array in the tensor operation. On top of this, we need to determine each element in the operand tensor corresponds to only one memory address in the tensor program when mapping to the tensorized instruction. To map a tensor program to a tensorized instruction, we need to know which loop levels are tensorized. We enumerate the loop levels to be tensorized, and these loop levels will be mapped to loops in the tensorized instruction. Note that only loops with the same annotation (data parallel or reduction) can be mapped to each other. Then we check if this enumerated mapping is feasible, by scanning each pair of operand correspondence determined in the last paragraph. If the operand in the tensor program is a constant, we just skip it222If it is a constant, the correspondence was already checked in the last section. This register corresponds to this constant.. If the operand is a memory operation, we inspect the index expressions of both memory operations in the operation and instruction. We define: * • $A$ is the set of loop variables to be mapped to the tensorized instruction. * • $B$ is the set of loop variables of the tensorized instruction. * • $f:A\mapsto B$ is the mapping we enumerate. * • $S(u):=\\{x\|x\text{ is loop variable in the index expression }u\\}$ * • $S^{\prime}(u):=\\{f(x)\|x\in S(u)\cap A\\}$ A mapping is considered feasible, if every pair of memory operation’s index expressions $(u,v)$, where $u$ is from the operation and $v$ is from the instruction, holds $S^{\prime}(u)\subseteq S(v)$. Figure 5(b).2 shows an example of inspection. If $S^{\prime}(u)$ is a subset of $S(v)$, this means the data loaded by the tensor operation should be broadcast along with the loop variables that do not exist in $S(v)$ to fill all the register lanes. If not, this means each register lane corresponds to multiple memory addresses under this mapping, which is not realistic for code generation, so we should try another enumeration. If there are multiple feasible mappings, we leave this as a dimension of code tuning space. Once this mapping is determined, it will guide the further loop transformation and code generation. ### III-C Code Transformation - Rewriter There are three phases in the code transformation: loop reorganization, tensorized instruction replacement, and tuning. #### III-C1 Loop Reorganization As discussed in Subsection III-B, the inspector selects the loop levels to be executed by the given instruction. To get poised for code generation, as shown in Figure 5(c), we need to tile these loops and reorder them to the innermost loop levels so that those innermost loops perform exactly the same semantics as the instruction. As we explained, tensor DSL provides the capability to reorganize the loops nests easily. #### III-C2 Tensorized Instruction Replacement After identifying the code region to be replaced by a tensorized instruction, the code generator should prepare each operand of this instruction. It is difficult to fully automate the operand preparation for different platforms because of their diverse execution models and assembly formats. Therefore, we formalize a unified programming interface to compiler developers to manually specify the rule of operand generation. In this interface, each loop variable to be replaced, and their coefficients in the index expression are exposed. For example, as shown in Figure 5(c), by analyzing the strides and trip count of ki, and ci, the array access c[x,y,c] will be transformed to a 16-lane vector; a[x,y,rc] will be vectorized along with c by 4, and broadcast along with ki by 16; b[r,s,k,c] will be vectorized along with ci by 4, and unrolled and concatenated along with ki. #### III-C3 Tuner All the other loop levels that are not involved in instruction rewriting can be reorganized to tune the performance. Here, we develop strategies to optimize the performance of tensor programs on both CPU and GPU. The generic philosophy is to exploit both fine- and coarse-grained parallelism. We also developed specialized strategies because of the different execution models and memory hierarchy. CPU Tuning: On CPU, data-parallel loops are distributed to multiple threads to achieve coarse-grained parallelism. On the other hand, the loop-carried dependence in reduction loops introduces RAW hazards in the execution pipeline. To avoid this penalty, and achieve instruction-level parallelism, we reorder and unroll a small degree of data parallel loops below the innermost reduction loop. The tuning space of CPU involves two dimensions, the degree of unrolling and parallelization. We enumerate these two parameters and profile the execution time to search for the best one. If the unrolling degree is too small, there will not be enough independent instructions to fill in the idle penalty cycles caused by RAW hazards. If it is too large, it will cause I-cache misses. Similarly, the number of threads can neither be too few or too many. If it is too few, the computing cores would have insufficient utilization and memory latency would not be hidden. Too many threads introduce context switching overhead. We rely on the tuning process to look for the best combination. Figure 6: Accumulating a p$\times$p “square window” avoids loop-carried data dependences, and reuses buffered submatrices. GPU Tuning: On GPU, coarse-grained parallelism is achieved by distributing the data parallel loops across the streaming multiprocessors. Similar to CPU, fine-grained parallelism is also achieved by reordering and unrolling a small degree of data parallel loops to avoid the pipeline penalty caused by loop- carried dependences. Moreover, on GPU, data reuse is explicitly managed by the software. Therefore, as it is shown in Figure 6, we adopt an outer-product style matrix multiply accumulation to reuse the buffered submatrices. Besides the generic optimization, we also developed optimization mechanisms specialized for DNN kernels. Among popular DNN models, there are many layers with relatively small width and height and deep channels. We apply _dimension fusion_ to layers with small width and height – these two dimensions are fused into one to save the redundant padding. In addition, we apply _split reduction_ to layers with deep channels. For a reduction loop with large trip count, we can split it and parallelize each split segment on threadIdx. After all the segments are done, we synchronize the threads and reduce the splitted segments in the shared memory. ## IV Implementation In this section, we will discuss technical details in our implementation. UNIT is implemented by extending Apache TVM [10], a full-stack deep learning compiler, with tensorized instruction support. We leverage TVM’s tensor DSL, tensor Op, tensor IR infrastructure, and the tuning infrastructure mechanisms [11, 23] to generate high performance kernels. In addition, implementing UNIT on top of TVM enables end-to-end model inference with other optimizations such as operator fusion, in addition to tensorization. ### IV-A Inspector The inspector pass is implemented by analyzing TVM’s ComputeOp data structure. This matches the expression tree of both the instruction and program and enumerates mappings between the loop variables. We enumerate the loops from the tensor’s innermost dimension to outermost dimension, and greedily return the first eligible one because of the better potential data locality for inner dimensions. The enumerated mapping provides us with the correspondence of loop variables between the instructions and the tensor operations. ### IV-B Rewriter These following steps will be performed by the rewriter: 1. 1. According to the loop correspondence analyzed by the inspector, we reorganize the loops to be tensorized by tiling these loops by the trip counts of the corresponding loops in the instruction, and reorder them to be the innermost loops. These loops will be annotated by a tensorize pragma to hint the instruction injection. 2. 2. Based on the strategies discussed in Section III-C, we reorganize the loops above not involved in instruction rewriting to tune the performance. 3. 3. We lower the manipulated loop nest to the tensor IR, and replace the loop body annotated with the tensorize pragma with the target instructions, as shown in Figure 5(c). Steps 1 and 2 are achieved by invoking TVM scheduling primitives on the tensor DSL level, and step 3 is a tensor IR transformation pass. Figure 7: The code sketch of CPU tuning. Next, we discuss the implementation of the tuning strategies discussed in the last section. CPU Tuning: The code sketch of tuned CPU code is shown in Figure 7. To implement the tuning we discussed in Section III-C, we enumerate two breaking points on the data parallel loop nest, which define how the loop levels are parallelized and unrolled. A breaking point is defined by a _loop level_ and _tiling factor_ , giving more flexibility to the division. Loops before the first breaking point, will be fused and parallelized. Loops between these two points will be executed in serialized order. Loops after the second breaking point will be reordered to the innermost and unrolled. GPU Tuning: As it is discussed in the last paragraph of Section III-C, both coarse-grained and fine-grained parallelism optimizations are applied on data- parallel loops, so there is a tradeoff between them: data reuse is increased by increasing the unrolling degree (each buffered submatrix is reused p times), but the coarse-grained parallelism is decreased. Also, a large unrolling degree may overwhelm the register resources. Therefore, the key to generic optimization is to choose a proper unrolling degree. On the other hand, greedily applying each specialized optimization does not always improve the performance. Though dimension fusion may save the memory traffic, it also introduces software overhead on data rearrangement. Similarly, though splitting the reduction loop introduces more parallelism, it also introduces thread synchronization overhead and register pressure. We enumerate each parameter, including the degree of reduction parallelization and whether to fuse the width and height dimensions, and then apply these transformations to the program and profile the performance to determine which transformation leads to the best performance. Figure 8: Quantized network inference (bs=1) accelerated by Intel VNNI. Figure 9: Mixed precision network inference (bs=1) accelerated by Tensor Core. ## V Methodology ### V-A Target Hardware Platforms We assess UNIT on three hardware platforms: Intel x86 CPU: We use Amazon EC2 C5.12xlarge instance as our x86 platform with 24-core Intel Xeon Platinum 8275CL CPU @3.00GHz (codename: Cascade Lake) and 96GB memory. ARM CPU: We use Amazon EC2 M6g.8xlarge instance as our ARM platform with AWS Graviton2 CPU, which features 32-core ARM Cortex-A72 CPU @2.30GHz and 128GB memory. Nvidia GPU: We use Amazon EC2 P3.2xlarge instance as our GPU platform with Nvidia Tesla V100 SXM2 GPU that has 16GB host memory. ### V-B Software Frameworks Code Generation: All programs implemented in Apache TVM are emitted to LLVM IR for code generation. We choose LLVM-10 as our backend, and to be compatible, we use CUDA-10.0 as the NVPTX linker and runtime. Baseline: We use vendor-provided libraries for baseline performance of operators whenever possible. Specifically, Intel oneDNN v1.6.1 and Nvidia cuDNN 7.6.5 are used as our CPU and GPU baselines, respectively. For end-to- end model inference, we looked for the best available solutions with those libraries, which was MXNet integrated with oneDNN for CPU and TVM integrated with cuDNN for GPU. Another set of baselines is the manually written implementation. To this end, we use the existing TVM solutions for Intel and ARM CPUs, which involve heavy engineering effort to carefully write intrinsics to use Intel VNNI and ARM DOT instructions. We did not find a manually written Tensor Core implementation that covers our evaluated workloads. ### V-C Workloads DNN Models: All DNN models are from the MXNet Model Zoo and converted to TVM’s graph IR, Relay [32], for quantization [19], layout transformation, and data padding. All these models adopt NCHW[x]c data layout [23] for the data and KCRS[y]k[x]c for the kernel. Here N denotes the batch size, C denotes the input channels, H and W are the width and height of the input image, and [x]c denotes that the original C is split by x. Similarly, K denotes the number of output channels, R and S are the height and width of the kernel, and [y]k denotes the original dimension K is split by y. [x] equals to the number of lanes of the instruction output, and [y] equals to the width of reduction. In the evaluation, we target the N=1 cases, because it is hard to optimize but critical for inference use cases. Comparing with batched cases where N>1, we cannot reuse the kernel tensor across samples, or exploit the parallelism brought by the data-parallel batching dimension. ## VI Evaluation Our evaluation of UNIT attempts to answer these questions: 1. 1. What is the performance of the end-to-end deep learning model inference powered by _tensorized_ instructions? 2. 2. How does each optimization technique that UNIT uses impact the performance? 3. 3. Can UNIT be extended to support new hardware platforms and tensor operations? ### VI-A End-to-End Performance In this subsection, we show the UNIT end-to-end effectiveness on Intel x86 and Nvidia GPU processors for tensorizing mixed precision instructions. For Intel x86 experiments, we use MXNet integrated with Intel oneDNN (referred to as MXNet-oneDNN) as the baseline. Another comparison of ours is TVM with manually written schedules using Intel’s VNNI instruction. The findings of this experiment are shown in Figure 9. We observe that UNIT achieves significant speedup compared to MXNet-oneDNN. Note that Intel oneDNN has access to manually written schedules that have been aggressively optimized and tuned by domain experts. We also observe that TVM overall achieves better performance than MXNet-oneDNN, but has suboptimal performance on resnet50 and resnet50b, which were heavily tuned by oneDNN engineers. On the other hand, UNIT outperforms both baselines, by 1.3$\times$ over MXNet-oneDNN and by 1.18$\times$ over TVM. Next, we test the efficacy of UNIT on utilizing Nvidia Tensor Core instructions for Nvidia GPUs. For the baseline, we integrate TVM with cuDNN, which has access to manually written aggressively tuned Tensor Core schedules. The findings of this experiment are shown in Figure 9. We observe that UNIT consistently achieves better performance than cuDNN with a mean speedup of 1.75$\times$ and up to 2.2$\times$. ### VI-B Optimization Implications In this subsection, we focus on the convolution operators of the DNN models to perform an in-depth analysis of the impact of different optimization techniques used by UNIT’s Rewriter. This is essentially an ablation study, showing how important different parts of UNIT are. There are 148 different convolution workloads (i.e., convolution with different feature map sizes, kernel sizes, strides, etc.) in the models, out of which we choose 16 representative convolution layers. These kernels cover diverse input shapes and strides. Other workloads behave similarly in the ablation study. We summarize the characteristics, namely, convolution attributes, like shapes, strides, etc., of the selected workloads in Table I. Intel x86 servers: As we discussed in Section III-C, we have two breaking points in CPU scheduling. The loop nests before the first breaking point are parallelized and the loop nests after the second breaking point are unrolled, while the ones in between the breaking point are executed serially. As loop nests can either be parallelized or unrolled (remaining one is serialized), we have a search space represented by the tuning pairs. Rewriter tunes this search space to generate a high-performance kernel. In this experiment, we incrementally measure the performance improvements brought by parallelizing, unrolling and tuning. The findings of this experiment are shown in Figure 11, normalizing the speedup to Intel oneDNN execution latency. First we fuse outer loop nests such that the loop bound of the fused loop nest is $<$ 3000, and measure the latency of the resulting kernel (shown by _Parallel_). Then, we take the remaining loop nests, and tile and unroll them such the unrolling factor is $<$ 8, and measure this performance (shown by _+Unroll_). Finally, instead of setting the limits as 3000 and 8, we tune the search space and measure performance (shown by _+Tune_), getting the final latency UNIT achieves. We observe that Parallel and Unroll together is responsible for most of the speedup. The additional speedup introduced by Tuning is quite small. It turns out that more than half of the kernels get the optimal performance on the first tuning pair (i.e. 3000 and 8), and more than 95% of the kernels get the optimal performance within the first 8 tuning pairs. CPU does poorly on workloads #1 and #4, because their output shapes (OH/OW) can neither be perfectly tiled nor fully unrolled. Inherited from TVM, loop residues are handled the by guarding it with a likely clause, which results in an if-branch that harms the performance. Nvidia GPU servers: As discussed in Section III-C, we employ three optimizations on GPU: generic coarse- and fine-grained parallelism, fusing width and height to save memory bandwidth, and parallelizing the reduction dimension. In this subsection, we study the impact of these optimizations on the performance. We show the findings in Figure 11, normalizing the speedup to Nvidia cuDNN. According to our evaluation, any unrolling degree (p in Figure 6) larger than 2 may overwhelm the registers, so we use p=2 to apply the generic optimization. The generic optimization already beat cuDNN in most cases (shown by _Generic_). Then, depending on the height and width values, Rewriter fuses the height and width dimensions to save memory bandwidth (shown by _+FuseDim_). Then, we split the reduction dimension K by 64 and measure the performance (_+SplitK_). Finally, we let Rewriter to choose the sizes for these 3 optimizations and measure performance (shown by _+Tune_). We observe that SplitK leads to the maximal speedup, as it leads to significant parallelism and keeps the Tensor Cores busy. More than 70% of the kernels can get high performance by employing fusion and parallelizing the reduction dimension. Similar to CPUs, the additional speedup by tuning is small. UNIT cannot outperform cuDNN on #1 and #15, because the strided data accesses lead to less data locality. However, since these adversarial cases (both CPU and GPU) only occupy a very small portion among all these models, we can still outperform vendor-provided libraries because of the generality of our optimization. Figure 10: The performance impact of the code space exploration. Figure 11: The performance impact of the code space exploration. TABLE I: Characteristics of the selected convolution layers. \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 \| 13 \| 14 \| 15 \| 16 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- C \| 288 \| 160 \| 1056 \| 80 \| 128 \| 192 \| 256 \| 1024 \| 128 \| 576 \| 96 \| 1024 \| 576 \| 64 \| 64 \| 608 IHW \| 35 \| 9 \| 7 \| 73 \| 16 \| 16 \| 16 \| 14 \| 16 \| 14 \| 16 \| 14 \| 14 \| 29 \| 56 \| 14 K \| 384 \| 224 \| 192 \| 192 \| 128 \| 192 \| 256 \| 512 \| 160 \| 192 \| 128 \| 256 \| 128 \| 96 \| 128 \| 192 R=S \| 3 \| 3 \| 1 \| 3 \| 3 \| 3 \| 3 \| 1 \| 3 \| 1 \| 3 \| 1 \| 1 \| 3 \| 1 \| 1 Stride \| 2 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 2 \| 1 OHW \| 17 \| 7 \| 7 \| 71 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 27 \| 28 \| 14 ### VI-C Extensibility We evaluate the extensibility of UNIT in two aspects: to new hardware platforms and to new deep learning tensor operations. We observe that by just representing the semantics of the new tensorized instruction in tensor DSL, UNIT can easily extend to new tensorized instructions and tensor operations. New Hardware Platforms: To demonstrate the capability of extending to new hardware platforms, we apply UNIT to an ARM CPU supporting the ARM DOT instruction. To the best of our knowledge, there is a lack of a deep learning framework with well-integrated ARM backend library support. In the absence of a framework baseline, we choose TVM compiling to ARM Neon assembly as the baseline (shown by TVM-NEON). Additionally, we find that TVM has manually- written schedules using ARM DOT instructions, which forms our second comparison baseline (shown by TVM-Manual). Note that in contrast to UNIT’s automatic approach, this is a manually written schedule requiring intense engineering efforts. Finally, we represent the semantics of ARM DOT instruction in UNIT’s tensor DSL and use UNIT to compile the models. The findings of this experiment are shown in Figure 12, showing normalized speedup compared to the TVM-Neon baseline. The results show that UNIT consistently outperforms both TVM-NEON and TVM-Manual, proving UNIT’s effectiveness in extending to new hardware platforms. 3D Convolution: We test UNIT on 3D convolution operation for mapping Intel VNNI tensorized instructions. Note that this does not require any changes from UNIT perspective; we are just giving a new input (tensor-level IR for conv3d) to UNIT. To evaluate this extensibility, we take all the 2D convolutions from Resnet18 and manually convert them to 3D convolutions. We then apply UNIT on these kernels and show the speedup compared to oneDNN baseline in Figure 13. We observe that UNIT easily extends to 3D convolution, as it has comparable performance for many convolution kernels, with an average of 1.2$\times$ speedup. Figure 12: The performance of ARM on model inference. Figure 13: The performance of each layer on res18-3d. ## VII Related Work Compilation support for hardware intrinsics: There exists a large body of literature on compilation support for various hardware intrinsics [33, 20, 27, 29, 16, 22, 28, 15, 36, 35]. Existing production compilers such as GCC and LLVM implement auto-vectorization to leverage SIMD intrinsics. Prior works such as [20, 33] propose various approaches to further improve the performance of the auto-vectorizer. These approaches cannot be extended to support tensor computation intrinsics which introduce “horizontal computation” within each lane. TVM [10] implements an extensible interface to support new hardware intrinsics that are not limited to SIMD instructions. However, programmers need to transform the program to match the behavior of the intrinsics and declare the lowering rule for the intrinsics prior to compilation. TVM will match the computation and replace it with the code snippets that call the target hardware intrinsics. Compared to TVM, UNIT performs the code detection and transformation automatically. This achieves higher flexibility and productivity. There are some prior works that, similar to UNIT, also perform program transformation and code generation automatically for tensor computation [36, 35]. However, these are limited to one platform or certain intrinsics and hence are not as flexible as UNIT. Decoupled Computation and Data Access: The analysis pass of UNIT is inspired by the decoupled-access execute (DAE) architectures [21, 30, 26, 41, 13, 42, 40]. Computation and data access are decoupled and specialized separately. The computation is offloaded onto a programmable data path and data access is encoded in hardware intrinsics and executed on specialized address generation unit (AGU). UNIT adopts a reversed approach, it matches computation on a fixed data path, and analyzes data access fed to the data path. Polyhedral model: Many prior works have built program analysis and transformation frameworks based on the polyhedral model for tensor programs [20, 36, 35, 15, 37, 12, 25, 38]. Loop Tactics [9] is one representative work which matches the pre-defined computation patterns in the polyhedral IR and transforms the matched patterns to optimized programs. UNIT distinguishes itself from Loop Tactics in: 1) Compared with the schedule tree [39] in the polyhedral model, the tensor DSL provides more information such as loop reduction properties and operand types; 2) UNIT provides an end-to-end solution including auto-tuning to obtain the optimal performance, whereas Loop Tactics requires the optimized schedules to be provided manually. Deep learning frameworks: UNIT is complementary to the existing deep learning frameworks. Existing frameworks such as Tensorflow [8], PyTorch [7], and MXNet [3] rely on vendor-crafted libraries to support the new tensor intrinsics. TVM [10] requires code re-writing at the user side. UNIT is able to handle new operators which might not be covered by the vendor libraries and spare the user from having to perform manual re-writing. We have demonstrated the effectiveness of the methodology of UNIT based on TVM. Similar technique can be applied to other frameworks to further boost their performance. ## VIII Conclusion Deep learning has prompted hardware vendors to add specialized tensorized instructions for dense tensor operations. These instructions perform “horizontal reduction” accumulate elementwise computation. While promising, introducing this new idiom complicates its general purpose applicability, as one has to rely on hand-written kernels to gain high performance brought by these instructions. In this paper, we introduce UNIT, a unified compilation pipeline, that represents the tensorized instructions from different hardware platforms using the same IR, then automatically detects the applicability of the tensorized instructions in a given tensor operation, transforms the loop nest to enable easy mapping of the tensorized instruction, and finally rewrites the loop body with the tensorized instructions. UNIT enables automatic tensorized instruction compilation over a variety of hardware platforms like Intel/ARM CPUs and Nvidia GPUs. Our analysis shows that UNIT achieves 1.3$\times$ speedup over oneDNN (VNNI instruction), 1.75$\times$ over cuDNN (Tensor Core instruction), and 1.13$\times$ over the manually written ARM intrinsics in TVM (DOT instruction). ## Acknowledgements This work is supported by NSF grant CCF-1751400 and Mu Li’s team at Amazon Web Services. ## Appendix A Artifact Appendix ### A-A Abstract This guide describes how to set up _UNIT_ compilation infrastructure and run the workloads we discussed in Section VI. This guide provides instructions to: * • Set up the experiment environment for _UNIT_ through Docker. * • Run end-to-end inference model shown in Figure 9, 9, and 12. * • Run the experiments to demonstrate the effects of our tuning strategies shown in Figure 11, and 11. * • Run the 3D-convolution results shown in Figure 13. Our experiments are conducted on Amazon EC2 c5.12xlarge for Intel VNNI, p3.2xlarge for Nvidia TensorCore, and m6g.8xlarge for ARM VDOT. To download and install our infrastructure, approximately 32GB of disk is required. We provide Dockerfile to set up the environment, and scripts to automatically run the experiments and plot the figures. ### A-B Artifact Checklist * • Program: As it is demonstatrated in Section V, we use nine typical DNN models, including ResNet, ImageNet, and MobileNet. * • Compilation: We need specific versions of TVM to run our experiments and baselines. They are included in the zip release. * • Data set: The test data is included in our zip release. * • Runtime environment: We run our artifact all on Ubuntu 18.04. For GPU, Nvidia GPU driver and additional runtime for Docker should be installed. * • Hardware: We run our experiments on AWS EC2 instances — c5.12xlarge for Intel VNNI, p3.2xlarge for Nvidia TensorCore, and m6g.8xlarge for ARM DOT. * • Execution: We provide scripts to run our experiments discussed in Section VI. It takes 2 hours to compile the models in Figure 9, half an hour to compile the models in Figure 9, and 1.4 hours to compile the models in Figure 12. It takes half an hour to run the experiments in Figure 11, and 11. * • Output: Our scripts both run the experiments and plot the figures in PDF files. * • Experiments: The results reported in our paper are generated by a physical machine, but in this artifact evaluation they all run on a virtual machine in Docker. Performance fluctuation may happen because of the overhead of virtualization. ### A-C Description #### A-C1 How Delivered Download our Dockerfile, scripts, and model test data at https://doi.org/10.5281/zenodo.4420522. #### A-C2 Hardware Dependencies * • AVX512_VNNI: This is available on Intel CPUs with Cascadelake architecture. In this work, we use AWS EC2 c5.12xlarge. The CPU model is Intel(R) Xeon(R) Platinum 8275CL CPU @3.00GHz. The rate is $2.04/hour, and it takes approximately one hour to set up the environment and 5 hours to run all the related experiments. * • TensorCore: This is avaiable on Nvidia GPUs with TensorCore extension. In this work, we use AWS EC2 p3.2xlarge. The GPU model is Tesla V100. Please install the GPU driver. The rate is $3.06/hour, and it takes approximately 1 hour to set up the environment, and another one hour run all the related experiments. * • ARM VDOT: This is available on ARM CPU v8.2 with dotprod extension. In this work, we use AWS EC2 m6g.8xlarge. The CPU model is Amazon Graviton 2. The rate is $1.232/hour, and it takes 1 hour to set up the environment and run the experiments. #### A-C3 Software Dependencies All our software dependences are installed automatically in Docker. Refer to this link for Docker installation. When setting up the last step of the the package repository, do choose the proper tab for your CPU platform (x86 or ARM). Refer to this to install Docker that runs Nvidia GPU. Nvidia Docker requires GPU driver installed, use this command to install: ⬇ $ sudo apt-get install nvidia-driver-455 ### A-D Installation Unzip the downloaded file, and there are three sub-zips — tensorcore.zip, vnni.zip, and arm.zip to evaluate the three platform we discussed in this paper. ### A-E Experiment Workflow #### A-E1 GPU We run the TensorCore experiment on an AWS EC2 p3.2xlarge instance. * • After building the docker image, an image hash value will be generated in the console log: ⬇ $ unzip tensorcore.zip && cd tensorcore $ sudo docker build . # 20 mins to build $ sudo docker run -tid \--runtime=nvidia <image> $ sudo docker attach <container> * • After entering the container, the experiment scripts are all in $HOME directory: ⬇ $ cd $HOME * • To replicate experiments run in Figure 9, and 11: ⬇ $ bash run_e2e.sh # Fig.9: e2e.pdf $ bash run_ablation.sh # Fig.11: gpu-dse.pdf * • It takes half an our to run these two scripts. Both the experiments and data plotting are done in these two scripts. Use these commands to take the generated PDF out of the container and look at them: ⬇ $ <ctrl-p><ctrl-q> # Temporarily detach $ sudo docker cp <container>:/root/e2e.pdf gpu-e2e.pdf $ sudo docker cp <container>:/root/gpu-dse.pdf . #### A-E2 CPU We run the Intel VNNI experiment on an AWS EC2 c5.12xlarge instance. It is also used to cross-compile ARM target. * • After building the docker image, an image hash value will be generated in the console log: ⬇ $ unzip vnni.zip && cd vnni $ sudo docker build . $ sudo docker run -tid <image> $ sudo docker attach <container> * • After entering the container, the experiment scripts are all in $HOME directory: ⬇ $ cd $HOME * • To replicate experiments run in Figure 9, 11, and 13: ⬇ $ bash run_e2e.sh # Fig.8: e2e.pdf $ bash run_ablation.sh # Fig.10: cpu-dse.pdf $ bash run_3d.sh # Fig.13: conv3d.pdf * • It takes about 2.5 hours to run these experiments, and you can use the following commands to take out these plotted figures and look at them: ⬇ $ <ctrl-p><ctrl-q> # Temporarily detach $ sudo docker cp <container>:/root/e2e.pdf . $ mv e2e.pdf cpu-e2e.pdf # Avoid conflict $ sudo docker cp <container>:/root/gpu-dse.pdf . $ sudo docker cp <container>:/root/conv3d.pdf . * • Use the following script to run ARM target compilation: ⬇ $ bash run_arm.sh It takes about two hours to get all models compiled on ARM. The compiled models will be in $HOME/arm-base and $HOME/arm-unit. * • Copy the compiled model to the ARM machine: ⬇ $ scp -i key.pem -r arm-unit <arm-machine>:~ $ scp -i key.pem -r arm-base <arm-machine>:~ $ ssh -i key.pem <arm-machine> * • Set up the ARM environment and run the experiments on ARM machine: ⬇ $ unzip arm.zip && cd arm $ mv ../arm-unit . $ mv ../arm-base . $ sudo docker build . $ sudo docker run -tid <image> $ sudo docker attach <container> $ cd $HOME && bash run_e2e.sh <ctrl-p> <ctrl-q> $ sudo docker cp \ <container>:/root/baseline.result . $ sudo docker cp \ <container>:/root/tensorize.result . * • Bring these two .result files to a x86 machine, and plot the graph: ⬇ $ python plot_e‘2e.py baseline.result tensorize.result # Fig. 13 $ mv e2e.pdf arm-e2e.pdf ### A-F Evaluation and Expected Result Finally, we have these PDF files: * • Figure 9, 9, and 12 should be compared against cpu-e2e.pdf, gpu-e2e.pdf, and arm-e2e.pdf. * – The ARM results reported in this paper were generated by an old version of TVM. 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# Fire Threat Detection From Videos with Q-Rough Sets Debarati B. Chakraborty<EMAIL_ADDRESS>Vinay Detani <EMAIL_ADDRESS>Shah Parshv Jigneshkumar<EMAIL_ADDRESS>Dept. of Computer Science and Engineering, Indian Institute of Technology, Jodhpur, India ###### Abstract This article defines new methods for unsupervised fire region segmentation and fire threat detection from video stream. Fire in control serves a number of purposes to human civilization, but it could simultaneously be a threat once its spread becomes uncontrolled. There exists many methods on fire region segmentation and fire non-fire classification. But the approaches to determine the threat associated with fire is relatively scare, and no such unsupervised method has been formulated yet. Here we focus on developing an unsupervised method with which the threat of fire can be quantified and accordingly generate an alarm in automated surveillance systems in indoor as well as in outdoors. Fire region segmentation without any manual intervention/ labelled data set is a major challenge while formulating such a method. Here we have used rough approximations to approximate the fire region, and to manage the incompleteness of the knowledge base, due to absence of any prior information. Utility maximization of Q-learning has been used to minimize ambiguities in the rough approximations. The new set approximation method, thus developed here, is named as Q-rough set. It is used for fire region segmentation from video frames. The threat index of fire flame over the input video stream has been defined in sync with the relative growth in the fire segments on the recent frames. All theories and indices defined here have been experimentally validated with different types of fire videos, through demonstrations and comparisons, as superior to the state of the art. ###### keywords: Fire segmentation, video processing , rough sets , Q learning , fire threat detection , granular computing ## 1 Introduction Fire is both a threat and a necessity of mankind. We light fires to fulfill our necessity (e.g., for cooking, generating heat), simultaneously it could be a threat if it becomes uncontrolled. We know that there are some objects that are easily flammable (e.g., cloths, plastics) and some that burns out gradually (e.g., candle, wood or logs in the fireplace etc.). We burn the latter kind of elements to meet our need in a controlled environment, whereas fires concerning the former group of elements could be dangerous. Fire spreads quite fast in the flammable objects, and from one several other objects catch the fire. Therefore, measuring the spread of the fire could be a solution in automated fire detection systems to find out the possible threat of the visible fire. Here we tried to come up with a solution so that the fire threat can be detected automatically only with a surveillance camera, and no other sensors will be required. Besides, we tried to make the method fit for both indoor and outdoor surveillance. Proper classification of fire region(s) from the non-fire ones and automated segmentation of the fire region is a very important step while quantifying the threat of the fire. The aim of this work could be summarized as follows: i) to develop an unsupervised method for fire pixel classification, ii) to identify fire regions from different varieties of videos, and iii) to determine the threat associated with the fire flame over the input video stream. Here in this article we have formulated a new method of fire region segmentation in videos using hybridization of rough sets and Q-learning. Besides, we have also defined a new measure, namely, fire threat index, to quantify the threat associated with a fire from video streams. Before discussing the other methods on vision-based fire detection, here in this section we shall roughly discuss the key concepts of rough set theory and Q-learning relevant to this article. Theory of rough sets, as explained by Pawlak [1], has become a popular mathematical framework for granular computing. The focus of the theory is on the ambiguity caused by limited discernibility of objects in the domain of discourse. Its key concepts are those of object ’indiscernibility’ and ’set approximation’. Two major characteristics of the theory that have drawn the attention of applied researchers are uncertainty handling (using lower and upper approximations) and granular computing (using information granules). Theory of rough sets has been proven to be successful in many areas, like, feature extraction from video streams with information flow [2], on-line multi-label feature selection from streaming data [3], and information entropy based fast feature selection for big data analytics [4]. We have aimed to develop an unsupervised fire detection method here in this article. Since we have limited access to the knowledge-base while identifying fire in a video frame, we have approximated the fire region with lower and upper approximations of rough sets. Q-learning is a reinforcement learning algorithm [5]. The primary difference between Q-learning and other reinforcement learning techniques is that it is model-free [6], i.e., here the agent does not need any state transition model either for learning or for action selection. Q-learning prefers the best Q-value from the state reached in the observed transition [7]. The actual policy, being followed by the agent, is not taken into account in Q-learning, that is why, it is also known as off-policy control. In Q-learning, the agent learns the action utility function (or Q-function) to estimate the expected utility of choosing a certain action when it is in a particular state. Here in this work we are not considering any prior information on the fire region or possible spread of the fire pixels. All information is to be determined by the unsupervised process itself. Therefore, the proposed solution has an initial rough approximation of fire regions, and then employs a Q-agent to minimize the ambiguities that is present is the rough approximation, with utility maximization. This is the broad idea underlying Q-rough set with utility-based agent. States, in which the Q-agents are supposed to be in, are different granules (clump of data points [8]) and actions are insertion or deletion of those granules from fire regions. The same Q-agents are then employed over the video stream to extract out the fire regions in each frame. The fire threat index is then formulated using the segmented fire regions over the video stream. The novelty of the technique described in the article can be summarized as: i) definition and application of the Q-rough set (that use unsupervised set estimations) to minimize ambiguities in rough approximations, ii) segmentation of fire regions from a video frame with Q-rough set, iii) employing Q-agents over a sequence of frames to extract fire regions in each frame, and iv) formulation of a fire threat index to quantify the threat of fire flame in a video stream. All the theoretical formulations have been experimentally verified and proven to be effective over the state of the art methods. The rest of the article is organised as follows. A few state of the art methods for fire detection are discussed in Section 2. The underlying steps of the proposed work is described in Section 3. Q-rough set is defined in Section 4 alongwith brief descriptions of rough sets and Q-learning. The method of fire region segmentation with Q-rough set is developed in Section 5. The quantification of the threat of fire flame is carried out in Section6. The qualitative and quantitative experimental results of the proposed methods are given in Section 7 alongwith suitable comparative studies. The overall conclusions of this article are drawn in Section 8. ## 2 Related Work The problem of fire detection from images is being addressed over a decade [9]. Here we are going to discuss a few bench-mark methods. The problem was first addressed by Chen _et al._ [10] where fire was detected with RGB values and rule-base. Ferneds et al. then developed a method for forest fire detection by classifying lidar signals with neural network committee machine in [11]. Flame detection by modelling the RGB distributions of fire -pixels with mixture of Gaussian (MoG) model [12], and with motion-information and hidden Markov model (HMM) [13] were then done by Toreyin _et al._ Fire pixel identification in videos was then carried out by Celik [14] where CIE Lab color space was used to identify fire pixels. The method of forest fire detection by incorporating static and dynamic features in videos in HSV color space was proposed by Zhao _et al._ in [15]. Chino _et al._ then developed a way of fire detection from still images by combining color and texture features in [16]. The method of fire detection with spatio-temporal consistency energy of each candidate fire region was estimated by Dimitropoulos _et al._ [17] with prior knowledge about the possible existence of fire in neighboring blocks from the current and previous video frames, and an SVM-based classification of fire-non fire regions was also executed there. Recently a couple of fire detection method has been developed with deep learning techniques [18]. Muhammad _et al._ [19] came up with a solution by incorporationg deep features of CNNs in fire-detection and high-priority cameras based on cognitive radio networks were developed. Kim and Lee [20] developed a method of detecting fire with faster region grow method using deep CNN (convolutional neural network). Cai et al. [21] formulated an improved deep CNN that uses the global average pooling layer instead of the full connected layer to fuse the acquired depth features and detect fire. All the methods, we have discussed so far, either requires some initial information about fire pattern or needs initial manual labelling for training. Most of the methods are focused on some specific applications, such as, forest fire, indoor fire, outdoor fire, etc. The method that we developed here is completely unsupervised and does not need any prior information/ manual intervention. Besides, it is very effective in classifying fire pixels in any types of video frame, that is it’s a general method for fire segmentation, which could be applicable to anywhere for fire detection. The method of quantification of fire threat is also new to literature. We are going to describe the basic steps of the method in the following section. ## 3 Proposed Work Our work can be subdivided into two parts, viz. i) detection of fire from videos and ii) determination of spreading of fire or quantifying the threat associated with the fire. The step-wise formulations of the two methods are shown in Figs. 1(a) and 1(b). Figure 1: Block diagram of (a) fire region segmentation and (b) threat detection methods In Fig. 1(a) proposed fire-flame detection method is shown, that will accumulate information from different color space and collate the information judiciously to classify the fire pixels correctly from the video frames. Here Q-learning is rough because the state-action policy is determined with the lower approximated regions (obvious fire region here), and the ambiguities between the two approximations get minimized by utility maximization of the Q-agents. The same Q-agents are then employed over the rest of the video sequence to have faster decision making over known states. This part is described in details in Section 5. In the second part of the work we aim to quantify the threat associated with the fire flame. It is shown in Fig. 1(b). Here the relative growth of fire over recent past of the video stream is considered. The average region of the fire flame throughout the video stream and over a few recent frames of video stream are being computed in this part. The fire threat index ($\mathcal{T_{F}}$) is then quantified with these information. This part is explained in Section 6. ## 4 Formulation of Q-Rough Sets This section discusses the formulation of Q-rough sets. Prior to that formulation, the basics of rough set theory and Q-learning have been briefly described. ### 4.1 Rough Sets Let there be an information system $S=(\mathcal{U},A)$, where $\mathcal{U}$ is the universe and $A$ is the set of attributes. For any set $B\subseteq A$, there is an equivalance relation $IND(B)$ such that $IND(B)=\\{(x,y)\in\mathcal{U}^{2}\|\forall p\in B,p(x)=p(y)\\}$, where $p(x)$ function returns the value of the attribute $p$ for data point $x$. The relation $IND(B)$ is called B-indiscernibility relation and any two points $(x,y)\in IND(B)$, i.e., satisfying the B-indiscernibility relation, indicate that $x$ and $y$ can not be distiguishable using the attribute set $B$. Let the equivalence class of B-indiscernibility relation be denoted by $[x]_{B}$, and $\mathcal{U}\|B$ denotes all such equivalence classes. Here $[x]_{B}$ is called a ’granule’ around the data point $x$, created by B-indiscernibility relation. (As stated before, a granule is a clump of object which can not be discriminated with a given attribute set.) Let us denote this granulated information system with $S_{B}=(\mathcal{U},A,[x]_{B})$. Let $X$ be a set in the universe $\mathcal{U}$ ($X\subseteq\mathcal{U}$) to be approximated based on the equivalence classes $[x]_{B}$ (i.e., granules) defined over $B$. Then, $X$ can be approximated in terms of granules from inner and outer sides as _B-lower approximation_ $\underline{B}X$ and _B-upper approximation_ $\overline{B}X$ respectively. They are defined as follows: $\underline{B}X=\\{x\in\mathcal{U}:[x]_{B}\subseteq X\\}$ (1) $\overline{B}X=\\{x\in\mathcal{U}:[x]_{B}\cap X\neq\emptyset\\}$ (2) $\underline{B}X$ represents the granules definitely belonging to $X$, while $\overline{B}X$ means granules definitely and possibly belonging to $X$. That means all the elements in $\underline{B}X$ can be certainly classified as member of $X$ on the basis of the knowledge in $B$, while some objects in $\overline{B}X$ can only be classified as possible members of $X$ on the basis of $B$. ### 4.2 Q-Learning Let the state and action in a given environment be denoted as $s$ and $a$ respectively, and the $Q$-value of doing action $a$ in state $s$ be denoted as $Q(s,a)$. The relation between the direct utility of the state ($U(s)$) and Q-value is as follows. $U(s)=\max_{a}Q(s,a)$ (3) If the Q-values are computed correctly, the following equation (Eqn. 4) will then only reach to an equilibrium. That is, $LHS=RHS$, iff Q-value is correct in Eqn. (4). $Q(s,a)=R(s)+\gamma\sum_{s^{\prime}}P(s^{\prime}\|s,a)\max_{a^{\prime}}Q(s^{\prime},a^{\prime})$ (4) In Eqn. 4 $R(s)$ represents the reward value associated with the state $s$; $\gamma$ is the discount factor that determines the importance of future rewards; $P(s^{\prime}\|s,a)$ determines the probability of reaching to the state $s^{\prime}$ from state $s$ given action $a$, it is determined from the state-transition model; and $Q(s^{\prime},a^{\prime})$ is the $Q$-value of future action $a^{\prime}$ in future state $s^{\prime}$. As we can see from Eqn. 4 an estimated state transition model is required here, therefore, it is not completely model-free. The temporal difference approach or TD Q-learning is the actually model-free one, i.e., state-transition model is not required here. The Q-value updation in TD Q-learning is carried out as follows. $Q(s,a)\leftarrow Q(s,a)+\alpha(R(s)+\gamma(Q(s^{\prime},a^{\prime})-Q(s,a))$ (5) $\alpha$ is the learning rate in Eqn. 5. ### 4.3 Q-Rough Set: Definition Here in this article we have focused on unsupervised estimation of a set. The method is envisioned to be completely unsupervised and only some basic set of features of the set can be provided apriori. Let us assume the set to be estimated be denoted as $X:X\in\mathcal{U}^{2}$. The given set of features be $B:B\subseteq A$, where $A$ is the complete set of features. The lower and upper approximations of the set: $\underline{B}X$ and $\overline{B}X$ can now be estimated according to Eqns. 1 and 2 respectively. However, the exact region of the set cannot be estimated through these equations. To extract the exact set and minimize the vagueness in the approximation here we employ a Q-learning agent. Here the set $\underline{B}X$ is used to determine the value function, policy and the state-transition model in Q-learning. The states ($s$) are treated to be the clump of similar data points (granules $[x]_{B}$) present in the set $\overline{B}X$, the set of actions, is of cardinality equals to 2 and it involves: $a=\\{$updation of $\underline{B}X:\rightarrow\underline{B}X\cup[x]_{B}$ and moving to the next connected boundary granule to $[x]_{B}$, keeping $\underline{B}X$ the same and moving to the next boundary granule connected to $\underline{B}_{X}$$\\}$. The reward function $R(s)$ is chosen based on the similarities between $[x]_{B}\leavevmode\nobreak\ s$ and $\underline{B}X$ over $n$-dimensional feature space. The similarities ($D$) between the state ($s\equiv[x]_{B}$) and model ($M\equiv\underline{B}X$) and reward for that granule ($R([x]_{B})$) are now defined as per the following equations. $\displaystyle D(M,s)=\|\|M-s\|\|_{n}$ (6) $\displaystyle R([x]_{B})=1-2D(M,s).$ (7) In Eqn (6) $R([x]_{B})$ is defined in such that the value of the reward should be within the range of $[-1,1]$. Maximum reward (+1) will be generated upon finding total similarity and minimum reward (-1) will be generated upon finding no similarities. While determining the Q-value of a boundary granule ($s\equiv[x]_{B}$), it can be observed that, here the state-action model will lead to two different future states ($s^{\prime}\equiv[x]_{B}^{\prime}$) based on two different actions. But the there is a single possible state for a particular action. Let $[x]_{B1}^{\prime}$ and $[x]_{B2}^{\prime}$ be the two possible future states that the agent can reach on taking the actions $a_{1}$ and $a_{2}$ respectively. Therefore we are going to get a simple binarized state transition model as follows. $\displaystyle P([x]_{B1}^{\prime}\|[x]_{B},a1)=1$ $\displaystyle P([x]_{B2}^{\prime}\|[x]_{B},a1)=0$ $\displaystyle P([x]_{B1}^{\prime}\|[x]_{B},a2)=0$ $\displaystyle P([x]_{B2}^{\prime}\|[x]_{B},a2)=1$ (8) Therefore, Eqn. (4) reduces to Eqn. (9) as follows. $Q([x]_{B},a)=R([x]_{B})+\gamma\max_{a^{\prime}}Q([x]_{B}^{\prime},a^{\prime})$ (9) The action, that is maximizing the $Q$-value will be selected. Upon completion of the same process over all the boundary granules, the final set $\underline{B}X$, that we are getting is named as Q-rough approximation of the set $X$. This is how the uncertainty present in the rough approximation of the set $X$ can be minimized, and exact is found out in the incomplete knowledge- base. ## 5 Q-Rough Set in Fire Detection Let the fire region $F$ be approximated in the input frame $f_{t}$, given a set of features ($B:B\in A$). We have used the set of rules as defined in [10], for R-G-B and Y-Cr-Cb color space to segment out initial fire regions from a video frame. Since our method is unsupervised, we are initially approximating the fire-region in a video frame with these rules. The lower approximation of fire region is done deploying the information of segmented regions in Y-Cr-Cb color space and the upper approximation is done considering the information of segmented region in RGB-color space on the top of the lower approximated region. The final approximation of the fire region in a frame is carried out with Q-rough set. A visual illustration of fire segmentation with Q-rough set is shown in Fig. 2. The process of set formation and approximation are shown in the following sections. Figure 2: Pictorial Representation of Fire Segmentation with Q-Rough Set ### 5.1 Formation of Granules Formation of proper granules plays an important role in approximation in decision making systems. Here while defining Q-rough set for fire segmentation we aim to granulate the image frame based on spatio-color similarities. This is how the separation will be closer to the natural ones. Therefore, we have used spatio-color granulation technique defined in [2]. That is, a granule around ($\aleph$) an point $p_{i}$ in the frame $f_{t}$ is formed according to Eqn. (10). $\displaystyle\aleph_{sp-clr}(p_{i})$ $\displaystyle=\cup p_{j}\in U$ (10) where $p_{i}\quad and\quad p_{j}$ are binary connected over the condition, $\|RGB(p_{j})-RGB(p_{i})\|<Thr$. The upper and lower approximations of the fire region ($F$) are carried out over these granules. ### 5.2 Lower Approximation of Fire-Region in a Frame The lower approximated region will consume the segmented region in Y-Cr-Cb color space. The rule-base mentioned in [15] for fire segmentation is used here in Y-Cr-Cb. As we know, in Y-Cr-Cb color space [22], ‘ Y ’ represents the luma component(very similar to the grayscale conversion of the original image ), whereas ‘ Cb ’ and ’Cr’ represent chroma components of blue-difference red- difference respectively. The rule-base that is followed for fire segmentation is in Y-Cr-Cb feature space is as follows. A pixel $p_{i}$, with Y-Cr-Cb values as $Y(p_{i})$, $Cr(p_{i})$ and $Cb(p_{i})$ will be treated as a fire pixel if it follows any of the following rules. * • Rule 1: $Y(p_{i})\geq Cb(p_{i})$ and $Cr(p_{i})\geq Cb(p_{i})$ * • Rule 2: $Y(x)>Y_{mean}$ and $Cr(p_{i})>Cr_{mean}$ and $Cb(p_{i})>Cb_{mean}$ * • Rule 3: $Cb(p_{i})\leq 120$ and $Cr(p_{i})>150$ $Y_{mean}$, $Cr_{mean}$, and $Cb_{mean}$ are the mean values of Y, Cr and Cb on that image frame. Let, $F_{YCrCb}$ be the fire-segmented region in the frame $f_{t}$. Then the lower approximation of the fire region ($\underline{B}F$) is defined according to Eqn. (11). This results in spatio- color granules that are a subset of $F_{YCrCb}$ are considered to be in the lower approximation of the set $F$. $\underline{B}F=\\{\aleph(p):\aleph(p)\in F_{YCrCb}\\}$ (11) ### 5.3 Upper Approximation of Fire Region in a Frame The upper approximated region will consume both the segmented outputs of Y-Cr- Cb feature space and RGB feature space. The rules that are defined in [10, 14, 15] for fire detection in RGB feature space as as follows. A pixel $p_{i}$ will be detected as fire pixel if it follows the following rules. * • Rule 1: $R(p_{i})\geq R_{mean}$ * • Rule 2: $R(p_{i})>G(p_{i})>B(p_{i})$ $R_{mean}$ is the mean value of R over the image frame. Let, $F_{RGB}$ be the segemnted region of fire in RGB feature space. The upper approximated fire region will be derived from $F_{RGB}\cup F_{YCrCb}$ region. The upper approximated fire region ($\overline{B}F$) is defined as according to Eqn. (12). $\overline{B}F=\\{\aleph(p):\aleph(p)\cap(F_{RGB}\cup F_{YCrCb})\neq\emptyset\\}$ (12) ### 5.4 Computation of Reward Function Proper computation of reward function is another major concern while defining Q-rough sets. As defined in Eqn. (6), the distance between the lower approximated region and a boundary granule is to be computed to determine the reward function. We know that the no. of data points in a $\underline{B}F$ and a boundary granule $\aleph(p)$ can never be the same. Therefore, instead of computing point-to-point distance between these two sets, we are computing the distance between their mean values in different feature space. Therefore, Eqn. (6) can be re-written here as follows. $\displaystyle D(\underline{B}F,\aleph(p))=\|\|mean(\underline{B}F)-mean(\aleph(p))\|\|_{RGBYCrCb}$ (13) $\displaystyle R(\aleph(p))=1-2D(\underline{B}F,\aleph(p)).$ (14) In Eqn. (13) normalized Euclidean Distance is considered as a distance metric. The Q-values will now be updated according to Eqn. (15). $Q(\aleph(p),a)=R(\aleph(p))+\gamma\max_{a^{\prime}}Q(\aleph(p)^{\prime},a^{\prime})$ (15) ### 5.5 Algorithm for Fire Detection with Q-Rough Set We are going to describe the proposed method step-wise for fire detection from a video frame $f_{t}$ with Q rough set. The theoretical details are presented in the preceding sections. The detailed methodology is summarized in the form of an algorithm in Algorithm 1. Algorithm 1 Fire Detection from A Frame with Q-Rough Set INPUT: $f_{t}$ OUTPUT: $f_{t}$ with segmented fire region $F$ INITIALIZE: $\underline{B}F=\overline{B}F$ $\Leftarrow\emptyset$ 1: Granulate the frame $f_{t}$ as described in Section 5.1 2: Segment out $F_{YCrCb}$ and $F_{RGB}$ as described in Sections 5.2 and 5.3 respectively. 3: Define $\underline{B}F$ and $\overline{B}F$ following Eqns. (11) and (12). 4: For a boundary granule $\aleph{p}\in\\{\overline{B}F-\underline{B}F\\}$ do the following. i) Compute $R(\aleph(p))$ with Eqn. (13) ii) Compute $Q(\aleph(p),a_{1})$ and $Q(\aleph(p),a_{2})$, with Eqn. (15) if $Q(\aleph(p),a_{1})>Q(\aleph(p),a_{2})$ then Set $\underline{B}F=\underline{B}F\cup\aleph(p)$ and move to the next connected granule of $\aleph(p)$ else Remove $\aleph(p)$ from from $\overline{B}X$ and moving to the granule connected to $\underline{B}X$. end if 5: Repeat Step 4 for the next boundary granule. 6: Repeat Step 4 and 5 till $\\{\overline{B}F-\underline{B}F\\}=\emptyset$ 7: Set $F=\underline{B}F$ ### 5.6 Fire Segmentation Over The Video Sequence So far we have discussed the method of fire segmentation with Q-rough sets over a video frame. Since we are dealing with video sequence, we do not need to repeat the entire process for each frame. Rather, we will employ our trained Q-agents to explore the fire region in the upcoming frames. That is, the Q-agents are going to check the spatio-color granules and in rest of the sequence, and similar granules will automatically be included in the lower approximated fire region. Let fire region in the $t^{th}$ frame ($f_{t}$) be approximated as $F_{t}$ with Algorithm 1. Let $\aleph(p_{t+1})$ be a granule under consideration in the frame $f_{t+1}$. The action over $\aleph(p_{t+1})$ will get automatically decided by the Q-agent in $f_{t+1}$, since it already knows which action maximizes the utility from $f_{t}$. The computation of Q-value will only be required in $f_{t+1}$ if $\aleph(p_{t+1})$ is an unknown state to the Q agent. ## 6 Determination of Threat of Fire After determination of fire regions in video sequences, we focus to determine the the threat associated with the fire. Fire is a necessity as well as a threat in human civilization. Fire is used for light, for heat, to cook etc. But fire is desired to be used keeping it in control, once it becomes uncontrolled, it creates threat and damages. Here we plan to develop a methodology to determine whether the fire is in use or it is becoming a threat. We have observed that, fire is generally in a controlled position while the flames are within a certain space, but it becomes a threat when the flame starts to spread. The faster the flames get spread, greater is the threat. Here we aim to quantify this phenomenon and accordingly generate an alarm with the possible threat. Here we have estimated relative spread in fire regions in a video sequence to quantify the threat. If the flame is in control, the fire regions may change from frame to frame, but there will be a flicker effect. That is, the fire regions may remain within some limits throughout the sequence. But if the fire becomes uncontrolled, the fire region will spread rapidly compared to its previous frames. This is why relative spread is considered for the threat computation. Let $F_{t}$ be the segmented fire region in frame $f_{t}$. Let the information from $P$ number of previous frames be used to determine the threat. The average fire segments ($F_{\mu}$) throughout the sequence and the and recent average fire segments ($F_{\mu P}$) are computed based on the following equations. $\displaystyle F_{\mu}=\frac{\sum_{i=1}^{N}F_{i}}{N}$ (16) $\displaystyle F_{\mu P}=\frac{\sum_{i=N-P}^{N}F_{i}}{P}:P<N$ (17) The threat index of fire ($\mathcal{T_{F}}$) is now defined with relative increment of spread in recent frames. It is computed with Eqn. (18). $\mathcal{T_{F}}=\frac{F_{\mu P}-F_{\mu}}{F_{\mu}}$ (18) Please note that a signed difference is considered in the numerator of right hand side of in Eqn. (18). This is because, the threat will only be positive if there if a relative spread in fire region, if the region is concentrated or the fire is extinguishing, the threat will become negative. Thereby eliminating occurrences of false alarms. Selection of the value of $P$ plays a major role in this index. Here we have decided to consider the relative growth of fire over the last one second. Therefore, $P$ is chosen based on the frame rate of video acquisition per second (fps). That is, if the video is acquired with a frame rate of 30 fps, $P$ is set to be $30$. ## 7 Experimental Results Here in this section we have experimentally demonstrated the effectiveness of the proposed methods in identifying the followings. The unsupervised fire segmentation method, developed here is proven to be superior in terms of qualitative and quantitative results over different types of input videos, from, a few state of the art methods. The threat detection method is also found to be effective in quantifying the threat associated with the fire. We have conducted our experiment with 30 different types of videos including more than 25000 video frames, with fire flame. Ten video sequences out of the thirty considered videos contain spreading fire or uncontrolled fire, and the rest contain controlled fire. Ten video sequence sets were acquired from the sets from ’Firesence’ database [23], and twenty video sequences from different links freely available on ’YouTube’. To limit the siz of the article we have shown the qualitative and quantative results obtained over seven different video sequences. Out of which we have selected only two videos where the fire region remained to be within a range, fire is either rapidly growing, or slowly spreading in the rest of the five videos shown here. The video sequences, over which we have shown the results are described as follows. ’PosVideo1’ [23] shows a part of a bus that is set on fire, the fire is gradually growing here. ’PosVideo2’ [23] shows fire is set into a kitchen and it is growing gradually. In ’PosVideo4’ and ’PosVideo5’ [23] fire is under control, two different burning fire-places are captured in these videos. Two men are setting fire in a forest region and fire start to grow is the content of ’PosVideo6’ [23]. In ’WeddingDress’ [24] video, fire is initially spreading over the wedding dresses and then becomes constant, whereas in ’ChristmasTree’ [25] video a Christmas tree catches fire that spreads rapidly. The effectiveness of Q-rough sets in segmenting out the fire regions is shown in the next section. ### 7.1 Fire Region Segmentation with Q-rough Set Here in this section we are going to demonstrate the effectiveness of Q-Rough sets in identifying the fire segments throughout the video sequences. The visual segmented outputs for four different frames over seven different sequences, described above, are shown in Figs. 3(1) to 4(7). It can be concluded from the visual results that the fire regions are well segmented out with Q-rough sets. The segmentation accuracy is quantitatively demonstrated in Table 1. Please note that, there was no ground truth available for the video sequences used here for the experimentations. Therefore, we have manually annotated ten frames, selected randomly from each video sequence, and performed the quantitative analysis. The average $Falsepositive\leavevmode\nobreak\ (FP)$, $FalseNegative\leavevmode\nobreak\ (FN)$, $Precision$, and $Recall$ values are given here in Table 1. Table 1: Fire Segmentation Accuracy of Q-Rough Sets Sequence \| $FP(\%)$ \| $FN(\%)$ \| $Precision$ \| $Recall$ ---\|---\|---\|---\|--- PosVideo1 \| 0.5 \| 15 \| 0.99 \| 0.87 PosVideo2 \| 3 \| 5 \| 0.97 \| 0.95 PosVideo4 \| 4 \| 1 \| 0.96 \| 0.99 PosVideo5 \| 3 \| 2 \| 0.97 \| 0.98 PosVideo11 \| 2 \| 10 \| 0.98 \| 0.9 WeddingDress \| 2 \| 8 \| 0.98 \| 0.91 ChristmasTree \| 23 \| 5 \| 0.77 \| 0.93 From Table 1 it can be observed that, the Q-rough set method is found to be quite efficient in segmenting out the fire regions almost in every video except ChistmasTree. The fire regions are well segmented even the ones that are small in size (PosVideo11 and PosVideo1), continuously burning (PosVideo4 and PosVideo5) or spreading rapidly (PosVideo2 and WeddingDress). In the ChristmassTree video the fire gets reflected in the wall and the wall reflects almost of similar color as that of the fire. Therefore, some part of the wall also gets segmented out as fire. (1) (2) (3) (4) Figure 3: Fire segmentation results with Q-rough sets for frame nos. (1) 15, 45, 75, 105 of ’PosVideo1’ sequence (2) 25, 45, 85, 105 of ’PosVideo2’ sequence (3) 25, 45, 85, 105 of ’PosVideo4’ sequence, (4) 25, 45, 85, 105 of ’PosVideo5’ sequence (5) (6) (7) Figure 4: Continuation of Fig. 3 (5) 25, 45, 85, 105 of ’PosVideo11’ sequence, (6) 25, 45, 85, 105 of ’WeddingDress’ sequence, and (7) 25, 45, 85, 105 of ’CristmassTree’ sequence; (i) input, (ii) segmented ### 7.2 Comparative Study Here in this section we have demonstrated qualitative and quantitative comparison of our fire segmentation method with four state of the art methods. The methods, with which the comparative studies are performed include: i) RGB and motion feature based method with HMM (RGB-M) [13], ii) HSV model based method developed for forest fire detection with static and dynamic features (HSV-SD) [15], iii) spatio-temporal flame modelling method (ST-F) [16], and iv) fire detection with improved deep learning (F-IDL) [21] method. The visual comparison over a frame of the seven different video sequences on with the five different methods are shown in Fig. 5. From the visual results it can be seen that the Q-rough set method is segmenting out the fire region most efficiently. (1) (2) (3) (4) (5) (6) (7) Figure 5: Visual comparison on fire segmentation on (1) PosVideo1 sequence, (2) PosVideo2 sequence, (3) PosVideo4 sequence, (4) PosVideo5 Sequence, (5) PosVideo11 sequence, (6) WeddingDress sequence, and (7) ChristmassTree Sequence; (a) input, (b) RGB-M method, (c) HSV-SD method, (d) ST-F method, (e) F-IDL method, and (f) Q-rough set (proposed) method Please note that, none of the methods used in the comparative study focus on classifying each fire pixel to the non-fire ones, therefore, the measures of classification accuracy, like, true positive, false negative is not going to be a fair parameter here for the comparative study. Therefore, a measure, namely, RMSE (root mean square error) is taken under consideration here for quantitative comparison. The root mean square error between the four corner pixels of the bounding boxes, covering the ground truth regions of fire, and the segmented regions of fire are considered here for the measurement. The average RMSE values for the seven videos sequences with five methods are given in Table 2 Table 2: Fire Region Segmentation: Comparison with Average RMSE Sequence \| $RGB-M$ \| $HSV-SD$ \| $ST-F$ \| $F-IDL$ \| $Q-RoughSet$ ---\|---\|---\|---\|---\|--- PosVideo1 \| 52.3 \| 8.2 \| 6.6 \| 13.4 \| 5.8 PosVideo2 \| 46.8 \| 11.1 \| 7.2 \| 5.8 \| 6.9 PosVideo4 \| 9.2 \| 6.3 \| 5.9 \| 6.1 \| 5.5 PosVideo5 \| 8.8 \| 5.2 \| 4.7 \| 5 \| 4.9 PosVideo11 \| 48.6 \| 11.2 \| 9.5 \| 7.3 \| 4.9 WeddingDress \| 19.4 \| 17.8 \| 8.3 \| 12.2 \| 5.1 ChristmasTree \| 34.6 \| 13.5 \| 12.6 \| 21.2 \| 16.3 It can be observed from Fig. 5 and Table 2 that the proposed unsupervised method performs superior or equally well to those of the state of the art methods. Besides, it can also be seen from the visual results (Fig. 5), that Q-rough set method can classify the fire pixels from the non-fire ones better than the other methods in comparison. ### 7.3 Effectiveness of Fire Threat Index $\mathcal{T_{F}}$ The effectiveness of $\mathcal{T_{F}}$ index in identifying the threat of fire is validated here with extensive experiments. The values of $\mathcal{T_{F}}$ for each frame throughout the seven video sequences are plotted here in Fig. 6. The $\mathcal{T_{F}}$ values for all the five methods, over which the comparative studies are performed, are computed here and plotted accordingly in Fig. 6. The red line represents the values of $\mathcal{T_{F}}$ indices obtained by Q-rough set method, the green line indicates $RGB-M$ method, blue line indicated $HSV-SD$ method, gray ’- -’ line indicates $ST-F$ method, and the solid black line indicates $F-IDL$ method in Fig. 6. The increament or decrement of fire flame is well reflected by the proposed $\mathcal{T_{F}}$ index. For example, fire is initially increasing and then decreasing in $PosVideo1$-sequence which is reflected in the curve in Fig. 6(1) almost by all the methods. The increasing threat of fire of $PosVideo2$-sequence can be inferred from Fig. 6(2) with the increasing values of $\mathcal{T_{F}}$. Fire flames have a flicker effect, and no threat in $PosVideo4$ and $PosVideo5$ sequences. It is best reflected by Q rough set method in Figs. 6(3) and 6(4), where the $\mathcal{T_{F}}$ values are withing a fixed range closed to the value zero. The gradually spreading fire in $PosVideo11$-sequence can be inferred from 6(5) with Q-rough set method. Initial spread and then concentration of fire in $WeddingDress$-sequence is reflected well in Fig. 6(6). The sudden spread of fire in $ChristmasTree$-sequence can be well detected in Fig. 6(7). Therefore, the proposed $\mathcal{T_{F}}$ index is successful in identifying the threat of fire correctly. (1) (2) (3) (4) (5) (6) (7) Figure 6: $\mathcal{T_{F}}$ index values over 1) PosVideo1 sequence, (2) PosVideo2 sequence, (3) PosVideo4 sequence, (4) PosVideo5 Sequence, (5) PosVideo11 sequence, (6) WeddingDress sequence, and (7) ChristmassTree Sequence ## 8 Conclusions and Future Work In this article we have primarily addressed two tasks related to fire threat detection from videos. The first one is unsupervised fire region segmentation with Q-rough sets and the second one is definition of fire threat index. The unsupervised method of fire region segmentation with Q-rough set has been proven to be effective visually and quantitatively in classifying fire pixels correctly. The performance of Q-rough set based fire segmentation has been proven to be superior in identifying the correct fire region over different types of video sequences with respect to different state of the art methods. The fire threat index is proven to be effective in reflecting the spread of the fire quickly. This index works well with all the fire segmentation methods, but it reflects the threat the best with Q-rough set method. Therefore it can be concluded that the proposed methods can be integrated in video surveillance systems to generate quick fire alarm. The Q-rough set based segmentation method could also be used in the other areas of video analysis, like, object tracking since the Q-agents could learn and explore the object of interest by themselves alongwith the input stream data. Besides, the lower- upper approximations with Q-rough set, defined here, could be used other areas to deal with the incompleteness of knowledge base and stream data. 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# Segmenting Transparent Object in the Wild with Transformer Enze Xie1 , Wenjia Wang2, Wenhai Wang3, Peize Sun1, Hang Xu1, Ding Liang2, Ping Luo1 1The University of Hong Kong 2Sensetime Research 3Nanjing University ###### Abstract This work presents a new fine-grained transparent object segmentation dataset, termed Trans10K-v2, extending Trans10K-v1, the first large-scale transparent object segmentation dataset. Unlike Trans10K-v1 that only has two limited categories, our new dataset has several appealing benefits. (1) It has 11 fine-grained categories of transparent objects, commonly occurring in the human domestic environment, making it more practical for real-world application. (2) Trans10K-v2 brings more challenges for the current advanced segmentation methods than its former version. Furthermore, a novel transformer-based segmentation pipeline termed Trans2Seg is proposed. Firstly, the transformer encoder of Trans2Seg provides the global receptive field in contrast to CNN’s local receptive field, which shows excellent advantages over pure CNN architectures. Secondly, by formulating semantic segmentation as a problem of dictionary look-up, we design a set of learnable prototypes as the query of Trans2Seg’s transformer decoder, where each prototype learns the statistics of one category in the whole dataset. We benchmark more than 20 recent semantic segmentation methods, demonstrating that Trans2Seg significantly outperforms all the CNN-based methods, showing the proposed algorithm’s potential ability to solve transparent object segmentation. Code is available in github.com/xieenze/Trans2Seg. ## 1 Introduction Modern robots, mainly mobile robots and mechanical manipulators, would benefit a lot from the efficient perception of the transparent objects in residential environments since the environments vary drastically. The increasing utilization of glass wall and transparent door in the building interior and the glass cups and bottles in residential rooms has resulted in the wrong detection in various range sensors. In robotic research, most systems perceive the environment by multi-data sensor fusion via sonars or lidars. The sensors are relatively consistent in detecting opaque objects but are still affected by the scan mismatching due to transparent objects. The unique feature of reflection, refraction, and light projection from the transparent objects may confuse the sensors. Thus a reliable vision-based method, which is much cheaper and more robust than high-precision sensors, would be efficient. (a) Selected images and corresponding high-quality masks. (b) Performance comparison on Trans10K-v2. Figure 1: (a) shows the high diversity of our dataset and high-quality annotations. (b) is Comparisons between Trans2Seg and other CNN-based semantic segmentation methods. All methods are trained on Trans10K-v2 with same epochs. mIoU is chosen as the metric. Deeper color bar indicates methods with larger FLOPS. Our Trans2Seg significantly surpasses other methods with lower flops. Although some transparent objects dataset Xu et al. (2015); Chen et al. (2018a); Mei et al. (2020) were proposed, there are some obvious problems. (1) Limited dataset scale. These datasets often have less than 1K images captured from the real-world and less than 10 unique objects. (2) Poor diversity. The scene of these datasets is monotonous. (3) Fewer classes. All these datasets have only two classes, background and transparent objects. They lack fine- grained categories, which limited their practicality. Recently, Xie et al. (2020) proposed a large-scale and high-diversity dataset termed Trans10K, which divide transparent objects as ‘Things’ and ‘Stuff’. The dataset is high diversity, but it also lacks fine-grained transparent categories. In this paper, we proposes a fine-grained transparent object segmentation dataset termed Trans10K-v2 with more elaborately defined categories. The images are inherit from Trans10K-v1 Xie et al. (2020). We annotate the 10428 images with 11 fine-grained categories: shelf, jar, freezer, window, glass door, eyeglass, cup, glass wall, glass bowl, water bottle, storage box. In Trans10K-v1, transparent things are defined to be grabbed by the manipulators and stuff are for robot navigation. Though two basic categories can partially help robots to interact with transparent objects, the provided fine-grained classes in Trans10K-v2 can provide more. We analyze these objects’ functions and how robots interact with them in appendix. Based on this challenging dataset, we design Trans2Seg, introducing Transformer into segmentation pipeline for its encoder-decoder architecture. First, the transformer encoder provides a global receptive field via self- attention. Larger receptive field is essential for segmenting transparent objects because transparent objects often share similar textures and context with its surroundings. Second, the decoder stacks successive layers to interact query embedding with transformer encoder output. To facilitate the robustness of transparent objects, we carefully design a set of learnable class prototype embeddings as the query for transformer decoder and the key is the feature map from the transformer encoder. Compared with convolutional paradigm, where the class prototypes is the fixed parameters of convolution kernel weight, our design provides a dynamic and context-aware implementation. As shown in Figure. 1(b), we train and evaluate 20 existing representative segmentation methods on Trans10K-v2, and found that simply applying previous methods to this task is far from sufficient. By successfully introducing Transformer into this task, our Trans2Seg significantly surpasses the best TransLab Xie et al. (2020) by a large margin (72.1 vs. 69.0 on mIoU). In summary, our main contributions are three-fold: * • We propose the largest glass segmentation dataset (Trans10K-v2) with 11 fine- grained glass image categories with a diverse scenario and high resolution. All the images are elaborately annotated with fine-shaped masks and function- oriented categories. * • We introduce a new transformer-based network for transparent object segmentation with transformer encoder-decoder architecture. Our method provides a global receptive field and is more dynamic in mask prediction, which shows excellent advantages. * • We evaluate more than 20 semantic segmentation methods on Trans10K-v2, and our Trans2Seg significantly outperforms these methods. Moreover, we show this task is largely unsolved. Thus more research is needed. ## 2 Related Work Semantic Segmentation. In deep learning era, convolutional neural network (CNN) puts forwards the development of semantic segmentation in various datasets, such as ADE20K, CityScapes and PASCAL VOC. One of the pioneer works approaches, FCN Long et al. (2015), transfers semantic segmentation into an end-to-end fully convolutional classification network. For improving the performance, especially around object boundaries, Chen et al. (2017); Lin et al. (2016); Zheng et al. (2015) propose to use structured prediction module, conditional random fields (CRFs) Chen et al. (2014), to refine network output. Dramatic improvements in performance and inference speed have been driven by aggregating features at multiples scales, for example, PSPNet Zhao et al. (2017) and DeepLab Chen et al. (2017, 2018b), and propagating structured information across intermediate CNN representations Gadde et al. (2016); Liu et al. (2017); Wang et al. (2018). Transparent Object Datasets. Xu et al. (2015) introduces TransCut dataset which only contain 49 images of 7 unique objects. To generate the segmentation result, Xu et al. (2015) optimized an energy function based on LF-linearity which also need to utilize the light-field cameras. Chen et al. (2018a) proposed TOM-Net. It contains 876 real images and 178K synthetic images which are generated by POV-Ray. However, only 4 unique objects are used in synthesizing the training data. Recnetly, Xie et al. (2020) introduce a first large-scale real-world transparent object segmentation dataset, termed Trans10K. It has 10K+ images. However, there are two categories in this dataset, which limits its practical use. In this work, our Trans10K-v2 inherited the data and annotates 11 fine-grained categories. Figure 2: Images in Trans10K-v2 dataset are carefully annotated with high quality. The first row shows sample images and the second shows the segmentation masks. The color scheme which encodes the object categories are listed on the right of the figure. Zoom in for best view. Trans10Kv2 \| shelf \| door \| wall \| box \| freezer \| window \| cup \| bottle \| jar \| bowl \| eyeglass ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- image num \| 280 \| 1572 \| 3059 \| 603 \| 90 \| 501 \| 3315 \| 1472 \| 997 \| 340 \| 410 CMCC \| 3.36 \| 5.19 \| 5.61 \| 2.57 \| 3.36 \| 4.27 \| 1.97 \| 1.82 \| 1.99 \| 1.31 \| 2.56 pixel ratio(%) \| 2.49 \| 9.23 \| 38.42 \| 3.67 \| 1.02 \| 4.28 \| 22.61 \| 6.23 \| 6.75 \| 3.67 \| 0.78 Table 1: Statistic information of Translabv2. ‘CMCC’ denotes Mean Connected Components of each category. ‘image num’ denotes the image number. ‘pixel ratio’ is the pixel number of a certain category accounts in all the pixels of transparent objects in Trans10K-v2. Transformer in Vision Tasks. Transformer Vaswani et al. (2017) has been successfully applied in both high-level vision and low-level vision Han et al. (2020). In ViT Dosovitskiy et al. (2020), Transformer is directly applied to sequences of image patches to complete image classification. In object detection areas Carion et al. (2020); Zhu et al. (2020), DETR reasons about the relations of the object queries and the global image context via Transformer and outputs the final set of predictions in parallel without non- maximum suppression(NMS) procedures and anchor generation. SETR Zheng et al. (2020) views semantic segmentation from a sequence-to-sequence perspective with Transformer. IPT Chen et al. (2020) applies Transformer model to low- level computer vision task, such as denoising, super-resolution and deraining. In video processing, Transformer has received significantly growing attention. VisTR Wang et al. (2020) accomplishes instance sequence segmentation by Transformer. Multiple-object tracking Sun et al. (2020); Meinhardt et al. (2021) employs Transformers to decode object queries and feature queries of the previous frame into bounding boxes of the current frame, and merged by Hungarian Algorithm or NMS. ## 3 Trans10K-v2 Dataset Dataset Introduction. Our Trans10K-v2 dataset is based on Trans10K dataset Xie et al. (2020). Following Trans10K, we use 5000, 1000 and 4428 images in training, validation and testing respectively. The distribution of the images is abundant in occlusion, spatial scales, perspective distortion. We further annotate the images with more fine-grained categories due to the functional usages of different objects. Trans10K-v2 dataset contains 10,428 images, with two main categories and 11 fine-grained categories: (1) Transparent Things containing cup, bottle, jar, bowl and eyeglass. (2) Transparent Stuﬀ containing windows, shelf, box, freezer, glass walls and glass doors. In respect to ﬁne-grained categories and high diversity, Trans10K-v2 is very challenging, and have promising potential in both computer vision and robotic researches. Annotation Principle. The transparent objects are manually labeled by expert annotators with professional labeling tool. The annotators were asked to provide more than 100 points when they trace the boundaries of each transparent object, which ensures the high-quality outline of the mask shapes. The way of annotation is mostly the same with semantic segmentation datasets such as ADE20K. We set the background with 0, and the 11 categories from 1 to 11. We also provide the scene environment of each image locates at. The annotators are asked to strictly following principles when they label the images: (I) Only highly transparent pixels are annotated as masks, other semi- transparent and non-transparent pixels are ignored. Highly transparent objects no matter made of glass, plastics or crystals should also be annotated. (II) When occluded by opaque objects, the pixels will be cropped from the masks. (III) The setting of all 11 fine-grained categories are elaborately observed and induced from the point of function. We analyze firstly how the robots need to deal with the transparent objects as avoiding or grasping or manipulating, then categorize the objects similar in shape and function into a fine-grained category. The detailed principle of how we categorize the objects is listed in appendix. Dataset Statistics. The statistic information of CMCC, imaga number, pixel proportion are listed in Table 1 in detail. From Table1, the sum of all the image numbers is larger than 10428 since some image has multiple category of objects. CMCC denotes Mean Connected Components of each category. It is caculated by dividing the connected components number of a certain category by the image number. The number of connected components are counted by the boundary of the masks. It represents the complexity of the transparent objects. Evaluation Metrics. Results are reported in three metrics that are widely used in semantic segmentation to benchmark the performance of fine-grained transparent object segmentation. (1) Pixel Accuracy indicates the proportion of correctly classified pixels. (2) Mean IoU indicates mean intersection over union. (3) Category IoU indicates the intersection over union of each category. ## 4 Method ### 4.1 Overall Pipeline The overall Trans2Seg architecture contains a CNN backbone, an encoder-decoder transformer, and a small convolutional head, as shown in Figure 3. For an input image of $(H,W,3)$, * • The CNN backbone generates image feature map of $(\frac{H}{16},\frac{W}{16},C)$. * • The encoder takes in the summation of flattened feature of $(\frac{H}{16}\frac{W}{16},C)$ and positional embedding of $(\frac{H}{16}\frac{W}{16},C)$, and outputs encoded feature of $(\frac{H}{16}\frac{W}{16},C)$. * • The decoder interacts the learned class prototypes of $(N,C)$ with encoded feature, and generates attention map of $(N,M,\frac{H}{16}\frac{W}{16})$, where $N$ is number of categories, $M$ is number of heads in multi-head attention. * • The small convolutional head up-samples the attention map to $(N,M,\frac{H}{4},\frac{W}{4})$, fuses it with high-resolution feature map Res2 and outputs attention map of $(N,\frac{H}{4},\frac{W}{4})$. The final segmentation is obtained by pixel-wise argmax operation on the output attention map. Figure 3: The whole pipeline of our hybrid CNN-Transformer architecture. First, the input image is fed to CNN to extract features $F$. Second, for transformer encoder, the features and position embedding are flatten and fed to transformer for self-attention, and output feature($F_{e}$) from transformer encoder. Third, for transformer decoder, we specifically define a set of learnable class prototype embeddings($E_{cls}$) as query, $F_{e}$ as key, and calculate the attention map with $E_{cls}$ and $F_{e}$. Each class prototype embedding corresponds to a category of final prediction. We also add a small conv head to fuse attention map and Res2 feature from CNN backbone. Details of transformer decoder and small conv head refer to Figure 4. Finally, we can get the predict results by doing pixel-wise argmax on the attention map. For example, in this figure, the segmentation mask of two categories (Bottle and Eyeglass) corresponds to two class prototypes with same colors. Figure 4: Detail of Transformer Decoder and small conv head. Input: The learnable category prototypes as query, features from transformer encoder as key and value. The inputs are fed to transformer decoder, which consists of several decoder layers. The attention map from last decoder layer and the Res2 feature from CNN backbone are combined and fed to a small conv head to get final prediction result. We also provide the Pseudo Code of small conv head for better understanding. ### 4.2 Encoder The Transformer encoder takes a sequence as input, so the spatial dimensions of the feature map $(\frac{H}{16},\frac{W}{16},C)$ is flattened into one dimension$(\frac{H}{16}\frac{W}{16},C)$. To compensate missing spatial dimensions, positional embedding Gehring et al. (2017) is supplemented to one dimension feature to provide information about the relative or absolute position of the feature in the sequence. The positional embedding has the same dimension $(\frac{H}{16}\frac{W}{16},C)$ with the flattened feature. The encoder is composed of stacked encoder layers, each of which consists of a multi-head self-attention module and a feed forward network Vaswani et al. (2017). ### 4.3 Decoder The Transformer decoder takes input a set of learnable class prototype embeddings as query, denoted by $E_{cls}$, the encoded feature as key and value, denoted by $F_{e}$, and output the attention map followed by Small Conv Head to obtain final segmentation result, as shown in Figure 4. The class prototype embeddings are learned category prototypes, updated iteratively by a series of decoder layers through multi-head attention mechanisms. We denoted iterative update rule by $\bigodot$, then the class prototype in each decoder layer is: $\rm{E_{cls}^{s}}=\bigodot_{i=0,..,s-1}\rm{softmax}(E_{cls}^{i}F_{e})F_{e}$ (1) In the final decoder layer, the attention map is extracted out to into small conv head: $\rm{attention\ map}=E_{cls}^{s}F_{e}$ (2) The pseudo code of small conv head is shown in shown in Figure 4. The attention map from Transformer decode is the shape of $(N,M,\frac{H}{16}\frac{W}{16})$, where $N$ is number of categories, $M$ is number of heads in multi-head attention. It is up-sampled to $(N,M,\frac{H}{4},\frac{W}{4})$, then fused with high-resolution feature map Res2 in the second dimension to $(N,M+C,\frac{H}{4},\frac{W}{4})$, and finally transformed into output attention map of $(N,\frac{H}{4},\frac{W}{4})$. The final segmentation is obtained by pixel-wise argmax operation on the output attention map. ### 4.4 Discussion The most related work with Trans2Seg is SETR and DETR Zheng et al. (2020); Carion et al. (2020). In this section we discuss the relations and differences in details. SETR. Trans2Seg and SETR are both segmentation pipelines. Their key difference is reflected in the design of the decoder. In SETR, the decoder is simple several convolutional layers, which is similar with most previous methods. However, the decoder of Trans2Seg is also transformer, which fully utilize the advantages of attention mechanism in semantic segmentation. DETR. Trans2Seg and DETR share similar components in the pipeline, including CNN backbone, Transformer encoder and decoder. The biggest difference is the definition of query. In DETR, the decoder’s queries represents $N$ learnable objects because DETR is designed for object detection. However, in Trans2Seg, the queries represents $N$ learnable class prototypes, where each query represents one category. We could see that the minor change on query design could generalize Transformer architecture to apply to diverse vision tasks, such as object detection and semantic segmentation. ## 5 Experiments ### 5.1 Implementation Details. We implement Trans2Seg with Pytorch. The ResNet-50 He et al. (2016) with dilation convolution at last stage. is adoped as the CNN extractor. For loss optimization, we use Adam optimizer with epsilon 1e-8 and weight decay 1e-4. Batch size is 8 per GPU. We set learning rate 1e-4 and decayed by the poly strategy Yu et al. (2018) for 50 epochs. We use 8 V100 GPUs for all experiments. For all CNN based methods, we random scale and crop the image to $480\times 480$ in training, and resize image to $513\times 513$ in inference, following common setting on PASCAL VOC Everingham and Winn (2011). For our Trans2Seg, we adopt transformer architecture and need to keep the shape of learned position embedding same in training/inference, so we directly resize the image to $512\times 512$. Code has been released for community to follow. ### 5.2 Ablation Studies. We use the FCN Long et al. (2015) as our baseline. FCN is a fully convolutional network with very simple design, and it is also a very classic semantic segmentation method. First, we demonstrate that transformer encoder can build long range attention between pixels, which has much larger receptive field than CNN filters. Second, we remove the CNN decoder in FCN and replace by our Transformer decoder, we design a set of learnable class prototypes as queries and show that this design further helps improve the accuracy. Third, we verify our method with transformer at different scales. id Trans. Enc. Trans. Dec. CNN Dec. mIoU 0 $\times$ $\times$ ✓ 62.7 1 ✓ $\times$ ✓ 68.8 2 ✓ ✓ $\times$ 72.1 Table 2: Effectiveness of Transformer encoder and decoder. ‘Trans.’ indicates Transformer. ‘Enc.’ and ‘Dec.’ means encoder and decoder. Scale hyper-param. GFlops MParams mIoU small e128-n1-m2 40.9 30.5 69.2 medium e256-n4-m3 49.0 56.2 72.1 large e768-n12-m4 221.8 327.5 70.3 Table 3: Performance of Transformer at different scales. ‘e{a}-n{b}-m{c}’ means the transformer with number of ‘a’ embedding dims, ‘b’ layers and ‘c’ mlp ratio. Self-Attention of Transformer Encoder. As shown in Figure 3, the FCN baseline without transformer encoder achieves 62.7% mIoU, when adding transformer encoder, the mIoU directly improves 6.1%, achieving 66.8% mIoU. It demonstrates that the self-attention module in transformer encoder provides global receptive filed, which is better than CNN’s local receptive field in transparent object segmentation. Category Prototypes of Transformer Decoder. In Figure 3, we verify the effectiveness of learnable category prototypes in transformer decoder. In column 2, with traditional CNN decoder, the mIoU is 68.8%. However, with our transformer decoder, the mIoU boosts up to 72.1% with 3.3% improvement. The strong performance benefits from the flexible representation that learnable category prototypes as queries to find corresponding pixels in feature map. Scale of Transformer. The scale of transformer is mainly influenced by three hyper-parameters: (1) embedding dim of feature. (2) number of attention layers. (3) mlp ratio in feed forward layer. We are interested in whether enlarge the model size can continuously improve performance. So we set three combinations, as shown in Figure 3. We can find that with the size of transformer increase, the mIoU first increase then decrease. We argue that if without massive data to pretrain, e.g. BERT Devlin et al. (2019) used large- scale nlp data, the transformer size is not the larger the better for our task. Figure 5: Visual comparison of Trans2Seg to other CNN-based semantic segmentation methods. Our Trans2Seg clearly outperforms others thanks to the transformer’s global receptive field and attention mechanism, especially in dash region. Zoom in for best view. Refer to supplementary materials for more visualized results. ### 5.3 Comparison to the state-of-the-art. Method \| FLOPs \| ACC $\uparrow$ \| mIoU $\uparrow$ \| Category IoU $\uparrow$ ---\|---\|---\|---\|--- \| \| bg \| shelf \| Jar \| freezer \| window \| door \| eyeglass \| cup \| wall \| bowl \| bottle \| box FPENet \| 0.76 \| 70.31 \| 10.14 \| 74.97 \| 0.01 \| 0.00 \| 0.02 \| 2.11 \| 2.83 \| 0.00 \| 16.84 \| 24.81 \| 0.00 \| 0.04 \| 0.00 ESPNetv2 \| 0.83 \| 73.03 \| 12.27 \| 78.98 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 6.17 \| 0.00 \| 30.65 \| 37.03 \| 0.00 \| 0.00 \| 0.00 ContextNet \| 0.87 \| 86.75 \| 46.69 \| 89.86 \| 23.22 \| 34.88 \| 32.34 \| 44.24 \| 42.25 \| 50.36 \| 65.23 \| 60.00 \| 43.88 \| 53.81 \| 20.17 FastSCNN \| 1.01 \| 88.05 \| 51.93 \| 90.64 \| 32.76 \| 41.12 \| 47.28 \| 47.47 \| 44.64 \| 48.99 \| 67.88 \| 63.80 \| 55.08 \| 58.86 \| 24.65 DFANet \| 1.02 \| 85.15 \| 42.54 \| 88.49 \| 26.65 \| 27.84 \| 28.94 \| 46.27 \| 39.47 \| 33.06 \| 58.87 \| 59.45 \| 43.22 \| 44.87 \| 13.37 ENet \| 2.09 \| 71.67 \| 8.50 \| 79.74 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 22.25 \| 0.00 \| 0.00 \| 0.00 HRNet_w18 \| 4.20 \| 89.58 \| 54.25 \| 92.47 \| 27.66 \| 45.08 \| 40.53 \| 45.66 \| 45.00 \| 68.05 \| 73.24 \| 64.86 \| 52.85 \| 62.52 \| 33.02 HardNet \| 4.42 \| 90.19 \| 56.19 \| 92.87 \| 34.62 \| 47.50 \| 42.40 \| 49.78 \| 49.19 \| 62.33 \| 72.93 \| 68.32 \| 58.14 \| 65.33 \| 30.90 DABNet \| 5.18 \| 77.43 \| 15.27 \| 81.19 \| 0.00 \| 0.09 \| 0.00 \| 4.10 \| 10.49 \| 0.00 \| 36.18 \| 42.83 \| 0.00 \| 8.30 \| 0.00 LEDNet \| 6.23 \| 86.07 \| 46.40 \| 88.59 \| 28.13 \| 36.72 \| 32.45 \| 43.77 \| 38.55 \| 41.51 \| 64.19 \| 60.05 \| 42.40 \| 53.12 \| 27.29 ICNet \| 10.64 \| 78.23 \| 23.39 \| 83.29 \| 2.96 \| 4.91 \| 9.33 \| 19.24 \| 15.35 \| 24.11 \| 44.54 \| 41.49 \| 7.58 \| 27.47 \| 3.80 BiSeNet \| 19.91 \| 89.13 \| 58.40 \| 90.12 \| 39.54 \| 53.71 \| 50.90 \| 46.95 \| 44.68 \| 64.32 \| 72.86 \| 63.57 \| 61.38 \| 67.88 \| 44.85 DenseASPP \| 36.20 \| 90.86 \| 63.01 \| 91.39 \| 42.41 \| 60.93 \| 64.75 \| 48.97 \| 51.40 \| 65.72 \| 75.64 \| 67.93 \| 67.03 \| 70.26 \| 49.64 DeepLabv3+ \| 37.98 \| 92.75 \| 68.87 \| 93.82 \| 51.29 \| 64.65 \| 65.71 \| 55.26 \| 57.19 \| 77.06 \| 81.89 \| 72.64 \| 70.81 \| 77.44 \| 58.63 FCN \| 42.23 \| 91.65 \| 62.75 \| 93.62 \| 38.84 \| 56.05 \| 58.76 \| 46.91 \| 50.74 \| 82.56 \| 78.71 \| 68.78 \| 57.87 \| 73.66 \| 46.54 OCNet \| 43.31 \| 92.03 \| 66.31 \| 93.12 \| 41.47 \| 63.54 \| 60.05 \| 54.10 \| 51.01 \| 79.57 \| 81.95 \| 69.40 \| 68.44 \| 78.41 \| 54.65 RefineNet \| 44.56 \| 87.99 \| 58.18 \| 90.63 \| 30.62 \| 53.17 \| 55.95 \| 42.72 \| 46.59 \| 70.85 \| 76.01 \| 62.91 \| 57.05 \| 70.34 \| 41.32 Translab \| 61.31 \| 92.67 \| 69.00 \| 93.90 \| 54.36 \| 64.48 \| 65.14 \| 54.58 \| 57.72 \| 79.85 \| 81.61 \| 72.82 \| 69.63 \| 77.50 \| 56.43 DUNet \| 123.69 \| 90.67 \| 59.01 \| 93.07 \| 34.20 \| 50.95 \| 54.96 \| 43.19 \| 45.05 \| 79.80 \| 76.07 \| 65.29 \| 54.33 \| 68.57 \| 42.64 UNet \| 124.55 \| 81.90 \| 29.23 \| 86.34 \| 8.76 \| 15.18 \| 19.02 \| 27.13 \| 24.73 \| 17.26 \| 53.40 \| 47.36 \| 11.97 \| 37.79 \| 1.77 DANet \| 198.00 \| 92.70 \| 68.81 \| 93.69 \| 47.69 \| 66.05 \| 70.18 \| 53.01 \| 56.15 \| 77.73 \| 82.89 \| 72.24 \| 72.18 \| 77.87 \| 56.06 PSPNet \| 187.03 \| 92.47 \| 68.23 \| 93.62 \| 50.33 \| 64.24 \| 70.19 \| 51.51 \| 55.27 \| 79.27 \| 81.93 \| 71.95 \| 68.91 \| 77.13 \| 54.43 Trans2Seg \| 49.03 \| 94.14 \| 72.15 \| 95.35 \| 53.43 \| 67.82 \| 64.20 \| 59.64 \| 60.56 \| 88.52 \| 86.67 \| 75.99 \| 73.98 \| 82.43 \| 57.17 Table 4: Evaluated state-of-the-art semantic segmentation methods. Sorted by FLOPs. Our proposes Trans2Seg surpasses all the other methods in pixel accuracy and mean IoU, as well as most of the category IoUs (8 in 11). Method \| #Param (M) \| GFLOPs \| mIoU (%) ---\|---\|---\|--- R50-SemanticFPN \| 28.5 \| 45.6 \| 36.7 R50-d8+DeeplabV3+ \| 26.8 \| 120.5 \| 41.5 R50-d16+DeeplabV3+ \| 26.8 \| 45.5 \| 40.6 R50-d16+Trans2Seg \| 56.1 \| 79.3 \| 39.7 Table 5: Performance of Trans2Seg on ADE20K dataset. Trans2Seg also works well on general semantic segmentation tasks. “d8” and “d16” means dilation 8 and 16, respectively. “R50” means ResNet-50 backbone. We select more than 20 semantic segmentation methods Xie et al. (2020); Chen et al. (2018c); Li et al. (2019a); Zhao et al. (2017); Yuan and Wang (2018); Yang et al. (2018); Long et al. (2015); Ronneberger et al. (2015); Yu et al. (2018); Lin et al. (2017); Chao et al. (2019); Wang et al. (2019a); Poudel et al. (2019, 2018); Wang et al. (2019b); Jin et al. (2019); Zhao et al. (2018); Li et al. (2019a); Liu and Yin (2019); Li et al. (2019b); Fu et al. (2019); Mehta et al. (2019) to evaluate on our Trans10K-v2 dataset, the methods selection largely follows the benchmark of TransLab Xie et al. (2020). For fair comparsion, we train all the methods with 50 epochs. Table 4 reports the overall quantitative comparison results on test set. Our Trans2Seg achieves state-of-the-art 72.15% mIoU and 94.14% pixel ACC, significant outperforms other pure CNN-based methods. For example, our method is 2.1% higher than TransLab, which is the previous SOTA method. We also find that our method tend to performs much better on small objects, such as ‘bottle’ and ’eyeglass’ (10.0% and 5.0% higher than previous SOTA). We consider that the transformer’s long range attention benefits the small transparent object segmentation. In Figure 5, we visualize the mask prediction of Trans2Seg and other CNN-based methods. We can find that benefit from transformer’s large receptive field and attention mechanism, our method can distinguish background and different categories transparent objects much better than other methods, especially when multiple objects with different categories occurs in one image. Moreover, our method can obtain high quality detail information,e.g. boundary of object, and tiny transparent objects, while other CNN-based methods fail to do so. More results are shown in supplementary material. ### 5.4 General Semantic Segmentation We try to transfer Trans2Seg on general semantic segmentation and it also achieves satisfied performance. Experiment Settings. We choose ADE20K Zhou et al. (2017), a challenging scene parsing benchmark for semantic segmentation. ADE20K contains 150 fine-grained semantic categories, where there are 20210, 2000, and 3352 images for training, validation and, testing, respectively. We set learning rate to 2.5e-5 for ADE20K experiments. We train all models with 40k iterations with 8 images/GPU and 8 GPUs, and use single-scale test in inference. The data augmentation is same as DeeplabV3+ Chen et al. (2018c). Results. As shown in Table 5, compared with Semantic FPN and DeeplabV3+, our Trans2Seg achieves 39.7 mIoU, which is a satisfied performance. Our Trans2Seg verifies robust transfer ability on challenging general segmentation dataset. Please note that we do not carefully tuned the hyper-parameters of Trans2Seg on ADE20K dataset. We are highly interested to design a better transformer- based general semantic segmentation pipeline in the future. ## 6 Conclusion In this paper, we present a new fine-grained transparent object segmentation dataset with 11 common categories, termed Trans10K-v2, where the data is based on the previous Trans10K. We also discuss the challenging and practical of the proposed dataset. Moreover, we propose a transformer-based pipeline, termed Trans2Seg, to solve this challenging task. In Trans2Seg, the transformer encoder provides global receptive field, which is essential for transparent objects segmentation. In the transformer decoder, we model the segmentation as dictionary look up with a set of learnable queries, where each query represents one category. Finally, we evaluate more than 20 mainstream semantic segmentation methods and shows our Trans2Seg clearly surpass these CNN-based segmentation methods. In the future, we are interested in exploring our Transformer encoder-decoder design on general segmentation tasks, such as Cityscapes and PASCAL VOC. We will also put more effort to solve transparent object segmentation task. ## 7 Appendix ### 7.1 Detailed Dataset Information #### 7.1.1 More Visualized Demonstration of Trans10K-v2. In this section we show more visualized demonstrations to show the diversity and quality of Trans10K-v2. In Figure 6 and Figure 7, we show more cropped objects to illustrate the high-diversity of the objects. We also show more images and ground-truth masks in Figure 8. All images and transparent objects in Trans10K-v2 are selected from complex real-world scenarios that have large variations such as scale, viewpoint, contrast, occlusion, categories and transparency. From Figure 8, we can also find that it is challenging for current semantic segmentation methods. Figure 6: Cropped objects of 5 kinds of transparent things: cup, jar, bottle, bowl, eyeglass. Zoom in for the best view. Figure 7: Cropped objects of 6 kinds of transparent stuff: wall, freezer, box, door, shelf, window. Zoom in for the best view. Figure 8: More images and corresponding high-quality masks in Trans10K-v2. Our dataset is high diversity in scale, categories, pose, contrast, occlusion, and transparency. Zoom in for the best view. #### 7.1.2 Scene information We also provide each image with a scene label that represents where the objects located in. As shown in the upper part of Table 6, we list the statistics of the distribution in different scenarios of each category in detail. The distribution highly follows the distribution of our residential environments. For example, the cups, bowls, and bottles are mostly placed on the desk, while glass walls are often located in mega-malls or office buildings. The visualized demonstration of our diverse scene distribution is shown in Figure 9. Trans10k-v2 contains abundant scenarios and we induce them into 13 categories: on the desk, mega-mall, store, bedroom, sitting room, kitchen, bathroom, windowsill, office, office building, outdoor, in the vehicle, study- room. This information is mainly used to demonstrate our abundant image distribution which could cover most of the common real-life scenarios. Each image is provided with a scene label. Figure 9: The image number distribution and selected images of different scenes in Trans10K-v2. For better demonstration, the image number in vertical axis is listed as logarithmic. #### 7.1.3 How Robots Deal with Transparent Objects Transparent objects are widespread in human residential environments, so the human-aiding robots find ways to deal with transparent objects. Some former robotic research illustrates the substantial value of solving this problem, mainly from grasping and navigation. This research primarily focuses on modifying the algorithm to deal with optical signals reflected from the transparent objects. For the manipulator grasping, previous work mainly focuses on grabbing water cups. Klank et al. (2011) propose an approach to reconstruct an approximate surface of the transparent cups and bottles by the internal sensory contradiction from two ToF (time of flight) images captured from an SR4k camera. The robot arm could grasp and manipulate the objects. Spataro et al. (2015) set up a BCI-robot platform to help patients suffering from limb muscle paralysis by grasping a glass cup for the patients. Starting from the point that the usual glass material absorbs light in specific wavelengths, Zhou et al. (2018) propose the Depth Likelihood Volume (DLV), which uses a Monte Carlo object localization algorithm to help the Michigan Progress Fetch robot localize and manipulate translucent objects. For the mobile robot navigation, some work also finds ways to exclude the side-effect of transparent stuff in residential scenarios. Foster et al. (2013) modify the standard occupancy grid algorithm during the procedure of autonomous-mapping robot localize transparent objects from certain angles. Kim and Chung (2016) design a novel scan matching algorithm by comparing all candidate distances scanned by the laser range finder penetrate and reflected from the glass walls. Singh et al. (2018) use information fusion by combining a laser scanner and a sonar on an autonomous-mapping mobile robot to reduce the uncertainty caused by glass. We analyze how robots deal with transparent objects from previous work and grade them into 4 patterns: navigation, grasping, manipulation, human-aiding. Navigation and grasping are the two fundamental interactions between robots and objects. Manipulation happens on complex objects like windows, doors, or bottles with lids. Human-aiding is the highest level of robot mission, and this kind of interaction always involve human, especially disabled patients. From these 4 patterns, we can then analyze and categorize the transparent objects in respect to functions. #### 7.1.4 Categorization Principle The 11 fine-grained categories are based on how the robots need to deal with transparent objects like avoiding or grasping or manipulating. For example, the goblet and cup are both open-mouthed and mainly used to drink water. These objects need to be grasped carefully since they do not have lids. They have different interactive actions with the robots. So they are both categorized as cup. We show the detailed demonstration of each category: (1) Shelf. Containing bookshelf, showcase, cabinet, etc. They mostly have sliding glass doors and are used to store goods. (2) Freezer. Containing vending machine, horizontal freezer, etc. They are electrical equipment and are used to storing drinks and food. (3) Door. Containing automatic glass door, standard glass door, etc. The doors are located in mega-mall, bathroom or office building. They are highly transparent and extensive. They could be used in navigation and helping disabled people pass through. (4) Wall. Glass walls look like doors. However, walls can not be opened. This clue should be perceived during mobile robots’ mapping procedure. Glass walls are common in mega-mall and office buildings. (5) Window. Windows could be opened like glass doors but should not be traveled through. (6) Box. Large boxes may not need to be grasped, but the manipulator robot needs to open the box and search for specific items. (7) Cup. We category all open-mouthed cups like goblets and regular cups into this category. Cups are used for drinking water. The manipulators need to grasp a cup carefully and be able to assist disabled people to drink water. (8) Bottle. Bottles are also used to drink water. But bottles have lids, so they need careful manipulation. (9) Eyeglass. Eyeglasses need careful grasping and manipulation to help disable people wear the eyeglasses. (10) Jar. This category contains jars, kettles and other transparent containers used to hold water, flavoring and food. (11) Bowl. Bowls are usually used to contain water or food. Different from jars, they do not have lids and need careful grasping. The sample objects of these categories could be find in Figure 8. We show the most common type of different categories by cropping the objects through masks. As shown in the lower part of Table 6, we analyze and list the interactive patterns of all the 11 fine-grained categories of objects. Navigation is the basic interactive pattern of stuff and grasping is the basic interactive pattern of things. All the objects with some complex interactions need to be manipulated like the robots helping people open the shelf or window. Human- aiding is the highest level of interaction and it always involves patients. The patients need robots to help with opening the door, or feeding water by a cup or bottle. (a) Occlusion and Crowd. (b) Extreme Transparency. Figure 10: Failure cases analysis. Our Trans2Seg fails to segment transparent objects in some complex scenarios. ### 7.2 More Visual Results Comparison. In this section, we visualize more test examples produced by our Trans2Seg and other CNN-based methods on Trans10K-v2 dataset in Figure 11. From these results, we can easily observe that our Trans2Seg outputs very high-quality transparent object segmentation masks than other methods. Such strong results mainly benefit from the successfully introducing Transformer into transparent object segmentation, which is the lack in other CNN-based methods. $\frac{Scene/Category}{Interaction}$ \| Stuff \| \| Things ---\|---\|---\|--- \| shelf \| freezer \| door \| wall \| window \| box \| \| cup \| bottle \| eyeglass \| jar \| bowl on the desk \| 3 \| 0 \| 0 \| 2 \| 4 \| 227 \| \| 1946 \| 834 \| 239 \| 302 \| 117 mega-mall \| 219 \| 35 \| 450 \| 1762 \| 76 \| 128 \| \| 169 \| 36 \| 75 \| 94 \| 14 store \| 13 \| 36 \| 5 \| 19 \| 3 \| 75 \| \| 444 \| 111 \| 1 \| 175 \| 57 bedroom \| 6 \| 0 \| 4 \| 9 \| 23 \| 2 \| \| 23 \| 33 \| 6 \| 6 \| 1 living room \| 10 \| 0 \| 7 \| 14 \| 19 \| 52 \| \| 310 \| 167 \| 25 \| 139 \| 67 kitchen \| 0 \| 8 \| 6 \| 4 \| 4 \| 19 \| \| 79 \| 23 \| 0 \| 46 \| 66 bathroom \| 0 \| 0 \| 33 \| 31 \| 8 \| 4 \| \| 5 \| 3 \| 4 \| 0 \| 2 windowsill \| 0 \| 0 \| 0 \| 31 \| 209 \| 4 \| \| 17 \| 8 \| 8 \| 17 \| 2 office room \| 15 \| 7 \| 25 \| 43 \| 12 \| 84 \| \| 298 \| 235 \| 51 \| 158 \| 2 office building \| 8 \| 3 \| 1021 \| 1107 \| 131 \| 5 \| \| 1 \| 5 \| 0 \| 2 \| 0 outdoor \| 0 \| 0 \| 13 \| 20 \| 2 \| 0 \| \| 0 \| 2 \| 0 \| 0 \| 0 in the vehicle \| 0 \| 0 \| 2 \| 0 \| 1 \| 0 \| \| 4 \| 0 \| 0 \| 0 \| 0 study-room \| 4 \| 0 \| 3 \| 2 \| 4 \| 1 \| \| 4 \| 1 \| 0 \| 2 \| 0 navigation \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| \| \| \| \| \| grasping \| \| \| \| \| \| \| \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ manipulation \| ✓ \| ✓ \| ✓ \| \| ✓ \| ✓ \| \| \| ✓ \| ✓ \| ✓ \| human-aiding \| \| \| ✓ \| \| \| \| \| ✓ \| ✓ \| ✓ \| \| ✓ Table 6: The upper part of this table: the number of the scene. 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# Programming Boundary Deformation Patterns in Active Networks Zijie Qu1, Jialong Jiang1, Heun Jin Lee2, Rob Phillips1,2,3 Shahriar Shadkhoo1 <EMAIL_ADDRESS>Matt Thomson1<EMAIL_ADDRESS>1Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA, USA. 2Department of Applied Physics, California Institute of Technology, Pasadena, CA, USA. 3Department of Physics, California Institute of Technology, Pasadena, CA, USA. ###### Abstract Active materials take advantage of their internal sources of energy to self- organize in an automated manner. This feature provides a novel opportunity to design micron-scale machines with minimal required control. However, self- organization goes hand in hand with predetermined dynamics that are hardly susceptible to environmental perturbations. Therefore utilizing this feature of active systems requires harnessing and directing the macroscopic dynamics to achieve specific functions; which in turn necessitates understanding the underlying mechanisms of active forces. Here we devise an optical control protocol to engineer the dynamics of active networks composed of microtubules and light-activatable motor proteins. The protocol enables carving activated networks of different shapes, and isolating them from the embedding solution. Studying a large set of shapes, we observe that the active networks contract in a shape-preserving manner that persists over the course of contraction. We formulate a coarse-grained theory and demonstrate that self-similarity of contraction is associated with viscous-like active stresses. These findings help us program the dynamics of the network through manipulating the light intensity in space and time, and maneuver the network into bending in specific directions, as well as temporally alternating directions. Our work improves understanding the active dynamics in contractile networks, and paves a new path towards engineering the dynamics of a large class of active materials. The rich and exotic dynamical behavior of active systems originates from energy consumption at the level of their constituents, which drives them out of equilibrium and endows them with the capability of self-organizing into micron-scale machines [1, 2, 3, 4]. A central goal is to harness the internally-generated dynamics and programming active stresses to accomplish desired tasks, through modulating the system boundaries and forces at macroscopic scales. Biology has served as the major source of inspiration in designing synthetic active systems [5, 6, 7]. In cells, cross-linked polymer networks mediate the active forces that are generated by motor proteins through hydrolyzing ATP. In vitro experiments with cell extracts and reconstituted networks of Microtubules (MTs) and kinesin motor proteins show self-organization into structures including asters and contractile/extensile networks [8, 9, 10, 11]. Mechanical properties of active networks have been extensively studied, experimentally [12, 13, 14, 15, 11, 16, 17, 18] as well as theoretically [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Important questions to be answered include: What modes of dynamics can potentially be probed in a controllable way, and how do we accomplish that? In this paper we address these questions in MT-motor proteins active networks. The interactions of such networks can be categorized into active and passive internal interactions, and network–environment interactions. The latter depend on the specific instrumentation of the experiments, often in an uncontrollable manner. Here, we develop an optical control protocol to activate the motor proteins within a region of illumination, form active MT-motor networks, and isolate them from the surrounding solution. Our strategy utilizes a recently developed optical experimental system to form and isolate active networks of different geometries. Dynamics of the isolated networks are dominated by active stresses with negligible fluid drag; Fig. (1a) [31, 32]. For a large set of distinct geometries we demonstrate that the active networks undergo shape-preserving contractions. Using a hydrodynamic model we demonstrate that the shape preservation is the direct consequence of viscous-like active stresses. The model teaches us how to program active stresses by modulating the light pattern and intensity. Specifically, we design protocols for spatiotemporal modulations of light intensity to achieve static bending, as well as temporally-alternating bending directions in the network. Figure 1: Optical-control protocol first activates cross-linking motor proteins to form the MT networks and isolates the network from embedding solution, allowing them to contract self-similarly. (a) An initial pulse of light activates motor proteins within a region of illumination. Activated motor proteins crosslink the MTs and form a contractile network. Isolation of the network from the solutions requires a second pulse at around $\sim 50-80$s. (b) shows the macroscopic (top row) and microscopic (second row) snapshots of the network, from left to right: during the activation and network formation, at the time of isolation, and shape preserving contraction. The colored dots in the second row track the loci of four distinct microscopic asters in time. The bottom panel shows the profile of the contracting network in time (horizontal axis). The major three phases of the dynamics are separated by vertical lines. (c) For three different shapes the self-similar contraction of networks is portrayed by overlaying the networks’ boundaries as they shrink in time. Figure 2: Self-similarity persists over time and is the consequence of linear scaling of velocity with radius. (a) depicts the algorithm for evaluating self-similarity between two timepoints. The boundaries of the two shapes are extracted. The larger shape (earlier time) is scaled down to match the linear scale of the smaller one. The self-similarity is then found by calculating the correlation between the two shapes. (b) For various geometries (and initial sizes) the deviation from self-similarity $(\delta)$ is measured between $t=100$s and later timepoints $t\leq 360$s (end of contraction). The bars start from zero deviation at $t=100$s and reach their maximum at $t=360$s. All shapes retain their initial shape up to at least $90\%$ accuracy. (c) The magnitude of radial velocity of the contracting networks, at different times. While the radial component increases linearly with distance from the center (hence the cones), the slopes ($\tau_{\xi}^{-1}=\|v_{r}(r)\|/r$) decrease in time. This linearity is assessed in (d): top panel shows the scatter plots of $v_{r}(r,t)$ vs. $r$. The Pearson correlation coefficient $\varrho(r,v_{r})$ is calculated at different timepoints, which varies in time between $0.999$ and $0.952$. The bottom panel shows the relative contribution of the radial and angular components of the velocity. The least contribution of radial component remains above $(v_{r}/v)^{2}\simeq 0.65$. ### Activity preserves the shape memory of contracting networks Performing experiments on several distinct geometries reveals striking universal dynamics that shed light on the underlying active mechanism. We first studied contracting circles as well as polygonal networks (squares, triangles, and hexagons) of different sizes; $460,\,600,\,750$ and $900{\mu}{\text{m}}$. We used a combination of microscopy and image analysis to track and infer network dynamics using labeled MTs. We found that across a wide range of geometries the MT-motor networks generate a contraction that is self-similar, i.e. shape preserving. We realized that the dynamics of networks consist of three phases: (I) Formation of MT-motor contractile networks, the shapes of which are determined by the region of illumination. The activated network is isolated from the background solution by the end of this phase. (II) Contraction phase during which the area of the network decreases over time while density of cross-linked network increases. (III) Deceleration of contraction as the density of filaments, and thus the MT-MT steric interactions increase. During the contractile phases (II and III), the network retains the initial shape of the light pattern. In order to assess self-similarity, we first segment images to find the regions occupied by the networks at different times. Next, for two shapes at timepoints $t_{1}$ and $t_{2}>t_{1}$, with areas $A_{1}$ and $A_{2}<A_{1}$, we scale down the larger shape by $\sqrt{A_{2}/A_{1}}$, and align the centers of the two shapes. Self-similarity is defined as the ratio of the bitwise overlap (AND operator) area, and $A_{2}$ (Fig. (2a)). To account for stochastic rigid rotations of each network around its center of mass, we maximize the self- similarity with respect to relative rotations over the range of $(-20\,,+20)$ degrees. The deviation from self-similarity, $\delta(t_{1},t_{2})$, is calculated by subtracting the self-similarity from unity. Across all networks examined we found that $\delta\lesssim 10\%$ over the entire course of the dynamics; Fig. (2b). The self-similar scaling of the network boundary over time is strongly suggestive of an underlying contractile mechanism that is distinct from those in passive systems. In a passive system, competition between bulk and boundary energies, along with the dissipative drag forces induced by the fluid, lead to distortions in the curvature of the initial network boundary that increase in time. The absence of these “equilibrating” (stress releasing) deformations in our system is indicative of strongly activity-driven dynamics, counteracting the dissipative effects. In comparison to convex shapes that are identified by uniformly positive boundary curvature, the richer geometric features of concave shapes (arcs of positive and negative curvatures) make the deviations from self-similar contraction easier to detect. Furthermore, boundary deformations in concave shapes are more probable to occur due to the bulk-boundary couplings, making the dynamics of concave shapes more informative from a physical perspective. Passive systems with free boundaries equilibrate to round shapes to minimize the sum of the bulk and boundary free energy, and perturbing the boundaries induces stresses in the bulk. Therefore, probing concave active networks provides a more stringent test for verifying the activity-dominated and drag- free contraction. We prepared networks in two concave geometries: hexagrams and cardioids, and found that these shapes contract with self-similarities indistinguishable from those generated in convex networks. In Figure (2b), we show for all shapes, the maximum deviation from self-similarity over the course of contraction measured with respect to the reference time $t=100$s. The deviation from self- similar contraction remains below $10\%$ for all convex shapes—in many cases below $5\%$. Between the two concave shapes, the cardioid shows a very small deviation of $2\%$, the hexagram reaches almost $10\%$ deviation, comparable to triangles and rectangles. The absence of such effects in concave shapes of active networks indicates that the contractile motion of our system is stress- free. More precisely, the contraction corresponds to uniform scaling of the intrinsic metric, in accord with the uniform velocity gradient. Figure 3: Comparison and agreement between experiments and theory supports the role of activity-induced viscous interaction in shape-preserving contraction. (a) For three shapes of hexagram, triangle and ellipse the velocity vector fields are shown as extracted via PIV in experiments (red) and simulated (blue); note the radial velocity in all cases. (b) The theoretical counterpart of the previously shown $v_{r}$ vs. $v$ experimental data in Fig. (2d). (c) For closer comparison the velocity along $x-$axis is plotted at different times for a contracting square. (d) The linear length scale $\xi_{\text{th.}}$ in time as predicted by theory, for different initial sizes. In (e) the fractional contraction $\lambda-\lambda_{\text{eq.}}$ for theoretical results collapse onto a single curve that is compared with those extracted from experiments. The curves decay exponentially over timescale $\tau_{c}$. ### Persistent self-similarity suggests linear radial velocity field High degree of persisting shape preservation suggests spatially-uniform and isotropic contraction of the networks. In accord with self-similarity, we found that the contracting networks generate a velocity field, as inferred from Particle Image Velocimetry (PIV), that remains linear throughout the dynamics across all network shapes and sizes. Specifically, Fig. (2c) shows the radial component of velocity field in the $x-y$ plane, generated by a contracting circle at different times. Similarly in Fig. (2d) top panel, the radial velocity is plotted as a function of distance $r$ from the center of mass. Linearity of velocity is evident from the Pearson correlation coefficient $\varrho(r,v_{r})$ which remains very close to unity. The slope of $v_{r}$ vs. $r$, corresponding to $\tau_{\xi}^{-1}$ in Fig. (2c), changes as a function of time and size of the network, hence the subscript $\xi$. The inverse of this slope ($\tau_{\xi}$) can be interpreted as the time it takes for a network of size $\xi$ to shrink to zero, if the contraction would not decelerate. However the contraction of the network leads to accumulation of mass which slows down the contraction, and $\tau_{\xi}$ diverges at an equilibrium density. For a system with free boundary conditions, locally uniform and isotropic contraction implies zero angular velocities. To verify this, we measured the contributions of radial and angular velocity components; Fig. (2d) bottom panel, and observed that the contribution of angular velocity remains very low for almost the entire course of contraction. In Fig. (2c), for visual clarity, we only show the velocity cones for a circle. However, the linearity of velocity as a function of distance, and the decrease of the slopes in time, hold true across all networks with different shapes and sizes. ### Hydrodynamic model reveals mechanism of universality of self-similar contraction Programming active contractile networks requires quantitative understanding of the response of the system to the external probes, e.g. light in our experiments. To understand how self-similar contractions emerge in response to internally generated stress, we developed and analyzed a coarse-grained hydrodynamic model of active networks. Our phenomenology draws on the following experimentally grounded postulates: (1) Isotropicity: the initially randomly oriented MTs organize small asters that are connected to each other via some intermediate MTs. The asters are, however, connected in random directions. Therefore for length scales of multiple asters size isotropicity seems to be a reasonable assumption; see the zoomed panels in Fig. (1b). (2) Activated motor proteins induce contractile stress. (3) Steric interactions become progressively stronger as the network contracts, and balance out the contractile stress at an equilibrium density of the network. The hydrodynamics of the system is governed by the conservation laws of total mass and momentum, where total refers to the MT network and the fluid. Mass conservation demands $\partial_{t}(\rho_{n}+\rho_{f})=-\nabla\cdot(\rho_{n}\mathbf{v}_{n}+\rho_{f}\mathbf{v}_{f})=0$, where $\partial_{t}$ denotes the partial time derivative, and $\rho_{n/f}$ are network/fluid densities. We drop the network’s subscript hereafter. Neglecting the inertial terms on macroscopic time scales, momentum conservation (force balance) for the network requires $\nabla\cdot\bm{\sigma}^{p}=\gamma(\mathbf{v}-\mathbf{v}_{f}).$ Here $\nabla\cdot\bm{\sigma}^{p}$ is the passive external force exerted from the surrounding fluid on the network, and $\gamma$ is the effective drag coefficient. On the other hand, the viscoelastic response of the network to the total stress reads $\bm{\sigma}^{p}+\bm{\sigma}^{a}=\eta\nabla\mathbf{v}$, in which $\bm{\sigma}^{a}$ is the active stress, and $\eta$ is the effective network viscosity. Under the assumption of $\|\mathbf{v}_{f}\|\ll\|\mathbf{v}\|$, we get $\displaystyle\partial_{t}\,\rho+\nabla\cdot(\rho\mathbf{v})=0,$ (1a) $\displaystyle\eta\nabla^{2}\mathbf{v}-\gamma\mathbf{v}=\nabla\cdot\bm{\sigma}^{a}.$ (1b) The dependency of the active stress on the intensity of light is crucial to programming the dynamics of network. In order to understand this dependency we simulate the dynamics of contractile networks based on the following assumptions, and assess their validity by comparing the results against experiments. Active stress is assumed to be isotropic, namely proportional to the identity matrix $\mathbb{1}$. In 2D we have $\bm{\sigma}^{a}=\frac{1}{2}\text{tr}({\bm{\sigma}^{a}})\mathbb{1}\equiv\sigma^{a}\mathbb{1}$. The active stress can be decomposed into two opposing terms: a contractile term $(\propto\rho)$, and an expansile steric term $(\propto\rho^{2})$. Strictly speaking, steric interactions are not intrinsically active, but emerge due to the activity-induced compression. The proportionality constants are assumed to increase linearly with the density of activated motor proteins, in turn an increasing function of the light intensity. The competition between the contractile and the steric interactions vanishes at an equilibrium density $\rho_{\text{eq.}}$, corresponding to the final size of the network when the contraction stops $\xi_{\text{eq.}}=\xi(t\to\infty)$. Simulating the network contraction over a range of convex and concave shapes we observe self-similar contractions across all geometries. In the activity dominated regime, the model yields a linear velocity field whose magnitude scales linearly with the distance from the network’s center of mass. Specifically, the ratio $\gamma\xi^{2}/\eta$ specifies the relative magnitude of passive and active forces over the longest contractile mode of contraction. In the high-activity regime, the model asymptotically reduces to $\eta\nabla\mathbf{v}=\bm{\sigma}^{a}$, and the velocity field can be solved given a MT network density. For a network of instantaneous size $\xi(t)$, with uniform MT density and free boundary conditions, the solutions of Eqs. (1) are radially symmetric vector fields with constant radial gradient of the form $\|\nabla\mathbf{v}(t)\|=\|v(r=\xi(t))\|/\xi(t)=\|\sigma^{a}(t)\|/\eta$. The linearity of the radial vector field $\mathbf{v}=-\sigma^{a}\,\mathbf{r}/\eta$ persists over the course of dynamics, when the two Eqs. (1) are solved simultaneously. As such, the velocity field generates angle-preserving dynamics: given points $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ in material (Lagrangian) coordinates of the network, their relative position vectors in Eulerian description is scaled by a factor that depends on the time points $t,s$, such that $\left[\mathbf{r}(\mathbf{x}_{1},t)-\mathbf{r}(\mathbf{x}_{2},t)\right]\propto\left[\mathbf{r}(\mathbf{x}_{1},s)-\mathbf{r}(\mathbf{x}_{2},s)\right]$. Thus, the linear velocity field generated in the activity-dominated regime, induces a self-similar, distance scaling map. Figure 4: Contraction of the network can be programmed via modulating the pattern of illumination in space and time. (a)/(b) show purely-spatial modulations of light, where the top segment of the rectangle is illuminated more/less strongly. Greater intensity of light activates larger number of motor proteins and thus generates larger active stresses, which leads to larger and faster contraction on the brighter side, and causes bending. In (c) the pattern of illumination varies in time to interpolate between the two static patterns of (a) and (b). Using this dynamic modulation we manage to change the bending direction as the network contracts. Linear velocity field with density-dependent gradient leads to universal dynamics. The density dependence of the velocity field appears through the stress tensor which determines the instantaneous slope. Active stress $\sigma^{a}$ is proportional to (a) $\rho-\rho_{\text{eq.}}$, and (b) activity $a_{\mathcal{I}}$, determined by the concentration of activated motor proteins, assumed to be proportional to the light intensity $\mathcal{I}$. Together with continuity equation, our model suggests a universality in the velocity field across different shapes and initial conditions. For linear contractions, the density of the MT network remains uniform during the initial phases of the contraction. From continuity equation the density of the network can be expressed as a function of contracted area as $\rho(t)\xi^{2}(t)=\rho_{0}\xi_{0}^{2}=\rho_{\text{eq.}}\xi_{\text{eq.}}^{2}$. Here $t_{0}$ is a reference time at which the density equals $\rho_{0}$. Combined with momentum conservation which determines the velocity field, we obtain: $\rho(t)=\rho_{\text{eq.}}+(\rho_{0}-\rho_{\text{eq.}})\exp(-2(t-t_{0})/\tau_{c})$, where $\tau_{c}$ is the contraction timescale and can be expressed in terms of model parameters $\tau_{c}=\eta/a_{\mathcal{I}}$; i.e. inversely proportional to activity. Correspondingly the linear size of the network can be expressed as $\xi(t)=\xi_{\text{eq.}}+(\xi_{0}-\xi_{\text{eq.}})\exp(-(t-t_{0})/\tau_{c})$. The normalized fractional contraction thus follows an exponential decay of the form $\lambda(t)-\lambda_{\text{eq.}}=\exp(-(t-t_{0})/\tau_{c})$. The results of the simulations for all shapes reproduce the same dynamics as observed in experiments, specifically for $\mathbf{v}(r,t)$ and $\lambda(t)$. The velocity field extracted by PIV from contracting networks, and those obtained from simulations are both linear and radial over shapes and over time; see Fig. (3a) for qualitative comparison. Consistent with experiments, we observe a linear velocity field over complete contractile dynamics for all shapes analyzed, and the divergence/slope of the velocity field decreases with decreasing size, or equivalently increasing density; Fig. (3b,c). In our experiments, we held the initial MT density constant across networks of different sizes and shapes. The fractional contraction for experiments on several shapes, as well as those obtained from the theory, as plotted in Fig. (3d), collapse onto the an exponential curve with decay time of $\tau_{c}$ which is inversely proportional to the activity. Given that activity is an increasing function of the light intensity, we expect the contraction to speed up upon cranking up the intensity. ### Programming deformation through spatial and temporal modulation of activity The hydrodynamic model suggests a simple strategy for programming the mechanical properties of MT networks through spatial-temporal modulation of activity. In our hydrodynamic model, the divergence/slope of the contractile velocity field depends on MT density and the activity $a_{\mathcal{I}}$, which sets the magnitude of stress and thus the contraction timescale. Activity can be modulated experimentally in time and space with light, providing a mechanism to modulate the mechanical behavior of the networks. Spatially- uniform illumination induces uniform activity and isotropic stress which leads to shape-preserving contraction. However, modulation of light pattern in space can generate a nonuniform stress tensor which leads to network regions that contract at different rates. By modulating light levels we modulate the relative local contraction rates which no longer preserve the shape of the network. Specifically, we generated networks where spatially distinct regions experience two different light intensities and, thus, generate two different contractile fields in close proximity. Differing contractile forces along the boundary between the two networks lead to deformation and bending. Thus, by modulating relative activity, we can induce deformation along the boundary of a network and program novel mechanical behaviors that deviate from the self- similar contractions observed in networks at uniform activity. We created a series of light patterns that modulate the relative activity to induce bending deformations. For example, we created a hinge pattern where distinct contractile networks are separated by a joint region, and in the joint region differences in activity lead to relative differences in contractile velocity fields and network bending; Fig. (4a). In a complement hinge pattern, we induce bending along the opposite direction by switching the orientation of the joint, see Fig. (4b). In addition to generating static deformations, spatial and temporal modulation of light patterns allow the generation of dynamical contraction and deformation through temporal modulation of relative activity. In particular, we temporally modulated the relative light intensity in the two regions of the hinge according to the following protocol. First we shine a light pattern that induces downward bending. The light pattern is subsequently swapped to the complementary pattern at around $t=100$s after the initial illumination. The differential intensities lead to reversal of the bending direction. The rates of the bending and reversal depend on the relative sizes of the two regions of illumination, relative light intensities, and the time at which swapping to complementary pattern takes place. Here we chose a relatively straightforward protocol with the same intensities and densities of MTs as chosen in the previously discussed case of self-similar contractions. Broadly, these experiments show that both spatial-temporal modulation of light intensity allows us to induce programmed patterns of mechanical deformation into active MT networks. In this way, the natural shape preservation property of active MT networks can be simply modulated through relative differences in activity in distinct parts of an induced network. This controllability of MT networks allows us to program units of networks in which different possess engineered mechanical properties and can perform work in a programmed and predetermined manner through internal couplings. ### Discussion Active networks are ubiquitous in biology, and their non-equilibrium properties are poorly understood. Our work reveals signature of activity in the mechanical properties at macroscopic scales. The self-similar contraction is intrinsically related to the non-equilibrium nature of the system, which preserves a geometric memory, unlike in passive systems where equilibration increases entropy and erases the memory of the initial state. This memory preservation property makes the behavior of the system more controllable without the need to tuning the microscopic degrees of freedom. Previous works analyzed active contractions in networks of MT and actin in cell extracts, where the contracting network is embedded in a viscous solution, thus subjected to drag forces. Our optical control strategies allow us to isolate the networks from passive boundaries while using light to modulate the shape and activity. Further, in conventional materials altering mechanical properties requires changing the microscopic structure of the material, for example, through doping. These changes are generically irreversible (plastic), and are hard to be modulated at the microscopic level. In our systems, the degree of linking of the network and the active stresses can be tuned in space and time, enabling a separate strategy for the programming and control over material mechanics. Activity induced deformations provide a strategy for engineering novel behaviors at micron length scales. ###### Acknowledgements. The authors are grateful to Inna-Marie Strazhnik for making illustrations, and to John Brady, Dominik Schildknecht and Enrique Amaya Perez for useful discussions. MT was supported by Packard Foundation, Rosen Center for Bioengineering, and Heritage Medical Research Institute. RP was supported by NIH grant number 1R35 GM118043-01. RP and MT would like to thank Foundational Questions Institute and Fetzer Franklin Fund through FQXi 1816 for funding the research. ## Appendix A Instrumentation and Imaging ### A.1 Active Matter System and Sample Chambers The system consists of stabilized microtubule, kinesin motors (constructed with light-induced hetero-dimer system) and an energy mix. All ingredients and buffer preparation protocol are documented in a previous paper by Ross et. al.[32], and we follow the exact same procedure in our study. The sample chambers are made by sandwiching pre-cut Parafilm M by coated slides and coverslips [32, 33]. The measured depth of the chamber is approximately 80$\mu$m. ### A.2 Microscope Instrumentation The experiment is conducted on an epifluorescence microscope (Nikon Ti2) with 10X magnification (Nikon Plan Apo $\lambda$ 10X). We customize the system by adding a programmable digital light projector (EKB Technologies DLP LightCrafter E4500 MKII Fiber Couple), which is used to image the light pattern activating the dimerization of kinesin motors. The DLP chip is illuminated by the four-channeled LED (ThorLabs LED4D067) at the wavelength of 470nm. Fluorescently labeled microtubules are illuminated by 660nm and imaged with digital camera (Hamamatsu orca-flash 4.0). The system is controlled with Micro-Manager on PC. ### A.3 Control Strategy for Isolating the Contracting Network When the light patterns are constantly projected onto the reaction sample, the contraction is accompanied by the formation of canals at the sharp corners of the pattern (e.g. vertices of polygons). These canals pave paths for the background solution—containing floating Mts—to pour into the region of illumination. These MTs get cross-linked upon entering this region, and form a steady state of flow; hence coupling of network and the background fluid. To isolate the cross-linked network from the ambient solution, we decrease the size of the projected pattern to prevent new MT-solution mix from flowing in. As shown in Fig.(1a), we first projected a pattern at full size to initiate the network cross-linking. After 80s, a shrunken pattern is projected, with the same geometry and light intensity, but with 70% linear size of the initial pattern. After this initial phase, the sample is constantly illuminated every 10s with 30ms duration, during which the light pattern is further decreased to 50% original linear size gradually over the course of contraction which stops at 5min. ## Appendix B Image Processing ### B.1 Segmentation and Detection of the Network The time lapse images of contracting network is segmented and isolated from the background solution utilizing a few built-in function of MATLAB Image Processing toolbox. During the contraction (phase II) the boundaries of the network is well separated from the solution that allows for segmentation. The steps are as follows: We first subtract the local background intensity using imflatfield function over regions of sizes of $\sim 300\,({\mu}m)$. This is required to remove artificial shadows. Next we use the watershed algorithm to separate the network from the background fluid. ### B.2 Measuring Velocity and Density Fields Velocity field is extracted at different time points using the built-in MATLAB function imregtform. This function estimates the displacement field $\mathbf{D}_{1\to 2}$ that warps the images at times $t_{1}$ onto the image at $t_{2}$. In the Lagrangian picture for a point labeled by $\mathbf{p}$, we get $\mathbf{r}(\mathbf{p},t_{2})=\mathbf{r}(\mathbf{p},t_{1})+\mathbf{D}_{1\to 2}(\mathbf{p})$. The displacement field is then converted to our units using the pixel value of $0.65\mu$m. We define the velocity field in terms of $\overline{\mathbf{r}}$ and $\overline{t}$, where $\displaystyle\overline{\mathbf{r}}=\frac{1}{2}\,\bigg{(}\mathbf{r}(\mathbf{p},t_{2})+\mathbf{r}(\mathbf{p},t_{1})\bigg{)},\qquad\overline{t}=\frac{1}{2}\,\big{(}t_{1}+t_{2}\big{)}.$ (2) The velocity field reads $\mathbf{v}(\overline{\mathbf{r}},\overline{t})=\mathbf{D}_{1\to 2}(\mathbf{p})\times 0.65/(t_{2}-t_{1})$ (3) In order to measure the velocity and density as a function of distance from center of mass (CoM), the center of mass of the network is found at each time point. Under the assumption that the local density of the network $\rho(\mathbf{r})$, is proportional to the intensity of light captured in gray-scale images $\mathcal{I}(\mathbf{r})$, the center of mass is obtained by ${}\mathbf{R}(t)=\frac{\int_{\text{net.}}d^{2}r\;\mathbf{r}\,\mathcal{I}(\mathbf{r})}{\int_{\text{net.}}d^{2}r\;\mathcal{I}(\mathbf{r})},$ (4) where $\int_{\text{net.}}d^{2}r$ integrates over the area of the network. For later time points when the intensity is saturated; hence not proportional to density, an alternative method is to use the velocity field of the network to estimate the position (and velocity) of the CoM. From Eq. (4), we have: $\mathbf{V}_{\text{CoM}}(t)=M^{-1}\,\int_{\text{net.}}d^{2}r\;\mathbf{v}\,\rho(\mathbf{r}).$ (5) Here $M$ is the total mass of the network—assumed to be conserved during the course of contraction; thus calculable from earlier time points when the density is safely assumed to be proportional to intensity. Redefining the position vector and velocities relative to those of the CoM we get $\mathbf{r}\equiv\mathbf{r}-\mathbf{R}$; and $\mathbf{v}(\mathbf{r},t)\equiv\mathbf{v}(\mathbf{r},t)-\mathbf{v}(\mathbf{R},t)$. Note that although on average the CoM is stationary, on the short timescales it is subject to small and fast random fluctuations due to the noisy background flows. The redefinition of velocity field ensures $\int_{\text{net.}}d^{2}r\,\mathbf{v}(\mathbf{r})\,\mathcal{I}(\mathbf{r})=0$. Therefore the CoM is now determined as the point at which the relative velocity vanishes. To find the velocity as a function of $\|\mathbf{r}\|$ (from CoM), the magnitude of the relative velocity is averaged over all points at a radius. ## References * [1] M Cristina Marchetti, Jean-François Joanny, Sriram Ramaswamy, Tanniemola B Liverpool, Jacques Prost, Madan Rao, and R Aditi Simha. Hydrodynamics of soft active matter. Reviews of Modern Physics, 85(3):1143, 2013. * [2] Sriram Ramaswamy. 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# Momentum dependent mean-fields of (anti)hyperons T. Gaitanos, A. Chorozidou Department of Theoretical Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece email: <EMAIL_ADDRESS> ###### Abstract We investigate the in-medium properties of hyperons and anti-hyperons in the framework of the Non-Linear Derivative (NLD) model. We focus on the momentum dependence of in-medium strangeness optical potentials. The NLD model is based on the simplicity of the well-established Relativistic Mean-Field (RMF) approximation, but it incorporates an explicit momentum dependence on a field- theoretical level. The extension of the NLD model to the (anti)baryon-octet is formulated in the spirit of SU(6) and G-parity arguments. It is shown that with an appropriate choice of momentum cut-offs the $\Lambda$, $\Sigma$ and $\Xi$ optical potentials are consistent with recent studies of the chiral effective field theory and Lattice-QCD calculations over a wide momentum region. In addition, we present NLD predictions for the in-medium momentum dependence of $\overline{\Lambda}$-, $\overline{\Sigma}$\- and $\overline{\Xi}$-hyperons. This work is important for future experimental studies such as CBM, PANDA at the Facility for Antiproton and Ion Research (FAIR). It is relevant for nuclear astrophysics too. ###### keywords: Equations of state of hadronic matter, optical potential, in-medium hyperon potentials. ## 1 Introduction Astrophysical observations on particularly massive neutron stars [1, 2, 3] have driven the nuclear physics and astrophysics communities to detailed investigations of the nuclear equation of state (EoS) under conditions far beyond the ordinary matter [4]. On one hand, theoretical and experimental studies on heavy-ion collisions over the last few decades concluded a softening of the high-density EoS in agreement with phenomenological and microscopic models [5, 6, 7]. On the other hand, the observations of two-solar mass pulsars [1, 2, 3] together with additional constraints on the high- density limit of the speed of sound [8] gave some controversial insights on the EoS of compressed baryonic matter. They provide an upper limit for the neutron star mass by excluding soft-type hadronic EoS’s at high baryon densities. Compressed baryonic matter may consist not only of nucleons. It can include fractions of heavier baryons, when their production is energetically allowed. These are the hyperons $\Lambda,~{}\Sigma$ and $\Xi$ as a part of the irreducible representations of SU(3). While the nucleon-nucleon (NN) interaction is very well known, the hyperon interactions are still not fully understood. Indeed, there are many experimental data for NN-scattering in free space and inside hadronic media (finite nuclei, heavy-ion collisions, hadron- induced reactions) allowing a precise determination of the NN-interaction. Concerning the strangeness sector (hyperon-nucleon (YN) or hyperon-hyperon (YY) interactions), there exist phenomenological and microscopic models with predictions for the in-medium hyperon properties at matter densities close to saturation and at higher densities. However, the experimental access to the strangeness sector is still scarce. A common prediction of theoretical models is a considerable softening of the hadronic EoS at high densities by adding to a system more degrees of freedom such as strangeness particles. The inclusion of hyperons into nuclear approaches made many of them, which were successfully applied to nuclear systems (nuclear matter, finite nuclei, nuclear reactions), incompatible with the astrophysical observations of two-solar mass pulsars [1, 2]. This is the so-called hyperon-puzzle [9, 10]. This puzzle has received recently theoretical attraction by a new observation of a quite massive neutron star [3]. A comprehensive theoretical view concerning the microscopic descriptions of in-medium properties of the baryon-octet is given in Ref. [11]. There exist also theoretical reviews based on the RMF approximation, see for instance Refs. [12, 13, 14]. It is thus of great interest to address the in-medium behaviour of hyperons in nuclear matter, as we do in this work. We use an alternative RMF approach based on the fact, that compressed matter consists of particles with high relative momenta. Therefore, not only the density dependence, but the momentum dependence of the in-medium interactions is important too. The reason for doing so is that conventional RMF-models do not explain the empirical saturation of the in-medium interactions of high-momenta (anti)nucleons. In terms of SU(6) this issue appears for high-momenta (anti)hyperons too. This is the Non-Linear Derivative (NLD) model [15]. It retains the basic RMF Lagrangian formulation, but it includes higher-order derivatives in the NN- interaction Lagrangians. It has been demonstrated that this Ansatz corrects the high-momentum behaviour of the interaction, makes the EoS softer at densities just above saturation, but at the same time it reproduces the two- solar mass pulsars at densities far beyond saturation [15]. Here we extend the NLD approach by including strangeness into the nuclear matter and discuss the momentum dependence of the in-medium hyperon potentials. ## 2 The NLD Model for the baryon octet In this section we briefly introduce the non-linear derivative (NLD) model and extend it to the baryon octet. A detailed description of the NLD model for nucleons can be found in Ref. [15]. The NLD-Lagrangian is based on the conventional Relativistic Hadro-Dynamics (RHD) [16] and it reads as $\displaystyle{\cal L}=$ $\displaystyle\frac{1}{2}\sum_{B}\left[\overline{\Psi}_{B}\gamma_{\mu}i\overrightarrow{\partial}^{\mu}\Psi_{B}-\overline{\Psi}_{B}i\overleftarrow{\partial}^{\mu}\gamma_{\mu}\Psi_{B}\right]-\sum_{B}m_{B}\overline{\Psi}_{B}\Psi_{B}$ $\displaystyle-$ $\displaystyle\frac{1}{2}m^{2}_{\sigma}\sigma^{2}+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-U(\sigma)$ $\displaystyle+$ $\displaystyle\frac{1}{2}m^{2}_{\omega}\omega_{\mu}\omega^{\mu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ $\displaystyle+$ $\displaystyle\frac{1}{2}m^{2}_{\rho}\vec{\rho}\,_{\mu}\vec{\rho}\,^{\mu}-\frac{1}{4}\vec{G}\,_{\mu\nu}\vec{G}\,^{\mu\nu}-\frac{1}{2}m^{2}_{\delta}\vec{\delta}\,^{2}+\frac{1}{2}\partial_{\mu}\vec{\delta}\,\,\partial^{\mu}\vec{\delta}\,$ $\displaystyle+$ $\displaystyle{\cal L}_{int}^{\sigma}+{\cal L}_{int}^{\omega}+{\cal L}_{int}^{\rho}+{\cal L}_{int}^{\delta}\,.$ (1) The sum over $B$ runs over the baryonic octet $\displaystyle\Psi_{B}=$ $\displaystyle(\Psi_{N},\Psi_{\Lambda},\Psi_{\Sigma},\Psi_{\Xi})^{T}$ (2) with $\displaystyle\Psi_{N}=$ $\displaystyle(\psi_{p},\psi_{n})^{T},~{}~{}\Psi_{\Lambda}=\psi_{\Lambda}$ (3) $\displaystyle\Psi_{\Sigma}=$ $\displaystyle(\psi_{\Sigma^{+}},\psi_{\Sigma^{0}},\psi_{\Sigma^{-}})^{T},~{}~{}\Psi_{\Xi}=(\psi_{\Xi^{0}},\psi_{\Xi^{-}})^{T}$ (4) for the isospin-doublets $\Psi_{N}$ and $\Psi_{\Xi}$, isospin-triplet $\Psi_{\Sigma}$ and the neutral $\Psi_{\Lambda}$. The interactions between the nucleon fields are described by the exchange of meson fields. These are the scalar $\sigma$ and vector $\omega^{\mu}$ mesons in the isoscalar channel, as well as the scalar $\vec{\delta}\,$ and vector $\vec{\rho}\,^{\mu}$ mesons in the isovector channel. Their corresponding Lagrangian densities are of the Klein-Gordon and Proca types, respectively. The term $U(\sigma)=\frac{1}{3}b\sigma^{3}+\frac{1}{4}c\sigma^{4}$ contains the usual selfinteractions of the $\sigma$ meson. The notations for the masses of fields in Eq. (1) are obvious. The field strength tensors are defined as $F^{\mu\nu}=\partial^{\mu}\omega^{\nu}-\partial^{\nu}\omega^{\mu}$, $\vec{G}\,^{\mu\nu}=\partial^{\mu}\vec{\rho}\,^{\nu}-\partial^{\nu}\vec{\rho}\,^{\mu}$ for the isoscalar and isovector fields, respectively. In the following we restrict to a minimal set of interaction degrees of freedom. In the iso-scalar sector, the $\sigma$\- and $\omega$-fields are obviously considered. In the iso-vector channel, we keep the vector, iso-vector $\rho$-meson field and neglect the $\delta$-field. The NLD interaction Lagrangians contain the conventional RHD combinations between the bilinear baryon- and linear meson-fields, however, they are extended by the inclusion of non-linear derivative operators $\overrightarrow{{\cal D}},\overleftarrow{{\cal D}}$ for each baryon species $B$: ${\cal L}_{int}^{\sigma}=\sum_{B}\frac{g_{\sigma B}}{2}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\Psi_{B}\sigma+\sigma\overline{\Psi}_{B}\,\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (5) ${\cal L}_{int}^{\omega}=-\sum_{B}\frac{g_{\omega B}}{2}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\gamma^{\mu}\Psi_{B}\omega_{\mu}+\omega_{\mu}\overline{\Psi}_{B}\gamma^{\mu}\,\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (6) ${\cal L}_{int}^{\rho}=-\sum_{B}\frac{g_{\omega\rho}}{2}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\gamma^{\mu}\vec{\tau}\Psi_{B}\vec{\rho}\,_{\mu}+\vec{\rho}\,_{\mu}\overline{\Psi}_{B}\vec{\tau}\gamma^{\mu}\,\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (7) for the isoscalar-scalar, isoscalar-vector and isovector-vector vertices, respectively. The arrows on the non-linear operator ${\cal D}_{B}$ indicate the direction of their action. The only difference with respect to the conventional RHD Lagrangian is the presence of additional operator functions $\overrightarrow{{\cal D}}_{B},~{}\overleftarrow{{\cal D}}_{B}$. As we will see, they will regulate the high momentum component of hyperons. For this reason we will call them as regulators too. The operator functions (or regulators) $\overrightarrow{{\cal D}}_{B},~{}\overleftarrow{{\cal D}}_{B}$ are hermitian and generic functions of partial derivative operator. That is, $\overrightarrow{{\cal D}}_{B}:={\cal D}\left(\overrightarrow{\xi}_{B}\right)$ and $\overleftarrow{{\cal D}}_{B}:={\cal D}\left(\overleftarrow{\xi}_{B}\right)$ with the operator arguments $\overrightarrow{\xi}_{B}=-\zeta_{B}^{\alpha}i\overrightarrow{\partial}_{\alpha},~{}\overleftarrow{\xi}_{B}=i\overleftarrow{\partial}_{\alpha}\zeta_{B}^{\alpha}$. The four vector $\zeta_{B}^{\mu}=v^{\mu}/\Lambda_{B}$ contains the cut-off $\Lambda_{B}$ and $v^{\mu}$ is an auxiliary vector. These regulators are assumed to act on the baryon spinors $\Psi_{B}$ and $\overline{\Psi}_{B}$ by a formal Taylor expansion with respect to the operator argument. The functional form of the regulators is constructed such that in the limit $\Lambda_{B}\to\infty$ the original RHD Lagrangians are recovered, that is, $\overrightarrow{{\cal D}}_{B}=\overleftarrow{{\cal D}}^{\dagger}_{B}\to 1$. The presence of higher-order partial derivatives in the Lagrangian mediate a modification of the field-theoretical prescriptions. As discussed in detail in the original work of Ref. [15], the generalized Euler-Lagrange equations as well as the Noether-currents contain additional infinite terms of higher-order partial derivative contributions. However, the main advantage of the NLD approach relies on the fact that these terms can be resummed to compact expressions. From the generalized Euler-Lagrange formalism we obtain the equations of motion for the degrees of freedom in the NLD model. The meson field equations of motion read $\displaystyle\partial_{\alpha}\partial^{\alpha}\sigma+m_{\sigma}^{2}\sigma+\frac{\partial U}{\partial\sigma}=\frac{1}{2}\sum_{B}g_{\sigma B}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\Psi_{B}+\overline{\Psi}_{B}\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (8) $\displaystyle\partial_{\mu}F^{\mu\nu}+m_{\omega}^{2}\omega^{\nu}=\frac{1}{2}\sum_{B}g_{\omega B}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\gamma^{\nu}\Psi_{B}+\overline{\Psi}_{B}\gamma^{\nu}\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (9) $\displaystyle\partial_{\mu}G^{\mu\nu}+m_{\rho}^{2}\vec{\rho}\,^{\nu}=\frac{1}{2}\sum_{B}g_{\rho B}\left[\overline{\Psi}_{B}\,\overleftarrow{{\cal D}}_{B}\gamma^{\nu}\vec{\tau}\Psi_{B}+\overline{\Psi}_{B}\vec{\tau}\gamma^{\nu}\overrightarrow{{\cal D}}_{B}\Psi_{B}\right]\,,$ (10) for the isoscalar-scalar, isoscalar-vector and isovector-vector exchange mesons, respectively. Each baryon-field obeys a Dirac-equation of the following type $\left[\gamma_{\mu}(i\partial^{\mu}-\Sigma^{\mu}_{B})-(m_{B}-\Sigma_{sB})\right]\psi_{B}=0\;,$ (11) with the selfenergies $\Sigma^{\mu}_{B}$ and $\Sigma_{sB}$ defined as $\displaystyle\Sigma^{\mu}_{B}$ $\displaystyle=$ $\displaystyle g_{\omega_{B}}\omega^{\mu}\overrightarrow{{\cal D}}_{B}+g_{\rho B}\vec{\tau}\,_{B}\cdot\vec{\rho}\,^{\mu}\overrightarrow{{\cal D}}_{B}~{},$ (12) $\displaystyle\Sigma_{sB}$ $\displaystyle=$ $\displaystyle g_{\sigma B}\sigma\overrightarrow{{\cal D}}_{B}\;.$ (13) Both Lorentz-components of the selfenergy, $\Sigma^{\mu}$ and $\Sigma_{s}$, show an explicit linear behaviour with respect to the meson fields $\sigma$, $\omega^{\mu}$ and $\vec{\rho}\,^{\mu}$ as in the standard RHD. However, they contain an additional dependence on the regulators. General expressions for the Noether-current and energy-momentum tensor can also be derived. We give them below in the RMF approximation. The RMF application of the NLD formalism to static hadronic matter follows the same procedure as in the conventional RHD. The spatial components of the meson fields in Minkowski- and isospin-spaces vanish, $\omega^{\mu}\to(\omega^{0},~{}\vec{0}\,)$ and $\vec{\rho}\,^{\mu}\to(\rho^{0}_{3},~{}\vec{0}\,_{3})$. For simplicity, we denote in the following the remaining isospin component of the isovector fields as $\rho^{\mu}$. The solutions of the RMF equations start with the usual plane wave ansatz $\psi_{B}(s,\vec{p}\,)=u_{B}(s,\vec{p}\,)e^{-ip^{\mu}x_{\mu}}$ where $B$ stands for the various isospin states of the baryons and $p^{\mu}=(E,\vec{p}\,)$ is the single baryon 4-momentum. The application of the non-linear derivative operator ${\cal D}_{B}$ to the plane wave Ansatz of the spinor fields results in regulators ${\cal D}_{B}$ which are now functions of the scalar argument $\xi_{B}=-\frac{v_{\alpha}p^{\alpha}}{\Lambda_{B}}$. That is, they depend explicitly on the single baryon momentum $p$ (with an appropriate choice of the auxiliary vector $v^{\alpha}$) and on the cut-off $\Lambda_{B}$, which may differ for each baryon type $B$. Each baryon fulfils a Dirac equation with the same form as in Eq. (11) and with corresponding explicitly momentum dependent scalar and vector selfenergies. Their vector components are given by $\displaystyle\Sigma^{\mu}_{p}=$ $\displaystyle g_{\omega N}\,\omega^{\mu}\,{\cal D}_{N}+g_{\rho N}\,\rho^{\mu}\,{\cal D}_{N}~{},$ (14) $\displaystyle\Sigma^{\mu}_{n}=$ $\displaystyle g_{\omega N}\,\omega^{\mu}\,{\cal D}_{N}-g_{\rho N}\,\rho^{\mu}\,{\cal D}_{N}~{},$ (15) $\displaystyle\Sigma^{\mu}_{\Lambda}=$ $\displaystyle g_{\omega\Lambda}\,\omega^{\mu}\,{\cal D}_{\Lambda}~{},$ (16) $\displaystyle\Sigma^{\mu}_{\Sigma^{+}}=$ $\displaystyle g_{\omega\Sigma}\,\omega^{\mu}\,{\cal D}_{\Sigma}+g_{\rho\Sigma}\,\rho^{\mu}\,{\cal D}_{\Sigma}~{},$ (17) $\displaystyle\Sigma^{\mu}_{\Sigma^{-}}=$ $\displaystyle g_{\omega\Sigma}\,\omega^{\mu}\,{\cal D}_{\Sigma}-g_{\rho\Sigma}\,\rho^{\mu}\,{\cal D}_{\Sigma}~{},$ (18) $\displaystyle\Sigma^{\mu}_{\Sigma^{0}}=$ $\displaystyle g_{\omega\Sigma}\,\omega^{\mu}\,{\cal D}_{\Sigma}~{},$ (19) $\displaystyle\Sigma^{\mu}_{\Xi^{-}}=$ $\displaystyle g_{\omega\Xi}\,\omega^{\mu}\,{\cal D}_{\Xi}-g_{\rho\Xi}\,\rho^{\mu}\,{\cal D}_{\Xi}~{},$ (20) $\displaystyle\Sigma^{\mu}_{\Xi^{0}}=$ $\displaystyle g_{\omega\Xi}\,\omega^{\mu}\,{\cal D}_{\Xi}+g_{\rho\Xi}\,\rho^{\mu}\,{\cal D}_{\Xi}\,.$ (21) Similar expressions result for the scalar selfenergies. In the following the scalar and time-like component of the baryon selfenergy will be denoted as $S_{B}$ and $V_{B}$, respectively. Note that the selfenergies are explicitly momentum dependent due to the regulators ${\cal D}_{B}={\cal D}_{B}(p)$ as specified below. The solutions of the Dirac equation are the standard Dirac- spinors with a proper normalization $N_{B}$ $u_{B}(s,\vec{p}\,)=N_{B}\left(\begin{array}[]{c}\varphi_{s}\\\ \\\ \displaystyle\frac{\vec{\sigma}\,\cdot\vec{p}\,}{E^{}_{B}+m^{}_{B}}\varphi_{s}\\\ \end{array}\right)\;,$ (22) but now for quasi-free baryons $B$ with an in-medium energy $E^{}_{B}:=E_{B}-V_{B}(p)~{},$ (23) and a Dirac mass $m^{}_{B}:=m_{B}-S_{B}(p)~{}.$ (24) At a given momentum the single particle energy $E$ is obtained from the in- medium on-shell relation (23). These expressions are needed for evaluation of expectation values, for instance, the source terms of the meson-field equations. For the definition of the nuclear matter we need a conserved nucleon density. It is obtained from the time-like component of the Noether- current $J^{\mu}$ defined as $\displaystyle J^{\mu}=\frac{\kappa}{(2\pi)^{3}}\,\sum_{B=p,n}\,\int\limits_{\|\vec{p}\,\|\leq p_{F_{B}}}\\!\\!\\!\\!\\!\\!d^{3}p\,\frac{\Pi^{\mu}_{B}}{\Pi^{0}_{B}}$ (25) with the generalized $4$-momentum $\displaystyle\Pi^{\mu}_{B}=p^{\mu}_{B}+m^{}_{B}\Big{(}\partial_{p}^{\mu}S_{B}\Big{)}-\Big{(}\partial_{p}^{\mu}\Sigma^{\beta}_{B}\Big{)}p^{}_{B\beta}$ (26) and the usual effective $4$-momentum $p^{\mu}_{B}=p^{\mu}-\Sigma^{\mu}_{B}\,.$ (27) The EoS (Equation of State) is obtained from the time-like components of the energy-momentum tensor. In nuclear matter the resummation procedure of the NLD model results in the following expression $\displaystyle T^{\mu\nu}=\sum_{B}\frac{\kappa}{(2\pi)^{3}}\int\limits_{\|\vec{p}\,\|\leq p_{F_{B}}}\\!\\!\\!\\!\\!\\!d^{3}p\,\frac{\Pi^{\mu}_{B}p^{\nu}}{\Pi^{0}_{B}}-g^{\mu\nu}\langle{\cal L}\rangle\,,$ (28) from which the energy density $\varepsilon\equiv T^{00}$ and the pressure $P$ can be calculated, see for details Ref. [15]. Finally, the NLD meson-field equations in the RMF approach to nuclear matter can be resummed to the following forms $\displaystyle m_{\sigma}^{2}\sigma+\frac{\partial U}{\partial\sigma}=$ $\displaystyle\sum_{B}g_{\sigma B}\,\Big{<}\overline{\psi}_{B}{\cal D}_{B}\psi_{B}\Big{>}=\sum_{B}g_{\sigma B}\,\rho_{sB}~{},$ (29) $\displaystyle m_{\omega}^{2}\omega=$ $\displaystyle\sum_{B}g_{\omega B}\,\Big{<}\overline{\Psi}_{B}\gamma^{0}{\cal D}_{B}\Psi_{B}\Big{>}=\sum_{B}g_{\omega B}\,\rho_{0B}\,,$ (30) with the scalar and vector density sources $\rho_{sB}=\frac{\kappa}{(2\pi)^{3}}\int\limits_{\|\vec{p}\,\|\leq p_{F_{B}}}\\!\\!\\!\\!\\!\\!d^{3}p\,\frac{m^{}_{B}}{\Pi^{0}_{B}}\,{\cal D}_{B}(p)~{},$ (31) $\rho_{0B}=\frac{\kappa}{(2\pi)^{3}}\int\limits_{\|\vec{p}\,\|\leq p_{F_{B}}}\\!\\!\\!\\!\\!\\!d^{3}p\,\frac{E^{}_{B}}{\Pi^{0}_{B}}\,{\cal D}_{B}(p)\,.$ (32) The isovector densities are calculated through the standard isospin relations. For a hyperon with a given momentum relative to nuclear matter at rest (at a given nucleon density and isospin asymmetry) the mesonic sources contain only nucleons, that is $B=p,n$. The meson-field equations of motion show a similar structure as those of the standard RMF approximation. However, the substantial difference between NLD and other conventional RMF models appears in the source terms which now contain in addition the momentum-dependent regulators ${\cal D}_{B}$. This is an important feature of the NLD model. The cut-off leads naturally to a particular suppression of the vector field at high densities or high Fermi- momenta in agreement with phenomenology, as discussed in detail in the previous work [15]. This feature is absent in conventional RHD approaches, except if one introduces by hand additional scalar/vector self-interactions. The key observable for general discussions related to momentum or energy dependencies of in-medium hadronic potentials is the Schroedinger-equivalent optical potential $U_{opt}$, which is a complex quantity. The imaginary part describes the scattering processes of a given particle, e.g., a hyperon, with a nucleon of the nuclear matter. The real part of the optical potential is related to the mean-field that a particle, e.g., a hyperon with a given momentum, experiences in the nuclear medium at a given density and isospin- asymmetry. The imaginary part of $U_{opt}$ cannot be calculated within a conventional RMF prescription. In RMF models one is usually interested in the real part of an optical potential that can be then examined in more realistic systems, for instance, in heavy-ion collisions or hadron-induced reactions within a relativistic transport theory. The missing imaginary part is then modelled within a collision term in terms of cross sections for elastic, quasi-elastic and inelastic channels with a proper counting of Pauli-Blocking effects. In the NLD model one cannot calculate precisely the imaginary part of $U_{opt}$. However, the NLD approach contains an explicit momentum dependence of the mean-fields, and thus, of the optical potential. This particular feature allow us to give, at least, estimations for the imaginary part of an optical potential too. This will be discussed in the case of the anti- hyperons, and we will mainly focus the study here on the real part of the optical potentials. The real part of the Schroedinger-equivalent optical potential for hyperons is obtained from a non-relativistic reduction of the Dirac-equation and reads $\displaystyle U_{opt}^{B}=-S_{B}+\frac{E_{B}}{m_{B}}V_{B}+\frac{1}{2m_{B}}\left(S_{B}^{2}-V_{B}^{2}\right)\,.$ (33) It describes the in-medium interaction of a baryon species $B$, e.g., a hyperon, with a momentum $p$ (or single-particle energy $E_{B}=E_{B}(p)$, see Eq. (23)) relative to nuclear matter at rest at a given density and isospin asymmetry. We will use Eq. (33) to compare the NLD results with the microscopic calculations from $\chi$-EFT and Lattice-QCD for the hyperon in- medium potentials. ## 3 Results and discussion ### 3.1 Nucleonic sector NLD parameters \| $\Lambda_{sN}$ \| $\Lambda_{vN}$ \| $g_{\sigma N}$ \| $g_{\omega N}$ \| $g_{\rho N}$ \| $b$ \| $c$ ---\|---\|---\|---\|---\|---\|---\|--- $[\mbox{$\,{\rm GeV}$}]$ \| $[\mbox{$\,{\rm GeV}$}]$ \| \| \| \| $[\frac{1}{fm}]$ \| $0.95$ \| $1.125~{}~{}$ \| $~{}~{}10.08$ \| $10.13$ \| $3.50$ \| $15.341$ \| $-14.735~{}~{}$ Bulk saturation properties \| $\rho_{sat}$ \| $E_{b}$ \| $K$ \| $a_{sym}$ \| \| \| $[\frac{1}{fm^{3}}]$ \| $[\frac{MeV}{A}]$ \| $[\mbox{$\,{\rm MeV}$}]$ \| $[\mbox{$\,{\rm MeV}$}]$ \| \| \| $0.156$ \| $-15.30$ \| $251$ \| $30$ \| \| \| Table 1: (Top) NLD parameters: meson-nucleon couplings $g_{mN},~{}(m=\sigma,\omega,\rho)$, $\sigma$ self-interaction constants $b,c$, and NLD cut-off for scalar ($\Lambda_{sN}$) and vector ($\Lambda_{vN}$) meson- nucleon isoscalar vertices. The isovector meson-nucleon cut-off is the same as the isoscalar-vector one. (Bottom) Bulk saturation properties of nuclear matter: saturation density $\rho_{sat}$, binding energy per nucleon $E_{b}$, compression modulus $K$ and asymmetry parameter $a_{sym}$ in the NLD model. See Ref. [15] for more details. We briefly give the status of the NLD model for the in-medium nucleons, before starting the discussion on the in-medium hyperon potentials. As in detail discussed in [15], a momentum dependent monopole form ${\cal D}(p)=\frac{\Lambda^{2}}{\Lambda^{2}+\vec{p\,}^{2}}$ (34) for the regulators turned out to be very effective for a simultaneous description of the low and high density nuclear matter properties. An example is shown in table 1 for the extracted saturation properties together with the model parameters. It is seen that the NLD model leads to a very good description of the empirical values at saturation. The NLD EoS is rather soft and similar to the density dependence of Dirac-Brueckner-Hartree-Fock microscopic calculations. At high densities, however, the NLD EoS becomes stiff. This feature makes a prediction of the maximum mass of neutron stars of $2M_{\odot}$ possible even with a soft compression modulus. Note that the NLD model gives a correct description of the Schroedinger-equivalent optical potential for in-medium protons and antiprotons simultaneously by imposing G-parity only [15]. ### 3.2 Strangeness sector For the strangeness sector we consider again nuclear matter at rest, at a given density, isospin-asymmetry and at zero temperature, in which hyperons ($\Lambda,\Sigma,\Xi$) are situated at a given momentum relative to the nuclear matter at rest. The quantity of interest will be the optical potential $U_{opt}$ of the in-medium hyperons , see Eq. (33). Since there is no experimental information on the momentum dependence of the in-medium hyperonic potentials, we use for our comparisons the recent microscopic calculations from Refs. [17] (see also Ref. [18]) and [19] as a guidance. They are based on the $\chi$-EFT approach in Next-To-Leading (NLO) order and to Lattice-QCD. In the NLD calculations we assume for the in-medium hyperon interactions no additional parameters except of the strangeness cut-off of the hyperons. That is, the various hyperon-nucleon couplings are fixed from the corresponding nucleon-nucleon ones by means of SU(6). The hyperon cut-offs retain their monopole form as in Eq. (34). In particular, they take the form ${\cal D}_{Y}(p)=\frac{\Lambda^{2}_{\gamma_{1}}}{\Lambda^{2}_{\gamma_{2}}+\vec{p\,}^{2}}\,,$ (35) with $\gamma=\sigma,~{}\omega,~{}\rho$ indicating the cut-off values for the hyperon-nucleon $\sigma,~{}\omega$\- and $\rho$-vertices, respectively, and $Y=\Lambda,\Sigma,\Xi$ denotes the hyperon type. In principle, one could use a single cut-off $\Lambda_{\gamma_{1}}=\Lambda_{\gamma_{2}}=\Lambda_{\gamma}$ for each meson-hyperon vertex. However, in order to describe the non-trivial momentum dependence of the microscopic calculations as precise as possible we allow for different cut-off values for the vector-isoscalar $\omega$\- and vector-isovector $\rho$-hyperon vertices, as shown in Eq. (35). For the isoscalar meson-hyperon interactions a single cut-off $\Lambda_{\sigma}=\Lambda_{\sigma_{1}}=\Lambda_{\sigma_{2}}$ for each hyperon type is used. This prescription was found to be the most appropriate one when comparing to the microscopic calculations. In fact, the scalar-like interactions are in any case better controlled with increasing density (respectively momentum) by $m^{\star}/E^{\star}$-suppression factors while the vector-like vertices do not include them, besides the NLD-regulators in the source terms of the meson-field equations (31,32). Note that $\Pi^{0}=E^{\star}$ for momentum-dependent regulators $\Pi^{0}=E^{\star}$ and for each baryon type B. Similar studies concerning the peculiar role of the vector $\omega$-meson exist in the literature. For instance, in Refs. [20, 21, 22] non-linear quadratic $\omega$-field contributions were considered as an alternative approach for the vector-like interaction Lagrangian leading to more complex density dependencies of their mean-fields. In the NLD model all higher-order non-linear terms are summed up into regulators. The novel feature of NLD is that these regulators mediate a non-linear density and, at the same time, a non-linear momentum dependence of in-medium potentials not only for nucleons, but for hyperons too. This will become clear in the following discussions. Figure 1: Optical potential of $\Lambda$-hyperons as function of their momentum $p$ in symmetric nuclear matter at saturation density. The NLD- results (thick-solid curve) are compared with $\chi$-EFT microscopic calculations (taken from [17]) at different orders LO (band with closed dashed borders) and NLO (band with closed solid borders) [17]. Further microscopic calculations from the Jülich group (dot-dashed curve) are shown too [23]. At first, the cut-offs of the hyperons have to be determined. The strangeness-$S=1$ cut-offs are adjusted to the corresponding hyperonic optical potentials at saturation density of symmetric and cold nuclear matter from $\chi$-EFT calculations. This is shown in Fig. 1 for the optical potential of $\Lambda$-hyperons. The gray bands correspond to the microscopic calculations at different orders in $\chi$-EFT, while the solid curve represents the NLD result. At low momenta the $\Lambda$ in-medium interaction is attractive, but it becomes repulsive at high momenta. The non-trivial momentum dependence in NLD arises from the explicitly momentum dependent regulators which show up twice: in the scalar and vector selfenergies and in the source terms of the meson fields. As a consequence, the cut-off regulates the $\Lambda$-potential not only at zero momentum, but particularly over a wide momentum region. The in-medium $\Lambda$-potential does not diverge with increasing $p$-values (not shown here), but it saturates. Furthermore, the in-medium $\Lambda$-potential at zero kinetic energy leads to a value of $U_{opt}^{\Lambda}\simeq-28~{}\mbox{$\,{\rm MeV}$}$, which is consistent with the NLO-calculations and also consistent with phenomenology. Therefore it exists an appropriate choice of cut-off regulators that do reproduce the microscopic calculations over a wide momentum range up to $p\simeq 1~{}\mbox{$\,{\rm GeV}$}$ very well. A similar picture occurs for the in- medium potential of $\Sigma$-hyperons, as shown in Fig. 2. The NLD cut-off for the $\Sigma$-particles can be regulated in such way to reproduce a repulsive potential at vanishing momentum with a weak momentum dependence at finite $\Sigma$-momentum. Again, the NLD calculations are able to describe the microscopic $\chi$-EFT results in NLO very well. The corresponding values for the strangeness cut-offs are tabulated in 2. Even if the origin of the cut- offs is different between the NLD model and the microscopic calculations, it may be interesting to note that these NLD cut-off values are close to the region between 500 and 650 $\,{\rm GeV}$ used in the $\chi$-EFT calculations. $\Lambda$ cut-off \| $\Sigma$ cut-off \| $\Xi$ cut-off ---\|---\|--- $\Lambda_{\sigma}$ \| $\Lambda_{\omega_{1}}$ \| $\Lambda_{\omega_{2}}$ \| $\Lambda_{\rho_{1}}$ \| $\Lambda_{\rho_{2}}$ \| $\Lambda_{\sigma}$ \| $\Lambda_{\omega_{1}}$ \| $\Lambda_{\omega_{2}}$ \| $\Lambda_{\rho_{1}}$ \| $\Lambda_{\rho_{2}}$ \| $\Lambda_{\sigma}$ \| $\Lambda_{\omega_{1}}$ \| $\Lambda_{\omega_{2}}$ \| $\Lambda_{\rho_{1}}$ \| $\Lambda_{\rho_{2}}$ 0.7 \| 0.85 \| 0.79 \| – \| – \| 0.67 \| 0.95 \| 0.79 \| 0.47 \| 0.47 \| 0.6 \| 0.8 \| 0.71 \| 1.3 \| 1.2 \| \| \| \| \| \| \| \| 0.63 \| 0.5 \| \| \| \| \| Table 2: $\Lambda$, $\Sigma$ and $\Xi$ cut-offs for $\sigma$\- ($\Lambda_{\sigma}$), $\omega$\- ($\Lambda_{\omega_{1,2}}$) and $\rho$-hyperon-nucleon ($\Lambda_{\rho_{1,2}}$) vertices in units of $\,{\rm GeV}$. In the cases for $\Sigma$ and $\Xi$ the isospin cut-offs ($\Lambda_{r1,2}$) are relevant for the charged particles only. For the $\Sigma$-hyperon different cut-off values $\Lambda_{\rho_{1,2}}$ are used for $\Sigma^{-}$ (upper line) and for $\Sigma^{+}$ (bottom line). Figure 2: Same as in Fig. 1, but for $\Sigma$-hyperon. We emphasize again the non-trivial momentum dependence of the in-medium hyperon-potentials, as manifested in the $\chi$-EFT calculations at different orders, see for instance Ref. [17]. This prescription modifies the momentum dependencies in such a complex way, which cannot be reproduced in standard RMF models by imposing SU(6) arguments. Furthermore, any standard RMF model leads to a divergent behaviour of optical potentials at high momenta. Note that a weak repulsive character of the $\Sigma$-potential, as proposed by the microscopic calculations, cannot be achieved in conventional RMF. The momentum-dependent NLD model resolves these issues effectively through momentum cut-offs of natural hadronic scale. Since we are dealing with hadronic matter, values of hadronic scale in the $\,{\rm GeV}$-regime for the NLD regulators seem to be an adequate choice. So far we have discussed the momentum dependence of the $\Lambda$ and $\Sigma$ hyperons at saturation density (Figs. 1 and 2). These comparisons served also as a guideline for the NLD cut-offs for the $\Lambda$ and $\Sigma$ baryons. Now we discuss the predictive power of the NLD approach by comparing in more detail the density and momentum dependence of the NLD formalism with the microscopic $\chi$-EFT calculations. This is shown in Figs. 3 and 4, where the momentum dependence of the $\Lambda$ (Fig. 3) and $\Sigma$ (Fig. 4) particles is displayed again, but now at various densities of symmetric nuclear matter. At first, the $\Lambda$ and $\Sigma$ optical potentials become more repulsive with increasing nuclear matter density in NLD. However, the non-trivial momentum and density dependence, as manifested in the NLD selfenergies and the meson-field sources, weakens the in-medium potentials with increasing momentum. In particular, the NLD model predicts astonishingly well the complex microscopic behaviours in momentum and at various densities of symmetric nuclear matter. In asymmetric matter besides the standard iso-scalar and iso-vector vertices ($\sigma$ and $\omega$ meson fields, respectively) the iso-vector and Lorentz- vector $\rho$-meson must be taken into account. In NLD we assume again a monopole form for the $\rho$-meson coupling to the hyperons too by using the coupling constant of table 1 and the cut-off values of table 2 for the isospin sector. Relevant are the cut-off values $\Lambda_{\rho_{1,2}}$ for the charged $\Sigma^{\pm}$-hyperons. They have been fixed from the corresponding $\chi$-EFT calculations for $\Sigma^{-}$ and $\Sigma^{+}$ at saturation density. The NLD calculations for the neutral $\Lambda$\- and $\Sigma^{0}$-hyperons are free of parameters here. The results for pure neutron matter at three different baryon densities are summarized in Fig. 5. The NLD model does predict the general microscopic trends. In particular, in the case of the neutral hyperons ($\Lambda$ and $\Sigma^{0}$), where within the RMF approximation the $\rho$-meson does not appear at all, one would expect identical results between symmetric and pure neutron matter (at same total baryon density and momentum). This is in fact not the case. There is an inherent isospin dependence in the source terms of the meson-field equations, see Eqs. (30) even for the $\sigma$\- and $\omega$-fields. The upper limits in those integrals (31, 32) are different for protons and neutrons between symmetric and asymmetric nuclear matter at the same total density. This leads to a different cut value in the regulators ${\cal D}_{p,n}$ and thus to a different result between symmetric and asymmetric matter. This NLD feature induces a hidden isospin dependence which is qualitatively consistent with the microscopic calculations at the three total densities as indicated in Fig. 5 for the ”isospin-blind” hyperons. Concerning the charged $\Sigma^{\pm}$-hyperons, the comparison between NLD and $\chi$-EFT calculations is obviously at best for densities close to saturation. In general, the NLD predictions follow satisfactorily the details of the microscopic in-medium potentials as function of momentum and matter density. Finally we discuss the in-medium properties of the cascade-hyperons as shown in Figs. 6 and 7 for symmetric nuclear matter (SNM) and pure neutron matter (PNM). Here we apply for comparison the latest microscopic calculations from Lattice-QCD. The same NLD scheme with appropriate monopole-type regulators leads to the results in Fig. 6 for symmetric nuclear matter at saturation density. It is seen that a simple monopole-like regulator with hadronic cut- off values can explain the microscopic Lattice calculations. Indeed, a soft attractive potential for in-medium $\Xi$-hyperons is obtained in the NLD model over a wide momentum range. The prediction of NLD is then displayed in Fig. 7 for pure neutron matter but at the same total density at saturation as in the previous figure. The hidden isospin-dependence modifies slightly the momentum dependence of the neutral $\Xi^{0}$-hyperon. In this case the Lattice calculations are reproduced only qualitatively by the NLD model, while for the charged cascade partner ($\Xi^{-}$) the comparison between NLD and Lattice is very well for pure neutron matter at saturation and over a broad region in single-particle cascade-momentum. In the future experiments such as those at FAIR the in-medium properties of anti-hadrons will be investigated too. We thus give predictions for anti- hyperon in-medium potentials too. We recall the novel feature of the NLD formalism [15], that is, a parameter free predictions for anti-baryon optical potentials in the spirit of G-parity. In fact, once the cut-off parameters are fixed from saturation properties, the application of NLD to anti-matter gave very successful results by imposing G-Parity only. Note that in conventional RMF models one has to introduce by hand additional scaling factors in order to reproduce the weak attractiveness of the anti-proton optical potential at vanishing momenta [24]. We therefore use the same NLD formalism for the description of anti-hyperons too and performed additional calculations for the $\overline{\Lambda}$, $\overline{\Sigma}$ and $\overline{\Xi}$ optical potentials as function of momentum and density. These results are shown in Fig. 8 for anti-$\Lambda$ (left), anti-$\Sigma$ (middle) and anti-$\Xi$ optical potentials versus their momentum at three densities of symmetric nuclear matter. Due to the negative sign in the Lorentz-vector component of the hyperon self-energy these potentials are in general attractive over a wide momentum range. Compared to the anti-proton potential at saturation these potentials are less attractive with a similar dependence on single-particle momentum. Since for anti-hyperons we make predictions and for anti-particles in general one may expect significant contributions to the imaginary part of $U_{opt}$ too, we briefly discuss the imaginary part of the anti-hyperon optical potentials too. An exact treatment of the imaginary part of the optical potential is not possible within an RMF model. However, within the NLD approach one can estimate the strength of $Im~{}U_{opt}$ from dispersion relations [15]. This prescription was successfully applied to the antiproton case in a previous work (see Ref. [15]), thus we apply it here for the anti- hyperons too. The results for $Im~{}U_{opt}$ are shown in the same figure 8 by the thin curves. One generally observes a strong contribution to the in-medium anti-hyperon interactions from the imaginary parts of the optical potentials too. These contributions are quite similar to the imaginary potential of antiprotons with a value around -150 $\,{\rm MeV}$ at very low kinetic energies (see for instance in [15] the second citation of 2015). However, in the antiproton-case the imaginary potential is rather strong relative to its real part, while for anti-hyperons both parts of the potential are sizeable. Even if the NLD results for the $Im~{}U_{opt}$ are only estimations, we can give a physical interpretation. In antinucleon-nucleon scattering annihilation can occur through the production of light pions. On the other hand, the interaction of anti-hyperons with nucleons can happen via the production of the heavier kaons due to strangeness conservation, which may influence the imaginary potential at low energies. This might be one reason why the imaginary part of the anti-hyperon optical potential is comparable with its corresponding real part particularly at very low energies. These calculations can be applied to anti-hadron induced reactions in the spirit of relativistic transport theory and can be tested in the future experiments at FAIR. Figure 3: Optical potential of $\Lambda$-hyperons versus their momentum at various densities of symmetric nuclear matter, as indicated by the Fermi- momenta in units of 1/fm-1. The NLD calculations at these three Fermi-momenta (thick-solid, thick-dashed and thick-dot-dashed curves) are compared to the $\chi$-EFT calculations at NLO [17]. Figure 4: Same as in Fig. 3, but for the $\Sigma$-hyperons. Figure 5: Optical potentials for hyperons (as indicated) versus their momentum for pure neutron matter. Solid curves with symbols indicate the NLD calculations while pure curves without symbols are the microscopic $\chi$-EFT results at NLO from Ref. [17]. Green pairs (circles- solid for NLD and solid for $\chi$-EFT) refer to low density of $p_{F}=1$ fm-1, red pairs (diamonds-dashed for NLD and dashed for $\chi$-EFT) refer to saturation density of $p_{F}=1.35$ fm-1 and blue pairs (triangles-dot-dashed for NLD and dot-dashed for $\chi$-EFT) refer to a density of $p_{F}=1.53$ fm-1. Figure 6: Optical potential of cascade hyperons versus their momentum for symmetric nuclear matter (SNM) at saturation density. The solid curve indicates the NLD predictions while the dashed curve and the gray band refer to recent Lattice calculations from Refs. [19] (Lattice2016 and Lattice2019). Figure 7: Same as in Fig. 6, but for pure neutron matter (PNM). The curves and bands belonging to $\Xi^{-}$ and $\Xi^{0}$ are indicated in this figure. Figure 8: Optical potentials for anti-hyperons versus their kinetic energy for symmetric nuclear matter (SNM) at various densities, as indicated. The NLD predictions for saturation density $\rho_{0}$ (thick-solid) and higher densities of $2\rho_{0}$ (thick-dashed) and $3\rho_{0}$ (thick-dashed-dot) are shown. For the anti-hyperons we show estimates for the imaginary part of their optical potentials too at saturation density $\rho_{0}$ (thin-solid), at $2\rho_{0}$ (thin-dashed) and at $3\rho_{0}$ (thin-dashed-dot). ## 4 Summary We have investigated the properties of strangeness particles inside nuclear matter in the framework of the NLD approach. The NLD model is based on the simplicity of the relativistic mean-field theory, but it includes the missing momentum dependence in a manifestly covariant fashion. This is realized by the introduction of non-linear derivative series in the interaction Lagrangian. In momentum space this prescription leads to momentum dependent regulators, which are determined by a cut-off. The NLD approach does not only resolve the optical potential issues of protons and antiprotons at high momenta, but it affects the density dependence. That is, the cut-off regulators make the EoS softer at densities close to saturation and stiffer at very high densities relevant for neutron stars. Because of the successful application of the NLD model to infinite nuclear matter (and to finite nuclei [25]), it is a natural desire to extend this approach to hadronic matter by taking strangeness degrees of freedom into account. This is realized in the spirit of SU(6) symmetry. We applied the NLD model to the description of in-medium hyperon interactions for ordinary nuclear matter. It was found that the strangeness cut-off regulates the momentum dependence of the optical potentials of hyperons in multiple ways. At first, the optical potentials do not diverge with increasing hyperon momentum. Furthermore, the NLD model predicts an attractive $\Lambda$-optical potential at low momenta, which becomes repulsive at high energies and finally saturates. In particular, it is possible to predict a weak and repulsive in- medium interaction for $\Sigma$-hyperons inside nuclear matter at saturation density. These results are in consistent agreement with calculations based on the chiral effective field theory. Regarding $\Xi$-hyperons, the NLD predictions turned out to be in agreement with recent Lattice-QCD calculations. In symmetric nuclear matter the cascade optical potential is attractive and it follows the Lattice-QCD results. In pure neutron matter the isospin-separation as predicted by the NLD model agrees with the Lattice-QCD behaviours qualitatively. While the potential of the neutral cascade particle remains attractive, the $\Xi^{-}$-hyperon shows a weak repulsion in neutron matter. The weak repulsion of those hyperons may likely effect to a stiffer EoS for neutron star matter. We briefly discussed the imaginary part of $U_{opt}$ of anti-hyperons too. These estimations indicate a significant contribution of the imaginary part to the anti-hyperon dynamics that could be explored in anti-hadron induced reactions. For instance, the present calculations can be tested in anti-proton induced reactions and in reactions with secondary $\Xi$-beams, as they are planned at FAIR in the future PANDA experiment. Obviously this study is relevant not only for hadron physics, but also for nuclear astrophysics. The application of the NLD approach to $\beta$-equilibrated compressed matter is under progress, in order to investigate the hyperon-puzzle in neutron stars. Another interesting application concerns the dynamics of neutron star binaries. To do so, an extension to hot and compressed hadronic matter is necessary and under progress too. 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# Collaborative Teacher-Student Learning via Multiple Knowledge Transfer Liyuan Sun Jianping Gou<EMAIL_ADDRESS>School of Computer Science and Telecommunication Engineering, Jiangsu University, Zhenjiang, 212013, China Baosheng Yu Lan Du Dacheng Tao Faculty of information technology, Monash University, Australia UBTECH Sydney AI Centre, School of Computer Science, Faculty of Engineering, The University of Sydney, Darlington, NSW 2008, Australia. ###### Abstract Knowledge distillation (KD), as an efficient and effective model compression technique, has been receiving considerable attention in deep learning. The key to its success is to transfer knowledge from a large teacher network to a small student one. However, most of the existing knowledge distillation methods consider only one type of knowledge learned from either instance features or instance relations via a specific distillation strategy in teacher-student learning. There are few works that explore the idea of transferring different types of knowledge with different distillation strategies in a unified framework. Moreover, the frequently used offline distillation suffers from a limited learning capacity due to the fixed teacher-student architecture. In this paper we propose a collaborative teacher-student learning via multiple knowledge transfer (CTSL-MKT) that prompts both self-learning and collaborative learning. It allows multiple students learn knowledge from both individual instances and instance relations in a collaborative way. While learning from themselves with self-distillation, they can also guide each other via online distillation. The experiments and ablation studies on four image datasets demonstrate that the proposed CTSL-MKT significantly outperforms the state-of-the-art KD methods. ###### keywords: ## 1 Introduction Deep neural networks have achieved state-of-the-art performance on many applications such as computer vision, natural language processing, and speech recognition in recent years. The remarkable performance of deep learning relies on designing deeper or wider network architectures with many layers and millions of parameters to enhance the learning capacity. However, it is almost impossible to deploy the large-scale networks on platforms with limited computation and storage resources, e.g., mobile devices and embedded systems. Thus, the model compression and acceleration techniques mainly including network pruning [2, 3], model quantization [34, 35] and knowledge distillation [20, 21, 39, 40, 36] are proposed for training lightweight deep models. Among compressing methods, knowledge distillation, which carries out knowledge transfer from a high-capacity teacher network to a low-capacity student one, has received increasing interest recently since it was first introduced in [20]. In knowledge distillation, the type of knowledge, the distillation strategy and the teacher-student architecture are three crucial factors that determine the KD performance [1]. As pointed out in [1], there are three kinds of knowledge, i.e., the response-based, the feature-based and the relation-based knowledge. Generally, most of KD methods distill the response-based knowledge (e.g., soft logits of the output layer) from a large teacher network and transfer it to a small student [20, 32, 22]. To overcome the limitation of knowledge from the output layer of teacher, the feature-based knowledge from the middle layers of teacher is also used to train the student [21, 18, 19]. Unlike both the response-based and the feature-based knowledge from individual instances, the relation-based knowledge from instance relations is modelled for improving student learning [37, 38, 15, 16, 17]. Each kind of knowledge can provide student training with an informative teacher guidance, and they can also compensate each other to enrich learning. However, most existing KD methods only consider either knowledge from individual instance features to maintain instance consistency between teacher and student or knowledge from instance relations to preserve the instance correlation consistency. There are a few works that consider more than one kind of knowledge in knowledge distillation [37, 14] at the same time and explore the efficacy of each kind of knowledge. Transferring different types of knowledge can be implemented with different distillation methods, e.g., offline distillation, online distillation and self-distillation [1]. Most of the KD methods employ offline distillation, which is one-way knowledge transfer from a pre-trained large teacher to a small student [20, 22, 13]. In offline distillation, the capacity gap caused by a fixed teacher-student architecture and the requirement of a large dataset for pre-training the teacher often result in a degraded performance [22]. Thus, finding a proper teacher-student architecture in offline distillation is challenging. In contrast, online distillation provides a one-phase end-to-end training scheme via teacher-student collaborative learning on a peer-network architecture instead of a fixed one [25, 33, 36, 32, 28, 12]. Self- distillation performs online distillation within the same network to reduce model over-fitting [23, 24]. Online distillation and self-distillation are promising methods for knowledge distillation as they bridge the capacity gap via avoiding the need of a large teacher network, leading to an improved performance. However, both KD methods used individually are limited to knowledge distillation from a single source, i.e., individual instances, online distillation could further suffer from the poor instance consistency between peer networks caused by the discrepancy in their network outputs. Figure 1: The overview diagram of CTSL-MKT. Consequently, it is desirable to have a unified framework that can integrate the advantages of different KD methods and make efficient use of different types of knowledge. Inspired by the idea of knowledge distillation via multiple distillation strategies to transfer more than one types of knowledge, we propose a collaborative teacher-student learning via multiple knowledge transfer (CTSL-MKT), which fuses self-distillation and online distillation in such a way that the former transfers the response-based knowledge within each peer network and the latter bidirectionally transfers both the response-based knowledge and the relation-based knowledge between peer networks. CTSL-MKT can overcome the aforementioned issued faced by existing KD methods that often use only one distillation strategy to transfer a single type of knowledge. The overview framework of CTSL-MKT is illustrated in Figure 1. To our knowledge, this is the first framework that integrates different distillation strategies together to transfer more than one type of knowledge simultaneously. In CTSL-MKT, each pear network conducts self-learning via self-distillation. Meanwhile, they carry out teacher-student collaborative learning to mutually teach each other. CTSL-MKT can also adopt a variety of peer network architectures, where the two peer networks can either share the same network architecture or have different ones. We believe that multiple knowledge transfer can provide much more informative knowledge to guide each peer network so that they can obtain better performance with a better generalization ability. We conduct a set of image classification experiments on four commonly-used datasets i.e., CIFAR-10, CIFAR-100, Tiny-ImageNet, and Market-1501. Experimental results demonstrate the superior performance of the proposed CTSL-MKT over the state-of-the-art KD methods. The main contributions in our works can be summarized as follows: * • A new teacher-student mutual learning framework effectively fuses the knowledge from individual instances and the knowledge from instance relationships. * • A self-learning enhanced collaborative learning integrates the advantages of both self-learning and online learning. * • The extensive experiments on a variety of the peer teacher-student networks that compare CTSL-MKT with the state-of-the-art methods to validate its effectiveness in image classification tasks. * • A set of ablation studies of different combinations of knowledge and distillation methods provides insights into how multiple knowledge transfer contribute to knowledge distillation. ## 2 Related Work ### 2.1 Self-Distillation Self-distillation is a novel training scheme for knowledge transfer [23, 24, 27, 11, 10]. In self-distillation, the teacher and student networks are identical and knowledge transfer is carried out within the same network. Yuan et al. empirically analyzed the performance of normal, reversed and defective KD methods, and showed that a weak teacher can strengthen the student and vice-versa [23]. A teacher-free knowledge distillation method (Tf-KD) instead makes student model conduct self-learning. To enhance the generalization and overcome over-fitting, class-wise self-knowledge distillation makes use of soft logits of different intra-class samples within a model [27]. Phuong and Lampert [11] proposed a distillation-based training method to reduce time complexity, where the output of later exit layer supervises the early exit layer via knowledge transfer. Rather than at the layer level, snapshot distillation [10] transfers knowledge from earlier to later epochs while training a deep model. Overall, self-distillation can overcome the issue of over-fitting and the capacity gap on the teacher-student architectures, improve the generalization ability and reduce the inference time of a deep model. However, the self- distillation performance could be limited by the one-sided response-based knowledge from model itself. To further improve knowledge distillation, we integrate both online and self-distillation into CTSL-MKT with more informative relation-based knowledge. ### 2.2 Collaborative Learning Recently, there are many new online distillation methods that train a teacher and a student simultaneously during knowledge transfer. Collaborative learning is the one used most often [25, 29, 32, 8, 9, 7], where the teacher and the student as peer networks collaboratively teach and learn from each other, and the peer network architectures can be different. In particular, Zhang et al. [25] proposed a deep mutual learning method (DML) for online distillation using the response-based knowledge. DML uses an ensemble of soft logits as knowledge and transfers it among arbitrary peer networks via collaborative learning [9]. Yao and Sun [8] further extended DML with dense cross-layer mutual-distillation, which learns both the teacher and the student collaboratively from scratch. Unlike the ensemble of peer networks, the advantage of a mutual knowledge distillation method is that it can fuse features of peer networks to collaboratively learn a powerful classifier [32]. However, the knowledge distilled by those online mutual distillation methods is limited to the response-based knowledge from individual instance features. In contrast, our work can further make use of the relation-based knowledge from instance relationships to further enrich the transferred knowledge. ### 2.3 Structural Knowledge Most of knowledge distillation approaches adopt the output logits of a deep model from individual samples as knowledge and make the logits of the teacher and student match each other. However, such response-based knowledge ignores the structural knowledge from the mutual relations of data examples, known as relation-based knowledge. In recent years, there are some newly proposed knowledge distillation methods based on structural relations of data samples [26, 30, 31, 6, 5, 4]. Park et al. [26] proposed a relational knowledge distillation method (RKD), which transfers the instance relation knowledge from a teacher to a student. Chen et al. [4] borrowed the idea of manifold learning to design a novel knowledge distillation method, in which the student preserves the feature embedding similarities of samples from the teacher. Peng et al. [31] designed a knowledge transfer method that makes sure the student matches the instance correlation consistently with the teacher. However, those structural knowledge distillation methods often ignore the knowledge directly from individual samples. Our proposed CTSL-MKT instead considers the knowledge from both individual instances and instance relationships, and the bidirectional knowledge transfer is carried out between peer networks via collaborative learning. ## 3 The Proposed CTSL-MKT Method Figure 2: The framework of CTSL-MKT with two peer networks. Note that the losses $L_{KL}(\emph{{p}}_{1},\emph{{p}}_{2})$ and $L_{KL}(\emph{{p}}_{2},\emph{{p}}_{1})$ are for mutual learning via the response-based knowledge transfer, $L_{RD}$ for mutual learning via the relation-based knowledge transfer, $L_{SD}^{k}(\emph{{p}}_{k}^{t},\bar{\emph{{p}}}_{k}^{t})$ for self-learning via the response-based knowledge transfer. CTLS-MKT unifies student self-learning and teacher-student mutual learning under one framework in such a way that it can utilise multiple types of knowledge and more than one distillation methods during the teacher-student learning. The teacher-student architectures used in CTSL-MKT are peer networks, such as ResNet [41] and MobilenetV2 [42]. Different from previous works, CTSL-MKT distills both the response-based knowledge from individual instance features and the relation-based knowledge from instance relationships. During teacher-student mutual learning, peer networks trained collaboratively can teach each other via online distillation with the two kinds of knowledge. Meanwhile, each peer network can also self-learn via self- distillation with the response-based knowledge. The two leaning processes working together can complement each other to explore different knowledge space and enhance the learning. Our CTSL-MKT can be seen as a new model compression technique that generalises the existing self-distillation and online distillation methods and enables fast computations and improves the generalization ability. The overall framework of CTSL-MKT with two peer networks is shown in Figure 2 as an example, and notations are summarized in Table 1. Notations Descriptions $X=\\{x_{1},x_{2},\cdots,x_{n}\\}$ $n$ input samples from $m$ classes $\emph{{y}}=\\{y^{1},y^{2},\cdots,y^{m}\\}$ the one-hot label vector for $x\in X$ $\emph{{z}}_{k}(x)=\\{z_{k}^{1},z_{k}^{2},\ldots,z_{k}^{m}\\}$ logits of a network $N_{k}$ for $x\in X$ where $z_{k}^{i}$ is the logit for class $i$ $\sigma_{i}(\emph{{z}}_{k}(x),t)$ softmax function with temperature $t$ $p_{k}^{it}=\sigma_{i}(\emph{{z}}_{k}(x),t)$ output of softmax for $z_{k}^{i}$ $\emph{{p}}_{k}^{t}=\\{p_{k}^{1t},p_{k}^{2t},\ldots,p_{k}^{mt}\\}$ predictions of $N_{k}$ with $t$ $\emph{{p}}_{k}=\\{p_{k}^{1},p_{k}^{2},\ldots,p_{k}^{m}\\}$ predictions of $N_{k}$ when $t=1$ $f(s_{1}^{k},s_{2}^{k},\cdots,s_{n}^{k})$ similarity loss of $n$ samples in $N_{k}$ Table 1: Notations used in CTSL-MKT. ### 3.1 Teacher-Student Mutual Learning Teacher-student mutual learning contains the response-based knowledge transfer and the relation-based knowledge transfer among peer network architectures. Response-Based Knowledge Transfer: The response-based knowledge (i.e., the output of a peer network) is learned from individual instances. Given a peer network $N_{k}$ and its output $\emph{{p}}_{k}$ with temperature parameter $t=1$, the collaborative response-based knowledge transfer makes the student network $N_{k}$ imitate the teacher network $N_{k^{\prime}}$ ($k\neq k^{\prime}$) with the following Kullback-Leibler (KL) divergence loss, $\small L_{KL}(\emph{{p}}_{k},\emph{{p}}_{k^{\prime}})=\sum_{x\in X}\sum_{i=1}^{m}\sigma_{i}(\emph{{z}}_{k^{\prime}}(x),1)log\frac{\sigma_{i}(\emph{{z}}_{k^{\prime}}(x),1)}{\sigma_{i}(\emph{{z}}_{k}(x),1)}.$ (1) Similarly, the loss that the student network $N_{k^{\prime}}$ uses to learn from the teacher network $N_{k}$ is $L_{KL}(\emph{{p}}_{k^{\prime}},\emph{{p}}_{k})$. During the collaborative learning for a classification task, each peer network $N_{k}$ will then be trained with both the KL divergence loss (Eq (1)) and the cross-entropy (CE) loss (Eq (2)). $L_{CE}(\emph{{y}},\emph{{p}}_{k})=-\sum_{x\in X}\sum_{i=1}^{m}y^{i}log(\sigma_{i}(\emph{{z}}_{k}(x),1))~{}.$ (2) Take two peer networks in Figure 2 as example, the losses used to train $N_{1}$ and $N_{2}$ will be $L_{CE}(\emph{{y}},\emph{{p}}_{1})+L_{KL}(\emph{{p}}_{1},\emph{{p}}_{2})$ and $L_{CE}(\emph{{y}},\emph{{p}}_{2})+L_{KL}(\emph{{p}}_{2},\emph{{p}}_{1})$, respectively. Relation-Based Knowledge Transfer: CTSL-MKT further integrates the relation- based knowledge learned from the instance relationships via the teacher- student mutual leaning in order to enrich the transferred knowledge and enhance the teacher guidance. Let $s_{j}^{k}=\phi_{k}(x_{j})$ (where $\phi_{k}(.)$ is a feature mapping function of $N_{k}$) be the output of any layer of the network $N_{k}$ for $x_{j}$, and $\chi^{\tau}$ denote a set of $\tau$-tuples of different samples. A set of $2$-tuples and a set of $3$-tuples thus correspond to $\chi^{2}=\left\\{(x_{u},x_{v})\|u\neq v\right\\}$ and $\chi^{3}=\left\\{x_{u},x_{v},x_{w})\|u\neq v\neq w\right\\}$, respectively. As in [26], the relation-based knowledge learned by the network $N_{k}$ can be modelled jointly by a distance-wise function and an angle-wise function. Given $N_{k}$, the distance-wise function captures the similarities between two samples in a $2$-tuple, which is defined as $f(s_{u}^{k},s_{v}^{k})=\frac{1}{\pi}\|\|s_{u}^{k}-s_{v}^{k}\|\|_{2}~{},$ (3) where $\pi=\frac{1}{\|\chi^{2}\|}\sum_{(x_{u},x_{v})\in\chi^{2}}\|\|s_{u}^{k}-s_{v}^{k}\|\|_{2}$ is a normalization constant. Accordingly, the instance relationships between any two peer networks $N_{k}$ and $N_{k^{\prime}}$ are transferred by the following distance-wise distillation loss $L_{DD}(x_{u},x_{v})=\sum_{(x_{u},x_{v})\in\chi^{2}}R\big{(}f(s_{u}^{k},s_{v}^{k}),f(s_{u}^{k^{\prime}},s_{v}^{k^{\prime}})\big{)}~{},$ (4) where $R(.)$ is Huber loss that reflects instance relationships and is defined as $R(a,b)=\left\\{\begin{array}[]{lr}\frac{1}{2}(a-b)^{2},\quad if~{}\|a-b\|\leq 1&\\\ \|a-b\|-\frac{1}{2},\quad otherwise&\end{array}\right.~{}.$ (5) Furthermore, the similarities between samples in a $3$-tuple are measured by an angle-wise function $f(s_{u}^{k},s_{v}^{k},s_{w}^{k})=\cos\angle s_{u}^{k}s_{v}^{k}s_{w}^{k}=<e^{uv},e^{wv}>~{},$ (6) where $e^{uv}=\frac{s_{u}^{k}-s_{v}^{k}}{\|\|s_{u}^{k}-s_{v}^{k}\|\|_{2}}$ and $e^{wv}=\frac{s_{w}^{k}-s_{v}^{k}}{\|\|s_{w}^{k}-s_{v}^{k}\|\|_{2}}$. The instance relationships are transferred between any two peer networks $N_{k}$ and $N_{k^{\prime}}$ with the angle-wise distillation loss, defined as $\displaystyle L_{AD}(x_{u},x_{v},x_{w})$ (7) $\displaystyle=$ $\displaystyle\sum_{(x_{u},x_{v},x_{w})\in\chi^{3}}R\big{(}f(s_{u}^{k},s_{v}^{k},s_{w}^{k}),f(s_{u}^{k^{\prime}},s_{v}^{k^{\prime}},s_{w}^{k^{\prime}})\big{)}~{}.$ It has been shown that the relation-based knowledge transfer can be more effective if the distance-wise function is used jointly with the angle-wise function [26], as they capture different degrees of similarities between samples. We formulate the instance relation distillation loss used in the collaborative learning between peer networks as $L_{RD}=L_{DD}(x_{u},x_{v})+\beta_{1}L_{AD}(x_{u},x_{v},x_{w})~{},$ (8) where $\beta_{1}$ is a tuning parameter that controls the balance between loss terms. Consequently, the mutual distillation loss with both the response-based and the relation-based knowledge between two peer networks ($N_{k}$ and $N_{k^{\prime}}$) is defined as: for network $N_{k}$, we have $L_{MD}^{k}=L_{RD}+\beta_{2}L_{KL}(\emph{{p}}_{k},\emph{{p}}_{k^{\prime}})~{},$ (9) where $\beta_{2}$ is a tuning parameter; for network $N_{k^{\prime}}$, we have $L_{MD}^{k^{\prime}}=L_{RD}+\beta_{2}L_{KL}(\emph{{p}}_{k^{\prime}},\emph{{p}}_{k})~{}.$ (10) Algorithm 1 The proposed CSL-MKT 0: Input samples $X$ with labels, learning rate $\eta$, hyperparameters $\alpha$, $\beta$, $\gamma$, $\beta_{1}$ and $\beta_{2}$. 1: Initialize: Initialize peer networks $N_{1}$ and $N_{2}$ to different conditions. 2: Stage 1: Pre-train $N_{1}$ and $N_{2}$ for use of the process of self- learning. 3: for k=1 to 2 do 4: Repeat: 5: Compute stochastic gradient of $L_{CE}$ in Eq. (2) and update $N_{k}$: 6: $N_{k}\leftarrow\text{$N_{k}$}$+$\eta$$\frac{\partial L_{CE}}{\partial N_{k}}$. 7: Until: $L_{CE}$ converges. 8: end for 9: Stage 2: Train $N_{1}$ and $N_{2}$ collaboratively. 10: Repeat: 11: for k=1 to 2 do 12: Compute stochastic gradient of $L_{KD}^{k}$ in Eq. (12) and update $N_{k}$: 13: $N_{k}\leftarrow\text{$N_{k}$}$+$\eta$$\frac{\partial L_{KD}^{k}}{\partial N_{k}}$. 14: end for 15: Until: $L_{KD}^{k}$ converges. 16: return _$N_{k}$_. ### 3.2 Student Self-learning During the collaborative learning between peer networks, if the outputs of peer networks are very diverse, the mutual knowledge transfer could become poor. Since the self-learning via self-distillation can improve the power of knowledge transfer [23], CTSL-MKT further introduces the self-learning of each peer network into collaborative learning via the response-based knowledge self-distillation. To conduct self-learning for each peer network $N_{k}$, we use the outputs $\bar{\emph{{p}}}_{k}^{t}$ of the pre-trained network $N_{k}$ to supervise itself with the following self-distillation loss: $\small L_{SD}^{k}(\emph{{p}}_{k}^{t},\bar{\emph{{p}}}_{k}^{t})=\sum_{x\in X}\sum_{i=1}^{m}\sigma_{i}^{t}(\bar{\emph{{z}}}_{k}(x),t)log\frac{\sigma_{i}^{t}(\bar{\emph{{z}}}_{k}(x),t)}{\sigma_{i}^{t}(\emph{{z}}_{k}(x),t)}~{}.$ (11) ### 3.3 The CSL-MKT Algorithm Finally, CSL-MKT conducts mutual learning and self-learning simultaneously in a unified framework, as shown in Algorithm 1. Its objective function for each pear network is defined as $L_{KD}^{k}=\alpha L_{CE}^{k}+\beta L_{MD}^{k}+\gamma L_{SD}^{k}~{},$ (12) where $\alpha$, $\beta$ and $\gamma$ are the tuning parameters, which balance the contribution of each loss in the collaborative learning, $L_{CE}^{k}$ is defined in Eq. (2), $L_{MD}^{k}$ for two peer networks in Eqs. (9) or (10), and $L_{SD}^{k}$ in Eq. (11). ## 4 Experiments Network \| Parameter size \| Baseline ---\|---\|--- (CIFAR-100) \| B_10 \| B_100 \| B_Tiny ResNet14 \| 6.50M \| 94.94 \| 76.12 \| - ResNet18 \| 11.22M \| 95.13 \| 75.77 \| 62.90 ResNet34 \| 21.33M \| 95.39 \| 77.66 \| - VGG19 \| 139.99M \| 92.83 \| 69.42 \| - MobileNetV2 \| 2.37M \| 90.97 \| 68.23 \| - ShuffleNetV2 \| 1.36M \| 91.03 \| 70.10 \| - AlexNet \| 57.41M \| - \| - \| 50.25 SqueezeNet \| 0.77M \| - \| - \| 43.68 Table 2: The parameter size of each peer network on CIFAR-100 and its classification performance on three datasets. Note that B_10, B_100, and B_Tiny denote Top-1 accuracy (%) achieved by each peer network on CIFAR-10, CIFAR-100 and Tiny-ImageNet, respectively. We conducted extensive experiments to verify the effectiveness of CTSL-MKT on image classification tasks using datasets including CIFAR-10 [47], CIFAR-100 [47], Tiny-ImageNet [48] and Market-1501 [49]. The peer network architectures were chosen from ResNet [41], MobileNet [42], ShuffleNet [43], VGG [45], AlexNet [44] and SqueezeNet [46]. CTSL-MKT was compared to the state-of-the- art KD methods, which are DML [25], Tf-KD [23] and RKD [26]. For a fair comparison, RKD uses online distillation with the peer networks. In all the experiments, the relation-based knowledge was modelled by the final feature embedding outputs by the peer networks. ### 4.1 Datasets and Settings CIFAR-10 and CIFAR-100. Both datasets have 60,000 $32\times 32$ images, where 50,000 images are for training and the other 10,000 images are for testing. The number of classes is 10 for CIFAR-10 and 100 for CIFAR-100, and each class has the same numbers of samples in both the training and the testing sets. On each dataset, data augmentation with random crops and horizontal flips was used to change the zero-padded $40\times 40$ images to $32\times 32$ ones. The peer networks were trained for 200 epochs with batch size 128 and initial learning rate 0.1 which is then multiplied by 0.2 at 60, 120, and 160 epochs. Temperature parameter was set to 10 for CIFAR-10 and 3 for CIFAR-100. Tiny-ImageNet. Tiny-ImageNet contains 100,000 training and 10,000 testing $64\times 64$ images from 200 classes, each of which has the same number of samples. Each image was randomly resized to $224\times 224$. The peer networks were trained for 90 epochs with batch size 64 and initial learning rate 0.1 which is then multiplied by 0.1 at 30, 60, 80 epochs. Temperature parameter was set to 2. Market-1501. Market-1501 includes 32,688 images taken from 1,501 identities under condition of six camera views. It has 751 identities for training and 750 ones for testing. Each image was zero-padded by 10 pixels on each side, and data augmentation with random crops and horizontal flips was used to change the zero-padded images to $256\times 128$ ones. The peer networks were trained for 60 epochs with batch size 32 and initial learning rate 0.05 which is then multiplied by 0.2 at 40 epochs. Temperature parameter was set to 6. We used the SGD optimizer for training the peer networks with momentum 0.9 and weight decay 5e-4, and all input images were normalized by Mean-Std normalization. All the hyper-parameters were greedily searched and set as follows. The hyper-parameters used on CIFAR and Tiny-ImageNet were set as $\alpha=0.1$, $\beta=0.05$ and $\gamma=0.9$ for MobileNet and ShuffleNet, and $\alpha=0.4$, $\beta=0.4$ and $\gamma=0.6$ for the other networks. On Market-1501, the hyper-parameters were set as $\alpha=1$, $\beta=0.9$ and $\gamma=1$ for all the networks. Besides, both $\beta_{1}$ and $\beta_{2}$ were set to 2. In all the experiments, we considered a pair of peer networks that have the same architecture or different architectures. Table 2 shows the parameter size of each network on CIFAR-100 and its top-1 accuracy on the two CIFAR datasets and the Tiny-ImageNet dataset, which serves as a baseline. ### 4.2 Results on CIFAR-10 Table 3 reports the average Top-1 accuracy of CTSL-MKT and the state-of-the- art competitors on CIFAR-10. It is not surprising that those knowledge distillation methods perform better than the corresponding single network due to the knowledge transfer, except for DML and RKD with ResNet14 and ResNet18. The possible reason for the slightly poor performance of DML and RKD with ResNet14 or ResNet18 could be that the discrepancies between outputs of small peer networks for individual instances hinder mutual learning. Among those knowledge distillation methods, CTSL-MKT performs the best with a significant improvement, which indicates that our idea of collaborative learning with multiple knowledge transfer is effective. Meanwhile, CTSL-MKT outperforms all the corresponding baselines shown in Table 2 with a noticeable margin. For example, CTSL-MKT with ShuffleNetV2-MobileNetV2 has increased the Top-1 accuracy by 1.68% and 1.18%, compared with the corresponding signal network baselines, i.e., ShuffleNetV2 and MobileNetV2 respectively. Moreover, although CTSL-MKT, DML, and RKD collaboratively learn the two peer networks, which can have the same network structures or different ones, each peer network in CTSL- MKT (i.e., CTSL-MKT_$N_{1}$ or CTSL-MKT_$N_{2}$) performs much better than its counterpart in DML and RKD, due to the multiple knowledge transfer. Network $N_{1}$ Network $N_{2}$ Tf-KD DML_$N_{1}$ DML_$N_{2}$ RKD_$N_{1}$ RKD_$N_{2}$ CTSL-MKT_$N_{1}$ CTSL-MKT_$N_{2}$ ResNet14 ResNet14 95.08$\pm$0.01 94.78$\pm$0.02 94.92$\pm$0.01 94.95$\pm$0.01 94.83$\pm$0.02 95.28$\pm$0.01 95.22$\pm$0.01 ResNet18 ResNet18 95.20$\pm$0.01 94.88$\pm$0.01 94.99$\pm$0.01 94.98$\pm$0.04 94.92$\pm$0.01 95.29$\pm$0.04 95.33$\pm$0.03 ResNet34 ResNet34 95.41$\pm$0.01 95.42$\pm$0.01 95.32$\pm$0.01 95.45$\pm$0.01 95.45$\pm$0.01 95.69$\pm$0.03 95.59$\pm$0.01 MobileNetV2 MobileNetV2 91.72$\pm$0.01 91.19$\pm$0.07 91.32$\pm$0.04 91.12$\pm$0.03 90.71$\pm$0.06 92.12$\pm$0.02 92.12$\pm$0.02 ShuffleNetV2 ShuffleNetV2 92.47$\pm$0.01 91.97$\pm$0.03 91.92$\pm$0.01 92.08$\pm$0.01 91.59$\pm$0.01 92.64$\pm$0.01 92.49$\pm$0.01 ResNet18 ResNet34 - 95.09$\pm$0.01 95.41$\pm$0.03 95.12$\pm$0.01 95.31$\pm$0.01 95.24$\pm$0.01 95.60$\pm$0.01 ResNet18 VGG19 - 95.11$\pm$0.03 93.49$\pm$0.02 95.03$\pm$0.01 93.50$\pm$0.01 95.16$\pm$0.01 93.91$\pm$0.01 ShuffleNetV2 MobileNetV2 - 91.78$\pm$0.03 91.25$\pm$0.08 91.73$\pm$0.01 90.72$\pm$0.01 92.71$\pm$0.01 92.15$\pm$0.01 Table 3: The average Top-1 accuracy (%) over three individual runs on CIFAR-10. ### 4.3 Results on CIFAR-100 Table 4 reports the average Top-1 accuracy of all the competing knowledge distillation methods with various network architectures on CIFAR-100. We have similar observations as those on CIRFAR-10. Overall, each competing method improves on the performance of the corresponding baseline, and CTSL-MKT gains the largest improvement. Compare to Tf-KD, DML and RKD, Top-1 accuracy of each peer network in CTSL-MKT has been improved by about 1% on average. For example, the accuracy of the two MobileNetV2 networks in CTSL-MKT has been increased by 2.46% and 2.68% respectively, compared to those in DML, and increased by 3.08% and 2.96% Top-1 accuracy compared to those in RKD. Network $N_{1}$ Network $N_{2}$ Tf-KD DML_$N_{1}$ DML_$N_{2}$ RKD_$N_{1}$ RKD_$N_{2}$ CTSL-MKT_$N_{1}$ CTSL-MKT_$N_{2}$ ResNet14 ResNet14 76.67$\pm$0.02 75.97$\pm$0.01 76.16$\pm$0.11 76.36$\pm$0.03 76.30$\pm$0.01 77.00$\pm$0.05 76.85$\pm$0.04 ResNet18 ResNet18 77.04$\pm$0.12 76.10$\pm$0.10 76.27$\pm$0.07 76.43$\pm$0.01 76.09$\pm$0.01 77.43$\pm$0.10 77.46$\pm$0.01 ResNet34 ResNet34 77.93$\pm$0.01 77.88$\pm$0.12 77.61$\pm$0.03 77.63$\pm$0.03 77.65$\pm$0.05 78.58$\pm$0.01 78.24$\pm$0.02 MobileNetV2 MobileNetV2 70.82$\pm$0.02 68.98$\pm$0.01 68.58$\pm$0.18 68.36$\pm$0.01 68.30$\pm$0.01 71.44$\pm$0.06 71.26$\pm$0.08 ShuffleNetV2 ShuffleNetV2 71.79$\pm$0.02 70.47$\pm$0.15 70.29$\pm$0.04 70.24$\pm$0.01 69.98$\pm$0.03 72.13$\pm$0.02 71.69$\pm$0.05 ResNet18 ResNet34 - 76.15$\pm$0.10 77.71$\pm$0.01 76.41$\pm$0.05 77.83$\pm$0.01 77.61$\pm$0.08 78.15$\pm$0.12 ResNet18 VGG19 - 76.51$\pm$0.02 68.80$\pm$3.74 76.29$\pm$0.02 68.28$\pm$0.87 77.23$\pm$0.02 72.72$\pm$0.06 ShuffleNetV2 MobileNetV2 - 70.47$\pm$0.13 68.83$\pm$0.14 70.50$\pm$0.28 67.87$\pm$0.01 72.46$\pm$0.15 71.34$\pm$0.09 Table 4: The average Top-1 accuracy (%) over three individual runs on CIFAR-100. (a) ShuffleNetV2 (b) MobileNetV2 Figure 3: The values of training loss over epochs on the CIFAR-100 training dataset. (a) ShuffleNetV2 (b) MobileNetV2 Figure 4: The values of Top-1 accuracy over epochs on the CIFAR-100 testing dataset. To further illustrate the learning process of the peer networks in CTSL-MKT, Figure 3 plots the training loss of ShuffleNetV2 and MobileNetV2 as a function of epochs, compared to Tf-KD, DML and RKD. It shows that CTSL-MKT and Tf-KD converge better than DML and RKD. The possible reason is that each network can self-learn in CTSL-MKT and Tf-KD to overcome the discrepancy in the outputs of peer networks in DML and RKD during learning. Although CTSL-MKT with multiple knowledge transfer introduces extra hyper-parameters, it can still converge faster, achieving comparable training loss with Tf-KD. The loss becomes stable around 120 epochs in general. Furthermore, Figure 4 displays the corresponding Top-1 accuracy of each peer network after each epoch on the testing dataset. It shows that the proposed CTSL-MKT outperforms the others after the convergence, its performance improves along with the decrement of the training loss. Overall, the patterns show that two peer networks in CTSL-MKT can work collaboratively, via teaching and learning from each other at each epoch, and each network gradually improves itself to achieve a better performance. ### 4.4 Results on Tiny-ImageNet Table 5 shows the average Top-1 accuracy of the competing methods with five various peer network architectures on Tiny-ImageNet. From the comparative results, it can be seen that CTSL-MKT significantly outperforms the baselines, DML, RKD and Tf-KD. However, on these five peer network architectures, some peer networks in DML, RKD and Tf-KD achieve poor performance, compared to their baselines. The possible reason is that the used peer networks are smaller with less informative knowledge in knowledge transfer and the outputs of peer networks for the same individual instances migth be different, which degrades the effectiveness of mutual learning. With multiple kinds of knowledge and distillation strategies, our CTSL-MKT can well improve the performance via mutual learning. Network $N_{1}$ Network $N_{2}$ Tf-KD DML_$N_{1}$ DML_$N_{2}$ RKD_$N_{1}$ RKD_$N_{2}$ CTSL-MKT_$N_{1}$ CTSL-MKT_$N_{2}$ ResNet18 ResNet18 63.29$\pm$0.02 62.30$\pm$0.01 62.39$\pm$0.03 62.80$\pm$0.01 62.42$\pm$0.08 63.63$\pm$0.08 63.64$\pm$0.02 AlexNet AlexNet 49.78$\pm$0.01 44.47$\pm$0.01 44.80$\pm$0.01 43.54$\pm$0.01 42.97$\pm$0.01 51.39$\pm$0.01 51.28$\pm$0.01 SqueezeNet SqueezeNet 41.66$\pm$0.01 47.16$\pm$0.03 46.95$\pm$0.19 48.22$\pm$0.01 48.55$\pm$0.09 48.60$\pm$0.30 48.86$\pm$0.03 AlexNet SqueezeNet - 44.35$\pm$0.51 46.15$\pm$0.30 44.66$\pm$1.87 46.86$\pm$0.41 50.98$\pm$0.08 47.99$\pm$0.03 ResNet18 AlexNet - 62.62$\pm$0.11 43.53$\pm$0.62 62.37$\pm$0.01 46.64$\pm$0.03 63.37$\pm$0.01 51.56$\pm$0.02 Table 5: The average Top-1 accuracy (%) over three individual runs on Tiny- ImageNet. ### 4.5 Results on Market-1501 We further compared those methods on Market-1501, which is used for a re- identification (re-id) task. In this set of experiments, we adopted ResNet50 that is usually used on this dataset to form a peer network architecture. Figure 5 shows the performance of Tf-KD, DML, RKD and CTSL-MKT, measured by Rank-1, Rank-5, Rank-10 and mAP. Note that these results were computed for only one peer network. DML and RKD with collaborative learning consistently perform better than Tf-KD via self-learning. Our CTSL-MKT outperforms both DML and RKD across all the metrics. Specifically, in terms of mAP, the improvement of CTSL-MKT over DML, RKD and Tf-KD is 1.22%, 0.8% and 4.69%, respectively. Figure 5: Comparative results (%) on Market-1501. ### 4.6 Ablation Study CTSL-MKT contains three knowledge distillation strategies, i.e., mutual learning via response-based knowledge transfer from individual instances (MLI), mutual learning via relation-based knowledge transfer from instance relationships (MLR) and self-learning via response-based knowledge transfer from individual instances (SLI). To study how each strategy contributes to the model performance, we consider the following four variants of CTSL-MKT: 1. _A_) the full model using the three strategies altogether, where we used both online distillation and self-distillation with the two kinds of knowledge; 2. _B_) the model using online distillation only with both the response-based knowledge (MLI) and the relation-based knowledge (MLR); 3. _C_) the model using online distillation with the relation-based knowledge (MLR) and self-distillation with the response-based knowledge (SLI); 4. _D_) the model using both online distillation and self-distillation with only the response-based knowledge, corresponding to MLI + SLI. Table 6 reports the average Top-1 accuracy of these four variations with different pairs of peer network architectures. We have the following observations: 1) Variant A (i.e., the full model) outperforms the other variants where one knowledge distillation strategy has been removed. It implies that the use of multiple types of knowledge with both online distillation and self-distillation plays a crucial role in the performance gain. 2) Variant B without self-distillation has the largest performance drop, compared with variants C and D. It indicates that self-distillation contributes substantially to the overall performance as it could offset the diversity issue caused by mutual learning and further enhance the knowledge distillation efficiency. 3) DML, Tf-KD and RKD can be seen as a special case of CTSL-MKT using only one knowledge distillation strategy. Jointly looking at Tables 6 and 4 reveals that knowledge distillation methods with two or more strategies almost always outperform those using only one strategy. Therefore, it is clear that knowledge distillation via proper multiple knowledge transfer is very beneficial for improving the performance of model compression. Case MLI MLR SLI ResNet14 ResNet14 ResNet18 ResNet18 MobileNetV2 MobileNetV2 A ✓ ✓ ✓ 77.00$\pm$0.05 76.85$\pm$0.04 77.43$\pm$0.10 77.46$\pm$0.01 71.44$\pm$0.06 71.26$\pm$0.08 B ✓ ✓ ✗ 76.57$\pm$0.04 76.37$\pm$0.02 76.67$\pm$0.04 76.66$\pm$0.09 69.10$\pm$0.01 69.23$\pm$0.04 C ✗ ✓ ✓ 76.69$\pm$0.02 76.70$\pm$0.04 77.35$\pm$0.04 77.29$\pm$0.08 71.18$\pm$0.05 71.10$\pm$0.08 D ✓ ✗ ✓ 76.73$\pm$0.03 76.70$\pm$0.08 77.26$\pm$0.07 77.12$\pm$0.04 71.14$\pm$0.02 71.04$\pm$0.10 (a) Ablation experiments on the same peer network architectures Case MLI MLR SLI ResNet14 ResNet18 ResNet18 ResNet34 ShuffleNetV2 MobileNetV2 A ✓ ✓ ✓ 77.07$\pm$0.03 77.28$\pm$0.04 77.61$\pm$0.08 78.15$\pm$0.12 72.46$\pm$0.15 71.34$\pm$0.09 B ✓ ✓ ✗ 76.35$\pm$0.01 76.53$\pm$0.07 76.53$\pm$0.02 77.83$\pm$0.01 70.57$\pm$0.03 68.69$\pm$0.01 C ✗ ✓ ✓ 76.69$\pm$0.23 77.06$\pm$0.02 77.12$\pm$0.01 77.99$\pm$0.03 72.06$\pm$0.04 71.10$\pm$0.11 D ✓ ✗ ✓ 76.68$\pm$0.03 77.13$\pm$0.02 77.39$\pm$0.01 77.68$\pm$0.01 72.20$\pm$0.09 71.05$\pm$0.15 (b) Ablation experiments on the different peer network architectures Table 6: Ablation study of CTSL-MKT in terms of the average Top-1 accuracy over three individual runs on CIFAR-100. ### 4.7 Experiment Discussion The experimental results reported above have demonstrated the effectiveness of the proposed CTSL-MKT, while being compared with several state-of-the-art knowledge distillation methods. We have the following remarks: * • Collaborative learning can make peer networks teach and learn from each other, and iteratively improve themselves. * • Self-learning of each peer network can further enhance the ability of mutual learning among peer networks by compensating the loss caused by the diversity issue. * • Multiple knowledge transfer with more than one types of knowledge and distillation strategies can significantly improve the KD performance. * • Various peer network architectures (i.e., teacher-student architectures) can be easily adopted for knowledge transfer via collaborative learning. ## 5 Conclusions In this paper, we propose a novel knowledge distillation method called collaborative teacher-student learning via multiple knowledge transfer (CTSL- MKT). 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# A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems Carsten Carstensen, Rekha Khot and Amiya K. Pani22footnotemark: 2 Department of Mathematics, Humboldt-Universität zu Berlin, 10099 Berlin, Germany. Email: <EMAIL_ADDRESS>of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076. Email<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain $\Omega$ is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space $H^{1}_{0}(\Omega)$ by a right-inverse $J$ of the interpolation operator $I_{h}$. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the $H^{1}$ and $L^{2}$ norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh- refinement in comparison to the uniform mesh-refinement. Keywords: second-order linear indefinite elliptic problems, virtual elements, nonconforming, polytopes, enrichment, stability, a priori error estimates, a residual-based a posteriori error estimate, adaptive mesh-refinement. AMS subject classifications: 65N12, 65N15, 65N30, 65N50. ## 1 Introduction The nonconforming virtual element method approximates the weak solution $u\in H^{1}_{0}(\Omega)$ to the second-order linear elliptic boundary value problem $\displaystyle{\cal L}u:=-\text{div}(\textbf{A}\nabla u+\textbf{b}u)+\gamma u=f\quad\mbox{in}\quad\Omega$ (1.1) for a given $f\in L^{2}(\Omega)$ in a bounded polygonal Lipschitz domain $\Omega\subset{\mathbb{R}}^{2}$ subject to homogeneous Dirichlet boundary conditions. ### 1.1 General introduction The virtual element method (VEM) introduced in [4] is one of the well-received polygonal methods for approximating the solutions to partial differential equations (PDEs) in the continuation of the mimetic finite difference method [7]. This method is becoming increasingly popular [1, 6, 5, 16, 17, 3] for its ability to deal with fairly general polygonal/polyhedral meshes. On the account of its versatility in shape of polygonal domains, the local finite- dimensional space (the space of shape functions) comprises non-polynomial functions. The novelty of this approach lies in the fact that it does not demand for the explicit construction of non-polynomial functions and the knowledge of degrees of freedom along with suitable projections onto polynomials is sufficient to implement the method. Recently, Beirão da Veiga et al. discuss a conforming VEM for the indefinite problem (1.1) in [6]. Cangiani et al. [17] develop a nonconforming VEM under the additional condition $\displaystyle 0\leq\gamma-\frac{1}{2}\text{div}(\textbf{b}),$ (1.2) which makes the bilinear form coercive and significantly simplifies the analysis. The two papers [6, 17] prove a priori error estimates for a solution $u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ in a convex domain $\Omega$. The a priori error analysis for the nonconforming VEM in [17] can be extended to the case when the exact solution $u\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ with $\sigma>1/2$ as it is based on traces. This paper shows it for all $\sigma>0$ and circumvents any trace inequality. Huang et al. [31] discuss a priori error analysis of the nonconforming VEM applied to Poisson and Biharmonic problems for $\sigma>0$. An a posteriori error estimate in [16] explores the conforming VEM for (1.1) under the assumption (1.2). There are a few contributions [16, 9, 34] on residual-based a posteriori error control for the conforming VEM. This paper presents a priori and a posteriori error estimates for the nonconforming VEM without (1.2), but under the assumption that the Fredholm operator ${\cal L}$ is injective. ### 1.2 Assumptions on (1.1) This paper solely imposes the following assumptions (A1)-(A3) on the coefficients $\textbf{A},\textbf{b},\gamma$ and the operator ${\cal L}$ in (1.1) with $f\in L^{2}(\Omega)$. 1. (A1) The coefficients $\textbf{A}_{jk},\textbf{b}_{j},\gamma$ for $j,k=1,2$ are piecewise Lipschitz continuous functions. For any decomposition $\cal{T}$ (admissible in the sense of Subsection $2.1$) and any polygonal domain $P\in\cal{T}$, the coefficients $\textbf{A},\textbf{b},\gamma$ are bounded pointwise a.e. by $\\|\textbf{A}\\|_{\infty},\\|\textbf{b}\\|_{\infty},\\|\gamma\\|_{\infty}$ and their piecewise first derivatives by $\|\textbf{A}\|_{1,\infty},\|\textbf{b}\|_{1,\infty},\|\gamma\|_{1,\infty}$. 2. (A2) There exist positive constants $a_{0}$ and $a_{1}$ such that, for a.e. $x\in\Omega$, $\textbf{A}(x)$ is SPD and $\displaystyle a_{0}\|\xi\|^{2}\leq\sum_{j,k=1}^{2}\textbf{A}_{jk}(x)\xi_{j}\xi_{k}\leq a_{1}\|\xi\|^{2}\quad\text{for all}\;\xi\in{\mathbb{R}}^{2}.$ (1.3) 3. (A3) The linear operator ${\cal L}:H^{1}_{0}(\Omega)\to H^{-1}(\Omega)$ is injective, i.e., zero is not an eigenvalue of ${\cal L}$ . Since the bounded linear operator ${\cal L}$ is a Fredholm operator [30, p. 321], (A3) implies that ${\cal L}$ is bijective with bounded inverse ${\cal L}^{-1}:H^{-1}(\Omega)\to H^{1}_{0}(\Omega)$. The Fredholm theory also entails the existence of a unique solution to the adjoint problem, that is, for every $g\in L^{2}(\Omega)$, there exists a unique solution $\Phi\in H^{1}_{0}(\Omega)$ to $\displaystyle{\cal L}^{}\Phi:=-\text{div}(\textbf{A}\nabla\Phi)+\textbf{b}\cdot\nabla\Phi+\gamma\Phi=g.$ (1.4) The bounded polygonal Lipschitz domain $\Omega$, the homogeneous Dirichlet boundary conditions, and (A1)-(A2) lead to some $0<\sigma\leq 1$ and positive constants $C_{\text{reg}}$ and $C^{}_{\text{reg}}$ (depending only on $\sigma,\Omega$ and coefficients of ${\cal L}$) such that, for any $f,g\in L^{2}(\Omega)$, the unique solution $u$ to (1.1) and the unique solution $\Phi$ to (1.4) belong to $H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ and satisfy $\displaystyle\\|u\\|_{1+\sigma,\Omega}\leq C_{\text{reg}}\\|f\\|_{L^{2}(\Omega)}\;\text{ and}\;\;\\|\Phi\\|_{1+\sigma,\Omega}\leq C^{}_{\text{reg}}\\|g\\|_{L^{2}(\Omega)}.$ (1.5) (The restriction $\sigma\leq 1$ is for convenience owing to the limitation to first-order convergence of the scheme.) ### 1.3 Weak formulation Given the coefficients $\textbf{A},\textbf{b},\gamma$ with (A1)-(A2), define, for all $u,v\in V:=H^{1}_{0}(\Omega)$, $a(u,v):=(\textbf{A}\nabla u,\nabla v)_{L^{2}(\Omega)},\hskip 14.22636ptb(u,v):=(u,\textbf{b}\cdot\nabla v)_{L^{2}(\Omega)},\hskip 14.22636ptc(u,v):=(\gamma u,v)_{L^{2}(\Omega)}$ (1.6) and $B(u,v):=a(u,v)+b(u,v)+c(u,v)$ (1.7) (with piecewise versions $a_{\mathrm{pw}},b_{\mathrm{pw}},c_{\mathrm{pw}}$ and $B_{\mathrm{pw}}$ for $\nabla$ replaced by the piecewise gradient $\nabla_{\mathrm{pw}}$ and local contributions $a^{P},b^{P},c^{P}$ defined in Subsection 3.1 throughout this paper). The weak formulation of the problem (1.1) seeks $u\in V$ such that $B(u,v)=(f,v)\quad\text{for all}\;v\in V.$ (1.8) Assumptions (A1)-(A3) imply that the bilinear form $B(\cdot,\cdot)$ is continuous and satisfies an inf-sup condition [11] $\displaystyle 0<\beta_{0}:=\inf_{0\neq v\in V}\sup_{0\neq w\in V}\frac{B(v,w)}{\\|v\\|_{1,\Omega}\\|w\\|_{1,\Omega}}.$ (1.9) ### 1.4 Main results and outline Section $2$ introduces the VEM and guides the reader to the first-order nonconforming VEM on polygonal meshes. It explains the continuity of the interpolation operator and related error estimates in detail. Section $3$ starts with the discrete bilinear forms and their properties, followed by some preliminary estimates for the consistency error and the nonconformity error. The nonconformity error uses a new conforming companion operator resulting in the well-posedness of the discrete problem for sufficiently fine meshes. Section $4$ proves the discrete inf-sup estimate and optimal a priori error estimates. Section $5$ discusses both reliability and efficiency of an explicit residual-based a posteriori error estimator. Numerical experiments in Section $6$ for three computational benchmarks illustrate the performance of an error estimator and show the improved convergence rate in adaptive mesh- refinement. ### 1.5 Notation Throughout this paper, standard notation applies to Lebesgue and Sobolev spaces $H^{m}$ with norm $\\|\cdot\\|_{m,\cal{D}}$ (resp. seminorm $\|\cdot\|_{m,\cal{D}}$) for $m>0$, while $(\cdot,\cdot)_{L^{2}({\cal D})}$ and $\\|\cdot\\|_{L^{2}({\cal D})}$ denote the $L^{2}$ scalar product and $L^{2}$ norm on a domain ${\cal D}$. The space $C^{0}(\cal D)$ consists of all continuous functions vanishing on the boundary of a domain ${\cal D}$. The dual space of $H^{1}_{0}(\Omega)$ is denoted by $H^{-1}(\Omega)$ with dual norm $\\|\cdot\\|_{-1}$. An inequality $A\lesssim B$ abbreviates $A\leq CB$ for a generic constant $C$, that may depend on the coefficients of ${\cal L}$, the universal constants $\sigma$, $\rho$ (from (M2) below), but that is independent of the mesh-size. Let $\mathcal{P}_{k}({\cal D})$ denote the set of polynomials of degree at most $k\in\mathbb{N}_{0}$ defined on a domain ${\cal D}$ and let $\Pi_{k}$ denote the piecewise $L^{2}$ projection on $\mathcal{P}_{k}({\cal T})$ for any admissible partition $\mathcal{T}\in\mathbb{T}$ (hidden in the notation $\Pi_{k}$). The notation $H^{s}(P):=H^{s}(\text{int}P)$ for a compact polygonal domain $P$ means the Sobolev space $H^{s}$ [30] defined in the interior $\text{int}(P)$ of $P$ throughout this paper. The outward normal derivative is denoted by $\frac{\partial\;\bullet}{\partial\textbf{n}_{P}}=\textbf{n}_{P}\cdot\nabla\bullet$ for the exterior unit normal vector $\textbf{n}_{P}$ along the boundary $\partial P$ of the domain $P$. ## 2 First-order virtual element method on a polygonal mesh This section describes class of admissible partitions of $\Omega$ into polygonal domains and the lowest-order nonconforming virtual element method for the problem (1.1) [17, 3]. ### 2.1 Polygonal meshes A polygonal domain $P$ in this paper is a non-void compact simply-connected set $P$ with polygonal boundary $\partial P$ so that $\text{int}(P)$ is a Lipschitz domain. The polygonal boundary $\partial P$ is a simple closed polygon described by a finite sequence of distinct points. The set ${\cal N}(\partial P)=\\{z_{1},z_{2},\dots,z_{J}\\}$ of nodes of a polygon $P$ is enumerated with $z_{J+1}:=z_{1}$ such that $E(j):=\text{conv}\\{z_{j},z_{j+1}\\}$ defines an edge and all $J$ edges cover the boundary $\partial P=E(1)\cup\dots\cup E(J)$ with an intersection $E(j)\cap E(j+1)=\\{z_{j+1}\\}$ for $j=1,\dots,J-1$ and $E(J)\cap E(1)={z_{1}}$ with $\text{dist}(E(j),E(k))>0$ for all distinct indices $j\neq k$. Let $\mathbb{T}$ be a family of partitions of $\overline{\Omega}$ into polygonal domains, which satisfies the conditions (M1)-(M2) with a universal positive constant $\rho$. 1. (M1) Admissibility. Any two distinct polygonal domains $P$ and $P^{\prime}$ in $\mathcal{T}\in\mathbb{T}$ are disjoint or share a finite number of edges or vertices. Figure 2.1: 2. (M2) Mesh regularity. Every polygonal domain $P$ of diameter $h_{P}$ is star-shaped with respect to every point of a ball of radius greater than equal to $\rho h_{P}$ and every edge $E$ of $P$ has a length $\|E\|$ greater than equal to $\rho h_{P}$. Here and throughout this paper, $h_{\mathcal{T}}\|_{P}:=h_{P}$ denotes the piecewise constant mesh-size and $\mathbb{T}(\delta):=\\{\mathcal{T}\in\mathbb{T}:h_{\text{max}}\leq\delta\leq 1\\}$ with the maximum diameter $h_{\text{max}}$ of the polygonal domains in $\mathcal{T}$ denotes the subclass of partitions of $\overline{\Omega}$ into polygonal domains of maximal mesh-size $\leq\delta$. Let $\|P\|$ denote the area of polygonal domain $P$ and $\|E\|$ denote the length of an edge $E$. With a fixed orientation to a polygonal domain $P$, assign the outer unit normal $\textbf{n}_{P}$ along the boundary $\partial P$ and $\textbf{n}_{E}:=\textbf{n}_{P}\|_{E}$ for an edge $E$ of $P$. Let $\mathcal{E}$ (resp. $\widehat{\mathcal{E}}$) denote the set of edges $E$ of $\mathcal{T}$ (resp. of $\widehat{{\cal T}}$) and $\mathcal{E}(P)$ denote the set of edges of polygonal domain $P\in\mathcal{T}$. For a polygonal domain $P$, define $\displaystyle\text{mid}(P):=\frac{1}{\|P\|}\int_{P}x\,dx\quad\text{and}\quad\text{mid}(\partial P):=\frac{1}{\|\partial P\|}\int_{\partial P}x\,ds.$ Let $\mathcal{P}_{k}({\cal T}):=\\{v\in L^{2}(\Omega):\forall P\in\mathcal{T}\quad v\|_{P}\in\mathcal{P}_{k}(P)\\}$ for $k\in\mathbb{N}_{0}$ and $\Pi_{k}$ denote the piecewise $L^{2}$ projection onto $\mathcal{P}_{k}({\cal T})$. The notation $\Pi_{k}$ hides its dependence on $\mathcal{T}$ and also assume $\Pi_{k}$ applies componentwise to vectors. Given a decomposition ${\cal T}\in\mathbb{T}$ of $\Omega$ and a function $f\in L^{2}(\Omega)$, its oscillation reads $\displaystyle\mathrm{osc}_{k}(f,P):=\\|h_{P}(1-\Pi_{k})f\\|_{L^{2}(P)}\quad\text{and}\quad\mathrm{osc}_{k}(f,{\cal T}):=\left(\sum_{P\in{\cal T}}\\|h_{P}(1-\Pi_{k})f\\|_{L^{2}(P)}^{2}\right)^{\displaystyle\nicefrac{{1}}{{2}}}$ with $\mathrm{osc}(f,\bullet):=\mathrm{osc}_{0}(f,\bullet)$. ###### Remark 1 (consequence of mesh regularity assumption). There exists an interior node $c$ in the sub-triangulation $\widehat{{\cal T}}(P):=\\{T(E)=\text{conv}(c,E):E\in\mathcal{E}(P)\\}$ of a polygonal domain $P$ with $h_{T(E)}\leq h_{P}\leq C_{\text{sr}}h_{T(E)}$ as illustrated in Figure 2.2. Each polygonal domain $P$ can be divided into triangles so that the resulting sub-triangulation $\widehat{{\cal T}}\|_{P}:=\widehat{{\cal T}}(P)$ of $\mathcal{T}$ is shape-regular. The minimum angle in the sub- triangulation solely depends on $\rho$ [13, Sec. 2.1]. (a) (b) Figure 2.2: (a) Polygon $P$ and (b) its sub-triangulation $\widehat{{\cal T}}(P)$. ###### Lemma 2.1 (Poincaré-Friedrichs inequality). There exists a positive constant $C_{\mathrm{PF}}$, that depends solely on $\rho$, such that $\displaystyle\\|f\\|_{L^{2}(P)}\leq C_{\mathrm{PF}}h_{P}\|f\|_{1,P}$ (2.1) holds for any $f\in H^{1}(P)$ with $\sum_{j\in J}\int_{E(j)}f\,ds=0$ for a nonempty subset $J\subseteq\\{1,\dots,m\\}$ of indices in the notation $\partial P=E(1)\cup\dots\cup E(m)$ of Figure 2.2. The constant $C_{\mathrm{PF}}$ depends exclusively on the number $m:=\|\mathcal{E}(P)\|$ of the edges in the polygonal domain $P$ and the quotient of the maximal area divided by the minimal area of a triangle in the triangulation $\widehat{{\cal T}}(P)$. Some comments on $C_{\mathrm{PF}}$ for anisotropic meshes are in order before the proof gives an explicit expression for $C_{\mathrm{PF}}$. ###### Example 2.1. Consider a rectangle $P$ with a large aspect ratio divided into four congruent sub-triangles all with vertex $c=\text{mid}(P)$. Then, $m=4$ and the quotient of the maximal area divided by the minimal area of a triangle in the criss- cross triangulation $\widehat{{\cal T}}(P)$ is one. Hence $C_{\mathrm{PF}}\leq 1.4231$ (from the proof below) is independent of the aspect ratio of $P$. ###### Proof of Lemma 2.1. The case $J=\\{1,\dots,m\\}$ with $f\in H^{1}(P)$ and $\int_{\partial P}f\,ds=0$ is well-known cf. e.g. [13, Sec. 2.1.5], and follows from the Bramble-Hilbert lemma [14, Lemma 4.3.8] and the trace inequality [13, Sec. 2.1.1]. The remaining part of the proof shows the inequality (2.1) for the case $J\subseteq\\{1,\dots,m\\}$. The polygonal domain $P$ and its triangulation $\widehat{{\cal T}}(P)$ from Figure 2.2 has the center $c$ and the nodes $z_{1},\dots,z_{m}$ for the $m:=\|\mathcal{E}(P)\|=\|\widehat{{\cal T}}(P)\|$ edges $E(1),\dots,E(m)$ and the triangles $T(1),\dots,T(m)$ with $T(j)=T(E(j))=\text{conv}\\{c,E(j)\\}=\text{conv}\\{c,z_{j},z_{j+1}\\}$ for $j=1,\dots,m$. Here and throughout this proof, all indices are understood modulo $m$, e.g., $z_{0}=z_{m}$. The proof uses the trace identity $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(x-c)\cdot\nabla f(x)\,dx$ (2.2) for $f\in H^{1}(P)$ as in the lemma. This follows from an integration by parts and the observation that $(x-c)\cdot\textbf{n}_{F}=0$ on $F\in\mathcal{E}(T(j))\backslash E(j)$ and the height $(x-c)\cdot\textbf{n}_{E(j)}=\frac{2\|T(j)\|}{\|E(j)}$ of the edge $E(j)$ in the triangle $T(j)$, for $x\in E(j)$; cf. [24, Lemma 2.1] or [25, Lemma 2.6] for the remaining details. Another version of the trace identity (2.2) concerns $\text{conv}\\{z_{j},c\\}=:F(j)=\partial T(j-1)\cap\partial T(j)$ and reads $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{F(j)}f\,ds$ $\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j-1)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j-1)}(x-z_{j-1})\cdot\nabla f(x)\,dx$ $\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx+\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j+1})\cdot\nabla f(x)\,dx$ (2.3) in $T(j-1)$ and $T(j)$. The three trace identities in (2.2)-(2.3) are rewritten with the following abbreviations, for $j=1,\dots m$, $\displaystyle x_{j}:=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds,\quad f_{j}:=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}f\,dx,\quad a_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(x-c)\cdot\nabla f(x)\,dx,$ $\displaystyle b_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j})\cdot\nabla f(x)\,dx,\quad c_{j}:=\frac{1}{2}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(x-z_{j+1})\cdot\nabla f(x)\,dx.$ Let $t_{\text{min}}=\min_{T\in\widehat{{\cal T}}(P)}\|T\|$ and $t_{\text{max}}=\max_{T\in\widehat{{\cal T}}(P)}\|T\|$ abbreviate the minimal and maximal area of a triangle in $\widehat{{\cal T}}(P)$ and let $\widehat{\Pi}_{0}f\in\mathcal{P}_{0}(\widehat{{\cal T}}(P))$ denote the piecewise integral means of $f$ with respect to the triangulation $\widehat{{\cal T}}(P)$. The Poincaré inequality in a triangle with the constant $C_{\text{P}}:=1/j_{1,1}$ and the first positive root $j_{1,1}\approx 3.8317$ of the Bessel function $J_{1}$ from [24, Thm. 2.1] allows for $\displaystyle\\|f-\widehat{\Pi}_{0}f\\|_{L^{2}(T(j))}\leq C_{\text{P}}h_{T(j)}\|f\|_{1,T(j)}\quad\text{for}\;j=1,\dots,m.$ Hence $\\|f-\widehat{\Pi}_{0}f\\|_{L^{2}(P)}\leq C_{\text{P}}h_{P}\|f\|_{1,P}$. This and the Pythagoras theorem (with $f-\widehat{\Pi}_{0}f\perp\mathcal{P}_{0}(\widehat{{\cal T}}(P))$ in $L^{2}(P)$) show $\displaystyle\\|f\\|^{2}_{L^{2}(P)}=\\|\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P))}+\\|f-\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P))}\leq\\|\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P))}+C_{\text{P}}^{2}h_{P}^{2}\|f\|^{2}_{1,P}.$ (2.4) It remains to bound the term $\\|\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P))}$. The assumption on $f$ reads $\sum_{j\in J}\int_{E(j)}f\,ds=\sum_{j\in J}\|E(j)\|x_{j}=0$ for a subset $J\subset\\{1,\dots,m\\}$ so that $0\in\text{conv}\\{\|E(1)\|x_{1},\dots,\|E(m)\|x_{m}\\}$. It follows $0\in\text{conv}\\{x_{1},\dots,x_{m}\\}$ and it is known that this implies $\displaystyle\sum_{k=1}^{m}x_{k}^{2}\leq{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}$ (2.5) for a constant ${\cal{M}}=\frac{1}{2(1-\cos(\pi/m))}$ that depends exclusively on $m$ [25, Lemma 4.2]. Recall (2.2) in the form $x_{j}=f_{j}+a_{j}$ to deduce from a triangle inequality and (2.5) that $\displaystyle\frac{1}{2}\sum_{j=1}^{m}f_{j}^{2}\leq\sum_{k=1}^{m}x_{k}^{2}+\sum_{\ell=1}^{m}a_{\ell}^{2}\leq{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}+\sum_{\ell=1}^{m}a_{\ell}^{2}.$ This shows that $\displaystyle t_{\text{max}}^{-1}\\|\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P)}=t_{\text{max}}^{-1}\sum_{j=1}^{m}\|T(j)\|f_{j}^{2}\leq\sum_{j=1}^{m}f_{j}^{2}\leq 2{\cal{M}}\sum_{k=1}^{m}(x_{k}-x_{k-1})^{2}+2\sum_{\ell=1}^{m}a_{\ell}^{2}.$ Recall (2.2)-(2.3) in the form $f_{j}-f_{j-1}=b_{j-1}-c_{j}$ and $x_{j}-x_{j-1}=f_{j}-f_{j-1}+a_{j}-a_{j-1}=b_{j-1}-a_{j-1}+a_{j}-c_{j}$ for all $j=1,\dots,m$. This and the Cauchy-Schwarz inequality imply the first two estimates in $\displaystyle 2\|x_{j}-x_{j-1}\|$ $\displaystyle=\bigg{\|}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j-1)}(c-z_{j-1})\cdot\nabla f(x)\,dx+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T(j)}(z_{j+1}-c)\cdot\nabla f(x)\,dx\bigg{\|}$ $\displaystyle\leq\max\\{\|c-z_{j-1}\|,\|c-z_{j+1}\|\\}\Big{(}\|T(j-1)\|^{-1/2}\|f\|_{1,T(j-1)}+\|T(j)\|^{-1/2}\|f\|_{1,T(j)}\Big{)}$ $\displaystyle\leq h_{P}t_{\text{min}}^{-1/2}\|f\|_{1,T(j-1)\cup T(j)}$ with the definition of $h_{P}$ and $t_{\text{min}}$ in the end. The inequality $\int_{T(j)}\|x-c\|^{2}\;dx\leq\frac{1}{2}h^{2}_{T(j)}\|T(j)\|$ [25, Lemma 2.7] and the Cauchy-Schwarz inequality show, for $j=1,\dots,m$, that $\displaystyle\|a_{j}\|\leq 2^{-3/2}h_{T(j)}\|T(j)\|^{-1/2}\|f\|_{1,\|T(j)\|}\leq 2^{-3/2}h_{P}t_{\text{min}}^{-1/2}\|f\|_{1,\|T(j)\|}.$ The combination of the previous three displayed estimates result in $\displaystyle 4h_{P}^{-2}(t_{\text{min}}/t_{\text{max}})\\|\widehat{\Pi}_{0}f\\|^{2}_{L^{2}(P)}\leq 2{\cal{M}}\sum_{k=1}^{m}\|f\|^{2}_{T(k-1)\cup T(k)}+\sum_{\ell=1}^{m}\|f\|^{2}_{1,T(\ell)}=(4{\cal{M}}+1)\|f\|^{2}_{1,P}.$ This and (2.4) conclude the proof with the constant $C_{\mathrm{PF}}^{2}=({\cal{M}}+1/4)(t_{\text{max}}/t_{\text{min}})+C_{\text{P}}^{2}$. ∎ In the nonconforming VEM, the finite-dimensional space $V_{h}$ is a subset of the piecewise Sobolev space $H^{1}(\mathcal{T}):=\\{v\in L^{2}(\Omega):\forall P\in\mathcal{T}\quad v\|_{P}\in H^{1}(P)\\}\equiv\prod_{P\in\mathcal{T}}H^{1}(P).$ The piecewise $H^{1}$ seminorm (piecewise with respect to $\mathcal{T}$ hidden in the notation for brevity) reads $\|v_{h}\|_{1,\text{pw}}:=\bigg{(}\sum_{P\in\mathcal{T}}\|v_{h}\|_{1,P}^{2}\bigg{)}^{1/2}\quad\text{for any}\;v_{h}\in H^{1}(\mathcal{T}).$ ### 2.2 Local virtual element space The first nonconforming virtual element space [3] is a subspace of harmonic functions with edgewise constant Neumann boundary values on each polygon. The extended nonconforming virtual element space [1, 17] reads $\displaystyle\widehat{V}_{h}(P):=\begin{rcases}\begin{dcases}v_{h}\in H^{1}(P):&\Delta v_{h}\in\mathcal{P}_{1}(P)\quad\text{and}\quad\forall E\in\mathcal{E}(P)\quad{\frac{\partial v_{h}}{\partial\textbf{n}_{P}}}\Big{\|}_{E}\in\mathcal{P}_{0}(E)\end{dcases}\end{rcases}.$ (2.6) ###### Definition 2.2 (Ritz projection). Let $\Pi^{\nabla}_{1}$ be the Ritz projection from $H^{1}(P)$ onto the affine functions $\mathcal{P}_{1}(P)$ in the $H^{1}$ seminorm defined, for $v_{h}\in H^{1}(P)$, by $\displaystyle(\nabla\Pi^{\nabla}_{1}v_{h}-\nabla v_{h},\nabla\chi)_{L^{2}(P)}=0\quad\text{for all}\;\chi\in\mathcal{P}_{1}(P)\quad\text{and}\quad\int_{\partial P}\Pi^{\nabla}_{1}v_{h}\,ds=\int_{\partial P}v_{h}\,ds.$ (2.7) ###### Remark 2 (integral mean). For $P\in\mathcal{T}$ and $f\in H^{1}(P)$, $\nabla\Pi^{\nabla}_{1}f=\Pi_{0}\nabla f$. (This follows from (2.7.a) and the definition of the $L^{2}$ projection operator $\Pi_{0}$ (acting componentwise) onto the piecewise constants $\mathcal{P}_{0}(P;\mathbb{R}^{2})$.) ###### Remark 3 (representation of $\Pi^{\nabla}_{1}$). For $P\in\mathcal{T}$ and $f\in H^{1}(P)$, the Ritz projection $\Pi^{\nabla}_{1}f$ reads $\displaystyle(\Pi^{\nabla}_{1}f)(x)=\frac{1}{\|P\|}\Big{(}\int_{\partial P}f\textbf{n}_{P}\,ds\Big{)}\cdot\Big{(}x-\text{mid}(\partial P)\Big{)}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{\partial P}f\,ds\quad\text{for}\;x\in P.$ (2.8) (The proof of (2.8) consists in the verification of (2.7): The equation (2.7.a) follows from Remark 2 with an integration by parts. The equation (2.7.b) follows from the definition of $\text{mid}(\partial P)$ as the barycenter of $\partial P$. ∎) The enhanced virtual element spaces [1, 17] are designed with a computable $L^{2}$ projection $\Pi_{1}$ onto $\mathcal{P}_{1}(\mathcal{T})$. The resulting local discrete space under consideration throughout this paper reads $\displaystyle V_{h}(P):=\begin{rcases}\begin{dcases}v_{h}\in\widehat{V}_{h}(P):v_{h}-\Pi^{\nabla}_{1}v_{h}\perp\mathcal{P}_{1}(P)\quad\text{in}\;L^{2}(P)\end{dcases}\end{rcases}.$ (2.9) The point in the selection of $V_{h}(P)$ is that the Ritz projection $\Pi^{\nabla}_{1}v_{h}$ coincides with the $L^{2}$ projection $\Pi_{1}v_{h}$ for all $v_{h}\in V_{h}(P)$. The degrees of freedom on $P$ are given by $\displaystyle\text{dof}_{E}(v)=\frac{1}{\|E\|}\int_{E}v\,ds\quad\textrm{for all}\;E\in\mathcal{E}(P)\;\text{and}\;v\in V_{h}(P).$ (2.10) ###### Proposition 2.3. $(a)$ The vector space $\widehat{V}_{h}(P)$ from (2.6) is of dimension $3+\|\mathcal{E}(P)\|$. $(b)$ $V_{h}(P)$ from (2.9) is of dimension $\|\mathcal{E}(P)\|$ and the triplet $(P,V_{h}(P),\text{dof}_{E}:E\in\mathcal{E}(P))$ is a finite element in the sense of Ciarlet [28]. ###### Proof. Let $E(1),\dots,E(m)$ be an enumeration of the edges $\mathcal{E}(P)$ of the polygonal domain $P$ in a consecutive way as depicted in Figure 2.2.a and define $W(P):=\mathcal{P}_{1}(P)\times\mathcal{P}_{0}(E{(1)})\times\dots\times\mathcal{P}_{0}(E{(m)})$. Recall $\widehat{V}_{h}(P)$ from (2.6) and identify the quotient space $\widehat{V}_{h}(P)/\mathbb{R}\equiv\left\\{f\in\widehat{V}_{h}(P):\right.\\\ \left.\int_{\partial P}f\,ds=0\right\\}$ with all functions in $\widehat{V}_{h}(P)$ having zero integral over the boundary $\partial P$ of $P$. Since the space $\widehat{V}_{h}(P)$ consists of functions with an affine Laplacian and edgewise constant Neumann data, the map $\displaystyle S:\widehat{V}_{h}(P)/\mathbb{R}\to W(P),\quad\quad f\mapsto\left(-\Delta f,\frac{\partial f}{\partial\textbf{n}_{P}}\Big{\|}_{E{(1)}},\dots,\frac{\partial f}{\partial\textbf{n}_{P}}\Big{\|}_{E{(m)}}\right)$ is well-defined and linear. The compatibility conditions for the existence of a solution of a Laplacian problem with Neumann data show that the image of $S$ is equal to $\displaystyle\mathcal{R}(S)=\left\\{(f_{1},g_{1},\dots,g_{m})\in W(P):\int_{P}f_{1}dx+\sum_{j=1}^{m}g_{j}\|E(j)\|=0\right\\}.$ (The proof of this identity assumes the compatible data $(f_{1},g_{1},\dots,g_{m})$ from the set on the right-hand side and solves the Neumann problem with a unique solution $\widehat{u}$ in $\widehat{V}_{h}(P)/\mathbb{R}$ and $S\widehat{u}=(f_{1},g_{1},\dots,g_{m})$.) It is known that the Neumann problem has a unique solution up to an additive constant and so $S$ is a bijection and the dimension $m+2$ of $\widehat{V}_{h}(P)/\mathbb{R}$ is that of $\mathcal{R}(S)$. In particular, dimension of $\widehat{V}_{h}(P)$ is $m+3$. This proves $(a)$. Let $\Lambda_{0},\Lambda_{1},\Lambda_{2}:H^{1}(P)\to\mathbb{R}$ be linear functionals $\displaystyle\Lambda_{0}f:=\Pi_{0}f,\quad\Lambda_{j}f:={\cal M}_{j}((\Pi^{\nabla}_{1}-\Pi_{1})f)$ with ${\cal M}_{j}f:=\Pi_{0}((x_{j}-c_{j})f)$ for $j=1,2$ and $f\in H^{1}(P)$ that determines an affine function $p_{1}\in\mathcal{P}_{1}(P)$ such that $(P,\mathcal{P}_{1}(P),(\Lambda_{0},\Lambda_{1},\Lambda_{2}))$ is a finite element in the sense of Ciarlet. For any edge $E(j)\in\mathcal{E}(P)$, define $\Lambda_{j+2}f=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E(j)}f\,ds$ as integral mean of the traces of $f$ in $H^{1}(P)$ on $E(j)$. It is elementary to see that $\Lambda_{0},\dots,\Lambda_{m+2}$ are linearly independent: If $f$ in $\widehat{V}_{h}(P)$ belongs to the kernel of all the linear functionals, then $\Pi^{\nabla}_{1}f=0$ from (2.8) with $\Lambda_{j}f=0$ for each $j=3,\dots,2+m$. Since the functionals $\Lambda_{j}f=0$ for $j=1,2$, $(x_{j}-c_{j})(\Pi^{\nabla}_{1}-\Pi_{1})f=0$ and $\Pi^{\nabla}_{1}f=0$ imply $\Pi_{1}f=0$. An integration by parts leads to $\displaystyle\\|\nabla f\\|_{L^{2}(P)}^{2}=(-\Delta f,f)_{L^{2}(P)}+\Big{(}f,\frac{\partial f}{\partial\textbf{n}_{P}}\Big{)}_{L^{2}(\partial P)}=0.$ This and $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{\partial P}f\,ds=0$ show $f\equiv 0$. Consequently, the intersection $\cap_{j=0}^{m+2}\text{Ker}(\Lambda_{j})$ of all kernels Ker$(\Lambda_{0}),\dots,\text{Ker}(\Lambda)_{m+2}$ is trivial and so that the functionals $\Lambda_{0},\dots,\Lambda_{m+2}$ are linearly independent. Since the number of the linear functionals is equal to the dimension of $\widehat{V}_{h}(P)$, $(P,\widehat{V}_{h}(P),\\{\Lambda_{0},\dots,\Lambda_{m+2}\\})$ is a finite element in the sense of Ciarlet and there exists a nodal basis $\psi_{0},\dots,\psi_{m+2}$ of $\widehat{V}_{h}(P)$ with $\displaystyle\Lambda_{j}(\psi_{k})=\delta_{jk}\quad\text{for all}\;j,k=0,\dots,m+2.$ The linearly independent functions $\psi_{3},\dots,\psi_{m+2}$ belong to $V_{h}(P)$ and so dim$(V_{h}(P))\geq m$. Since $V_{h}(P)\subset\widehat{V}_{h}(P)$ and three linearly independent conditions $(1-\Pi^{\nabla}_{1})v_{h}\perp\mathcal{P}_{1}(P)$ in $L^{2}(P)$ are imposed on $\widehat{V}_{h}(P)$ to define $V_{h}(P)$, dim$(V_{h}(P))\leq m$. This shows that dim$(V_{h}(P))=m$ and hence, the linear functionals $\text{dof}_{E}=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}\bullet\,ds$ for $E\in\mathcal{E}(P)$ form a dual basis of $V_{h}(P)$. This concludes the proof of $(b)$. ∎ ###### Remark 4 (stability of $L^{2}$ projection). The $L^{2}$ projection $\Pi_{k}$ for $k=0,1$ is $H^{1}$ and $L^{2}$ stable in $V_{h}(P)$, in the sense that any $v_{h}$ in $V_{h}(P)$ satisfies $\displaystyle\\|\Pi_{k}v_{h}\\|_{L^{2}(P)}\leq\\|v_{h}\\|_{L^{2}(P)}\;\text{and}\;\\|\nabla(\Pi_{k}v_{h})\\|_{L^{2}(P)}\leq\\|\nabla v_{h}\\|_{L^{2}(P)}.$ (2.11) (The first inequality follows from the definition of $\Pi_{k}$. The orthogonality in (2.9) and the definition of $\Pi_{1}$ imply that the Ritz projection $\Pi^{\nabla}_{1}$ and the $L^{2}$ projection $\Pi_{1}$ coincide on the space $V_{h}(P)$ for $P\in\mathcal{T}$. This with the definition of the Ritz projection $\Pi^{\nabla}_{1}$ verifies the second inequality. ∎) ###### Definition 2.4 (Fractional order Sobolev space [14]). Let $\alpha:=(\alpha_{1},\alpha_{2})$ denote a multi-index with $\alpha_{j}\in\mathbb{N}_{0}$ for $j=1,2$ and $\|\alpha\|:=\alpha_{1}+\alpha_{2}.$ For a real number $m$ with $0<m<1$, define $\displaystyle H^{1+m}(\omega):=\left\\{v\in H^{1}(\omega):\frac{\|v^{\alpha}(x)-v^{\alpha}(y)\|}{\|x-y\|^{(1+m)}}\in L^{2}(\omega\times\omega)\quad\text{for all}\;\|\alpha\|=1\right\\}$ with $v^{\alpha}$ as the partial derivative of $v$ of order $\alpha$. Define the seminorm $\|\cdot\|_{1+m}$ and Sobolev-Slobodeckij norm $\\|\cdot\\|_{1+m}$ by $\displaystyle\|v\|_{1+m,\omega}^{2}=\sum_{\|\alpha\|=1}\int_{\omega}\int_{\omega}\frac{{\|v^{\alpha}(x)-v^{\alpha}(y)\|}^{2}}{\|x-y\|^{2(1+m)}}\,dx\,dy\quad\text{and}\quad\\|v\\|_{1+m,\omega}^{2}=\\|v\\|^{2}_{1,\omega}+\|v\|_{1+m,\omega}^{2}.$ ###### Proposition 2.5 (approximation by polynomials [29, Thm. 6.1]). Under the assumption (M2), there exists a positive constant $C_{\mathrm{apx}}$ (depending on $\rho$ and on the polynomial degree $k$) such that, for every $v\in H^{m}(P)$, the $L^{2}$ projection $\Pi_{k}(P)$ on $\mathcal{P}_{k}$ for $k\in\mathbb{N}_{0}$ satisfies $\displaystyle\\|v-\Pi_{k}v\\|_{L^{2}(P)}+h_{P}\|v-\Pi_{k}v\|_{1,P}\leq C_{\mathrm{apx}}h_{P}^{m}\|v\|_{m,P}\quad\text{for}\;1\leq m\leq k+1.$ (2.12) ### 2.3 Global virtual element space Define the global nonconforming virtual element space, for any $\mathcal{T}\in\mathbb{T}$, by $\displaystyle V_{h}:=\left\\{v_{h}\in H^{1}(\mathcal{T}):\forall P\in\mathcal{T}\quad v_{h}\|_{P}\in V_{h}(P)\quad\text{and}\quad\forall E\in\mathcal{E}\quad\int_{E}[v_{h}]_{E}\,ds=0\right\\}.$ (2.13) Let $[\cdot]_{E}$ denote the jump across an edge $E\in\mathcal{E}$: For two neighboring polygonal domains $P^{+}$ and $P^{-}$ sharing a common edge $E\in\mathcal{E}(P^{+})\cap\mathcal{E}(P^{-})$, $[v_{h}]_{E}:=v_{h\|P^{+}}-v_{h\|P^{-}}$, where $P^{+}$ denote the adjoint polygonal domain with $\textbf{n}_{P^{+}\|E}=\textbf{n}_{E}$ and $P^{-}$ denote the polygonal domain with $\textbf{n}_{P^{-}\|E}=-\textbf{n}_{E}$. If $E\subset\partial\Omega$ is a boundary edge, then $[v_{h}]_{E}:=v_{h}\|_{E}$. ###### Example 2.2. If each polygonal domain $P$ is a triangle, then the finite-dimensional space $V_{h}$ coincides with CR-FEM space. (Since the dimension of the vector space $V_{h}(P)$ is three and $\mathcal{P}_{1}(P)\subset V_{h}(P)$, $V_{h}(P)=\mathcal{P}_{1}(P)$ for $P\in\mathcal{T}$.) ###### Lemma 2.6. There exists a universal constant $C_{\mathrm{F}}$ (that depends only on $\rho$ from (M2)) such that, for all ${\cal T}\in\mathbb{T}$, any $v_{h}\in V_{h}$ from (2.13) satisfies $\displaystyle\\|v_{h}\\|_{L^{2}(\Omega)}\leq C_{\mathrm{F}}\|v_{h}\|_{1,\mathrm{pw}}.$ (2.14) ###### Proof. Recall from Remark 1 that $\widehat{{\cal T}}$ is a shape regular sub- triangulation of $\mathcal{T}$ into triangles. Since $V_{h}\subset H^{1}(\widehat{{\cal T}})$ and the Friedrichs’ inequality holds for all functions in $H^{1}(\widehat{{\cal T}})$ [14, Thm. 10.6.16], there exists a positive constant $C_{\text{F}}$ such that the (first) inequality holds in $\displaystyle\\|v_{h}\\|_{L^{2}(\Omega)}\leq C_{\text{F}}\left(\sum_{T\in\widehat{{\cal T}}}\\|\nabla v_{h}\\|_{L^{2}(T)}^{2}\right)^{1/2}=C_{\text{F}}\|v_{h}\|_{1,\mathrm{pw}}.$ The (second) equality follows for $v_{h}\in H^{1}(P)$ with $P\in\mathcal{T}$. ∎ Lemma 2.6 implies that the seminorm $\|\cdot\|_{1,\mathrm{pw}}$ is equivalent to the norm $\\|\cdot\\|_{1,\mathrm{pw}}:=\\|\cdot\\|^{2}_{L^{2}(\Omega)}+\|\cdot\|^{2}_{1,\mathrm{pw}}$ in $V_{h}$ with mesh-size independent equivalence constants. ### 2.4 Interpolation ###### Definition 2.7 (interpolation operator). Let $(\psi_{E}:E\in\mathcal{E})$ be the nodal basis of $V_{h}$ defined by $\text{dof}_{E}(\psi_{E})=1$ and $\text{dof}_{F}(\psi_{E})=0$ for all other edges $F\in\mathcal{E}\setminus\\{E\\}$. The global interpolation operator $I_{h}:H^{1}_{0}(\Omega)\to V_{h}$ reads $\displaystyle I_{h}v:=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v\,ds\Big{)}\psi_{E}\quad\text{for}\;v\in V.$ Since a Sobolev function $v\in V$ has traces and the jumps $[v]_{E}$ vanish across any edge $E\in\mathcal{E}$, the interpolation operator $I_{h}$ is well- defined. Recall $\rho$ from (M2), $C_{\mathrm{PF}}$ from Lemma 2.1, and $C_{\text{apx}}$ from Proposition 2.5. ###### Theorem 2.8 (interpolation error). 1. $\left(a\right)$ There exists a positive constant $C_{\mathrm{Itn}}$ (depending on $\rho$) such that any $v\in H^{1}(P)$ and its interpolation $I_{h}v\in V_{h}(P)$ satisfy $\displaystyle\\|\nabla I_{h}v\\|_{L^{2}(P)}\leq C_{\mathrm{Itn}}\\|\nabla v\\|_{L^{2}(P)}.$ 2. $\left(b\right)$ Any $P\in\mathcal{T}\in\mathbb{T}$ and $v\in H^{1}(P)$ satisfy $\|v-I_{h}v\|_{1,P}\leq(1+C_{\mathrm{Itn}})\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}$ and $\displaystyle h_{P}^{-1}\\|(1-\Pi_{1}I_{h})v\\|_{L^{2}(P)}+\|(1-\Pi_{1}I_{h})v\|_{1,P}\leq(1+C_{\mathrm{PF}})\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}.$ 3. $\left(c\right)$ The positive constant $C_{\mathrm{I}}:=C_{\mathrm{apx}}(1+C_{\mathrm{Itn}})(1+C_{\mathrm{PF}})$, any $0<\sigma\leq 1$, and any $v\in H^{1+\sigma}(P)$ with the local interpolation $I_{h}v\|_{P}\in V_{h}(P)$ satisfy $\displaystyle\\|v-I_{h}v\\|_{L^{2}(P)}+h_{P}\|v-I_{h}v\|_{1,P}\leq C_{\mathrm{I}}h^{1+\sigma}_{P}\|v\|_{1+\sigma,P}.$ (2.15) ###### Proof of $(a)$. The boundedness of the interpolation operator in $V_{h}(P)$ is mentioned in [17] with a soft proof in its appendix. The subsequent analysis aims at a clarification that $C_{\text{I}}$ depends exclusively on the parameter $\rho$ in (M2). The elementary arguments apply to more general situations in particular to 3D. Given $I_{h}v\in V_{h}(P)$, $q_{1}:=-\Delta I_{h}v\in\mathcal{P}_{1}(P)$ is affine and $\int_{E}(v-I_{h}v)\,ds=0$. Since $\frac{\partial I_{h}v}{\partial\textbf{n}_{P}}$ is edgewise constant, this shows $\int_{E}{\frac{\partial I_{h}v}{\partial\textbf{n}_{P}}}\|_{E}(v-I_{h}v)\,ds=0$ for all $E\in\mathcal{E}(P)$ and so $\big{\langle}\frac{\partial I_{h}v}{\partial\textbf{n}_{P}},v-I_{h}v\big{\rangle}_{\partial P}=0$. An integration by parts leads to $\displaystyle(\nabla I_{h}v,\nabla(I_{h}v-v))_{L^{2}(P)}=(q_{1},I_{h}v-v)_{L^{2}(P)}=(q_{1},\Pi^{\nabla}_{1}I_{h}v-v)_{L^{2}(P)}$ with $q_{1}\in\mathcal{P}_{1}(P)$ and $\Pi_{1}v_{h}=\Pi^{\nabla}_{1}v_{h}$ for $v_{h}\in V_{h}(P)$ in the last step. Consequently, $\displaystyle\\|\nabla I_{h}v\\|_{L^{2}(P)}^{2}$ $\displaystyle=(\nabla I_{h}v,\nabla(I_{h}v-v))_{L^{2}(P)}+(\nabla I_{h}v,\nabla v)_{L^{2}(P)}$ $\displaystyle=(q_{1},\Pi^{\nabla}_{1}I_{h}v-v)_{L^{2}(P)}+(\nabla I_{h}v,\nabla v)_{L^{2}(P)}$ $\displaystyle\leq\\|q_{1}\\|_{L^{2}(P)}\\|v-\Pi^{\nabla}_{1}I_{h}v\\|_{L^{2}(P)}+\\|\nabla I_{h}v\\|_{L^{2}(P)}\\|\nabla v\\|_{L^{2}(P)}$ (2.16) with the Cauchy inequality in the last step. Remark 2 and 3 on the Ritz projection, and the definition of $I_{h}$ show $\displaystyle\Pi_{0}\nabla v=\nabla\Pi^{\nabla}_{1}v=\|P\|^{-1}\int_{\partial P}v\,\textbf{n}_{P}\,ds=\|P\|^{-1}\int_{\partial P}I_{h}v\textbf{n}_{P}\,ds=\Pi_{0}\nabla I_{h}v=\nabla\Pi^{\nabla}_{1}I_{h}v.$ (2.17) The function $f:=v-\Pi^{\nabla}_{1}I_{h}v\in H^{1}(P)$ satisfies $\int_{\partial P}f\,ds=\int_{\partial P}(v-I_{h}v)\,ds=0$ and the Poincaré- Friedrichs inequality from Lemma 2.1.a shows $\displaystyle\\|v-\Pi^{\nabla}_{1}I_{h}v\\|_{L^{2}(P)}\leq C_{\text{PF}}h_{P}\\|\nabla(v-\Pi^{\nabla}_{1}I_{h}v)\\|_{L^{2}(P)}=C_{\text{PF}}h_{P}\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}$ (2.18) with (2.17) in the last step. Let $\phi_{c}\in S^{1}_{0}(\widehat{{\cal T}}(P)):=\\{w\in C^{0}(P):w\|_{T(E)}\in\mathcal{P}_{1}(T(E))\quad\text{for all}\;E\in\mathcal{E}(P)\\}$ denote the piecewise linear nodal basis function of the interior node $c$ with respect to the triangulation $\widehat{{\cal T}}(P)=\\{T(E):E\in\mathcal{E}(P)\\}$ (cf. Figure 2.2.b for an illustration of $\widehat{{\cal T}}(P)$). An inverse estimate $\displaystyle\\|f_{1}\\|_{L^{2}(T(E))}\leq C_{1}\\|\phi_{c}^{1/2}f_{1}\\|_{L^{2}(T(E))}\quad\text{for all}\;f_{1}\in\mathcal{P}_{1}(\widehat{{\cal T}}(P))$ on the triangle $T(E):=\text{conv}(E\cup\\{c\\})$ holds with the universal constant $C_{1}$. A constructive proof computes the mass matrices for $T$ with and without the weight $\phi_{c}$ to verify that the universal constant $C_{1}$ does not depend on the shape of the triangle $T(E)$. This implies $\displaystyle C_{1}^{-1}\\|q_{1}\\|_{L^{2}(P)}^{2}\leq(\phi_{c}q_{1},q_{1})_{L^{2}(P)}=(-\Delta I_{h}v,\phi_{c}q_{1})=(\nabla I_{h}v,\nabla(\phi_{c}q_{1}))_{L^{2}(P)}$ (2.19) with an integration by parts for $\phi_{c}q_{1}\in H^{1}_{0}(P)$ and $I_{h}v$ in the last step. The mesh-size independent constant $C_{2}$ in the standard inverse estimate $\displaystyle h_{T(E)}\\|\nabla q_{2}\\|_{L^{2}(T(E))}\leq C_{2}\\|q_{2}\\|_{L^{2}(T(E))}\quad\text{for all}\;q_{2}\in\mathcal{P}_{2}(T(E))$ depends merely on the angles in the triangle $T(E),E\in\mathcal{E}(P),$ and so exclusively on $\rho$. With $C^{-1}_{\text{sr}}h_{P}\leq h_{T(E)}$ from Remark 1, this shows $\displaystyle C_{2}^{-1}C_{\text{sr}}^{-1}h_{P}\\|\nabla\phi_{c}q_{1}\\|_{L^{2}(P)}\leq\\|\phi_{c}q_{1}\\|_{L^{2}(P)}\leq\\|q_{1}\\|_{L^{2}(P)}.$ This and (2.19) lead to $\displaystyle\\|q_{1}\\|_{L^{2}(P)}\leq C_{1}C_{2}C_{\text{sr}}h_{P}^{-1}\\|\nabla I_{h}v\\|_{L^{2}(P)}.$ (2.20) The combination with (2.16)-(2.18) proves $\displaystyle\\|\nabla I_{h}v\\|_{L^{2}(P)}^{2}$ $\displaystyle\leq(C_{1}C_{2}C_{\text{sr}}C_{\text{PF}}\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}+\\|\nabla v\\|_{L^{2}(P)})\\|\nabla I_{h}v\\|_{L^{2}(P)}$ $\displaystyle\leq(1+C_{1}C_{2}C_{\text{sr}}C_{\text{PF}})\\|\nabla v\\|_{L^{2}(P)}\\|\nabla I_{h}v\\|_{L^{2}(P)}.\qed$ ###### Proof of $(b)$. The identity (2.17) reads $\Pi_{0}\nabla(1-I_{h})v=0$ and the triangle inequality results in $\displaystyle\|v-I_{h}v\|_{1,P}$ $\displaystyle=\\|(1-\Pi_{0})\nabla(1-I_{h})v\\|_{L^{2}(p)}$ $\displaystyle\leq\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}+\\|(1-\Pi_{0})\nabla I_{h}v\\|_{L^{2}(P)}.$ (2.21) Since $I_{h}$ is the identity in $\mathcal{P}_{1}(P)$, it follows $(1-\Pi_{0})\nabla I_{h}v=(1-\Pi_{0})\nabla I_{h}(v-\Pi^{\nabla}_{1}v).$ This and the boundedness of the interpolation operator $I_{h}$ lead to $\displaystyle\\|(1-\Pi_{0})\nabla I_{h}v\\|_{L^{2}(P)}$ $\displaystyle\leq\\|\nabla I_{h}(1-\Pi^{\nabla}_{1})v\\|_{L^{2}(P)}$ $\displaystyle\leq C_{\mathrm{Itn}}\\|\nabla(1-\Pi^{\nabla}_{1})v\\|_{L^{2}(P)}=C_{\mathrm{Itn}}\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}$ (2.22) with Remark 2 in the last step. The combination of (2.21) and (2.22) proves the first part of $(b)$. The identity $\|(1-\Pi_{1}I_{h})v\|_{1,P}=\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}$ follows from (2.17). Since $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$ and $\int_{\partial P}v\,ds=\int_{\partial P}I_{h}v\,ds=\int_{\partial P}\Pi^{\nabla}_{1}I_{h}v\,ds$, the Poincaré-Friedrichs inequality $\\|(1-\Pi_{1}I_{h})v\\|_{L^{2}(P)}\leq C_{\text{PF}}h_{P}\|(1-\Pi_{1}I_{h})v\|_{1,P}$ follows from Lemma 2.1.a. This concludes the proof of $(b)$. ∎ ###### Proof of $(c)$. This is an immediate consequence of the part $(b)$ with (2.12) and the Poincaré-Friedrichs inequality for $v-I_{h}v$ (from above) in Lemma 2.1.a. ∎ ## 3 Preliminary estimates This subsection formulates the discrete problem along with the properties of the discrete bilinear form such as boundedness and a G$\mathring{a}$rding-type inequality. ### 3.1 The discrete problem Denote the restriction of the bilinear forms $a(\cdot,\cdot),\hskip 2.84526ptb(\cdot,\cdot)$ and $c(\cdot,\cdot)$ on a polygonal domain $P\in\mathcal{T}$ by $a^{P}(\cdot,\cdot),\hskip 2.84526ptb^{P}(\cdot,\cdot)$ and $c^{P}(\cdot,\cdot)$. The corresponding local discrete bilinear forms are defined for $u_{h},v_{h}\in V_{h}(P)$ by $\displaystyle a_{h}^{P}(u_{h},v_{h})$ $\displaystyle:=(\textbf{A}\nabla\Pi_{1}u_{h},\nabla\Pi_{1}v_{h})_{L^{2}(P)}+S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h}),$ (3.1) $\displaystyle b_{h}^{P}(u_{h},v_{h})$ $\displaystyle:=\ (\Pi_{1}u_{h},\textbf{b}\cdot\nabla\Pi_{1}v_{h})_{L^{2}(P)},$ (3.2) $\displaystyle c_{h}^{P}(u_{h},v_{h})$ $\displaystyle:=(\gamma\Pi_{1}u_{h},\Pi_{1}v_{h})_{L^{2}(P)},$ (3.3) $\displaystyle B_{h}^{P}(u_{h},v_{h})$ $\displaystyle:=a_{h}^{P}(u_{h},v_{h})+b_{h}^{P}(u_{h},v_{h})+c_{h}^{P}(u_{h},v_{h}).$ (3.4) Choose the stability term $S^{P}(u_{h},v_{h})$ as a symmetric positive definite bilinear form on $V_{h}(P)\times V_{h}(P)$ for a positive constant $C_{s}$ independent of $P$ and $h_{P}$ satisfying $C_{s}^{-1}a^{P}(v_{h},v_{h})\leq S^{P}(v_{h},v_{h})\leq C_{s}a^{P}(v_{h},v_{h})\quad\text{for all}\hskip 2.84526ptv_{h}\in V_{h}(P)\hskip 2.84526pt\text{with}\hskip 2.84526pt\Pi_{1}v_{h}=0.$ (3.5) For some positive constant approximation $\overline{\textbf{A}}_{P}$ of A over $P$ and the number $N_{P}:=\|\mathcal{E}(P)\|$ of the degrees of freedom (2.10) of $V_{h}(P)$, a standard example of a stabilization term from [4],[36, Sec. 4.3] with a scaling coefficient $\overline{\textbf{A}}_{P}$ reads $\displaystyle S^{P}(v_{h},w_{h}):=\overline{\textbf{A}}_{P}\sum_{r=1}^{N_{P}}\text{dof}_{r}(v_{h})\text{dof}_{r}(w_{h})\quad\text{for all}\;v_{h},w_{h}\in V_{h}.$ (3.6) Note that an approximation $\overline{\textbf{A}}_{P}$ is a positive real number (not a matrix) and can be chosen as $\sqrt{a_{0}a_{1}}$ with the positive constants $a_{0}$ and $a_{1}$ from (A2). For $f\in L^{2}(\Omega)$ and $v_{h}\in V_{h}$, define the right-hand side functional $f_{h}$ on $V_{h}$ by $\displaystyle(f_{h},v_{h})_{L^{2}(P)}$ $\displaystyle:=(f,\Pi_{1}v_{h})_{L^{2}(P)}.$ (3.7) The sum over all the polygonal domains $P\in\mathcal{T}$ reads $\displaystyle a_{h}(u_{h},v_{h})$ $\displaystyle:=\sum_{P\in\mathcal{T}}a_{h}^{P}(u_{h},v_{h}),\hskip 14.22636ptb_{h}(u_{h},v_{h}):=\sum_{P\in\mathcal{T}}b_{h}^{P}(u_{h},v_{h}),$ $\displaystyle c_{h}(u_{h},v_{h})$ $\displaystyle:=\sum_{P\in\mathcal{T}}c_{h}^{P}(u_{h},v_{h}),\hskip 14.22636pts_{h}(u_{h},v_{h}):=\sum_{P\in\mathcal{T}}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h}),$ $\displaystyle B_{h}(u_{h},v_{h})$ $\displaystyle:=\sum_{P\in\mathcal{T}}B_{h}^{P}(u_{h},v_{h}),\hskip 14.22636pt(f_{h},v_{h}):=\sum_{P\in\mathcal{T}}(f_{h},v_{h})_{P}\quad\text{for all}\;u_{h},v_{h}\in V_{h}.$ The discrete problem seeks $u_{h}\in V_{h}$ such that $\displaystyle B_{h}(u_{h},v_{h})=(f_{h},v_{h})\quad\text{for all}\;v_{h}\in V_{h}.$ (3.8) ###### Remark 5 (polygonal mesh with small edges). The conditions (M1)-(M2) are well established and apply throughout the paper. The sub-triangulation $\widehat{{\cal T}}$ may not be shape-regular without the edge condition $\|E\|\geq\rho h_{P}$ for an edge $E\in\mathcal{T}(P)$ and $P\in\mathcal{T}$, but satisfies the maximal angle condition and the arguments employed in the proof of [8, Lemma 6.3] can be applied to show (2.20) in Theorem 2.8.a. For more general star-shaped polygon domains with short edges, the recent anisotropic analysis [8, 15, 18] indicates that the stabilization term has to be modified as well to avoid a logarithmic factor in the optimal error estimates. ### 3.2 Properties of the discrete bilinear form The following proposition provides two main properties of the discrete bilinear form $B_{h}$. ###### Proposition 3.1. There exist positive universal constants $M,\alpha$ and a universal nonnegative constant $\beta$ depending on the coefficients $\textbf{A},\textbf{b},\gamma$ such that 1. $\left(a\right)$ Boundedness: $\|B_{h}(u_{h},v_{h})\|\leq M\|u_{h}\|_{1,\mathrm{pw}}\|v_{h}\|_{1,\mathrm{pw}}\quad\text{for all}\hskip 2.84526ptu_{h},v_{h}\in V_{h}.$ 2. $\left(b\right)$ G$\mathring{a}$rding-type inequality: $\alpha\|v_{h}\|^{2}_{1,\mathrm{pw}}-\beta\\|v_{h}\\|^{2}_{L^{2}(\Omega)}\leq B_{h}(v_{h},v_{h})\quad\text{for all}\hskip 2.84526ptv_{h}\in V_{h}.$ ###### Proof of $\left(a\right)$. The upper bound of the coefficients from the assumption (A1), the Cauchy- Schwarz inequality, the stability (2.11) of $\Pi_{1}$, and the definition (3.5) of the stabilization term imply the boundedness of $B_{h}$ with $M:=(1+C_{s})\\|\textbf{A}\\|_{\infty}+C_{\mathrm{F}}\\|\textbf{b}\\|_{\infty}+C_{\mathrm{F}}^{2}\\|\gamma\\|_{\infty}$. The details of the proof follow as in [6, Lemma 5.2] with the constant $C_{\mathrm{F}}$ from Lemma 2.6. ∎ ###### Proof of $\left(b\right)$. The first step shows that $a_{h}(\cdot,\cdot)$ is coercive. For $v_{h}\in V_{h}(P)$, $\Pi_{1}v_{h}=\Pi^{\nabla}_{1}v_{h}$ and $\nabla\Pi_{1}v_{h}\perp\nabla(v_{h}-\Pi^{\nabla}_{1}v_{h})$ in $L^{2}(P;\mathbb{R}^{2})$. This orthogonality, the assumption (A2), and the definition of the stability term (3.5) with the constant $C_{s}^{-1}\leq 1$ imply for $\alpha_{0}=a_{0}C_{s}^{-1}$ that $\displaystyle\alpha_{0}\|v_{h}\|_{1,\mathrm{pw}}^{2}\leq a_{0}\\|\nabla_{\mathrm{pw}}\Pi_{1}v_{h}\\|^{2}_{L^{2}(\Omega)}+a_{0}C_{s}^{-1}\\|\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h}\\|^{2}_{L^{2}(\Omega)}$ $\displaystyle\quad\leq\left(\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}v_{h},\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+C_{s}^{-1}(\textbf{A}\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h},\nabla_{\mathrm{pw}}(1-\Pi_{1})v_{h}\right)_{L^{2}(\Omega)}$ $\displaystyle\quad\leq(\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}v_{h},\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})v_{h},(1-\Pi_{1})v_{h})=a_{h}(v_{h},v_{h}).$ (3.9) The Cauchy-Schwarz inequality, (2.11), and the Young inequality lead to $\displaystyle\|b_{h}(v_{h},v_{h})+c_{h}(v_{h},v_{h})\|$ $\displaystyle\leq\\|\textbf{b}\\|_{\infty}\\|\Pi_{1}v_{h}\\|_{L^{2}(\Omega)}\\|\nabla_{\mathrm{pw}}\Pi_{1}v_{h}\\|_{L^{2}(\Omega)}+\\|\gamma\\|_{\infty}\\|\Pi_{1}v_{h}\\|_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq\\|\textbf{b}\\|_{\infty}\\|v_{h}\\|_{L^{2}(\Omega)}\|v_{h}\|_{1,\mathrm{pw}}+\\|\gamma\\|_{\infty}\\|v_{h}\\|_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq\frac{\\|\textbf{b}\\|^{2}_{\infty}}{2\alpha_{0}}\\|v_{h}\\|_{L^{2}(\Omega)}^{2}+\frac{\alpha_{0}}{2}\|v_{h}\|^{2}_{1,\mathrm{pw}}+\\|\gamma\\|_{\infty}\\|v_{h}\\|_{L^{2}(\Omega)}^{2}.$ (3.10) The combination of (3.9)-(3.10) proves $\displaystyle\frac{\alpha_{0}}{2}\|v_{h}\|^{2}_{1,\mathrm{pw}}-\left(\frac{\\|\textbf{b}\\|^{2}_{\infty}}{2\alpha_{0}}+\\|\gamma\\|_{\infty}\right)\\|v_{h}\\|^{2}_{L^{2}(\Omega)}\leq B_{h}(v_{h},v_{h}).$ This concludes the proof of $\left(b\right)$ with $\alpha=\frac{\alpha_{0}}{2}$ and $\beta=\frac{\\|\textbf{b}\\|^{2}_{\infty}}{2\alpha_{0}}+\\|\gamma\\|_{\infty}$. ∎ ###### Remark 6 ($\\|\cdot\\|_{h}\approx\|\cdot\|_{1,\mathrm{pw}}$). The discrete space $V_{h}$ of the nonconforming VEM is endowed with the natural norm $\\|\cdot\\|_{h}:=a_{h}(\cdot,\cdot)^{1/2}$ induced by the scalar product $a_{h}$. The boundedness of $a_{h}$ is proven in $(a)$, while (3.9) shows the converse estimate in the equivalence $\\|\cdot\\|_{h}\approx\|\cdot\|_{1,\mathrm{pw}}$ in $V_{h}$, namely $\alpha_{0}\|v_{h}\|^{2}_{1,\mathrm{pw}}\leq a_{h}(v_{h},v_{h})\leq\\|\textbf{A}\\|_{\infty}(1+C_{s})\|v_{h}\|_{1,\mathrm{pw}}^{2}\quad\text{for all}\;v_{h}\in V_{h}.$ ### 3.3 Consistency error This subsection discusses the consistency error between the continuous bilinear form $B$ and the corresponding discrete bilinear form $B_{h}$. Recall the definition $B^{P}(\cdot,\cdot)\equiv a^{P}(\cdot,\cdot)+b^{P}(\cdot,\cdot)+c^{P}(\cdot,\cdot)$ and $B_{h}^{P}(\cdot,\cdot)\equiv a_{h}^{P}(\cdot,\cdot)+b_{h}^{P}(\cdot,\cdot)+c_{h}^{P}(\cdot,\cdot)$ for a polygonal domain $P\in\mathcal{T}$ from Subsection 2.1. ###### Lemma 3.2 (consistency). $(a)$ There exists a positive constant $C_{\text{cst}}$ (depending only on $\rho$) such that any $v\in H^{1}(\Omega)$ and $w_{h}\in V_{h}$ satisfy $\displaystyle B^{P}(\Pi_{1}v,w_{h})-B_{h}^{P}(\Pi_{1}v,w_{h})\leq C_{\mathrm{cst}}\,h_{P}\\|v\\|_{1,P}\|w_{h}\|_{1,P}\quad\text{for all}\;P\in\mathcal{T}.$ (3.11) $(b)$ Any $f\in L^{2}(\Omega)$ and $f_{h}:=\Pi_{1}f$ satisfy $\displaystyle\\|f-f_{h}\\|_{V_{h}^{}}:=\sup_{0\neq v_{h}\in V_{h}}\frac{(f-f_{h},v_{h})}{\\|v_{h}\\|_{1,\mathrm{pw}}}\leq C_{\mathrm{PF}}\,\mathrm{osc}_{1}(f,\mathcal{T}).$ (3.12) ###### Proof. Observe that $S^{P}((1-\Pi_{1})\Pi_{1}v,(1-\Pi_{1})w_{h})=0$ follows from $(1-\Pi_{1})\Pi_{1}v=0$. The definition of $B^{P}$ and $B_{h}^{P}$ show $\displaystyle B^{P}(\Pi_{1}v,w_{h})-B_{h}^{P}(\Pi_{1}v,w_{h})=:T_{1}+T_{2}+T_{3}.$ (3.13) The term $T_{1}$ in (3.13) is defined as the difference of the contributions from $a^{P}$ and $a^{P}_{h}$. Their definitions prove the equality (at the end of the first line below) and the definition of $\Pi_{1}$ prove the next equality in $\displaystyle T_{1}$ $\displaystyle:=a^{P}(\Pi_{1}v,w_{h})-a_{h}^{p}(\Pi_{1}v,w_{h})=(\textbf{A}\nabla\Pi_{1}v,\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}$ $\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\nabla\Pi_{1}v),\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}\leq h_{P}\|\textbf{A}\|_{1,\infty}\|v\|_{1,P}\|w_{h}\|_{1,P}.$ The last inequality follows from the Cauchy-Schwarz inequality, the Lipschitz continuity of A, and the stabilities $\\|\nabla\Pi_{1}v_{h}\\|_{L^{2}(P)}\leq\\|\nabla v_{h}\\|_{L^{2}(P)}$ and $\\|\nabla(1-\Pi_{1})w_{h}\\|_{L^{2}(P)}\leq\\|\nabla w_{h}\\|_{L^{2}(P)}$ from Remark 4. Similar arguments apply to $T_{2}$ from the differences of $b^{P}$ and $b^{P}_{h}$, and $T_{3}$ from those of $c^{P}$ and $c_{h}^{P}$ in (3.13). This leads to $\displaystyle T_{2}$ $\displaystyle:=b^{P}(\Pi_{1}v,w_{h})-b_{h}^{P}(\Pi_{1}v,w_{h})$ $\displaystyle=((\textbf{b}-\Pi_{0}\textbf{b})\Pi_{1}v,\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}+((\Pi_{0}\textbf{b})(1-\Pi_{0})(\Pi_{1}v),\nabla(1-\Pi_{1})w_{h})_{L^{2}(P)}$ $\displaystyle\leq(\|\textbf{b}\|_{1,\infty}+C_{\mathrm{apx}}\\|\textbf{b}\\|_{\infty})h_{P}\\|v\\|_{1,P}\|w_{h}\|_{1,P},$ $\displaystyle T_{3}$ $\displaystyle:=c^{P}(\Pi_{1}v,w_{h})-c_{h}^{P}(\Pi_{1}v,w_{h})=(\gamma\Pi_{1}v,(1-\Pi_{1})w_{h})_{L^{2}(P)}\leq C_{\mathrm{PF}}\,\\|\gamma\\|_{\infty}h_{P}\\|v\\|_{L^{2}(P)}\|w_{h}\|_{1,P}.$ The inequality for the last step in $T_{2}$ follows from the Cauchy-Schwarz inequality, the Lipschitz continuity of b, the estimate $\\|(1-\Pi_{0})\Pi_{1}v\\|_{L^{2}(P)}\leq\\|(1-\Pi_{0})v\\|_{L^{2}(P)}\leq C_{\text{apx}}h_{P}\|v\|_{1,P}$ from (2.12), and the above stabilities $\\|\nabla\Pi_{1}v_{h}\\|_{L^{2}(P)}\leq\\|\nabla v_{h}\\|_{L^{2}(P)}$ and $\\|\nabla(1-\Pi_{1})w_{h}\\|_{L^{2}(P)}\leq\\|\nabla w_{h}\\|_{L^{2}(P)}$. The inequality for the last step in $T_{3}$ follows from the Cauchy-Schwarz inequality, $\\|\Pi_{1}v\\|_{L^{2}(P)}$ $\leq\\|v\\|_{L^{2}(P)}$ from (2.11) and the Poincaŕe-Friedrichs inequality in Lemma 2.1.a for $w_{h}-\Pi_{1}w_{h}$ with $\int_{\partial P}(w_{h}-\Pi_{1}w_{h})\,ds=0$ from $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$. The combination of the above estimates shows (3.11). The proof of (3.12) adapts the arguments in the above analysis of $T_{3}$ and the definition of $\mathrm{osc}_{1}(f,\mathcal{T})$ in Subsection 2.1 for the proof of $\displaystyle(f-f_{h},w_{h})_{L^{2}(P)}=(f-\Pi_{1}f,w_{h}-\Pi_{1}w_{h})_{L^{2}(P)}\leq C_{\text{PF}}\|w_{h}\|_{1,P}\,\mathrm{osc}_{1}(f,P).$ This concludes the proof. ∎ ### 3.4 Nonconformity error Enrichment operators play a vital role in the analysis of nonconforming finite element methods [12]. For any $v_{h}\in V_{h},$ the objective is to find a corresponding function $Jv_{h}\in H_{0}^{1}(\Omega)$. The idea is to map the VEM nonconforming space into the Crouzeix-Raviart finite element space $\displaystyle\text{CR}_{0}^{1}(\widehat{{\cal T}}):=\\{v\in{\cal P}_{1}(\widehat{{\cal T}}):\hskip 2.84526pt$ $\displaystyle\forall\;E\in\widehat{{\cal E}}\quad v\hskip 2.84526pt\text{is continuous at mid}(E)\quad\text{and}$ $\displaystyle\forall\;E\in{\cal E}(\partial\Omega)\quad v(\text{mid}(E))=0\\}$ with respect to the shape-regular triangulation $\widehat{{\cal T}}$ from Remark 1. Let $\psi_{E}$ be the edge-oriented basis functions of CR${}_{0}^{1}(\widehat{{\cal T}})$ with $\psi_{E}(\text{mid}\hskip 2.84526ptE)=1$ and $\psi_{E}(\text{mid}\hskip 2.84526ptF)=0$ for all other edges $F\in\widehat{{\cal E}}\setminus\\{E\\}.$ Define the interpolation operator $I_{\text{CR}}:V_{h}\to\text{CR}_{0}^{1}(\widehat{{\cal T}})$, for $v_{h}\in V_{h}$, by $\displaystyle I_{\text{CR}}v_{h}=\sum_{F\in\widehat{{\cal E}}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{F}v_{h}\,ds\right)\psi_{F}.$ (3.14) The definition of $V_{h}$ implies $\int_{F}[v_{h}]\,ds=0$ for $v_{h}\in V_{h}$ and for all $F\in\mathcal{E}$. Since $v_{h}\|_{P}\in H^{1}(P)$, it follows $\int_{F}[v_{h}]\,ds=0$ for all $F\in\widehat{{\cal E}}\setminus\mathcal{E}$. This shows $\int_{F}v_{h\|T^{\pm}}\,ds$ is unique for all edges $F=\partial T^{+}\cap\partial T^{-}\in\widehat{{\cal E}}$ and, consequently, $I_{\text{CR}}v_{h}$ is well-defined (independent of the choice of traces selected in the evaluation of $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{F}v_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{F}v_{h}\|_{T^{+}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{F}v_{h}\|_{T^{-}}\,ds$). The approximation property of $I_{\text{CR}}$ on each $T\in\widehat{{\cal T}}$ reads $\displaystyle h_{T}^{-1}\\|v_{h}-I_{\text{CR}}v_{h}\\|_{L^{2}(T)}+\|v_{h}-I_{\text{CR}}v_{h}\|_{1,T}\leq 2\|v_{h}\|_{1,T}$ (3.15) (cf. [23, Thm 2.1] or [21, Thm 4] for explicit constants). Define an enrichment operator $E_{h}:\text{CR}_{0}^{1}(\widehat{{\cal T}})\to H_{0}^{1}(\Omega)$ by averaging the function values at each interior vertex $z$, that is, $\displaystyle E_{h}v_{\text{CR}}(z)=\frac{1}{\|\widehat{{\cal T}}(z)\|}\sum_{T\in\widehat{{\cal T}}(z)}{v_{\text{CR}}}\|_{T}(z)$ (3.16) and zero on boundary vertices. In (3.16) the set $\widehat{{\cal T}}(z):=\\{T\in\widehat{{\cal T}}\hskip 2.84526pt\|\hskip 2.84526ptz\in T\\}$ of neighboring triangles has the cardinality $\|\widehat{{\cal T}}(z)\|\geq 3$. The following lemma describes the construction of a modified companion operator $J:V_{h}\to H_{0}^{1}(\Omega)$, which is a right-inverse of the interpolation operator $I_{h}$ from Definition 2.7. ###### Lemma 3.3 (conforming companion operator). There exists a linear map $J:V_{h}\to H^{1}_{0}(\Omega)$ and a universal constant $C_{\mathrm{J}}\lesssim 1$ such that any $v_{h}\in V_{h}$ satisfies $I_{h}Jv_{h}=v_{h}$ and 1. (a) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds$ for any edge $E\in\widehat{{\cal E}},$ 2. (b) $\displaystyle\nabla_{\mathrm{pw}}(v_{h}-Jv_{h})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ in $L^{2}(\Omega;\mathbb{R}^{2}),$ 3. (c) $\displaystyle v_{h}-Jv_{h}\perp\mathcal{P}_{1}(\mathcal{T})$ in $L^{2}(\Omega),$ 4. (d) $\\|h_{\mathcal{T}}^{-1}(v_{h}-Jv_{h})\\|_{L^{2}(\Omega)}+\|v_{h}-Jv_{h}\|_{1,\mathrm{pw}}\leq C_{\mathrm{J}}\|v_{h}\|_{1,\mathrm{pw}}.$ ###### Design of $J$ in Lemma 3.3. ∎ Given $v_{h}\in V_{h}$, let $v_{\text{CR}}:=I_{\text{CR}}v_{h}\in\text{CR}^{1}_{0}(\widehat{{\cal T}})$. There exists an operator $J^{\prime}:\text{CR}_{0}^{1}(\widehat{{\cal T}})\to H_{0}^{1}(\Omega)$ from [22, Prop. 2.3] such that any $v_{\text{CR}}\in\text{CR}_{0}^{1}(\widehat{{\cal T}})$ satisfies 1. (a’) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}J^{\prime}v_{CR}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v_{CR}\,ds$ for any edge $E\in\widehat{{\cal E}},$ 2. (b’) $\displaystyle\int_{P}\nabla_{\mathrm{pw}}(v_{\text{CR}}-J^{\prime}v_{\text{CR}})\,dx=0$ for all $P\in\mathcal{T}$, 3. (c’) $\displaystyle\\|h_{\widehat{{\cal T}}}^{-1}(v_{\text{CR}}-J^{\prime}v_{\text{CR}})\\|_{L^{2}(\Omega)}+\|v_{\text{CR}}-J^{\prime}v_{\text{CR}}\|_{1,\mathrm{pw}}\leq C_{\mathrm{J^{\prime}}}\min_{v\in H^{1}_{0}(\Omega)}\|v_{\text{CR}}-v\|_{1,\mathrm{pw}}$ with a universal constant $C_{\mathrm{J^{\prime}}}$ from [25]. Set $v:=J^{\prime}I_{\text{CR}}v_{h}\in V:=H^{1}_{0}(\Omega)$. Recall that $\widehat{{\cal T}}(P)$ is a shape-regular triangulation of $P$ into a finite number of triangles. For each $T\in\widehat{{\cal T}}(P)$, let $b_{T}\in W_{0}^{1,\infty}(T)$ denote the cubic bubble-function $27\lambda_{1}\lambda_{2}\lambda_{3}$ for the barycentric co-ordinates $\lambda_{1},\lambda_{2},\lambda_{3}\in\mathcal{P}_{1}(T)$ of $T$ with $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{T}b_{T}\,dx=9/20$ and $\\|\nabla b_{T}\\|_{L^{2}(T)}\lesssim h_{T}^{-1}\|T\|^{1/2}\approx 1.$ Let $b_{T}$ be extended by zero outside $T$ and, for $P\in\mathcal{T}$, define $\displaystyle b_{P}:=\frac{20}{9}\sum_{T\in\widehat{{\cal T}}(P)}b_{T}\in W_{0}^{1,\infty}(P)\subset W_{0}^{1,\infty}(\Omega)$ (3.17) with $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{P}b_{P}\,dx=1$ and $\\|\nabla b_{P}\\|_{L^{2}(P)}\lesssim h_{P}^{-1}\|P\|^{1/2}\approx 1$. Let $v_{P}\in\mathcal{P}_{1}(\mathcal{T})$ be the Riesz representation of the linear functional $\mathcal{P}_{1}(\mathcal{T})\to\mathbb{R}$ defined by $w_{1}\mapsto(v_{h}-v,w_{1})_{L^{2}(\Omega)}$ for $w_{1}\in\mathcal{P}_{1}(\mathcal{T})$ in the Hilbert space $\mathcal{P}_{1}(\mathcal{T})$ endowed with the weighted $L^{2}$ scalar product $(b_{P}\bullet,\bullet)_{L^{2}(P)}$. Hence $v_{P}$ exists uniquely and satisfies $\Pi_{1}(v_{h}-v)=\Pi_{1}(b_{P}v_{P})$. Given the bubble-functions $(b_{P}:P\in\mathcal{T})$ from (3.17) and the above functions $(v_{P}:P\in\mathcal{T})$ for $v_{h}\in V_{h}$, define $\displaystyle Jv_{h}:=v+\sum_{P\in\mathcal{T}}v_{P}b_{P}\in V.$ (3.18) ###### Proof of (a). Since $b_{P}$ vanishes at any $x\in E\in\mathcal{E}$, it follows for any $E\in\widehat{\mathcal{E}}$ that $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}J^{\prime}v_{\text{CR}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v_{\text{CR}}\,ds=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds,$ where the definition of $v=J^{\prime}v_{\text{CR}}$, (a’), and $v_{\text{CR}}=I_{\text{CR}}v_{h}$ lead to the second, third, and fourth equality. This proves (a). ∎ ###### Proof of (b). An integration by parts and (a) show, for all $v_{h}\in V_{h}$ with $Jv_{h}$ from (3.18), that $\displaystyle\int_{P}\nabla Jv_{h}\,dx=\int_{\partial P}Jv_{h}\textbf{n}_{P}\,ds=\sum_{E\in\mathcal{E}(P)}\Big{(}\int_{E}Jv_{h}\textbf{n}_{E}\,ds\Big{)}=\sum_{E\in\mathcal{E}(P)}\Big{(}\int_{E}v_{h}\textbf{n}_{E}\,ds\Big{)}=\int_{P}\nabla v_{h}\,dx.$ Since this holds for all $P\in\mathcal{T}$, it proves (b). ∎ ###### Proof of (c). This is $\Pi_{1}v_{h}=\Pi_{1}Jv_{h}$ and guaranteed by the design of $J$ in (3.18). ∎ ###### Proof of (d). This relies on the definition of $J$ in (3.18) and $J^{\prime}$ with (c’). Since (a) allows for $\int_{\partial P}(v_{h}-Jv_{h})\,ds=0$, the Poincaré- Friedrichs inequality from Lemma 2.1.a implies $\displaystyle h_{P}^{-1}\\|v_{h}-Jv_{h}\\|_{L^{2}(P)}\leq C_{\text{PF}}\|v_{h}-Jv_{h}\|_{1,P}.$ Hence it remains to prove $\|v_{h}-Jv_{h}\|_{1,\mathrm{pw}}\lesssim\|v_{h}\|_{1,\mathrm{pw}}.$ Triangle inequalities with $v_{h},Jv_{h},v=J^{\prime}v_{\text{CR}}$ and $v_{\text{CR}}=I_{\text{CR}}v_{h}$ show the first and second inequality in $\displaystyle\|v_{h}-Jv_{h}\|_{1,\mathrm{pw}}-\|v-Jv_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|v-v_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|v_{h}-I_{\text{CR}}v_{h}\|_{1,\mathrm{pw}}+\|v_{\text{CR}}-J^{\prime}v_{\text{CR}}\|_{1,\mathrm{pw}}\leq(1+C_{\mathrm{J^{\prime}}})\|v_{h}\|_{1,\mathrm{pw}}$ (3.19) with (b’) for $\|v_{\text{CR}}\|_{1,\mathrm{pw}}=\\|\Pi_{0}\nabla_{\mathrm{pw}}v_{h}\\|_{L^{2}(\Omega)}\leq\\|\nabla_{\mathrm{pw}}v_{h}\\|_{L^{2}(\Omega)}=\|v_{h}\|_{1,\mathrm{pw}}$ in the last step. The equivalence of norms in the finite-dimensional space $\mathcal{P}_{1}(P)$ assures the existence of a positive constant $C_{b}$, independent of $h_{P}$, such that any $\chi\in\mathcal{P}_{1}(P)$ satisfies the inverse inequalities $\displaystyle C_{b}^{-1}\\|\chi\\|^{2}_{L^{2}(P)}\leq$ $\displaystyle(b_{P},\chi^{2})_{L^{2}(P)}\leq C_{b}\\|\chi\\|^{2}_{L^{2}(P)},$ (3.20) $\displaystyle C_{b}^{-1}\\|\chi\\|_{L^{2}(P)}\leq\\|b_{P}\chi\\|_{L^{2}(P)}$ $\displaystyle+h_{P}\\|\nabla(b_{P}\chi)\\|_{L^{2}(P)}\leq C_{b}\\|\chi\\|_{L^{2}(P)}.$ (3.21) These estimates are completely standard on shape-regular triangles [2, p. 27] or [37]; so they hold on each $T\in\widehat{{\cal T}}$ and, by definition of $b_{P}$, their sum is (3.20)-(3.21). The analysis of the term $\|v-Jv_{h}\|_{1,\mathrm{pw}}$ starts with one $P\in\mathcal{T}$ and (3.18) for $\displaystyle\|v-Jv_{h}\|_{1,P}=\|v_{P}b_{P}\|_{1,P}\leq C_{b}h_{P}^{-1}\\|v_{P}\\|_{L^{2}(P)}$ (3.22) with (3.21) in the last step. The estimate (3.20) leads to the first inequality in $\displaystyle C_{b}^{-1}\\|v_{P}\\|^{2}_{L^{2}(P)}\leq(b_{P}v_{P},v_{P})_{L^{2}(P)}=(v_{h}-v,v_{P})_{L^{2}(P)}\leq\\|v_{h}-v\\|_{L^{2}(P)}\\|v_{P}\\|_{L^{2}(P)}.$ The equality results from $\Pi_{1}(v_{h}-v)=\Pi_{1}(v_{P}b_{P})$ and $v_{P}\in\mathcal{P}_{1}(\mathcal{T})$, while the last step is the Cauchy- Schwarz inequality. Consequently, $\\|v_{P}\\|_{L^{2}(P)}\leq C_{b}\\|v_{h}-v\\|_{L^{2}(P)}$. This and (3.22) show $\displaystyle\|v-Jv_{h}\|_{1,\mathrm{pw}}\leq C_{b}^{2}\\|h^{-1}_{\mathcal{T}}(v-v_{h})\\|_{L^{2}(\Omega)}\leq C_{b}^{2}C_{\mathrm{PF}}\|v-v_{h}\|_{1,\mathrm{pw}}$ with $\int_{\partial P}(v-v_{h})\,ds=0$ from $(a)$ and hence the Poincaré- Friedrichs inequality for $v-v_{h}$ from Lemma 2.1.a in the last step. Recall $\|v-v_{h}\|_{1,\mathrm{pw}}\lesssim\|v_{h}\|_{1,\mathrm{pw}}$ from (3.19) to conclude $\|v-Jv_{h}\|_{1,\mathrm{pw}}\lesssim\|v_{h}\|_{1,\mathrm{pw}}$ from the previous displayed inequality. This concludes the proof of (d). ∎ ###### Proof of $I_{h}J=\text{id}\;\text{ in}\;V_{h}$. Definition 2.7 and Lemma 3.3.a show, for all $v_{h}\in V_{h}$, that $\displaystyle I_{h}Jv_{h}=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}Jv_{h}\,ds\Big{)}\psi_{E}=\sum_{E\in\mathcal{E}}\Big{(}\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}v_{h}\,ds\Big{)}\psi_{E}=v_{h}.$ This concludes the proof of Lemma 3.3. ∎ Since $V_{h}$ is not a subset of $H^{1}_{0}(\Omega)$ in general, the substitution of discrete function $v_{h}$ in the weak formulation leads to a nonconformity error. ###### Lemma 3.4 (nonconformity error). There exist positive universal constants $C_{\mathrm{NC}},C^{}_{\mathrm{NC}}$ (depending on the coefficients $\textbf{A},\textbf{b}$ and the universal constants $\rho,\sigma$) such that all $f,g\in L^{2}(\Omega)$ and all $\mathcal{T}\in\mathbb{T}(\delta)$ (with the assumption $h_{\text{max}}\leq\delta\leq 1$) satisfy $(a)$ and $(b)$. $(a)$ The solution $u\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ to $(\ref{1})$ satisfies $\displaystyle\sup_{0\neq v_{h}\in V_{h}}\frac{\|B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})_{L^{2}(\Omega)}\|}{\\|v_{h}\\|_{1,\mathrm{pw}}}\leq C_{\mathrm{NC}}h_{\text{max}}^{\sigma}\\|f\\|_{L^{2}(\Omega)}.$ (3.23) $(b)$ The solution $\Phi\in H^{1+\sigma}(\Omega)\cap H^{1}_{0}(\Omega)$ to the dual problem $(\ref{5})$ satisfies $\displaystyle\sup_{0\neq v_{h}\in V_{h}}\frac{\|B_{\mathrm{pw}}(v_{h},\Phi)-(g,v_{h})_{L^{2}(\Omega)}\|}{\\|v_{h}\\|_{1,\mathrm{pw}}}\leq C^{}_{\mathrm{NC}}h_{\text{max}}^{\sigma}\\|g\\|_{L^{2}(\Omega)}.$ (3.24) ###### Proof of $(a)$. Given $v_{h}\in V_{h}$, define $Jv_{h}\in V$ and the piecewise averages $\overline{\textbf{A}}:=\Pi_{0}(\textbf{A}),\overline{\textbf{b}}:=\Pi_{0}(\textbf{b})$, and $\overline{\gamma}:=\Pi_{0}(\gamma)$ of the coefficients $\textbf{A},\textbf{b}$, and $\gamma$. The choice of test function $v:=Jv_{h}\in V$ in the weak formulation (1.8) having extra properties provides the terms with oscillations in the further analysis. Abbreviate $\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u$. The weak formulation (1.8), Lemma 3.3.b-c, and the Cauchy-Schwarz inequality reveal that $\displaystyle B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})_{L^{2}(\Omega)}=B_{\mathrm{pw}}(u,v_{h}-Jv_{h})-(f,v_{h}-Jv_{h})_{L^{2}(\Omega)}$ $\displaystyle\leq\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(\Omega)}\\|\nabla_{\mathrm{pw}}(1-J)v_{h}\\|_{L^{2}(\Omega)}+\\|h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma u)\\|_{L^{2}(\Omega)}\\|h_{\mathcal{T}}^{-1}(1-J)v_{h}\\|_{L^{2}(\Omega)}.$ (3.25) The first term on the right-hand side of (3.25) involves the factor $\displaystyle\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(\Omega)}$ $\displaystyle\leq\\|\textbf{A}\nabla u-\Pi_{0}(\textbf{A}\nabla u)\\|_{L^{2}(\Omega)}+\\|\textbf{b}u-\Pi_{0}(\textbf{b}u)\\|_{L^{2}(\Omega)}$ $\displaystyle\leq\\|(\textbf{A}-\overline{\textbf{A}})\nabla u+\overline{\textbf{A}}(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}+\\|(\textbf{b}-\overline{\textbf{b}})u+\overline{\textbf{b}}(1-\Pi_{0})u\\|_{L^{2}(\Omega)}$ $\displaystyle\leq\Big{(}h_{\text{max}}(\|\textbf{A}\|_{1,\infty}+\|\textbf{b}\|_{1,\infty})+C_{\text{apx}}(h_{\text{max}}^{\sigma}\\|\textbf{A}\\|_{\infty}+h_{\text{max}}\\|\textbf{b}\\|_{\infty})\Big{)}\;\\|u\\|_{1+\sigma,\Omega}.$ The last inequality follows from the Lipschitz continuity of the coefficients A and b, and the estimate (2.12). Lemma 3.3.d leads to the estimates $\\|\nabla_{\mathrm{pw}}(1-J)v_{h}\\|_{L^{2}(\Omega)}\leq C_{J}\|v_{h}\|_{1,\mathrm{pw}}$ and $\displaystyle\\|h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma u)\\|_{L^{2}(\Omega)}\\|h_{\mathcal{T}}^{-1}(1-J)v_{h}\\|_{L^{2}(\Omega)}\leq\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})C_{J}\|v_{h}\|_{1,\mathrm{pw}}.$ The substitution of the previous estimates in (3.25) with $h_{\mathrm{max}}\leq 1$ (from $\delta\leq 1$ by assumption) and the regularity (1.5) show $\displaystyle B_{\mathrm{pw}}(u,v_{h})-(f,v_{h})\leq C_{\mathrm{NC}}h_{\text{max}}^{\sigma}\\|f\\|_{L^{2}(\Omega)}\\|v_{h}\\|_{1,\mathrm{pw}}$ with $C_{\mathrm{NC}}:=C_{J}\Big{(}(\|\textbf{A}\|_{1,\infty}+\|\textbf{b}\|_{1,\infty}+C_{\text{apx}}(\\|\textbf{A}\\|_{\infty}+\\|\textbf{b}\\|_{\infty})+\\|\gamma\\|_{\infty})C_{\text{reg}}+1\Big{)}$. This concludes the proof of Lemma 3.4.a. ∎ ###### Proof of $(b)$. The solution $\Phi\in V$ to (1.4) satisfies $B(v,\Phi)=(g,v)_{L^{2}(\Omega)}$ for all $v\in V.$ This implies $B_{\mathrm{pw}}(v_{h},\Phi)-(g,v_{h})_{L^{2}(\Omega)}=B_{\mathrm{pw}}(v_{h}-Jv_{h},\Phi)-(g,v_{h}-Jv_{h})_{L^{2}(\Omega)}.$ The arguments in the proof of $(a)$ lead to the bound (3.24) with $C^{}_{\mathrm{NC}}:=C_{J}\Big{(}(\|\textbf{A}\|_{1,\infty}+C_{\text{apx}}\\|\textbf{A}\\|_{\infty}+\\|\textbf{b}\\|_{\infty}+\\|\gamma\\|_{\infty})C^{}_{\text{reg}}+1\Big{)}.$ The remaining analogous details are omitted in the proof of Lemma 3.4.b for brevity. ∎ ## 4 A priori error analysis This section focuses on the stability, existence, and uniqueness of the discrete solution $u_{h}$. The a priori error analysis uses the discrete inf- sup condition. ### 4.1 Existence and uniqueness of the discrete solution ###### Theorem 4.1 (stability). There exist positive constants $\delta\leq 1$ and $C_{\mathrm{stab}}$ (depending on $\alpha,\beta,\sigma,\rho,$ and $C_{\mathrm{F}}$) such that, for all $\mathcal{T}\in\mathbb{T}(\delta)$ and for all $f\in L^{2}(\Omega)$, the discrete problem (3.8) has a unique solution $u_{h}\in V_{h}$ and $\displaystyle\|u_{h}\|_{1,\mathrm{pw}}\leq C_{\mathrm{stab}}\\|f_{h}\\|_{V_{h}^{}}.$ ###### Proof. In the first part of the proof, suppose there exists some solution $u_{h}\in V_{h}$ to the discrete problem (3.8) for some $f\in L^{2}(\Omega)$. (This is certainly true for all $f\equiv 0\equiv u_{h}$, but will be discussed for all those pairs at the end of the proof and shall lead to the uniqueness of discrete solutions.) Since $u_{h}$ satisfies a G$\mathring{a}$rding-type inequality in Proposition 3.1.b, $\displaystyle\alpha\|u_{h}\|_{1,\mathrm{pw}}^{2}$ $\displaystyle\leq\beta\\|u_{h}\\|^{2}_{L^{2}(\Omega)}+B_{h}(u_{h},u_{h})=\beta\\|u_{h}\\|^{2}_{L^{2}(\Omega)}+(f_{h},u_{h})_{L^{2}(\Omega)}.$ This, (2.14), and the definition of the dual norm in (3.12) lead to $\displaystyle\alpha\|u_{h}\|_{1,\mathrm{pw}}\leq\beta C_{\text{F}}\\|u_{h}\\|_{L^{2}(\Omega)}+\\|f_{h}\\|_{V_{h}^{}}.$ (4.1) Given $g:=u_{h}\in L^{2}(\Omega)$, let $\Phi\in V\cap H^{1+\sigma}(\Omega)$ solve the dual problem ${\cal L}^{}\Phi=g$ and let $I_{h}\Phi\in V_{h}$ be the interpolation of $\Phi$ from Subsection 2.4. Elementary algebra shows $\displaystyle\\|u_{h}\\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=\Big{(}(g,u_{h})_{L^{2}(\Omega)}-B_{\mathrm{pw}}(u_{h},\Phi)\Big{)}+B_{\mathrm{pw}}(u_{h},\Phi- I_{h}\Phi)$ $\displaystyle\quad+\Big{(}B_{\mathrm{pw}}(u_{h},I_{h}\Phi)-B_{h}(u_{h},I_{h}\Phi)\Big{)}+(f_{h},I_{h}\Phi)_{L^{2}(\Omega)}.$ (4.2) Rewrite a part of the third term corresponding to diffusion on the right-hand side of (4.2) as $\displaystyle a^{P}(u_{h},I_{h}\Phi)-a_{h}^{P}(u_{h},I_{h}\Phi)=(\textbf{A}\nabla u_{h},\nabla(1-\Pi_{1})I_{h}\Phi)_{L^{2}(P)}$ $\displaystyle+(\nabla(1-\Pi_{1})u_{h},(\textbf{A}-\Pi_{0}\textbf{A})(\nabla\Pi_{1}I_{h}\Phi))_{L^{2}(P)}-S^{P}\big{(}(1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Phi\big{)}.$ The Cauchy-Schwarz inequality in the semi-scalar product $S^{P}(\bullet,\bullet)$, and (3.5) with the upper bound $\\|\textbf{A}\\|_{\infty}$ for the coefficient A in $a^{P}(\bullet,\bullet)$ lead to the estimate $\displaystyle C_{s}^{-1}$ $\displaystyle S^{P}\big{(}(1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Phi\big{)}\leq\|(1-\Pi_{1})u_{h}\|_{1,P}\|(1-\Pi_{1})I_{h}\Phi\|_{1,P}$ $\displaystyle\qquad\leq\\|\textbf{A}\\|_{\infty}\|u_{h}\|_{1,P}\Big{(}\\|\nabla(I_{h}\Phi-\Phi)\\|_{L^{2}(P)}+\\|\nabla(1-\Pi_{1}I_{h})\Phi\\|_{L^{2}(P)}\Big{)}$ $\displaystyle\qquad\leq\\|\textbf{A}\\|_{\infty}C_{\text{apx}}\Big{(}2+C_{\mathrm{PF}}+C_{\text{Itn}}\Big{)}h_{P}^{\sigma}\|u_{h}\|_{1,P}\|\Phi\|_{1+\sigma,P}$ (4.3) with Theorem 2.8.b followed by (2.12) in the final step. This and Theorem 2.8 imply that $\displaystyle\|a^{P}(u_{h},I_{h}\Phi)-a_{h}^{P}(u_{h},I_{h}\Phi)\|$ $\displaystyle\leq h_{P}^{\sigma}\|u_{h}\|_{1,P}\\|\Phi\\|_{1+\sigma,P}$ $\displaystyle\quad\times\Big{(}\\|\textbf{A}\\|_{\infty}C_{\text{apx}}(2+C_{\mathrm{PF}}+C_{\text{Itn}})(1+C_{s})+\|\textbf{A}\|_{1,\infty}C_{\text{Itn}}\Big{)}.$ The terms $b^{P}-b_{h}^{P}$ and $c^{P}-c_{h}^{P}$ are controlled by $\displaystyle\|b^{P}(u_{h},I_{h}\Phi)-b_{h}^{P}(u_{h},I_{h}\Phi)\|+\|c^{P}(u_{h},I_{h}\Phi)-c_{h}^{P}(u_{h},I_{h}\Phi)\|$ $\displaystyle\leq h_{P}^{\sigma}\\|\Phi\\|_{1+\sigma,P}\big{(}\\|\textbf{b}\\|_{\infty}(C_{\text{apx}}(2+C_{\mathrm{PF}}+C_{\text{Itn}})\\|u_{h}\\|_{L^{2}(P)}+C_{\mathrm{Itn}}C_{\mathrm{PF}}\|u_{h}\|_{1,P})$ $\displaystyle\hskip 71.13188pt+\\|\gamma\\|_{\infty}C_{\mathrm{PF}}(C_{\text{Itn}}\\|u_{h}\\|_{L^{2}(P)}+\|u_{h}\|_{1,P})\big{)}.$ The combination of the previous four displayed estimates with Lemma 2.6 leads to an estimate for $P$. The sum over all polygonal domains $P\in\mathcal{T}$ reads $\displaystyle B_{\mathrm{pw}}(u_{h},I_{h}\Phi)-B_{h}(u_{h},I_{h}\Phi)\leq C_{d}h_{\text{max}}^{\sigma}\|u_{h}\|_{1,\mathrm{pw}}\\|\Phi\\|_{1+\sigma,\Omega}$ (4.4) with a universal constant $C_{d}$. The bound for (4.2) results from Lemma 3.4.b for the first term, the boundedness of $B_{\mathrm{pw}}$ (with a universal constant $M_{b}:=\\|\textbf{A}\\|_{\infty}+C_{\mathrm{F}}\\|\textbf{b}\\|_{\infty}+C_{\mathrm{F}}^{2}\\|\gamma\\|_{\infty}$) and (2.15) for the second term, (4.4) for the third term, and Theorem 2.8.a for the last term on the right-hand side of (4.2). This shows $\displaystyle\\|u_{h}\\|^{2}_{L^{2}(\Omega)}$ $\displaystyle\leq\Big{(}C^{}_{\mathrm{NC}}+C_{\text{I}}M_{b}+C_{d}\Big{)}h_{\text{max}}^{\sigma}\|u_{h}\|_{1,\mathrm{pw}}\\|\Phi\\|_{1+\sigma,\Omega}+C_{\text{Itn}}\\|f_{h}\\|_{V_{h}^{}}\\|\Phi\\|_{1,\Omega}.$ This and the regularity estimate (1.5) lead to $C_{3}=C^{}_{\mathrm{NC}}+C_{\text{I}}M_{b}+C_{d}$ in $\displaystyle\\|u_{h}\\|_{L^{2}(\Omega)}\leq C_{3}\,C^{}_{\text{reg}}h_{\text{max}}^{\sigma}\|u_{h}\|_{1,\mathrm{pw}}+C_{\text{Itn}}\\|f_{h}\\|_{V_{h}^{}}.$ The substitution of this in (4.1) proves $\displaystyle\alpha\|u_{h}\|_{1,\mathrm{pw}}\leq\beta C_{\text{F}}C_{3}C^{}_{\text{reg}}h_{\text{max}}^{\sigma}\|u_{h}\|_{1,\mathrm{pw}}+(\beta C_{\text{F}}C_{\text{Itn}}+1)\\|f_{h}\\|_{V_{h}^{}}.$ (4.5) For all $0<h_{\text{max}}\leq\delta:=(\frac{\alpha}{2\beta C_{\text{F}}C_{3}C^{}_{\text{reg}}})^{1/\sigma}$, the constant $\overline{c}=(1-\frac{\beta}{\alpha}C_{\text{F}}C_{3}C^{}_{\text{reg}}h_{\text{max}}^{\sigma})$ is positive and $C_{\text{stab}}:=\frac{\beta C_{\text{F}}C_{\text{Itn}}+1}{\alpha-\beta C_{\mathrm{F}}C_{3}C^{}_{\text{reg}}h_{0}^{\sigma}}$ is well-defined. This leads in (4.5) to $\displaystyle\|u_{h}\|_{1,\mathrm{pw}}\leq C_{\text{stab}}\\|f_{h}\\|_{V_{h}^{}}.$ (4.6) In the last part of the proof, suppose $f_{h}\equiv 0$ and let $u_{h}$ be any solution to the resulting homogeneous linear discrete system. The stability result (4.6) proves $u_{h}\equiv 0$. Hence, the linear system of equations (3.8) has a unique solution and the coefficient matrix is regular. This proves that there exists a unique solution $u_{h}$ to (3.8) for any right-hand side $f_{h}\in V_{h}^{}$. The combination of this with (4.6) concludes the proof. ∎ An immediate consequence of Theorem 4.1 is the following discrete inf-sup estimate. ###### Theorem 4.2 (discrete inf-sup). There exist $0<\delta\leq 1$ and $\overline{\beta}_{0}>0$ such that, for all $\mathcal{T}\in\mathbb{T}(\delta)$, $\displaystyle\overline{\beta}_{0}\leq\inf_{0\neq u_{h}\in V_{h}}\sup_{0\neq v_{h}\in V_{h}}\frac{B_{h}(u_{h},v_{h})}{\|u_{h}\|_{1,\mathrm{pw}}\|v_{h}\|_{1,\mathrm{pw}}}.$ (4.7) ###### Proof. Define the operator ${\cal L}_{h}:V_{h}\to V_{h}^{},$ $v_{h}\mapsto B_{h}(v_{h},\bullet)$. The stability Theorem 4.1 can be interpreted as follows: For any $f_{h}\in V_{h}^{}$ there exists $u_{h}\in V_{h}$ such that ${\cal L}_{h}u_{h}=f_{h}$ and $\displaystyle\overline{\beta}_{0}\|u_{h}\|_{1,\mathrm{pw}}\leq\\|f_{h}\\|_{V_{h}^{}}=\sup_{0\neq v_{h}\in V_{h}}\frac{(f_{h},v_{h})}{\|v_{h}\|_{1,\mathrm{pw}}}=\sup_{0\neq v_{h}\in V_{h}}\frac{B_{h}(u_{h},v_{h})}{\|v_{h}\|_{1,\mathrm{pw}}}.$ The discrete problem $B_{h}(u_{h},\bullet)=(f_{h},\bullet)$ has a unique solution in $V_{h}$. Therefore, $f_{h}$ and $u_{h}$ are in one to one correspondence and the last displayed estimate holds for any $u_{h}\in V_{h}$. The infimum over $u_{h}\in V_{h}$ therein proves (4.7) with $\overline{\beta}_{0}=C_{\text{stab}}^{-1}$. ∎ ### 4.2 A priori error estimates This subsection establishes the error estimate in the energy norm $\|\cdot\|_{1,\mathrm{pw}}$ and in the $L^{2}$ norm. The discrete inf-sup condition allows for an error estimate in the $H^{1}$ norm and an Aubin- Nitsche duality argument leads to an error estimate in the $L^{2}$ norm. Recall $u\in H^{1}_{0}(\Omega)$ is a unique solution of (1.8) and $u_{h}\in V_{h}$ is a unique solution of (3.8). Recall the definition of the bilinear form $s_{h}(\cdot,\cdot)$ from Section 3.1 and define the induced seminorm $\|v_{h}\|_{\mathrm{s}}:=s_{h}(v_{h},v_{h})^{1/2}$ for $v_{h}\in V_{h}$ as a part of the norm $\\|\cdot\\|_{h}$ from Remark 6. ###### Theorem 4.3 (error estimate). Set $\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u\in H(\text{div},\Omega)$. There exist positive constants $C_{4},C_{5},$ and $\delta$ such that, for all $\mathcal{T}\in\mathbb{T}(\delta)$, the discrete problem (3.8) has a unique solution $u_{h}\in V_{h}$ and $\displaystyle\|u-u_{h}\|_{1,\mathrm{pw}}+\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+h_{\mathrm{max}}^{-\sigma}(\\|u-u_{h}\\|_{L^{2}(\Omega)}+\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)})+\|u_{h}\|_{\mathrm{s}}+\|I_{h}u-u_{h}\|_{\mathrm{s}}$ $\displaystyle\quad\leq C_{4}\Big{(}\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)}+\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})\Big{)}\leq C_{5}h_{\mathrm{max}}^{\sigma}\\|f\\|_{L^{2}(\Omega)}.$ (4.8) ###### Proof. Step 1 (initialization). Let $I_{h}u\in V_{h}$ be the interpolation of $u$ from Definition 2.7. The discrete inf-sup condition (4.7) for $I_{h}u-u_{h}\in V_{h}$ leads to some $v_{h}\in V_{h}$ with $\|v_{h}\|_{1,\mathrm{pw}}\leq 1$ such that $\displaystyle\overline{\beta}_{0}\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}=B_{h}(I_{h}u-u_{h},v_{h}).$ Step 2 (error estimate for $\|u-u_{h}\|_{1,\mathrm{pw}}$). Rewrite the last equation with the continuous and the discrete problem (1.8) and (3.8) as $\displaystyle\overline{\beta}_{0}\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}=B_{h}(I_{h}u,v_{h})-B(u,v)+(f,v)_{L^{2}(\Omega)}-(f_{h},v_{h})_{L^{2}(\Omega)}.$ This equality is rewritten with the definition of $B(u,v)$ in (1.7), the definition of $B_{h}(I_{h}u,v_{h})$ in Section 3.1, and with $f_{h}=\Pi_{1}f$. Recall $v:=Jv_{h}\in V$ from Lemma 3.3 and recall $\nabla_{\mathrm{pw}}\Pi_{1}I_{h}u=\Pi_{0}\nabla u$ from (2.17). This results in LHS $\displaystyle:=\overline{\beta}_{0}\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}-s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})v_{h})$ $\displaystyle=(\textbf{A}\Pi_{0}\nabla u+\textbf{b}\Pi_{1}I_{h}u,\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}+(\gamma\Pi_{1}I_{h}u,\Pi_{1}v_{h})_{L^{2}(\Omega)}-(\bm{\sigma},\nabla v)_{L^{2}(\Omega)}$ $\displaystyle\qquad+(f-\gamma u,v)_{L^{2}(\Omega)}-(f,\Pi_{1}v_{h})_{L^{2}(\Omega)}.$ Abbreviate $w:=v-\Pi_{1}v_{h}$ and observe the orthogonalities $\nabla_{\mathrm{pw}}w\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ in $L^{2}(\Omega;\mathbb{R}^{2})$ and $w\perp\mathcal{P}_{1}(\mathcal{T})$ in $L^{2}(\Omega)$ from Lemma 3.3.b-c and the definition of $\Pi_{1}$ with $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$. Lemma 3.3.d, the bound $\|(1-\Pi^{\nabla}_{1})v_{h}\|_{1,\mathrm{pw}}\leq\|v_{h}\|_{1,\mathrm{pw}}\leq 1$, and the Poincaré-Friedrichs inequality for $v_{h}-\Pi^{\nabla}_{1}v_{h}$ from Lemma 2.1.a lead to $\displaystyle\|w\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|v-v_{h}\|_{1,\mathrm{pw}}+\|v_{h}-\Pi_{1}v_{h}\|_{1,\mathrm{pw}}\leq C_{\mathrm{J}}+1,$ (4.9) $\displaystyle\\|h_{\mathcal{T}}^{-1}w\\|_{L^{2}(\Omega)}$ $\displaystyle\leq\\|h_{\mathcal{T}}^{-1}(v-v_{h})\\|_{L^{2}(\Omega)}+\\|h_{\mathcal{T}}^{-1}(v_{h}-\Pi_{1}v_{h})\\|_{L^{2}(\Omega)}\leq C_{\mathrm{J}}+C_{\mathrm{PF}}.$ (4.10) Elementary algebra and the above orthogonalities prove that LHS $\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\Pi_{0}-1)\nabla u+\textbf{b}(\Pi_{1}I_{h}u-u),\nabla_{\mathrm{pw}}\Pi_{1}v_{h})_{L^{2}(\Omega)}-((1-\Pi_{0})\bm{\sigma},\nabla_{\mathrm{pw}}w)_{L^{2}(\Omega)}$ $\displaystyle\qquad+(\gamma(\Pi_{1}I_{h}u-u),\Pi_{1}v_{h})_{L^{2}(\Omega)}+(h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma u),h_{\mathcal{T}}^{-1}w)_{L^{2}(\Omega)}$ $\displaystyle\leq\Big{(}\|\textbf{A}\|_{1,\infty}+(1+C_{\mathrm{PF}})(\\|\textbf{b}\\|_{\infty}+C_{\mathrm{F}}\\|\gamma\\|_{\infty})\Big{)}h_{\text{max}}\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}$ $\displaystyle\quad+(C_{\mathrm{J}}+1)\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)}+(C_{\mathrm{J}}+C_{\text{PF}})\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})$ (4.11) with the Lipschitz continuity of A, Lemma 2.8.b, the stabilities of $\Pi_{1}$ from (2.11), and (4.9)-(4.10) in the last step. The definition of stability term (3.5) and Theorem 2.8.b lead to $\displaystyle C_{s}^{-1}s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})v_{h})$ $\displaystyle\leq\\|\textbf{A}\\|_{\infty}\|(1-\Pi_{1})I_{h}u\|_{1,\mathrm{pw}}\|(1-\Pi_{1})v_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\\|\textbf{A}\\|_{\infty}(\|I_{h}u-u\|_{1,\mathrm{pw}}+\|u-\Pi_{1}I_{h}u\|_{1,\mathrm{pw}})\|v_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\\|\textbf{A}\\|_{\infty}(2+C_{\text{Itn}}+C_{\mathrm{PF}})\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}\|v_{h}\|_{1,\mathrm{pw}}.$ (4.12) The triangle inequality, the bound (2.15) for the term $\|u-I_{h}u\|_{1,\mathrm{pw}}$, and (4.11)-(4.12) for the term $\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}$ conclude the proof of (4.8) for the term $\|u-u_{h}\|_{1,{\mathrm{pw}}}$. Step $3$ (duality argument). To prove the bound for $u-u_{h}$ in the $L^{2}$ norm with a duality technique, let $g:=I_{h}u-u_{h}\in L^{2}(\Omega)$. The solution $\Phi\in H^{1}_{0}(\Omega)\cap H^{1+\sigma}(\Omega)$ to the dual problem (1.4) satisfies the elliptic regularity (1.5), $\displaystyle\\|\Phi\\|_{1+\sigma,\Omega}\leq C^{}_{\text{reg}}\\|I_{h}u-u_{h}\\|_{L^{2}(\Omega)}.$ (4.13) Step $4$ (error estimate for $\\|u-u_{h}\\|_{L^{2}(\Omega)}$). Let $I_{h}\Phi\in V_{h}$ be the interpolation of $\Phi$ from Definition 2.7. Elementary algebra reveals the identity $\displaystyle\\|g\\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=((g,g)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(g,\Phi))+B_{\mathrm{pw}}(g,\Phi- I_{h}\Phi)$ $\displaystyle\quad+(B_{\mathrm{pw}}(g,I_{h}\Phi)-B_{h}(g,I_{h}\Phi))+B_{h}(g,I_{h}\Phi).$ (4.14) The bound (4.4) with $g$ as the first argument shows $\displaystyle B_{\mathrm{pw}}(g,I_{h}\Phi)-B_{h}(g,I_{h}\Phi)\leq C_{d}h_{\text{max}}^{\sigma}\|g\|_{1,\mathrm{pw}}\\|\Phi\\|_{1+\sigma,\Omega}.$ This controls the third term in (4.14), Lemma 3.4.b controls the first term, the boundedness of $B_{\mathrm{pw}}$ and the interpolation error estimate (2.15) control the second term on the right-hand side of (4.14). This results in $\displaystyle\\|I_{h}u-u_{h}\\|^{2}_{L^{2}(\Omega)}\leq(C^{}_{\mathrm{NC}}+C_{\mathrm{I}}M_{b}+C_{d})h_{\mathrm{max}}^{\sigma}\|g\|_{1,\mathrm{pw}}\\|\Phi\\|_{1+\sigma,\Omega}+B_{h}(g,I_{h}\Phi).$ (4.15) It remains to bound $B_{h}(g,I_{h}\Phi)$. The continuous and the discrete problem (1.8) and (3.8) imply $\displaystyle B_{h}(g,I_{h}\Phi)=B_{h}(I_{h}u,I_{h}\Phi)-B(u,\Phi)+(f,\Phi)_{L^{2}(\Omega)}-(f_{h},I_{h}\Phi)_{L^{2}(\Omega)}.$ The definition of $B_{h}$ and $\Pi_{0}$ lead to $\displaystyle B_{h}(g,I_{h}\Phi)-s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})I_{h}\Phi)$ $\displaystyle=((\textbf{A}-\Pi_{0}\textbf{A})(\Pi_{0}-1)\nabla u+\textbf{b}(\Pi_{1}I_{h}u-u),\nabla_{\mathrm{pw}}\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}+(\gamma(\Pi_{1}I_{h}u-u),\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}$ $\displaystyle\qquad-((1-\Pi_{0})\bm{\sigma},\nabla_{\mathrm{pw}}(1-\Pi_{1}I_{h})\Phi)_{L^{2}(\Omega)}+(f-\gamma u,\Phi-\Pi_{1}I_{h}\Phi)_{L^{2}(\Omega)}.$ (4.16) The bound for the stability term as in (4.12) is $\displaystyle s_{h}((1-\Pi_{1})I_{h}u,(1-\Pi_{1})I_{h}\Phi)$ $\displaystyle\leq C_{s}\\|\textbf{A}\\|_{\infty}\|(1-\Pi_{1})I_{h}u\|_{1,\mathrm{pw}}\|(1-\Pi_{1})I_{h}\Phi\|_{1,\mathrm{pw}}$ $\displaystyle\leq C_{s}\\|\textbf{A}\\|_{\infty}(2+C_{\mathrm{Itn}}+C_{\mathrm{PF}})^{2}C_{\text{apx}}h_{\text{max}}^{\sigma}\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}\|\Phi\|_{1+\sigma,\Omega}.$ (4.17) Step $5$ (oscillation). The last term in (4.16) is of optimal order $O(h_{\text{max}}^{1+\sigma})$, but the following arguments allow to write it as an oscillation. Recall the bubble-function $b_{\mathcal{T}}\|_{P}:=b_{P}\in H^{1}_{0}(P)$ from (3.17) extended by zero outside $P$. Given $\Psi:=\Phi-\Pi_{1}I_{h}\Phi$, let $\Psi_{1}\in\mathcal{P}_{1}(\mathcal{T})$ be the Riesz representation of the linear functional $\mathcal{P}_{1}(\mathcal{T})\to\mathbb{R}$ defined by $w_{1}\mapsto(\Psi,w_{1})_{L^{2}(\Omega)}$ in the Hilbert space $\mathcal{P}_{1}(\mathcal{T})$ endowed with the weighted scalar product $(b_{\mathcal{T}}\bullet,\bullet)_{L^{2}(\Omega)}$. That means $\Pi_{1}(b_{\mathcal{T}}\Psi_{1})=\Pi_{1}\Psi$. The identity $(f-\gamma u,b_{\mathcal{T}}\Psi_{1})_{L^{2}(\Omega)}=(\bm{\sigma},\nabla(b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}$ follows from (1.8) with the test function $b_{\mathcal{T}}\Psi_{1}\in H^{1}_{0}(\Omega)$. The $L^{2}$ orthogonalities $\Psi- b_{\mathcal{T}}\Psi_{1}\perp\mathcal{P}_{1}(\mathcal{T})$ in $L^{2}(\Omega)$ and $\nabla(b_{\mathcal{T}}\Psi_{1})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ in $L^{2}(\Omega;\mathbb{R}^{2})$ allow the rewriting of the latter identity as $\displaystyle(f-\gamma u,\Psi)_{L^{2}(\Omega)}=(h_{\mathcal{T}}(1-\Pi_{1})(f-\gamma u),h_{\mathcal{T}}^{-1}(\Psi- b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}+((1-\Pi_{0})\bm{\sigma},\nabla(b_{\mathcal{T}}\Psi_{1}))_{L^{2}(\Omega)}$ $\displaystyle\qquad\leq\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})\\|h_{\mathcal{T}}^{-1}(\Psi- b_{\mathcal{T}}\Psi_{1})\\|_{L^{2}(\Omega)}+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)}\|b_{\mathcal{T}}\Psi\|_{1,\mathrm{pw}}.$ (4.18) It remains to control the terms $\\|h_{\mathcal{T}}^{-1}(\Psi- b_{\mathcal{T}}\Psi_{1})\\|_{L^{2}(\Omega)}$ and $\|b_{\mathcal{T}}\Psi\|_{1,\mathrm{pw}}$. Since the definition of $I_{h}$ and the definition of $\Pi^{\nabla}_{1}$ with $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$ imply $\int_{\partial P}\Psi\,ds=\int_{\partial P}(\Phi-\Pi_{1}I_{h}\Phi)\,ds=0$, this allows the Poincaré-Friedrichs inequality for $\Psi$ from Lemma 2.1.a on each $P\in\mathcal{T}$. This shows $\displaystyle\\|h_{\mathcal{T}}^{-1}\Psi\\|_{L^{2}(\Omega)}\leq C_{\mathrm{PF}}\|\Psi\|_{1,\mathrm{pw}}\leq C_{\mathrm{PF}}C_{\text{apx}}h_{\text{max}}^{\sigma}\|\Phi\|_{1+\sigma,\Omega}$ (4.19) with Theorem 2.8.b and (2.12) in the last inequality. Since $b_{P}\Psi_{1}\in H^{1}_{0}(P)$ for $P\in\mathcal{T}$, the Poincaré-Friedrichs inequality from Lemma 2.1.a leads to $\displaystyle\\|h_{P}^{-1}(b_{P}\Psi_{1})\\|_{L^{2}(P)}\leq C_{\mathrm{PF}}\|b_{P}\Psi_{1}\|_{1,P}.$ (4.20) The first estimate in (3.20), the identity $\Pi_{1}(b_{\mathcal{T}}\Psi_{1})=\Pi_{1}\Psi$, and the Cauchy-Schwarz inequality imply $\displaystyle C_{b}^{-1}\\|h_{P}^{-1}\Psi_{1}\\|_{L^{2}(P)}^{2}\leq\\|h_{P}^{-1}b_{P}^{1/2}\Psi_{1}\\|_{L^{2}(P)}^{2}=(h_{P}^{-1}\Psi_{1},h_{P}^{-1}\Psi)_{L^{2}(P)}\leq\\|h_{P}^{-1}\Psi_{1}\\|_{L^{2}(P)}\\|h_{P}^{-1}\Psi\\|_{L^{2}(P)}.$ This proves $\\|h_{P}^{-1}\Psi_{1}\\|_{L^{2}(P)}\leq C_{b}\\|h_{P}^{-1}\Psi\\|_{L^{2}(P)}$. The second estimate in (3.21) followed by the first estimate in (3.20) leads to the first inequality and the arguments as above lead to the second inequality in $\displaystyle C_{b}^{-3/2}\|b_{P}\Psi_{1}\|_{1,P}\leq\\|h_{P}^{-1}b_{P}^{1/2}\Psi_{1}\\|_{L^{2}(P)}\leq\\|h_{P}^{-1}\Psi_{1}\\|_{L^{2}(P)}^{1/2}\\|h_{P}^{-1}\Psi\\|_{L^{2}(P)}^{1/2}$ $\displaystyle\leq C_{b}^{1/2}\\|h_{P}^{-1}\Psi\\|_{L^{2}(P)}$ with $\\|h_{P}^{-1}\Psi_{1}\\|_{L^{2}(P)}^{1/2}\leq C_{b}^{1/2}\\|h_{P}^{-1}\Psi\\|_{L^{2}(P)}^{1/2}$ from above in the last step. The combination of the previous displayed estimate and (4.18)-(4.20) results with $C_{6}:=C_{\mathrm{PF}}C_{\text{apx}}(1+C_{b}^{2}(1+C_{\mathrm{PF}}))$ in $\displaystyle(f-\gamma u,\Psi)_{L^{2}(\Omega)}\leq C_{6}(\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)})h_{\text{max}}^{\sigma}\|\Phi\|_{1+\sigma,\Omega}.$ (4.21) Step $6$ (continued proof of estimate for $\\|u-u_{h}\\|_{L^{2}(\Omega)}$). The estimate in Step 2 for $\|g\|_{1,\mathrm{pw}}$, (4.15)-(4.17), and (4.21) with the regularity (4.13) show $\displaystyle\\|I_{h}u-u_{h}\\|_{L^{2}(\Omega)}\lesssim h_{\mathrm{max}}^{\sigma}\Big{(}\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})\Big{)}.$ (4.22) Rewrite the difference $u-u_{h}=(u-I_{h}u)+(I_{h}u-u_{h})$, and apply the triangle inequality with (2.15) for the first term $\\|u-I_{h}u\\|_{L^{2}(\Omega)}\leq C_{\mathrm{I}}h_{\text{max}}^{1+\sigma}\|u\|_{1+\sigma,\Omega}.$ This and (4.22) for the second term $I_{h}u-u_{h}$ conclude the proof of the estimate for the term $h_{\text{max}}^{-\sigma}\\|u-u_{h}\\|_{L^{2}(\Omega)}$ in (4.8) . Step $7$ (stabilisation error $\|u_{h}\|_{\mathrm{s}}$ and $\|I_{h}u-u_{h}\|_{\mathrm{s}}$). The triangle inequality and the upper bound of the stability term (3.5) lead to $\displaystyle\|u_{h}\|_{\mathrm{s}}\leq\|I_{h}u-u_{h}\|_{\mathrm{s}}+\|I_{h}u\|_{\mathrm{s}}\leq C_{s}^{1/2}\\|\textbf{A}\\|_{\infty}^{1/2}(\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}+\|(1-\Pi_{1})I_{h}u\|_{1,\mathrm{pw}})$ with $\|(1-\Pi_{1})(I_{h}u-u_{h})\|_{1,\mathrm{pw}}\leq\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}$ in the last inequality. The arguments as in (4.12) prove that $\|(1-\Pi_{1})I_{h}u\|_{1,\mathrm{pw}}\leq(2+C_{\text{Itn}}+C_{\mathrm{PF}})\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}$. This and the arguments in Step 2 for the estimate of $\|I_{h}u-u_{h}\|_{1,\mathrm{pw}}$ show the upper bound in (4.8) for the terms $\|u_{h}\|_{\mathrm{s}}$ and $\|I_{h}u-u_{h}\|_{\mathrm{s}}$. Step $8$ (error estimate for $u-\Pi_{1}u_{h}$). The VEM solution $u_{h}$ is defined by the computed degrees of freedom given in (2.10), but the evaluation of the function itself requires expansive additional calculations. The later are avoided if $u_{h}$ is replaced by the Ritz projection $\Pi_{1}u_{h}$ in the numerical experiments. The triangle inequality leads to $\displaystyle\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}\leq\|u-u_{h}\|_{1,\mathrm{pw}}+\|u_{h}-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}.$ (4.23) A lower bound of the stability term (3.5) and the assumption (A2) imply $\displaystyle\|u_{h}-\Pi_{1}u_{h}\|_{1,P}\leq a_{0}^{-1/2}C_{s}^{1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}.$ (4.24) This shows that the second term in (4.23) is bounded by $\|u_{h}\|_{\mathrm{s}}$. Hence Step 2 and Step 7 prove the estimate for $\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}$. Since $\int_{\partial P}(u_{h}-\Pi_{1}u_{h})\,ds=0$ from the definition of $\Pi^{\nabla}_{1}$ and $\Pi_{1}=\Pi^{\nabla}_{1}$ in $V_{h}$, the combination of Poincaré-Friedrichs inequality for $u_{h}-\Pi_{1}u_{h}$ from Lemma 2.1.a and (4.24) result in $\displaystyle C_{\mathrm{PF}}^{-1}a_{0}^{1/2}C_{s}^{-1/2}\\|u_{h}-\Pi_{1}u_{h}\\|_{L^{2}(P)}\leq h_{P}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}.$ (4.25) The analogous arguments for $\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}$, (4.25), and the estimate for $\|u_{h}\|_{\mathrm{s}}$ prove the bound (4.8) for the term $h_{\text{max}}^{-\sigma}\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}$. This concludes the proof of Theorem 4.3. ∎ ## 5 A posteriori error analysis This section presents the reliability and efficiency of a residual-type a posteriori error estimator. ### 5.1 Residual-based explicit a posteriori error control Recall $u_{h}\in V_{h}$ is the solution to the problem (3.8), and the definition of jump $[\cdot]_{E}$ along an edge $E\in\mathcal{E}$ from Section 2. For any polygonal domain $P\in\mathcal{T}$, set $\displaystyle\eta_{P}^{2}:=h_{P}^{2}\\|f-\gamma\Pi_{1}u_{h}\\|_{L^{2}(P)}^{2}$ \| (volume residual), ---\|--- $\displaystyle\zeta_{P}^{2}:=S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})$ \| (stabilization), $\displaystyle\Lambda_{P}^{2}:=\\|(1-\Pi_{0})(\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h})\\|_{L^{2}(P)}^{2}$ \| (inconsistency), $\displaystyle\Xi_{P}^{2}:=\sum_{E\in\mathcal{E}(P)}\|E\|^{-1}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}^{2}$ \| (nonconformity). These local quantities $\bullet\|_{P}$ form a family ($\bullet\|_{P}:P\in\mathcal{T}$) over the index set $\mathcal{T}$ and their Euclid vector norm $\bullet\|_{\mathcal{T}}$ enters the upper error bound: $\eta_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\eta_{P}^{2})^{1/2}$, $\zeta_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\zeta_{P}^{2})^{1/2}$, $\Lambda_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\Lambda_{P}^{2})^{1/2}$, and $\Xi_{\mathcal{T}}:=(\sum_{P\in\mathcal{T}}\Xi_{P}^{2})^{1/2}$. The following theorem provides an upper bound to the error $u-u_{h}$ in the $H^{1}$ and the $L^{2}$ norm. Recall the elliptic regularity (1.5) with the index $0<\sigma\leq 1$, and recall the assumption $h_{\text{max}}\leq 1$ from Subsection 2.1. ###### Theorem 5.1 (reliability). There exist positive constants $C_{\text{rel}1}$ and $C_{\text{rel}2}$ (both depending on $\rho$) such that $\displaystyle C_{\mathrm{rel}1}^{-2}\|u-u_{h}\|_{1,\mathrm{pw}}^{2}\leq\eta_{\mathcal{T}}^{2}+\zeta_{\mathcal{T}}^{2}+\Lambda_{\mathcal{T}}^{2}+\Xi_{\mathcal{T}}^{2}$ (5.1) and $\displaystyle\\|u-u_{h}\\|_{L^{2}(\Omega)}^{2}\leq C_{\mathrm{rel}2}^{2}\sum_{P\in\mathcal{T}}\Big{(}h_{P}^{2\sigma}(\eta_{P}^{2}+\zeta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2})\Big{)}.$ (5.2) The proof of this theorem in Subsection 5.3 relies on a conforming companion operator elaborated in the next subsection. The upper bound in Theorem 5.1 is efficient in the following local sense, where $\omega_{E}:=\textrm{int}(\cup\mathcal{T}(E))$ denotes the patch of an edge $E$ and consists of the one or the two neighbouring polygons in the set $\mathcal{T}(E):=\\{P^{\prime}\in\mathcal{T}:E\subset\partial P^{\prime}\\}$ that share $E$. Recall $\bm{\sigma}=\textbf{A}\nabla u+\textbf{b}u$ from Subsection 4.2 and the data-oscillation $\mathrm{osc}_{1}(f,P):=\\|h_{P}(1-\Pi_{1})f\\|_{L^{2}(P)}$ from Subsection 2.1. ###### Theorem 5.2 (local efficiency up to oscillation). The quantities $\eta_{P},\zeta_{P},\Lambda_{P},$ and $\Xi_{P}$ from Theorem 5.1 satisfy $\displaystyle\zeta^{2}_{P}$ $\displaystyle\lesssim\|u-u_{h}\|^{2}_{1,P}+\|u-\Pi_{1}u_{h}\|^{2}_{1,P}$ (5.3) $\displaystyle\eta_{P}^{2}$ $\displaystyle\lesssim\\|u-u_{h}\\|^{2}_{1,P}+\|u-\Pi_{1}u_{h}\|^{2}_{1,P}+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(P)}^{2}+\mathrm{osc}_{1}^{2}(f-\gamma u,P),$ (5.4) $\displaystyle\Lambda_{P}^{2}$ $\displaystyle\lesssim\\|u-u_{h}\\|^{2}_{1,P}+\|u-\Pi_{1}u_{h}\|^{2}_{1,P}+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(P)}^{2},$ (5.5) $\displaystyle\Xi_{P}^{2}$ $\displaystyle\lesssim\sum_{E\in\mathcal{E}(P)}\sum_{P^{\prime}\in\omega_{E}}(\\|u-u_{h}\\|^{2}_{1,P^{\prime}}+\|u-\Pi_{1}u_{h}\|^{2}_{1,P^{\prime}}).$ (5.6) The proof of Theorem 5.2 follows in Subsection 5.4. The reliability and efficiency estimates in Theorem 5.1 and 5.2 lead to an equivalence up to the approximation term $\text{apx}:=\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma u,\mathcal{T}).$ Recall the definition of $\|u_{h}\|_{\mathrm{s}}$ from Subsection 4.2. In this paper, the norm $\|\cdot\|_{1,\mathrm{pw}}$ in the nonconforming space $V_{h}$ has been utilised for simplicity and one alternative is the norm $\\|\cdot\\|_{h}$ from Remark 6 induced by $a_{h}$. Then it appears natural to have the total error with the stabilisation term as $\text{total error}:=\|u-u_{h}\|_{1,\mathrm{pw}}+\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+h_{\text{max}}^{-\sigma}\\|u-u_{h}\\|_{L^{2}(\Omega)}+h_{\text{max}}^{-\sigma}\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}+\|u_{h}\|_{\mathrm{s}}.$ The point is that Theorem 4.3 assures that total error $+$ apx converges with the expected optimal convergence rate. ###### Corollary 5.3 (equivalence). The $\mathrm{estimator}:=\eta_{\mathcal{T}}+\zeta_{\mathcal{T}}+\Lambda_{\mathcal{T}}+\Xi_{\mathcal{T}}\approx\mathrm{total\;error}+\mathrm{apx}$. ###### Proof. ∎Theorem 5.2 motivates apx and shows $\mathrm{estimator}\lesssim\\|u-u_{h}\\|_{1,\mathrm{pw}}+\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(\Omega)}+\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})+\|u_{h}\|_{\mathrm{s}}\leq\mathrm{total\;error}+\mathrm{apx}.$ This proves the first inequality $\lesssim$ in the assertion. Theorem 5.1, the estimates in Subsection 5.3.3.1, and the definition of $\|u_{h}\|_{s}$ show $\text{total error}\lesssim\text{estimator}$. The first of the terms in apx is $\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(\Omega)}\leq\\|\bm{\sigma}-\Pi_{0}\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}\leq\\|\bm{\sigma}-\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}+\\|(1-\Pi_{0})\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}.$ The definition of $\bm{\sigma}$ and $\bm{\sigma}_{h}$ plus the triangle and the Cauchy-Schwarz inequality show $\displaystyle\\|\bm{\sigma}-\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}\leq\\|\textbf{A}\\|_{\infty}\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+\\|\textbf{b}\\|_{\infty}\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}\lesssim\\|u-\Pi_{1}u_{h}\\|_{1,\mathrm{pw}}.$ The upper bound is $\lesssim$ estimator as mentioned above. Since the term $\\|(1-\Pi_{0})\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}=\Lambda_{\mathcal{T}}$ is a part of the estimator, $\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(\Omega)}\lesssim\mathrm{estimator}$. The other term in apx is $\displaystyle\mathrm{osc}_{1}(f-\gamma u,\mathcal{T})$ $\displaystyle\leq\mathrm{osc}_{1}(f-\gamma\Pi_{1}u_{h},\mathcal{T})+\\|h_{\mathcal{T}}\gamma(u-\Pi_{1}u_{h})\\|_{L^{2}(\Omega)}$ $\displaystyle\leq\eta_{\mathcal{T}}+\\|\gamma\\|_{\infty}h_{\text{max}}\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}\lesssim\text{estimator}.\qed$ Section 5 establishes the a posteriori error analysis of the nonconforming VEM. Related results are known for the conforming VEM and the nonconforming FEM. ###### Remark 7 (comparison with nonconforming FEM). Theorem 5.1 generalizes a result for the nonconforming FEM in [19, Thm. 3.4] from triangulations into triangles to those in polygons (recall Example 2.2). The only difference is the extra stabilization term that can be dropped in the nonconforming FEM. ###### Remark 8 (comparison with conforming VEM). The volume residual, the inconsistency term, and the stabilization also arise in the a posteriori error estimator for the conforming VEM in [16, Thm. 13]. But it also includes an additional term with normal jumps compared to the estimator (5.1). The extra nonconformity term in this paper is caused by the nonconformity $V_{h}\not\subset V$ in general. ### 5.2 Enrichment and conforming companion operator The link from the nonconforming approximation $u_{h}\in V_{h}$ to a global Sobolev function in $H^{1}_{0}(\Omega)$ can be designed with the help of the underlying refinement $\widehat{{\cal T}}$ of the triangulation $\mathcal{T}$ (from Section 2). The interpolation $I_{\text{CR}}:V+V_{h}\to\textrm{CR}^{1}_{0}(\widehat{{\cal T}})$ in the Crouzeix-Raviart finite element space $\textrm{CR}^{1}_{0}(\widehat{{\cal T}})$ from Subsection 3.4 allows for a right-inverse $J^{\prime}$. A companion operator $J^{\prime}\circ I_{\text{CR}}:V_{h}\to H^{1}_{0}(\Omega)$ acts as displayed $V_{h}$$\text{CR}^{1}_{0}(\widehat{{\cal T}})$$H^{1}_{0}(\Omega)$$J^{\prime}$$I_{\text{CR}}$ Define an enrichment operator $E_{\mathrm{pw}}:\mathcal{P}_{1}(\widehat{{\cal T}})\to S^{1}_{0}(\widehat{{\cal T}})$ by averaging nodal values: For any vertex $z$ in the refined triangulation $\widehat{{\cal T}}$, let $\widehat{{\cal T}}(z)=\\{T\in\widehat{{\cal T}}:z\in T\\}$ denote the set of $\|\widehat{{\cal T}}(z)\|\geq 1$ many triangles that share the vertex $z$, and define $\displaystyle E_{\mathrm{pw}}v_{1}(z)=\frac{1}{\|\widehat{{\cal T}}(z)\|}\sum_{T\in\widehat{{\cal T}}(z)}{v_{1}}\|_{T}(z)$ for an interior vertex $z$ (and zero for a boundary vertex $z$ according to the homogeneous boundary conditions). This defines $E_{\mathrm{pw}}v_{1}$ at any vertex of a triangle $T$ in $\widehat{{\cal T}}$, and linear interpolation then defines $E_{\mathrm{pw}}v_{1}$ in $T\in\widehat{{\cal T}}$, so that $E_{\mathrm{pw}}v_{1}\in S^{1}_{0}(\widehat{{\cal T}})$. Huang et al. [31] design an enrichment operator by an extension of [32] to polygonal domains, while we deduce it from a sub-triangulation. The following lemma provides an approximation property of the operator $E_{\mathrm{pw}}$. ###### Lemma 5.4. There exists a positive constant $C_{En}$ that depends only on the shape regularity of $\widehat{{\cal T}}$ such that any $v_{1}\in\mathcal{P}_{1}(\mathcal{T})$ satisfies $\displaystyle\\|h_{\mathcal{T}}^{-1}(1-E_{\mathrm{pw}})v_{1}\\|_{L^{2}(\Omega)}+\|(1-E_{\mathrm{pw}})v_{1}\|_{1,\mathrm{pw}}\leq C_{\mathrm{En}}\left(\sum_{E\in{\mathcal{E}}}\|E\|^{-1}\\|[v_{1}]_{E}\\|_{L^{2}(E)}^{2}\right)^{1/2}.$ (5.7) ###### Proof. There exists a positive constant $C_{En}$ independent of $h$ and $v_{1}$ [32, p. 2378] such that $\displaystyle\\|h_{\widehat{{\cal T}}}^{-1}(1-E_{\mathrm{pw}})v_{1}\\|_{L^{2}(\Omega)}+\left(\sum_{T\in\widehat{{\cal T}}}\\|\nabla(1-E_{\mathrm{pw}})v_{1}\\|_{L^{2}(T)}^{2}\right)^{1/2}\leq C_{\mathrm{En}}\left(\sum_{E\in\widehat{\mathcal{E}}}\|E\|^{-1}\\|[v_{1}]_{E}\\|_{L^{2}(E)}^{2}\right)^{1/2}.$ Note that any edge $E\in\mathcal{E}$ is unrefined in the sub-triangulation $\widehat{{\cal T}}$. Since $v_{1\|P}\in H^{1}(P)$ is continuous in each polygonal domain $P\in\mathcal{T}$ and $h_{T}\leq h_{P}$ for all $T\in\widehat{{\cal T}}(P)$, the above inequality reduces to (5.7). This concludes the proof. ∎ Recall the $L^{2}$ projection $\Pi_{1}$ onto the piecewise affine functions $\mathcal{P}_{1}(\mathcal{T})$ from Section 2. An enrichment operator $E_{\mathrm{pw}}\circ\Pi_{1}:V_{h}\to H^{1}_{0}(\Omega)$ acts as displayed $V_{h}$$\mathcal{P}_{1}(\mathcal{T})\hookrightarrow\mathcal{P}_{1}(\widehat{{\cal T}})$$H^{1}_{0}(\Omega)$$E_{\mathrm{pw}}$$\Pi_{1}$ ### 5.3 Proof of Theorem 5.1 #### 5.3.1 Reliable $H^{1}$ error control Define $E_{1}u_{h}:=E_{\mathrm{pw}}\Pi_{1}u_{h}\in H^{1}_{0}(\Omega)$ so that $u-E_{1}u_{h}\in H^{1}_{0}(\Omega)$. The inf-sup condition (1.9) leads to some $v\in H^{1}_{0}(\Omega)$ with $\\|v\\|_{1,\Omega}\leq 1$ and $\displaystyle\beta_{0}\\|u-E_{1}u_{h}\\|_{1,\Omega}=B(u-E_{1}u_{h},v)=((f,v)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},v))+B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{1}u_{h},v)$ (5.8) with $B(u,v)=(f,v)$ from (1.8) and the piecewise version $B_{\mathrm{pw}}$ of $B$ in the last step. The definition of $B_{h}$ from Subsection 3.1 and the discrete problem (3.8) with $v_{h}=I_{h}v$ imply $\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h},\Pi_{1}I_{h}v)+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)=B_{h}(u_{h},I_{h}v)=(f,\Pi_{1}I_{h}v)_{L^{2}(\Omega)}.$ (5.9) Abbreviate $w:=v-\Pi_{1}I_{h}v$ and $\bm{\sigma}_{h}:=\textbf{A}\nabla_{\mathrm{pw}}\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$. This and (5.9) simplify $\displaystyle(f,v)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},v)=(f,w)_{L^{2}(\Omega)}-B_{\mathrm{pw}}(\Pi_{1}u_{h},w)+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$ $\displaystyle=(f-\gamma\Pi_{1}u_{h},w)_{L^{2}(\Omega)}-((1-\Pi_{0})\bm{\sigma}_{h},\nabla_{\mathrm{pw}}w)_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$ (5.10) with $\int_{P}\nabla w\,dx=0$ for any $P\in\mathcal{T}$ from (2.17) in the last step. Recall the notation $\eta_{P},\Lambda_{P}$, and $\zeta_{P}$ from Subsection 5.1. The Cauchy-Schwarz inequality and Theorem 2.8.b followed by $\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(\Omega)}\leq\|v\|_{1,\Omega}\leq 1$ in the second step show $\displaystyle(f-\gamma\Pi_{1}u_{h},w)_{L^{2}(P)}$ $\displaystyle\leq\eta_{P}h_{P}^{-1}\\|w\\|_{L^{2}(P)}\leq(1+C_{\mathrm{PF}})\eta_{P},$ (5.11) $\displaystyle((1-\Pi_{0})\bm{\sigma}_{h},\nabla w)_{L^{2}(P)}$ $\displaystyle\leq\Lambda_{P}\|w\|_{1,P}\leq(1+C_{\mathrm{PF}})\Lambda_{P}.$ (5.12) The upper bound $\\|\textbf{A}\\|_{\infty}$ of the coefficient A, (3.5), and the Cauchy-Schwarz inequality for the stabilization term lead to the first inequality in $\displaystyle C_{s}^{-1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}v)$ $\displaystyle\leq\\|\textbf{A}\\|_{\infty}^{1/2}S^{P}((1-\Pi_{1})u_{h},(1-\Pi_{1})u_{h})^{1/2}\|(1-\Pi_{1})I_{h}v\|_{1,P}$ $\displaystyle\leq\\|\textbf{A}\\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}})\zeta_{P}.$ (5.13) The second inequality in (5.13) follows as in (4.3) and with $\\|(1-\Pi_{0})\nabla v\\|_{L^{2}(P)}\leq 1$. Recall the boundedness constant $M_{b}$ of $B_{\mathrm{pw}}$ from Subsection 4.1 and deduce from (5.7) and the definition of $\Xi_{\mathcal{T}}$ from Subsection 5.1 that $\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{1}u_{h},v)\leq M_{b}\|\Pi_{1}u_{h}-E_{1}u_{h}\|_{1,\mathrm{pw}}\leq M_{b}C_{\mathrm{En}}\Xi_{\mathcal{T}}.$ (5.14) The substitution of (5.10)-(5.14) in (5.8) reveals that $\displaystyle\\|u-E_{1}u_{h}\\|_{1,\Omega}$ $\displaystyle\leq C_{7}(\eta_{\mathcal{T}}+\Lambda_{\mathcal{T}}+\zeta_{\mathcal{T}}+\Xi_{\mathcal{T}})$ (5.15) with $\beta_{0}C_{7}=1+C_{\mathrm{PF}}+C_{s}^{1/2}\\|\textbf{A}\\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}})+M_{b}C_{\mathrm{En}}.$ The combination of (4.24), (5.15) and (5.7) leads in the triangle inequality $\displaystyle\|u-u_{h}\|_{1,\mathrm{pw}}\leq\|u-E_{1}u_{h}\|_{1,\Omega}+\|E_{1}u_{h}-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+\|\Pi_{1}u_{h}-u_{h}\|_{1,\mathrm{pw}}$ to (5.1) with $C_{\text{rel}1}/2=C_{7}+C_{\mathrm{En}}+a_{0}^{-1/2}C_{s}^{1/2}$. ∎ #### 5.3.2 Reliable $L^{2}$ error control Recall $I_{\text{CR}}$ from (3.14) and $J^{\prime}$ from the proof of Lemma 3.3, and define $E_{2}u_{h}:=J^{\prime}I_{\text{CR}}u_{h}\in H^{1}_{0}(\Omega)$ from Subsection 5.2. Let $\Psi\in H^{1}_{0}(\Omega)\cap H^{1+\sigma}(\Omega)$ solve the dual problem $B(v,\Psi)=(u-E_{2}u_{h},v)$ for all $v\in V$ and recall (from (1.5)) the regularity estimate $\displaystyle\\|\Psi\\|_{1+\sigma,\Omega}\leq C^{}_{\text{reg}}\\|u-E_{2}u_{h}\\|_{L^{2}(\Omega)}.$ (5.16) The substitution of $v:=u-E_{2}u_{h}\in V$ in the dual problem shows $\displaystyle\\|u-E_{2}u_{h}\\|^{2}_{L^{2}(\Omega)}=B(u-E_{2}u_{h},\Psi).$ The algebra in (5.8)-(5.10) above leads with $v=\Psi$ to the identity $\displaystyle\\|u-E_{2}u_{h}\\|^{2}_{L^{2}(\Omega)}-s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})I_{h}\Psi)=(f-\gamma\Pi_{1}u_{h},\Psi-\Pi_{1}I_{h}\Psi)_{L^{2}(\Omega)}$ $\displaystyle\quad\quad-((1-\Pi_{0})\bm{\sigma}_{h},\nabla_{\mathrm{pw}}(\Psi-\Pi_{1}I_{h}\Psi))_{L^{2}(\Omega)}+B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi).$ (5.17) The definition of $I_{\text{CR}}$ and $J^{\prime}$ proves the first and second equality in $\int_{E}u_{h}\,ds=\int_{E}I_{\text{CR}}u_{h}\,ds=\int_{E}E_{2}u_{h}\,ds\quad\text{for all}\;E\in\mathcal{E}.$ This and an integration by parts imply $\int_{P}\nabla(u_{h}-E_{2}u_{h})\,dx=0$ for all $P\in\mathcal{T}$. Hence Definition 2.2 of Ritz projection $\Pi^{\nabla}_{1}=\Pi_{1}$ in $V_{h}$ shows $\int_{P}\nabla(\Pi_{1}u_{h}-E_{2}u_{h})\,ds=0$ for all $P\in\mathcal{T}$. This $L^{2}$ orthogonality $\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h})\perp\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ and the definition of $B_{\mathrm{pw}}$ in the last term of (5.17) result with elementary algebra in $\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)=((\textbf{A}-\Pi_{0}\textbf{A})\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h}),\nabla\Psi)_{L^{2}(\Omega)}$ $\displaystyle\;+(\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h}),(\Pi_{0}\textbf{A})(1-\Pi_{0})\nabla\Psi)_{L^{2}(\Omega)}+(\Pi_{1}u_{h}-E_{2}u_{h},\textbf{b}\cdot\nabla\Psi+\gamma\Psi)_{L^{2}(\Omega)}.$ (5.18) The triangle inequality and (c’) from the proof of Lemma 3.3 imply the first inequality in $\displaystyle\|\Pi_{1}u_{h}-E_{2}u_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}\|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}\min_{v\in V}\|I_{\text{CR}}u_{h}-v\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}\|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}\|I_{\text{CR}}u_{h}-E_{1}u_{h}\|_{1,\mathrm{pw}}$ $\displaystyle\leq\|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}\|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}(\|I_{\text{CR}}u_{h}-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+\|\Pi_{1}u_{h}-E_{1}u_{h}\|_{1,\mathrm{pw}})$ $\displaystyle\leq(1+C_{\mathrm{J^{\prime}}})\|u_{h}-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}+C_{\mathrm{J^{\prime}}}\|\Pi_{1}u_{h}-E_{1}u_{h}\|_{1,\mathrm{pw}}.$ (5.19) The second estimate in (5.19) follows from $E_{1}u_{h}\in V$, the third is a triangle inequality, and eventually $\|\Pi_{1}u_{h}-I_{\text{CR}}u_{h}\|_{1,\mathrm{pw}}\leq\|u_{h}-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}$ results from the orthogonality $\nabla_{\mathrm{pw}}(u_{h}-I_{\text{CR}})\perp\mathcal{P}_{0}(\widehat{{\cal T}};\mathbb{R}^{2})$ and $\Pi_{1}u_{h}\in\mathcal{P}_{1}(\mathcal{T})$. The Cauchy-Schwarz inequality, the Lipschitz continuity of A, and the approximation estimate $\\|(1-\Pi_{0})\nabla\Psi\\|_{L^{2}(P)}\leq C_{\text{apx}}h_{P}^{\sigma}\|\Psi\|_{1+\sigma,P}$ in (5.18) lead to the first inequality in $\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)$ $\displaystyle\leq\sum_{P\in\mathcal{T}}\Big{(}(h_{P}\|\textbf{A}\|_{1,\infty}+\\|\textbf{A}\\|_{\infty}C_{\text{apx}}h_{P}^{\sigma})\|\Pi_{1}u_{h}-E_{2}u_{h}\|_{1,P}$ $\displaystyle\qquad+\\|\Pi_{1}u_{h}-E_{2}u_{h}\\|_{L^{2}(P)}(\\|\textbf{b}\\|_{\infty}+\\|\gamma\\|_{\infty})\Big{)}\\|\Psi\\|_{1+\sigma,P}$ $\displaystyle\leq\sum_{P\in\mathcal{T}}\Big{(}h_{P}\|\textbf{A}\|_{1,\infty}+\\|\textbf{A}\\|_{\infty}C_{\text{apx}}h_{P}^{\sigma}+C_{\mathrm{PF}}(\\|\textbf{b}\\|_{\infty}+\\|\gamma\\|_{\infty})h_{P}\Big{)}\|\Pi_{1}u_{h}-E_{2}u_{h}\|_{1,P}\\|\Psi\\|_{1+\sigma,P}$ $\displaystyle\leq C_{8}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}((1+C_{\mathrm{J^{\prime}}})\|u_{h}-\Pi_{1}u_{h}\|_{1,P}+C_{\mathrm{J^{\prime}}}\|\Pi_{1}u_{h}-E_{1}u_{h}\|_{1,P})\\|\Psi\\|_{1+\sigma,P}.$ (5.20) The second inequality in (5.20) follows from the Poincaré-Friedrichs inequality in Lemma 2.1.a for $\Pi_{1}u_{h}-E_{2}u_{h}$ with $\int_{\partial P}(\Pi_{1}u_{h}-E_{2}u_{h})\,ds=0$ (from above); the constant $C_{8}:=\|\textbf{A}\|_{1,\infty}+C_{\text{apx}}\\|\textbf{A}\\|_{\infty}+C_{\mathrm{PF}}(\\|\textbf{b}\\|_{\infty}+\\|\gamma\\|_{\infty})$ results from (5.19) and $h_{P}\leq h_{P}^{\sigma}$ (recall $h_{\text{max}}\leq 1$). Lemma 5.4 with $v_{1}=\Pi_{1}u_{h}$ and (4.24) in (5.20) show $\displaystyle B_{\mathrm{pw}}(\Pi_{1}u_{h}-E_{2}u_{h},\Psi)$ $\displaystyle\leq C_{8}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}\zeta_{P}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}\Xi_{P})\\|\Psi\\|_{1+\sigma,P}.$ (5.21) Rewrite (5.11)-(5.13) with $w=\Psi-\Pi_{1}I_{h}\Psi$ and $h_{P}^{-1}\\|w\\|_{L^{2}(P)}+\|w\|_{1,P}\leq(1+C_{\mathrm{PF}})\\|(1-\Pi_{0})\nabla\Psi\\|_{L^{2}(P)}\leq C_{\text{apx}}(1+C_{\mathrm{PF}})h_{P}^{\sigma}\|\Psi\|_{1+\sigma,P}$ from (2.12). This and (5.21) lead in (5.17) to $\displaystyle\\|u-E_{2}u_{h}\\|_{L^{2}(\Omega)}^{2}\leq C_{9}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}(\eta_{P}+\zeta_{P}+\Lambda_{P}+\Xi_{P})\\|\Psi\\|_{1+\sigma,P}$ for $C_{9}:=C_{\text{apx}}(1+C_{\mathrm{PF}}+C_{s}^{1/2}\\|\textbf{A}\\|_{\infty}^{1/2}(2+C_{\mathrm{PF}}+C_{\text{Itn}}))+C_{8}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}).$ This and the regularity (5.16) result in $\displaystyle\\|u-E_{2}u_{h}\\|_{L^{2}(\Omega)}\leq C_{9}C^{}_{\text{reg}}\sum_{P\in\mathcal{T}}h_{P}^{\sigma}(\eta_{P}+\zeta_{P}+\Lambda_{P}+\Xi_{P}).$ (5.22) The arguments in the proof of (5.20)-(5.21) also lead to $\displaystyle\\|E_{2}u_{h}-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}\leq C_{\mathrm{PF}}((1+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}})\sum_{P\in\mathcal{T}}h_{P}(\zeta_{P}+\Xi_{P}).$ (5.23) The combination of (4.25), (5.22)-(5.23) and the triangle inequality $\\|u-u_{h}\\|_{L^{2}(\Omega)}\leq\\|u-E_{2}u_{h}\\|_{L^{2}(\Omega)}+\\|E_{2}u_{h}-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}+\\|\Pi_{1}u_{h}-u_{h}\\|_{L^{2}(\Omega)}$ lead to (5.2) with $C_{rel2}/2=C_{9}C^{}_{\text{reg}}+C_{\mathrm{PF}}\big{(}(2+C_{\mathrm{J^{\prime}}})a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{J^{\prime}}}C_{\mathrm{En}}\big{)}.$ This concludes the proof of the $L^{2}$ error estimate in Theorem 5.1. ∎ #### 5.3.3 Comments ##### 5.3.3.1 Estimator for $u-\Pi_{1}u_{h}$ The triangle inequality with (5.1) and (4.24) provide an upper bound for $H^{1}$ error $\displaystyle\frac{1}{2}\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}^{2}\leq\|u-u_{h}\|_{1,\mathrm{pw}}^{2}+\|(1-\Pi_{1})u_{h}\|_{1,\mathrm{pw}}^{2}\leq 2C_{\text{rel}1}^{2}(\eta_{\mathcal{T}}^{2}+\zeta_{\mathcal{T}}^{2}+\Lambda_{\mathcal{T}}^{2}+\Xi_{\mathcal{T}}^{2}).$ The same arguments for an upper bound of the $L^{2}$ error in Theorem 5.1 show that $\displaystyle\frac{1}{2}\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq\\|u-u_{h}\\|_{L^{2}(\Omega)}^{2}+\\|(1-\Pi_{1})u_{h}\\|_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq C_{\text{rel}2}^{2}\sum_{P\in\mathcal{T}}h_{P}^{2\sigma}(\eta_{P}^{2}+2\zeta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2}).$ The numerical experiments do not display $C_{\text{rel}1}$ and $C_{\text{rel}2}$, and directly compare the error $H1e:=\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}$ in the piecewise $H^{1}$ norm and the error $L2e:=\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}$ in the $L^{2}$ norm with the upper bound $H1\mu$ and $L2\mu$ (see, e.g., Figure 6.3). ##### 5.3.3.2 Motivation and discussion of apx We first argue that those extra terms have to be expected and utilize the abbreviations $\bm{\sigma}:=\textbf{A}\nabla u+\textbf{b}u$ and $g:=f-\gamma u$ for the exact solution $u\in H^{1}_{0}(\Omega)$ to (1.8), which reads $\displaystyle(\bm{\sigma},\nabla v)_{L^{2}(\Omega)}=(g,v)_{L^{2}(\Omega)}\quad\text{for all}\;v\in H^{1}_{0}(\Omega).$ (5.24) Recall the definition of $s_{h}(\cdot,\cdot)$ from Subsection 3.1. The discrete problem (3.8) with the discrete solution $u_{h}\in V_{h}$ assumes the form $\displaystyle(\bm{\sigma}_{h},\nabla\Pi_{1}v_{h})_{L^{2}(\Omega)}+s_{h}((1-\Pi_{1})u_{h},(1-\Pi_{1})v_{h})=(g_{h},\Pi_{1}v_{h})_{L^{2}(\Omega)}\quad\text{for all}\;v_{h}\in V_{h}$ (5.25) for $\bm{\sigma}_{h}:=\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$, and $g_{h}:=f-\gamma\Pi_{1}u_{h}$. Notice that $\bm{\sigma}_{h}$ and $g_{h}$ may be replaced in (5.25) by $\Pi_{0}\bm{\sigma}_{h}$ and $\Pi_{1}g_{h}$ because the test functions $\nabla\Pi_{1}v_{h}$ and $\Pi_{1}v_{h}$ belong to $\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2})$ and $\mathcal{P}_{1}(\mathcal{T})$ respectively. In other words, the discrete problems (3.8) and (5.25) do not see a difference of $\bm{\sigma}_{h}$ and $g_{h}$ compared to $\Pi_{0}\bm{\sigma}_{h}$ and $\Pi_{1}g_{h}$ and so the errors $\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}$ and $g_{h}-\Pi_{1}g_{h}$ may arise in a posteriori error control. This motivates the a posteriori error term $\\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}\\|_{L^{2}(\Omega)}=\Lambda_{\mathcal{T}}$ as well as the approximation terms $\bm{\sigma}-\Pi_{0}\bm{\sigma}$ and $g-\Pi_{1}g$ on the continuous level. The natural norm for the dual variable $\bm{\sigma}$ is $L^{2}$ and that of $g$ is $H^{-1}$ and hence their norms form the approximation term apx as defined in Subsection 5.1. ###### Example 5.1 ($\textbf{b}=0$). The term $(1-\Pi_{0})\bm{\sigma}$ may not be visible in case of no advection $\textbf{b}=0$ at least if A is piecewise constant. Suppose $\textbf{A}\in\mathcal{P}_{0}(\mathcal{T};\mathbb{R}^{2\times 2})$ and estimate $\\|(1-\Pi_{0})(\textbf{A}\nabla u)\\|_{L^{2}(\Omega)}\leq\\|\textbf{A}\\|_{\infty}\\|(1-\Pi_{0})\nabla u\\|_{L^{2}(\Omega)}\lesssim\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}.$ If A is not constant, there are oscillation terms that can be treated properly in adaptive mesh-refining algorithms, e.g., in [27]. ###### Example 5.2 ($\gamma$ piecewise constant). While the data approximation term $\mathrm{osc}_{1}(f,\mathcal{T})$ [10] is widely accepted as a part of the total error in the approximation of nonlinear problems, the term $\mathrm{osc}_{1}(\gamma u,\mathcal{T})=\\|\gamma h_{\mathcal{T}}(u-\Pi_{1}u)\\|_{L^{2}(\Omega)}\lesssim h_{\text{max}}^{1+\sigma}\\|f\\|_{L^{2}(\Omega)}$ is of higher order and may even be absorbed in the overall error analysis for a piecewise constant coefficient $\gamma\in\mathcal{P}_{0}(\mathcal{T})$. In the general case $\gamma\in L^{\infty}(\Omega)\backslash\mathcal{P}_{0}(\mathcal{T})$, however, $\mathrm{osc}_{1}(u,\mathcal{T})$ leads in particular to terms with $\\|\gamma-\Pi_{0}\gamma\\|_{L^{\infty}(\Omega)}$. ##### 5.3.3.3 Higher-order nonconforming VEM The analysis applied in Theorem 5.1 can be extended to the nonconforming VEM space of higher order $k\in\mathbb{N}$ (see [17, Sec. 4] for the definition of discrete space). Since the projection operators $\nabla\Pi_{k}^{\nabla}$ and $\Pi_{k-1}\nabla$ are not the same for general $k$, and the first operator does not lead to optimal order of convergence for $k\geq 3$, the discrete formulation uses $\Pi_{k-1}\nabla$ (cf. [6, Rem. 4.3] for more details). The definition and approximation properties of the averaging operator $E_{\mathrm{pw}}$ extend to the operator $E^{k}:\mathcal{P}_{k}(\widehat{{\cal T}})\to H^{1}_{0}(\Omega)$ (see [32, p. 2378] for a proof). The identity (5.9) does not hold in general, but algebraic calculations lead to $\displaystyle\eta_{P}^{2}$ $\displaystyle:=h_{P}^{2}\\|f-\gamma\Pi_{k}u_{h}\\|_{L^{2}(P)}^{2},\qquad\Lambda_{P}^{2}:=\\|(1-\Pi_{k-1})(\textbf{A}\Pi_{k-1}\nabla u_{h}+\textbf{b}\Pi_{k}u_{h})\\|_{L^{2}(P)}^{2}$ $\displaystyle\zeta_{P}^{2}$ $\displaystyle:=S^{P}((1-\Pi_{k})u_{h},(1-\Pi_{k})u_{h}),\qquad\Xi_{P}^{2}:=\sum_{E\in\mathcal{E}(P)}\|E\|^{-1}\\|[\Pi_{k}u_{h}]_{E}\\|^{2}_{L^{2}(E)}.$ The analysis developed for the upper bound of $L^{2}$ norm also extends to the general case. The model problem is chosen in 2D for the simplicity of the presentation. The results of this work can be extended to the three- dimensional case with appropriate modifications. The present analysis holds for any higher regularity index $\sigma>0$ and avoids any trace inequality for higher derivatives. This is possible by a medius analysis in the form of companion operators [26]. ##### 5.3.3.4 Inhomogeneous boundary data The error estimator for general Dirichlet condition $u\|_{\partial\Omega}=g\in H^{1/2}(\partial\Omega)$ can be obtained with some modifications of [33] in Theorem 5.1. The only difference is in the modified jump contributions of the boundary edges in the nonconformity term $\displaystyle\Xi_{\mathcal{T}}^{2}=\sum_{E\in\mathcal{E}(\Omega)}\|E\|^{-1}\\|[\Pi_{1}u_{h}]\\|_{L^{2}(E)}^{2}+\sum_{E\in\mathcal{E}(\partial\Omega)}\|E\|^{-1}\\|g-\Pi_{1}u_{h}\\|_{L^{2}(E)}^{2}.$ ### 5.4 Proof of Theorem 5.2 Recall the notation $\bm{\sigma}=\textbf{A}\nabla u+\textbf{b}u$ and $\bm{\sigma}_{h}=\textbf{A}\nabla\Pi_{1}u_{h}+\textbf{b}\Pi_{1}u_{h}$ from Subsection 5.3. ###### Proof of (5.3). The upper bound (3.5) for the stabilisation term and the triangle inequality show $\displaystyle\zeta_{P}^{2}\leq C_{s}\|(1-\Pi_{1})u_{h}\|^{2}_{1,P}\leq 2C_{s}(\|u-u_{h}\|^{2}_{1,P}+\|u-\Pi_{1}u_{h}\|^{2}_{1,P}).$ This concludes the proof of (5.3). ∎ ###### Proof of (5.5). The definition of $\Lambda_{P},\Pi_{0}$, and the triangle inequality lead to $\displaystyle\Lambda_{P}=\\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}_{h}\\|_{L^{2}(P)}$ $\displaystyle\leq\\|\bm{\sigma}_{h}-\Pi_{0}\bm{\sigma}\\|_{L^{2}(P)}$ $\displaystyle\leq\\|\textbf{A}\nabla(\Pi_{1}u_{h}-u)+\textbf{b}(\Pi_{1}u_{h}-u)\\|_{L^{2}(P)}+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(P)}.$ (5.26) The upper bound $\\|\textbf{A}\\|_{\infty}$ and $\\|\textbf{b}\\|_{\infty}$ for the coefficients and the triangle inequality lead to $\displaystyle\Lambda_{P}-\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(P)}\leq(\\|\textbf{A}\\|_{\infty}+\\|\textbf{b}\\|_{\infty})\\|\Pi_{1}u_{h}-u\\|_{1,P}$ $\displaystyle\qquad\leq(\\|\textbf{A}\\|_{\infty}+\\|\textbf{b}\\|_{\infty})(\\|u_{h}-\Pi_{1}u_{h}\\|_{1,P}+\\|u-u_{h}\\|_{1,P})\leq C_{10}(\zeta_{P}+\\|u-u_{h}\\|_{1,P})$ (5.27) with $\\|u_{h}-\Pi_{1}u_{h}\\|_{1,P}\leq(1+h_{P}C_{\mathrm{PF}})a_{0}^{-1/2}C_{s}^{1/2}\zeta_{P}$ from (4.24)-(4.25) and with $C_{10}:=(\\|\textbf{A}\\|_{\infty}+\\|\textbf{b}\\|_{\infty})((1+h_{P}C_{\mathrm{PF}})a_{0}^{-1/2}C_{s}^{1/2}+1)$. This followed by (5.3) concludes the proof of (5.5). ∎ Recall the bubble-function $b_{\mathcal{T}}\|_{P}=b_{P}$ supported on a polygonal domain $P\in\mathcal{T}$ from (3.17) as the sum of interior bubble- functions supported on each triangle $T\in\widehat{{\cal T}}(P)$. ###### Proof of (5.4). Rewrite the term $\displaystyle f-\gamma\Pi_{1}u_{h}=\Pi_{1}(f-\gamma\Pi_{1}u_{h})+(1-\Pi_{1})(f-\gamma\Pi_{1}u_{h})=:R+\theta,$ (5.28) and denote $R_{P}:=R\|_{P}$ and $\theta_{P}:=\theta\|_{P}$. The definition of $B_{\mathrm{pw}}(u-\Pi_{1}u_{h},v)$ and the weak formulation $B(u,v)=(f,v)$ from (1.8) for any $v\in V$ imply $\displaystyle B_{\mathrm{pw}}(u-\Pi_{1}u_{h},v)+(\bm{\sigma}_{h},\nabla v)_{L^{2}(\Omega)}$ $\displaystyle=(f-\gamma\Pi_{1}u_{h},v)_{L^{2}(\Omega)}=(R+\theta,v)_{L^{2}(\Omega)}.$ (5.29) Since $b_{P}R_{P}$ belongs to $H^{1}_{0}(\Omega)$ (extended by zero outside $P$), $v:=b_{P}R_{P}\in V$ is admissible in (5.29). An integration by parts proves that $(\Pi_{0}\bm{\sigma}_{h},\nabla(b_{P}R_{P}))_{L^{2}(P)}=0$. Therefore, (5.29) shows $\displaystyle(R_{P},b_{P}R_{P})_{L^{2}(P)}=B^{P}(u-\Pi_{1}u_{h},b_{P}R_{P})-(\theta_{P},b_{P}R_{P})_{L^{2}(P)}+((1-\Pi_{0})\bm{\sigma}_{h},\nabla(b_{P}R_{P}))_{L^{2}(P)}.$ The substitution of $\chi=R_{P}=\Pi_{1}(f-\gamma\Pi_{1}u_{h})\|_{P}\in\mathcal{P}_{1}(P)$ in (3.20) and the previous identity with the boundedness of $B$ and the Cauchy-Schwarz inequality lead to the first two estimates in $\displaystyle C_{b}^{-1}\\|R_{P}\\|_{L^{2}(P)}^{2}\leq(R_{P},b_{P}R_{P})_{L^{2}(P)}$ $\displaystyle\quad\leq\Big{(}M_{b}\|u-\Pi_{1}u_{h}\|_{1,P}+\\|(1-\Pi_{0})\bm{\sigma}_{h}\\|_{L^{2}(P)}\Big{)}\|b_{P}R_{P}\|_{1,P}+\\|\theta_{P}\\|_{L^{2}(P)}\\|b_{P}R_{P}\\|_{L^{2}(P)}$ $\displaystyle\quad\leq C_{b}\Big{(}M_{b}\|u-\Pi_{1}u_{h}\|_{1,P}+\Lambda_{P}+h_{P}\\|\theta_{P}\\|_{L^{2}(P)}\Big{)}h_{P}^{-1}\\|R_{P}\\|_{L^{2}(P)}.$ The last inequality follows from the definition of $\Lambda_{P}$, and (3.21) with $\chi=R_{P}$. This proves that $C_{b}^{-2}h_{P}\\|R_{P}\\|_{L^{2}(P)}\leq M_{b}\|u-\Pi_{1}u_{h}\|_{1,P}+\Lambda_{P}+h_{P}\\|\theta_{P}\\|_{L^{2}(P)}.$ Recall $\eta_{P}$ from Subsection 5.1 and $\eta_{P}=h_{P}\\|f-\gamma\Pi_{1}u_{h}\\|_{L^{2}(P)}\leq h_{P}\\|R_{P}\\|_{L^{2}(P)}+h_{P}\\|\theta_{P}\\|_{L^{2}(P)}$ from the split in (5.28) and the triangle inequality. This and the previous estimate of $h_{P}\\|R_{P}\\|_{L^{2}(P)}$ show the first estimate in $\displaystyle\eta_{P}$ $\displaystyle\leq C_{b}^{2}(M_{b}\|u-\Pi_{1}u_{h}\|_{1,P}+\Lambda_{P})+(C_{b}^{2}+1)h_{P}\\|\theta_{P}\\|_{L^{2}(P)}$ $\displaystyle\leq(C_{b}^{2}+1)\Big{(}M_{b}\|u-\Pi_{1}u_{h}\|_{1,P}+\Lambda_{P}+h_{P}\\|(f-\gamma\Pi_{1}u_{h})-\Pi_{1}(f-\gamma u)\\|_{L^{2}(P)}\Big{)}$ $\displaystyle\leq(C_{b}^{2}+1)\Big{(}(M_{b}+h_{P}\\|\gamma\\|_{\infty})\\|u-\Pi_{1}u_{h}\\|_{1,P}+\Lambda_{P}+\mathrm{osc}_{1}(f-\gamma u,P)\Big{)}.$ The second step results from the definition of $\theta_{P}=(1-\Pi_{1})(f-\gamma\Pi_{1}u_{h})\|_{P}$ in (5.28) followed by the $L^{2}$ orthogonality of $\Pi_{1}$, and the last step results from an elementary algebra with the triangle inequality and $\mathrm{osc}_{1}(f-\gamma u,P)=h_{P}\\|(1-\Pi_{1})(f-\gamma u)\\|_{L^{2}(P)}$ from Subsection 5.1. The triangle inequality for the term $u-\Pi_{1}u_{h}$ and the estimate of $\\|u_{h}-\Pi_{1}u_{h}\\|_{1,P}$ as in (5.27) lead to $\displaystyle C_{11}^{-1}\eta_{P}\leq\\|u-u_{h}\\|_{1,P}+\zeta_{P}+\Lambda_{P}+\mathrm{osc}_{1}(f-\gamma u,P)$ with $C_{11}:=(C_{b}^{2}+1)(M_{b}+h_{P}\\|\gamma\\|_{\infty})((1+h_{P}C_{\mathrm{PF}})a_{0}^{-1/2}C_{s}^{1/2})+1)$. The combination of (5.3) and (5.5) in the last displayed estimate concludes the proof of (5.4). ∎ ###### Proof of (5.6). Recall for $u\in H^{1}_{0}(\Omega)$ and $u_{h}\in V_{h}$ that $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}u\,ds$ and $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}u_{h}\,ds$ are well defined for all edges $E\in\mathcal{E}$, and so the constant $\alpha_{E}:=\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\\!\int_{E}(u-u_{h})\,ds$ is uniquely defined as well. Since the jump of $u-\alpha_{E}$ across any edge $E\in\mathcal{E}$ vanishes, $[\Pi_{1}u_{h}]_{E}=[\Pi_{1}u_{h}-u+\alpha_{E}]_{E}$. Recall $\omega_{E}=\text{int}(P^{+}\cup P^{-})$ for $E\in\mathcal{E}(\Omega)$ and $\omega_{E}=\text{int}(P)$ for $E\in\mathcal{E}(\partial\Omega)$ from Subsection 5.1. The trace inequality $\\|v\\|^{2}_{L^{2}(E)}\leq C_{T}(\|E\|^{-1}\\|v\\|^{2}_{L^{2}(\omega_{E})}+\|E\|\;\\|\nabla v\\|^{2}_{L^{2}(\omega_{E})})$ (cf. [13, p. 554]) leads to $\displaystyle\|E\|^{-1/2}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}\leq C_{T}\left(\|E\|^{-1}\\|\Pi_{1}u_{h}-u+\alpha_{E}\\|_{L^{2}(\omega_{E})}+\\|\nabla_{\mathrm{pw}}(\Pi_{1}u_{h}-u)\\|_{L^{2}(\omega_{E})}\right).$ This and the triangle inequality show the first estimate in $\displaystyle\|E\|^{-1/2}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}$ $\displaystyle\leq C_{T}\Big{(}\|E\|^{-1}(\\|u_{h}-\Pi_{1}u_{h}\\|_{L^{2}(\omega_{E})}+\\|u_{h}-u+\alpha_{E}\\|_{L^{2}(\omega_{E})})$ $\displaystyle\qquad+\\|\nabla_{\mathrm{pw}}(u_{h}-\Pi_{1}u_{h})\\|_{L^{2}(\omega_{E})}+\\|\nabla_{\mathrm{pw}}(u-u_{h})\\|_{L^{2}(\omega_{E})}\Big{)}.$ (5.30) The estimates (4.24)-(4.25) control the term $\\|u_{h}-\Pi_{1}u_{h}\\|_{1,P}$ as in (5.27), and the Poincaré-Friedrichs inequality from Lemma 2.1.b for $u_{h}-u+\alpha_{E}$ with $\int_{E}(u_{h}-u+\alpha_{E})\,ds=0$ (by the definition of $\alpha_{E}$) implies that $\\|u_{h}-u+\alpha_{E}\\|_{L^{2}(P)}\leq C_{\mathrm{PF}}h_{P}\|u_{h}-u\|_{1,P}$. This with the mesh assumption $h_{P}\leq\rho^{-1}\|E\|$ and (5.30) result in $\displaystyle\|E\|^{-1/2}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}\leq C_{T}((C_{\mathrm{PF}}\rho^{-1}+1)a_{0}^{-1/2}C_{s}^{1/2}+C_{\mathrm{PF}}+1)\sum_{P^{\prime}\in\omega_{E}}(\Lambda_{P^{\prime}}+\|u-u_{h}\|_{1,P^{\prime}}).$ Since this holds for any edge $E\in\mathcal{E}(P)$, the sum over all these edges and the bound (5.3) in the above estimate conclude the proof of (5.6). ∎ ###### Remark 9 (convergence rates of $L^{2}$ error control for $0<\sigma\leq 1$). The efficiency estimates (5.4)-(5.6) with a multiplication of $h_{P}^{2\sigma}$ show that the local quantity $h_{P}^{2\sigma}(\eta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2})$ converges to zero with the expected convergence rate. ###### Remark 10 (efficiency up to stabilisation and oscillation for $L^{2}$ error control when $\sigma=1$). For convex domains and $\sigma=1$, there is even a local efficiency result that is briefly described in the sequel: The arguments in the above proof of (5.4)-(5.5) lead to $\displaystyle h_{P}^{2}\eta_{P}^{2}$ $\displaystyle\lesssim\\|u-u_{h}\\|^{2}_{L^{2}(P)}+h_{P}^{2}(\zeta_{P}^{2}+\mathrm{osc}_{1}^{2}(f-\gamma u,P)+\\|(1-\Pi_{0})\bm{\sigma}\\|_{L^{2}(P)}^{2}),$ $\displaystyle h_{P}^{2}\Lambda_{P}^{2}$ $\displaystyle\lesssim\\|u-u_{h}\\|^{2}_{L^{2}(P)}+h_{P}^{2}(\zeta_{P}^{2}+\\|\textbf{A}-\Pi_{0}\textbf{A}\\|_{L^{\infty}(P)}^{2}\\|f\\|^{2}_{L^{2}(\Omega)}+\\|(1-\Pi_{0})\textbf{b}u\\|_{L^{2}(P)}^{2}).$ The observation $[\Pi_{1}u_{h}]_{E}=[\Pi_{1}u_{h}-u]_{E}$ for the term $\Xi_{P}$, the trace inequality, and the triangle inequality show, for any $E\in\mathcal{E}$, that $\displaystyle\|E\|^{1/2}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}$ $\displaystyle\leq C_{T}\left(\\|u_{h}-\Pi_{1}u_{h}\\|_{L^{2}(\omega_{E})}+\\|u-u_{h}\\|_{L^{2}(\omega_{E})}\right.$ $\displaystyle\quad\left.+\|E\|(\\|\nabla\Pi_{1}(u-u_{h})\\|_{L^{2}(\omega_{E})}+\\|\nabla(u-\Pi_{1}u)\\|_{L^{2}(\omega_{E})})\right).$ The bound (4.25) for the first term and the inverse estimate $\\|\nabla\chi\\|_{L^{2}(P)}\leq C_{\text{inv}}h_{P}^{-1}\\|\chi\\|_{L^{2}(P)}$ for $\chi\in\mathcal{P}_{k}(P)$ for the third term result in $\displaystyle\|E\|^{1/2}\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}\lesssim\\|u-u_{h}\\|_{L^{2}(\omega_{E})}+\|E\|\sum_{P^{\prime}\in\omega_{E}}\Big{(}\\|\nabla(1-\Pi_{1})u\\|_{L^{2}(P^{\prime})}+\Lambda_{P^{\prime}}\Big{)}.$ The mesh assumption (M2) implies that $h_{P}^{2}\Xi_{P}^{2}\leq\rho^{-1}\sum_{E\in\mathcal{E}(P)}\|E\|\;\\|[\Pi_{1}u_{h}]_{E}\\|_{L^{2}(E)}^{2}$. This and the above displayed inequality prove the efficiency estimate for $h_{P}^{2}\Xi_{P}^{2}$. ## 6 Numerical experiments This section manifests the performance of the a posteriori error estimator and an associated adaptive mesh-refining algorithm with D$\ddot{o}$rfler marking [37]. The numerical results investigate three computational benchmarks for the indefinite problem (1.1). ### 6.1 Adaptive algorithm Input: initial partition ${\cal T}_{0}$ of $\Omega$. For $\ell=0,1,2,\dots$ do 1. 1. SOLVE. Compute the discrete solution $u_{h}$ to (3.8) with respect to $\mathcal{T}_{\ell}$ for $\ell=0,1,2\dots$ (cf. [5] for more details on the implementation). 2. 2. ESTIMATE. Compute all the four terms $\eta_{\ell}:=\eta_{\mathcal{T}_{\ell}},\zeta_{\ell}:=\zeta_{\mathcal{T}_{\ell}},\Lambda_{\ell}:=\Lambda_{\mathcal{T}_{\ell}}$ and $\Xi_{\ell}:=\Xi_{\mathcal{T}_{\ell}}$, which add up to the upper bound (5.1). 3. 3. MARK. Mark the polygons $P$ in a subset ${\cal M}_{\ell}\subset$ ${\cal T}_{\ell}$ with minimal cardinality and $\displaystyle{H1\mu}_{\ell}^{2}:=H1\mu^{2}({\cal T}_{\ell}):=\eta_{\ell}^{2}+\zeta_{\ell}^{2}+\Lambda_{\ell}^{2}+\Xi_{\ell}^{2}\leq 0.5\sum_{P\in{\cal M}_{\ell}}(\eta_{P}^{2}+\zeta_{P}^{2}+\Lambda_{P}^{2}+\Xi_{P}^{2}).$ 4. 4. REFINE \- Refine the marked polygon domains by connecting the mid-point of the edges to the centroid of respective polygon domains and update ${\cal T}_{\ell}$. (cf. Figure 6.1 for an illustration of the refinement strategy.) Figure 6.1: Refinement of a polygon into quadrilaterals end do Output: The sequences $\mathcal{T}_{\ell}$, and the bounds $\eta_{\ell},\zeta_{\ell},\Lambda_{\ell},\Xi_{\ell}$, and $H1\mu_{\ell}$ for $\ell=0,1,2,\dots$. The adaptive algorithm is displayed for mesh adaption in the energy error $H^{1}$. Replace estimator $H1\mu_{\ell}$ in the algorithm by $L2\mu_{\ell}$ (the upper bound in (5.2)) for local mesh-refinement in the $L^{2}$ error. Both uniform and adaptive mesh-refinement run to compare the empirical convergence rates and provide numerical evidence for the superiority of adaptive mesh-refinement. Note that uniform refinement means all the polygonal domains are refined. In all examples below, $\overline{\textbf{A}}_{P}=1$ in (3.6). The numerical realizations are based on a MATLAB implementation explained in [35] with a Gauss-like cubature formula over polygons. The cubature formula is exact for all bivariate polynomials of degree at most $2n-1$, so the choice $n\geq(k+1)/2$ leads to integrate a polynomial of degree $k$ exactly. The quadrature errors in the computation of examples presented below appear negligible for the input parameter $n=5$. ### 6.2 Square domain (smooth solution) This subsection discusses the problem (1.1) with the coefficients $\textbf{A}=I,\textbf{b}=(x,y)$ and $\gamma=x^{2}+y^{3}$ on a square domain $\Omega=(0,1)^{2}$, and the exact solution $\displaystyle u=16x(1-x)y(1-y)\arctan(25x-100y+50)$ with $f={\cal L}u$. Since $\gamma-\frac{1}{2}\text{div}(\textbf{b})=x^{2}+y^{3}-1$ is not always positive on $\Omega$, this is an indefinite problem. Initially, the error and the estimators are large because of an internal layer around the line $25x-100y+50=0$ with large first derivative of $u$ resolved after few refinements as displayed in Figure 6.2. Figure 6.2: Output $\mathcal{T}_{1},\mathcal{T}_{8},\mathcal{T}_{15}$ of the adaptive algorithm (a) (b) Figure 6.3: Convergence history plot of estimator $\mu$ and error $e:=u-\Pi_{1}u_{h}$ in the (a) piecewise $H^{1}$ norm (b) $L^{2}$ norm vs number ndof of degrees of freedom for both uniform and adaptive refinement ### 6.3 L-shaped domain (non-smooth solution) This subsection shows an advantage of using adaptive mesh-refinement over uniform meshing for the problem (1.1) with the coefficients as $\textbf{A}=I,\textbf{b}=(x,y)\hskip 2.84526pt\text{and}\hskip 2.84526pt\gamma=-4$ on a L-shaped domain $\Omega=(-1,1)^{2}\backslash[0,1)\times(-1,0]$ and the exact solution $\displaystyle u=r^{2/3}\sin\left(\frac{2\theta}{3}\right)$ with $f:={\cal L}u$. Since the exact solution is not zero along the boundary $\partial\Omega$, the error estimators are modified according to Subsection 5.3.3.4. Since $\gamma-\frac{1}{2}\text{div}(\textbf{b})=-5<0$, the problem is non-coercive. Observe that with increase in number of iterations, refinement is more at the singularity as highlighted in Figure 6.4. Since the exact solution $u$ is in $H^{(5/3)-\epsilon}(\Omega)$ for all $\epsilon>0$, from a priori error estimates the expected order of convergence in $H^{1}$ norm is $1/3$ and in $L^{2}$ norm is at least $2/3$ with respect to number of degrees of freedom for uniform refinement. Figure 6.5 shows that uniform refinement gives the sub-optimal convergence rate, whereas adaptive refinement lead to optimal convergence rates ($1/2$ for $H^{1}$ norm and $5/6$ in $L^{2}$ norm). Figure 6.4: Output $\mathcal{T}_{1},\mathcal{T}_{10},\mathcal{T}_{15}$ of the adaptive refinement (a) (b) Figure 6.5: Convergence history plot of estimator $\mu$ and error $e:=u-\Pi_{1}u_{h}$ in the (a) piecewise $H^{1}$ norm (b) $L^{2}$ norm vs number ndof of degrees of freedom for both uniform and adaptive refinement ### 6.4 Helmholtz equation This subsection considers the exact solution $u=1+\tanh(-9(x^{2}+y^{2}-0.25))$ to the problem $\displaystyle-\Delta u-9u=f\quad\quad\text{in}\quad\Omega=(-1,1)^{2}.$ There is an internal layer around the circle centered at $(0,0)$ and of radius $0.25$ where the second derivatives of $u$ are large because of steep increase in the solution resulting in the large error at the beginning, and this gets resolved with refinement as displayed in Figure 6.6. Figure 6.6: Output $\mathcal{T}_{1},\mathcal{T}_{5},\mathcal{T}_{11}$ of the adaptive refinement (a) (b) Figure 6.7: Convergence history plot of estimator $\mu$ and error $e:=u-\Pi_{1}u_{h}$ in the (a) piecewise $H^{1}$ norm (b) $L^{2}$ norm vs number ndof of degrees of freedom for both uniform and adaptive refinement ### 6.5 Conclusion The three computational benchmarks provide empirical evidence for the sharpness of the mathematical a priori and a posteriori error analysis in this paper and illustrate the superiority of adaptive over uniform mesh-refining. The empirical convergence rates in all examples for the $H^{1}$ and $L^{2}$ errors coincide with the predicted convergence rates in Theorem 4.3, in particular, for the non-convex domain and reduced elliptic regularity. The a posteriori error bounds from Theorem 5.1 confirm these convergence rates as well. The ratio of the error estimator $\mu_{\ell}$ by the $H^{1}$ error $e_{\ell}$, sometimes called efficiency index, remains bounded up to a typical value 6; we regard this as a typical overestimation factor for the residual- based a posteriori error estimate. Recall that the constant $C_{\text{reg}}$ has not been displayed so the error estimator $\mu_{\ell}$ does not provide a guaranteed error bound. Figure 6.8 and 6.9 display the four different contributions volume residual $(\sum_{P}\eta_{P}^{2})^{1/2}$, stabilization term $(\sum_{P}\zeta_{P}^{2})^{1/2}$, inconsistency term $(\sum_{P}\Lambda_{P}^{2})^{1/2}$ and the nonconformity term $(\sum_{P}\Xi_{P}^{2})^{1/2}$ that add up to the error estimator $\mu_{\ell}$. We clearly see that all four terms converge with the overall rates that proves that none of them is a higher-order term and makes it doubtful that some of those terms can be neglected. The volume residual clearly dominates the a posteriori error estimates, while the stabilisation term remains significantly smaller for the natural stabilisation (with undisplayed parameter one). The proposed adaptive mesh-refining algorithm leads to superior convergence properties and recovers the optimal convergence rates. This holds for the first example with optimal convergence rates in the large pre-asymptotic computational range as well as in the second with suboptimal convergence rates under uniform mesh-refining according to the typical corner singularity and optimal convergence rates for the adaptive mesh-refining. The third example with the Helmholtz equation and a moderate wave number shows certain moderate local mesh-refining in Figure 6.6 but no large improvement over the optimal convergence rates for uniform mesh-refining. The adaptive refinement generates hanging nodes because of the way refinement strategy is defined, but this is not troublesome in VEM setting as hanging node can be treated as a just another vertex in the decompostion of domain. However, an increasing number of hanging nodes with further mesh refinements may violate the mesh assumption (M2), but numerically the method seems robust without putting any restriction on the number of hanging nodes. The future work on the theoretical investigation of the performance of adaptive mesh-refining algorithm is clearly motivated by the successful numerical experiments. The aforementioned empirical observation that the stabilisation terms do not dominate the a posteriori error estimates raises the hope for a possible convergence analysis of the adaptive mesh-refining strategy with the axioms of adaptivity [20] towards a proof of optimal convergence rates: The numerical results in this section support this conjecture at least for the lowest-order VEM in 2D for indefinite non-symmetric second-order elliptic PDEs. Figure 6.8: Estimator components corresponding to the error $H1e=\|u-\Pi_{1}u_{h}\|_{1,\mathrm{pw}}$ of the adaptive refinement presented in Subsection 6.2-6.4 Figure 6.9: Estimator components corresponding to the error $L2e=\\|u-\Pi_{1}u_{h}\\|_{L^{2}(\Omega)}$ of the adaptive refinement presented in Subsection 6.2-6.4 Acknowledgements The authors sincerely thank one anonymous referee for suggestions that led to Remark 5. 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# Online Streaming End-to-End Neural Diarization Handling Overlapping Speech and Flexible Numbers of Speakers ###### Abstract We propose a streaming diarization method based on an end-to-end neural diarization (EEND) model, which handles flexible numbers of speakers and overlapping speech. In our previous study, the speaker-tracing buffer (STB) mechanism was proposed to achieve a chunk-wise streaming diarization using a pre-trained EEND model. STB traces the speaker information in previous chunks to map the speakers in a new chunk. However, it only worked with two-speaker recordings. In this paper, we propose an extended STB for flexible numbers of speakers, FLEX-STB. The proposed method uses a zero-padding followed by speaker-tracing, which alleviates the difference in the number of speakers between a buffer and a current chunk. We also examine buffer update strategies to select important frames for tracing multiple speakers. Experiments on CALLHOME and DIHARD II datasets show that the proposed method achieves comparable performance to the offline EEND method with 1-second latency. The results also show that our proposed method outperforms recently proposed chunk-wise diarization methods based on EEND (BW-EDA-EEND). Index Terms: online speaker diarization, EEND, overlapping speech, flexible numbers of speakers ## 1 Introduction Speaker diarization, a challenging technique that responds to the question “who spoke when” [1, 2, 3, 4, 5, 6], assigns speaker labels to audio regions. Diarization produces outcomes that downstream tasks can utilize. For example, it can provide the turn-taking information and build a pre-processing pipeline for automatic speech recognition in meetings [7, 8, 9, 10], call-center telephone conversations [11, 12, 13], and home environments [14, 15, 16]. The three challenging aspects that current speaker diarization systems should fulfill are overlapping speech, unknown number of speakers, and online operation. However, it is still an open problem to solve these conditions at once. Conventional clustering-based systems primarily focus on clustering algorithms and speaker embeddings such as Gaussian mixture models (GMM) [17, 18], i-vector [19, 20, 21], d-vector [22, 23], and x-vector [24, 25]. However, most clustering-based systems assume that there is only one speaker per segment. As a result, these systems cannot deal with the overlapping speech in general except for a few studies, e.g., [26]. To solve the overlapping issue, an end-to-end neural diarization model (EEND) was proposed [27]. EEND directly minimizes the diarization error by mapping the multi-speaker mixture recording to joint speech activities using a single neural network. The model estimates the speech activity using a dedicated stream for every speaker; hence, EEND inherently assigns two or more labels to the overlapping regions. EEND has already shown significant performance improvement on overlapping speech, especially after adopting the self- attention mechanism (SA-EEND) [28], and with a fixed number of speakers. Table 1: Comparison of speaker diarization methods. Method \| Online \| Overlapping \| Flexible #speakers ---\|---\|---\|--- x-vector+clustering [24] \| – \| – \| ✓ UIS-RNN [22, 23] \| ✓ \| – \| ✓ EEND/SA-EEND [27, 29, 28] \| – \| ✓ \| – EEND-EDA/SC-EEND [30, 31] \| – \| ✓ \| ✓ RSAN [32, 33] \| ✓ \| ✓ \| ✓ BW-EDA-EEND [34] \| ✓ \| ✓ \| ✓ This work \| ✓ \| ✓ \| ✓ To deal with overlapping speech and flexible numbers of speakers, Horiguchi et al. introduced the encoder-decoder based attractor (EDA) module to SA-EEND [30], and Fujita et al. extended the SA-EEND to speaker-wise conditional EEND (SC-EEND) [31, 35]. Both extensions have only been evaluated in offline mode. To cope with online applications, the speaker-tracing buffer (STB) [36] was proposed to trace the speaker permutation information across chunks which enables the offline pre-trained SA-EEND model to work in an online manner. The original STB achieved comparable diarization accuracy to the offline EEND with $1\text{\,}\mathrm{s}$ chunk size but this method was limited to two-speaker recordings. In [34], Han et al. proposed the block-wise-EDA-EEND (BW-EDA-EEND) which makes the EDA-EEND work in an online fashion. Motivated by Transformer- XL [37], this approach utilizes the previous hidden states of the transformer encoder as input to the EDA-EEND. To satisfy all the three requirements together, among the existing diarization methods as shown in Table 1, the Recurrent Selective Attention Network (RSAN) [32, 33] and the block-wise-EDA-EEND (BW-EDA-EEND) stand out. However, due to the speech separation-based training objective, RSAN is hard to adapt to real recordings, and the evaluations under real scenarios are not reported. On the other hand, although BW-EDA-EEND [34] conducted online experiments on $10\text{\,}\mathrm{s}$ chunk size conditions, which cause large latency. In this paper, we consider more realistic streaming applications with a smaller chunk size such as $1\text{\,}\mathrm{s}$. In this work, we extend the inference algorithm of existing offline model (e.g., EEND-EDA) to operate in an online mode using the speaker-tracing buffer for flexible numbers of speakers (FLEX-STB) without re-training the offline model. FLEX-STB is designed to deal with variable numbers of speakers using a zero-padding mechanism with reasonable latency. Four frame selection strategies are also proposed to contain the speaker permutation information in FLEX-STB. The proposed diarization system can operate in an online mode handling overlapping speech and flexible number of speakers, and working in real scenarios such as CALLHOME and DIHARD II with $1\text{\,}\mathrm{s}$ chunk size. ## 2 Preliminary In this section, we briefly explain two key elements: EEND for flexible numbers of speakers and the original STB that enables the offline SA-EEND systems to work online. ### 2.1 EEND for flexible numbers of speakers Given a $T$-length sequence of $D$-dimensional log-scaled Mel-filterbank-based acoustic features $\mathbf{X}\in\mathbb{R}^{D\times T}$, a neural network- based function $\mathrm{EEND}:\mathbb{R}^{D\times T}\rightarrow(0,1)^{S\times T}$ calculates posterior probabilities of speech activities at each time frame $\hat{\mathbf{Y}}=(\hat{\mathbf{y}}_{t})_{t=1}^{T}\in(0,1)^{S\times T}$ as follows: $\hat{\mathbf{Y}}=\mathrm{EEND}(\mathbf{X}),$ (1) Here, $\hat{\mathbf{y}}_{t}\coloneqq\left[\hat{y}_{1,t},\dots,\hat{y}_{S,t}\right]^{\mathsf{T}}$ is the posterior of speech activities calculated for each speaker $s\in\\{1,\dots,S\\}$ independently, where $\left(\cdot\right)^{\mathsf{T}}$ denotes the matrix transpose and $S$ is the number of speakers. Diarization results $\tilde{\mathbf{Y}}=(\tilde{y}_{s,t})_{s,t}\in\\{0,1\\}^{S\times T}$ are obtained by applying a threshold value $\theta$ (e.g., 0.5) to the posteriors $\hat{\mathbf{Y}}$. If $\tilde{y}_{s,t}=\tilde{y}_{s^{\prime},t}=1~{}(s\neq s^{\prime})$, it means that both speakers $s$ and $s^{\prime}$ are estimated to have spoken at time $t$, which is regarded as the overlapping region. If $\forall s\in\\{1,\dots,S\\},~{}\tilde{y}_{s,t}=0$, it indicates that no speaker is estimated to have spoken at time $t$. Note that EEND used permutation invariant training [27] so that there is no condition to decide the order of output speakers. While the original EEND [27, 29] fixes the number of speakers $S$ by its network structure, variants of EEND [30, 31, 35] have been proposed to estimate the number of speakers $\hat{S}$. However, these methods perform only in the offline setting. ### 2.2 Speaker-tracing buffer for fixed number of speakers One of the straightforward online extensions of EEND is to perform diarization process for each chunk of acoustic features and concatenated diarization results across the chunk. However, this cannot obtain a consistent speaker permutation of the whole recording. This is because the EEND used permutation invariant training [27] so that there is no condition to decide the order of output speakers. We call this speaker permutation problem. To solve the speaker permutation problem, we have proposed speaker-tracing buffer (STB) [36] for the original EENDs, which assume that the number of speakers was known as prior. Let $\mathbf{X}_{i}\in\mathbb{R}^{D\times\Delta}$ represents the subsequence of $\mathbf{X}$ at chunk $i\in\left\\{1,\dots,I\right\\}$ with a fixed chunk length $\Delta$, i.e., $\mathbf{X}=\left[\mathbf{X}_{1},\dots,\mathbf{X}_{i},\dots,\mathbf{X}_{I}\right]$. The $\mathrm{EEND}:\mathbb{R}^{D\times T}\rightarrow(0,1)^{S\times T}$ function accepts the input features of flexible length $T$ and produces the posteriors of speech activities of the same length for each speaker. Note that the number of speakers $S$ is fixed in this section. #### 2.2.1 Initialization The STB possesses two matrices: acoustic features $\mathbf{X}^{(\text{buf})}_{i}\in\mathbb{R}^{D\times L_{i}}$ and the corresponding posteriors $\mathbf{Y}_{i}^{\text{(buf)}}\in\mathbb{R}^{S\times L_{i}}$ from $\mathrm{EEND}\left(\cdot\right)$, where $L_{i}$ is the buffer length after $i$-th update. The matrices are initialized at the first chunk as follows: $\displaystyle\mathbf{X}_{1}^{\text{(buf)}}$ $\displaystyle=\mathbf{X}_{1},$ (2) $\displaystyle\mathbf{Y}_{1}^{\text{(buf)}}$ $\displaystyle=\hat{\mathbf{Y}}_{1}=\mathrm{EEND}(\mathbf{X}_{1}).$ (3) As we assume that the chunk size $\Delta$ is smaller than the maximum number of frames $L_{\text{max}}$ in the buffer, all the inputs and outputs of the first chunk can be fed into STB. #### 2.2.2 Chunk-wise processing handling speaker permutation From the second chunk, posteriors $\hat{\mathbf{Y}}_{i}$ are computed using the STB. Firstly, an input concatenated with the the buffer is fed into $\mathrm{EEND}\left(\cdot\right)$: $\left[\mathbf{\hat{Y}}_{i-1}^{\text{(buf)}},\mathbf{\hat{Y}}_{i}\right]=\mathrm{EEND}\left(\left[\mathbf{X}_{i-1}^{\text{(buf)}},\mathbf{X}_{i}\right]\right)\in(0,1)^{S\times(L_{i-1}+\Delta)}.$ (4) Next, the optimal speaker permutation for the current chunk is calculated as follows: $\psi=\operatorname{arg\,max}_{\phi\in\mathrm{Perm}(S_{i})}\mathrm{Corr}\left(\mathbf{Y}_{i-1}^{\text{(buf)}},\mathbf{P}_{\phi}\mathbf{\hat{Y}}^{\text{(buf)}}_{i-1}\right),$ (5) where $\mathbf{P}_{\phi}\in[0,1]^{S\times S}$ is a permutation matrix for the $\phi$-th permutation in $\mathrm{Perm}(S_{i})$, which is all the possible permutations of the sequence $\left(1,\dots,S\right)$. $\mathrm{Corr}\left(\mathbf{A},\mathbf{B}\right)$ calculates the correlation between two matrices $\mathbf{A}=\left(a_{ij}\right)_{jk}$ and $\mathbf{B}=\left(b_{jk}\right)_{ij}$ defined as $\displaystyle\mathrm{Corr}\left(\mathbf{A},\mathbf{B}\right)\coloneqq\sum_{i,j}\left(a_{jk}-\bar{a}\right)\left(b_{jk}-\bar{b}\right),$ (6) where $\bar{a}$ and $\bar{b}$ are the mean values of $\mathbf{A}$’s and $\mathbf{B}$’s elements, respectively. Finally, the posterior probabilities of the $i$-th chunk are calculated with the permutation matrix that gives the highest correlation as follows: $\mathbf{Y}_{i}=\mathbf{P}_{\psi}\mathbf{\hat{Y}}_{i}.$ (7) If the length of $\left[\mathbf{Y}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}\right]$ is larger than the predetermined maximum buffer length $L_{\text{max}}$, we select frames to be kept in the STB, which are used to solve the speaker permutation problem occurred by the future inputs. In the paper [36], four selection strategies have been proposed. The STB is a solution to the online diarization problem; however, it cannot be directly applied to EEND for unknown and flexible numbers of speakers. One reason is because the number of speakers may be different across chunks so that we cannot calculate correlation using Eq. (missing) 6. The other reason is that the most promising selection strategy used the absolute difference of probabilities of two speakers’ speech activities; thus, the method is limited to two-speaker EENDs. ## 3 Proposed method In this paper, we proposed the FLEX-STB which extends the STB coping with the two obstacles to use it with EEND for unknown numbers of speakers [30, 31]. The FLEX-STB deals with the varying number of speakers across chunks by increasing the number of speaker slots in the speaker-tracing buffer with the zero-padding in Section 3.1. When the system detects new speakers, it adds new zero-speaker-activity slots to the speaker buffer. We also propose four selection strategies to update the buffer, each of which are not limited by the number of speakers, in Section 3.2. Figure 1: Proposed speaker-tracing buffer for unknown numbers of speakers. Zero-padding is applied to mitigate the different number of speakers between $\mathbf{Y}^{\text{(buf)}}_{i-1}$ and $\mathbf{\hat{Y}}^{\text{(buf)}}_{i-1}$. ### 3.1 Speaker-tracing buffer for flexible numbers of speakers (FLEX-STB) In this section, we assume that EEND estimates not only speech activities but also the number of speakers $S$, i.e., $\mathrm{EEND}:\mathbb{R}^{D\times T}\rightarrow(0,1)^{S\times T}$. Firstly, to alleviate the different number of speakers between the buffer $\mathbf{Y}_{i-1}^{\text{(buf)}}$ and the current chunk’s output $\mathbf{\hat{Y}}_{i}$, the posterior of the no-speech-activity speaker is considered as zero so that the zero-padding function is applied as follows: $\displaystyle\mathbf{Z}^{\text{(buf)}}_{i-1}$ $\displaystyle=\mathsf{ZeroPadding}\left(\mathbf{Y}^{\text{(buf)}}_{i-1},S_{i}\right),$ (8) $\displaystyle\left[\mathbf{\hat{Z}}_{i-1}^{\text{(buf)}},\mathbf{\hat{Z}}_{i}\right]$ $\displaystyle=\mathsf{ZeroPadding}\left(\left[\mathbf{\hat{Y}}^{\text{(buf)}}_{i-1},\hat{\mathbf{Y}}_{i}\right],S_{i}\right),$ (9) where $S_{i}=\max(S_{i-1},S_{i})$ and $\mathsf{ZeroPadding}(\mathbf{A},S)$ appends row zero vectors to $\mathbf{A}$ so that the first dimension becomes $S$. Next, the speaker permutation $\mathbf{P}_{\psi}$ for the current chunk is calculated between $\mathbf{Z}_{i-1}^{\text{(buf)}}$ and $\mathbf{\hat{Z}}^{\text{(buf)}}_{i-1}$ using Eq. (missing) 5. Then, the output for the current chunk is permuted as follows: $\mathbf{Y}_{i}=\mathbf{P}_{\psi}\mathbf{\hat{Z}}_{i},$ (10) where $\mathbf{Y}_{i}$ is the final diarization result of the chunk $i$. After that, at most $L_{\text{max}}$ time indexes $\mathcal{T}\subseteq\left\\{1,\dots,L_{i-1}+\Delta\right\\}$ are selected based on the concatenated outputs $\left[\mathbf{Z}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}\right]\in(0,1)^{S_{i}\times(L_{i-1}+T)}$, and the FLEX-STB is updated as follows: $\displaystyle\mathbf{X}^{\text{(buf)}}_{i}=\left[\mathbf{x}_{\tau}\mid\tau\in\mathcal{T}\right],\;\mathbf{Y}^{\text{(buf)}}_{i}=\left[\mathbf{y}_{\tau}\mid\tau\in\mathcal{T}\right],$ (11) where $\mathbf{x}_{\tau}$ is the $\tau$-th column vector of $[\mathbf{X}^{\text{(buf)}}_{i-1},\mathbf{X}_{i}]$, $\mathbf{y}_{\tau}$ is the $\tau$-th column vector of $[\mathbf{Z}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}]$. The frame selection strategies are described in Section 3.2. ### 3.2 Selection strategy When the number of accumulated features becomes larger than the buffer size $L_{\text{max}}$, a selection strategy is needed to keep relevant features that contain the speaker permutation information from $\left[\mathbf{X}^{\text{(buf)}}_{i-1},\mathbf{X}_{i}\right]$ and $\left[\mathbf{Z}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}\right]$. In this section, four selection functions are proposed for flexible numbers of speakers. • Uniform sampling: Uniform distribution sampling is applied to extract $L_{\text{max}}$ frames. * • First-in-first-out (FIFO): The most recent $L_{\text{max}}$ features and the corresponding diarization results are stored in the buffer, which follows the first-in-first-out manner. * • Kullback-Leibler divergence based selection: We utilize the Kullback-Leibler divergence (KLD) to measure the difference between two probability distributions: the speaker activities distribution and the uniform distribution at time $t$, which can be represented as follows: $\displaystyle\text{KLD}_{t}$ $\displaystyle=\sum_{s=1}^{S_{i}}p_{s,t}\log{\frac{p_{s,t}}{q_{s,t}}},$ (12) $\displaystyle p_{s,t}$ $\displaystyle=\frac{r_{s,t}}{\sum_{s^{\prime}=1}^{S_{i}}r_{s^{\prime},t}},$ (13) $\displaystyle q_{s,t}$ $\displaystyle=\frac{1}{S_{i}},$ (14) where $\left[\mathbf{Z}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}\right]=(r_{s,t})_{\begin{subarray}{c}1\leq t\leq(L_{i-1}+\Delta)\\\ 1\leq s\leq{S_{i}}\end{subarray}}$ is the posteriors from EEND with FLEX-STB and $q_{s,t}$ is the uniform distribution. Top $L_{\text{max}}$ samples with the highest KLD values are selected from $\left[\mathbf{Z}^{\text{(buf)}}_{i-1},\mathbf{Y}_{i}\right]$ and the corresponding $\left[\mathbf{X}^{\text{(buf)}}_{i-1},\mathbf{X}_{i}\right]$. * • Weighted sampling using KLD: The combination of uniform sampling and KLD based selection. $L_{\text{max}}$ features are randomly selected with the probabilities which are proportional to $\text{KLD}_{t}$. Table 2: DERs (%) of online EEND-EDA with chunk size $\Delta=$1\text{\,}\mathrm{s}$$ using FLEX-STB and offline EEND-EDA with chunk size $\Delta=\infty$. Note that all results are based on the estimated number of speakers, including the overlapping regions without oracle SAD. \| Online ($\Delta=$1\text{\,}\mathrm{s}$$) \| \| ---\|---\|---\|--- \| CALLHOME \| DIHARD II \| Offline ($\Delta=\infty$) \| $L_{\text{max}}=$10\text{\,}\mathrm{s}$$ \| $L_{\text{max}}=$50\text{\,}\mathrm{s}$$ \| $L_{\text{max}}=$100\text{\,}\mathrm{s}$$ \| $L_{\text{max}}=$10\text{\,}\mathrm{s}$$ \| $L_{\text{max}}=$50\text{\,}\mathrm{s}$$ \| $L_{\text{max}}=$100\text{\,}\mathrm{s}$$ \| CALLHOME \| DIHARD II FLEX-STB with selection strategy \| \| \| \| \| \| \| \| Uniform sampling \| 27.6 \| 20.2 \| 19.3 \| 52.4 \| 39.3 \| 36.8 \| - \| - FIFO \| 29.5 \| 19.4 \| 19.1 \| 57.2 \| 41.1 \| 37.0 \| - \| - KLD selection \| 30.0 \| 22.3 \| 20.9 \| 52.6 \| 40.8 \| 37.7 \| - \| - Weighted sampling using KLD \| 26.6 \| 20.0 \| 19.5 \| 50.3 \| 37.9 \| 36.0 \| - \| - Without FLEX-STB \| - \| - \| - \| - \| - \| - \| 15.3 \| 32.9 ## 4 Experiment ### 4.1 Data We generated 100k simulated mixtures of one to four speakers following the procedure in [30] using Switchboard-2 (Phase I, II, III), Switchboard Cellular (Part 1, 1), and the NIST Speaker Recognition Evaluation datasets (SRE). Additionally, we added noises from the MUSAN corpus [38] and room impulse responses (RIRs) from the Simulated Room Impulse Response Database [39]. These simulated mixtures were used for training the EEND-based model. Two real conversation datasets: the CALLHOME [11] and the DIHARD II [3] were prepared for evaluation. ### 4.2 Experiment setting In this paper, we evaluated the proposed method on the offline EEND-EDA model. The EEND-EDA model was trained with four Transformer encoder blocks and 256 attention units containing four heads [30]. We firstly trained the model using a two-speaker dataset for 100 epochs and then finetuned with the concatenation of one- to four-speaker simulated datasets for 25 epochs. Finally, EEND-EDA model was finetuned using a development set of CALLHOME, or DIHARD II, respectively. We evaluated all systems with the diarization error rate (DER) metric in both overlapping and non-speech regions. A collar tolerance of 250 ms was applied at the start and end of each segment for the CALLHOME dataset. Following the regulation of the second DIHARD challenge [3], we did not use collar tolerance for the DIHARD II dataset. ### 4.3 Results #### 4.3.1 Effect of selection strategies and buffer size Table 2 shows the effect of the selection strategies and the buffer size of the FLEX-STB on the EEND-EDA model in the left part. Experiment conditions varied from four selection methods with buffer sizes equal to $10\text{\,}\mathrm{s}$, $50\text{\,}\mathrm{s}$ and $100\text{\,}\mathrm{s}$ but fixed the chunk size $\Delta$ to $1\text{\,}\mathrm{s}$. All results were calculated with the estimated number of speakers including the overlapping regions without oracle sound activity detection (SAD). It is shown that incremental buffer size which provides more input information improved the accuracy regardless of the selection strategies. Regarding the selection strategies, on most cases weighted sampling using KLD outperformed other strategies on both datasets. The best results from online system are $19.1\text{\,}\mathrm{\char 37\relax}$ and $36.0\text{\,}\mathrm{\char 37\relax}$ for CALLHOME and DIHARD II, respectively. #### 4.3.2 Comparison with the offline EEND-EDA system We also compared the performance of our proposed online and baseline offline systems in Table 2. The input of the offline EEND-EDA system is the whole recording during inference while that for the online system is the $1\text{\,}\mathrm{s}$ chunk. Compared with the offline system, DERs of the online system increases by $3.8\text{\,}\mathrm{\char 37\relax}$ and $3.1\text{\,}\mathrm{\char 37\relax}$ on these two datasets, which would be acceptable degradation by considering the benefit of streaming diarization. The performance degradation is supposed to come from the mismatch between the offline model which was trained with fixed large chunk size and the online mechanism whose input sizes are incrementally increased. #### 4.3.3 Comparison with other online diarization systems First, we compared our method with the recently proposed BW-EDA-EEND [34] on the CALLHOME dataset. In order to compare with BW-EDA-EEND in the same condition, we evaluated our method with a $10\text{\,}\mathrm{s}$ chunk size. As shown in Table 3, in a $10\text{\,}\mathrm{s}$ chunk size and estimated SAD condition, our proposed method outperforms the BW-EDA-EEND on all speaker- number cases on the CALLHOME dataset. Next, we compared our proposed method with other systems in more realistic scenario, i.e., DIHARD II. For a fair comparison with other online methods, we follow the DIHARD II track 1, where the oracle SAD information is provided. We used the oracle SAD information to filter out non-speech frames of the estimated diarization result. Table 4 shows the comparison with other systems. The proposed online EEND-EDA with FLEX-STB achieved a DER of $25.8\text{\,}\mathrm{\char 37\relax}$, which outperformed the UIS-RNN-SML, and is comparable to the offline DIHARD II baseline. #### 4.3.4 Real-time factor and latency Our experiment was conducted on one NVIDIA Tesla P100 GPU. To calculate the average computing time of one buffer, we filled the buffer with dummy values for the first iteration to keep the buffer size always the same among chunks. The real-time factor was equal to 0.13 when we applied FLEX-STB to EEND-EDA with chunk size equal to $1\text{\,}\mathrm{s}$, and a buffer size of $100\text{\,}\mathrm{s}$. This means that the average computation duration of a $1\text{\,}\mathrm{s}$ chunk was $0.13\text{\,}\mathrm{s}$ which is acceptable for the online processing. Table 3: DERs (%) of each number of speakers on the CALLHOME dataset with $10\text{\,}\mathrm{s}$ chunk size. Both estimated the number of speakers and included the overlapping regions without using oracle SAD. \| Number of speakers ---\|--- Method \| 2 \| 3 \| 4 BW-EDA-EEND [34] \| 11.8 \| 18.3 \| 26.0 EEND-EDA w/ FLEX-STB \| 10.0 \| 14.0 \| 21.1 Table 4: DERs (%) on DIHARD II dataset computed by using oracle SAD including overlapping regions. Online systems with STB were evaluated in a $1\text{\,}\mathrm{s}$ chunk size $\Delta$ and $100\text{\,}\mathrm{s}$ buffer size $L_{\mathrm{max}}$. Method \| DER ---\|--- DIHARD-2 baseline (offline) [3] \| 26.0 UIS-RNN-SML [23] \| 27.3 EEND-EDA w/ FLEX-STB \| 25.8 ## 5 Conclusion In this paper, we proposed an online streaming speaker diarization method that handles overlapping speech and flexible numbers of speakers. A speaker tracing buffer for flexible numbers of speakers was proposed to mitigate the different number of speakers among chunks. 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# Sharp upper bounds for moments of quadratic Dirichlet $L$-functions Peng Gao School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China<EMAIL_ADDRESS> ###### Abstract. We establish unconditional sharp upper bounds of the $k$-th moments of the family of quadratic Dirichlet $L$-functions at the central point for $0\leq k\leq 2$. Mathematics Subject Classification (2010): 11M06 Keywords: moments, quadratic Dirichlet $L$-functions, upper bounds ## 1\. Introduction Moments of central values of families of $L$-functions have attracted much attention in research as they can be applied to address the non-vanishing issue of these values, which in turn carry significant arithmetic implications. Much progress has been made in late years that largely enhances our understanding of these moments. A conjecture concerning asymptotic expressions for the moments of various families of $L$-functions has been made by J. P. Keating and N. C. Snaith in [Keating-Snaith02], in connection with random matrix theory. Another conjecture of J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith in [CFKRS] gives more precise predictions by including lower order terms on the asymptotic behaviors of the moments for certain families of $L$-functions. In [R&Sound] and [R&Sound1], Z. Rudnick and K. Soundararajan developed a simple and powerful method towards establishing lower bounds for rational moments of families of $L$-functions of the conjectured order of magnitude. The method is further extended by M. Radziwiłł and K. Soundararajan in [Radziwill&Sound1] to all moments larger than the first. On the other hand, an approach due to K. Soundararajan in [Sound01] enables one to obtain the corresponding upper bounds under the generalized Riemann hypothesis (GRH). This approach was further sharpened by A. J. Harper [Harper] to give upper bounds of desired order of magnitude for moments of $L$-functions in many families conditionally. In [Radziwill&Sound], M. Radziwiłł and K. Soundararajan developed a new principle that allows one to seek upper bounds of all smaller moments with the knowledge of an upper bound for a particular moment. This principle is further implemented by W. Heap and K. Soundararajan in [H&Sound] to treat the case of lower bounds. In this paper, we are interested in the family of quadratic Dirichlet $L$-functions. Asymptotic formulas for the first two moments of this family was obtained by M. Jutila in [Jutila] with the error terms being subsequently improved in [ViTa, DoHo, Young1, sound1, Sono]. For the purpose of this paper, we are interested in the family given by $\\{L(s,\chi_{8d})\\}$ for $d$ being odd and square-free integers. Here $\chi_{8d}=\left(\frac{8d}{\cdot}\right)$ is the Kronecker symbol. The study of the above family was initiated by K. Soundararajan in [sound1], who obtained mollified first two moments of the family to show that at least $87.5\%$ of the members of this family have non- vanishing central values. The third moment of the above family is also obtained for the first time in [sound1]. The error term was improved by M. P. Young in [Young2] for a smoothed version. In [Shen], Q. Shen obtained the fourth moment of the above family under GRH. Asymptotic formulas for all positive real moments of $\\{L(\tfrac{1}{2},\chi_{8d})\\}$ are conjectured by J. C. Andrade and Keating, J. P. in [Andrade-Keating01]. Combining the above mentioned work of [Harper, Sound01, R&Sound, R&Sound1, Radziwill&Sound1], we now know that $\displaystyle X(\log X)^{\frac{k(k+1)}{2}}\ll_{k}\sideset{}{{}^{}}{\sum}_{\begin{subarray}{c}0<d<X\\\ (d,2)=1\end{subarray}}\|L(\tfrac{1}{2},\chi_{8d})\|^{k}\ll_{k}X(\log X)^{\frac{k(k+1)}{2}},$ where the lower bound above holds for all real $k\geq 1$ unconditionally and the upper bound above holds for all $k\geq 0$ under GRH. Here and throughout the paper, we denote $\sideset{}{{}^{}}{\sum}$ for the sum over square-free integers. It is our goal in this paper to apply the principal of obtaining upper bounds given in [Radziwill&Sound] by M. Radziwiłł and K. Soundararajan in our setting. Before we state our result, we would like to recall the principle. Although the principle works for general $L$-functions, we only adapt it for the family $\\{L(s,\chi_{8d})\\}$ for simplicity. For this, we recall that a result of B. Hough [Hough] shows that under GRH, the quantity $\log\|L(\tfrac{1}{2},\chi_{8d})\|$ with $X<d\leq 2X$ for large $X$ is normally distributed with mean $\tfrac{1}{2}\log\log d$ and variance $\log\log d$. On the other hand, it is shown in [Hough] that the sum (1.1) $\displaystyle\sum_{n\leq z}\frac{\Lambda(n)\chi_{8d}(n)}{n^{\frac{1}{2}}}$ is similarly distributed as $\log\|L(\tfrac{1}{2},\chi_{8d})\|$ for $z=X^{1/(\log\log X)^{2}}$, say. Here $\Lambda(n)$ is the von Mangoldt function. As the contribution of prime powers $\geq 3$ is negligible in the above sum and the contribution of prime squares is about $\log\log X$, we see that the difference between the expression given in (1.1) and $\log\log X$ is mainly determined by $\displaystyle{\mathcal{P}}(d)=\sum_{p\leq z}\frac{1}{\sqrt{p}}\chi_{8d}(p).$ Taking exponentials, this implies that the quantity $\|L(\tfrac{1}{2},\chi_{8d})\|(\log\|d\|)^{1/2}\exp(-{\mathcal{P}}(d))$ is usually small. Thus, for any given real numbers $n>0,0<k<1$, we may estimate the quantity $(\|L(\tfrac{1}{2},\chi_{8d})\|(\log\|d\|^{1/2})^{nk}$ by writing it as $\displaystyle\Big{(}\|L(\tfrac{1}{2},\chi_{8d})\|(\log\|d\|)^{1/2}\Big{)}^{nk}=\|L(\tfrac{1}{2},\chi_{8d})\|^{nk}(\log\|d\|)^{nk/2}\exp(-nk(1-k){\mathcal{P}}(d))\cdot\exp(nk(1-k){\mathcal{P}}(d)).$ We now recall that Young’s inequality asserts that we have $ab\leq a^{p}/p+b^{q}/q$ for real numbers $a,b\geq 0$ and real numbers $p,q\geq 1$ satisfying $1/p+1/q=1$. Applying this with $a=\|L(\tfrac{1}{2},\chi_{8d})\|^{nk}(\log\|d\|)^{nk/2}\exp(-nk(1-k){\mathcal{P}}(d)),b=\exp(nk(1-k){\mathcal{P}}(d))$ and $p=1/k,q=1/(1-k)$, we see that (1.2) $\displaystyle\Big{(}\|L(\tfrac{1}{2},\chi_{8d})\|(\log\|d\|)^{1/2}\Big{)}^{nk}\leq k\|L(\tfrac{1}{2},\chi_{8d})\|^{n}(\log\|d\|)^{n/2}\exp(-n(1-k){\mathcal{P}}(d))+(1-k)\exp(kn{\mathcal{P}}(d)).$ As we expect $\|L(\tfrac{1}{2},\chi_{8d})\|^{n}(\log\|d\|)^{n/2}\exp(-n{\mathcal{P}}(d))$ to be small most of the time, the right side of (1.2) should be bounded above by $\exp(kn{\mathcal{P}}(d))$ on average. As expanding $\exp(kn{\mathcal{P}}(d))$ into Euler products leads to a too long Dirichlet polynomial to estimate, we approximate it by taking a suitably long Taylor expansion to achieve our goal since ${\mathcal{P}}(d)$ is often small in size. We note that the bound given in (1.2) may be modified to establish similar bounds for general $L$-functions. If we further know that the corresponding $L$-function satisfies $L(\tfrac{1}{2})\geq 0$, then we may replace $\|L(\tfrac{1}{2})\|^{n}$ on the right side of (1.2) by $L(\tfrac{1}{2})^{n}$ to see that if we have a good understanding of the $n$-th moment (twisted by another character) of the corresponding family of $L$-functions at the central value, then we shall be able to obtain sharper upper bounds for every $m$-th moment of the same family of $L$-functions with $0\leq m\leq n$. This is precisely what has been carried out by Radziwiłł and Soundararajan in [Radziwill&Sound] to treat the moments of quadratic twists of $L$-functions attached to elliptic curves, since in this case it is known that the corresponding $L$-functions have non-negative values at $1/2$ and the first moment of the family can be evaluated. For the family of quadratic twists of Dirichlet $L$-functions, although our current stage of knowledge is short of determining whether $L(\tfrac{1}{2},\chi_{8d})\geq 0$ is always valid, we do know that these values are real so that $L(\tfrac{1}{2},\chi_{8d})^{2}\geq 0$ is always true. Thanks to the work of K. Soundararajan in [sound1], we also have a good knowledge of the twisted second moment of the same family. Combining these with the above mentioned principle of Radziwiłł and Soundararajan, we are therefore able to establish unconditionally the correct order of magnitude of every moment less than the second of the family of quadratic Dirichlet $L$-functions. This is given in the following result. ###### Theorem 1.1. Unconditionally, for every $0\leq k\leq 2$, we have (1.3) $\displaystyle\sideset{}{{}^{}}{\sum}_{\begin{subarray}{c}0<d<X\\\ (d,2)=1\end{subarray}}\|L(\tfrac{1}{2},\chi_{8d})\|^{k}\ll_{k}X(\log X)^{\frac{k(k+1)}{2}}.$ We notice that the twisted third moment of the quadratic family of Dirichlet $L$-functions has been evaluated by M. P. Young in [Young2]. Thus, if one assumes that $L(\tfrac{1}{2},\chi_{8d})\geq 0$ for all $d$ under consideration (which follows from GRH), then we are able to obtain the following result. ###### Theorem 1.2. Assume that $L(\tfrac{1}{2},\chi_{8d})\geq 0$ for all odd, square-free $d$. Then the bound given in (1.3) is valid for every $0\leq k\leq 3$. In particular, this is true under GRH. We omit the proof of Theorem 1.2 in the paper as it is similar to that of Theorem 1.1. We also notice that using the approach of Radziwiłł and Soundararajan in [Radziwill&Sound], we may be able to give an alternative proof of a result of B. Hough [Hough, Corollary 1.1] that says the distribution of logarithms of central values of $L(\tfrac{1}{2},\chi_{8d})$ is bounded above by the Gaussian distribution. We would like to point out here that in order for the principle of Radziwiłł and Soundararajan on obtaining upper bounds for moments of $L$-functions to work, one in general needs to evaluate a given moment of $L$-functions twisted by certain character instead of just evaluating the moment itself. Previously, it is known that these twisted moments play vital roles when using mollifiers to study the non-vanishing issues of central values of $L$-functions. They are also necessary in the work of M. P. Young [Young1, Young2] to use a recursive method to reduce the sizes of error terms in the moments of $L$-functions at the central point. Now, the principle of Radziwiłł and Soundararajan provides a further evidence on the importance of studying these twisted moments. In the last section of this paper, we propose a variant of the above mentioned principle of Radziwiłł and Soundararajan for obtaining upper bounds of moments of $L$-functions. This may potentially lead to a slightly simpler treatment (at least in some cases) when acquiring such upper bounds. ## 2\. Preliminaries We include here some tools needed in our proof of Theorem 1.1 together with an initial treatment of the proof. ### 2.1. Tools From now on, we reserve the letter $p$ for a prime number and we recall the following well-known Mertens’ formula (see [MVa1, Theorem 2.7]) and a consequence of it (via partial summation). ###### Lemma 2.2. Let $x\geq 2$. We have, for some constant $b$, $\sum_{p\leq x}\frac{1}{p}=\log\log x+b+O\Big{(}\frac{1}{\log x}\Big{)}.$ Also, for any integer $j\geq 1$, we have $\sum_{p\leq x}\frac{(\log p)^{j}}{p}=\frac{(\log x)^{j}}{j}+O((\log x)^{j-1}).$ Now, we denote $\Phi$ for a smooth, non-negative function compactly supported on $[1/2,5/2]$ with $\Phi(x)=1$ for $x\in[1,2]$, and define, for any complex number $s$, ${\widehat{\Phi}}(s)=\int_{0}^{\infty}\Phi(x)x^{s}\frac{dx}{x}.$ We define $\delta_{n=\square}$ to be $1$ when $n=\square$ and $0$ otherwise, where we write $\square$ for a perfect square. Similar to the proof of [Radziwill&Sound, Proposition 1], we have the following result concerning a smoothed sum of quadratic characters. ###### Lemma 2.3. For large $X$ and any odd positive integer $n$, we have $\displaystyle\sideset{}{{}^{}}{\sum}_{\begin{subarray}{c}(d,2)=1\end{subarray}}\chi_{8d}(n)\Phi\Big{(}\frac{d}{X}\Big{)}=\displaystyle\delta_{n=\square}{\widehat{\Phi}}(1)\frac{2X}{3\zeta(2)}\prod_{p\|n}\Big{(}\frac{p}{p+1}\Big{)}+O(X^{\frac{1}{2}+\epsilon}\sqrt{n}).$ We denote further $d(n)$ for the divisor function and $\sigma(n)$ for the sum of the positive divisors of $n$. Also define $\Lambda_{j}(n)$ for all integers $j\geq 0$ to be the coefficient of $n^{-s}$ in the Dirichlet series expansion of $(-1)^{j}\zeta^{(j)}(s)/\zeta(s)$. Note that this implies that $\Lambda_{1}(n)=\Lambda(n)$ and that $\Lambda_{j}(n)$ is supported on integers having at most $j$ distinct prime factors such that $\Lambda_{j}(n)\ll_{j}n^{j}$. Combining Proposition 1.1 and Proposition 1.3 in [sound1] and setting $Y=X^{1/4},M=1$ there, we readily deduce the following asymptotic result concerning the twisted second moment of quadratic Dirichlet $L$-functions. ###### Lemma 2.4. Writing any odd $l$ as $l=l_{1}l^{2}_{2}$ with $l_{1}$ square-free, we have for any $\varepsilon>0$, $\displaystyle\begin{split}\sideset{}{{}^{}}{\sum}_{(d,2)=1}L(\tfrac{1}{2},\chi_{8d})^{2}\chi_{8d}(l)\Phi(\frac{d}{X})=&\frac{D\widehat{\Phi}(1)}{36\zeta(2)}\frac{d(l_{1})}{\sqrt{l_{1}}}\frac{l_{1}}{\sigma(l_{1})h(l)}X\Big{(}\log^{3}\Big{(}\frac{X}{l_{1}}\Big{)}-3\sum_{\begin{subarray}{c}p\|l_{1}\end{subarray}}\log^{2}p\log\Big{(}\frac{X}{l_{1}}\Big{)}+O(l)\Big{)}+O\left(X^{\frac{3}{4}+\varepsilon}l^{\tfrac{1}{2}+\varepsilon}_{1}\right),\end{split}$ where $D=\frac{1}{8}\displaystyle\prod_{\begin{subarray}{c}p\geq 3\end{subarray}}\left(1-\frac{1}{p}\right)h(p)$ and $h$ is the multiplicative function defined on prime powers by $\displaystyle h(p^{k})=1+\frac{1}{p}+\frac{1}{p^{2}}-\frac{4}{p(p+1)},\quad k\geq 1.$ Also, $\displaystyle O(l)=$ $\displaystyle\sum^{3}_{j,k=0}\sum_{\begin{subarray}{c}m\|l_{1}\end{subarray}}\sum_{\begin{subarray}{c}n\|l_{1}\end{subarray}}\frac{\Lambda_{j}(m)}{m}\frac{\Lambda_{k}(n)}{n}D(m,n)Q_{j,k}\Big{(}\log\frac{X}{l_{1}}\Big{)}-3\Big{(}A+B\frac{\widehat{\Phi}^{\prime}(1)}{\widehat{\Phi}(1)}\Big{)}\sum_{\begin{subarray}{c}p\|l\end{subarray}}\log^{2}p,$ where $A$ and $B$ are absolute constants and $D(m,n)\ll 1$ uniformly for all $m$ and $n$. The $Q_{j,k}$ are polynomials of degree $\leq 2$ whose coefficients involve only absolute constants and linear combinations of $\frac{\widehat{\Phi}^{(j)}(1)}{\widehat{\Phi}(1)}$ for $1\leq j\leq 3$. Lastly, we define for any non-negative integer $\ell$ and any real number $x$, (2.1) $E_{\ell}(x)=\sum_{j=0}^{\ell}\frac{x^{j}}{j!}.$ We recall the following two key inequalities from [Radziwill&Sound, Lemma 1, Lemma 2]. ###### Lemma 2.5. Let $\ell\geq 0$ be an even integer. The function $E_{\ell}(x)$ is positive, convex and satisfies $E_{\ell}(x)\geq e^{x}$ for $x\leq 0$. Moreover, for $x\leq\ell/e^{2}$, we have $e^{x}\leq\Big{(}1+\frac{e^{-\ell}}{16}\Big{)}E_{\ell}(x).$ ###### Lemma 2.6. Let $x_{1}$, $\ldots$, $x_{R}$ be real numbers and let $C=\exp((e^{-\ell_{1}}+\ldots+e^{-\ell_{R}})/16)$, where $\ell_{1}$, $\ldots,\ell_{R}$ are positive even integers. Then we have for any $y\geq 0$ and $0\leq k\leq 1$, $\displaystyle y^{k}\leq$ $\displaystyle Cky\prod_{j=1}^{R}E_{\ell_{j}}((k-1)x_{j})+C(1-k)\prod_{j=1}^{R}E_{\ell_{j}}(kx_{j})$ $\displaystyle+\sum_{r=0}^{R-1}\Big{(}Cky\prod_{j=1}^{r}E_{\ell_{j}}((k-1)x_{j})+C(1-k)\prod_{j=1}^{r}E_{\ell_{j}}(kx_{j})\Big{)}\Big{(}\frac{e^{2}x_{r+1}}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}.$ ### 2.7. Initial treatment of the proof of Theorem 1.1 Let $X$ be a large number and $\\{\ell_{j}\\}_{1\leq j\leq R}$ be a sequence of even natural numbers defined by $\ell_{1}=2\lceil 100\log\log X\rceil$ and $\ell_{j+1}=2\lceil 100\log\ell_{j}\rceil$ for $j\geq 1$. Here we choose $R$ to be the largest natural number such that $\ell_{R}>10^{4}$ and observe that we have $\ell_{j}>\ell_{j+1}^{2}$ for all $1\leq j\leq R-1$. Let ${P}_{1}$ be the set of odd primes below $X^{1/\ell_{1}^{2}}$. For $2\leq j\leq R$, we define ${P_{j}}$ to be the set of primes lying in the interval $(X^{1/\ell_{j-1}^{2}},X^{1/\ell_{j}^{2}}]$. We also define ${\mathcal{P}}_{j}(d)=\sum_{p\in P_{j}}\frac{1}{\sqrt{p}}\chi_{8d}(p).$ Given a real number $\alpha$, we denote (2.2) $\displaystyle{\mathcal{N}}_{j}(d,\alpha)=E_{\ell_{j}}(\alpha{\mathcal{P}}_{j}(d)),\quad\mathcal{N}(d,\alpha)=\prod_{j=1}^{R}{\mathcal{N}}_{j}(d,\alpha).$ We further set for two real numbers $n,k$ satisfying $0\leq k\leq 1$, ${\mathcal{A}}_{j}(d)={\mathcal{N}}_{j}(d,(k-1)n),\quad{\mathcal{B}}_{j}(d)={\mathcal{N}}_{j}(d,nk).$ We apply Lemma 2.6 with $y=\|L(\frac{1}{2},\chi_{8d})\|^{n}(\log d)^{-\frac{n}{2}}$ and $x_{j}=n{\mathcal{P}}_{j}(d)$ to obtain the following bounds for the moments of $L(\frac{1}{2},\chi_{8d})$. ###### Proposition 2.8. With notations as above, we have (2.3) $\displaystyle\begin{split}\Big{(}\|L(\frac{1}{2},\chi_{8d})\|^{n}(\log d)^{-\frac{n}{2}}\Big{)}^{k}\leq&Ck\|L(\frac{1}{2},\chi_{8d})\|^{n}(\log d)^{-\frac{n}{2}}\Big{(}\prod_{j=1}^{R}{\mathcal{A}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\\\ &+C(1-k)\Big{(}\prod_{j=1}^{R}{\mathcal{B}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{B}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}.\end{split}$ Arguing similar to [Radziwill&Sound, Section 3.4], we see that in order to prove Theorem 1.1, it suffices to show that the right side of (2.3) averaged over $d$ is $\ll X(\log X)^{\frac{(nk)^{2}}{2}}$. We close this section by giving such an estimation for the terms involving with ${\mathcal{B}}_{j}(d)$. ###### Proposition 2.9. With notations as above, we have $\sideset{}{{}^{}}{\sum}_{(d,2)=1}\Big{(}\prod_{j=1}^{R}{\mathcal{B}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{B}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\Phi\Big{(}\frac{d}{X}\Big{)}\ll X(\log X)^{\frac{(nk)^{2}}{2}}.$ ###### Proof. Let $w(n)$ be the multiplicative function defined by $w(p^{\alpha})=\alpha!$ for prime powers $p^{\alpha}$ and let $\Omega(n)$ denote the number of distinct prime powers dividing $n$. We also define functions $b_{j}(n),p_{j}(n)$ for $1\leq j\leq R$ such that $b_{j}(n),p_{j}(n)=0$ or $1$, and we have $b_{j}(n)=1$ ($p_{j}(n)=1$) if and only if $n$ is composed of at most (exactly, counted with multiplicity) $\ell_{j}$ primes, all from the interval $P_{j}$ . Using these notations, we see that (2.4) ${\mathcal{B}}_{j}(d)=\sum_{n_{j}}\frac{1}{\sqrt{n_{j}}}\frac{(nk)^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\chi_{8d}(n_{j}),\quad\frac{1}{\ell_{j}!}{\mathcal{P}}_{j}(d)^{\ell_{j}}=\sum_{n_{j}}\frac{1}{w(n_{j})\sqrt{n_{j}}}p_{j}(n_{j})\chi_{8d}(n_{j}),\quad 1\leq j\leq R.$ We note here that both ${\mathcal{B}}_{j}(d)$ and ${\mathcal{P}}_{j}(d)^{\ell_{j}}$ are short Dirichlet polynomials since $b_{j}(n_{j}),p_{j}(n_{j})=0$ unless $n_{j}\leq(X^{1/\ell_{j}^{2}})^{\ell_{j}}=X^{1/\ell_{j}}$. It follows that the expressions $\prod_{j=1}^{R}{\mathcal{B}}_{j}(d),\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)$ are all short Dirichlet polynomials of length at most $X^{1/\ell_{1}+\ldots+1/\ell_{R}}<X^{1/1000}$. We expand the term $\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)$ for some $0\leq r\leq R-1$ using (2.4) and apply Lemma 2.3 to estimate it. By doing so, we may ignore the error term in Lemma 2.3 as both ${\mathcal{B}}_{j}$ and ${\mathcal{P}_{j}}^{\ell_{j}}(d)$ are short Dirichlet polynomials. Considering the main term contributions from Lemma 2.3, we see that $\displaystyle\prod_{j=1}^{r}{\mathcal{B}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}\ll X$ $\displaystyle\prod_{j=1}^{r}\Big{(}\sum_{n_{j}=\square}\frac{1}{\sqrt{n_{j}}}\frac{(nk)^{\Omega(n_{j})}}{w(n_{j})}\prod_{p\|n_{j}}\Big{(}\frac{p}{p+1}\Big{)}b_{j}(n_{j})\Big{)}$ $\displaystyle\times\Big{(}\ell_{r+1}!\sum_{n_{r+1}=\square}\frac{1}{w(n_{r+1})\sqrt{n_{r+1}}}\prod_{p\|n_{r+1}}\Big{(}\frac{p}{p+1}\Big{)}p_{r+1}(n_{r+1})\Big{)}.$ The proof of the proposition now follows by arguing in the same way as in the proof of Proposition 4 in [Radziwill&Sound]. ∎ ## 3\. Proof of Theorem 1.1 In view of our discussions in the previous section, it remains to show that the right side of (2.3) averaged over $d$ for the terms involving with ${\mathcal{A}}_{j}(d)$ is also $\ll X(\log X)^{\frac{(nk)^{2}}{2}}$ for $n=2$. As our approach here may be applied to treat other values of $n$, we shall retain the symbol $n$ in most of the places in the rest of the section instead of specitying it to be $2$. Thus, to conclude the proof of Theorem 1.1, it suffices to establish the following result. ###### Proposition 3.1. With notations as above, we have for $n=2$, (3.1) $\displaystyle\sideset{}{{}^{}}{\sum}_{(d,2)=1}\|L(\frac{1}{2},\chi_{8d})\|^{n}\Big{(}\prod_{j=1}^{R}{\mathcal{A}}_{j}(d)+\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Big{)}\Phi\Big{(}\frac{d}{X}\Big{)}\ll X(\log X)^{\frac{(nk)^{2}+n}{2}}.$ In the remaining of this section, we give a proof of Proposition 3.1. First note that we may replace $\|L(\frac{1}{2},\chi_{8d})\|^{n}$ by $L(\frac{1}{2},\chi_{8d})^{n}$ when $n=2$ since $L(\frac{1}{2},\chi_{8d})$ is real. As the arugments are similar, it suffices to show that (3.2) $\displaystyle\sideset{}{{}^{}}{\sum}_{(d,2)=1}$ $\displaystyle L(\tfrac{1}{2},\chi_{8d})^{n}\sum_{r=0}^{R-1}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d)\Big{(}\frac{e^{2}n{\mathcal{P}}_{r+1}(d)}{\ell_{r+1}}\Big{)}^{\ell_{r+1}}\Phi\Big{(}\frac{d}{X}\Big{)}\ll X(\log X)^{\frac{(nk)^{2}+n}{2}}.$ Note first that we have ${\mathcal{A}}_{j}(d)=\sum_{n_{j}}\frac{1}{\sqrt{n_{j}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\chi_{8d}(n_{j}),\quad 1\leq j\leq R.$ Analogue to our discussions above, the product $\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{{r+1}}^{\ell_{r+1}}(d)$ for all $0\leq r\leq R-1$ are short Dirichlet polynomials of length at most $X^{1/1000}$. We now apply Lemma 2.4 to evaluate $\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}$(d) for some $0\leq r\leq R-1$ by expanding it into Dirichlet series. Once again we may focus only on the main term to see that, upon writing $n_{j}=(n_{j})_{1}(n_{j})_{2}^{2}$ with $(n_{j})_{1}$ being square-free, (3.3) $\displaystyle\begin{split}&\sideset{}{{}^{}}{\sum}_{(d,2)=1}L(\tfrac{1}{2},\chi_{8d})^{n}\prod_{j=1}^{r}{\mathcal{A}}_{j}(d){\mathcal{P}}_{r+1}^{\ell_{r+1}}(d)\Phi\Big{(}\frac{d}{X}\Big{)}\\\ \ll&X\sum_{n_{1},\cdots,n_{r+1}}\Big{(}\prod_{j=1}^{r}\frac{1}{\sqrt{n_{j}(n_{j})_{1}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\frac{d((n_{j})_{1})(n_{j})_{1}}{\sigma((n_{j})_{1})h(n_{j})}\Big{)}\Big{(}\frac{\ell_{r+1}!}{\sqrt{n_{r+1}(n_{r+1})_{1}}}\frac{p_{r+1}(n_{r+1})}{w(n_{r+1})}\frac{d((n_{r+1})_{1})(n_{r+1})_{1}}{\sigma((n_{r+1})_{1})h(n_{r+1})}\Big{)}\\\ &\times\Big{(}\log^{3}\Big{(}\frac{X}{(n_{1})_{1}\cdots(n_{r+1})_{1}}\Big{)}-3\sum_{\begin{subarray}{c}p\|(n_{1})_{1}\cdots(n_{r+1})_{1}\end{subarray}}\log^{2}p\log\Big{(}\frac{X}{(n_{1})_{1}\cdots(n_{r+1})_{1}}\Big{)}+O(n_{1}\cdots n_{r+1})\Big{)}.\end{split}$ As the estimations are similar, we may consider the above sums involving with the terms $\log^{3}(X/((n_{1})_{1}\cdots(n_{r+1})_{1}))=(\log X-\log((n_{1})_{1}\cdots(n_{r+1})_{1}))^{3}$ only. Upon expanding, we observe that we can write $\log^{3}(X/((n_{1})_{1}\cdots(n_{r+1})_{1}))$ as linear combinations of the sum: (3.4) $\displaystyle C(m_{0},\ldots,m_{r+1})(\log X)^{m_{0}}\sum_{\begin{subarray}{c}p_{i}\|(n_{i})_{1}\\\ 1\leq i\leq r+1\end{subarray}}\prod_{1\leq i\leq r+1}(\log p_{i})^{m_{i}},$ where $m_{j},0\leq j\leq r+1$ are non-negative integers satisfying $\sum_{0\leq j\leq r+1}m_{j}=3$ and where $C(m_{0},\ldots,m_{r+1})$ are bounded constants. Without loss of generality, we may group terms to consider the total contribution to (3.3) from all terms of the above form corresponding to $m_{i_{1}}=m_{i_{2}}=m_{i_{3}}=1$ for some $1\leq i_{1}<i_{2}<i_{3}\leq r+1$. For example, when the corresponding $p_{i_{1}}\in P_{1},p_{i_{2}}\in P_{2},p_{i_{3}}\in P_{r+1}$, the contribution is (3.5) $\displaystyle\begin{split}\ll&\sum_{l_{1},l_{2},l_{3}\geq 0}\prod^{3}_{s=1}\Big{(}\frac{\log p_{i_{s}}}{p^{l_{s}+1}_{i_{s}}}\frac{\|(n(k-1))\|^{2l_{s}+1}}{(2l_{s}+1)!}\frac{np_{i_{s}}}{(p_{i_{s}}+1)h(p^{2l_{s}+1}_{i_{s}})}\Big{)}\\\ &\times\prod_{j=1}^{r}\Big{(}\sum_{(n_{j},p_{i_{1}}p_{i_{2}})=1}\frac{1}{\sqrt{n_{j}(n_{j})_{1}}}\frac{(n(k-1))^{\Omega(n_{j})}}{w(n_{j})}\widetilde{b}_{j,l_{1},l_{2}}(n_{j})\frac{d((n_{j})_{1})(n_{j})_{1}}{\sigma((n_{j})_{1})h(n_{j})}\Big{)}\\\ &\times\Big{(}\ell_{r+1}!\sum_{(n_{r+1},p_{i_{3}})=1}\frac{1}{\sqrt{n_{r+1}(n_{r+1})_{1}}}\frac{p_{r+1}(n_{r+1}p^{2l_{3}+1}_{i_{3}})}{w(n_{r+1})}\frac{d((n_{r+1})_{1})(n_{r+1})_{1}}{\sigma((n_{r+1})_{1})h(n_{r+1})}\Big{)},\end{split}$ where we define $\widetilde{b}_{j,l_{1},l_{2}}(n_{j})=b_{j}(n_{j}p^{l_{j}}_{i_{j}})$ for $j=1,2$ and $\widetilde{b}_{j,l_{1},l_{2}}(n_{j})=b_{j}(n_{j})$ otherwise. Let us consider the sum over $n_{1}$ in (3.5). If we replace the factor $\widetilde{b}_{l,l_{1},l_{2}}(n_{1})$ by $1$, then the sum becomes (3.6) $\displaystyle\begin{split}&\prod_{\begin{subarray}{c}p\in P_{1}\\\ (p,p_{i_{1}})=1\end{subarray}}\Big{(}\sum_{j=0}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}}{(2j)!h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{(n(k-1))^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}\\\ \ll&\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\ (p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in P_{1}\end{subarray}}\frac{1}{p}\Big{)},\end{split}$ where for some constant $A$ independent of $p$, $\displaystyle\begin{split}C(p)=&\exp(\frac{A}{p^{2}})\Big{(}1-\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\frac{1}{p}\Big{)}\Big{(}\sum_{j=0}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}}{(2j)!h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{(n(k-1))^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}.\end{split}$ Note here that $C(p)$ is well-defined as one checks readily checked that each factor in the above product is positive for $n=2,0\leq k\leq 1$ and $p\geq 3$. Meanwhile, we note that the left side of (3.6) is also (3.7) $\displaystyle\begin{split}\gg\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in P_{1}\end{subarray}}\frac{1}{p}\Big{)}.\end{split}$ On the other hand, using Rankin’s trick by noticing that $2^{\Omega(n_{1})-\ell_{1}}\geq 1$ if $\Omega(n_{1})>\ell_{1}$, we see that the error introduced by replacing $\widetilde{b}_{l,l_{1},l_{2}}(n_{1})$ with $1$ does not exceed (3.8) $\displaystyle\begin{split}&\sum_{n_{1}}\frac{1}{\sqrt{n_{1}(n_{1})_{1}}}\frac{\|n(k-1)\|^{\Omega(n_{1})}}{w(n_{1})}2^{\Omega(n_{1})-\ell_{1}}\frac{d((n_{1})_{1})(n_{1})_{1}}{\sigma((n_{1})_{1})h(n_{1})}\\\ \leq&2^{-\ell_{1}}\prod_{\begin{subarray}{c}p\in P_{1}\\\ (p,p_{i_{1}})=1\end{subarray}}\Big{(}1+\sum_{j=1}^{\infty}\frac{1}{p^{j}}\frac{(n(k-1))^{2j}2^{2j}}{(2j)!}\frac{1}{h(p^{2j})}+\sum_{j=0}^{\infty}\frac{1}{p^{j+1}}\frac{\|n(k-1)\|^{2j+1}2^{2j+1}}{(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}\\\ \ll&2^{-\ell_{1}}\exp\Big{(}\big{(}2(n(k-1))^{2}+2n^{2}(1-k)\big{)}\sum_{p\in P_{1}}\frac{1}{p}\Big{)}.\end{split}$ We deduce from (3.6), (3.7) and (3.8) that the error term is (3.9) $\displaystyle\ll$ $\displaystyle 2^{-\ell_{1}}\exp\Big{(}\big{(}\frac{3}{2}(n(k-1))^{2}+3n^{2}(1-k)\big{)}\sum_{p\in P_{1}}\frac{1}{p}\Big{)}\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\ (p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in P_{1}\end{subarray}}\frac{1}{p}\Big{)}.$ Note that Lemma 2.2 implies that for all $1\leq j\leq R$, we have $\sum_{p\in P_{j}}1/p\leq 2\log\ell_{j-1}\leq\ell_{j}/36$ from our definition on $\ell_{j}$. We obtain from this and (3.9) that for $n=2$, we have (3.10) $\displaystyle\begin{split}&\sum_{(n_{1},p_{i_{1}})=1}\frac{1}{\sqrt{n_{1}(n_{1})_{1}}}\frac{(n(k-1))^{\Omega(n_{1})}}{w(n_{1})}\widetilde{b}_{1,l_{1},l_{2}}(n_{1})\frac{d((n_{1})_{1})(n_{1})_{1}}{\sigma((n_{1})_{1})h(n_{1})}\\\ \ll&(1+O(2^{-\ell_{1}/2}))\Big{(}\prod_{\begin{subarray}{c}p\in P_{1}\\\ (p,p_{i_{1}})=1\end{subarray}}C(p)\Big{)}\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in P_{1}\end{subarray}}\frac{1}{p}\Big{)}.\end{split}$ We may also establish similar estimations for sums over $n_{j},2\leq j\leq r$. Next, we apply Rankin’s trick again to see that the contribution of the $n_{r+1}$ terms in (3.5) is $\leq\ell_{r+1}!10^{-\ell_{r+1}}\prod_{p\in P_{r+1}}\Big{(}\sum_{j=0}^{\infty}\frac{10^{2j}}{p^{j}(2j)!h(p^{2j})}+\sum_{j=1}^{\infty}\frac{10^{2j+1}}{p^{j+1}(2j+1)!}\frac{np}{(p+1)h(p^{2j+1})}\Big{)}.$ We apply Lemma 2.2, the estimation $\ell_{r+1}!\leq\ell_{r+1}(\ell_{r+1}/e)^{\ell_{r+1}}$ and the definition of $\ell_{r+1}$ to see that the above is $\ll\ell_{r+1}\Big{(}\frac{\ell_{r+1}}{10e}\Big{)}^{\ell_{r+1}}\exp\Big{(}70\sum_{p\in P_{r+1}}\frac{1}{p}\Big{)}\ll\ell_{r+1}\Big{(}\frac{\ell_{r+1}}{10e}\Big{)}^{\ell_{r+1}}\exp(\tfrac{5}{7}\ell_{r+1}).$ Combining the this with (3.5) and (3.10), we conclude that the contribution from all terms of the forms given in (3.4) corresponding to $m_{i_{1}}=m_{i_{2}}=m_{i_{3}}=1$ for some $1\leq i_{1}<i_{2}<i_{3}\leq r+1$ to the left side of (3.2) is $\displaystyle\ll$ $\displaystyle{X}e^{-\ell_{r+1}/7}\prod_{1\leq j\leq r}\Big{(}1+O(2^{-\ell_{j}/2})\Big{)}\prod_{\begin{subarray}{c}p\in\bigcup_{j=1}^{r}P_{j}\end{subarray}}C(p)\times\exp\Big{(}\big{(}\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)\big{)}\sum_{\begin{subarray}{c}p\in\bigcup_{j=1}^{r}P_{j}\end{subarray}}\frac{1}{p}\Big{)}$ $\displaystyle\times\Big{(}\sum_{p\in\bigcup_{j=1}^{r}P_{j}}\sum_{l\geq 0}\frac{\log p}{p^{l+1}}\frac{\|(n(k-1))\|^{2l+1}}{(2l+1)!}\frac{np}{(p+1)h(p^{2l+1})}\Big{)}^{3}$ $\displaystyle\ll$ $\displaystyle e^{-\ell_{r+1}/7}X(\log X)^{\frac{(n(k-1))^{2}}{2}+n^{2}(k-1)+3},$ by using Lemma 2.2 while noting that $\displaystyle\prod_{p}C(p)\ll 1$ and $\displaystyle\prod_{1\leq j\leq r}\Big{(}1+O(2^{-\ell_{j}/2})\Big{)}\ll 1$ since $\ell_{j}>\ell_{j+1}^{2}$. Summing over $r$, we deduce that the estimation in (3.2) is valid and this completes the proof of the proposition. ## 4\. A further remark In this section we describe a variant on the principle of Radziwiłł and Soundararajan mentioned in Section 1 for obtaining upper bounds of moments of $L$-functions. This is based on the following simple observation. ###### Lemma 4.1. Let $\ell$ be a non-negative even integer. For all real number $x$, let $E_{\ell}(x)$ be defined in (2.1). We have $E_{\ell}(x)E_{\ell}(-x)\geq 1.$ ###### Proof. We expand out the product $f_{\ell}(x):=E_{\ell}(x)E_{\ell}(-x)-1$ as a polynomial to see that it suffices to show that the coefficients of $x^{i},0\leq j\leq 2\ell$ are all non-negative. Note that the function $f_{\ell}(x)$ is even so that only even powers of $x$ appear in the expansion. Furthermore, it follows from the binomial theorem that the only non-zero even powers of $x$ involved are those of the form $x^{2j}$ for $2j>l$. We then set $\ell=2k$ to see that the coefficient of $x^{2(k+j)}$ for some integer $j>0$ is given by $\frac{1}{(2(k+j))!}\sum^{2k}_{i=2j}\binom{2(k+j)}{i}(-1)^{i}=-\frac{2}{(2(k+j))!}\sum^{2j-1}_{i=0}\binom{2(k+j)}{i}(-1)^{i}\geq 0,$ where the last inequality above follows from [FI10, (6.6)] and this completes the proof. ∎ Now, for two real numbers $n,k$ with $0<k<1$, we apply Lemma 4.1 to see that $\displaystyle\sideset{}{{}^{}}{\sum}_{(d,2)=1}\|L(\tfrac{1}{2},\chi_{8d})\|^{nk}\Phi(\frac{d}{X})\leq\sideset{}{{}^{}}{\sum}_{(d,2)=1}\|L(\tfrac{1}{2},\chi_{8d})\|^{nk}{\mathcal{N}}(d,n(k-1))^{k}{\mathcal{N}}(d,n(1-k))^{k}\Phi(\frac{d}{X}),$ where we recall that the definition of ${\mathcal{N}}(d,\alpha)$ is given in (2.2). We further apply Hölder’s inequality to the right side expression above to deduce that $\displaystyle\sideset{}{{}^{}}{\sum}_{(d,2)=1}\|L(\tfrac{1}{2},\chi_{8d})\|^{nk}\Phi(\frac{d}{X})\leq\Big{(}\sideset{}{{}^{}}{\sum}_{(d,2)=1}\|L(\tfrac{1}{2},\chi_{8d})\|^{n}{\mathcal{N}}(d,n(k-1))\Phi(\frac{d}{X})\Big{)}^{k}\Big{(}\sideset{}{{}^{}}{\sum}_{(d,2)=1}{\mathcal{N}}(d,n(1-k))^{k/(1-k)}\Phi(\frac{d}{X})\Big{)}^{1-k}.$ This approach can be applied to obtain upper bounds for the $k$-th moment of $L$-functions. In particalur, it is convenient to study the $\tfrac{1}{2}$-th moment as it suffices to evaluate the average over $d$ of $\|L(\tfrac{1}{2},\chi_{8d})\|{\mathcal{N}}(d,-\tfrac{1}{2})$ and ${\mathcal{N}}(d,\tfrac{1}{2})$ by taking $n=1,k=1/2$ above. If we assume that $L(\tfrac{1}{2},\chi_{8d})\geq 0$ (which follows from GRH), then we just need to consider $L(\tfrac{1}{2},\chi_{8d}){\mathcal{N}}(d,-\tfrac{1}{2})$, which is relatively simpler compared to our work above. The same method applies equally to the study on the moments of quadratic twists of $L$-functions attached to elliptic curves (as we know in this case the corresponding $L$-functions have non-negative values at the central point). More generally, when $k=1/m$ with $m>2$ a positive integer, we denote $\\{L(s,f)\\}_{f\in\mathcal{F}}$ for a general family of $L$-functions, we may apply Lemma 4.1 (with a suitable adjustment on the defintion of ${\mathcal{N}}$) to see that $\displaystyle\sum_{f\in\mathcal{F}}\|L(\tfrac{1}{2},f)\|^{k}\leq\sum_{f\in\mathcal{F}}\|L(\tfrac{1}{2},f)\|^{k}{\mathcal{N}}(f,k-1)^{k}\Big{(}\prod^{m-2}_{i=1}\big{(}{\mathcal{N}}(f,1-ik){\mathcal{N}}(f,(i+1)k-1)\big{)}^{k}\Big{)}{\mathcal{N}}(f,1-(m-1)k)^{k}.$ Applying Hölder’s inequality $m-1$ times to the right side expression above, we deduce that $\displaystyle\sum_{f\in\mathcal{F}}\|L(\tfrac{1}{2},f)\|^{k}\leq$ $\displaystyle\Big{(}\sum_{f\in\mathcal{F}}\|L(\tfrac{1}{2},f)\|{\mathcal{N}}(f,k-1)\Big{)}^{k}\prod^{m-2}_{i=1}\Big{(}\sum_{f\in\mathcal{F}}{\mathcal{N}}(f,1-ik){\mathcal{N}}(f,(i+1)k-1)\Big{)}^{k}\Big{(}\sum_{f\in\mathcal{F}}{\mathcal{N}}(f,1-(m-1)k)\Big{)}^{k}.$ It follows that in order to obtain upper bounds for the $1/m$-th moment of the corresponding family, we only need to be able to evaluate the average over $f\in\mathcal{F}$ of $\|L(\tfrac{1}{2},f)\|{\mathcal{N}}(f,k-1)$ and quantities involving with products of at most two copies of ${\mathcal{N}}$. The above approach can also be adapted to treat moments of the Riemann zeta function on the critical line. We shall however not go any further in this direction here. Acknowledgments. P. G. is supported in part by NSFC grant 11871082. ## References
# Critical behaviors of the $O(4)$ and $Z(2)$ symmetries in the QCD phase diagram Yong-rui Chen School of Physics, Dalian University of Technology, Dalian, 116024, P.R. China Rui Wen School of Physics, Dalian University of Technology, Dalian, 116024, P.R. China Wei-jie Fu<EMAIL_ADDRESS>School of Physics, Dalian University of Technology, Dalian, 116024, P.R. China ###### Abstract In this work we have studied the QCD phase structure and critical dynamics related to the 3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes in the two-flavor quark-meson low energy effective theory within the functional renormalization group approach. We have employed the expansion of Chebyshev polynomials to solve the flow equation for the order-parameter potential. The chiral phase transition line of $O(4)$ symmetry in the chiral limit, and the $Z(2)$ line of critical end points related to the explicit chiral symmetry breaking are depicted in the phase diagram. Various critical exponents related to the order parameter, chiral susceptibilities and correlation lengths have been calculated for the 3-$d$ $O(4)$ and $Z(2)$ universality classes in the phase diagram, respectively. We find that the critical exponents obtained in the computation, where a field-dependent mesonic nontrivial dispersion relation is taken into account, are in quantitative agreement with results from other approaches, e.g., the conformal bootstrap, Monte Carlo simulations and $d=3$ perturbation expansion, etc. Moreover, the size of the critical regime in the QCD phase diagram is found to be very small. ††preprint: ## I Introduction Significant progress has been made in studies of QCD phase structure over the last decade, both from the experimental and theoretical sides; see, e.g. Stephanov (2006); Friman _et al._ (2011); Luo and Xu (2017); Andronic _et al._ (2018); Fischer (2019); Bzdak _et al._ (2020); Fu _et al._ (2020); Bazavov _et al._ (2020); Borsanyi _et al._ (2020); Fu _et al._ (2021). One of the most prominent features of the QCD phase structure is the probable presence of a second order critical end point (CEP) in the phase diagram spanned by the temperature $T$ and baryon chemical potential $\mu_{B}$ or densities, which separates the first order phase transition at high $\mu_{B}$ from the continuous crossover at low $\mu_{B}$ Stephanov (2006). The existence and location of CEP are, however, still open questions, whose answers would definitely help us to unravel the most mysterious veil related to the properties of strongly interacting matter under extreme conditions. The Beam Energy Scan (BES) Program at the Relativistic Heavy Ion Collider (RHIC) is aimed at searching for and locating the critical end point, where fluctuation observables sensitive to the critical dynamics, e.g., high-order cumulants of net-proton, net-charge, net-kaon multiplicity distributions, have been measured Adamczyk _et al._ (2014a, b); Luo (2015); Adamczyk _et al._ (2018). Notably, a non-monotonic dependence of the kurtosis of the net-proton multiplicity distribution on the beam energy with $3.1\sigma$ significance in central collisions has been reported by the STAR collaboration recently Adam _et al._ (2020). On the other hand, lattice QCD simulations have provided us with a plethora of knowledge about the QCD phase structure, e.g., the crossover nature of the chiral phase transition at finite $T$ and vanishing $\mu_{B}$ with physical current quark mass Aoki _et al._ (2006), pseudo-critical temperature Borsanyi _et al._ (2014); Bazavov _et al._ (2014), curvature of the phase boundary Bazavov _et al._ (2019); Borsanyi _et al._ (2020), etc. Because of the notorious sign problem at finite chemical potential, the reliability regime of lattice calculations is restricted to be $\mu_{B}/T\lesssim 2\sim 3$, where no CEP has been found. Free from the sign problem, the first-principle functional approaches, e.g, the functional renormalization group (fRG) and Dyson- Schwinger equations (DSE), could potentially extend the regime of reliability to $\mu_{B}/T\sim 4$ Fischer (2019); Fu _et al._ (2020). With benchmark tests of observables at finite $T$ and low $\mu_{B}$ in comparison to lattice calculations, e.g., the quark condensate, curvature of the phase boundary, etc., functional approaches, both fRG and DSE, have predicted a CEP located in a region of $450\,\mathrm{MeV}\lesssim\mu_{B}\lesssim 650\,\mathrm{MeV}$ Fischer (2019); Fu _et al._ (2020); Isserstedt _et al._ (2019); Gao and Pawlowski (2020a, b) recently. An alternative method used to circumvent the possible location of CEP, is to determine the critical temperature $T_{c}$ of the chiral phase transition in the chiral limit, more specifically, i.e., massless light up and down quarks and a physical strange quark mass. Since it is believed that the value of $T_{c}$ sets an upper bound for the temperature of CEP Halasz _et al._ (1998); Buballa and Carignano (2019). Very recently, the critical temperature $T_{c}$ in the chiral limit has been investigated and its value is extrapolated from both lattice simulations Ding _et al._ (2019) and functional approach Braun _et al._ (2020). Moreover, further lattice calculations indicate that axial anomaly remains manifested at $T\approx 1.6\,T_{c}$, which implies that the chiral phase transition of QCD in the chiral limit is of 3-$d$ $O(4)$ universality class Ding _et al._ (2020); see, e.g., Pisarski and Wilczek (1984) for more discussions about the relation between the axial anomaly and the symmetry universality classes. In this work, we would like to study the QCD phase structure in the chiral limit and finite current quark mass, i.e., with a finite pion mass, in the two-flavor quark-meson low energy effective theory (LEFT) within the fRG approach. For more discussions about the fRG approach, see, e.g., QCD related reviews Berges _et al._ (2002); Pawlowski (2007); Schaefer and Wambach (2008); Gies (2012); Rosten (2012); Braun (2012); Pawlowski (2014); Dupuis _et al._ (2020). In contrast with the lattice simulation and the first- principle fRG-QCD calculation Ding _et al._ (2019); Braun _et al._ (2020), the chiral limit could be accessed strictly in the LEFT. Furthermore, we would also like to study the critical behaviors of the 3-$d$ $O(4)$ and $Z(2)$ universality classes, including various critical exponents, which belong to the second-order chiral phase transitions in the chiral limit and at the critical end point with finite quark mass, respectively. To that end, we expand the effective potential of order parameter as a sum of Chebyshev polynomials in the computation of fRG flow equations; see Risch (2013) for more details. The Chebyshev expansion of solutions to a set of integrodifferential equations is, in fact, a specific formalism of more generic pseudo-spectral methods Boyd (2000), and see also, e.g., Borchardt and Knorr (2015, 2016); Knorr (2020) for applications of pseudo-spectral methods in the fRG. In fact, another two numerical methods are more commonly used in solving the flow equation for the effective potential: one is the Taylor expansion of the effective potential around some value Pawlowski and Rennecke (2014); Yin _et al._ (2019), and the other discretization of the effective potential on a grid Schaefer and Wambach (2005). The (dis)advantages of these two methods are distinct. The former is liable to implementation of the numerical calculations, but short of global properties of the effective potential, that is, however, indispensable to studies of chiral phase transition in the chiral limit or around CEP; the latter is encoded with global information on the potential, but it loses numerical accuracy near the phase transition point which is necessary especially for the computation of critical exponents. The Chebyshev expansion used in this work combines the merits from both approaches, i.e., the global potential and the numerical accuracy, and thus it is very suitable for the studies of critical behaviors in the QCD phase diagram. Remarkably, a discontinuous Galerkin scheme has been applied in the context of fRG recently Grossi and Wink (2019), which is well-suited for studies of the first-order phase transition. This paper is organized as follows: In Sec. II we briefly introduce the flow equations in the quark-meson LEFT and the method of the Chebyshev expansion for the effective potential. The obtained phase diagram and QCD phase structure are presented and discussed in Sec. III. In Sec. IV scaling analyses for the the 3-$d$ $O(4)$ and $Z(2)$ universality classes are performed, and various critical exponents are obtained. We also discuss the size of the critical regime there. In Sec. V we give a summary and conclusion. Some threshold functions and anomalous dimension in the flow equations, and some relations for the Chebyshev polynomials are collected in Appendix A and Appendix B, respectively. ## II Functional renormalization group and the low energy effective theories Thanks to the Wilson’s idea of the renormalization group (RG), see, e.g., Wilson and Kogut (1974), it has been well known that usually the active degrees of freedom are quite different, when the energy scale of a system evolves from a hierarchy into another. The relevant dynamics in different hierarchies are connected with each other through the evolution of RG equations. To be more specific, in QCD the partonic degrees of freedom, i.e., the quarks and gluons, in the high energy perturbative regime are transformed into the collective hadronic ones in the nonperturbative region of low energy, with the RG scale evolving from the ultraviolet (UV) to infrared (IR) limits Weinberg (1979), and see also, e.g., Gies and Wetterich (2002, 2004); Pawlowski (2007); Floerchinger and Wetterich (2009); Braun _et al._ (2016); Mitter _et al._ (2015); Cyrol _et al._ (2018a); Eser _et al._ (2018); Fu _et al._ (2020) for recent development of the relevant ideas within the fRG approach. When the momentum or RG scale is below, say $\sim 1$ GeV, which is related to a narrow transition region from the perturbative to nonperturbative QCD, calculated results of Yang-Mills theory and QCD in Landau gauge indicate that the gluons develop a finite mass gap and decouple from the system, and see, e.g. Mitter _et al._ (2015); Cyrol _et al._ (2016); Fu _et al._ (2020); Huber (2020) for more details. As a consequence, contributions to the flow equations of effective action from the glue sector could be safely neglected, if the initial evolution scale is set at a UV scale $\Lambda\lesssim 1$ GeV. Hence, within the fRG approach, one is left with the flow equation for the low energy effective theory, which reads $\displaystyle\partial_{t}\Gamma_{k}[\Phi]=$ $\displaystyle-\mathrm{Tr}\Big{(}G_{q\bar{q},k}\partial_{t}R_{q,k}\Big{)}+\frac{1}{2}\mathrm{Tr}\Big{(}G_{\phi\phi,k}\partial_{t}R_{\phi,k}\Big{)}\,,$ (1) with the RG scale $k$ and the RG time defined as $t=\ln(k/\Lambda)$. Apparently, Eq. (1) is an ordinary differential equation for the $k$-dependent effective action, $\Gamma_{k}[\Phi]$, the arguments $\Phi=(q,\bar{q},\phi)$ of which are the quark and mesonic fields in the LEFT. The equation in Eq. (1), which describes the evolution of the effective action with the RG scale, is also well known as the Wetterich equation Wetterich (1993), see also Ellwanger (1994); Morris (1994). The flow receives contributions from both the quark and mesonic degrees of freedom, as shown on the r.h.s. of Eq. (1), where $G_{q\bar{q},k}$ and $G_{\phi\phi,k}$ are the $k$-dependent full quark and meson propagators, respectively, and are related to the quadratic derivatives of $\Gamma_{k}[\Phi]$ with respect to their respective fields, viz. $\displaystyle G_{\phi\phi/q\bar{q}}[\Phi]=\left(\frac{1}{\frac{\delta^{2}\Gamma_{k}[\Phi]}{\delta\Phi^{2}}+R_{\Phi,k}}\right)_{\phi\phi/q\bar{q}}\,.$ (2) where $R_{q,k}$ and $R_{\phi,k}$ as well as in Eq. (1) are the IR regulators, which are employed to suppress quantum fluctuations of momenta $q\lesssim k$, and their explicit expressions used in the work are given in Eqs. (60) and (61). Moreover, interested readers could refer to QCD related fRG review articles Berges _et al._ (2002); Pawlowski (2007); Schaefer and Wambach (2008); Gies (2012); Rosten (2012); Braun (2012); Pawlowski (2014); Dupuis _et al._ (2020) for more details about the formalism of fRG, and also Braun _et al._ (2010); Braun (2009); Braun _et al._ (2011a); Mitter _et al._ (2015); Braun _et al._ (2016); Cyrol _et al._ (2016, 2018a, 2018b); Fu _et al._ (2020); Braun _et al._ (2020); Fu _et al._ (2021) for recent progress on relevant studies. In this work, we adopt a truncation for the effective action in Eq. (1) as follows $\displaystyle\Gamma_{k}[\Phi]=$ $\displaystyle\int_{x}\bigg{\\{}Z_{q,k}\bar{q}\big{(}\gamma_{\mu}\partial_{\mu}-\gamma_{0}\hat{\mu}\big{)}q+\frac{1}{2}Z_{\phi,k}(\rho)\big{(}\partial_{\mu}\phi\big{)}^{2}$ $\displaystyle+h_{y,k}\bar{q}\big{(}T^{0}\sigma+i\gamma_{5}\vec{T}\cdot\vec{\pi}\big{)}q+V_{k}(\rho)-c\sigma\bigg{\\}}\,,$ (3) with the shorthand notation $\int_{x}=\int_{0}^{1/T}dx_{0}\int d^{3}x$, where the quark field $q=(u\,,d)^{T}$ and the meson field $\phi=\left(\sigma,\vec{\pi}\right)$ are in the fundamental and adjoint representations of $SU(N_{f})$ in the flavor space with $N_{f}=2$, respectively. They interact with each other via a Yukawa coupling with a coupling strength $h_{Y,k}$, where the subscript Y is used to distinguish it from the reduced external field $h$ in Eq. (15). Here $T^{i}$ ($i=1\,,2\,,3$) are the generators of $SU(2)$ with $\operatorname{Tr}(T^{i}T^{j})=\frac{1}{2}\delta^{ij}$ and $T^{0}=\frac{1}{\sqrt{2N_{f}}}\mathbb{1}_{N_{f}\times N_{f}}$. Note that both the effective potential $V_{k}(\rho)$ and the mesonic wave function renormalization $Z_{\phi,k}(\rho)$ in Eq. (3) depend on the meson field by means of $\rho=\phi^{2}/2$, which are $O(4)$ invariant. $Z_{q,k}$ is the quark wave function renormalization. Notice that the term linear in the order parameter field, i.e., $-c\sigma$ in Eq. (3), breaks the chiral symmetry explicitly, and thus here $c$ is essentially an external “magnetic” field in the language of magnetization. Moreover, $\hat{\mu}=\mathrm{diag}(\mu_{u},\mu_{d})$ is the matrix of quark chemical potentials in the flavor space, and $\mu=\mu_{u}=\mu_{d}$ is assumed throughout this work, which is related to the baryon chemical potential via $\mu=\mu_{B}/3$. For more discussions about the quark-meson LEFT in Eq. (3) or its extensions, e.g., Polyakov-loop quark-meson LEFT, QCD assisted LEFT, etc., and their applications in calculations of QCD thermodynamics and phase structure, fluctuations and correlations of conserved charges, etc., see, e.g., Schaefer and Wambach (2005); Schaefer _et al._ (2007); Skokov _et al._ (2010); Herbst _et al._ (2011); Skokov _et al._ (2011); Karsch _et al._ (2011); Morita _et al._ (2011); Skokov _et al._ (2012); Haas _et al._ (2013); Herbst _et al._ (2013, 2014); Fu and Pawlowski (2016, 2015); Fu _et al._ (2016); Sun _et al._ (2018); Fu _et al._ (2018, 2019); Wen _et al._ (2019); Wen and Fu (2019); Yin _et al._ (2019); Hansen _et al._ (2020); Fu _et al._ (2021). ### II.1 Flow equations Substituting the effective action in Eq. (3) into the Wetterich equation in Eq. (1), one readily obtains the flow equation of the effective potential as follows $\displaystyle\partial_{t}V_{k}(\rho)=$ $\displaystyle\frac{k^{4}}{4\pi^{2}}\bigg{[}\big{(}N^{2}_{f}-1\big{)}l^{(B,4)}_{0}(\bar{m}^{2}_{\pi,k},\eta_{\phi,k};T)$ $\displaystyle+l^{(B,4)}_{0}(\bar{m}^{2}_{\sigma,k},\eta_{\phi,k};T)$ $\displaystyle-4N_{c}N_{f}l^{(F,4)}_{0}(\bar{m}^{2}_{q,k},\eta_{q,k};T,\mu)\bigg{]}\,,$ (4) with the threshold functions $l^{(B,4)}_{0}$ and $l^{(F,4)}_{0}$ given in Eq. (64) and Eq. (65), respectively. Here, the scale-dependent meson and quark masses read $\displaystyle\bar{m}^{2}_{\pi,k}$ $\displaystyle=\frac{V^{\prime}_{k}(\rho)}{k^{2}Z_{\phi,k}}\,,\qquad\bar{m}^{2}_{\sigma,k}=\frac{V^{\prime}_{k}(\rho)+2\rho V^{\prime\prime}_{k}(\rho)}{k^{2}Z_{\phi,k}}\,,$ (5) $\displaystyle\bar{m}^{2}_{q,k}$ $\displaystyle=\frac{h_{y,k}^{2}\rho}{2k^{2}Z^{2}_{q,k}}\,,$ (6) which are RG invariant and dimensionless. The meson and quark anomalous dimensions in the threshold functions in Eq. (4) are defined as follows $\displaystyle\eta_{\phi,k}$ $\displaystyle=-\frac{\partial_{t}Z_{\phi,k}}{Z_{\phi,k}}\,,\quad\eta_{q,k}=-\frac{\partial_{t}Z_{q,k}}{Z_{q,k}}\,,$ (7) where the meson anomalous dimension is obtained by projecting the flow equation in Eq. (1) onto the inverse pion propagator, to wit, $\displaystyle\eta_{\phi,k}(\rho)$ $\displaystyle=-\frac{1}{3Z_{\phi,k}}\delta_{ij}\frac{\partial}{\partial(\|\bm{p}\|^{2})}\frac{\delta^{2}\partial_{t}\Gamma_{k}}{\delta\pi_{i}(-p)\delta\pi_{j}(p)}\Bigg{\|}_{\begin{subarray}{c}p_{0}=0\\\ \bm{p}=0\end{subarray}}\,,$ (8) the explicit expression of which is presented in Eq. (68). Note that $\eta_{\phi,k}$ is dependent on the meson field via $\rho$. In comparison to the effects of the meson wave function renormalization on the chiral phase transition at finite temperature and density, it has been found that those of quark wave function renormalization and the running Yukawa coupling are relatively milder, see, e.g., Pawlowski and Rennecke (2014); Fu and Pawlowski (2015); Yin _et al._ (2019). Therefore, in this work we adopt the simplification as follows $\displaystyle\eta_{q,k}$ $\displaystyle=0\,,\quad\quad\partial_{t}\bar{h}_{y,k}=0\,,$ (9) with the renormalized Yukawa coupling given in Eq. (69), and use two different truncations: one is the usual local potential approximation (LPA), where the mesonic anomalous dimension is vanishing as well, and the $k$-dependent term in Eq. (3) is just the effective potential; the other is the truncation with the field-dependent mesonic anomalous dimension in Eq. (8) taken into account besides the potential, which is denoted as LPA′ in this work. Note that the notation LPA′ in literatures, e.g., Helmboldt _et al._ (2015); Fu and Pawlowski (2015), usually stands for the truncation with a field-independent mesonic anomalous dimension which is, strictly speaking, different from the case in this work. Figure 1: Dependence of the mesonic wave function renormalization $Z_{\phi}$ on the order-parameter field $\bar{\sigma}$ at vanishing baryon chemical potential $\mu_{B}=0$ and several values of temperature $T=\Delta T+T_{c}$. See text for more details. As an illustrative example, we show the mesonic wave function renormalization $Z_{\phi}\equiv Z_{\phi,k=k_{\mathrm{IR}}}$ as a function of the renormalized sigma field $\bar{\sigma}=Z_{\phi}^{1/2}\sigma$ obtained in LPA′ in Fig. 1, where $k_{\mathrm{IR}}$ is the RG scale in the IR limit, and one would has $k_{\mathrm{IR}}\rightarrow 0$ in principle, which, however, is impossible to realize in numerical calculations. In our calculation the value of $k_{\mathrm{IR}}$ is reduced as small as possible, and we find the convergence is obtained when $k_{\mathrm{IR}}=1$ MeV. Note that the mesonic wave function renormalization at the scale of UV cutoff $\Lambda$, see Sec. III in the following, is assumed to be identical to unity, i.e., $Z_{\phi,k=\Lambda}=1$. In Fig. 1, we choose several values of temperature $T=\Delta T+T_{c}$ at and above the critical temperature that is $T_{c}=143.6$ MeV in the chiral limit and at vanishing $\mu_{B}$. One observes that with the increase of the temperature, the peak structure of $Z_{\phi}$ as a function of the renormalized sigma field $\bar{\sigma}$ becomes smoother. ### II.2 Chebyshev expansion of the effective potential Figure 2: Phase diagrams in the plane of $T$ and $\mu_{B}$, obtained in the quark-meson low energy effective theory within the fRG approach. Two truncations for the fRG calculations have been employed: one is the local potential approximation (LPA) and the other is that beyond the LPA, in which a field-dependent mesonic wave function renormalization is taken into account, i.e., the truncation LPA′, and see text for more details. The relevant results are presented in the left and right panels, respectively. The black dashed lines in both panels denote the $O(4)$ chiral phase transition in the chiral limit, and the black circles indicate the location of the tricritical point. The solid lines of different colors in the left panel denote the first-order phase transitions with different pion masses in the vacuum, i.e. different values of $c$ in Eq. (3), and the solid one in the right panel is the first-order phase transition line in the chiral limit. The red dashed lines in both panels stand for line composed of critical end points (CEP) corresponding to continuously varying pion masses, which belong to the $Z(2)$ symmetry class. The star in the left panel indicates the location of CEP with physical pion mass. In both phase diagrams we use red and blue crosses to label the locations where critical exponents in Sec. IV are calculated for the $O(4)$ and $Z(2)$ universality classes, respectively. In this work we solve the flow equation in Eq. (4) by expanding the effective potential as a sum of Chebyshev polynomials up to an order $N_{v}$, to wit, $\displaystyle\bar{V}_{k}(\bar{\rho})$ $\displaystyle=\sum^{N_{v}}_{n=1}c_{n,k}T_{n}(\bar{\rho})+\frac{1}{2}c_{0,k}\,,$ (10) with $\bar{V}_{k}(\bar{\rho})=V_{k}(\rho)$, $\bar{\rho}=Z_{\phi,k}\rho$, where quantities with a bar denote renormalized variables. The Chebyshev polynomial $T_{n}(\bar{\rho})$ is given in Eq. (78), and the superscript $[0,\bar{\rho}_{\mathrm{max}}]$ in Eq. (78) denoting the interval of $\bar{\rho}$ is neglected for brevity here. Differentiating Eq. (10) with respect to the RG time $t$ with $\rho$ fixed, one is led to $\displaystyle\partial_{t}\big{\|}_{\rho}\bar{V}_{k}(\bar{\rho})=$ $\displaystyle\sum^{N_{v}}_{n=1}\Big{(}\partial_{t}c_{n,k}-d_{n,k}\eta_{\phi,k}(\bar{\rho})\bar{\rho}\Big{)}T_{n}(\bar{\rho})$ $\displaystyle+\frac{1}{2}\Big{(}\partial_{t}c_{0,k}-d_{0,k}\eta_{\phi,k}(\bar{\rho})\bar{\rho}\Big{)}\,,$ (11) where we have used the Chebyshev expansion for the derivative of the effective potential as shown in Eq. (82) and coefficients $d_{n,k}$’s are the respective expanding coefficients. Employing the discrete orthogonality relation in Eq. (76) by summing up the $N+1$ zeros of $T_{N+1}(\bar{\rho})$ in Eq. (79), one arrives at $\displaystyle\partial_{t}c_{m,k}=$ $\displaystyle\frac{2}{N+1}\sum^{N}_{i=0}\Big{(}\partial_{t}\big{\|}_{\rho}\bar{V}_{k}(\bar{\rho}_{i})\Big{)}T_{m}(\bar{\rho}_{i})$ $\displaystyle+\frac{2}{N+1}\sum^{N_{v}}_{n=1}\sum^{N}_{i=0}d_{n,k}T_{m}(\bar{\rho}_{i})T_{n}(\bar{\rho}_{i})\eta_{\phi,k}(\bar{\rho}_{i})\bar{\rho}_{i}$ $\displaystyle+\frac{1}{N+1}d_{0,k}\sum^{N}_{i=0}T_{m}(\bar{\rho}_{i})\eta_{\phi,k}(\bar{\rho}_{i})\bar{\rho}_{i}\,,$ (12) which is the flow equation for the expansion coefficients in Eq. (10). ## III Phase diagram It is left to specify the parameters in the LEFT, prior to presenting our calculated results. The UV cutoff of flow equations in the LEFT is chosen to be $\Lambda=500$ MeV, and the effective potential in Eq. (3) at $k=\Lambda$ reads $\displaystyle V_{\Lambda}(\rho)$ $\displaystyle=\frac{\lambda_{\Lambda}}{2}\rho^{2}+\nu_{\Lambda}\rho\,,$ (13) with $\lambda_{\Lambda}=20$ and $\nu_{\Lambda}=0$ . The Yukawa coupling is $k$-independent as shown in Eq. (9) and is given by $\bar{h}_{y}=6.4$. Concerning the Chebyshev expansion, we choose $N=81$ for the number of zeros and $N_{v}=21$ for the maximal order of Chebyshev polynomials. We have also checked that there is no difference when the value of $N_{v}$ is increased. Moreover, the upper bound of $\bar{\rho}$ is chosen to be $\bar{\rho}_{\mathrm{max}}=9\times 10^{3}\,\mathrm{MeV}^{2}$, well above the value of minimum of the potential in the IR. In the LPA, these values of parameters lead to the pion decay constant $f_{\pi}=$87 MeV and the constituent quark mass $m_{q}=$278.4 MeV in the vacuum and in the chiral limit. While if the explicit breaking strength of the chiral symmetry in Eq. (3) is increased up to $c=1.85\times 10^{-3}\,(\mathrm{GeV})^{3}$, one obtains the physical pion mass $m_{\pi}=$138 MeV, as well as $f_{\pi}=$93 MeV and $m_{q}=$297.6 MeV in the vacuum. Note that in order to facilitate the comparison between the calculation with the truncation LPA and that with LPA′, we use the same values of parameters above in the LPA′ computation as in LPA. In Fig. 2 we show the phase diagrams of LEFT in the $T\\!-\\!\mu_{B}$ plane, calculated within the fRG approach with the truncations LPA and LPA′, in the left and right panels, respectively. The black dashed lines in both panels denote the second-order $O(4)$ chiral phase transition of $N_{f}=2$ flavor in the chiral limit. The black circles indicate the location of the tricritical point, beyond which the second-order phase transition evolves into a discontinuous first-order one, which are shown by the solid lines. Note that the solid lines of different colors in the left panel denote the first-order phase transitions with different pion masses in the vacuum, i.e. different values of $c$ in Eq. (3), and in the right panel, we only give the first-order phase transition line in the chiral limit, since numerical calculations become quite difficult in the region of high $\mu_{B}$ and low $T$ with the truncation LPA′. The red dashed lines in both panels are the trajectories of the critical end points with the change of the strength of explicit chiral symmetry breaking $c$, which belong to the 3-$d$ $Z(2)$ Ising university class. The critical temperature at vanishing baryon chemical potential is found to be $T_{c}=144$ MeV in LPA and 143 MeV in LPA′ in the chiral limit. The tricritical point is located at $(T_{\mathrm{tri}},{\mu_{B}}_{\mathrm{tri}})_{{}_{\tiny{\mathrm{LPA}}}}=(50,764)$ MeV in the LPA and $(T_{\mathrm{tri}},{\mu_{B}}_{\mathrm{tri}})_{{}_{\tiny{\mathrm{\mathrm{LPA}^{\prime}}}}}=(47,687)$ MeV in the LPA′, which are shown in the phase diagrams by the black circles. The location of CEP corresponding to the physical pion mass in the LPA, shown in the left panel of Fig. 2 by the star, is $(T_{{}_{\tiny{\mathrm{CEP}}}},{\mu_{B}}_{{}_{\tiny{\mathrm{CEP}}}})_{{}_{\tiny{\mathrm{LPA}}}}=(8,885)$ MeV. In both phase diagrams in Fig. 2 we also use red and blue crosses to label the locations where critical exponents in Sec. IV would be calculated for the 3-$d$ $O(4)$ and $Z(2)$ universality classes, respectively. The calculated points for the $O(4)$ and $Z(2)$ phase transition in the LPA are given by $(T_{{}_{O(4)}},{\mu_{B}}_{{}_{O(4)}})_{{}_{\tiny{\mathrm{LPA}}}}=(144,0)$ MeV and $(T_{{}_{Z(2)}},{\mu_{B}}_{{}_{Z(2)}})_{{}_{\tiny{\mathrm{LPA}}}}=(38,795)$ MeV, respectively; and the relevant values in the LPA′ read $(T_{{}_{O(4)}},{\mu_{B}}_{{}_{O(4)}})_{{}_{\tiny{\mathrm{LPA}^{\prime}}}}=(143,0)$ MeV and $(T_{{}_{Z(2)}},{\mu_{B}}_{{}_{Z(2)}})_{{}_{\tiny{\mathrm{LPA}^{\prime}}}}=(41,702)$ MeV. ## IV Critical behavior and critical exponents A variety of scaling analysis has been performed for the $O(4)$ universality class, e.g., in the $O(N)$ model Toussaint (1997); Engels and Mendes (2000); Parisen Toldin _et al._ (2003); Engels _et al._ (2003); Braun and Klein (2008); Engels and Vogt (2010) and two-flavor quark-meson model Berges _et al._ (1999); Schaefer and Pirner (1999); Bohr _et al._ (2001); Stokic _et al._ (2010). The dynamics of a system in the critical regime near a second- order critical point is governed by long-wavelength fluctuations, and the correlation length tends to be divergent as the system moves towards the critical point. Critical exponents play a pivotal role in studies of the critical dynamics, which are independent of micro interactions, but rather universal for the same symmetry class, dimension of the system, etc., and see Stokic _et al._ (2010); Braun and Klein (2008) for more details. In the following, we follow the standard procedure and give our notations for the relevant various critical exponents. To begin with, from the effective action in Eq. (3) one readily obtains the thermodynamic potential density, which reads $\displaystyle\Omega\big{(}T,\mu_{B},\,c\big{)}$ $\displaystyle=V_{k=0}(\rho)-c\sigma\,,$ (14) where the order parameter field $\sigma\equiv\langle\sigma\rangle$ or $\rho=\sigma^{2}/2$ is on its equation of motion. We then introduce the reduced temperature and reduced external “magnetic” field as follows $\displaystyle t$ $\displaystyle=\frac{T-T_{c}}{T_{0}}\,,\qquad h=\frac{c}{c_{0}}\,,$ (15) where $T_{c}$ is the critical temperature, and they are normalized by $T_{0}$ and $c_{0}$, i.e., some appropriate values of $T$ and $c$. In the language of magnetization under an external magnetic field, the order parameter $\sigma$ here is just the corresponding magnetization density, i.e., $M\equiv\sigma$, and the explicit chiral symmetry breaking parameter $c$ is equivalent to the magnetic field strength $H\equiv c$. We will not distinguish them in the following any more. In the critical regime the thermodynamic potential in Eq. (14) is dominated by its singular part $f_{s}$, i.e., $\displaystyle\Omega\big{(}t,h\big{)}$ $\displaystyle=f_{s}(t,h)+f_{reg}(t,h)\,,$ (16) where the second term on the r.h.s. is the regular one, and the notation for the baryon chemical potential is suppressed. In what follows we adopt the notations in Braun _et al._ (2011b), and the scaling function $f_{s}(t,h)$ on the r.h.s. of Eq. (16) satisfies the scale relation to leading order, viz. $\displaystyle f_{s}(t,h)$ $\displaystyle=\ell^{-d}f_{s}(t\,\ell^{y_{t}},\,h\,\ell^{y_{h}})\,,$ (17) where $\ell$ is a dimensionless rescaling factor. The scaling function in Eq. (17) leads us to a variety of relations for various critical exponents Berges _et al._ (1999); Tetradis (2003); Schaefer and Pirner (1999); Braun and Klein (2008), e.g., $\displaystyle y_{t}$ $\displaystyle=\frac{1}{\nu}\,,\quad\\!\\!y_{h}=\frac{\beta\delta}{\nu}\,,\quad\\!\\!\beta=\frac{\nu}{2}(d-2+\eta)\,,\quad\\!\\!\gamma=\beta(\delta-1)\,,$ $\displaystyle\gamma$ $\displaystyle=(2-\eta)\nu\,,\quad\delta=\frac{d+2-\eta}{d-2+\eta}\,,\quad\nu d=\beta(1+\delta)\,,$ (18) with the spacial dimension $d$. The critical exponents $\beta$ and $\delta$ describe the critical behavior of the order parameter in the direction of $t$ or $h$, respectively, i.e., $\displaystyle M(t,h=0)$ $\displaystyle\sim(-t)^{\beta}\quad\mathrm{with}\quad t<0\,,$ (19) $\displaystyle M(t=0,h)$ $\displaystyle\sim h^{1/\delta}\,.$ (20) The exponent $\gamma$ is related to the susceptibility of order parameter $\chi$, and $\nu$ to the correlation length $\xi$, which reads $\displaystyle\chi$ $\displaystyle\sim\|t\|^{-\gamma}\,,\quad\mathrm{and}\quad\xi\sim\|t\|^{-\nu}\,,$ (21) The scaling relation in Eq. (17) allows us to readily obtain the critical behavior for various observables. For instance, the order parameter and its susceptibilities read $\displaystyle M$ $\displaystyle=-\frac{\partial f_{s}}{\partial H}\,,\quad\chi_{\sigma}=\frac{\partial M}{\partial H}\,,\quad\chi_{\pi}=\frac{M}{H}\,,$ (22) where $\chi_{\sigma}\equiv\chi_{l}$ and $\chi_{\pi}\equiv\chi_{t}$ are also called as the longitudinal and transverse susceptibilities, respectively. Choosing an appropriate value of the rescaling factor such that $h\,\ell^{y_{h}}=1$ in Eq. (17), one is led to $\displaystyle f_{s}(t,h)$ $\displaystyle=h^{d/y_{h}}f_{s}(z,1)\,,$ (23) with the scaling variable $z=t/h^{1/(\beta\delta)}$. Inserting Eq. (23) into the first equation in Eq. (22), one arrives at $\displaystyle M$ $\displaystyle=h^{1/\delta}f(z)\,,$ (24) where we have introduced $\displaystyle f(z)$ $\displaystyle\equiv\frac{1}{H_{0}}\Big{[}\frac{z}{\beta\delta}\frac{\partial f_{s}(z,1)}{\partial z}-\frac{d\nu}{\beta\delta}f_{s}(z,1)\Big{]}\,,$ (25) which is a scaling function dependent only on $z$. With appropriate values of $H_{0}$ and $T_{0}$ in Eq. (15), it can be shown that the scaling function in Eq. (25) has the properties $f(0)=1$ and $f(z)\simeq(-z)^{\beta}$ with $z\rightarrow-\infty$ Braun _et al._ (2011b). Consequently, it is straightforward to express the longitudinal and transverse susceptibilities in Eq. (22) in terms of the scaling function $f(z)$, to wit, $\displaystyle\chi_{\sigma}$ $\displaystyle=\frac{1}{H_{0}}h^{1/\delta-1}f_{\chi}(z)\,,$ (26) with $\displaystyle f_{\chi}(z)$ $\displaystyle\equiv\frac{1}{\delta}\Big{[}f(z)-\frac{z}{\beta}f^{\prime}(z)\Big{]}\,,$ (27) and $\displaystyle\chi_{\pi}$ $\displaystyle=\frac{1}{H_{0}}h^{1/\delta-1}f(z)\,.$ (28) Alternative to the choice of $h\,\ell^{y_{h}}=1$ in Eq. (17), one can also employ $t\,\ell^{y_{t}}=1$, which is equivalent to the Widom-Griffiths parametrization Widom (1965); Griffiths (1967) of the equation of state by means of the scaling variables, as follows $\displaystyle x$ $\displaystyle\equiv\frac{t}{M^{1/\beta}}\,,\qquad y\equiv\frac{h}{M^{\delta}}\,,$ (29) which are obviously related to the other parametrization by the relations which read $\displaystyle z$ $\displaystyle=\frac{x}{y^{1/(\beta\delta)}}\,,\qquad f(z)=\frac{1}{y^{1/\delta}}\,.$ (30) Hence the scaling function $y(x)$ has the properties $y(0)=1$ and $y(-1)=0$. In the same way, one readily obtains the expressions of susceptibilities in this parametrization, which read $\displaystyle\chi_{\sigma}$ $\displaystyle=\frac{1}{H_{0}M^{\delta-1}}\Big{[}\delta y(x)-\frac{1}{\beta}xy^{\prime}(x)\Big{]}^{-1}\,,$ (31) $\displaystyle\chi_{\pi}$ $\displaystyle=\frac{1}{H_{0}M^{\delta-1}}\frac{1}{y}\,.$ (32) ### IV.1 Order parameter Figure 3: Logarithm of the reduced order parameter $\tilde{\sigma}$ in Eq. (33) as a function of $\ln(-t)$ (left panel) or $\ln(h)$ (right panel) for the second-order $O(4)$ and $Z(2)$ phase transitions with truncations LPA and LPA′, where the phase transition points are chosen to be the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The solid lines represent linear fits to the calculated discrete data points, from which values of the critical exponents $\beta$ and $\delta$ are extracted. $T_{c}-T$ (MeV) \| ($10^{-4}$, $5\\!\times\\!10^{-3}$) \| ($10^{-2}$, 0.1) \| (0.1, 0.5 ) \| (0.5, 1) \| (1, 5) ---\|---\|---\|---\|---\|--- $\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ \| 0.3989(41) \| 0.5164(65) \| 0.4374(36) \| 0.4077(44) \| 0.3921(43) $\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ \| 0.3352(12) \| 0.2830(26) \| 0.2724(18) \| 0.2689(17) \| 0.247(17) Table 1: Values of the critical exponent $\beta$ extracted from different ranges of temperature, which are denoted by their distances to the corresponding critical temperature, i.e., $T_{c}-T$. The calculations are performed with the truncation LPA, and the phase transition points are chosen to be the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The flow equation of effective potential in Eq. (4) is solved by the use of the Chebyshev expansion as discussed in Sec. II.2, i.e., evolving the flow equations of the expansion coefficients in Eq. (12) from the UV cutoff $\Lambda$ to the infrared limit $k\rightarrow 0$, and then the expectation value of the order parameter $\sigma$ is determined by minimizing the thermodynamic potential in Eq. (14). Note that two different truncations, i.e., LPA and LPA′ as shown in Sec. II.1, are employed in the calculations. The critical exponents $\beta$ and $\delta$ are given in Eqs. (19) and (20), which are related to the scaling behavior of the order parameter as the phase transition is approached towards in the temperature or external field direction, respectively. Note, however, that in the case of $Z(2)$ phase transition as indicated by the blue cross in the phase diagram in Fig. 2, the order parameter should be modified slightly and we introduce the reduced order parameter which reads $\displaystyle\tilde{\sigma}$ $\displaystyle=\frac{\sigma-\sigma^{\prime}}{f_{\pi}}\,,$ (33) where $f_{\pi}$ is the pion decay constant in the vacuum and $\sigma^{\prime}$ is the expectation value of sigma field at the phase transition point, which is nonvanishing on the red dashed lines of $Z(2)$ in the phase diagrams in Fig. 2. Correspondingly, the reduced external field in Eq. (15) is modified into $\displaystyle h$ $\displaystyle=\frac{c-c^{\prime}}{c_{0}}\,,$ (34) where $c^{\prime}$ is the $\sigma^{\prime}$-related external field on the $Z(2)$ phase transition line. Notice that both $c^{\prime}$ and $\sigma^{\prime}$ are vanishing on the $O(4)$ phase transition line, viz., the black dashed lines in Fig. 2. In our calculations below, the normalized external field strength $c_{0}$ in Eq. (34) is chosen to be the value corresponding to the physical pion mass, and the normalized temperature in Eq. (15) is to be the critical one $T_{0}=T_{c}$. In Fig. 3 we show the log-log plots of the reduced order parameter $\tilde{\sigma}$ versus the reduced temperature $-t$ or external field $h$ for the second-order $O(4)$ and $Z(2)$ phase transitions. The calculations are performed in the quark-meson LEFT with the fRG in both LPA and LPA′. The phase transition points are chosen to be the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. A linear relation is used to fit the calculated discrete data points in Fig. 3, and as shown in Eq. (19) and Eq. (20), one could extract the values of the critical exponents $\beta$ and $\delta$ from the slope of these linear curves. This leads us to $\displaystyle\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.3989(41)\,,\qquad\beta^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.3832(31)\,,$ (35) for the $O(4)$ universality class in LPA and LPA′, respectively. In the case of the $Z(2)$ universality class, one arrives at $\displaystyle\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.3352(12)\,,\qquad\beta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.3259(01)\,.$ (36) In the same way, the values of $\delta$ are obtained as follows $\displaystyle\delta^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=4.975(57)\,,\qquad\delta^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=4.859(37)\,,$ (37) $\displaystyle\delta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=4.941(22)\,,\qquad\delta^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=4.808(14)\,.$ (38) It is found that the critical exponents $\beta$ and $\delta$ of the $O(4)$ and $Z(2)$ phase transitions in 3-$d$ systems calculated in this work are consistent with previous results, e.g., Monte Carlo simulation of spin model Kanaya and Kaya (1995) and $d=3$ expansion for $Z(2)$Zinn-Justin (2001). Comparing the relevant results in LPA and LPA′, one observes that both $\beta$ and $\delta$ obtained in LPA′ are slightly smaller than those in LPA. ### IV.2 Preliminary assessment of the size of the critical region It is well known that critical exponents are universal for the same universality classes. The size of the critical region is, however, non- universal and depends on the interactions and other details of system concerned. Furthermore, there has been a longstanding debate on the size of the critical region in QCD. Lattice QCD simulations show that the chiral condensate, i.e., the order parameter in Eq. (24), for physical quark masses are well described by Eq. (24) plus a small analytic regular term Ejiri _et al._ (2009); Kaczmarek _et al._ (2011); Ding _et al._ (2019), which, in another word, implies that the size of the critical regime of QCD is large enough, such that QCD with physical quark mass is still in the chiral critical regime. On the contrary, it is found in Braun and Klein (2008); Braun _et al._ (2011b); Klein (2017) that the pion mass required to observe the scaling behavior is very small, at least one order of magnitude smaller than the physical pion mass. Moreover, it is also found that the critical region around the CEP in the QCD phase diagram is very small Schaefer and Wambach (2007). In Tab. 1 we present the values of the critical exponent $\beta$ extracted from different ranges of temperature. One observes that when the temperature range is away from the critical temperature larger than $0.01$ MeV, the value of $\beta$ deviates from its universal value pronouncedly. This applies for both the $O(4)$ and $Z(2)$ universality classes. Given the systematic errors in the computation of this work, one could safely conclude that our calculation indicates that the critical region in the QCD phase diagram is probably very small, and it is smaller than 1 MeV in the direction of temperature. ### IV.3 Chiral susceptibility Figure 4: Logarithm of the longitudinal susceptibility $\chi_{\sigma}$ as a function of $\ln(t)$ in the chiral symmetric phase. The calculation is done in the quark-meson LEFT within the fRG approach with truncations LPA and LPA′, where the phase transition points are chosen to be at the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The solid lines represent linear fits to the calculated discrete data points, from which value of the critical exponent $\gamma$ is extracted. Figure 5: Longitudinal susceptibility of the order parameter $\chi_{\sigma}$ as a function of the reduced temperature $t$ with several different values of the reduced external field $h$, calculated in the LPA (left panel) and LPA′ (right panel). The phase transition is chosen to be near the location of the red cross in the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class. Figure 6: Left panel: logarithm of the reduced pseudo-critical temperature $t_{pc}$, defined by the peak of the susceptibility $\chi_{\sigma}$ as shown in Fig. 5, as a function of the logarithm of the reduced external field strength $h$. Right panel: logarithm of the peak height of the susceptibility, $\chi_{\sigma}\big{\|}_{t_{pc}}$, versus the logarithm of the reduced pseudo-critical temperature. Calculations are done within the fRG approach with the truncations LPA and LPA′. The phase transition is chosen to be near the location of the red cross in the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class. Figure 7: Logarithms of the transverse (left panel) and longitudinal (right panel) susceptibilities as functions of the logarithm of $-t$ with a fixed value of the reduced external field $h=8.4\times 10^{-9}$ in the chiral broken phase near the coexistence line. Calculations are performed within the fRG approach with the truncations LPA and LPA′. The phase transition is chosen to be near the location of the red cross in the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class, where the baryon chemical potential is vanishing. Figure 8: Left panel: logarithm of the correlation length as a function of the logarithm of the reduced external field strength with $t=0$. Right panel: logarithm of the correlation length as a function of the logarithm of the reduced temperature with $h=0$. Both calculations are performed in the quark-meson LEFT within the fRG approach with truncations LPA and LPA′, where the phase transition points are chosen to be at the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The solid lines represent linear fits to the calculated discrete data points, from which values of the critical exponent $\nu_{c}$ and $\nu$ are yielded. According to Eq. (29), the reduced order parameter reads $\displaystyle\tilde{\sigma}$ $\displaystyle\sim h^{1/\delta}y(x)^{-1/\delta}\,.$ (39) Moreover, it has been shown in Griffiths (1967) that given $x>0$ and $M>M_{0}$ for some value $M_{0}$ in Eq. (29), the scaling function can be expanded as $\displaystyle y(x)$ $\displaystyle=\sum_{n=1}^{\infty}c_{n}x^{\gamma-2\beta(n-1)}$ $\displaystyle=x^{\gamma}\big{(}c_{1}+c_{2}x^{-2\beta}+c_{3}x^{-4\beta}+\dots\big{)}\,.$ (40) Inserting the leading term in Eq. (40) into Eq. (39) and utilizing the relation $\gamma=\beta(\delta-1)$ as shown in Eq. (18), one is led to the reduced order parameter with $t>0$ and $h\rightarrow 0$, which reads $\displaystyle\tilde{\sigma}$ $\displaystyle\sim t^{-\gamma}h\,.$ (41) Consequently, the longitudinal and transverse susceptibilities of the order parameter as defined in Eq. (22) are readily obtained as follows $\displaystyle\chi_{\sigma}$ $\displaystyle=\chi_{\pi}\sim t^{-\gamma}\,,$ (42) which is in agreement with Eq. (21) in the limit $h\rightarrow 0$ and in the symmetric phase, as it should be. Equation (41) also allows us to extract the value of the exponent $\gamma$, by directly investigating the scaling relation of $\tilde{\sigma}$ and $t$ in the chiral symmetric phase with a fixed, small value of $h$. In Fig. 4 we show the logarithm of the longitudinal susceptibility $\chi_{\sigma}$ versus that of the reduced temperature, where $h=3.5\times 10^{-10}$ is chosen in the calculations. We have checked that this value of $h$ is small enough to make sure that the value of $\gamma$ obtained from the linear fit of $\ln(\chi_{\sigma})$-$\ln(t)$ is convergent. In the same way, the flow equations of fRG are resolved with two truncations LPA and LPA′, and the phase transition points are chosen to be at the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. The values of the exponent $\gamma$ are obtained as follows $\displaystyle\gamma^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=1.5458(68)\,,\qquad\gamma^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=1.4765(76)\,,$ (43) $\displaystyle\gamma^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=1.3313(96)\,,\qquad\gamma^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=1.2362(77)\,.$ (44) Once more, one observes that these values, in particular those obtained in the LPA′, are in good agreement with the values of $\gamma$ for the $O(4)$ and $Z(2)$ symmetry universality classes, respectively; see, e.g., Kanaya and Kaya (1995); Zinn-Justin (2001). In Fig. 5 the longitudinal susceptibility of the order parameter $\chi_{\sigma}$, as shown in Eq. (22), is depicted versus the reduced temperature with several different values of the reduced external field. Here we only focus on the case of $O(4)$ symmetry, and thus choose the phase transition to be near the location of the red cross in the phase diagrams in Fig. 2, i.e., the phase transition with vanishing baryon chemical potential. When the external field $h$ that breaks the chiral symmetry explicitly is nonzero, the second-order phase transition becomes a continuous crossover, as shown in Fig. 5. One can define a pseudo-critical temperature $T_{pc}$, which is the peak position of the curve $\chi_{\sigma}$ versus $T$, and thus the reduced pseudo-critical temperature reads $\displaystyle t_{pc}$ $\displaystyle=\frac{T_{pc}-T_{c}}{T_{c}}\,.$ (45) One observes from Fig. 5 that with the increasing $h$, the peak height of the susceptibility decreases and the pseudo-critical temperature $t_{pc}$ increases. The rescaling relation between $t_{pc}$ and $h$ as well as that between the peak height of $\chi_{\sigma}$ and $t_{pc}$ reads $\displaystyle t_{pc}$ $\displaystyle\sim h^{1/(\gamma+\beta)}\,,\qquad\chi_{\sigma}\big{\|}_{t_{pc}}\sim t_{pc}^{-\gamma}\,,$ (46) and see, e.g., Pelissetto and Vicari (2002) for more details. In Fig. 6 we show the logarithm of the reduced pseudo-critical temperature versus the logarithm of the reduced external field strength, and the logarithm of the peak height of the susceptibility versus the logarithm of the reduced pseudo-critical temperature in the left and right panels, respectively. The phase transition is also chosen to be near the location of the red cross in the phase diagrams in Fig. 2 for the $O(4)$ symmetry universality class, where the baryon chemical potential is vanishing. Linear fitting to the calculated discrete data in Fig. 6 yields $\beta=0.403(19)$ and $\gamma=1.543(15)$ for the LPA, and $\beta=0.405(22)$ and $\gamma=1.454(17)$ for the LPA′, which are in agreement with the relevant values in Eq. (35) and Eq. (43) within errors for the $O(4)$ second-order phase transition in 3-$d$ space. In turn, the agreement of critical exponents obtained from different scaling relations also provides us with the necessary check for the inner consistency of computations. Note, however, that the critical exponents $\beta$ and $\gamma$ determined from the scaling relations in Eq. (46) are significantly less accurate than those in Eq. (35) and Eq. (43). As another check for the consistency, we consider the susceptibilities in the chiral broken phase near the coexistence line, i.e., $x=-1$, with $t<0$ and $h\rightarrow 0$. Inserting Eq. (39) into Eqs. (31) (32), one is led to $\displaystyle\chi_{\sigma}$ $\displaystyle\sim h^{1/\delta-1}\frac{\beta y(x)^{1-1/\delta}}{\beta\delta y(x)-xy^{\prime}(x)}\,,$ (47) $\displaystyle\chi_{\pi}$ $\displaystyle\sim h^{1/\delta-1}y(x)^{-1/\delta}\,.$ (48) when the system is near the coexistence line, one has $x\rightarrow-1$ and $y\sim h/(-t)^{\beta\delta}$. Hence, the transverse susceptibility is readily obtained as follows $\displaystyle\chi_{\pi}$ $\displaystyle\sim h^{-1}(-t)^{\beta}\,.$ (49) In order to obtain a similar expression for the longitudinal susceptibility, one needs further information on the equation of state $y(x)$. As the system is located in the broken phase near the coexistence line, the dynamics is dominated by Goldstone modes, which are massless in the chiral limit. The relevant critical behavior in this regime is governed by a Gaussian fixed point, and thus the corresponding exponents are as same as values of mean fields Wallace and Zia (1975); Brezin and Wallace (1973), which leaves us with $\displaystyle y(x)$ $\displaystyle\sim(1+x)^{2}\,,\qquad\mathrm{for}\qquad x\rightarrow-1\,,$ (50) and see, e.g., Braun and Klein (2008); Stokic _et al._ (2010) for more relevant discussions. Substituting equation above into Eq. (47), one arrives at $\displaystyle\chi_{\sigma}$ $\displaystyle\sim h^{-1/2}(-t)^{\beta-(\beta\delta/2)}\,.$ (51) As Eqs. (49) (51) show, the transverse and longitudinal susceptibilities are proportional to the external field with different powers in the broken phase, i.e., $-1$ and $-1/2$ for the former and latter, respectively. In Fig. 7 we show $\ln(\chi_{\pi})$ and $\ln(\chi_{\sigma})$ versus $\ln(-t)$ with a fixed value of the reduced external field $h=8.4\times 10^{-9}$ in the chiral broken phase near the coexistence line. Similarly, here we only consider the phase transition of $O(4)$ symmetry with $\mu_{B}=0$ in the phase diagrams in Fig. 2. As shown in Eqs. (49) (51), the ratios of the linear fitting to $\ln(\chi_{\pi})$-$\ln(-t)$ and $\ln(\chi_{\sigma})$-$\ln(-t)$ are just the values of $\beta$ and $\beta-(\beta\delta/2)$, respectively. Consequently, one arrives at $\beta=0.3979(41)$ and $\delta=4.984(74)$ in LPA and $\beta=0.3832(54)$ and $\delta=4.86(10)$ in LPA′, which agree very well with the relevant values in Eq. (35) and Eq. (37). ### IV.4 Correlation length \| Method \| $\beta$ \| $\delta$ \| $\gamma$ \| $\nu$ \| $\nu_{c}$ \| $\eta$ ---\|---\|---\|---\|---\|---\|---\|--- $O(4)$ QM LPA (this work) \| fRG Chebyshev \| 0.3989(41) \| 4.975(57) \| 1.5458(68) \| 0.7878(25) \| 0.3982(17) \| 0 $O(4)$ QM LPA′ (this work) \| fRG Chebyshev \| 0.3832(31) \| 4.859(37) \| 1.4765(76) \| 0.7475(27) \| 0.4056(19) \| 0.0252(91)* $Z(2)$ QM LPA (this work) \| fRG Chebyshev \| 0.3352(12) \| 4.941(22) \| 1.3313(96) \| 0.6635(17) \| 0.4007(45) \| 0 $Z(2)$ QM LPA′ (this work) \| fRG Chebyshev \| 0.3259(01) \| 4.808(14) \| 1.2362(77) \| 0.6305(23) \| 0.4021(43) \| 0.0337(38)* $O(4)$ scalar theories Tetradis and Wetterich (1994) \| fRG Taylor \| 0.409 \| 4.80* \| 1.556 \| 0.791 \| \| 0.034 $O(4)$ KT phase transition Von Gersdorff and Wetterich (2001) \| fRG Taylor \| 0.387* \| 4.73* \| \| 0.739 \| \| 0.047 $Z(2)$ KT phase transition Von Gersdorff and Wetterich (2001) \| fRG Taylor \| \| \| \| 0.6307 \| \| 0.0467 $O(4)$ scalar theories Litim and Pawlowski (2001) \| fRG Taylor \| 0.4022* \| 5.00* \| \| 0.8043 \| \| $O(4)$ scalar theories LPABraun and Klein (2008) \| fRG Taylor \| 0.4030(30) \| 4.973(30) \| \| 0.8053(60) \| \| $O(4)$ QM LPA Stokic _et al._ (2010) \| fRG Taylor \| 0.402 \| 4.818 \| 1.575 \| 0.787 \| 0.396 \| $O(4)$ scalar theories Bohr _et al._ (2001) \| fRG Grid \| 0.40 \| 4.79 \| \| 0.78 \| \| 0.037 $Z(2)$ scalar theories Bohr _et al._ (2001) \| fRG Grid \| 0.32 \| 4.75 \| \| 0.64 \| \| 0.044 $O(4)$ scalar theories De Polsi _et al._ (2020) \| fRG DE $\mathcal{O}(\partial^{4})$ \| \| \| \| 0.7478(9) \| \| 0.0360(12) $Z(2)$ scalar theories Balog _et al._ (2019); De Polsi _et al._ (2020) \| fRG DE $\mathcal{O}(\partial^{6})$ \| \| \| \| 0.63012(5) \| \| 0.0361(3) $O(4)$ CFTs Kos _et al._ (2015) \| conformal bootstrap \| \| \| \| 0.7472(87) \| \| 0.0378(32) $Z(2)$ CFTs Kos _et al._ (2014) \| conformal bootstrap \| \| \| \| 0.629971(4) \| \| 0.0362978(20) $O(4)$ spin model Kanaya and Kaya (1995) \| Monte Carlo \| 0.3836(46) \| 4.851(22) \| 1.477(18) \| 0.7479(90) \| 0.4019(71)* \| 0.025(24)* $Z(2)$ $d=3$ expansion Zinn-Justin (2001) \| summed perturbation \| 0.3258(14) \| 4.805(17)* \| 1.2396(13) \| 0.6304(13) \| 0.4027(23) \| 0.0335(25) Mean Field \| \| 1/2 \| 3 \| 1 \| 1/2 \| 1/3 \| 0 Table 2: Critical exponents for the $O(4)$ and $Z(2)$ symmetry universality classes in 3-$d$ space, obtained in the quark-meson LEFT within the fRG approach with truncations LPA and LPA′, where the effective potential is expanded as a sum of Chebyshev polynomials. Our calculated results are also in comparison to relevant results from previous fRG calculations, e.g., scalar theories with the effective potential expanded in a Taylor series Tetradis and Wetterich (1994); Von Gersdorff and Wetterich (2001); Litim and Pawlowski (2001); Braun and Klein (2008), or discretized on a grid Bohr _et al._ (2001), the quark-meson (QM) low energy effective theory with LPA Stokic _et al._ (2010), derivative expansions (DE) up to orders of $\mathcal{O}(\partial^{4})$ and $\mathcal{O}(\partial^{6})$ Balog _et al._ (2019); De Polsi _et al._ (2020). Moreover, results from other approaches, such as the conformal bootstrap for the 3-$d$ conformal field theories (CFTs) Kos _et al._ (2014, 2015), Monte Carlo simulation Kanaya and Kaya (1995), and $d=3$ perturbation expansion Zinn-Justin (2001), as well as the mean-field values of exponents are also presented. Note that values with an asterisk are obtained with scaling laws in Eq. (18). It is well known that the correlation length $\xi$, plays a pivotal role in the critical dynamics, since fluctuations of wavelength $\sim\xi$ are inevitably involved in the dynamics. As a system is approaching towards a second-order phase transition, the most relevant degrees of freedom are the long-wavelength modes of low energy, and the correlation length is divergent at the phase transition Landau and Lifshitz (1980). The critical behavior of correlation length is described by the critical exponent $\nu$, as shown in Eq. (21). In the symmetric phase $t>0$, it reads $\displaystyle\xi$ $\displaystyle\sim t^{-\nu}\,,\qquad\mathrm{with}\qquad h=0\,,$ (52) which illustrates the scaling relation between the correlation length and the reduced temperature. Moreover, one can also define another critical exponent $\nu_{c}$ related to the scaling relation between the correlation length and the reduced external field, to wit, $\displaystyle\xi$ $\displaystyle\sim h^{-\nu_{c}}\,,\qquad\mathrm{with}\qquad t=0\,.$ (53) In our setup in the quark-meson LEFT, cf. Sec. II, the correlation length is proportional to the inverse of the renormalized $\sigma$-meson mass, viz., $\displaystyle\xi$ $\displaystyle\sim\frac{1}{m_{\sigma}}\,,$ (54) where $m_{\sigma}$ is related to the dimensionless $k$-dependent sigma mass $\bar{m}_{\sigma,k}$ in Eq. (5) via the relation as follows $\displaystyle m_{\sigma}$ $\displaystyle=\bar{m}_{\sigma,k}(\sigma=\sigma_{{}_{\mathrm{EoM}}})k\,,\quad\mathrm{with}\quad k\rightarrow 0\,,$ (55) where the scale $k$ is chosen to be in the IR limit $k\rightarrow 0$, and the mass is calculated on the equation of motion of the order parameter field. In Fig. 8 we show the scale relation between the correlation length and the reduced external field strength, and that between the correlation length and the reduced temperature, respectively. In the same way, we adopt the two different truncations: LPA and LPA′. The phase transition points are also chosen to be at the locations of the red and blue crosses in the phase diagrams in Fig. 2 for the $O(4)$ and $Z(2)$ universality classes, respectively. By the use of the linear fitting to the calculated data, one obtains values of the critical exponent $\nu$ as follows $\displaystyle\nu^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.7878(25)\,,\qquad\nu^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.7475(27)\,,$ (56) $\displaystyle\nu^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.6635(17)\,,\qquad\nu^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.6305(23)\,,$ (57) as well as those of the critical exponent $\nu_{c}$, i.e., $\displaystyle{\nu_{c}}^{{}^{O(4)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.3982(17)\,,\qquad{\nu_{c}}^{{}^{O(4)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.4056(19)\,,$ (58) $\displaystyle{\nu_{c}}^{{}^{Z(2)}}_{{}_{\mathrm{LPA}}}$ $\displaystyle=0.4007(45)\,,\qquad{\nu_{c}}^{{}^{Z(2)}}_{{}_{\mathrm{LPA}^{\prime}}}=0.4021(43)\,.$ (59) Finally, we close Sec. IV with a summary of various critical exponents calculated in this work in Tab. 2. Respective results for the $O(4)$ and $Z(2)$ symmetry universality classes with truncation LPA or LPA′ are presented in the first several rows in Tab. 2. As we have discussed in Sec. II, the effective potential is expanded as a sum of Chebyshev polynomials in our calculations, which captures global properties of the order-parameter potential very well. In Tab. 2 we also present values of critical exponents obtained from other computations, e.g., scalar theories calculated within the fRG with the effective potential expanded in a Taylor series Tetradis and Wetterich (1994); Von Gersdorff and Wetterich (2001); Litim and Pawlowski (2001); Braun and Klein (2008), or discretized on a grid Bohr _et al._ (2001), quark-meson LEFT within the fRG in LPA Stokic _et al._ (2010), derivative expansion of the fRG up to orders of $\mathcal{O}(\partial^{4})$ and $\mathcal{O}(\partial^{6})$ Balog _et al._ (2019); De Polsi _et al._ (2020), the conformal bootstrap for the 3-$d$ conformal field theories Kos _et al._ (2014, 2015), Monte Carlo simulation Kanaya and Kaya (1995), and the $d=3$ perturbation expansion Zinn-Justin (2001). One observes that our calculated results are in good agreement with the relevant results from previous fRG calculations as well as those from the conformal bootstrap, Monte Carlo simulation, and the $d=3$ perturbation expansion. Remarkably, the calculation with the truncation LPA′ is superior to that with LPA, and the former has already provided us with quantitative reliability for the prediction of the critical exponents in comparison to other approaches. ## V summary QCD phase structure and related critical behaviors have been studied in the two-flavor quark-meson low energy effective theory within the fRG approach in this work. More specifically, we have expanded the effective potential as a sum of Chebyshev polynomials to solve its flow equation. Consequently, both the global properties of the effective potential and the numerical accuracy necessary for the computation of critical exponents are retained in our calculations. Moreover, we have employed two different truncations for the effective action: one is the usually used local potential approximation and the other is that beyond the local potential approximation, in which a field- dependent mesonic wave function renormalization is encoded. With the numerical setup within the fRG approach described above, we have obtained the phase diagram in the plane of $T$ and $\mu_{B}$ for the two- flavor quark-meson LEFT in the chiral limit, including the second-order phase transition line of $O(4)$, the tricritical point and the first-order phase transition line. Furthermore, we also show the $Z(2)$ line in the phase diagram, which is the trajectory of the critical end point moving with the successive variance of the strength of explicit chiral symmetry breaking, or the varying pion mass. In the phase diagram, we have performed detailed scaling analyses for the 3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes, and investigated the critical behaviors in the vicinity of phase transition both in the chiral symmetric and broken phases. Moreover, the transverse and longitudinal susceptibilities of the order parameter have been calculated in the chiral broken phase near the coexistence line. A variety of critical exponents related to the order parameter, chiral susceptibilities and correlation lengths have been calculated for the 3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes in the phase diagram, respectively. The calculated results are also compared with those from previous fRG calculations, either employing the Taylor expansion for the order-parameter potential or discretizing it on a grid, derivative expansion of the effective action, the conformal bootstrap, Monte Carlo simulations, and the $d=3$ perturbation expansion. We find that the critical exponents obtained in the quark-meson LEFT within the fRG approach, where the order-parameter potential is expanded in terms of Chebyshev polynomials and a field-dependent mesonic wave function renormalization is taken into account, are in quantitative agreement with results from approaches aforementioned. Furthermore, we have also investigated the size of the critical regime, and it is found that the critical region in the QCD phase diagram is probably very small, and it is smaller than 1 MeV in the direction of temperature. ###### Acknowledgements. We thank Jan M. Pawlowski for illuminating discussions. We also would like to thank other members of the fQCD collaboration Braun _et al._ (2021) for work on related subjects. The work was supported by the National Natural Science Foundation of China under Contract No. 11775041, and the Fundamental Research Funds for the Central Universities under Contract No. DUT20GJ212. ## Appendix A Threshold functions and anomalous dimensions We employ the $3d$ flat regulators Litim (2001, 2000) for quarks and mesons in this paper $\displaystyle R_{\phi,k}(q_{0},\bm{q})$ $\displaystyle=Z_{\phi,k}\bm{q}^{2}r_{B}(\bm{q}^{2}/k^{2})\,,$ (60) $\displaystyle R_{q,k}(q_{0},\bm{q})$ $\displaystyle=Z_{q,k}i\bm{\gamma}\cdot\bm{q}r_{F}(\bm{q}^{2}/k^{2})\,,$ (61) with $\displaystyle r_{B}(x)$ $\displaystyle=\left(\frac{1}{x}-1\right)\Theta(1-x)\,,$ (62) $\displaystyle r_{F}(x)$ $\displaystyle=\left(\frac{1}{\sqrt{x}}-1\right)\Theta(1-x)\,.$ (63) The threshold functions in Eq. (4) are given by $\displaystyle l_{0}^{(B,d)}(\bar{m}^{2}_{\phi,k},\eta_{\phi,k};T)$ $\displaystyle=$ $\displaystyle\frac{2}{d-1}\left(1-\frac{\eta_{\phi,k}}{d+1}\right)\frac{1}{\sqrt{1+\bar{m}^{2}_{\phi,k}}}$ $\displaystyle\times\bigg{(}\frac{1}{2}+n_{B}(\bar{m}^{2}_{\phi,k};T)\bigg{)}\,,$ (64) and $\displaystyle l_{0}^{(F,d)}(\bar{m}^{2}_{q,k},\eta_{q,k};T,\mu)$ $\displaystyle=$ $\displaystyle\frac{2}{d-1}\left(1-\frac{\eta_{q,k}}{d}\right)\frac{1}{2\sqrt{1+\bar{m}^{2}_{q,k}}}$ $\displaystyle\times\Big{(}1-n_{F}(\bar{m}^{2}_{q,k};T,\mu)-n_{F}(\bar{m}^{2}_{q,k};T,-\mu)\Big{)}\,.$ (65) with the bosonic and fermionic distribution functions reading $\displaystyle n_{B}(\bar{m}^{2}_{\phi,k};T)=$ $\displaystyle\frac{1}{\exp\bigg{\\{}\frac{k}{T}\sqrt{1+\bar{m}_{\phi,k}^{2}}\bigg{\\}}-1}\,,$ (66) and $\displaystyle n_{F}(\bar{m}^{2}_{q,k};T,\mu)=$ $\displaystyle\frac{1}{\exp\bigg{\\{}\frac{1}{T}\Big{[}k\sqrt{1+\bar{m}^{2}_{q,k}}-\mu\Big{]}\bigg{\\}}+1}\,,$ (67) respectively. The meson anomalous dimension in Eq. (8) is given by $\displaystyle\eta_{\phi,k}(\rho)$ $\displaystyle=\frac{1}{6\pi^{2}}\Bigg{\\{}\frac{4}{k^{2}}\bar{\rho}(\bar{V}^{\prime\prime}_{k}(\bar{\rho}))^{2}\mathcal{BB}_{(2,2)}(\bar{m}^{2}_{\pi,k},\bar{m}^{2}_{\sigma,k};T)$ $\displaystyle+N_{c}\bar{h}^{2}_{y,k}\bigg{[}\mathcal{F}_{(2)}(\bar{m}^{2}_{q,k};T,\mu)(2\eta_{q,k}-3)$ $\displaystyle-4(\eta_{q,k}-2)\mathcal{F}_{(3)}(\bar{m}^{2}_{q,k};T,\mu)\bigg{]}\Bigg{\\}}\,,$ (68) with $\displaystyle\bar{h}_{y,k}$ $\displaystyle=\frac{h_{y,k}}{Z_{q,k}(Z_{\phi,k})^{1/2}}\,.$ (69) Note that threshold functions $\mathcal{BB}_{(2,2)}$, $\mathcal{F}_{(2)}$ and $\mathcal{F}_{(3)}$ in Eq. (68) can be found in e.g., Fu and Pawlowski (2015); Yin _et al._ (2019). ## Appendix B Some relations for the Chebyshev polynomials In this appendix we collect some relations for the Chebyshev polynomials, which are used in solving the flow equation for the effective potential in Eq. (10). The Chebyshev polynomial of order $n$ reads $\displaystyle T_{n}(x)$ $\displaystyle=\cos\big{(}n\arccos(x)\big{)}\,,$ (70) with nonnegative integers $n$’s and $x\in[-1,1]$. The explicit expressions for the Chebyshev polynomials could be obtained by the recursion relation as follows $\displaystyle T_{n+2}(x)$ $\displaystyle=2xT_{n+1}(x)-T_{n}(x)\,,\quad n\geq 0\,,$ (71) with $T_{0}(x)=1$ and $T_{1}(x)=x$. The $N+1$ zeros of $T_{N+1}(x)$ in the region $-1\leq x\leq 1$ are given by $\displaystyle x_{k}$ $\displaystyle=\cos\left(\frac{\pi(k+\frac{1}{2})}{N+1}\right)\,,\quad k=0,\,1,\,\cdots N\,.$ (72) A discrete orthogonality relation is fulfilled by the Chebyshev polynomials, to wit, $\displaystyle\sum_{k=0}^{N}T_{i}(x_{k})T_{j}(x_{k})$ $\displaystyle=\left\\{\begin{array}[]{l}0\qquad\qquad\qquad i\neq j\\\\[4.30554pt] (N+1)/2\qquad i=j\neq 0\\\\[4.30554pt] N+1\qquad\quad\;\;\;i=j=0\end{array}\right.\,,$ (76) where $x_{k}$’s are the $N+1$ zeros of $T_{N+1}(x)$ in Eq. (72), and $i,\,j\leq N$. The interval $[-1,1]$ for $x$ could be extended to an arbitrary one $[y_{\mathrm{min}},y_{\mathrm{max}}]$ for $y$ via the linear relation as follows $\displaystyle x$ $\displaystyle=\frac{2y-(y_{\mathrm{max}}+y_{\mathrm{min}})}{y_{\mathrm{max}}-y_{\mathrm{min}}}\,,$ (77) and the generalized Chebyshev polynomials are defined by $\displaystyle T_{n}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)$ $\displaystyle\equiv T_{n}\big{(}x(y)\big{)}\,.$ (78) Therefore, the zeros in $y$ corresponding to Eq. (72) read $\displaystyle y_{k}$ $\displaystyle=\frac{y_{\max}-y_{\min}}{2}\cos\left(\frac{\pi(k+\frac{1}{2})}{N+1}\right)+\frac{y_{\max}+y_{\min}}{2}\,,$ (79) with $k=0,\,1,\,\cdots N$. Then, a function $f(y)$ with $y\in[y_{\mathrm{min}},y_{\mathrm{max}}]$ can be approximated as $\displaystyle f(y)$ $\displaystyle\approx\left[\sum_{i=1}^{N}c_{i}T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y)\right]+\frac{1}{2}c_{0}\,,$ (80) where the coefficients could be readily obtained by the use of the orthogonality relation in Eq. (76), which yields $\displaystyle c_{i}$ $\displaystyle=\frac{2}{N+1}\sum_{k=0}^{N}f(y_{k})T_{i}^{[y_{\mathrm{min}},y_{\mathrm{max}}]}(y_{k})\,,$ (81) with $i=0,\,1,\,\cdots N$. With the Chebyshev approximation of the function $f(y)$ in Eq. 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# bssm: Bayesian Inference of Non-linear and Non-Gaussian State Space Models in R by Jouni Helske and Matti Vihola ###### Abstract We present an R package bssm for Bayesian non-linear/non-Gaussian state space modelling. Unlike the existing packages, bssm allows for easy-to-use approximate inference based on Gaussian approximations such as the Laplace approximation and the extended Kalman filter. The package accommodates also discretely observed latent diffusion processes. The inference is based on fully automatic, adaptive Markov chain Monte Carlo (MCMC) on the hyperparameters, with optional importance sampling post-correction to eliminate any approximation bias. The package implements also a direct pseudo- marginal MCMC and a delayed acceptance pseudo-marginal MCMC using intermediate approximations. The package offers an easy-to-use interface to define models with linear-Gaussian state dynamics with non-Gaussian observation models, and has an Rcpp interface for specifying custom non-linear and diffusion models. ## Introduction State space models (SSM) are a flexible class of latent variable models commonly used in analysing time series data (cf. Durbin and Koopman, 2012). There are a number of packages available for state space modelling for R, especially for two special cases: a linear-Gaussian SSM (LGSSM) where both the observation and state densities are Gaussian with linear relationships with the states, and an SSM with discrete state space, which is sometimes called a hidden Markov model (HMM). These classes admit analytically tractable marginal likelihood functions and conditional state distributions (conditioned on the observations), making inference relatively straightforward. See for example (Petris and Petrone, 2011; Tusell, 2011; Helske, 2017; Helske and Helske, 2019) for review of some of the R packages dealing with these type of models. The present R package bssm is designed for Bayesian inference of general state space models with non-Gaussian and/or non-linear observational and state equations. The package primary aim is to provide easy-to-use and fast functions for fully Bayesian inference with common time series models such as basic structural time series model (Harvey, 1989) with exogenous covariates and simple stochastic volatility models. The package accomodates also custom non-linear models and discretised diffusion models. When extending the state space modelling to non-linear or non-Gaussian models, some difficulties arise. As the likelihood is no longer analytically tractable, computing the latent state distributions, as well as hyperparameter estimation of the model becomes more challenging. One general option is to use Markov chain Monte Carlo (MCMC) methods targeting the full joint posterior of hyperparameters and the latent states, for example by Gibbs sampling or Hamiltonian Monte Carlo. Unfortunately, the joint posterior is typically very high dimensional and due to the strong autocorrelation structures of the state densities, the efficiency of such methods can be relatively poor. Another asymptotically exact approach is based on the pseudo-marginal particle MCMC approach (Andrieu et al., 2010), where the likelihood function and the state distributions are estimated using sequential Monte Carlo (SMC) i.e. the particle filter (PF) algorithm. Instead of computationally demanding Monte Carlo methods, approximation-based methods such extended and unscented Kalman filters may be used, as well as Laplace approximations, which are provided for example by the INLA (Lindgren and Rue, 2015) R package. The latter are computationally appealing, but may lead to hard-to-quantify biases of the posterior. Some of the R packages suitable for Bayesian state space modelling include pomp (King et al., 2016), rbi (Jacob and Funk, 2020), nimbleSMC (Michaud et al., 2020; NIMBLE Development Team, 2020), and rstan (Stan Development Team, 2020). With the package pomp, user defines the model using R or C snippets for simulation from and evaluation of the latent state and observation level densities, allowing flexible model construction. The rbi package is an interface to LibBi (Murray, 2015), a standalone software with a focus on Bayesian state space modelling on high-performance computers. The pomp package provides several simulation-based inference methods mainly based on iterated filtering and maximum likelihood, whereas rbi is typically used for Bayesian inference via particle MCMC. For a more detailed comparison of differences of rbi/LibBi and pomp with examples, see (Funk and King, 2020). The nimbleSMC package contains some particle filtering algorithms which can be used in the general Nimble modelling system (de Valpine et al., 2017), whereas the rstan package provides an R interface to the Stan C++ package, a general statistical modelling platform (Carpenter et al., 2017). The key difference to the aforementioned packages and motivation behind the present bssm package is to combine the use of fast approximation-based methods with Monte Carlo correction step, leading to computationally efficient and unbiased (approximation error free) inference of the joint posterior of hyperparameters and latent states, as suggested in (Vihola et al., 2020). In a nutshell, the method uses MCMC which targets an approximate marginal posterior of the hyperparameters, and an importance sampling type weighting which provides asymptotically exact inference on the joint posterior of hyperparameters and the latent states. In addition to this two-stage procedure, the bssm supports also delayed acceptance pseudo-marginal MCMC (Christen and Fox, 2005) using the approximations, and direct pseudo-marginal MCMC. To our knowledge, importance sampling and delayed acceptance in this form are not available in other Bayesian state space modelling packages in R. ## Supported models We denote the sequence of observations $(y_{1},\ldots,y_{T})$ as $y$, and the sequence of latent state variables $(\alpha_{1},\ldots,\alpha_{T})$ as $\alpha$. The latent states $\alpha_{t}\in\mathbb{R}^{d}$ are typically vector-valued, whereas we focus mainly on scalar observations $y_{t}\in\mathbb{R}$ (vector-valued observations are also supported, assuming conditional independence (given $\alpha_{t}$) in case of non-Gaussian observations). A general state space model consists of two parts: observation level densities $g_{t}^{(\theta)}(y_{t}\|\alpha_{t})$ and latent state transition densities $\mu_{t}^{(\theta)}(\alpha_{t+1}\|\alpha_{t})$. Typically both $g_{t}^{(\theta)}$ and $\mu_{t}^{(\theta)}$ depend on unknown parameter vector $\theta$ for which we can define arbitrary prior $p(\theta)$. In a linear-Gaussian SSM, both $g_{t}^{(\theta)}$ and $\mu_{t}^{(\theta)}$ are Gaussian densities and they depend linearly on the current and previous state vectors, respectively. Section Models with linear-Gaussian state dynamics describes a common extension to these models supported by bssm, which relaxes the assumptions on observational density $g_{t}^{(\theta)}$, by allowing exponential family links, and stochastic volatility models. While the main focus of bssm is in state space models with linear-Gaussian state dynamics, there is also support for more general non-linear models, discussed briefly in Section Other state space models. Section Using the bssm package describes how arbitrary models based on these definitions are constructed in bssm. ### Models with linear-Gaussian state dynamics The primary class of models supported by bssm consists of SSMs with linear- Gaussian state dynamics of form $\displaystyle\alpha_{t+1}$ $\displaystyle=c_{t}+T_{t}\alpha_{t}+R_{t}\eta_{t},$ where $c_{t}\in\mathbb{R}^{d}$, $T_{t}\in\mathbb{R}^{d\times d}$, and $R_{t}\in\mathbb{R}^{d\times k}$ can depend on the unknown parameters $\theta$ and covariates. The noise terms $\eta_{t}\sim N(0,I_{k})$ and $\alpha_{1}\sim N(a_{1},P_{1})$ are independent. These state dynamics can be combined with the observational level density $g_{t}$ of form $g_{t}(y_{t}\|d_{t}+Z_{t}\alpha_{t},\phi,u_{t}),$ where parameters $\phi$ and the known vector $u_{t}$ are distribution specific and can be omitted in some cases. Currently, following observational level distributions are supported: * • Gaussian distribution: $y_{t}=d_{t}+Z_{t}\alpha_{t}+H_{t}\epsilon_{t}$ with $\epsilon_{t}\sim N(0,I)$. * • Poisson distribution: $g_{t}(y_{t}\|d_{t}+Z_{t}\alpha_{t},u_{t})=\textrm{Poisson}(u_{t}\exp(d_{t}+Z_{t}\alpha_{t}))$, where $u_{t}$ is the known exposure at time $t$. * • Binomial distribution: $g_{t}(y_{t}\|d_{t}+Z_{t}\alpha_{t},u_{t})=\textrm{B}(u_{t},\operatorname{logit}^{-1}(d_{t}+Z_{t}\alpha_{t}))$, where $u_{t}$ is the number of trials and $\operatorname{logit}^{-1}(d_{t}+Z_{t}\alpha_{t})$ is the probability of the success. * • Negative binomial distribution: $g_{t}(y_{t}\|d_{t}+Z_{t}\alpha_{t},\phi,u_{t})=\textrm{NB}(\exp(d_{t}+Z_{t}\alpha_{t}),\phi,u_{t})$, where $u_{t}\exp(d_{t}+Z_{t}\alpha_{t})$ is the expected value, $\phi$ is the dispersion parameter, and $u_{t}$ is a known offset term. * • Gamma distribution: $g_{t}(y_{t}\|d_{t}+Z_{t}\alpha_{t},\phi,u_{t})=\textrm{Gamma}(\exp(d_{t}+Z_{t}\alpha_{t}),\phi,u_{t})$, where $u_{t}\exp(d_{t}+Z_{t}\alpha_{t})$ is the expected value, $\phi$ is the shape parameter, and $u_{t}$ is a known offset term. * • Stochastic volatility model: $g_{t}(y_{t}\|Z_{t}\alpha_{t})=\exp(\alpha_{t}/2)\epsilon_{t}$, with $\epsilon_{t}\sim N(0,1)$. Here the state dynamics is also fixed as $\alpha_{t+1}=\mu+\rho(\alpha_{t}-\mu)+\sigma_{\eta}\eta_{t}$, with $\eta_{t}\sim N(0,1)$ and $\alpha_{1}\sim N(\mu,\sigma^{2}_{\eta}/(1-\rho^{2}))$. For multivariate models, these distributions can be combined arbitrarily, except the stochastic volatility model case which is currently handled separately. Also for fully Gaussian model, the observational level errors $\epsilon_{t}$ can be correlated across time series. ### Other state space models The general non-linear Gaussian model in the bssm has following form: $\displaystyle y_{t}$ $\displaystyle=Z(t,\alpha_{t},\theta)+H(t,\alpha_{t},\theta)\epsilon_{t},$ $\displaystyle\alpha_{t+1}$ $\displaystyle=T(t,\alpha_{t},\theta)+R(t,\alpha_{t},\theta)\eta_{t},$ $\displaystyle\alpha_{1}$ $\displaystyle\sim N(a_{1}(\theta),P_{1}(\theta)),$ with $t=1,\ldots,n$, $\epsilon_{t}\sim N(0,\textrm{I}_{p})$, and $\eta\sim N(0,\textrm{I}_{k})$. The bssm package also supports models where the state equation is defined as a continuous-time diffusion model of the form $\textrm{d}\alpha_{t}=\mu(\alpha_{t},\theta)\textrm{d}t+\sigma(\alpha_{t},\theta)\textrm{d}B_{t},\quad t\geq 0,$ where $B_{t}$ is a Brownian motion and where $\mu$ and $\sigma$ are scalar- valued functions, with the univariate observation density $p(y_{k}\|\alpha_{k})$ defined at integer times $k=1\ldots,n$. ## Inference methods The main goal of bssm is to facilitate easy-to-use full Bayesian inference of the joint posterior $p(\alpha,\theta\|y)$ for models discussed in Section Supported models. The inference methods implemented in bssm are based on a factorised approach where the joint posterior of hyperparameters $\theta$ and latent states $\alpha$ is given as $p(\alpha,\theta\|y)\propto p(\theta)p(\alpha,y\|\theta)=p(\theta)p(y\|\theta)p(\alpha\|y,\theta),$ where $p(y\|\theta)$ is the parameter marginal likelihood and $p(\alpha\|y,\theta)$ is the smoothing distribution. All the inference algorithms are based on a Markov chain Monte Carlo on the parameters $\theta$, whose single iteration may be summarised as follows: 1: Draw a proposal $\theta^{\prime}\sim N(\theta^{i-1},\Sigma_{i-1})$. 2: Calculate the (approximate) marginal likelihood $\hat{p}(y\|\theta^{\prime})$. 3: Accept the proposal with probability $\alpha:=\min\Big{\\{}1,\frac{p(\theta^{\prime})\hat{p}(y\|\theta^{\prime})}{p(\theta^{i-1})\hat{p}(y\|\theta^{i-1})}\Big{\\}}$. 4: If the proposal $\theta^{\prime}$ is accepted, set $\theta^{i}=\theta^{\prime}$. Otherwise, set $\theta^{i}=\theta^{i-1}$. 5: Adapt the proposal covariance matrix $\Sigma_{i-1}\to\Sigma_{i}$. Algorithm 1 One iteration of MCMC algorithm for sampling $p(\theta\|y)$. The adaptation step 5 in bssm currently implements the robust adaptive Metropolis algorithm (Vihola, 2012) with fixed target acceptance rate (0.234 by default) provided by the ramcmc package (Helske, 2016). The (approximate) marginal likelihood $\hat{p}(y\|\theta)$ takes different forms, leading to different inference algorithms, discussed below. ### Direct inference: marginal algorithm and particle MCMC The simplest case is with a linear-Gaussian SSM, where we can use the exact marginal likelihood $\hat{p}(y\|\theta)=p(y\|\theta)$, in which case Algorithm 1 reduces to (an adaptive) random-walk Metropolis algorithm targeting the posterior marginal of the parameters $\theta$. Inference from the full posterior may be done using the simulation smoothing algorithm (Durbin and Koopman, 2002) conditional to the sampled hyperparameters. The other ‘direct’ option, which can be used with any model, is using the bootstrap particle filter (BSF) (Gordon et al., 1993), which leads to a _random_ $\hat{p}(y\|\theta)$ which is an unbiased estimator of $p(y\|\theta)$. In this case, Algorithm 1 reduces to (an adaptive) particle marginal Metropolis-Hastings (Andrieu et al., 2010). Full posterior inference is achieved simultaneously, by picking particle trajectories based on their ancestries as in the filter-smoother algorithm (Kitagawa, 1996). Note that with BSF, the desired acceptance rate needs to be lower, depending on the number of particles used (Doucet et al., 2015). ### Approximate inference: Laplace approximation and the extended Kalman filter The direct BSF discussed above may be used with any non-linear and/or non- Gaussian model, but may be slow and/or poor mixing. To alleviate this, bssm provides computationally efficient (intermediate) approximate inference in case of non-Gaussian observation models in Section Models with linear-Gaussian state dynamics, and in case of non-linear dynamics in Section Other state space models. With non-Gaussian models of Section Models with linear-Gaussian state dynamics, we use an approximating Gaussian model $\tilde{p}(y,\alpha\|\theta)$ which is a Laplace approximation of $p(\alpha,y\|\theta)$ following (Durbin and Koopman, 2000). We write the likelihood as follows $\displaystyle p(y\|\theta)$ $\displaystyle=\int p(\alpha,y\|\theta)\textrm{d}\alpha=\tilde{p}(y\|\theta)E\left[\frac{p(y\|\alpha,\theta)}{\tilde{p}(y\|\alpha,\theta)}\right],$ where $\tilde{p}(y\|\theta)$ is the likelihood of the Laplace approximation and the expectation is taken with respect to its conditional $\tilde{p}(\alpha\|y,\theta)$ (Durbin and Koopman, 2012). Indeed, denoting $\hat{\alpha}$ as the mode of $\tilde{p}(\alpha\|\theta,y)$, we may write $\displaystyle\log p(y\|\theta)$ $\displaystyle=\log\tilde{p}(y\|\theta)+\log\frac{p(y\|\hat{\alpha},\theta)}{\tilde{p}(y\|\hat{\alpha},\theta)}+\log E\left[\frac{p(y\|\alpha,\theta)/p(y\|\hat{\alpha},\theta)}{\tilde{p}(y\|\alpha,\theta)/\tilde{p}(y\|\hat{\alpha},\theta)}\right].$ If $\tilde{p}$ resembles $p$ with typical values of $\alpha$, the latter logarithm of expectation is zero. We take $\hat{p}(y\|\theta)$ as the expression on the right, dropping the expectation. When $\hat{p}$ is approximate, the MCMC algorithm targets an approximate posterior marginal. Approximate full inference may be done analogously as in Section Direct inference: marginal algorithm and particle MCMC, by simulating trajectories conditional to the sampled parameter configurations $\theta^{i}$. We believe that approximate inference is often good enough for model development, but strongly recommend using post-correction as discussed in Section Post-processing by importance weighting to check the validity of the final inference. In addition to these algorithms, bssm also supports $\hat{p}(y\|\theta)$ based on the extended KF (EKF) or iterated EKF (IEKF) (Jazwinski, 1970) which can be used for models with non-linear dynamics (Section Other state space models). Approximate smoothing based on (iterated) EKF is also supported. It is also possible to perform direct inference as in Section Direct inference: marginal algorithm and particle MCMC, but instead of the BSF, employ particle filter based on EKF (Van Der Merwe et al., 2001). ### Post-processing by importance weighting The inference methods in Section Approximate inference: Laplace approximation and the extended Kalman filter are computationally efficient, but come with a bias. The bssm implements importance-sampling type post-correction as discussed in (Vihola et al., 2020). Indeed, having MCMC samples $(\theta^{i})$ from the approximate posterior constructed as in Section Approximate inference: Laplace approximation and the extended Kalman filter, we may produce (random) weights and latent states, such that the weighted samples form estimators which are consistent with respect to the true posterior $p(\alpha,\theta\|y)$. The primary approach which we recommend for post-correction is based on a “$\psi$-APF’ ’ — a particle filter using the Gaussian approximations of Section Approximate inference: Laplace approximation and the extended Kalman filter. In essence, this particle filter employs the dynamics and a look-ahead strategy coming from the approximation, which leads to low-variance estimators; see (Vihola et al., 2020) and package vignettes111https://cran.r-project.org/package=bssm/vignettes/psi_pf.html for more detailed description. Naturally $\psi$-APF can also be used in place of BSF in direct inference of Section Direct inference: marginal algorithm and particle MCMC. ### Direct inference using approximation-based delayed acceptance An alternative to approximate MCMC and post-correction, bssm also supports an analogous delayed acceptance method (Christen and Fox, 2005; Banterle et al., 2019) (here denoted by DA-MCMC). This algorithm is similar to 1, but in case of ‘acceptance’, leads to second-stage acceptance using the same weights as the post-correction would; see (Vihola et al., 2020) for details. Note that as in direct approach for non-Gaussian/non-linear models, the desired acceptance rate with DA-MCMC should be lower than the default 0.234. The DA-MCMC also leads to consistent posterior estimators, and often outperforms the direct particle marginal Metropolis-Hastings. However, empirical findings (Vihola et al., 2020) and theoretical considerations (Franks and Vihola, 2020) suggest that approximate inference with post- correction may often be preferable. The bssm supports parallelisation with post-correction using OpenMP, which may further promote the latter. ### Inference with diffusion state dynamics For general continuous-time diffusion models, the transition densities are intractable. The bssm uses Millstein time-discretisation scheme for approximate simulation, and inference is based on the corresponding BSF. Fine time-discretisation mesh gives less bias than the coarser one, with increased computational complexity. The DA and IS approaches can be used to speed up the inference by using coarse discretisation in the first stage and then using more fine mesh in the second stage. For comparison of DA and IS approaches in case of geometric Brownian motion model, see (Vihola et al., 2020). ## Using the bssm package Main functions of bssm related to the MCMC sampling, approximations, and particle filtering are written in C++, with help of Rcpp (Eddelbuettel and François, 2011) and RcppArmadillo (Eddelbuettel and Sanderson, 2014) packages. On the R side, the package uses S3 methods to provide a relatively unified workflow independent of the type of the model one is working with. The model building functions such as bsm_ng and svm are used to construct the model objects of same name which can be then passed to other methods, such as logLik and run_mcmc which compute the log-likelihood value and run MCMC algorithm respectively. We will now briefly describe the main functionality of bssm. For more detailed descriptions of different functions and their arguments, see the corresponding documentation in R and the package vignettes. ### Constructing the model For models with linear-Gaussian state dynamics, bssm includes some predefined models such as bsm_lg and bsm_ng for univariate Gaussian and non-Gaussian structural time series models with external covariates, for which user only needs to supply the data and priors for unknown model parameters. In addition, bssm supports general model building functions ssm_ulg, ssm_mlg for custom univariate and multivariate Gaussian models and ssm_ung, and ssm_mng for their non-Gaussian counterparts. For these models, users need to supply their own R functions for the evaluation of the log prior density and for updating the model matrices given the current value of the parameter vector $\theta$. It is also possible to avoid defining the matrices manually by leveraging the formula interface of the KFAS package (Helske, 2017) together with as_bssm function which converts KFAS model to a bssm equivalent model object. This is especially useful in case of complex multivariate models with covariates. As an example, consider a Gaussian local linear trend model of the form $\displaystyle y_{t}$ $\displaystyle=\mu_{t}+\epsilon_{t},$ $\displaystyle\mu_{t+1}$ $\displaystyle=\mu_{t}+\nu_{t}+\eta_{t},$ $\displaystyle\nu_{t+1}$ $\displaystyle=\nu_{t}+\xi_{t},$ with zero-mean Gaussian noise terms $\epsilon_{t},\eta_{t},\xi_{t}$ with unknown standard deviations. Using the time series of the mean annual temperature (in Fahrenheit) in New Haven, Connecticut, from 1912 to 1971 (available in the datasets package) as an example, this model can be built with bsm function as library("bssm")data("nhtemp", package = "datasets")prior <\- halfnormal(1, 10)bsm_model <\- bsm_lg(y = nhtemp, sd_y = prior, sd_level = prior, sd_slope = prior) Here we use helper function halfnormal which defines half-Normal prior distribution for the standard deviation parameters, with the first argument defining the initial value of the parameter, and second defines the scale parameter of the half-Normal distribution. Other prior options are normal, tnormal (truncated normal), gamma, and uniform. As an example of multivariate model, consider bivariate Poisson model with latent random walk model, defined as $\displaystyle y_{i,t}$ $\displaystyle\sim\textrm{Poisson}(\exp(x_{t})),\quad i=1,2,$ $\displaystyle x_{t+1}$ $\displaystyle=x_{t}+\eta_{t},$ with $\eta_{t}\sim N(0,\sigma^{2})$, and prior $\sigma\sim\textrm{Gamma}(2,0.01)$. This model can be built with ssm_mng function as # Generate observationsset.seed(1)x <\- cumsum(rnorm(50, sd = 0.2))y <\- cbind( rpois(50, exp(x)), rpois(50, exp(x)))# Log prior density functionprior_fn <\- function(theta) { dgamma(theta, 2, 0.01, log = TRUE)}# Model parameters from hyperparametersupdate_fn <\- function(theta) { list(R = array(theta, c(1, 1, 1)))}# define the modelmng_model <\- ssm_mng(y = y, Z = matrix(1,2,1), T = 1, R = 0.1, P1 = 1, distribution = "poisson", init_theta = 0.1, prior_fn = prior_fn, update_fn = update_fn) Here the user-defined functions prior_fn and update_fn define the log-prior for the model and how the model components depend on the hyperparameters $\theta$ respectively. For models where the state equation is no longer linear-Gaussian, we use pointer-based interface by defining all model components as well as functions defining the Jacobians of $Z(\cdot)$ and $T(\cdot)$ needed by the extended Kalman filter as C++ snippets. General non-linear Gaussian model can be defined with the function ssm_nlg. Discretely observed diffusion models where the state process is assumed to be continuous stochastic process can be constructed using the ssm_sde function, which takes pointers to C++ functions defining the drift, diffusion, the derivative of the diffusion function, and the log-densities of the observations and the prior. As an example of the latter, let us consider an Ornstein–Uhlenbeck process $\textrm{d}\alpha_{t}=\rho(\nu-\alpha_{t})\textrm{d}t+\sigma\textrm{d}B_{t},$ with parameters $\theta=(\phi,\nu,\sigma)=(0.5,2,1)$ and the initial condition $\alpha_{0}=1$. For observation density, we use Poisson distribution with parameter $\exp(\alpha_{k})$. We first simulate a trajectory $x_{0},\ldots,x_{n}$ using the sde.sim function from the sde package (Iacus, 2016) and use that for the simulation of observations $y$: library("sde")x <\- sde.sim(t0 = 0, T = 100, X0 = 1, N = 100, drift = expression(0.5 * (2 - x)), sigma = expression(1), sigma.x = expression(0))y <\- rpois(100, exp(x[-1])) We then compile and build the model as Rcpp::sourceCpp("ssm_sde_template.cpp")pntrs <\- create_xptrs()sde_model <\- ssm_sde(y, pntrs$drift, pntrs$diffusion, pntrs$ddiffusion, pntrs$obs_density, pntrs$prior, c(0.5, 2, 1), 1, FALSE) The templates for the C++ functions for SDE and non-linear Gaussian models can be found from the package vignettes on the CRAN222https://CRAN.R-project.org/package=bssm. ### Markov chain Monte Carlo in bssm The main purpose of the bssm is to allow computationally efficient MCMC-based inference for various state space models. For this task, a method run_mcmc can be used. The function takes a number of arguments, depending on the model class, but for many of these, default values are provided. For linear-Gaussian models, we only need to supply the number of iterations. Using the previously created local linear trend model for the New Haven temperature data of Section Constructing the model, we run an MCMC with 100,000 iterations where first 10,000 is discarded as a burn-in (burn-in phase is also used for the adaptation of the proposal distribution): mcmc_bsm <\- run_mcmc(bsm_model, iter = 1e5, burnin = 1e4) The print method for the output of the MCMC algorithms gives a summary of the results, and detailed summaries for $\theta$ and $\alpha$ can be obtained using summary function. For all MCMC algorithms, bssm uses so-called jump chain representation of the Markov chain $X_{1},\ldots,X_{n}$, where we only store each accepted $X_{k}$ and the number of steps we stayed on the same state. So for example if $X_{1:n}=(1,2,2,1,1,1)$, we present such chain as $\tilde{X}=(1,2,1)$, $N=(1,2,3)$. This approach reduces the storage space and makes it more computationally efficient to use importance sampling type correction algorithms. One drawback of this approach is that the results from the MCMC run correspond to weighted samples from the target posterior, so some of the commonly used postprocessing tools need to be adjusted. Of course, in case of other methods than IS-weighting, the simplest option is to just expand the samples to typical Markov chain using the stored counts $N$. This can be done using the function expand_sample which returns an object of class "mcmc" of the coda package (Plummer et al., 2006) (thus the plotting and diagnostic methods of coda can also be used). We can also directly transform the posterior samples to a "data.frame" object by using as.data.frame method for the MCMC output (for IS-weighting, the returned data frame contains additional column weights). This is useful for example for visualization purposes with the ggplot2 (Wickham, 2016) package: library("ggplot2")d <\- as.data.frame(mcmc_bsm, variable = "theta")ggplot(d, aes(x = value)) + geom_density(bw = 0.1, fill = "#9ebcda") + facet_wrap(~ variable, scales = "free") + theme_bw() Figure 1: Posterior densities of hyperparameters $\theta$ of the linear- Gaussian model for nhtemp data. Figure 1 shows the estimated posterior densities of the three standard deviation parameters of the model. The relatively large observational level standard deviation $\sigma_{y}$ suggests that the underlying latent temperature series is much smoother than the observed series, which can also be seen from Figure 2 which show the original observations (black dots) spread around the estimated temperature series (solid line). library("dplyr")d <\- as.data.frame(mcmc_bsm, variable = "states")summary_y <\- d %>% filter(variable == "level") %>% group_by(time) %>% summarise(mean = mean(value), lwr = quantile(value, 0.025), upr = quantile(value, 0.975))ggplot(summary_y, aes(x = time, y = mean)) + geom_ribbon(aes(ymin = lwr, ymax = upr), alpha = 0.25) + geom_line() + geom_point(data = data.frame(mean = nhtemp, time = time(nhtemp))) + theme_bw() + xlab("Year") + ylab("Mean annual temperature in New Haven") Figure 2: Observed annual average temperatures in New Haven (black dots) and predicted mean (solid line) with 95% prediction intervals (grey ribbon) from ‘bssm‘. For non-Gaussian models the default MCMC algorithm is approximate inference based on Laplace approximation combined with importance sampling post- correction (Section Post-processing by importance weighting). It is also possible to perform first approximate MCMC using the argument mcmc_type = "approx", and then perform the post-correction step using the results from the approximate MCMC. In doing so, we can also use the function suggest_N to find a suitable number of particles $N$ for $\psi$-APF in the spirit of Doucet et al. (2015): out_approx <\- run_mcmc(mng_model, mcmc_type = "approx", iter = 50000)est_N <\- suggest_N(mng_model, out_approx)out_exact <\- post_correct(mng_model, out_approx, particles = est_N$N) The function suggest_N computes the standard deviation of the logarithm of the post-correction weights (i.e. the random part of log-likelihood of $\psi$-APF) at the approximate MAP estimator of $\theta$ using a range of $N$ and returns a list with component N which is the smallest number of particles where the standard deviation was less than one. For small and moderate problems typically 10-20 particles is enough. ### Filtering and smoothing The bssm also offers separate methods for performing (approximate) state filtering and smoothing which may be useful in some custom settings. For LGSSM, methods kfilter and smoother perform Kalman filtering and smoothing. For non-Gaussian models with linear-Gaussian dynamics, approximate filtering and smoothing estimates can be obtained by calls to kfilter and smoother, in which case these functions first construct an approximating Gaussian model for which the Kalman filter/smoother is then applied. For non- linear models defined by nlg_ssm we can run approximate filtering using the extended Kalman filter with the function ekf, the unscented Kalman filter with the function ukf, or the iterated EKF (IEKF) by changing the argument iekf_iter of the ekf function. Function ekf_smoother can be used for smoothing based on EKF/IEKF. For particle filtering the bssm package supports general bootstrap particle filter for all model classes of the bssm (function bootstrap_filter). For nlg_ssm, extended Kalman particle filtering (Van Der Merwe et al., 2001) is also supported (function ekpf_filter). For particle smoothing, function particle_smoother with the smoothing based on BSF is available for all models. In addition, $\psi$-APF (using argument method = "psi") is available for all models except of ssm_sde class. Currently, only filter-smoother approach (Kitagawa, 1996) for particle smoothing is supported. ## Comparison of IS-MCMC and HMC Vihola et al. (2020) compared the computational efficiency of delayed acceptance MCMC and importance sampling type MCMC approaches in various settings. Here we make a small experiment comparing the generic Hamiltonian Monte Carlo using the NUTS sampler (Hoffman and Gelman, 2014) with rstan, and IS-MCMC with bssm. Given that the bssm is specialized for state space models whereas Stan is a general purpose tool suitable for wider range of problems, it is to be expected that bssm performs better in terms of computational efficiency. The purpose of this experiment is to illustrate this fact, i.e., that there is still demand for specialized algorithms for various types of statistical models. For complete code of the experiment, see Appendix: Code for section Comparison of IS-MCMC and HMC. We consider the case of a random walk with drift model with negative binomial observations and some known covariate $x_{t}$, defined as $\displaystyle y_{t}$ $\displaystyle\sim\textrm{NB}(\exp(\beta x_{t}+\mu_{t}),\phi)$ $\displaystyle\mu_{t+1}$ $\displaystyle=\mu_{t}+\nu_{t}+\eta_{t},$ $\displaystyle\nu_{t+1}$ $\displaystyle=\nu_{t},$ with zero-mean Gaussian noise term $\eta_{t}$ with unknown standard deviation $\sigma_{\mu}$. Based on this we simulate one realization of $y$ and $x$ with $n=200$, $\phi=5$, $\beta=-0.9$, $\nu=0.01$, $\sigma_{\mu}=0.1$. For the IS approach we use ng_bsm function for model building, with prior variances 100 and 0.01 for the initial states $\mu_{1}$ and $\nu_{1}$. For hyperparameters, we used fairly uninformative half-Normal distribution with standard deviation 0.5 for $\sigma_{\mu}$ and 0.1 for $\sigma_{\nu}$. We then ran the IS-MCMC algorithm with run_mcmc using a burn-in phase of length 10,000 and run 50,000 iterations after the burn-in, with 10 particles per SMC. Using the same set up, we ran the MCMC with rstan using 15,000 iterations (with first 5000 used for warm-up). Note that in order to avoid sampling problems, it was necessary to tweak the default control parameters of the sampler (see Appendix). Table 1 shows the results. We see both methods produce identical results (within the Monte Carlo error), but while rstan produces similar Monte Carlo standard errors with smaller amount of total iterations than bssm, the total computation time of rstan is almost 80 times higher than with bssm (58 minutes versus 45 seconds), which suggests that for these type of problems it is highly beneficial to take advantage of the known model structure and available approximations versus general Bayesian software such as Stan which makes no distinction between latent states $\alpha$ and hyperparameters $\theta$. Table 1: Estimates of posterior mean, standard deviation and Monte Carlo standard error of the mean for hyperparameters $\theta$ and latent states for last time point for the example model. \| bssm \| rstan ---\|---\|--- \| Mean \| SD \| MCSE \| Mean \| SD \| MCSE $\sigma_{\mu}$ \| $0.092$ \| $0.037$ \| $9\times 10^{-4}$ \| $0.090$ \| $0.036$ \| $9\times 10^{-4}$ $\sigma_{\nu}$ \| $0.003$ \| $0.003$ \| $5\times 10^{-5}$ \| $0.003$ \| $0.003$ \| $7\times 10^{-5}$ $\phi$ \| $5.392$ \| $0.910$ \| $2\times 10^{-2}$ \| $5.386$ \| $0.898$ \| $1\times 10^{-2}$ $\beta$ \| $-0.912$ \| $0.056$ \| $1\times 10^{-3}$ \| $-0.911$ \| $0.056$ \| $7\times 10^{-4}$ $\mu_{200}$ \| $6.962$ \| $0.346$ \| $5\times 10^{-3}$ \| $6.965$ \| $0.349$ \| $4\times 10^{-3}$ $\nu_{200}$ \| $0.006$ \| $0.020$ \| $3\times 10^{-4}$ \| $0.006$ \| $0.019$ \| $2\times 10^{-4}$ ## Conclusions State space models are a flexible tool for analysing a variety of time series data. Here we introduced the R package bssm for fully Bayesian state space modelling for a large class of models with several alternative MCMC sampling strategies. All computationally intensive parts of the package are implemented with C++ with parallel computation support for IS-MCMC making it an attractive option for many common models where relatively accurate Gaussian approximations are available. Compared to early versions of the bssm package, the option to define R functions for model updating and prior evaluation have lowered the bar for analysing custom models. The package is also written in a way that it is relatively easy to extend to new model types similar to current bsm_lg in future. The bssm package could be expanded to allow other proposal adaptation schemes such as adaptive Metropolis algorithm by Haario et al. 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URL https://ggplot2.tidyverse.org. ## Appendix: Code for section Comparison of IS-MCMC and HMC library("bssm")# Simulate the dataset.seed(123)n <\- 200sd_level <\- 0.1drift <\- 0.01beta <\- -0.9phi <\- 5level <\- cumsum(c(5, drift + rnorm(n - 1, sd = sd_level)))x <\- 3 + (1:n) * drift + sin(1:n + runif(n, -1, 1))y <\- rnbinom(n, size = phi, mu = exp(beta * x + level))# Construct model for bssmbssm_model <\- bsm_ng(y, xreg = x, beta = normal(0, 0, 10), phi = halfnormal(1, 10), sd_level = halfnormal(0.1, 1), sd_slope = halfnormal(0.01, 0.1), a1 = c(0, 0), P1 = diag(c(10, 0.1)^2), distribution = "negative binomial")# run the MCMCfit_bssm <\- run_mcmc(bssm_model, iter = 60000, burnin = 10000, particles = 10, seed = 1)# create the Stan modellibrary("rstan")stan_model <\- "data { int<lower=0> n; // number of data points int<lower=0> y[n]; // time series vector[n] x; // covariate}parameters { real<lower=0> sd_slope; real<lower=0> sd_level; real beta; real<lower=0> phi; // instead of working directly with true state variables // it is often suggested use standard normal variables in sampling // and reconstruct the true parameters in transformed parameters block // this should make sampling more efficient although coding the model // is less intuitive. vector[n] level_std; // N(0, 1) level noise vector[n] slope_std; // N(0, 1) slope noise}transformed parameters { vector[n] level; vector[n] slope; // construct the actual states level[1] = 10 * level_std[1]; slope[1] = 0.1 * slope_std[1]; slope[2:n] = slope[1] + cumulative_sum(sd_slope * slope_std[2:n]); level[2:n] = level[1] + cumulative_sum(slope[1:(n-1)]) + cumulative_sum(sd_level * level_std[2:n]);}model { beta ~ normal(0, 10); phi ~ normal(0, 10); sd_slope ~ normal(0, 0.1); sd_level ~ std_normal(); // standardised noise terms level_std ~ std_normal(); slope_std ~ std_normal(); y ~ neg_binomial_2_log(level + beta * x, phi);}"stan_data <\- list(n = n, y = y, x = x)stan_inits <\- list(list(sd_level = 0.1, sd_slope = 0.01, phi = 1, beta = 0))# need to increase adapt_delta and max_treedepth in order to avoid divergencesfit_stan <\- stan(model_code = stan_model, data = stan_data, iter = 15000, warmup = 5000, control = list(adapt_delta = 0.99, max_treedepth = 12), init = stan_inits, chains = 1, refresh = 0, seed = 1)d_stan <\- summary(fit_stan, pars = c("sd_level", "sd_slope", "phi", "beta", "level[200]", "slope[200]" ))$summary[,c("mean", "sd", "se_mean")]d_bssm <\- summary(fit_bssm, variable = "both", return_se = TRUE)# Parameter estimates:d_stand_bssm$thetad_bssm$states$Mean[200,]d_bssm$states$SD[200,]d_bssm$states$SE[200,]# Timings:sum(get_elapsed_time(fit_stan))fit_bssm$time[3] _Jouni Helske Department of Mathematics and Statistics University of Jyväskylä Finland ORCiD: 0000-0001-7130-793X <EMAIL_ADDRESS>_ _Matti Vihola Department of Mathematics and Statistics University of Jyväskylä Finland ORCiD: 0000-0002-8041-7222 <EMAIL_ADDRESS>_
††thanks: Present address: Samsung Electronics, Gyeonggi–do 16677, Republic of Korea # Characterization of a flux-driven Josephson parametric amplifier with near quantum-limited added noise for axion search experiments Çağlar Kutlu<EMAIL_ADDRESS>Korea Advanced Institute of Science and Technology, Daejeon 34051, Republic of Korea Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea Arjan F. van Loo Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351–0198, Japan Sergey V. Uchaikin Andrei N. Matlashov Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea Doyu Lee Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea Seonjeong Oh Jinsu Kim Korea Advanced Institute of Science and Technology, Daejeon 34051, Republic of Korea Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea Woohyun Chung Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea Yasunobu Nakamura Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351–0198, Japan Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Meguro–ku, Tokyo 153–8904, Japan Yannis K. Semertzidis Korea Advanced Institute of Science and Technology, Daejeon 34051, Republic of Korea Center for Axion and Precision Physics Research, Institute for Basic Science, Daejeon 34051, Republic of Korea ###### Abstract The axion, a hypothetical elementary pseudoscalar, is expected to solve the strong _CP_ problem of QCD and is also a promising candidate for dark matter. The most sensitive axion search experiments operate at millikelvin temperatures and hence rely on instrumentation that carries signals from a system at cryogenic temperatures to room temperature instrumentation. One of the biggest limiting factors affecting the parameter scanning speed of these detectors is the noise added by the components in the signal detection chain. Since the first amplifier in the chain limits the minimum noise, low-noise amplification is of paramount importance. This paper reports on the operation of a flux-driven Josephson parametric amplifier (JPA) operating at around $2.3\text{\,}\mathrm{GHz}$ with added noise approaching the quantum limit. The JPA was employed as a first stage amplifier in an experimental setting similar to the ones used in haloscope axion detectors. By operating the JPA at a gain of $19\text{\,}\mathrm{dB}$ and cascading it with two cryogenic amplifiers operating at $4\text{\,}\mathrm{K}$, noise temperatures as low as $120\text{\,}\mathrm{mK}$ were achieved for the whole signal detection chain. ## I Introduction Axions are spin-0 particles that emerge as a result of the Peccei-Quinn mechanism which was originally proposed as a solution to the strong _CP_ problem of quantum chromodynamics[1, 2]. They were also identified as viable candidates for all or a fraction of the cold dark matter in our universe[3, 4, 5]. It is possible to detect axions upon their conversion to microwave photons, using resonant cavities immersed in high magnetic fields[6]. Since the axion mass is unknown, these detectors employ a mechanism to scan different frequencies corresponding to different axion masses. The scanning rate of such detectors scales with $1/T_{\mathrm{sys}}^{2}$, where $T_{\mathrm{sys}}$ is the system noise background characterized in units of temperature. It can be decomposed as $T_{\mathrm{sys}}=T_{\mathrm{cav}}+T_{\mathrm{add}}$, where the first term denotes the noise temperature accompanying the signal itself and the second one denotes the noise added by the signal detection chain. Throughout this work, noise temperature refers to the added noise unless otherwise stated. In order to reduce $T_{\mathrm{cav}}$, the cavity is cooled to millikelvin temperatures. If the first amplifier has sufficiently high gain ($G_{1}$), its noise temperature ($T_{1}$) will be the dominant contribution to $T_{\mathrm{add}}$ as given by the well-known relation[7]: $T_{\mathrm{add}}=T_{1}+\frac{T_{\mathrm{rest}}}{G_{1}}$ where $T_{\mathrm{rest}}$ is the noise temperature of the whole chain except the first amplifier. Amplifiers based on Josephson junctions including microstrip superconducting quantum interference device amplifiers (MSA) and JPA have already been shown to be capable of gains higher than $30\text{\,}\mathrm{dB}$, and noise temperatures approaching the quantum limit[8, 9]. While an MSA has an internal shunt resistor used for biasing which hinders noise performance[10, 11], by design the JPA requires no resistive element to operate. Several experiments presently searching for dark matter axions have already adopted the JPA as the first amplifier[12, 13, 14]. In this work, the frequency coverage, gain and noise properties of a flux- driven JPA for use in an axion dark matter experiment operating around $2.3\text{\,}\mathrm{GHz}$ are investigated. The power spectral density of the noise accompanying a signal measured in an impedance matched environment can be given as [15] : $\displaystyle S_{n}(f,T)=hf\left[\frac{1}{\exp{\left(\frac{hf}{k_{B}T}\right)}-1}+\frac{1}{2}\right]$ (1) where $h$ is Planck’s constant and $k_{B}$ is Boltzmann’s constant. The first term in the brackets is the mean number of quanta at frequency $f$ at the bath temperature $T$ and the second term is the contribution from zero-point fluctuations. The lower limit on noise temperature for linear phase- insensitive amplifiers is given by[16] $T_{Q}=\lim_{T\rightarrow 0}S_{n}(f,T)/(k_{B})=hf/(2k_{B})$ which is about $55.2\text{\,}\mathrm{mK}$ at $2.3\text{\,}\mathrm{GHz}$. Using a $2.3\text{\,}\mathrm{GHz}$ flux-driven JPA $T_{\mathrm{add}}\approx$120\text{\,}\mathrm{mK}$$ is achieved. This corresponds to a $T_{\mathrm{sys}}\approx$190\text{\,}\mathrm{mK}$$ for an axion haloscope experiment running at a bath temperature of $50\text{\,}\mathrm{mK}$. The lower bound for $T_{\mathrm{sys}}$ is given by the standard quantum limit[17] $T_{\mathrm{SQL}}=2T_{Q}$ which is about $110\text{\,}\mathrm{mK}$ at $2.3\text{\,}\mathrm{GHz}$. ## II Flux-driven JPA The equivalent circuit diagram of the tested device is shown in Figure 1. It consists of a superconducting quantum interference device (SQUID) attached to the end of a coplanar waveguide $\lambda/4$ resonator that is coupled via a capacitor ($C_{c}$) to the transmission line for the signal input and output. The SQUID acts as a variable inductor whose value depends on the magnetic flux passing through its loop. In the setup, a superconducting coil is used to provide the necessary DC flux ($\phi$) through the SQUID loop in order to tune the resonance frequency ($f_{r}$). Parametric amplification is achieved by modulating the flux through the SQUID using a pump signal. The pump tone is provided by a separate transmission line inductively coupled to the SQUID. The JPA is operated in the three-wave mixing mode[18] where the pump ($f_{p}$), the signal ($f_{s}$), and the idler ($f_{i}$) frequencies satisfy the relation $f_{p}=f_{s}+f_{i}$. The signal input and output share the same port. A circulator is used to separate them. Since the $\lambda/4$ resonator only allows odd harmonics, there is no measurable pump leakage to the output line. This prevents the stronger pump tone from saturating the rest of the amplifiers in the chain[19]. Figure 2 shows a schematic for the axion search experimental setup. Figure 1: Equivalent circuit diagram of the JPA sample. The JPA was fabricated by photolithography of a Nb layer, deposited on a $0.3\text{\,}\mathrm{mm}$ thick Si substrate. The SQUID was placed on top of the Nb layer by E-beam lithography followed by shadow evaporation[20, 21]. The sample was attached to a printed circuit board (PCB) and the transmission lines were bonded with Al wires. The PCB was fixed onto a gold plated copper structure and placed inside a superconducting coil. The whole structure was covered tightly with a lead shield and attached to the mixing-chamber (MC) plate using a gold plated copper rod. ## III Measurements When there is no pump tone present, the JPA can be modeled as a resonator with a well-defined quality factor and resonance frequency which are functions of flux. The resonance frequency is estimated from the frequency domain phase response using a parameter fit[22]. The phase response is obtained by doing a transmission S-parameter measurement using a vector network analyzer (VNA) in the configuration as shown in Figure 2. The resonance frequency was measured as a function of the coil current (see Figure 3). It was found that the minimum observable resonance frequency was at $2.18\text{\,}\mathrm{GHz}$ and the maximum was $2.309\text{\,}\mathrm{GHz}$. The lower bound is due to the frequency band of the circulators which spans from $2.15\text{\,}\mathrm{GHz}$ to $2.60\text{\,}\mathrm{GHz}$. At the lower frequencies, the JPA becomes much more sensitive to flux noise due to a higher $\frac{\partial f_{r}}{\partial\phi}$. This work mainly focused on operation with frequencies above $2.2\text{\,}\mathrm{GHz}$. Figure 2: The experimental setup used in all the characterization measurements. SG, VNA, and SA stand for the signal generator, vector network analyzer, and spectrum analyzer, respectively. During this work, the switch that selects between the cavity and the noise source was always kept at the position shown in the figure. The ports IN2 and OUT were used to directly measure the JPA characteristics, bypassing the cavity. The microwave short element shown next to the JPA was used to bypass the JPA for calibration measurements. U1 and U2 are HEMT amplifiers with noise temperatures of $1.5\text{\,}\mathrm{K}$ and $5\text{\,}\mathrm{K}$, respectively. During the experiments, the MC plate temperature was stabilized at $50\pm{}1\text{\,}\mathrm{mK}$. With the temperature fixed, the frequency response of the JPA is determined by three experimental variables: the coil current ($i_{b}$), the pump frequency ($f_{p}$), and the pump power ($P_{p}$). The measurements shown in this work had $i_{b}$ confined to the region where the flux through the SQUID loop is given by $-0.5\phi_{0}<\phi<0$, where $\phi_{0}$ is the magnetic flux quantum. Therefore, $f_{r}$ can be unambiguously converted to $\phi$ or $i_{b}$. All experiments began with a transmission measurement, with the resonance frequency tuned to $2.18\text{\,}\mathrm{GHz}$. This becomes the baseline measurement to be used for the duration of the experiment. When the result was compared to a separate measurement, in which a microwave short was put in place of the JPA, it was found that the baseline obtained via such an off-resonance measurement was at most $0.2\text{\,}\mathrm{dB}$ lossier than an ideal mirror. The JPA gain ($G_{J}$) was estimated by dividing the transmission magnitude response with the baseline’s magnitude response. To investigate the gain behavior, a sweep over the parameters $i_{b}$, $f_{p}$, $P_{p}$ was made and the maximum gain was measured at each point. After each $i_{b}$ tuning step, the resonance frequency is estimated by performing a phase measurement and applying a parameter fit. With the detuning defined as $\delta=f_{p}/2-f_{r}$, the equigain contours had a minimum in necessary pump power around $\delta=0$, as shown in Figure 4(a). It was observed that for resonance frequencies above $2.299\text{\,}\mathrm{GHz}$ the minimum starts to shift to lower detunings which is attributed to pump-induced shifts in resonance frequency[22]. Figure 4(b) shows that the slice of $\delta=0$ can be used to achieve peak gains of up to $30\text{\,}\mathrm{dB}$ along the frequency range of the device. Figure 3: Resonance frequency versus flux obtained by sweeping the coil current and measuring the phase response at each step. One period corresponds to a current of $324.4\text{\,}\mathrm{\SIUnitSymbolMicro A}$. The inset shows the fit performed to estimate the resonance frequency for each applied flux. (a) (b) Figure 4: (a) Maximum gain measured as a function of detuning and pump power for a flux bias corresponding to $f_{r}=$2.29\text{\,}\mathrm{GHz}$$. (b) Maximum gain as a function of frequency and pump power with $f_{p}=2f_{r}$. To investigate noise temperature, a methodology similar to the well-known Y-factor method[23] was used. A $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ cryogenic microwave terminator was used as the noise source. A bias-tee was attached in front for improved thermalization of its inner conductor. These two components were fixed onto a gold-plated copper plate along with a ruthenium oxide temperature sensor and a $100\text{\,}\mathrm{\SIUnitSymbolOhm}$ resistor functioning as a heater. This plate was then fixed onto the MC plate so that the dominant thermalization was through a thin copper wire attached to the MC plate. The noise source was connected to the switch input using a superconducting coaxial cable, which provides thermal isolation while minimizing losses. Using a PID controller, the terminator temperature could be adjusted from $50\text{\,}\mathrm{mK}$ to $1\text{\,}\mathrm{K}$ without affecting the MC plate temperature. The noise power generated by the noise source was measured using a spectrum analyzer (SA) with $1\text{\,}\mathrm{kHz}$ resolution bandwidth after being amplified by the JPA and the rest of the signal detection chain. The power spectra were recorded at noise source temperatures ($T_{s}$) of $60120180\text{\,}\mathrm{mK}$. The power values were converted into power spectral densities (PSD) by dividing them with the noise bandwidth corresponding to the SA settings used. Before each PSD measurement, the JPA gain and passive resonance were measured. From these measurements, it was concluded that there were neither gain changes nor resonance shifts. From the obtained PSD values $S(T_{s})$, a fit was done to a function of the following form independently for each frequency bin (see Figure 5) : $\displaystyle S(T_{s})=(2G_{J}-1)\frac{G_{L}G_{\mathrm{tot}}}{G_{J}}(S_{n}(T_{s})+rk_{B}T_{n}+\gamma)$ (2) where $S_{n}(T_{s})$ is the noise PSD of the source, $G_{\mathrm{tot}}$ is the total gain seen from the reference plane, $G_{L}$ is the loss factor between the $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ terminator and the reference plane and $T_{n}$ is the noise temperature. The reference plane is at the end of the superconducting cable connected to the noise source (see Figure 2). Here, $r$ and $\gamma$ are factors that are explained in Appendix A. Figure 5: The upper plot shows the set of power spectra obtained during a noise temperature measurement performed for a tuning at $f_{r}=$2.305\text{\,}\mathrm{GHz}$$ with $f_{p}=2f_{r}$. The offset $\nu$ is defined as $\nu=f-f_{r}$ where $f$ is the center of the frequency bin at which the power was measured using the spectrum analyzer. $T_{s}$ is the temperature of the noise source. The inset shows three vertical slices which were fit with Equation 2. The lower plot shows the estimated noise temperature of the whole chain as a function of $\nu$. Since the amplifier needs to be tuned along with the cavity during the axion experiment, the noise temperature was investigated at different frequencies. The measurements were done in $5\text{\,}\mathrm{MHz}$ steps from $2.282.305\text{\,}\mathrm{GHz}$. At each step, the pump power and resonance frequency were tuned such that the JPA gain was about $20$ $\mathrm{dB}$. From these measurements (Figure 6) a minimum noise temperature of $120\text{\,}\mathrm{mK}$ was observed at $2.28\text{\,}\mathrm{GHz}$. Figure 6: Total gain and the noise temperature of the whole chain for 6 tuning points with $19.3\pm{}0.5\text{\,}\mathrm{dB}$ JPA gain. Both quantities were estimated from noise temperature measurements. The small variations along tuning frequencies were mainly attributed to the losses due to the microwave components before the JPA. Another important characteristic is the saturation that occurs when a narrowband signal is applied. For stable and predictable operation the JPA must be operated away from the effects of saturation. A common way to quantify the saturation of an amplifier is to determine the input power at which the gain is reduced by $1\text{\,}\mathrm{dB}$ ($P_{1\mathrm{dB}}$). The $P_{1\mathrm{dB}}$ was measured at $\delta=0$ for different frequencies and different pump powers corresponding to different gains. It is evident from the results (see Figure 7) that an axion-like signal with an expected power of $-180\text{\,}\mathrm{\milli}$ is far from saturating the device. While saturation from narrowband signals is avoidable to a certain extent, it was observed that thermal noise at the input can also saturate and alter the behavior of the device. For frequencies below $2.28\text{\,}\mathrm{GHz}$ with gains above $23\text{\,}\mathrm{dB}$, the device started showing saturated behavior with thermal noise when the noise source temperature was raised above $120\text{\,}\mathrm{mK}$, which was done to measure noise temperature. While this does not necessarily mean that the device is unusable below these frequencies, it renders the direct measurement of the noise temperature using a noise source unreliable for these frequency and gain regions. Figure 7: Saturation measurements for three different $f_{r}$. Each measurement was done by sweeping the signal powers from the VNA and observing at which input power the gain reduces by 1 dB. The horizontal axis corresponds to the unsaturated gain measured with the lowest signal power available from VNA. ## IV Conclusion In conclusion, a flux-driven JPA, tunable in the range $2.22.305\text{\,}\mathrm{GHz}$ was demonstrated and determined to be operational for use in axion search experiments. The added noise temperatures of the receiver chain were measured using a noise source at a location as close as possible to the origin of the axion signal. With an added noise temperature of $120\text{\,}\mathrm{mK}$ the system was shown to reach $T_{\mathrm{sys}}\approx 1.7T_{\mathrm{SQL}}$. This is the first record of $T_{\mathrm{sys}}$ below $2T_{\mathrm{SQL}}$ for an axion haloscope setup operating below $10\text{\,}\mathrm{GHz}$. The saturation input power for the JPA was observed to be more than adequate for an axion-like signal. Currently, the tested JPA is being used as part of a KSVZ[24, 25] sensitive axion search experiment at the Center for Axion and Precision Physics Research (CAPP). The system is taking physics data with a scanning speed that has been improved more than an order of magnitude. We expect that further optimization of the JPA design could result in improved instantaneous bandwidth and tuning range. This work was supported by the Institute for Basic Science (IBS-R017–D1–2021–a00) and JST ERATO (Grant No. JPMJER1601). A. F. van Loo is supported by a JSPS postdoctoral fellowship. ## Appendix A Noise Temperature Estimation Figure 8: The simplified model used for the noise temperature estimations conducted in this work. Bold letters denote the power gains of components. The reference plane marks the input of the detector chain. Arrows denote the flow of power entering the nodes shown with a small circle. $G_{L}$ is a composite gain factor for everything between the noise source and the reference plane. $G_{c}$ is the circulator gain factor. $G_{J}$ is the signal and $G_{I}$ is the idler gain for the JPA. The amplifier gain $G_{R}$ and noise temperature $T_{nR}$ contain the effects of all elements after the last circulator, including SA noise. For simplicity, circulators are assumed to have complete rotational symmetry with respect to their ports and to be completely identical to each other. The output PSD from a component with its input connected to a matched source can be written as : $\displaystyle S_{\mathrm{O}}=GS_{\mathrm{in}}+S_{\mathrm{added}}$ (3) where $S_{\mathrm{in}}$ is the source PSD, $G$ is the power gain of the component, and $S_{\mathrm{added}}$ is the noise added by it. The noise temperature ($T_{n}$) is a measure of the added noise at the output of a component. By convention, it is defined as if it is for noise entering the device itself: $T_{n}=S_{\mathrm{added}}/(k_{B}G)$. The entire detection chain (see Figure 8), from the reference plane to the spectrum analyzer, can be described as a single composite component with $G=G_{\mathrm{tot}}$ and noise temperature $T_{n}$. The noise temperature can be defined for a situation similar to the experimental one where a narrowband axion signal with power $A$ is present. This signal enters the chain from a source connected to the reference plane. Assuming the source is thermalized to the MC plate with the temperature $T_{f}$, then the defining relation for $T_{n}$ can be written as : $\displaystyle S_{O}=G_{\mathrm{tot}}(\underbrace{A\delta(f-f_{s})+S_{n}(T_{f})}_{S_{\mathrm{in}}}+k_{B}T_{n})$ (4) where $S_{O}$ is the PSD at the output, $G_{\mathrm{tot}}$ is the total power gain for the signal from the reference plane, $S_{n}$ is the noise coming from the source itself. The main idea here is that if one has a reliable estimate of $T_{n}$, and understands the source environment well ($S_{n}(T_{f})$), it is straightforward to estimate $A$ without the precise knowledge of $G_{\mathrm{tot}}$. This is possible since $S_{O}$ can be easily measured at two frequencies $f_{s}$ and $f_{s}^{\prime}$ using a spectrum analyzer. Provided that $\|f_{s}-f_{s}^{\prime}\|$ is small enough so that $T_{n}$ is approximately the same for both frequencies, $A$ can be estimated from these two measurements. This approach forms the basis of the analysis methods applied in axion dark matter search experiments[26, 27, 28, 29]. The detection chain consists of passive components, the JPA and the HEMT amplifiers. Each one of these adds noise in a different way. A passive component at physical temperature $T_{f}$ has $S_{\mathrm{added}}=(1-G)S_{n}(T_{f})$. The HEMT amplifier noise is usually estimated from measurements. The JPA adds noise by two main mechanisms. The first one is by amplifying the input noise at the idler mode onto the signal mode. The second one is via the losses or other dissipation mechanisms inside or before the sample. Ideally, the latter can be made zero, whereas the former will approach to the half-photon added noise in the limit of a $0\text{\,}\mathrm{K}$ bath temperature. Using the model shown in Figure 8, it is straightforward to write a relation for the output PSD. For clarity, the explicit frequency dependence of the thermal noise $S_{n}$ and of the gains will be omitted. Also, the approximation $S_{n}(f,T)\approx S_{n}(f_{p}-f,T)$ will be denoted with the shorthand $S_{nf}=S_{n}(f,T_{f})$. This approximation has less than 30 ppm error given that $\|2f-f_{p}\|<$100\text{\,}\mathrm{kHz}$$. Note that $100\text{\,}\mathrm{kHz}$ is the typical bandwidth for the JPA tested in this work. Furthermore, the transmission characteristics of the microwave components will be assumed to not vary on a scale of $100\text{\,}\mathrm{kHz}$. Using the gain symbols for components as shown in Figure 8, the power flow at each node in terms of their PSD is written as : $\displaystyle\begin{split}S_{B}&=G_{L}S_{A}+(1-G_{L})S_{nf}\\\ S_{C}&=G_{c}S_{B}+(1-G_{c})S_{nf}\\\ S_{D}&=G_{c}S_{C}+(1-G_{c})S_{nf}\\\ S_{E}&=G_{J}S_{D}+G_{I}S_{D}+G_{J}S_{j}\\\ S_{F}&=G_{c}S_{E}+(1-G_{c})S_{nf}\\\ S_{O}&=G_{R}(S_{F}+k_{B}T_{nR})\end{split}$ (5) The idler gain is denoted by $G_{I}$, and is substituted using $G_{I}=G_{J}-1$[30] in the following derivations. As shown in Equation 5, the idler contribution to the noise appears as $G_{I}S_{D}$. The symbol $S_{j}$ denotes an unknown noise density added at the JPA stage which does not contribute to the quantum limit but rather contains losses or other mechanisms of stationary noise. Note that the noise propagating back from the later stages is also included in $S_{j}$. The output $S_{O}$ can be written for two cases. In the first case, the noise source is operational at temperature $T_{s}$, and in the second case, a signal source at temperature $T_{f}$ is connected to the reference plane. The former case describes the measurement situation, whereas the latter case is only used to define $T_{n}$ in terms of the parameters in the model. For the first case, i.e. $S_{A}=S_{n}(T_{s})$, the output PSD can be written as : $\displaystyle S_{O}^{(1)}$ $\displaystyle=\underbrace{G_{R}G_{c}^{3}G_{L}(2G_{J}-1)}_{G_{\mathrm{noise}}}\left[S_{n}(T_{s})+S_{\alpha}\right]$ (6) $\displaystyle S_{\alpha}$ $\displaystyle=\lambda^{(1)}S_{nf}+\frac{G_{J}S_{j}}{(2G_{J}-1)G_{c}^{2}G_{L}}+\left.\frac{k_{B}T_{nR}}{G_{c}^{3}G_{L}(2G_{J}-1)}\right.$ (7) $\displaystyle\lambda^{(1)}$ $\displaystyle=\beta_{l}+\frac{\beta_{c}}{G_{L}}+\frac{\beta_{c}}{G_{c}G_{L}}+\frac{\beta_{c}}{G_{c}^{2}G_{L}}$ (8) $\displaystyle\beta_{\mathord{\color[rgb]{0,0,0}\bullet}}$ $\displaystyle\equiv\frac{1}{G_{\mathord{\color[rgb]{0,0,0}\bullet}}}-1$ (9) The output for the second case, where $S_{B}=A\delta(f-f_{s})+S_{nf}$, is written as : $\displaystyle S_{O}^{(2)}$ $\displaystyle=\underbrace{G_{J}G_{c}^{3}G_{R}}_{G_{\mathrm{tot}}}\left[S_{B}+k_{B}T_{n}\right]$ (10) $\displaystyle k_{B}T_{n}$ $\displaystyle=\lambda^{(2)}S_{nf}+\frac{S_{j}}{G_{c}^{2}}+k_{B}\frac{T_{nR}}{G_{c}^{3}G_{J}}$ (11) $\displaystyle\lambda^{(2)}$ $\displaystyle=\left[\frac{G_{J}-1}{G_{J}}+\left(\beta_{c}+\frac{\beta_{c}}{G_{c}}\right)\frac{2G_{J}-1}{G_{J}}+\frac{\beta_{c}}{G_{c}^{2}}\right]$ (12) Here, the unknowns are $S_{j}$, $T_{nR}$ and $G_{R}$. It is clear from Equations 11 and 12 that $T_{n}$ approaches $T_{Q}$ as expected in the limits of $G_{J}\gg 1$, $G_{c}\rightarrow 0$, and $G_{L}\rightarrow 0$. Using Equations 6, 7 and 11 $,S_{O}^{(1)}$ can be rewritten as : $\displaystyle\begin{split}S_{O}^{(1)}&=\frac{G_{\mathrm{tot}}}{r}(S_{n}(T_{s})+rk_{B}T_{n}+\gamma)\\\ r&=\frac{G_{J}}{G_{L}(2G_{J}-1)}\\\ \gamma&=\left(\lambda^{(1)}-r\lambda^{(2)}\right)S_{nf}\end{split}$ (13) This relation is used to perform a fit with $G_{\mathrm{tot}}$ and $T_{n}$ as the fit parameters. For the estimations, the parameters $G_{L}$ and $G_{c}$ were taken as $-0.05\text{\,}\mathrm{dB}$, and $-0.4\text{\,}\mathrm{dB}$ respectively. Some typical values for $r$ and $\gamma/k_{B}$ can be found in Table 1. 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# Tuning the performance of a micrometer-sized Stirling engine through reservoir engineering Niloyendu Roy 111 Corresponding author. email<EMAIL_ADDRESS>Chemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore - 560064, INDIA Nathan Leroux Unité Mixte de Physique CNRS/Thales, 91767 Palaiseau, France A K Sood Department of Physics, Indian Institute of Science, Bangalore- 560012, INDIA International Centre for Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore - 560064, INDIA Rajesh Ganapathy 222 Corresponding author. email<EMAIL_ADDRESS>International Centre for Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore - 560064, INDIA School of Advanced Materials (SAMat), Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore - 560064, INDIA ###### Abstract Colloidal heat engines are paradigmatic models to understand the conversion of heat into work in a noisy environment - a domain where biological and synthetic nano/micro machines function. While the operation of these engines across thermal baths is well-understood, how they function across baths with noise statistics that is non-Gaussian and also lacks memory, the simplest departure from equilibrium, remains unclear. Here we quantified the performance of a colloidal Stirling engine operating between an engineered memoryless non-Gaussian bath and a Gaussian one. In the quasistatic limit, the non-Gaussian engine functioned like an equilibrium one as predicted by theory. On increasing the operating speed, due to the nature of noise statistics, the onset of irreversibility for the non-Gaussian engine preceded its thermal counterpart and thus shifted the operating speed at which power is maximum. The performance of nano/micro machines can be tuned by altering only the nature of reservoir noise statistics. Experimental advances in nano/micro manipulation has made feasible the realization of mesoscale heat engines with only a single atom rossnagel2016single or colloidal particle blickle2012realization ; quinto2014microscopic ; martinez2016brownian ; ciliberto2017experiments ; martinez2017colloidal as the working fluid. Even while the functioning of these engines is strongly influenced by fluctuations in the local environment with parameters like work and efficiency becoming stochastic quantities, when operating between equilibrium heat baths, their cycle-averaged performance mirrors their macroscopic counterparts and standard thermodynamic relations apply sekimoto1998langevin ; sekimoto2010stochastic ; esposito2010efficiency ; seifert2012stochastic ; verley2014unlikely ; rana2014single . Recently, Krishnamurthy et al. krishnamurthy2016micrometre experimentally realized an active stochastic heat engine by replacing the isothermal branches of a Stirling cycle with isoactive ones. Here, a colloidal particle in a time- varying optical potential was periodically cycled across two bacterial reservoirs characterized by different levels of activity. Unlike in equilibrium thermal baths where the displacement distribution of the colloid, $\rho(x)$, is a Gaussian, in active reservoirs, it was non-Gaussian and heavy- tailed krishnamurthy2016micrometre ; wu2000particle . These rare large displacement events resulted in large work output and the efficiency of this active engine was found to surpass equilibrium engines; even those operating between thermal baths with an infinite temperature difference. Since the metabolic activity of the bacteria could not be altered rapidly, this engine was operated only in the quasistatic limit, i.e. for a cycle duration $\tau$ larger than the relaxation time of the colloid. Subsequent theoretical calculations for the $\tau\to\infty$ limit posited that a departure from equilibrium efficiencies requires noise not just with non-Gaussian statistics but also with memory, a feature typical of active baths due to the persistent motion of the particles zakine2017stochastic . In fact, when the bath noise is non-Gaussian and white, an effective temperature $T_{eff}$ defined through the variance of $\rho(x)$ is thought to act like a bona fide temperature zakine2017stochastic ; fodor2018non and engines operating between such baths are expected to perform like equilibrium ones in the quasistatic limit. Whether this similarity persists when $\tau$ is reduced and irreversibility begins to set in is not known and is worth exploring since real heat engines never operate in the quasistatic limit as here their power $P\to 0$. On the experimental front, memoryless non-Gaussian heat baths are yet to be realised and predictions even in the quasistatic limit remain untested. Here we engineered a memoryless non-Gaussian heat bath and then constructed and quantified the functioning of a colloidal Stirling heat engine operating between such a bath and a thermal one for different $\tau$. In the quasistatic limit, the performance of this non-Gaussian engine mirrored a classical Stirling engine operating between thermal/Gaussian baths in agreement with theoretical predictions. Strikingly, due primarily to differences in the noise statistics of the baths, the small $\tau$ behaviour of these engines was quite different. On lowering $\tau$, not only did the distribution of work done per cycle, $\rho(W_{\text{cyc}})$, by the non-Gaussian engine become increasingly negatively skewed, unlike the standard Stirling case where it remained Gaussian, the onset of irreversibility for these two engines was also different. Importantly, we demonstrate that even sans memory, changing the nature of noise statistics of the reservoirs between which an engine operates allows tuning its performance characteristics, specifically, the $\tau$ at which the power goes through a maximum. ## Results ### Reservoir engineering by flashing optical traps Our experimental scheme for reservoir engineering is elaborated in Figure 1a. A polystyrene colloidal particle of radius $R=2.5$ $\mu$m suspended in water is held in a harmonic optical potential, $U={1\over 2}k_{1}\langle x^{2}\rangle$, created by tightly focusing a laser beam (1064 nm ALS-IR-5-SF, Azur Light Systems France) through a microscope objective (Leica Plan Apochromat 100X, N.A. 1.4, oil) that is also used for imaging the particle (see methods). Here, $k_{1}$ is the stiffness of this primary trap, $x$ is the displacement of the colloid from the centre of the optical trap and $\langle\rangle$ denotes an average. At equilibrium, the trap stiffness can be determined through the equipartition relation ${1\over 2}k_{1}\langle x^{2}\rangle={1\over 2}k_{B}T$ where $k_{B}$ is the Boltzmann constant and $T$ is the bath temperature, which in our experiments is fixed at 300 K. As a first step, we attempted to engineer a reservoir that mimicked a thermal bath, i.e. with Gaussian noise statistics, but with a desired $T_{eff}$. To this end, we imposed an additional noise on the colloidal particle along one spatial dimension, here the $x-$axis (Figure 1a), from a second optical trap of fixed intensity but with a time-dependent centre that was flashed at a distance $\delta a(t)$ away from the primary one (Figure 1b). This was made possible by using a second laser (Excelsior 1064 nm, Spectra Physics USA) coupled to the microscope through a spatial light modulator (Boulder Nonlinear Systems USA) and the flashing frequency was held fixed at $34$ Hz (see Methods). Earlier reservoir engineering studies wherein the colloidal particle experienced only the potential from the flashing trap found that when $\delta a$ was drawn from a Gaussian distribution, the particle indeed behaved like one in a thermal bath but at a $T_{eff}>T$ and furthermore, when $\delta a(t)<R$, the trap stiffness also remained unaltered. berut2014energy ; chupeau2018thermal . Here we adhered to the same protocol and further ensured that the peak of the $\delta a$ distribution coincided with the centre of the primary trap. Thus, the effective trap stiffness in our experiments $k=k_{1}+k_{2}$, where $k_{2}$ is the stiffness of the flashing trap. Like in a thermal bath, $\rho(x)$ of the trapped colloidal particle was a Gaussian (solid circles in Figure 1d) and its power spectral density (PSD) a Lorentzian, allowing us to determine $k_{2}$ and hence $T_{eff}$ chupeau2018thermal (Supplementary Figure 1 and Supplementary Note 1). For the $\delta a(t)$ profile shown in Figure 1b, the particle experienced a $T_{eff}=1331$ K. Engineering a memoryless non-Gaussian reservoir involved only a small tweak to the manner in which the external noise was imposed on the colloidal particle. The instantaneous $\delta a$ was now drawn randomly from a distribution with zero mean and skew, as before, but with a high kurtosis (see Methods and Supplementary Figure 2). Such a distribution has a narrow central region with heavy tails. The flashing optical trap is thus mostly coincident with the primary trap, thereby confining the particle strongly, and is occasionally positioned a large distance away from the centre leading to a large excursion by the particle (Figure 1c and Supplementary Movie). The overall noise experienced by the particle is $\delta$-correlated as the thermal and imposed noise are individually $\delta$-correlated. Under the influence of such a noise, the corresponding $\rho(x)$ of the colloidal particle was also non- Gaussian (hollow squares in Figure 1d). The PSD of the particle could be fit to a Lorentzian over the dynamic range accessible and since all other experimental parameters are held fixed, the roll-off frequency of the PSD was also same as the Gaussian case (Supplementary Figure 3 and Supplementary Note 1). For an appropriate choice of the variance and kurtosis of the $\delta a$ distribution, we could engineer the $T_{eff}$ of the non-Gaussian bath, again defined through the variance of $\rho(x)$, to be nearly identical to that in a Gaussian bath (Figure 1d). ### Performing a Stirling cycle between engineered reservoirs Armed with the capability to engineer reservoirs, we built a colloidal Stirling engine operating between a hot non-Gaussian and a cold Gaussian bath held at temperatures $T_{eff}^{H}=1824$ K and $T_{eff}^{C}=1570$ K, respectively. We also compared the performance of this non-Gaussian engine with a standard Stirling engine operating between engineered Gaussian baths with similar effective temperature difference (see Supplementary Note 2). The Stirling cycle we executed with the trapped colloid (Figure 1e), like in previous studies blickle2012realization ; krishnamurthy2016micrometre ; schmiedl2007efficiency , comprised of an isothermal compression (path ① - ②) and expansion step (path ③ - ④) linked by two isochoric transitions (paths ② - ③ and ④ - ①). In the isothermal compression (expansion) steps, $k$ was increased (decreased) linearly from $k_{min}=$2.5 pN$\mu$m-1 to $k_{max}=$ 2.7 pN$\mu$m-1 by changing $k_{1}$ alone. The isochroric transitions were near instantaneous and occurred on millisecond time scales. We exploited the ability to rapidly alter $T_{eff}$ and also the nature of noise statistics through the SLM to explore engine performance over a range of $\tau$ which spanned from 2 s to 32 s (see Methods). ### Elucidating the origins of irreversibility in the non-Gaussian Stirling engine The framework of stochastic thermodynamics provides a prescription for calculating thermodynamic quantities like the work, power, and efficiency of mesoscopic machines sekimoto1998langevin ; sekimoto2010stochastic ; seifert2012stochastic ; schmiedl2007efficiency . The work done per cycle, $W_{cyc}$, by the particle due to a modulation in the stiffness of the trap is just the change in potential energy and is given by $W_{cyc}=\int_{t_{i}}^{t_{i}+\tau}\frac{\partial U}{\partial k}\circ dk\equiv\frac{1}{2}\int_{t_{i}}^{t_{i}+\tau}x^{2}\circ dk$. Here, the $\circ$ signifies that the product is taken in the Stratonovich sense and $t_{i}$ is the starting time of $i^{\text{th}}$ cycle. Owing to its stochastic nature, $W_{cyc}$ of the engine fluctuates from cycle-to-cycle and we quantified the nature of these fluctuations through the probability distribution function $\rho(W_{cyc})$. Figure 2a and b show $\rho(W_{cyc})$ at different $\tau$ for the thermal and non-Gaussian Stirling cycles, respectively. Focusing on the large cycle duration ($\tau=32$ s) first, we observed that $\rho(W_{cyc})$ is a Gaussian for the thermal and also for the non-Gaussian cycles (circles in Figure 2a and b). The experimentally calculated average work done per cycle, $\langle W_{cyc}\rangle$, is negative indicating that the engine extracts heat from the bath to perform work on the surroundings. Further, $\tau=32$ s corresponds to the quasistatic limit for both since the value of $\langle W_{cyc}\rangle$ is in excellent agreement with the theoretically calculated quasistatic Stirling work output, $W_{\infty}=k_{B}(T_{eff}^{C}-T_{eff}^{H})\ln\sqrt{\frac{k_{max}}{k_{min}}}$ (short solid horizontal lines in Figure 2c). On lowering $\tau$, $\rho(W_{cyc})$ for the thermal Stirling engine remained a Gaussian (Figure 2a) and $\langle W_{cyc}(\tau)\rangle\approx\langle W_{cyc}(\tau=32\text{ s})\rangle$ (hollow circles Figure 2c). As expected of such a distribution, $\langle W_{cyc}\rangle$ was the same as the most- probable work $W^{}$ \- the value of $W_{cyc}$ where $\rho(W_{cyc})$ is a maximum (solid circles Figure 2c). For the non-Gaussian engine on the other hand, on reducing $\tau$, $\rho(W_{cyc})$ became increasingly negatively skewed (Figure 2b) and $W^{}(\tau)$ also became increasingly positive (solid squares Figure 2c). $\langle W_{cyc}(\tau)\rangle$ however, was marginally smaller than $\langle W_{cyc}(\tau=32\text{ s})\rangle$. (hollow squares Figure 2c). We note that the work done by a thermal Stirling engine at a finite $\tau$ is given by the empirical relation schmiedl2007efficiency ; blickle2012realization $W(\tau)=W_{\infty}+W_{diss}\equiv W_{\infty}+\frac{\Sigma}{\tau}$ (1) where, $W_{diss}$ is the dissipative work which accounts for the particle's inability to fully explore the available phase space when $k$ is rapidly lowered during the hot isotherm and $\Sigma$ is a constant also called the irreversibility parameter. Since $W_{diss}$ is a positive quantity as per definition, at small enough $\tau$, the overall work done itself can be positive indicating the stalling of the engine. Clearly there is no buildup of irreversibility for the thermal engine as $\tau$ is lowered since $\langle W_{cyc}(\tau)\rangle\equiv W^{}(\tau)\approx W_{\infty}$, while for the non- Gaussian one, there is, even if only in the most-probable sense ($\langle W_{cyc}(\tau)\rangle\approx W_{\infty}<W^{}(\tau)$), and the engine stalls for $\tau\leq 10$ s. We also found excellent agreement between equation (1) and our data allowing us to determine $\Sigma=0.11\text{ }k_{B}T_{eff}^{C}$ (red solid line in Figure 2c). The observed behaviour of the non-Gaussian engine can be easily rationalized by analyzing the relaxation of the particle in the hot isotherm at the level of an individual cycle. For the particle to fully sample the statistical properties of the non-Gaussian hot reservoir, it should also experience the occasional large kicks that displaces it far from the center and not just the ones that predominantly keep it confined close to it. As $\tau$ is lowered, in most cycles, the probability that the particle encounters a large kick in the isothermal expansion step also becomes increasingly small. Due to the incomplete exploration of the available phase volume in these cycles, less useful work is performed and $W^{}(\tau)$ lifts off with decreasing $\tau$. In a few cycles, where these large kicks are present, anomalously large work is done by the engine and this results in $\rho(W_{cyc})$ being negatively skewed. When an adequate number of cycles, which has to be increased when $\tau$ is lowered, have been performed, all features of the noise are sampled and the engine operates like one in the quasistatic limit in an average sense with $\langle W_{cyc}(\tau)\rangle\to W_{\infty}$ (Figure 2c). This inference can be strengthened by quantifying the equilibration of the particle over a fixed, but limited, number of cycles for all $\tau$. In Figure 2d, we show ${k\langle x^{2}\rangle\over k_{B}T_{eff}^{H}}$ calculated over a small window in the middle of the hot isotherm and averaged over $N=50$ cycles for the thermal (squares) and the non-Gaussian engine (circles). Despite $N$ being small, ${k\langle x^{2}\rangle\over k_{B}T_{eff}^{H}}$ is close to 1 at all $\tau$ for the thermal engine implying that it is truly in the quasistatic limit, while for the non-Gaussian engine this is the case only at large $\tau$ with a clear violation of quasistaticity setting in for $\tau\leq 10$ s. Evidently, for a non-Gaussian engine $W^{}(\tau)$, and not $\langle W_{cyc}(\tau)\rangle$, is a more precise metric for performance. ### Tuning the performance of a Stirling engine through memoryless non- Gaussian noise We now examined how differences in the nature of noise-statistics affected the power output of our engines. In the quasistatic limit $P(\tau)=-{\langle W_{cyc}(\tau)\rangle\over\tau}\to 0$ since $\tau\to\infty$, while at high cycle frequencies $W_{diss}$ is large and $P$ is once again small. At intermediate $\tau$, however, these effects compete resulting in a maximum in $P$ and this is a feature of both macroscopic and mesoscopic engines blickle2012realization ; curzon1975efficiency . Figure 3a shows the most- probable power, $P^{}(\tau)={-W^{}(\tau)\over\tau}$, for the Gaussian Stirling engine (circles) and the non-Gaussian one (squares). Since for the thermal engine, over the range of $\tau$ studied $\Sigma=0$, $P^{}(\tau)$, which is same as same as $P(\tau)$, only increases monotonically on lowering $\tau$ and does not exhibit a maximum. Whereas for the non-Gaussian engine, on reducing $\tau$, $P^{}(\tau)$ first appears to increase slightly, crosses zero for $\tau<10$ s and then becomes more negative indicating stalling of the engine. We emphasize that for a Stirling cycle executed under conditions identical to that in Figure 1e but where the non-Gaussian reservoir is replaced by a Gaussian one with the same $T_{eff}^{H}$, the maximum in $P$ is expected to be at a $\tau$ that is lower than even the Gaussian engine studied here (see Supplementary Note 3). This clearly shows that, even sans memory, altering the statistical properties of the noise bath alone allows for tuning the performance characteristics of mesoscopic heat engines. For a complete understanding of the operation of the non-Gaussian engine, we calculated its efficiency at various $\tau$ and benchmarked it with the thermal engine. Conventionally, the efficiency, $\varepsilon={W_{cyc}\over Q}$, where $Q$ is the heat absorbed by the particle when it is in contact with the hot reservoir. $Q$ is the sum of the isochoric heat during the transition from state point ② to ③, $Q_{2\to 3}=-{1\over 2}k_{max}(T_{eff}^{H}-T_{eff}^{C})$ and the isothermal heat during transition from ③ to ④, $Q_{3\to 4}=\int_{(3)}^{(4)}\frac{\partial U}{\partial x}\dot{x}dt=W_{H}+Q_{boundary}$. Here, $W_{H}=\frac{1}{2}\int_{(3)}^{(4)}x^{2}\circ dk$ is the work done in the hot isotherm and $Q_{boundary}=-\frac{1}{2}[k(t)x^{2}(t)]_{(3)}^{(4)}$. For the non-Gaussian engine, we naturally chose $W^{}$ instead of $W_{cyc}$ and defined the most-probable efficiency $\varepsilon^{}={W^{}\over{\langle W_{H}\rangle+\langle Q_{boundary}\rangle+\langle Q_{isochoric}\rangle}}$ (See Supplementary Note 4). For the thermal engine, the experimentally determined $\varepsilon^{}$ (black circles in Figure 3b) hovers around the theoretically calculated saturation Stirling efficiency $\varepsilon_{Sat}=\varepsilon_{c}[1+{\varepsilon_{c}\over\ln(k_{max}/k_{min})}]^{-1}$ (solid blue line). Here, $\varepsilon_{c}=1-{T_{eff}^{C}\over T_{eff}^{H}}$ is the Carnot efficiency. Whereas for the non-Gaussian engine, $\varepsilon^{}(\tau)$ converges to $\varepsilon_{Sat}$ only at large $\tau$ (red squares in Figure 3a). When $\tau$ is reduced, $\varepsilon^{}(\tau)$ drops and becomes negative for $\tau<10$ s indicating stalling of the engine. Of particular importance in the operation of real heat engines is the efficiency at maximum power $\varepsilon_{Max}$, which for the non-Gaussian engine is at $\tau=10.3$ s with $\varepsilon_{Max}=0.025$. Most remarkably, this value in excellent agreement with theoretically predicted Curzon-Ahlborn efficiency, $\varepsilon_{CA}={\varepsilon_{Sat}\over 2-\alpha\varepsilon_{Sat}}=0.026$ curzon1975efficiency ; schmiedl2007efficiency . In our experiments, $\alpha\sim 0$ is a constant calculated from the irreversibility parameters corresponding to the work done in the hot and cold isotherms (Supplementary Figure 4 and Supplementary Note 5). While it is known that $\varepsilon_{Max}\approx\varepsilon_{CA}$ for both macro and mesoscopic thermal engines, ours is the first observation of this being the case even for a non-Gaussian engine. ## Discussion Collectively, our experiments show that a micrometer-sized Stirling engine operating between a Gaussian and a non-Gaussian bath, without memory, indeed performs like a conventional engine in the quasistatic limit as anticipated by theory. On lowering the cycle times, the buildup of irreversibility in the engine, due entirely to the non-Gaussian nature of noise, results in work distributions that become increasingly negatively skewed unlike a thermal engine where it remains Gaussian. Strikingly, this noise-induced enhancement of irreversibility modulates the performance characteristics of the non- Gaussian engine in a manner similar to predictions by Curzon and Ahlborn for thermal engines where irreversibility sets in purely due to the rapid change of the control parameter. Our experiments thus reveal a new strategy for optimizing the performance of a mesoscale engine by tuning only the nature of noise statistics. Importantly, the ease with which the noise can be engineered and also applied locally, i.e. on the particle scale, in our approach presents advantages over other reservoir engineering methods where this can prove to be difficult, if not impossible martinez2013effective ; martinez2017colloidal . This should now make feasible the experimental realization of new stochastic machines like the non-Gaussian and the Buttiker–Landauer ratchet luczka1997symmetric ; buttiker1987transport ; landauer1988motion . ## Methods ### Experimental set-up for Reservoir Engineering In order to impart additional noise into the trapped colloid, a secondary optical trap was flashed along a line passing through the time-averaged centre of the particle at variable distances from the same. This was achieved by coupling a second laser (Excelsior 1064 nm, Spectra Physics USA) to the microscope which is reflected from a Spatial Light Modulator (Boulder Nonlinear Systems USA). The Spatial Light Modulator (SLM) contains a $512\times 512$ array of shiny electrodes covered with a transparent liquid crystal layer so that an electric potential modulation across the electrodes imposes an additional phase pattern on the incident beam. We interfaced the SLM to a computer so that a series of desired phase patterns can be fed to the SLM at a fixed frequency of $34Hz$. This enabled us to dynamically reconfigure the position of the first order diffraction spot by applying a series of linear diffraction grating patterns with varying periodicity which is controlled through a computer. We blocked the zeroth order spot so that only the first order spot is incident on the back of the microscope objective resulting in a flickering optical trap in the vicinity of the tweezed colloidal particle. ### Image acquisition and processing Images of the trapped colloid was captured at $250$ Hz using a fast camera (Photron 500K-M3) attached to the microscope. Position of particle's centre in each frame was located at the subpixel level using the particle tracking codes by R. Parthasarathy parthasarathy2012rapid . This allowed us to find the particle's position within an accuracy of $5$ nm. ### Non-Gaussian Reservoir Engineering For engineering the non-Gaussian reservoir, $\delta a$ were chosen from a $\delta-correlated$ distribution with zero mean and skewness but an extremely high kurtosis of 50. One such distribution with standard deviation of $\sigma=0.28\mu m$ is represented in Supplementary Figure 2(b). To create this distribution, we first generate two highly asymmetric distributions $\delta a_{L}$ and $\delta a_{R}$ (Supplementary Figure 2(a)) with a standard deviation of $0.28\mu m$, a kurtosis of $60$ and a skewness of $-6.5$ for $\delta a_{L}$ and $+6.5$ for $\delta a_{R}$ through Pearson's protocol in MATLAB. Next we add/subtract a suitable number to $\delta a_{L}$ and $\delta a_{R}$ so that their peaks coincide at zero. Then we take union of $\delta a_{L}$ and $\delta a_{R}$ and randomly permute all the elements to finally obtain the set of $\delta a$. In order to realize a desired effective temperature with such a noise, the standard deviation of $\delta a$ is optimized. It should be noted that heavy tails rise due to extremely rare events that can only be captured with a huge statistics. Since we are limited by a flashing frequency of $34Hz$, it is not possible to completely sample the statistics with in one isotherm even for the largest $\tau$. To address this issue, the engine was cycled enough number of times (depending on $\tau$) so that the collection of all the hot isotherms exhausts all the rare events. ### Instantaneous isochoric transitions The isochoric transitions ②$\rightarrow$③ and ④$\rightarrow$① shown in Figure 1e of the main text is realised by changing the statistics and the variance of $\delta a$-distribution. The transition ②$\rightarrow$③ is realised by changing the $\delta a$ distribution from a Gaussian resulting in $T_{eff}=1570K$ to a non-Gaussian producing $T_{eff}=1824.3K$ while the transition ④$\rightarrow$① is realised by the reverse. Since the secondary laser is diffracted by a computer controlled SLM, the distribution from which $\delta a$s are chosen can be altered in $1/34$th of a second. Thus the particle is decoupled and coupled from one engineered reservoir to the other in less than $33ms$ which is almost negligible even in compared to the lowest cycle time and hence instantaneous. ## Acknowledgements N.R. thanks Dr. Sudeesh Krishnamurthy for fruitful discussions. N.R. thanks Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) for financial support. AKS thanks Department of Science and Technology (DST), Govt. of India for a Year of Science Fellowship. 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Figure 1: Experimental realization of a non-Gaussian Stirling heat engine. a The big red spot represents the primary optical trap and the small red spots represent the secondary flashing optical trap at different time instances $t_{1}$, $t_{2}$ and $t_{3}$. b and c show the distance $\delta a(t)$ from the primary trap at which the secondary trap was flashed as a function of $t$ for engineering a Gaussian and a non-Gaussian reservoir, respectively. d shows the probability distribution of particle displacements, $\rho(x)$, for the engineered Gaussian/thermal (solid blue circles) and the non-Gaussian reservoir (red hollow squares) for a nearly identical $T_{eff}$. e shows a quintessential Stirling cycle between a hot non-Gaussian bath at $T_{eff}^{H}=1824$ K and a cold Gaussian reservoir with $T_{eff}^{C}=1570$ K. The trap stiffness $k$ is varied linearly in the expansion/compression steps. Having a fixed primary trap and a second flashing optical trap, as opposed to just the latter, prevented the trapped particle from escaping the trap and allowed for long experiments. $\rho(x)$ of the particle measured at the four state points labeled ① to ④ is also shown. The black lines are Gaussian fits. Figure 2: Buildup of irreversibility in the non-Gaussian Stirling engine at finite $\tau$. a and b show the probability distribution of work done per cycle $\rho(W_{cyc})$ for the Gaussian and the non-Gaussian engine, respectively, for $\tau=32$ s (blue circles), $10.6$ s (red triangles) and $5.6$ s (black squares). Solid lines represent corresponding Gaussian fits to the data. c Red hollow and solid squares show the average work done per cycle $\langle W_{cyc}\rangle$ and the most-probable work $W$, respectively, for the non-Gaussian engine at various $\tau$. The red solid line is a fit to Equation 1. Black hollow and solid circles show $\langle W_{cyc}\rangle$ and $W^{}$ respectively for the thermal/Gaussian engine. At large $\tau$, the experimentally calculated work for these engines agrees with theoretically calculated quasistatic work $W_{\infty}$ indicated by the Red and Black short horizontal lines for the non-Gaussian and Gaussian engine respectively. d Red squares (black circles) represent the ratio $k\langle x^{2}\rangle/k_{B}T_{eff}^{H}$ calculated at the midpoint of the hot isotherm of the non-Gaussian (Gaussian) engine at various $\tau$. The horizontal line indicates the equilibrium condition, which is strongly violated inside the shaded grey region, in case of the non-Gaussian engine. Figure 3: Quantifying the performance of a non-Gaussian Stirling engine. In a, red squares (black circles) show the most-probable power $P^{}$ of the non-Gaussian (Gaussian) engine at various $\tau$. $P^{}$ increases slightly and then rapidly falls for the non-Gaussian engine for $\tau\leq 10.6$ s. The red solid line is calculated from the fit to Equation 1 and is overlaid on the experimental data. b Red squares (black circles) represent the most-probable efficiency $\varepsilon^{*}$ of the non-Gaussian (Gaussian) engine at various $\tau$. The blue solid lines indicate the theoretically calculated saturation Stirling saturation, $\varepsilon_{Sat}$. Efficiency $\varepsilon_{max}$ just before the rapid drop in power ($\tau=10.6$) of the non-Gaussian engine agrees with the Curzon-Ahlborn efficiency $\varepsilon_{CA}$. Note that the black vertical line through the first data point (smallest $\tau$) is a portion of a large error bar. The error bars at other $\tau$ values are smaller than the symbol size.
# triSurfaceImmersion: Computing volume fractions and signed distances from triangulated surfaces immersed in unstructured meshes Tobias Tolle<EMAIL_ADDRESS>Dirk Gründing<EMAIL_ADDRESS>darmstadt.de Dieter Bothe<EMAIL_ADDRESS>Tomislav Marić <EMAIL_ADDRESS>Mathematical Modeling and Analysis Institute, Mathematics department, TU Darmstadt, Alarich-Weiss-Straße 10, 64287 Darmstadt, Germany ###### Abstract We propose a numerical method that enables the calculation of volume fractions from triangulated surfaces immersed in unstructured meshes. First, the signed distances are calculated geometrically near the triangulated surface. For this purpose, the computational complexity has been reduced by using an octree space subdivision. Second, an approximate solution of the Laplace equation is used to propagate the inside/outside information from the surface into the solution domain. Finally, volume fractions are computed from the signed distances in the vicinity of the surface. The volume fraction calculation utilizes either geometrical intersections or a polynomial approximation based on signed distances. An adaptive tetrahedral decomposition of polyhedral cells ensures a high absolute accuracy. The proposed method extends the admissible shape of the fluid interface (surface) to triangulated surfaces that can be open or closed, disjoint, and model objects of technical geometrical complexity. Current results demonstrate the effectiveness of the proposed algorithm for two-phase flow simulations of wetting phenomena, but the algorithm has broad applicability. For example, the calculation of volume fractions is crucial for achieving numerically stable simulations of surface tension-driven two-phase flows with the unstructured Volume-of-Fluid method. The method is applicable as a discrete phase-indicator model for the unstructured hybrid Level Set / Front Tracking method. The implementation is available on GitLab [27]. This a pre-print of the accepted article https://doi.org/10.1016/j.cpc.2021.108249, when citing, please refer to the accepted article. ###### keywords: volume of fluid , triangular surface mesh , signed distances , unstructured mesh PROGRAM SUMMARY Program Title: argo/triSurfaceImmersion CPC Library link to program files: (to be added by Technical Editor) Developer’s repository link: https://gitlab.com/leia-methods/argo Code Ocean capsule: (to be added by Technical Editor) Licensing provisions: GPLv3 Programming language: C++ Nature of problem: Computing volume fractions and signed distances from triangulated surfaces immersed in unstructured meshes. Solution method: First, the algorithm computes minimal signed distances between mesh points (cell centers and cell corner-points) and the triangulated surface, in the close vicinity of the surface. The sign is computed with respect to the surface normal orientation. Afterwards, the sign is propagated throughout the unstructured volume mesh by an approximate solution of a diffusion equation. The bulk cells’ volume fractions are set, and interface cells are identified based on the signed distances. Volume fractions in cells intersected by the triangulated surface mesh are either computed by geometric intersections between surface triangles and a cell or by an approximation of the volume fraction approximation from signed distances, coupled with tetrahedral cell decomposition and refinement. Additional comments including restrictions and unusual features: The volume mesh can consist of cells of arbitrary shape. The surface mesh normal vectors need to be oriented consistently. ## 1 Introduction We present a new numerical algorithm that calculates initial conditions for simulations of two-phase flow problems for fluid interfaces of complex shapes. The initial conditions are calculated in the form of signed distances and volume fractions from fluid interfaces approximated as arbitrarily shaped triangular surfaces immersed into unstructured meshes. The signed distances are relevant as initial conditions for the Level Set method [48, 49] for multiphase flow simulation. Volume fractions on unstructured meshes are required for the unstructured Volume-of-Fluid (VOF) method (cf. [30] for a recent review). In fact, we have applied the proposed algorithms to model experimental fluid interfaces from wetting experiments [16], which was not possible using available contemporary approaches that model fluid interfaces using (compositions of) implicit functions or parameterized surfaces. The proposed algorithm approximates the surfaces using triangle meshes that are omnipresent in Computer-Aided Design (CAD) because of their versatility: they can approximate basic surfaces such as spheres and ellipsoids, but also surfaces of mechanical parts, disjoint surfaces in mechanical assemblies, or surfaces resulting from imaging scans. The overall simulation domain $\Omega\subset\mathbb{R}^{3}$ is separated into two subdomains $\Omega=\Omega^{+}(t)\cup\Omega^{-}(t)$, representing phase $1$ and phase $2$, respectively, as illustrated for a liquid drop on a surface in fig. 1. At the contact line $\Gamma:=\partial\Omega\cap\overline{\Omega^{+}}\cap\overline{\Omega^{-}}$, the liquid-gas interface $\Sigma$ encloses a contact angle $\theta$ with the solid surface $\partial\Omega_{\text{wall}}$. Furthermore, the normal vector $\boldsymbol{n}_{\Sigma}$ of the interface $\Sigma$ is oriented such that it points into the gas phase. $\Omega^{+}(t)$ --- $\Sigma(t)$ --- $\theta$ --- $\partial\Omega$ --- $\partial\Omega_{\text{wall}}$ --- $\Omega^{-}(t)$ --- $\Gamma$ --- $\boldsymbol{n}_{\Sigma}$ --- Figure 1: The different domains for a liquid ($-$) drop on a solid surface surrounded by a gas ($+$) phase. Typically, a continuum mechanical model is used for the description of such fluid mechanical problems. This description is often based on a sharp interface model, as depicted in fig. 1. With this model, the liquid-gas interface can be described using an indicator function $\chi(\mathbf{x},t):=\begin{cases}1,&\mathbf{x}\in\Omega^{-}\subset\mathbb{R}^{3}\\\ 0,&\text{otherwise}.\end{cases}$ (1) An approximate solution of this model requires a decomposition of the solution domain into volumes that have no volume overlaps, the closed _cells_ $\Omega_{c}$, denoted by $\Omega\approx\tilde{\Omega}=\\{\Omega_{c}\\}_{c\in C}$ (2) where $C=\\{1,2,3,\dots,N_{c}\\}$ is a set of indices to mesh cells. As can be seen in fig. 2, the mesh is a set of non-overlapping subsets (_cells_) $\Omega_{c}\subset\tilde{\Omega}$. With non-overlapping, we mean that the volume of an intersection between any two cells is zero. _Index sets_ represent the unstructured mesh data [15]. We consider a set of cell corner- points $P_{h}$ where each point in $P_{h}$ is an element of $\mathbb{R}^{3}$. Geometrically, each cell $\Omega_{c}$ is a volume bounded by polygons, so- called _faces_. A global set of faces $F_{h}$ is defined, and each face is a sequence of _indices_ of points in $P_{h}$. In this context, we define a cell set $C_{c}$ as a set of indices of faces in the set of mesh faces $F_{h}$. Therefore, when referring to a volume defined by the cell, we use $\Omega_{c}$ and its magnitude is then $\|\Omega_{c}\|$, and when we refer to the cell as an unordered index set, we use $C_{c}$ and its magnitude $\|C_{c}\|$ is the number of faces that bound the cell. Solutions of continuum mechanical problems in geometrically complex solution domains significantly benefit from unstructured meshes. For example, gradients of solution variables are resolved at geometrically complex boundaries by employing mesh boundary layers, strongly reducing the number of cells required to achieve specific accuracy. Hence, this approx reduces the overall required computational resources. As the phase indicator $\chi(\mathbf{x},t)$ given by eq. 1 contains a jump discontinuity, it poses difficulties for numerical simulations of two-phase flows. With Volume of fluid (VOF) methods, this non-continuous description is discretized by introducing the so-called _volume fraction_ $\alpha_{c}=\dfrac{1}{\|\Omega_{c}\|}\int_{\Omega_{c}}\chi(\mathbf{x},t)dx.$ (3) The unstructured VOF methods [30] rely on the volume fraction field $\alpha_{c}$ to track interface with the advecting velocity obtained from the solution of two-phase Navier-Stokes equations in a single-field formulation. All multiphase flow simulation methods that utilize the single-field formulation of Navier-Stokes equations approximate the phase-indicator function similarly to eq. 3. The phase-indicator approximation utilizes signed distances in the Level Set [48, 47, 49] method, the volume fractions approximate the phase indicator for the Volume-of-Fluid [8, 37, 17, 40] method. Various methods exist that compute the volume fraction $\alpha_{c}$ based on the exact phase indicator $\chi(\mathbf{x},t)$. The majority of methods calculate the integral in eq. 3 numerically, as schematically shown in fig. 2, using numerical quadrature. $\Sigma$ --- $\chi(\mathbf{x},t)=1$ --- $\chi(\mathbf{x},t)=0$ --- $\Sigma$ --- $\Omega_{c}$ --- $\Omega^{-}$ --- $h(x)$ --- $x$ --- $y$ --- $\Omega^{-}$ --- Figure 2: Calculating volume fractions of a circular interface by numerical integration. Different approaches are outlined below with increasing complexity in terms of admissible shapes of the fluid interface. The admissible shapes range from analytic descriptions of basic geometric shapes such as spheres and ellipsoids to implicit functions (or their combinations) and more general shapes approximated with volume meshes. Strobl et al. [46] propose an exact intersection between a sphere and a tetrahedron, a wedge, or a hexahedron. The proposed algorithm is exact and fast, though it is limited to the spherical interface shape. Fries and Omerović [13] represent the fluid interface as a level set and propose a higher-order quadrature for the integral on the right-hand side of eq. 3. The parametrization of the surface uses roots of the implicit function found by the closest-point algorithm. Results are presented for hexahedral and tetrahedral unstructured meshes that may also be strongly deformed. Fries and Omerović [13, fig. 52, fig. 53] also show results with higher-order ($>2$) convergence for the volume integration of an arbitrary non-linear function on hexahedral and tetrahedral meshes. However, the volume and area integration error is reported for a single function. While a relative global volume error between ${1}\mathrm{e}{-08}$ and ${1}\mathrm{e}{-06}$ is reported, no information about the required CPU times is provided. In the approach proposed by Fries and Omerović [13], fluid interfaces with complex shapes are modeled as a composition of implicit functions. Kromer and Bothe [25] propose an efficient third-order accurate quadrature for the eq. (3). Contrary to Jones et al. [21], who decompose cells into tetrahedrons, Kromer and Bothe [25] locally approximate the hypersurface by a paraboloid based on the principal curvatures. Applying the Gaussian divergence theorem to eq. (3) then yields contributions from the cell boundary and the approximated hypersurface patch. Using the surface divergence theorem, Kromer and Bothe [25] reformulate the contribution from the hypersurface patch into a set of line integrals, where the associated integrand emerges from the solution of a Laplace-Beltrami-type problem. The method of Kromer and Bothe [22] is directly applicable to unstructured meshes. However, locally, i.e., within a cell, the fluid interface must be $C^{2}$ and simply connected. Aulisa et al. [3] and Bnà et al. [5, 6] calculate the volume fraction by representing the indicator function as a height function inside cubic cells, using the structure of the underlying Cartesian mesh. Numerical integration of the height function is illustrated by fig. 2. However, extending this approach to unstructured meshes raises many questions. First, constructing a height function in a specific direction is complex and computationally expensive [38]. Second, the orientation of the interface in the chosen coordinate system may easily make the problem ill-conditioned. Finally, required mesh-search operations are complicated as the face normals of polyhedral cells are typically not aligned with the coordinate axes. The calculation of the volume fraction given by $\alpha_{c}=\frac{\|\Omega^{-}\cap\Omega_{c}\|}{\|\Omega_{c}\|}$ can be reformulated into the integration of a function $f=1$ within $\Omega^{-}\cap\Omega_{c}$. Since $\partial\Omega_{c}$ consists of piecewise- planar surfaces (faces), the complexity lies in the non-planar part of the surface $\partial\Omega^{-}\cap\Omega_{c}=\Sigma(t)\cap\Omega_{c}$. Trimmed isogeometric analysis can be used to integrate $f=1$ within $\Omega^{-}\cap\Omega_{c}$ by representing $\partial\Omega^{-}\cap\Omega_{c}$ using a trimmed NURBS surface, effectively resulting in $\alpha_{c}=\frac{\|\Omega^{-}\cap\Omega_{c}\|}{\|\Omega_{c}\|}$ for complex non- linear CAD surfaces. Although not yet applied to volume fraction calculation ($f=1$ integration), trimmed isogeometric analysis has been applied to solving PDEs in solution domains bounded by NURBS surfaces [24, 43, 36]. Similarly, the immersed isogeometric analysis (e.g. [11]) requires function integration in cut cells, where the integration of $f=1$ in the cut cell is equivalent to computing $\|\Omega^{-}\cap\Omega_{c}\|$ used in volume fraction calculation. Although it is a potentially interesting alternative approach for computing volume fractions from CAD surfaces, the isogeometric analysis requires NURBS trimming, octree refinement, and higher-order quadratures. These efforts are worthwhile for the goal of achieving higher-order solutions for PDEs in complex solution domains. However, as demonstrated in the results section, our proposed algorithms achieve sufficient accuracy for signed distances and volume fractions on unstructured meshes while relying on straightforward second-order accurate discretization. The signed distances in the Level Set Method require re-distancing (correction). The re-distancing methods are usually based on approximate solutions of Partial Differential Equations (PDEs) that ensure the signed- distance property [41]. Contrary to this approach, the unstructured Level Set / Front Tracking method [28, 51] _geometrically_ computes minimal signed distances from $\tilde{\Sigma}$. This calculation is relatively straightforward on structured meshes [44, 45], but significantly more complex on unstructured meshes [28, 51]. Here we significantly extend the calculation of signed distances from [28, 51] by introducing an efficient approximate propagation of the inside/outside information from $\tilde{\Sigma}$. Volume fraction calculation methods outlined so far model the fluid interface using exact functions and handle more complex interface shapes via combinations of these functions. A combination of exact functions cannot accurately capture the shape of the fluid interface in many cases. For example, when the interface shape is prescribed experimentally Hartmann et al. [16]. One approach exists that can handle arbitrarily complex interface shapes. In this approach, the fluid interface encloses a volumetric mesh as its boundary surface mesh. This mesh given by the fluid interface is intersected with a ”background” mesh that stores volume fractions. This approach is called _volume mesh intersection_. An example for such an intersection between $\tilde{\Omega}$ and cells from $\tilde{\Omega}^{-}$ is shown in fig. 3. In principle, this approach is relatively straightforward, provided an accurate geometrical intersection of tetrahedrons is available. However, geometrical operations based on floating-point numbers are not stable and can lead to severe errors [52, chap. 45]. $\tilde{\Sigma}$ --- $\Omega_{c}$ --- $\Omega_{c}$ --- $\tilde{\Omega}^{-}_{l}$ --- $\tilde{\Omega}^{-}_{l}$ --- Figure 3: Calculating volume fractions from a circular interface by volume mesh intersection. Ahn and Shashkov [1] have initialized volume fractions by volume mesh intersection as shown in fig. 3. In this approach, the approximated phase $\tilde{\Omega}^{-}(t)$ is decomposed into volumes (an unstructured mesh), equivalently to the decomposition $\tilde{\Omega}$ given by eq. 2. The boundary $\partial\Omega^{-}$ is the fluid interface $\Sigma(t)$, and it is approximated as a polygonal surface mesh, leading to $\Omega^{-}\approx\tilde{\Omega}^{-}:=\\{\tilde{\Omega}^{-}_{l}\\}_{l\in L},$ (4) i.e. an approximation of $\Omega^{-}$. Generally, as shown in the detail in fig. 3, a cell $\Omega_{c}$ of the background mesh $\tilde{\Omega}$ may overlap with multiple cells $\Omega_{l}$ from the $\tilde{\Omega}^{-}$ mesh, and vice versa. We define a set of indices $l$ of cells $\tilde{\Omega}^{-}_{l}$ in $\tilde{\Omega}^{-}$ that overlap with the cell $\Omega_{c}$: the so-called _cell stencil_ of $\Omega_{c}$ in $\tilde{\Omega}^{-}_{l}$, namely $\mathcal{S}(\Omega_{c},\tilde{\Omega}^{-})=\\{l\in L:\Omega_{c}\cap\tilde{\Omega}^{-}_{l}\neq\emptyset,\text{where}~{}\Omega_{c}\in\tilde{\Omega},\tilde{\Omega}^{-}_{l}\in\tilde{\Omega}^{-}\\},$ (5) where $L$ is an index set, containing indices of cells from $\tilde{\Omega}^{-}$. Volume fractions $\\{\alpha_{c}\\}_{c\in C}$ can then be calculated by performing the intersection $\alpha_{c}=\frac{\|\cup_{l\in\mathcal{S}(\Omega_{c},\tilde{\Omega}^{-})}\Omega_{c}\cap\tilde{\Omega}^{-}_{l}\|}{\|\Omega_{c}\|}.$ (6) Since each $\tilde{\Omega}^{-}_{l}$ overlaps with at least a one cell from $\tilde{\Omega}$, and we can approximate the number of cells from $\tilde{\Omega}$ that intersect each cell from $\tilde{\Omega}^{-}$ as $N(\tilde{\Omega}^{-},\tilde{\Omega})\approx\|\tilde{\Omega}^{-}\|\underset{l\in L}{\text{mean}}(\|\mathcal{S}(\tilde{\Omega}^{-}_{l},\tilde{\Omega})\|),$ (7) where $\|\tilde{\Omega}^{-}\|$ denotes the number of cells in the mesh $\tilde{\Omega}^{-}$. The average number of cells $\Omega_{c}$ overlapping $\tilde{\Omega}^{-}_{l}$, $\underset{l\in L}{\text{mean}}\|C(\tilde{\Omega}^{-}_{l},\tilde{\Omega})\|$, depends on the mesh densities of both meshes, $\tilde{\Omega}$ and $\tilde{\Omega}^{-}$. However, we do know that $\underset{l\in L}{\text{mean}}\|C(\tilde{\Omega}^{-}_{l},\tilde{\Omega})\|>1$. Next, we know that $\|\tilde{\Omega}^{-}\|$ grows quadratically in $2D$ and cubically in $3D$ with a uniform increase in mesh resolution, taken as the worst case scenario. It grows linearly in $2D$ and quadratically in $3D$ if $\tilde{\Omega}^{-}$ is refined only near the interface $\tilde{\Sigma}:=\partial\tilde{\Omega}^{-}$. Consequently, the computational complexity of the volume mesh intersection algorithm in terms of cell/cell intersections is quadratic in $2D$ and cubic in $3D$ in the worst case, and linear in $2D$ and quadratic in $3D$ if local refinement is used to increase the resolution of $\tilde{\Sigma}$. The quadratic complexity in $3D$ is a serious drawback of this algorithm, especially for large simulations where $\|\tilde{\Omega}^{-}\|$ easily reaches hundred thousand cells per CPU core. Menon and Schmidt [34] have extended the volume mesh intersection algorithm from Ahn and Shashkov [1] to perform a volume conservative remapping of variables in the collocated Finite Volume Method (FVM) with second-order accuracy on unstructured meshes. Their results confirm the polynomial computational complexity in terms of absolute CPU times for this volume mesh intersection algorithm [34, table 3]. López et al. [26] propose a volume truncation algorithm for non-convex cells and apply it to the initialization of volume fractions from exact functions on unstructured meshes. Cell-subdivision is introduced to handle cases for which the interface crosses an edge of a cell twice. Non-planar truncated volumes are triangulated [26, fig 18], and second-order accuracy is demonstrated in terms of the relative global volume error for a uniform resolution and a higher-order accuracy when locally refined sub-grid meshes are used. Ivey and Moin [18] initialize volume fractions on unstructured meshes using tetrahedral decomposition of non-convex cells and perform geometrical intersections with a similar approach as the approach from Ahn and Shashkov [1]. Unlike Ahn and Shashkov [1], Ivey and Moin [18] compute volume fractions of intersected tetrahedrons by intersecting them with exact signed distance functions that are used to model the fluid interface. Therefore, this algorithm cannot directly utilize arbitrarily shaped interfaces. However, their approach utilizes a linear interpolation of intersection points between the tetrahedron and the signed-distance function and yields second-order accuracy. Accuracy is further increased using adaptive mesh refinement. The approaches reviewed so far require an exact representation of the interface using explicit analytic expressions, which hinders the direct application of such algorithms to initial conditions resulting from experiments as these are typically not available as function compositions. The volume mesh intersection algorithm [1] is flexible but computationally expensive, and it requires highly accurate and robust geometrical intersections. The following sections outline the proposed algorithm that uses an unstructured surface mesh $\tilde{\Sigma}$ to compute signed distances and volume fractions on unstructured meshes. Relying on unstructured surface meshes retains the ability to handle arbitrary-shaped surfaces while avoiding computationally expensive cell/cell intersections. Of course, using surface meshes to approximate the fluid interface renders the proposed algorithm second-order accurate; however, sufficient absolute accuracy is achievable with second-order accurate methods using local mesh refinement on the background mesh [7, 12]. Applying local mesh refinement on the background mesh in the close vicinity of the triangulated surface increases the accuracy and limits it to the resolution of the surface mesh, not the background mesh that stores volume fractions and signed distances. The proposed algorithm geometrically computes signed distances near the fluid interface. These signed distances (so-called _narrow-band_ signed-distances) are then propagated throughout $\tilde{\Omega}$ by an approximate solution of a diffusion equation. The propagated signed distances determine the value of the phase indicator $\chi(\mathbf{x},t)$ in those cells that are either completely empty $(\alpha_{c}=0)$, or completely full $(\alpha_{c}=1)$. Finally, second-order accurate volume fraction values are calculated in intersected cells $(0<\alpha_{c}<1)$. This work enables the calculation of complex initial conditions for different multiphase simulation methods. These include in particular geometric [20, 18, 39, 29, 42] and algebraic VOF methods [54, 9]. The calculation of volume fractions from a surface mesh (marker points in 2D) was done in the mixed markers / VOF method by Aulisa et al. [2]: the proposed algorithm significantly extends this idea towards an accurate and fast volume fraction model for Front Tracking methods [53], as well as the hybrid Level Set / Front Tracking methods on structured [44, 45] or unstructured [28, 51] meshes. Signed distances and the respective inside-outside information from triangulated surfaces are available for unstructured Level Set and Immersed Boundary methods. ## 2 Surface mesh / cell intersection algorithm The calculation of volume fractions by the proposed Surface Mesh Cell Intersection/Approximation (SMCI/A) algorithm, outlined in fig. 4, requires signed distances to the interface at cell centres and cell corner points. As a naive computation is computationally expensive (section 2.2), we employ an octree based approach to the calculation of signed distances. Starting point of the octree based search is the calculation of search radii at the relevant points. $r_{p}$ --- $\mathbf{x}_{p}\in P_{h}$ --- $\mathbf{x}_{c}$ --- $r_{c}$ --- (a) Calculation of search radii. $\tilde{\Sigma}$ --- octree --- $\mathbf{n}_{\tilde{\Sigma}}$ --- (b) Octree sub-division of the surface mesh $\tilde{\Sigma}$ bounding-box. $\phi_{c}^{+}$ --- $\mathbf{n}_{\tilde{\Sigma}}$ --- $\phi_{c}^{-}$ --- $\phi_{p}^{+}$ --- $\phi_{p}^{-}$ --- (c) Narrow-band signed distances from the search radii and the octree. $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- (d) Positive and negative sign diffusion throughout $\tilde{\Omega}$. $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $\tilde{\chi}_{c}=1$ --- $\tilde{\chi}_{c}=0$ --- (e) Phase indicator $\tilde{\chi}$ is $1$ or $0$ in cells strictly inside/outside of $\tilde{\Sigma}$, respectively. $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $-$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $+$ --- $-$ --- $-$ --- $+$ --- $\tilde{\chi}_{c}=1$ --- $\tilde{\chi}_{c}=0$ --- (f) Computing the approximated phase indicator $\tilde{\chi}$ in intersected cells. Figure 4: Steps of the Surface Mesh Intersection / Approximation (SMCI/A) algorithms. ### 2.1 Calculation of search radii In the first step, a search radius $r_{c}$ and $r_{p}$ is calculated at each cell center and cell-corner point, respectively. This is illustrated in fig. 4(a). Here, the cell search radius $r_{c}$ is defined by $r_{c}=\lambda_{s}\operatorname{min}_{f\in F_{c}}\\|\mathbf{x}_{f,O}-\mathbf{x}_{f,N}\\|_{2},$ (8) where $\mathbf{x}_{c}$ is the cell center, $\lambda_{s}>0$ is the _search radius factor_ detailed below and $\mathbf{x}_{f,O}$, $\mathbf{x}_{f,N}$ are the cell centers of two cells that share the face with index $f$ of the cell $\Omega_{c}$ ($O$ for owner cell with a smaller cell index than the neighbor cell $N$). Here, the index set $F_{c}$ contains the indices of those faces that form the boundary of $\Omega_{c}$. Based on (8), the corner-point search radius $r_{p}$ is defined by $r_{p}=\lambda_{s}\operatorname{min}_{c\in C_{p}(\mathbf{x}_{p})}r_{c},$ (9) where $\mathbf{x}_{p}$ is the cell-corner point, while the _point-cell stencil_ is the index set $\mathcal{S}(\mathbf{x}_{p},\tilde{\Omega})$, that contains indices of all cells from $\tilde{\Omega}$ whose corner-point is $\mathbf{x}_{p}$. The search radii introduced above are used to define search balls in $3D$ (circles in $2D$), which are used to reduce the number of calculations to determine signed distances between the cell corner points $\mathbf{x}_{p}$ and the cell centers $\mathbf{x}_{c}$ with respect to the provided surface mesh $\tilde{\Sigma}$. ### 2.2 Octree decomposition of the surface mesh and signed distance calculation In contrast to various other approaches for volume fraction initialization, the fluid interface is not represented by the proposed algorithm using a function, but as a surface mesh, consisting of triangles. To define the interface $\tilde{\Sigma}$, we first denote the convex hull of a set of $n$ points $P^{n}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{n}\\},\mathbf{x}_{i}\in\mathbb{R}^{3}$ by $\operatorname{conv}(P^{n}):=\left\\{\mathbf{x}\in\mathbb{R}^{3}:\mathbf{x}=\sum_{\mathbf{x}_{i}\in P^{n}}\gamma_{i}\mathbf{x}_{i},\sum_{i=1}^{n}\gamma_{i}=1\right\\}.$ (10) Using this, a triangle is defined as the convex hull of a point triple: $\mathcal{T}:=\operatorname{conv}(P^{3})$. Consequently, the surface mesh is defined as $\tilde{\Sigma}:=\\{\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{n}\\}.$ (11) With the structure of $\tilde{\Sigma}$ in mind, we want to emphasize why an octree based approach is the key to obtaining reasonable computation times. Consider the case where a minimal distance between a point $\mathbf{x}$ and $\tilde{\Sigma}$ would be calculated for each cell center $\mathbf{x}_{c}$ and cell-corner point $\mathbf{x}_{p}$. The need for the spatial subdivision and search operations becomes obvious, as this would require a distance computation between each point of the interface mesh and each cell centers and cell corner points of the background mesh. Consequently, this would require $\|C\|\|\tilde{\Sigma}\|$ operations to compute the geometric signed distances at cell centers and additional computations for evaluating signed distances at cell-corner points. For our computations below, the number $\|C\|$ often reaches the order of $1e05$ per CPU core, while $\|\tilde{\Sigma}\|$ is typically on the order of $1e04$ per CPU core. Aiming at redistancing computations for a dynamic setting in multiphase flows where $\tilde{\Sigma}=\tilde{\Sigma}(t)$, such a large number of distance computations makes such a brute force redistancing approach prohibitively expensive. The first step of the signed distance calculation is the computation of an Axis-Aligned Bounding Box (AABB) from the surface mesh $\tilde{\Sigma}$. The AABB is used to build an octree data structure, illustrated as a $2D$ quadtree subdivision in fig. 4(b), which is used to access $\tilde{\Sigma}$. The octree data structure enables fast search queries involving cell centers and cell corner-points that are close to the surface mesh $\tilde{\Sigma}$, with a logarithmic computational complexity with respect to the number of vertices in $\tilde{\Sigma}$ [32, 33]. The structure of the octree depends on the ordering of vertices in $\tilde{\Sigma}$: since $\tilde{\Sigma}$ is an unstructured surface mesh, its vertices are generally sufficiently unordered, which makes the octree well-balanced. Once the octree has been constructed, it can be used to find the closest points $\mathbf{x}\in\tilde{\Sigma}$ to cell centres $\mathbf{x}_{c}$ and cell corner points $\mathbf{x}_{p}$. Note that this is only true for those $\mathbf{x}_{c},\mathbf{x}_{p}$ which are sufficiently close to $\tilde{\Sigma}$ in terms of their search radius $r_{c},r_{p}$. Thus, the search radii define a so-called _narrow band_ around $\tilde{\Sigma}$, where the nearest distances are calculated geometrically. We denote the narrow band of $\tilde{\Sigma}$ with $\mathcal{N}(\tilde{\Sigma})$, and the closed ball $\mathcal{B}(\mathbf{x}^{},r):=\\{\mathbf{x}\in\mathbb{R}^{3}\|\,\\|\mathbf{x}-\mathbf{x}^{}\\|_{2}\leq r\\}$ with a radius $r$ around a point $\mathbf{x}$. Then $\mathcal{N}(\tilde{\Sigma}):=\left\\{\mathbf{x}\in\mathbb{R}^{3}\|~{}\exists~{}\mathcal{T}\in\tilde{\Sigma}\text{ such that }\mathcal{T}\cap\mathcal{B}(\mathbf{x},r)\neq\emptyset\right\\},$ (12) where $r$ is either $r_{p}$ or $r_{c}$. For a point $\mathbf{x}\in\mathcal{N}(\tilde{\Sigma})$, the octree provides the closest point $\mathbf{x}_{\text{min}}\in\mathcal{T}_{\text{min}}$ for some $\mathcal{T}\in\tilde{\Sigma}$ and the corresponding triangle $\mathcal{T}_{\text{min}}$ itself. While the absolute distance can be directly computed as $\\|\mathbf{x}-\mathbf{x}_{\text{min}}\\|_{2}$, care must be taken when computing the sign with respect to the orientation of $\tilde{\Sigma}$. Directly using the triangle normals $\mathbf{n}_{\mathcal{T}}$ may lead to false signs and consequently, to erroneous volume fractions. Thus, we follow the work of [50, 4] and compute _angle weighted normals_ $\mathbf{n}_{\mathbf{x}_{v}}=\frac{\sum_{\mathcal{T}\in\text{ngh}(\mathbf{x}_{v})}\beta_{\mathcal{T}}\mathbf{n}_{\mathcal{T}}}{\sum_{\mathcal{T}\in\text{ngh}(\mathbf{x}_{v})}\beta_{\mathcal{T}}}$ (13) at the vertices $\mathbf{x}_{v}$ of $\tilde{\Sigma}$. Here, $\text{ngh}(\mathbf{x}_{v})$ denotes the set of all triangles containing $\mathbf{x}_{v}$, $\mathbf{n}_{\mathcal{T}}$ a triangle normal and $\beta_{\mathcal{T}}$ the inner angle of $\mathcal{T}$ at $\mathbf{x}_{v}$. Baerentzen and Aanaes [4] propose a classification of the point $\mathbf{x}_{\text{min}}$ whether it is located within a triangle, on an edge, or a vertex and base the choice of the normal on this classification. While such a classification is simple in theory, a robust implementation is difficult due to the limited precision of floating point arithmetic. Thus, we opt for a linear interpolation of $\mathbf{n}_{\mathbf{x}_{v}}$ within $\mathcal{T}_{\text{min}}$ to $\mathbf{x}_{\text{min}}$, denoted $\mathbf{n}_{I}(\mathbf{x}_{\text{min}},\mathcal{T}_{\text{min}})$. With this normal computation, the signed distance between $\mathbf{x}$ and $\mathbf{x}_{\text{min}}$ is calculated by $\phi^{g}(\mathbf{x},\tilde{\Sigma})=\text{sign}((\mathbf{x}-\mathbf{x}_{\text{min}})\cdot\mathbf{n}_{I}(\mathbf{x}_{\text{min}},\mathcal{T}_{\text{min}}))\\|\mathbf{x}-\mathbf{x}_{\text{min}}\\|_{2}.$ (14) where the supindex $g$ indicates a geometrical construction. This procedure is illustrated in fig. 4(c). The robustness of this approach with regard to inside/outside classification is demonstrated in section 4.3. Using the spatial subdivision provided by the octree, the computational complexity for finding the minimal distances between mesh points and $\tilde{\Sigma}$ is reduced severely, as the vast majority of cell centers $\mathbf{x}_{c}$ are not even considered for calculation as no triangle $\mathcal{T}\in\tilde{\Sigma}$ exists within the corresponding search ball. The closest triangles of those points $\mathbf{x}_{c}$, whose ball $\mathcal{B}(\mathbf{x}_{c},r_{c})$ intersects $\tilde{\Sigma}$ are found with logarithmic search complexity with respect to $\|\tilde{\Sigma}\|$. This significant reduction of complexity can potentially enable a future application of the proposed algorithm on moving interfaces $\tilde{\Sigma}(t)$ as a geometrically exact marker field model for unstructured Front Tracking methods. Therefore, it is crucial to understand that the $\operatorname{min}_{\mathcal{T}\in\tilde{\Sigma}}$ operation in eq. 14 throughout this text relies on the octree spatial subdivision and search queries. ### 2.3 Signed distance propagation After the calculation of geometric signed distances in the narrow band around $\tilde{\Sigma}$, the signed distances are propagated to the bulk of different phases, as shown in fig. 4(d). In [28, 51], the geometric signed distances are set to large positive numbers throughout the domain, and a graph-traversal algorithm is used to iteratively correct the signs of signed distances using face-cell and point-point graph connectivity provided by the unstructured mesh. Graph-traversal is computationally expensive and complicated to implement in parallel. Here we propose a straightforward alternative that instantaneously propagates signs of signed distances through the solution domain and is parallelized easily. We rely on the diffusion equation for the signed distances, namely $\displaystyle-\Delta\phi$ $\displaystyle=0,$ (15) $\displaystyle\nabla\phi$ $\displaystyle=0,\quad\text{for}\quad\mathbf{x}\in\partial\Omega$ and its discretization using the unstructured finite volume method in OpenFOAM [19, 23, 35], giving a linear system of equations. The key idea to sign propagation is to apply a few iterations ($<5$) of an iterative linear solver to this system. In our case a Conjugate Gradient approach with an incomplete lower upper preconditioner has been used. With the initial field set to $\displaystyle\phi(\mathbf{x})$ $\displaystyle=\begin{cases}\phi_{g}(\mathbf{x},\tilde{\Sigma}),&\quad\text{if }\mathbf{x}\in\mathcal{N}(\tilde{\Sigma})\\\ 0,&\quad\text{otherwise,}\end{cases}$ (16) this small number of iterations suffices to properly propagate $\text{sign}(\phi)$ with respect to the orientation of $\tilde{\Sigma}$ throughout $\tilde{\Omega}$. Prerequisite for this approach to work is that the narrow band has a certain minimum width in interface normal direction. At least four cells on each side of the interface are required to ensure a robust propagation. This is achieved by setting a global search radius factor $\lambda_{s}:=4$ in eq. 8 used to calculate $r_{c}$ at cell centers. Note that increasing $\lambda_{s}$ beyond this value only increases computational costs, and does not impact the accuracy of the proposed algorithm, as with a larger value of $\lambda_{s}$ the narrow band $\mathcal{N}(\Sigma)$ becomes wider and consequently the geometrical signed distances are calculated at more points $\mathbf{x}_{c},\mathbf{x}_{p}$, using eqs. 17 and 20, respectively. Two aspects have to be considered when solving the linear system of equations resulting from the discretization of eq. 15. First, cells for which $\mathbf{x}_{c}\in\mathcal{N}(\tilde{\Sigma})$ have to be excluded from the vector of unknowns as $\phi^{g}(\mathbf{x}_{c})$ is already known for those. Second, for cells away from $\mathcal{N}(\tilde{\Sigma})$ the only relevant information is $\text{sign}(\phi_{c})$ indicating $\Omega_{c}\in\Omega^{-}$ or $\Omega_{c}\in\Omega^{+}$, respectively. A few iterations of a linear solver suffice to reliably propagate $\text{sign}(\phi_{c})$ to the entire domain. The resulting field is $\phi_{c}=\begin{cases}\phi^{g}_{c},&\text{if }\mathbf{x}_{c}\in\mathcal{N}(\tilde{\Sigma}),\\\ \phi^{a}_{c},&\text{otherwise,}\end{cases}$ (17) with $\phi^{g}_{c}$ denoting geometric signed distances and $\phi^{a}_{c}$ approximate values from the solution of eq. 15 carrying inside/outside information but without geometric meaning. Once the cell-centered signed distances $\phi_{c}$ are computed, they are used to calculate the signed distances at cell corner-points via $\phi^{I}_{p}=\sum_{c\in C_{p}}w_{p,c}\phi_{c},$ (18) where $C_{p}$ is the index set of cells that contain the cell corner point $\mathbf{x}_{p}$ and the supindex $I$ indicating interpolation. Furthermore, $w_{p,c}$ is the _inverse-distance weighted_ (IDW) interpolation weight $w_{p,c}=\frac{\\|\mathbf{x}_{c}-\mathbf{x}_{p}\\|_{2}^{-1}}{\sum_{\tilde{c}\in C_{p}}\\|\mathbf{x}_{\tilde{c}}-\mathbf{x}_{p}\\|_{2}^{-1}}.$ (19) As with $\phi_{c}$, the accuracy of $\phi_{p}$ is irrelevant outside of the narrow band of $\tilde{\Sigma}$, only the sign of the signed distance is important in the bulk. To correct for the error introduced by the IDW- interpolation in eq. 18, signed distances at cell-corner points of intersected cells are calculated geometrically $\phi_{p}=\begin{cases}\phi^{g}_{p},&\quad\text{if }\mathbf{x}_{p}\in\mathcal{N}(\tilde{\Sigma}),\\\ \phi^{I}_{p},&\quad\text{otherwise.}\end{cases}$ (20) Equations 17 and 20 define the final signed distances at cell centers and cell-corner points, respectively. These quantities will have the value of a geometrical distance to $\tilde{\Sigma}$ in the narrow band, while outside of the narrow band only the correct sign resulting from the approximative solution of eq. 15 is relevant. ### 2.4 Volume fraction calculation Once the signed distances at cell centers $\\{\phi_{c}\\}_{c=1,2,\dots,\|\tilde{\Omega}\|}$ and cell corner points $\\{\phi_{p}\\}_{p=1,2,\dots\|P_{h}\|}$ are calculated as outlined in the previous section, the SMCI algorithm calculates the volume fractions in a straightforward way. The volume fraction calculation is shown schematically for the SMCI algorithm in fig. 5(b). Each cell is decomposed into tetrahedrons, using the cell centroid $\mathbf{x}_{c}$ as the base point of the tetrahedron, the centroid of the face $\mathbf{x}_{c,f}$, and two successive points from the cell-face, $\mathbf{x}_{c,f,i},\mathbf{x}_{c,f,i+1}$. The resulting tetrahedron has the distance $\phi_{c}$ associated to the cell centroid, the distance $\phi_{c,f}$ associated to the face centroid, and and $(\phi_{c,f,i},\phi_{c,f,i+1})$ pair of distances associated with a pair of points that belong to the cell-face $(c,f)$, as shown in fig. 5(b). If all the distances of the tetrahedron are negative, the tetrahedron lies in the negative halfspace with respect to $\tilde{\Sigma}$, and its total volume contributes to the sum of the volume of phase $1$ inside the volume $\Omega_{c}$. If a pair of distances in a tetrahedron has different signs, the tetrahedron is intersected by the interface approximated by the surface mesh $\tilde{\Sigma}$. The volume of this intersection is calculated by geometrically intersecting the tetrahedron with those triangles from $\tilde{\Sigma}$, that have a non-zero intersection with a ball $\mathcal{B}$ enclosing the tetrahedron. The center of the ball $\mathcal{B}_{c,f,i}:=\mathcal{B}(\mathbf{x}_{c,f,i},R{c,f,i})$ is the centroid of the tetrahedron $\mathbf{x}_{c,f,i}=0.25(\mathbf{x}_{c}+\mathbf{x}_{c,f}+\mathbf{x}_{c,f,i}+\mathbf{x}_{c,f,\text{mod}(i+1,\|F_{c,f}\|)})$, where $i=0,\dots,\|F_{c,f}\|-1$, and $F_{f}$ is the oriented set of indices of the points $\mathbf{x}$ (cf. fig. 5(b)) that belong to the face $f$ of the cell $\Omega_{c}$. The radius of the tetrahedron-ball $\mathcal{B}_{c,f,i}$ is then $R_{c,f,i}=\operatorname{max}(\\|\mathbf{x}_{c}-\mathbf{x}_{c,f,i}\\|,\\|\mathbf{x}_{c,f}-\mathbf{x}_{c,f,i}\\|,\\|\mathbf{x}_{c,f,j}-\mathbf{x}_{c,f,i}\\|,\\|\mathbf{x}_{c,f,\text{mod}(j+1,\|F_{c,f}\|)}-\mathbf{x}_{c,f,i}\\|),$ (21) $j=0,\dots,\|F_{c,f}\|-1$. This sub-set of $\tilde{\Sigma}$ is found using the octree data structure with logarithmic complexity with respect to $\tilde{\Sigma}$, as outlined in the previous section. For the example tetrahedron in the cell shown in fig. 5(b), the resulting intersection between the approximated interface $\tilde{\Sigma}$ and a tetrahedron from the cell $\Omega_{c}$ is shown as the shaded volume. The magnitude of this volume is computed by applying the Gauss divergence theorem using eq. 31. The phase- specific volumes from cell-tetrahedrons are summed up for the cell $\Omega_{c}$, into the total phase-specific volume of the phase $1$ within the cell $\Omega_{c}$, and the volume fraction is therefore computed as $\alpha_{c}=\dfrac{\sum_{f=0,\dots\|C_{c}\|-1}\sum_{i=0,\dots,\|F_{c,f}\|-1}\|T(\mathbf{x}_{c},\mathbf{x}_{c,f},\mathbf{x}_{c,f,{i}},\mathbf{x}_{c,f,\text{mod}(i+1,\|F_{c,f}\|)})\cap(\mathcal{B}_{c,f,i}\cap\tilde{\Sigma})\|}{\|\Omega_{c}\|}$ (22) with $T:=\\{\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\mathbf{x}_{4}\\}$ denoting a tetrahedron. $\mathbf{x}_{c}$ --- (a) A cell $\Omega_{c}$ intersected by $\tilde{\Sigma}$. $\mathbf{x}_{c},\phi(\mathbf{x}_{c})$ --- $\phi(\mathbf{x}_{c,f,i+1})$ --- $\phi(\mathbf{x}_{c,f,i})$ --- $\mathbf{x}_{c,f,i}$ --- $R_{c,f,i}$ --- $\mathbf{x}_{c,f},\phi(\mathbf{x}_{c,f})$ --- $\mathbf{x}_{c,f,i+1}$ --- $\mathbf{x}_{c,f,i}$ --- (b) Tetrahedral cell decomposition. Figure 5: Centroid decomposition of an interface cell into tetrahedra and calculation of $\alpha_{c}$ using the SMCI/A algorithms. The SMCI algorithm is summarized by algorithm 1. Algorithm 1 The Surface-Mesh / Cell Intersection Algorithm (SMCI) 1:$\alpha_{c}=0$, $\phi_{c,p}=0$ 2:Compute search radius for cell centers ${r_{c}}_{c\in C}$ using eq. 8. 3:for cell centroids $\\{\mathbf{x}_{c}\\}_{c\in C}$ do 4: Place the vertices of $\tilde{\Sigma}$ into an octree (section 2.2). 5: Find the triangle $\mathcal{T}_{n}\in\tilde{\Sigma}$ nearest to $\mathbf{x}_{c}$ within a ball $\mathcal{B}(\mathbf{x}_{c},r_{c})$. 6: Set $\phi^{g}_{c}:=\phi^{g}(\mathbf{x}_{c},\mathcal{T}_{n})$ using eq. 14. 7:end for 8:Approximately solve eq. 15 to propagate $sign(\phi_{c})$. 9:Compute search radius for cell corner points ${r_{p}}_{p\in P}$ using eq. 9. 10:Find all intersected cells $I=\\{c,\quad\phi_{c}\phi_{p}<0\text{ for at least one }$p$\\}$. 11:Use eq. 17 to correct $\phi_{c}$ within the narrow band. 12:Compute $\phi_{p}$ in the bulk using eq. 18. 13:Use eq. 20 to correct $\phi_{p}$ within the narrow band. 14:for cells $\\{\Omega_{c}\\}_{c\in C}$ do 15: if $\phi_{c}\leq 0$ and all corner-point distances $\phi_{p}\leq 0$ then $\triangleright$ Cell is inside the negative $\tilde{\Sigma}$-halfspace. 16: $\alpha_{c}=1$ 17: end if 18: if cell $\Omega_{c}$ is intersected, $c\in I$ then $\triangleright$ Cell is intersected by $\tilde{\Sigma}$. 19: $\alpha_{c}$ given by eq. 22. 20: end if 21:end for ## 3 Surface-Mesh / Cell Approximation algorithm This section presents an alternative approach to the computation of volume fractions presented in section 2.4. While section 2.4 details a method based on geometric intersections, this section introduces an algorithm based on volumetric reconstruction by adaptive mesh refinement. Detrixhe and Aslam [10] introduce a second order accurate approximation for the volume fraction of a triangle (2D) or a tetrahedron (3D). Their model is an algebraic expression taking the signed distances $\phi$ of the vertices as arguments. In contrast, we propose a volume fraction initialization algorithm that employs this model in combination with an adaptive tetrahedral cell decomposition and the octree- based signed distance calculation described in section 2. We term this algorithm _Surface-Mesh/Cell Approximation_ (SMCA) and it is outlined below. (a) Identify potential interface cells (marked grey) using bounding ball criterion. Shown are circles with radii $\|\phi_{c}\|$. (b) Adaptive, tetrahedral decomposition of interface cells. Compute $\phi$ at new vertices. (c) Compute the volume fraction $\alpha_{c}$ using the model of Detrixhe and Aslam [10] (detail view). Figure 6: Steps of the SMCA algorithm following signed distance computation and inside/outside propagation. The SMCA-algorithm is based on the signed distance results of the SMCI- algorithm introduced in section 2. The steps depicted in fig. 4(a) \- 4(d) of the SMCI/A are used to compute $\phi_{c},\phi_{p}$ in the narrow band and propagate inside/outside information in the rest of the mesh points. Subsequent steps for the computation of volume fractions are displayed in fig. 6. First, all cells intersected by $\tilde{\Sigma}$ are identified to reduce computational costs, as only these cells have intermediate values $0<\alpha_{c}<1$. This step is depicted in fig. 6(a). Each cell for which $\mathbf{x}_{c}\in\mathcal{N}(\tilde{\Sigma})$ is checked with the _bounding ball criterion_. We define a bounding ball (bb) for a point $\mathbf{x}_{\text{bb}}\in\Omega_{c}$ using $r_{bb}=\operatorname{max}_{\mathbf{x}\in\Omega_{c}}\\|\mathbf{x}-\mathbf{x}_{bb}\\|_{2}$. This ball is the smallest ball that contains all points of $\Omega_{c}$. We compare this bounding ball to $\mathcal{B}(\mathbf{x}_{\text{bb}},\|\phi(\mathbf{x}_{\text{bb}}))$. These balls are shown in fig. 7, where the bounding ball is illustrated by a dashed and the other ball by a continuous line. As a general observation, if the bounding ball is contained in the ball with the radius $\|\phi(\mathbf{x}_{bb})\|$, i.e. $\mathcal{B}(\mathbf{x}_{bb},r_{bb})\subseteq\mathcal{B}(\mathbf{x}_{bb},\|\phi(\mathbf{x}_{bb})\|)$, then such a cell is guaranteed to be a bulk cell. This cell can then be removed from the set of cells in the narrow band to reduce the number of cells which are considered for decomposition in the next step. If the criterion is not satisfied, the cell is considered an interface cell. Two remarks on this criterion: first, the existence of such a $\mathbf{x}_{\text{bb}}$ is not a necessary but a sufficient condition. Second, in a practical implementation evaluation of this criterion is only feasible for a small number of points when aiming to keep computational costs reasonable. Thus, the actual check is performed by evaluating $f_{\text{bb}}(\mathbf{x},\phi_{\mathbf{x}},\Omega_{c})=\begin{cases}1,\quad\operatorname{max}_{\mathbf{x}_{i}\in\Omega_{c}}\\|\mathbf{x}_{i}-\mathbf{x}\\|_{2}\leq\|\phi_{\mathbf{x}}\|,\\\ 0,\quad\text{otherwise}\end{cases}$ (23) with $\mathbf{x}\in\Omega_{c}$. The evaluation of the $\operatorname{max}$-operator is based on a comparison to the corner points $\mathbf{x}_{i}$ of the cell $\Omega_{c}$. For example, in our implementation this function is only evaluated at cell centres $\mathbf{x}_{c}$ (original mesh cells, see below) or cell corner points (tetrahedra resulting from decomposition). As a consequence, a few bulk cells are considered as interface cells (fig. 7(b)). We deem this acceptable as this only has a minor impact on the computational time, but not on the computed volume fractions. $\Sigma$ --- $r_{\text{bb}}$ --- $\|\phi_{c}\|$ --- $\mathbf{x}_{c}$ --- (a) Bulk cell: the ball $\mathcal{B}(\mathbf{x}_{c},\|\phi_{c}\|)$ contains the cell bounding ball $\mathcal{B}(\mathbf{x}_{c},r_{bb})$. $\Sigma$ --- $\|\phi_{c}\|$ --- $\mathbf{x}_{c}$ --- $r_{\text{bb}}$ --- (b) False positive: a bulk cell which is not detected by the bounding ball criterion as $\mathcal{B}(\mathbf{x}_{c},r_{bb})\nsubseteq\mathcal{B}(\mathbf{x}_{c},\|\phi_{c}\|).$ Figure 7: Illustration of the idea of the bounding ball criterion in 2D for clarity. The solid grey line represents $\mathcal{B}(\mathbf{x}_{c},\|\phi_{c}\|)$, the grey dashed one $\mathcal{B}(\mathbf{x}_{c},r_{bb})$. After identification of interface cells, the cell volume fractions are initialized according to the sign of $\phi_{c}$, $\alpha_{c}=\begin{cases}1,\quad\phi_{c}\leq 0,\\\ 0,\quad\text{otherwise}.\end{cases}$ (24) This gives correct volume fractions for bulk cells, while the values of interface cells are updated as described below. Each cell flagged as an interface cell by the method described above is decomposed into tetrahedra using its centroid and cell face centroids as shown in fig. 5. Each resulting tetrahedron is further refined in an adaptive manner such that resolution is only subsequently increased where a new tetrahedron is again intersected by the interface. To achieve this, a tetrahedron $T$ is checked with the bounding ball criterion eq. 23. The criterion is only evaluated at the vertex $\mathbf{x}_{\text{max}}\in T$ for which $\|\phi(\mathbf{x}_{\text{max}})\|=\operatorname{max}_{\mathbf{x}\in T}\|\phi(\mathbf{x})\|$. Only if $f_{\text{bb}}(\mathbf{x}_{\text{max}},\phi,T)~{}=~{}0$ (eq. 23), $T$ is considered for further decomposition. $\mathbf{x}_{1}$ --- $\mathbf{x}_{2}$ --- $\mathbf{x}_{3}$ --- $\mathbf{x}_{4}$ --- $\mathbf{x}_{12}$ --- $\mathbf{x}_{24}$ --- $\mathbf{x}_{23}$ --- $\mathbf{x}_{34}$ --- $\mathbf{x}_{13}$ --- $\mathbf{x}_{14}$ --- (a) Original tetrahedron with vertices ($\mathbf{x}_{i}$, black) and edge midpoints ($\mathbf{x}_{ij}$, grey). (b) Four tetrahedra are created by combining each vertex with its connected edge midpoints (indicated by dashed lines). $\mathbf{x}_{12}$ --- $\mathbf{x}_{23}$ --- $\mathbf{x}_{34}$ --- $\mathbf{x}_{24}$ --- $\mathbf{x}_{14}$ --- $\mathbf{x}_{13}$ --- (c) Decompose octahedron into four tetrahedra by combining each grey edge with the black line formed by two opposite points (here $\mathbf{x}_{12}$, $\mathbf{x}_{34}$). Figure 8: Decomposition of a tetrahedron into eight tetrahedra using edge midpoints. An obvious choice would be decomposition at the centroid of $T$. However, repeated application of this approach results in increasingly flattened tetrahedra. To avoid this problem, we apply the decomposition shown in fig. 8. First, from the vertices edge centres of the tetrahedron $\mathbf{x}_{ij}=\frac{1}{2}(\mathbf{x}_{i}+\mathbf{x}_{j}),\quad i,j\in\\{1,2,3,4\\},i\neq j$ (25) are computed (fig. 8(a)). By combining each vertex $\mathbf{x}_{i}$ with the three edge centres of the adjacent edges, four new tetrahedra are created (fig. 8(b)). The remainder of the original tetrahedron is an octahedron (fig. 8(b) grey dashed lines) constituted by the edge centres $\mathbf{x}_{ij}$. This octahedron is decomposed into four additional tetrahedra by choosing two opposite edge centres as shown by the black line in fig. 8(c). The indices of vertices of such a line are the numbers one to four. From the remaining four edge centres, point pairs are created such that $\\{\mathbf{x}_{mn},\mathbf{x}_{mo}\\}$ or $\\{\mathbf{x}_{mn},\mathbf{x}_{on}\\}$, yielding four pairs. Combining each pair with $\\{\mathbf{x}_{ij},\mathbf{x}_{kl}\\}$ (e.g. black edge in fig. 8(c)) gives the aforementioned four tetrahedra. Subsequently, $\phi$ is computed for the added vertices $\mathbf{x}_{ij}$. The decomposition is based on the pair of edge centres that have the smallest distance between each other. Refinement is completed when a maximum refinement level $l_{\text{max}}$ is reached. This can either be an arbitrary prescribed value or can be computed such that the edge length of the refined tetrahedra is comparable to the edge length of surface triangles. In the latter case, $l_{\text{max}}=\operatorname{min}_{l\in\mathbb{N}}\left(\frac{L_{\text{tet}}}{L_{\text{tri}}}<2^{l}\right)$ (26) with $L_{\text{tet}}$ and $L_{\text{tri}}$ being cell specific reference lengths for tetrahedra and surface triangles, respectively. Different choices for $L_{\text{tet}}$ and $L_{\text{tri}}$ are possible. We choose $\displaystyle L_{\text{tet}}=\frac{1}{n_{t}}\sum_{\mathbf{e}\in E_{\text{cdc}}}\|\mathbf{e}\|,$ $\displaystyle L_{\text{tri}}=\operatorname{min}_{\mathbf{e}\in E_{\tilde{\Sigma},c}}\|\mathbf{e}\|$ with $E_{\text{cdc}}$ denoting the set of edges resulting from tetrahedral decomposition of a cell $\Omega_{c}$ at its centroid, $n_{t}$ the number of edges in $E_{\text{cdc}}$ and $E_{\tilde{\Sigma},c}$ a subset of edges of $\tilde{\Sigma}$. The set $E_{\tilde{\Sigma},c}$ consists of all edges of $\mathcal{T}\in\tilde{\Sigma}$ for which $\mathcal{T}\cap\mathcal{B}(\mathbf{x}_{\text{cp}},r_{\text{cp}})\neq\emptyset$. Here, $\displaystyle\mathbf{x}_{\text{cp}}$ $\displaystyle=\frac{1}{\|P_{\text{cp}}\|}\sum_{\mathbf{x}_{i}\in P_{\text{cp}}},$ $\displaystyle P_{\text{cp}}$ $\displaystyle:=\\{\mathbf{x}\in\tilde{\Sigma}:\operatorname{min}_{\mathbf{x}_{i}\in\Omega_{c}}\\|\mathbf{x}-\mathbf{x}_{i}\\|_{2}\\}$ and the radius $r_{\text{cp}}=\operatorname{max}_{\mathbf{x}\in P_{\text{cp}}}\\|\mathbf{x}-\mathbf{x}_{\text{cp}}\\|_{2}$. Finally, after computing a tetrahedral decomposition of each interface cell, the volume fraction of a cell $\Omega_{c}$ is calculated as $\alpha_{c}=\frac{1}{\|\Omega_{c}\|}\sum_{T\in T_{c}}\alpha(T)\|\operatorname{conv}(T)\|$ (27) where $T_{c}$ denotes the set of tetrahedra resulting from the decomposition of $\Omega_{c}$ and $\|\operatorname{conv}(T)\|$ the volume of $T$. The volume fraction $\alpha(T)$ is computed with the approach of Detrixhe and Aslam [10] (eq. 7), repeated here $\alpha(T)=\left\\{\begin{aligned} &1,&&\phi_{4}\leq 0,\\\ &1-\frac{\phi_{4}^{3}}{(\phi_{4}-\phi_{1})(\phi_{4}-\phi_{2})(\phi_{4}-\phi_{3})},&&\phi_{3}\leq 0<\phi_{4},\\\ &1-\frac{\phi_{1}\phi_{2}(\phi_{3}^{2}+\phi_{3}\phi_{4}+\phi_{4}^{2})+\phi_{3}\phi_{4}(\phi_{3}\phi_{4}-(\phi_{1}+\phi_{2})(\phi_{3}+\phi_{4}))}{(\phi_{1}-\phi_{3})(\phi_{2}-\phi_{3})(\phi_{1}-\phi_{4})(\phi_{2}-\phi_{4})},&&\phi_{2}\leq 0<\phi_{3},\\\ &-\frac{\phi_{1}^{3}}{(\phi_{2}-\phi_{1})(\phi_{3}-\phi_{1})(\phi_{4}-\phi_{1})},&&\phi_{1}\leq 0<\phi_{2},\\\ &0&&\phi_{1}>0,\end{aligned}\right.$ (28) where $\phi_{4}\geq\phi_{3}\geq\phi_{2}\geq\phi_{1}$ are the signed distances at the vertices $\mathbf{x}_{i}$ of $T$. The overall approach is summarized in algorithm 2. Algorithm 2 The Surface-Mesh / Cell Approximation Algorithm (SMCA) 1:Follow algorithm 1 up to step 13. 2:Identify interface cells (eq. 23) 3:Set bulk $\alpha_{c}$ (eq. 24) 4:Centroid decomposition of cells into tetrahedra (fig. 5) 5:for $l\in\\{1,\ldots,l_{\text{max}}\\}$ do 6: Flag tetrahedra for further refinement (eq. 23) 7: Decompose flagged tetrahedra (fig. 8) 8: Compute $\phi$ for new points (eq. 14) 9:end for 10:Compute $\alpha_{c}$ for interface cells (eq. 27) ## 4 Results The software implementation is available on GitLab [27]: we refer to the specific version (git tag) used to generate results described below. Detailed information on how to build and use the software is provided in the README.md file in the root folder of the software repository. We use the difference between the total volume given by the volume fraction calculated from the surface on the unstructured mesh, and the exact volume bounded by the surface, namely $E_{v}=\frac{1}{V_{e}}\left\|V_{e}-\sum_{c\in C}\alpha_{c}\|\Omega_{c}\|\right\|,$ (29) as the measure of accuracy of the proposed algorithms. Here, $V_{e}$ is the volume given by the exact surface function, or the volume that is bounded by a given surface mesh if an exact surface function is not available, e.g. in sections 4.2 and 4.3. In these cases, we calculate $V_{e}$ using $V_{e}=\frac{1}{3}\left\|\int_{V_{e}}\nabla\cdot\mathbf{x}\,dV\right\|=\frac{1}{3}\left\|\int_{\partial V_{e}}\mathbf{x}\cdot\mathbf{n}\,dS\right\|$ (30) where $\partial V_{e}$ is the surface that bounds $V_{e}$. As this surface is triangluated, eq. 30 can be expanded further $\displaystyle V_{e}$ $\displaystyle=\frac{1}{3}\left\|\sum_{t\in{1..N_{\tilde{\Sigma}}}}\int_{T_{t}}\mathbf{x}\cdot\mathbf{n}\,dS\right\|=\frac{1}{3}\left\|\sum_{t\in{1..N_{\tilde{\Sigma}}}}\int_{T_{t}}(\mathbf{x}-\mathbf{x}_{t}+\mathbf{x}_{t})\cdot\mathbf{n}\,dS\right\|=\frac{1}{3}\left\|\sum_{t\in{1..N_{\tilde{\Sigma}}}}\mathbf{x}_{t}\cdot\mathbf{S}_{t}\right\|$ (31) where $N_{\tilde{\Sigma}}$ is the number of triangles in $\tilde{\Sigma}$, $T_{t}\in\tilde{\Sigma}$ are triangles that form the interface mesh, and $\mathbf{x}_{t},\mathbf{S_{t}}$ are their respective centroids and area normal vectors. Computing architecture \| ---\|--- CPU \| \| vendor_id : AuthenticAMD \| cpu family : 23 \| model : 49 \| model name : AMD Ryzen Threadripper 3990X 64-Core Processor \| frequency : 2.90 GHz Compiler \| \| version : g++ (Ubuntu 10.2.0-5ubuntu1 20.04) 10.2.0 \| optimization flags : -std=c++2a -O3 Table 1: Used computing architecture. Table 1 contains the details on the computing architectures used to report the absolute CPU times in the result section. We have fixed the CPU frequency to 2.9GHz to stabilize the CPU time measurements. ### 4.1 Sphere and ellipsoid Exact initialization algorithms for spheres are available on unstructured meshes [46, 25]. We use the sphere and ellipsoid test cases to confirm the second-order convergence of SMCI/A algorithms and their applicability as a volume fraction model for the unstructured Level Set / Front Tracking method [28, 51]. The sphere case consists of a sphere with a radius $R=0.15$, and the ellipsoid half-axes are $(0.4,0.3,0.2)$. Both the sphere and ellipsoid center are at $(0.5,0.5,0.5)$, in a unit box domain. Error convergence, CPU time and additional data are publicly available [31]. #### 4.1.1 SMCI Algorithm Figure 9 shows the expected second-order convergence of the global error $E_{v}$ given by eq. 29 on cubic fig. 9(a) and irregular hexahedral fig. 9(b) unstructured meshes. In fig. 9, $N_{c}$ is the number of cells used along each spatial dimension of $\tilde{\Omega}$ and $N_{T}$ is the number of triangles used to resolve the sphere. (a) Equidistant mesh. (b) Irregular hexahedral mesh. Figure 9: $E_{v}$ errors of the SMCI algorithm for the sphere. The grey dashed line indicates second order convergence. The CPU times reported in fig. 10 for the architecture A1 in table 1 show that the SMCI algorithm is a promising candidate for a volume fraction model for the unstructured Level Set / Front Tracking method. The complexity of the algorithm expressed in terms of the measured CPU time remains, linear for a constant ratio $\sqrt{N_{T}}/N_{c}$. The computational complexity increases to quadratic with an increasing number of triangles per cell $\sqrt{N_{T}}/N_{c}$: this happens when a very fine surface mesh is used to compute volume fractions on a very coarse volume mesh. An intersection between a highly resolved surface mesh and single cell of a relatively coarse mesh is shown in fig. 11(a). This configuration is relevant for accurate initialization of volume fractions on coarse meshes, but irrelevant for calculating the phase indicator for Front Tracking, where only a small number of triangles per multimaterial cell ($\leq 10$) is present. Therefore, linear complexity of the SMCI algorithm for small ratios $\sqrt{N_{T}}/N_{c}$ makes SMCI a potential candidate for a highly accurate geometrical volume fraction model for the unstructured Level Set / Front Tracking method. We will investigate this possibility in our future work. When considering the absolute CPU times, it is important to note that the SMCI algorithm has not yet been optimized for performance. Figure 10: CPU times of the SMCI algorithm for the sphere initialized on a cubic unstructured mesh. The volume error $E_{v}$ for a sphere is shown in fig. 9(b) for a perturbed hexahedral mesh. An example perturbed mesh from this parameter study is shown in fig. 11(b). The mesh is distorted by randomly perturbing cell corner points, using a length scale factor $\alpha_{e}\in[0,1]$ for the edges $e$ that surround the mesh point. We have used $\alpha_{e}=0.25$, resulting in perturbations that are of the size of $0.25\,\times$ the edge length. This results in a severe perturbation of the mesh shown in fig. 11(b), as well as non-planarity of the faces of hexahedral cells. Still, as shown in fig. 9(b), SMCI retains second-order convergence, which is also the case for the initialization of the ellipsoid on the equidistant fig. 12 and perturbed hexahedral mesh fig. 12(b). (a) SMCI: intersected cell. $\alpha_{c}$ --- 0 --- 0.5 --- 1 --- (b) SMCI: sphere and ellipsoid volume fractions. Figure 11: SMCI algorithm used with a sphere and an ellipsoid on an unstructured hexahedral mesh. (a) Equidistant mesh. (b) Irregular hexahedral mesh. Figure 12: $E_{v}$ errors of the SMCI algorithm for the ellipsoid. The grey dashed line indicates second order convergence. #### 4.1.2 SMCA algorithm First, the effectiveness of the local adaptivity employed in the SMCA algorithm is examined with a spherical interface as described in section 4.1. Resolution of the volume mesh is fixed to $N_{c}=16$ cells in each direction while the sphere is resolved with $\sqrt{N_{T}}\approx 410$ triangles. Maximum refinement levels $l_{\text{max}}$ from $0$ to $3$ are manually prescribed. In fig. 13, the resulting global volume errors $E_{v}$ are displayed. This test case confirms the expected second-order convergence of $E_{v}$ with adaptive refinement. Figure 13: $E_{v}$ errors of the SMCA algorithm using different refinement levels $l_{\text{max}}$ for a sphere. Resolution of volume and surface mesh are fixed to $N_{c}=16$ and $\sqrt{N_{T}}\approx 410$. The grey dashed line indicates second order convergence. An exemplary tetrahedral decomposition of a perturbed hex cell with a part of the the surface mesh is displayed in fig. 14. Figure 14: Tetrahedral decomposition of a perturbed hex cell used to approximate $\alpha_{c}$. Tetrahedra from different refinement levels are shown in different colors (level 1: blue, level 2: grey, level 3: red). Due to adaptivity, the highest refinement level is localized in the vicinity of the surface mesh.. It demonstrates that the adaptive refinement based on the bounding ball criterion eq. 23 works as intended. Refinement is localized to the vicinity around the interface. Yet, the approach ensures all tetrahedra intersected by the interface are actually refined. The effectiveness of the local adaptive refinement compared to a uniform one becomes apparent when comparing the resulting number of tetrahedra. Our adaptive approach yields around $2247$ tetrahedra per interface cell on average for the spherical interface with $\sqrt{N_{T}}\approx 410$, $N_{c}=16$ and $l_{\text{max}}=3$. A uniform decomposition, on the contrary, would result in $M_{i}\times M_{r}^{l_{\text{max}}}=24\times 8^{3}\approx 47.9\times 10^{3}$ tetrahedra, where $M_{i}$ denotes the number of tetrahedra from initial cell decomposition and $M_{r}$ the number of tetrahedra from refining a tetrahedron. Thus, the local adaptive refinement reduces the required overall number of tetrahedra by a factor of $5.5$ in comparison to a uniform refinement, without affecting the accuracy. Having verified the refinement procedure, accuracy of the SMCA algorithm and its convergence with respect to surface mesh resolution is assessed in the following. As for the SMCI algorithm, a sphere and an ellipsoid are used for this purpose. Results for the sphere in terms of the global volume error $E_{v}$ (eq. 29) are shown in fig. 15 for cubic cells (fig. 15(a)) and perturbed hexahedral cells (fig. 15(b)). Domain size, sphere centre and radius are identical to the SMCI setup as well as the perturbation factor $\alpha_{e}=0.25$. The maximum refinement level is computed according to eq. 26. Both mesh types yield nearly identical results and show second-order convergence. Resolution of the volume mesh $N_{c}$ has a minor influence for coarser surface meshes which vanishes for $\sqrt{N_{T}}>100$. (a) Equidistant mesh. (b) Irregular hexahedral mesh. Figure 15: $E_{v}$ errors of the SMCA algorithm for the sphere. The grey dashed line indicates second order convergence. For the ellipsoidal interface, the errors $E_{v}$ are shown in fig. 16. The results are qualitatively and quantitatively similar to those of the spherical interface. (a) Equidistant mesh. (b) Irregular hexahedral mesh. Figure 16: $E_{v}$ errors of the SMCA algorithm for the ellipsoid. The grey dashed line indicates second order convergence. Figure 17: CPU times of the SMCA algorithm for the sphere initialized on a cubic unstructured mesh. Absolute computational times required for the initialization of a sphere with the SMCA algorithm are displayed in fig. 17. Run times have been measured on the architecture listed in table 1. As the implementation SMCI algorithm, our implementation of the SMCA algorithm has not yet been optimized for performance. Because of the algebraic calculation of volume fractions from signed distances, the SMCA algorithm allows a direct comparison with volume fraction initialization methods on unstructured meshes that represent the fluid interface using function composition. Considering section 1, logical choices for the comparison are the methods of Ahn and Shashkov [1], Fries and Omerović [13], Jones et al. [21]. However, Ahn and Shashkov [1] do not provide convergence results for the 3D initialization and Fries and Omerović [13] integrate a function that is $\neq 1$ within their 3D surface, so the result of the quadrature does not correspond to the volume enclosed by the surface. We therefore provide a direct comparison with Jones et al. [21], specifically Jones et al. [21, table 3]. Absolute volume errors are computed for an octant of a sphere with radius $R=0.5$, placed at $(0,0,0)$ within a unit-length cubical domain, and are shown in fig. 18. Tetrahedral unstructured meshes are generated using the Delaunay algorithm in gmsh [14], by providing a discretization length that results in a number of mesh points comparable to Jones et al. [21, table 3, No Nodes]. As shown in fig. 18, the accuracy of the SMCA algorithm depends on the volume mesh resolution and the number of refinement levels when an implicit (exact) sphere is used as interface description. This is expected since both parameters influence the size of the refined tetrahedra which are used to approximate the volume fraction. Consequently, the achievable accuracy is not limited by the volume mesh resolution and can be controlled through the number of refinement levels. The lowest absolute errors are in the order of magnitude of $10^{-9}$, achieved by SMCA using $10$ refinement levels, and correspond to relative errors in the order of magnitude of $10^{-8}$, which is around $4$ orders of magnitude lower than minimal VOF advection errors reported so far in the literature [30], and are therefore admissible as initial volume fraction values. Even higher levels of absolute accuracy, comparable to Jones et al. [21, table 3, $\overline{\epsilon}_{6},\overline{\epsilon}_{9}$], can be achieved with further refinement, with substantially increased computational expense. However, such further increase in accuracy is without significance to the volume fraction advection [30]. Contrary to the implicit (exact) sphere, resolving a sphere using a triangular mesh is more challenging, as the absolute accuracy depends on the resolution of the surface mesh. Results for spheres triangulated using the Frontal Algorithm in gmsh [14] are shown in fig. 18. Doubling the resolution of the surface mesh, as expected, doubles the accuracy of SMCA with triangulated surfaces as input. This approach of course does not make sense for a sphere, whose implicit (exact) function is easily defined. For geometrically complex surfaces shown below, it is important to have in mind that the resolution of the surface mesh together with the refinement level determine the absolute accuracy and computational costs. Figure 18: Comparing the SMCA algorithm and Jones et al. [21, table 3] on tetrahedral meshes. ### 4.2 Surface of a fluid from an experiment Some methods that are surveyed in section 1 can initialize volume fractions from exact implicit surfaces, such as a sphere or an ellipsoid, analyzed in section 4.1. One novelty of SMCI/A algorithms is their ability to compute volume fractions from arbitrary surfaces on arbitrary unstructured meshes. For example, volume fractions given by an experimental surface were calculated by the SMCI algorithm in Hartmann et al. [16] for studying breakup dynamics of a capillary bridge on a hydrophobic stripe between two hydrophilic stripes. In [16], the experimental setup involves a liquid bridge that is formed between two larger droplets across a hydrophobic stripe. The hydrophobic stripe drives the collapse of this liquid bridge, that is observed experimentally and in a simulation in [16]. The quantitative comparison of the simulation and the experiment from [16] is shown in fig. 19(a). The experimental surface from Hartmann et al. [16], used to initialize volume fractions, is shown in fig. 19(b). The SMCI algorithm computes the volume vractions of the experimental fluid interface from [16] with the volume error $E_{v}=7.789e-06$. As shown in section 4.1, the accuracy of the initialization depends on the quality of the surface mesh, not on the resolution of the volume mesh, that is chosen in this case to appropriately resolve the hydrodynamics in [16]. (a) Qualitative comparison with experiment, image from [16]. $f$ --- $0.0$ --- $1.0$ --- $0.2$ --- $0.4$ --- $0.6$ --- $0.8$ --- (b) Initialization of volume fractions $f$ for the wetting experiment, image adapted from [16]. Figure 19: Simulation of the wetting experiment with the fluid interface given as a triangular surface mesh [16]. ### 4.3 CAD model To demonstrate that the SMCI/A algorithms are able to handle interfaces more complex than shown in section 4.1 and section 4.2, the surface mesh from a CAD model displayed in fig. 20(a) is used. (a) Surface mesh from a CAD model. (b) Cross section of the volume mesh with part of the surface mesh, colored by signed distance. Figure 20: Surface and volume mesh of the CAD model test case. In contrast to the previous interfaces, this one features sharp edges and geometric features of distinctly different sizes. The mesh for this test case has been generated with the _cartesianMesh_ tool of cfMesh [22]. Refinement is used in the vicinity of the interface. This meshing procedure is chosen to obtain a mesh that closer resembles that of an industrial application than a uniform cubic mesh. A cross section of the mesh is depicted in fig. 20(b). Before examining the computed volume fractions for this case, the signed distance calculation (section 2.2) and sign propagation (section 2.3) are verified. The presence of sharp edges (see fig. 20(a)) makes this test case more prone to false inside/outside classifications than the others shown so far. (a) Cells for which $\phi_{c}\geq 0$ (blue) overlayed with the surface mesh (grey). (b) Cross section through the mesh with cells colored by volume fraction. Figure 21: Inside/outside computation and resulting volume fractions for the CAD geometry. Yet our procedure yields the correct sign for the distance in all cells as shown in fig. 21(a). The enclosed volume of the surface mesh is considered as $\Omega^{+}$, thus $\phi>0$ for all points $\mathbf{x}\in\Omega^{+}$. As displayed in fig. 21(a) and confirmed by further manual inspection of the results, the proposed signed distance calculation correctly classifies all cells within the narrow band and robustly propagates this information to the entire domain. This is reflected in the volume fractions as computed, shown in fig. 21(b). Bulk cells are assigned values of either $1$ or $0$, depending on whether they are located in $\Omega^{+}$ or $\Omega^{-}$ and mixed cells with $0<\alpha_{c}<0$ are only found where the surface mesh is located. Accuracy- wise, the global errors $E_{v}$ depicted in fig. 22 have been obtained with the SMCA algorithm using different refinement levels. As for the spherical interface (see fig. 13), second-order convergence is achieved, even though the surface mesh approximates a non-smooth interface here. Figure 22: $E_{v}$ errors of the SMCA algorithm using different refinement levels $l_{\text{max}}$ for the CAD model with the reference volume $V_{e}$ computed by eq. 31. The grey dashed line indicates second order convergence. ## 5 Conclusions The proposed Surface-Mesh Cell Intersection / Approximation algorithms accurately compute signed distances from arbitrary surfaces intersecting arbitrary unstructured meshes. Geometrical calculations ensure the accuracy of signed distances near the discrete surface. The signed distances (actually their inside / outside information) are propagated into the bulk using the approximate solution of a Laplace equation. Once the signed distances are available in the full simulation domain, the SMCI algorithm computes volume fractions by intersecting arbitrarily-shaped mesh cells with the given surface mesh, while the SMCA algorithm approximates volume fractions using signed distances stored at cell corner points. Both algorithms are robust and show second-order convergence for exact surfaces and arbitrarily shaped surface meshes. The SMCI algorithm scales linearly with a small number of surface triangles per cut-cell. Since a small number of triangles per cell is a requirement for Front Tracking, this linear-complexity makes SMCI an interesting candidate for computing volume fractions in the 3D unstructured Level Set / Front Tracking method [28, 51], which will be the subject of future investigations. ## 6 Acknowledgments Calculations for this research were conducted on the Lichtenberg high performance computer of the TU Darmstadt. 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# SN 2017hpa: A Nearby Carbon-Rich Type Ia Supernova with a Large Velocity Gradient Xiangyun Zeng Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People’s Republic of China School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China Xiaofeng Wang<EMAIL_ADDRESS>Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China Beijing Planetarium, Beijing Academy of Science of Technology, Beijing 100044, People’s Republic of China Ali Esamdin<EMAIL_ADDRESS>Xinjiang Astronomical Observatory, Chinese Academy of Sciences, Urumqi, Xinjiang 830011, People’s Republic of China Craig Pellegrino Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117-5575, USA WeiKang Zheng Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Jujia Zhang Yunnan Observatories (YNAO), Chinese Academy of Sciences, Kunming 650216, People’s Republic of China Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, People’s Republic of China Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, People’s Republic of China Jun Mo Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China Wenxiong Li Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China The School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel D. Andrew Howell Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Alexei V. Filippenko Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Miller Senior Fellow, Miller Institute for Basic Research in Science, University of California, Berkeley, CA 94720, USA Han Lin Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China Thomas G. Brink Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Edward A. Baron Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, USA Jamison Burke Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA James M. DerKacy Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, USA Curtis McCully Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Daichi Hiramatsu Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Griffin Hosseinzadeh Center for Astrophysics \| Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA Las Cumbres Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 9311-5575, USA Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA Benjamin T. Jeffers Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Timothy W. Ross Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Benjamin E. Stahl Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Department of Physics, University of California, Berkeley, CA 94720-7300, USA Marc J. Staley Graduate Fellow Samantha Stegman Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA Department of Chemistry, University of Wisconsin, Madison, WI 53706, USA Stefano Valenti Department of Physics, University of California, Davis, CA 95616, USA Lifan Wang George P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, Texas A&M University, Department of Physics and Astronomy, 4242 TAMU, College Station, TX 77843, USA Danfeng Xiang Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China Jicheng Zhang Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing, 100084, People’s Republic of China Tianmeng Zhang Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, People’s Republic of China ###### Abstract We present extensive, well-sampled optical and ultraviolet photometry and optical spectra of the Type Ia supernova (SN Ia) 2017hpa. The light curves indicate that SN 2017hpa is a normal SN Ia with an absolute peak magnitude of $M_{\rm max}^{B}\approx$ $-19.12\pm 0.11$ mag and a post-peak decline rate ${\Delta}m_{15}(B)$ = $1.02\pm 0.07$ mag. According to the quasibolometric light curve, we derive a peak luminosity of $1.25\times 10^{43}$ erg s-1and a 56Ni mass of $0.63\pm 0.02\,M_{\odot}$. The spectral evolution of SN 2017hpa is similar to that of normal SNe Ia, while it exhibits unusually rapid velocity evolution resembling that of SN 1991bg-like SNe Ia or the high- velocity subclass of SNe Ia, with a post-peak velocity gradient of $\sim$ $130\pm 7$ km s-1 d-1. Moreover, its early spectra ($t<-7.9$ d) show prominent C ii $\lambda$6580 absorption feature, which disappeared in near-maximum-light spectra but reemerged at phases from $t\,\sim\,+8.7$ d to $t\,\sim\,+11.7$ d after maximum light. This implies that some unburned carbon may mix deep into the inner layer, and is supported by the low C ii $\lambda$6580 to Si ii $\lambda$6355 velocity ratio ($\sim 0.81$) observed in SN 2017hpa. The O i $\lambda$7774 line shows a velocity distribution like that of carbon. The prominent carbon feature, low velocity seen in carbon and oxygen, and large velocity gradient make SN 2017hpa stand out from other normal SNe Ia, and are more consistent with predictions from a violent merger of two white dwarfs. Detailed modelling is still needed to reveal the nature of SN 2017hpa. supernovae: individual: SN 2017hpa — supernovae: general: high velocity gradient ††software: SNooPy (Burns et al., 2011, 2014), SN-Spectral Evolution (https://github.com/mwvgroup/SN-Spectral-Evolution), Minim Code (Chatzopoulos et al., 2013), IRAF (Tody, 1986, 1993), DAOPHOT (Stetson, 1987), PyZOGY (Zackay et al., 2016; Guevel & Hosseinzadeh, 2017), lcogtsnpipe (Valenti et al., 2016), LOSS data-reduction pipeline (Ganeshalingam et al., 2010; Stahl et al., 2019, 2020), Astropy (Astropy Collaboration et al., 2013), Matplotlib (Hunter et al., 2007), Scipy (https://www.scipy.org/), Numpy (https://numpy.org/) ## 1 Introduction Type Ia supernovae (SNe Ia) are widely believed to arise from explosions of carbon-oxygen (CO) white dwarfs (WDs) in a binary system, which have a typical absolute $V$-band peak magnitude of $\sim-19$ mag (Phillips, 1993; Perlmutter et al., 1999; Wang et al., 2006). The relatively uniform stellar explosions of SNe Ia make them useful as standardizable candles in measuring extragalactic distances (Phillips, 1993; Riess et al., 1996; Wang et al., 2005; Guy et al., 2005; Howell, 2011; Burns et al., 2018), leading to the discovery of the accelerating of the Universe (Riess et al., 1998; Perlmutter et al., 1999). In recent years, larger samples of SNe Ia have been used to further constrain the nature of dark energy driving the acceleration (e.g., Betoule et al., 2014; Abbott et al., 2019). However, the progenitor systems and explosion mechanism of SNe Ia still remain controversial (e.g., Maoz et al., 2014). Two popular scenarios are the violent merger-triggered explosion of two WDs, known as the double-degenerate (DD) scenario (Webbink, 1984; Iben & Tutukov, 1984), and the accretion-triggered explosion of a WD with a nondegenerate companion, known as the single- degenerate (SD) scenario (Whelan & Iben, 1973; Nomoto, 1982; Nomoto et al., 1997). In general, the detection of signatures of circumstellar material (CSM) around some SNe Ia supports the SD scenario (Hamuy et al., 2003; Aldering et al., 2006; Patat et al., 2007; Sternberg et al., 2011; Dilday et al., 2012; Maguire et al., 2013; Silverman et al., 2013; Wang et al., 2019), though some theoretical studies show that the CSM could be also produced in the DD scenario (Shen et al., 2013; Raskin & Kasen, 2013). On the other hand, there is also evidence for nondetections of companion signatures for some SNe Ia, thus favoring the DD scenario (González Hernández et al., 2012; Schaefer & Pagnotta, 2012; Olling et al., 2015; Tucker et al., 2019). Popular explosion models of SNe Ia include the following cases. (1) The CO WD accretes material from the companion star until its mass nearly reaches the Chandrasekhar mass limit ($M_{\rm Ch}$, $\sim$1.4 $M_{\odot}$, Chandrasekhar, 1957) and compressional heating at the center causes the explosion (Piersanti et al., 2004). (2) The detonation of a thin layer of He on the surface of a WD (Kromer et al., 2010; Shen & Moore, 2014) triggers a second detonation in the WD center and hence the explosion of a sub-$M_{\rm Ch}$ mass C-O WD. (3) The violent merger or secular merger of two WDs, accompanied by radiation of gravitational waves (Röpke et al., 2012; García-Berro & Lorén-Aguilar, 2017). (4) In triple systems, oscillations of the third star cause a direct collision of two WDs and trigger the SN explosion (Thompson, 2011; Mazzali et al., 2018). In view of these explosion mechanisms, the delayed detonation model is one of the most suitable ones to account for the observed properties of SNe Ia, which initially involves a deflagration of a $M_{\rm Ch}$ CO WD and later a supersonic detonation (Khokhlov, 1991; Höflich et al., 2017). Nevertheless, the double detonation model of sub-$M_{\rm Ch}$ CO WDs shows many striking features, and can also explain the observed properties of some SNe Ia (Shen et al., 2018). Observationally, there is increasing evidence for spectroscopic and photometric diversity of SNe Ia. Most SNe Ia can be classified as spectroscopically normal ones, while a small fraction exhibit peculiar properties in some respects (e.g., Branch et al., 1993; Filippenko, 1997), such as the SN 1991T-like overluminous SNe (Filippenko et al., 1992a; Ruiz- Lapuente et al., 1992; Phillips, 1993), the SN 1991bg-like subluminous SNe (Filippenko et al., 1992b; Leibundgut et al., 1993), or the SN 2002cx-like subclasses (Filippenko, 2003; Li et al., 2003). Based on differences in Si ii velocity evolution, Benetti et al. (2005) divided SNe Ia into three subclasses: high velocity gradients (HVG), low velocity gradients (LVG), and FAINT. According to the equivalent width (EW) of Si ii $\lambda$6355 and Si ii $\lambda$5972 absorption lines near maximum brightness, Branch et al. (2006) divided SNe Ia into core normal (CN), broad line (BL), cool (CL), and shallow silicon (SS) subgroups. The sublumious SN 1991bg-like and overluminous SN 1991T-like SNe Ia have large overlap with the CL and SS subclasses, respectively (Branch et al., 2006). Based on the Si ii $\lambda$6355 velocity measured near the time of $B$-band maximum, Wang et al. (2009) classified SNe Ia into normal-velocity (NV) and high-velocity (HV) subsets. The HV subclass is found to share some common properties such as red $B-V$ color, slow decay in blue bands starting at $t\approx 40$ d from the peak, and abundant surrounding CSM (Wang et al., 2008, 2009, 2019; Foley et al., 2011; Mandel et al., 2014). Although asymmetric explosions have been proposed to explain the occurrence of HV and NV subclasses of SNe Ia (Maeda et al., 2010), it is difficult to account for the fact that these two subgroups have different birth environments (Wang et al., 2013). Early-time observations can place important constraints on the explosion physics of SNe Ia, including the size of the primary WD (Bloom et al., 2012), the radius of the companion star (Hosseinzadeh et al., 2017), the distribution of 56Ni in the ejecta, and the possible existence of CSM (Piro & Morozova, 2016). Therefore, clarifying the progenitor systems and explosion mechanisms affects our understanding of stellar evolution and precision cosmology. The unburned carbon detected in early-time spectra can provide important clues to the progenitor system and explosion mechanism of SNe Ia (Yamanaka et al., 2009; Silverman et al., 2011; Taubenberger et al., 2011; Thomas et al., 2011; Silverman & Filippenko, 2012; Hsiao et al., 2013; Li et al., 2019a). Previous studies show that nearly 30% of SNe Ia show signatures of C ii $\lambda$6580 absorption at $t\approx-4$ d (relative to the time of maximum light), while this fraction is over 40% when examining the $t\approx-10$ d spectra (Parrent et al., 2011; Thomas et al., 2011; Folatelli et al., 2012; Silverman & Filippenko, 2012; Maguire et al., 2014). These studies show that carbon-positive SNe Ia tend to be LVG subtypes (Folatelli et al., 2012) and have bluer optical colors around maximum light (Thomas et al., 2011; Silverman & Filippenko, 2012). Among those carbon-positive SNe Ia, there are two events which show carbon absorption lasting until 1–3 weeks after maximum light. One is SN 2002fk, which has detecable carbon absorption lines in the $t\approx 10$ d spectrum Cartier et al. (2014). Another example is SN 2018oh studied by Li et al. (2019a), the first Kepler-discovered SN Ia with a spectroscopic classification; the carbon feature can be detected even in the $t\approx 20.5$ d spectrum, representing the latest detection of carbon in SNe Ia. The origin of these carbon detections in post-maximum spectra still remains unclear. SN 2017hpa is the third SN Ia with persistent carbon feature; it exploded in the spiral galaxy UGC 3122 (see Fig. 1) at a distance of $\sim$ 65.6 Mpc (redshift $z\approx 0.0156$). The prominent carbon features and small distance of SN 2017hpa provide us with another excellent chance to study the observed diversity of SNe Ia. In this paper, the optical observations and data reduction are presented in Section 2. Section 3 discusses the light and color curves, while Section 4 shows the spectroscopic evolution. The quasibolometric light curve and origin of prominent carbon feature of SN 2017hpa are discussed in Section 5. We summarize in Section 6. ## 2 Observations and Data Reduction ### 2.1 Discovery and Host Galaxy SN 2017hpa was discovered at $\alpha=04^{h}39^{m}50^{s}.750$, $\delta=07^{\circ}03\arcmin 54\arcsec.90$ (J2000) on 2017 Oct. 25.35 (UT dates are adopted throughout this paper) during the Puckett Observatory World Supernova Search (POSS) Gagliano et al. (2017). Figure 1 shows a color image of SN 2017hpa. A spectrum taken $\sim 0.65$ d after the discovery classified it as a normal SN Ia (Floers et al., 2017). Figure 1: The left panel shows a color image synthesized from $gri$-band observations from PanSTARRS, and the faint star right beside the position of SN 2017hpa is totally covered in following observations. The right panel shows a color image synthesized from $gri$-band observations from TNT; SN 2017hpa is marked with a red circle while the reference stars are numbered. The host galaxy of SN 2017hpa is UGC 3122, which is classified as SAB(rc)s at $z=0.015631\pm 0.000005$ (Paturel et al., 2002; Springob et al., 2005). This redshift corresponds to a distance modulus $\mu=34.05\pm 0.38$ mag with a velocity uncertainty of 500 km s-1 (Willick et al., 1997), assuming a Hubble constant of 73.5 km s-1 Mpc-1 (Riess et al., 2018). ### 2.2 Photometry After the discovery of SN 2017hpa, we triggered follow-up photometric observations on several telescopes, including the 0.8 m Tsinghua-NAOC telescope (TNT; Huang et al., 2012; Zhang et al., 2015), the Las Cumbres Obervatory (LCO) Telescope network (Shporer et al., 2011; Brown et al., 2013), the 0.76 m Katzman Automatic Imaging Telescope (KAIT) at Lick Observatory (Li et al., 2001; Filippenko et al., 2001; Filippenko, 2005), and the 1 m Nickel reflector at Lick Observatory. The TNT, Nickel, and KAIT monitored SN 2017hpa in the $BVRI$ bands, while the LCO 1 m telescope sampled its light curves in the $BVgri$ bands. For photometric observations obtained from the LCO during the Global Supernova Project, PyZOGY (Zackay et al., 2016; Guevel & Hosseinzadeh, 2017) is employed for image subtraction while the $lcogtsnpipe$ (Valenti et al., 2016) is applied for measuring the SN flux. An ad hoc pipeline (based on the IRAF DAOPHOT package; Stetson, 1987) is applied to reduce images from the TNT and extract instrumental magnitudes of the SN. Photometric images from the Lick Observatory are reduced using the Lick Observatory Supernova Search (LOSS; Filippenko et al., 2001) data-reduction pipeline (Ganeshalingam et al., 2010; Stahl et al., 2019, 2020), while DAOPHOT (Stetson, 1987) is applied to implement the point-spread-function (PSF) photometry. For the TNT instrumental magnitudes, the $BV$-band images are calibrated using the APASS catalog (Henden et al., 2016) and the $gri$-band magnitudes are calibrated using the PanSTARRS catalog (Chambers et al., 2016; Waters et al., 2016; Flewelling et al., 2016; Magnier et al., 2016). The local standard stars with photometric magnitudes from APASS and PanSTARRS are listed in Table 1. The unfiltered instrumental magnitudes from LOSS are calibrated to the standard Landolt $R$-band magnitudes based on the transformed local standards of SDSS (Li et al., 2003; Zheng et al., 2017). The LOSS $BVRI$ instrumental magnitudes are calibrated to the Johnson system using a series of Landolt (Landolt, 1992) standard stars taken on a number of photometric nights. Table 1: Photometric Standards in the SN 2017hpa Field 1aaStandard stars used for calibration of instrumental magnitudes. Star \| $\alpha$(J2000) \| $\delta$(J2000) \| $B$ (mag) \| $V$ (mag) \| $g$ (mag) \| $r$ (mag) \| $i$ (mag) ---\|---\|---\|---\|---\|---\|---\|--- 1 \| 04:40:00.331 \| +07:02:17.167 \| 14.125(020) \| 13.368(015) \| 13.689(014) \| 13.114(034) \| 12.915(040) 2 \| 04:40:03.904 \| +07:01:18.451 \| 16.618(065) \| 15.697(037) \| 16.160(089) \| 15.398(107) \| 15.110(070) 3 \| 04:39:40.185 \| +07:02:54.100 \| 15.836(029) \| 15.046(049) \| 15.403(030) \| 14.779(034) \| 14.593(045) 4 \| 04:40:00.268 \| +07:06:44.968 \| 14.140(034) \| 13.228(016) \| 13.625(016) \| 12.887(036) \| 12.632(035) 5 \| 04:40:02.771 \| +07:00:42.491 \| 16.812(121) \| 16.013(077) \| 16.389(088) \| 15.752(053) \| 15.491(174) 6 \| 04:39:52.514 \| +07:01:50.567 \| 17.515(169) \| 16.242(070) \| 16.645(040) \| 15.957(026) \| 15.637(040) 7 \| 04:39:49.255 \| +07:04:03.612 \| 17.125(095) \| 16.130(077) \| 16.594(044) \| 15.802(069) \| 15.477(068) 8 \| 04:39:57.276 \| +07:04:14.560 \| 15.626(052) \| 14.823(023) \| 15.191(018) \| 14.556(044) \| 14.326(043) 9 \| 04:39:54.303 \| +07:01:52.794 \| 17.553(151) \| 16.262(008) \| 16.918(114) \| 15.686(037) \| 15.101(034) 10 \| 04:39:58.351 \| +07:00:56.540 \| 17.480(238) \| 16.547(158) \| 16.996(066) \| 16.230(041) \| 15.995(040) 11 \| 04:39:41.224 \| +07:05:43.976 \| 17.430(026) \| 16.468(078) \| 17.033(093) \| 16.288(066) \| 16.102(040) Optical and ultraviolet (UV) observations of SN 2017hpa were also obtained with the Neil Gehrels Swift Observatory (Gehrels et al., 2004). The Swift/UVOT observations started at relatively early phases in six bands including $uvw2$, $uvm2$, $uvw1$, $u$, $b$, and $v$ (Roming et al., 2005). The filters in lower case are used throughout this paper for photometric observations in UVOT bands. Using zeropoints extracted by Breeveld et al. (2011) in the Vega system, the data-reduction pipeline of the Swift Optical/Ultraviolet Supernova Archive (SOUSA; Brown et al., 2014) is applied to obtain the Swift optical/UV light curves of SN 2017hpa. The source counts are measured using a 3$\arcsec$ aperture and corrections are based on the average PSF. The template- subtraction technique has also been applied to the Swift images and the final uncertainty in the photometry is the combination of statistical uncertainties in galaxy subtraction count rates and a 2% systematic fluctuation at each pixel caused by differences in response sensitivity across the photon detector. The final observed Swift and ground-based light curves are shown in Figure 2, and the corresponding magnitudes are tabulated in Table 2 and Table 3. Table 2: Photometric Observations of SN 2017hpa by Ground-Based Telescopes MJD \| EpochaaRelative to the epoch of $B$-band maximum brightness (MJD = 58,066.6). Magnitudes are calibrated to AB magnitude system. \| $B$ (mag) \| $V$ (mag) \| $R$ (mag) \| $I$ (mag) \| $g$ (mag) \| $r$ (mag) \| $i$ (mag) \| $Clear$ (mag) \| Telescope ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 58053.34 \| -13.30 \| 17.235(035) \| 16.967(022) \| $\cdots$ \| $\cdots$ \| 17.054(081) \| 17.060(097) \| 17.311(143) \| $\cdots$ \| TNT 58053.54 \| -13.10 \| 16.971(031) \| 16.869(027) \| 16.741(023) \| 16.668(039) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 16.623(022) \| KAIT4 58053.82 \| -12.83 \| 16.946(041) \| 16.848(030) \| $\cdots$ \| $\cdots$ \| 16.981(025) \| 16.863(065) \| 17.078(043) \| $\cdots$ \| LCO 58054.35 \| -12.29 \| 16.897(033) \| 16.684(020) \| $\cdots$ \| $\cdots$ \| 16.896(167) \| 16.806(102) \| 17.122(136) \| $\cdots$ \| TNT 58054.36 \| -12.28 \| 16.741(012) \| 16.634(008) \| 16.539(010) \| 16.463(015) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| Nickel 58054.36 \| -12.28 \| 16.744(012) \| 16.642(011) \| 16.546(011) \| 16.463(016) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| KAIT4 58054.54 \| -12.11 \| 16.717(027) \| 16.614(021) \| 16.505(019) \| 16.442(190) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 16.355(024) \| KAIT4 58054.84 \| -11.80 \| 16.705(043) \| 16.659(042) \| $\cdots$ \| $\cdots$ \| 16.692(022) \| 16.614(058) \| 16.856(045) \| $\cdots$ \| LCO 58055.19 \| -11.45 \| 16.674(031) \| 16.484(017) \| $\cdots$ \| $\cdots$ \| 16.469(074) \| 16.507(089) \| 16.846(130) \| $\cdots$ \| TNT 58055.54 \| -11.10 \| 16.460(021) \| 16.409(017) \| 16.276(017) \| 16.217(030) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 16.147(016) \| KAIT4 $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ 58203.00 \| +136.35 \| 19.821(478) \| 18.920(192) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| TNT 58207.17 \| +140.52 \| 20.258(264) \| 19.730(330) \| 20.210(455) \| 19.657(309) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| Nickel 58207.17 \| +140.52 \| 20.281(285) \| 19.851(363) \| 20.501(477) \| 19.562(302) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| KAIT4 58210.16 \| +143.52 \| 20.056(159) \| 19.796(416) \| 19.774(295) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| Nickel 58210.16 \| +143.52 \| 20.021(159) \| 19.960(435) \| 19.991(337) \| 19.606(277) \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| KAIT4 Table 3: Swift UVOT Photometry of SN 2017hpa MJD \| EpochaaRelative to the epoch of $B$-band maximum (MJD = 58,066.6). Magnitudes are calibrated to Vega magnitude system. \| $uvw2$ (mag) \| $uvm2$ (mag) \| $uvw1$ (mag) \| $u$ (mag) \| $b$ (mag) \| $v$ (mag) ---\|---\|---\|---\|---\|---\|---\|--- 58053.19 \| -13.45 \| 19.73(16) \| 20.18(33) \| 18.87(13) \| 17.51(06) \| 17.06(04) \| 16.66(05) 58054.59 \| -12.06 \| 20.18(50) \| $\cdots$ \| 18.98(26) \| 16.92(11) \| 16.73(07) \| 16.47(11) 58055.71 \| -10.93 \| 19.41(17) \| 20.40(33) \| 18.25(12) \| 16.35(05) \| 16.29(04) \| 16.22(06) 58060.76 \| -5.89 \| 18.70(09) \| 19.72(24) \| 17.24(06) \| 15.43(03) \| 15.62(03) \| 15.59(04) 58067.35 \| 0.70 \| 18.30(13) \| 19.52(28) \| 17.24(12) \| 15.36(05) \| 15.42(04) \| 15.22(06) 58069.00 \| 2.35 \| 18.61(11) \| 19.63(20) \| 17.45(09) \| 15.42(03) \| 15.42(03) \| 15.25(04) 58071.00 \| 4.34 \| 18.68(10) \| 20.23(30) \| 17.69(10) \| 15.63(04) \| 15.54(03) \| 15.31(04) 58073.05 \| 6.41 \| 18.88(12) \| 19.59(20) \| 17.67(10) \| 15.87(04) \| 15.66(03) \| 15.34(04) 58074.59 \| 7.95 \| 18.97(12) \| 19.67(19) \| 17.90(11) \| 15.98(04) \| 15.79(03) \| 15.39(04) 58080.95 \| 14.31 \| 19.17(13) \| 20.00(24) \| 18.55(13) \| 16.85(06) \| 16.40(04) \| 15.77(04) 58082.35 \| 15.71 \| 19.12(19) \| 20.52(51) \| 18.62(18) \| 16.97(09) \| 16.56(05) \| 15.80(07) 58085.14 \| 18.49 \| 19.70(15) \| 19.94(18) \| 19.14(14) \| 17.36(06) \| 16.92(04) \| 16.00(04) 58088.60 \| 21.96 \| 19.96(20) \| 20.11(22) \| 19.50(17) \| 17.63(08) \| 17.21(05) \| 16.27(05) 58095.17 \| 28.52 \| 20.02(19) \| 20.29(23) \| 19.96(22) \| 18.38(12) \| 17.75(06) \| 16.46(06) Figure 2: The observed UV and optical light curves of SN 2017hpa. ### 2.3 Spectroscopy A total of 26 low-resolution optical spectra of SN 2017hpa have been obtained using different telescopes and equipment, including the AFOSC mounted on the Asiago Ekar telescope, the BFOSC mounted on the Xinglong 2.16 m telescope (XLT; Jiang et al., 1999; Zhang et al., 2016; Fan et al., 2016), the YFOSC on the Lijiang 2.4 m telescope (LJT; Chen et al., 2001; Zhang et al., 2012; Wang et al., 2019) of Yunnan Astronomical Observatories, the Kast spectrograph on the Lick 3 m Shane telescope (Miller & Stone, 1993; Stahl et al., 2020), and the LCO 2 m Faulkes Telescope North (FTN; Brown et al., 2013). The journal of spectroscopic observations is presented in Table 4, including one spectrum from the Transient Name Server111https://wis-tns.weizmann.ac.il/ and six spectra from Stahl et al. (2020). When no data-reduction pipeline for was available, we applied standard IRAF routines to reduce the spectra. Spectrophotometric standard stars observed at an airmass comparable to the target on the same night were used to calibrate the flux density of SN 2017hpa. The extinction curves of the various observatories are utilized to correct for atmospheric extinction, and spectra of the standard stars are used to eliminate the telluric absorption lines. Table 4: Spectroscopic Observations of SN 2017hpa MJD \| EpochaaRelative to the epoch of $B$-band maximum (MJD = 58,066.6). \| $\lambda_{\rm Start}$ \| $\lambda_{\rm End}$ \| Instrument ---\|---\|---\|---\|--- 58052.5 \| -14.1 \| 3387 \| 8249 \| Asiago (public) 58053.2 \| -13.5 \| 3496 \| 9173 \| LJT 58054.2 \| -12.4 \| 3502 \| 9171 \| LJT 58056.5 \| -10.1 \| 3622 \| 10400 \| Lick 3 m 58058.7 \| -7.9 \| 3399 \| 9999 \| LCO 58064.5 \| -2.1 \| 3249 \| 9999 \| LCO 58068.7 \| 2.0 \| 3249 \| 9999 \| LCO 58071.3 \| 4.7 \| 3746 \| 8840 \| XLT 58075.4 \| 8.7 \| 3250 \| 10000 \| LCO 58076.1 \| 9.5 \| 3744 \| 8839 \| XLT 58078.4 \| 11.7 \| 3630 \| 10400 \| Lick 3 m 58091.7 \| 25.1 \| 3500 \| 9100 \| LJT 58094.3 \| 27.7 \| 3299 \| 9999 \| LCO 58099.3 \| 32.6 \| 3299 \| 9999 \| LCO 58099.3 \| 32.6 \| 3630 \| 10400 \| Lick 3 m 58105.4 \| 38.7 \| 3632 \| 10400 \| Lick 3 m 58107.7 \| 41.1 \| 3500 \| 9100 \| LJT 58110.3 \| 43.7 \| 5896 \| 8182 \| Lick 3 m 58111.3 \| 44.7 \| 3249 \| 10000 \| LCO 58118.7 \| 52.1 \| 3500 \| 9100 \| LJT 58119.4 \| 52.8 \| 3300 \| 10000 \| LCO 58121.3 \| 54.7 \| 3299 \| 10000 \| LCO 58127.8 \| 61.1 \| 3500 \| 9100 \| LJT 58131.4 \| 64.7 \| 3632 \| 10400 \| Lick 3 m 58132.3 \| 65.7 \| 3400 \| 9300 \| LCO 58153.3 \| 86.6 \| 3300 \| 9299 \| LCO ## 3 Light Curves ### 3.1 Optical and Ultraviolet Light Curves The multiband UV/optical light curves of SN 2017hpa are shown in Figure 2; one can see that the observations in optical bands have nearly daily sampling, ranging from about 2 weeks before to over 100 d after $B$-band maximum light. The light curves of SN 2017hpa are similar to those of normal SNe Ia, reaching maximum slightly earlier in the $I/i$ and $UV$ bands than the $B$ band, and having a prominent shoulder in $R/r$ as well as a secondary maximum in $I/i$. The slight deviations between the $BVgri$ light curves of different telescopes are primarily due to different filter transmission functions, as shown in Figure 3. The transmission differences at the red edge of the $I$-band filters may cause the $I$-band discrepancies between LCO and TNT. Applying a polynomial fit to the $B$-band light curves around maximum light yields a peak of $15.48\pm 0.03$ mag on MJD = 58066.6 (UT 2017 November 9.6). The $V$-band light curve reached its peak of $15.35\pm 0.2$ mag on MJD = 58068.4, $\sim 1.8$ d after the $B$-band peak. Figure 3: The transmission curves of the TNT and LCO filters; each curve is normalized to the peak. Figures 4 and 5 compare the multiband UV/optical light curves of SN 2017hpa with those of several well-observed normal SNe Ia which have comparable ${\Delta}m_{15}(B)$, including SN 2003du(Stanishev et al., 2007), SN 2005cf (Wang et al., 2009), SN 2011fe (Maguire et al., 2013), SN 2012cg (Munari et al., 2013; Brown et al., 2014), SN 2013dy (Pan et al., 2015; Zhai et al., 2016), and SN 2018oh (Li et al., 2019a). The UV/optical light curves of the comparison SNe Ia have been normalized to SN 2017hpa. As can be seen from Figure 4, SN 2017hpa and other normal comparision SNe Ia have similar light- curve shapes near $B$-band maximum. Although the UV light curves of SN 2017hpa are similar to those of other comparison SNe Ia, they seem to show excess emission at early phases, especially the first two data points. This may suggest additional energy beyond the radioactive decay of centrally-located nickel, such as surface nickel mixing (Piro & Nakar, 2013) or interaction of SN ejecta with a companion star or with CSM (Kasen, 2010). The post-peak decline rate ${\Delta}m_{15}(B)$ of the $B$-band light curve is measured to be $1.02\pm 0.07$ mag, and the color stretch (Burns et al., 2014) is determined to be $s_{BV}$= $0.94\pm 0.03$. Figure 4: Comparison of the optical light curves of SN 2017hpa with other well-observed SNe Ia having similar decline rates. The light curves of the comparison SNe Ia have been normalized to match the observed peak magnitudes of SN 2017hpa. Figure 5: Comparison of the UV light curves of SN 2017hpa with other well-observed SNe Ia having similar decline rates. The light curves of the comparison SNe Ia have been normalized to match the peak magnitudes of SN 2017hpa. ### 3.2 Reddening and Color Curves Assuming $R_{V}=3.1$ (Cardelli et al., 1989), we obtain the line-of-sight Galactic extinction for SN 2017hpa to be $A_{V}=0.485$ mag (Schlegel et al., 1998; Schlafly & Finkbeiner, 2011), corresponding to a color excess of $E(B-V)_{\rm gal}=0.156$ mag. After removing the Galactic reddening, the $B-V$ color is found to be $0.002\pm 0.05$ mag at $t=0$ d and $1.08\pm 0.06$ mag at $t=35$ d relative to $B$ maximum, consistent with typical values of normal SNe Ia (Phillips et al., 1999; Wang et al., 2009). We applied SuperNovae in object-oriented Python (SNooPy; Burns et al., 2011, 2014) to fit the multiband light curves of SN 2017hpa, as shown in Figure 6. Both $EBV$ and $st$ models in SNooPy2 are adopted to estimate the host-galaxy extinction, and an average host reddening is derived to be $E(B-V)_{\rm host}$ = $0.06\pm 0.06$ mag. The relatively low host-galaxy reddening is consistent with the fact that the SN is located far away from the center of the host galaxy. Moreover, the spectra of SN 2017hpa show no detectable absorption feature of Na i D due to host galaxy. Figure 6: Best-fit light-curve model from SNooPy2. The light curves are shifted vertically for clarity. The dashed lines represent the 1$\sigma$ uncertainty of the best-fit light-curve templates. The optical intrinsic color evolution of SN 2017hpa is shown in Figure 7. At $t\gtrsim-10$ d, both the $B-V$ and $g-r$ color curves evolve toward the red until reaching the reddest color at 4–5 weeks after $B$ maximum. Both the $V-I$ and $g-i$ color curves show a short-term evolution from red to blue until $t\approx-10$ d; then they evolve redward and reach the red peak at $t\approx 35$ d. After that, the $V-I$ and $g-i$ color curves became progressively bluer. Figure 7: The $B-V$, $V-I$, $g-r$, and $g-i$ color curves of SN 2017hpa compared with those of SNe 2003du, 2005cf, 2011fe, 2012cg, 2013dy, and 2018oh. All light curves including those of SN 2017hpa have been dereddened using SNooPy2. Overall, the color-curve evolution of SN 2017hpa is similar to that of SN 2005cf and SN 2018oh, except it has a bluer color at very early phases (especially the $g-r$ color). Based on the near-UV (NUV) colors, SNe Ia can be classified into NUV-red and NUV-blue subgroups (Milne et al., 2013). Figure 8 shows the observed $uvw1-V$ color evolution of SN 2017hpa together with that of SNe 2005cf, 2011fe, and 2018oh. One can see that SN 2017hpa can be put into the NUV-red group. Figure 8: The $uvw1-v$ color of SN 2017hpa compared to the group of NUV-blue and NUV-red SNe. The pink shaded region represents the regions covered by SNe Ia that are classified as NUV-red and the blue shaded region represents the regions covered by SNe Ia that are classified as NUV-blue. The overplotted curve is the unreddened color evolution of SN 2017hpa (see Milne et al., 2013). ### 3.3 First-Light Time The rise time and first-light time can put additional constraints on the radius of the exploding star itself (Bloom et al., 2012; Piro & Nakar, 2013). The observation on Oct. 13, 2017 by Gagliano et al. (2017) provides a non- detection limit magnitude of 20.5 mag. However, this observation is about 12 days before the discovery and can not provide very useful constraint for the explosion date and hence the rise time of the light curves. We thus only utilized the discovery magnitude, i.e., 17.9 mag $\pm$ 0.3 mag in clear band (close to broadband $R$), obtained at $\sim$2.0 days before our multi-color observations when performing the rise time fitting. The ideal expanding fireball (Riess et al., 1999) model and broken-power-law (Zheng & Filippenko, 2017) model are both adopted to fit the $R$-band light curve of SN 2017hpa (as shown in Figure 9), and the first light time of the light curve is estimated as 58047.08 $\pm$ 0.73 days and 58049.65 $\pm$ 0.24 days, respectively. The mean fitted first-light time (FFLT) is adopted as 58048.37 $\pm$ 0.97 days. Figure 9: Fit to the observed $R$-band light curves using the analytic function from Zheng & Filippenko (2017) and the ideal fireball model (Riess et al., 1999). The black triangle represents the discovery magnitude, i.e., 17.9 mag $\pm$ 0.3 mag in clear band (close to broadband R), obtained at $\sim$2.0 days before our multi-color observations. The bottom panel shows the residual relative to the best-fit curves. The horizontal dashed line in the bottom panel represents zero residual. With the time of maximum light in $B$ and the derived FFLT, the rise time of SN 2017hpa is estimated to be $18.26\pm 0.97$ d, comparable to that of typical SNe Ia (Zheng et al., 2017). The first multi-color observation of SN 2017hpa is thus estimated to be $\sim 5$ d after the FFLT, $\sim 13$ d prior to $B$ maximum. ## 4 Optical Spectra ### 4.1 Temporal Evolution of the Spectra The evolution of the optical spectra of SN 2017hpa is displayed in Figure 10. The early-time spectra are characterized by prominent absorption lines of intermediate-mass elements (IMEs), such as Fe ii $\lambda\lambda$4404,5018, Mg ii $\lambda$4481, Si ii $\lambda$6355, S ii $\lambda\lambda$5468,5654, Ca ii NIR triplet and Ca ii H&K. At $t\sim$ 2 weeks before the $B$-band maximum, the absorption troughs near 4300 $\rm\AA$ and 4800 $\rm\AA$ could be attributed to Fe ii/Fe iii/Mg ii, while the distinct absorption notches near 6300 $\rm\AA$ and 7000 $\rm\AA$ could be due to C ii $\lambda$6580 and C ii $\lambda$7234, respectively. The C ii $\lambda$6580 absorption is relatively strong while the C ii $\lambda$7234 absorption is weaker. The Si ii $\lambda$6355 absorption lines at this phase display a perfect gaussian profile without invoking the high-velocity feature (HVF), while a Ca ii NIR HVF could be detected through multi-gaussian fitting (Zhao et al., 2015, 2016). After $t\sim$ 1 week before maximum light, both C ii $\lambda$6580 and C ii $\lambda$7234 absorptions are still prominent in the spectra of SN 2017hpa, and the absorption lines of “W”-shaped S ii and Si ii $\lambda$5972 start to emerge in the spectra of SN 2017hpa. With the decreasing of the expansion velocity of the photosphere, the absorption minimum of Si ii $\lambda$6355 line gradually shifted redward, and the absorption lines of iron group elements and sulfur gradually increase in its strength. At around $B$-band maximum, the spectra are primarily dominated by “W”-shaped S ii absorption features near 5400 $\rm\AA$, the blended absorption lines of Fe ii and Si ii/Si iii near 4500 $\rm\AA$ and Si ii $\lambda$6355, while the C II features become invisible at this phase. By $t\sim$ 0 days, the HVFs of Ca ii NIR triplet become relatively weak and the photospheric component started to emerge in the spectra. At $t\sim+10$ days after the $B$ maximum, the photospheric components of the Ca ii NIR continute to gain the strength and start to dominate the spectra features. Interestingly, the C ii $\lambda$6580 absorption feature seems to reemerge in the spectra of SN 2017hpa around this phase, which is rarely seen in other SNe Ia. At about one month after the $B$-band maximum, the Ca ii H&K lines and NIR triplet are the main spectral features. Meanwhile, the features of iron group element begin to dominate in the spectra when the SN enter the early nebular phase. Figure 11 compares spectra of SN 2017hpa at several epochs with those of well-observed SNe Ia with similar ${\Delta}m_{15}(B)$. Figure 10: Optical spectral evolution of SN 2017hpa. All of the spectra have been corrected for the redshift of the host galaxy and reddening. The epochs shown on the right side represent the phases in days relative to $B$-band maximum light. The dashed line marks the center of Si ii $\lambda$6355 line profile at +2.04 d from $B$-band maximum. The color of the spectrum stands for different instruments. The spectra have been shifted vertically for clarity. Figure 11: Spectra of SN 2017hpa at $t\approx-14$, $-8$, -2, and +32 d relative to $B$-band maximum, compared with spectra of SNe 2005cf (Garavini et al., 2007; Wang et al., 2009), 2011fe (Mazzali et al., 2014; Zhang et al., 2016), and 2013dy (Zheng et al., 2013; Pan et al., 2015; Zhai et al., 2016) at comparable phases. Correction of reddening and redshift of the host galaxy had been done for all of the given spectra. The spectra have been shifted vertically for clarity. For SN 2017hpa, the C ii $\lambda$6580 and C ii $\lambda$7234 absorptions appear stronger than those of the comparison SNe Ia in the early-phase spectra, as shown in Figure 11(a). Moreover, the O i $\lambda$7774 absorption line of SN 2017hpa seems to also be stronger than in the comparison SNe Ia except for SN 2011fe and SN 2012cg. The pseudo equivalent-width (pEW) of C ii $\lambda$6580 is measured to be 14.0$\pm$1.0 $\rm\AA$ for SN 2017hpa at $t\approx-13$ d, while those measured for SNe 2005cf, 2011fe and 2012cg are 8.0$\pm$1.1 $\rm\AA$, 0.8$\pm$0.2 $\rm\AA$ and 1.0$\pm$0.6 $\rm\AA$, respectively. No C ii absorption feature is detected in the spectra of SN 2013dy at similar phase. The corresponding pEW of O i $\lambda$7774 is measured as 52.0$\pm$6.3 $\rm\AA$ for SN 2017hpa at this epoch, comparable to that of SN 2011fe (48.2$\pm$0.4 $\rm\AA$), while those measured for SNe 2012cg and 2013dy are 16.2$\pm$1.4 $\rm\AA$ and 16.3$\pm$3.4 $\rm\AA$, respectively. No prominent O i $\lambda$7774 is detected in the spectra of SN 2005cf at similar epoch, which is consistent with the findings by Wang et al. (2009). Following the discovery by Zhao et al. (2016) that the velocity of O i $\lambda$7774 line shows a positive correlation with that of C ii $\lambda$6580\. We propose that more unburned carbon and oxygen may be kept in the explosion ejecta of SNe 2017hpa, 2011fe, and 2012cg, although the stronger O i $\lambda$7774 absorption observed in SN 2017hpa could result from higher oxygen abundance of the exploding white dwarf (Cui et al., 2020). A detached Ca ii NIR HVF could be detected through multi-gaussian fitting proposed by Zhao et al. (2015, 2016). Figure 11(b) shows the comparison at $\sim 1$ week before maximum light. All spectra show an increase in absorption strength of IMEs. The Si ii $\lambda$6355 velocity of SN 2017hpa derived from absorption minimum is $12,500\pm 180$ km s-1, which is comparable to that of the comparison sample. The C ii $\lambda$6580 absorption line remained visible in the red wing of Si ii $\lambda$6355 at this epoch. The spectra near maximum light are displayed in Figure 11(c). The absorption features due to IMEs such as Si ii at 4130 $\rm\AA$, Si iii at 4560 $\rm\AA$, and S ii at 5468, 5612, and 5654 $\rm\AA$ become prominent at this phase. The C ii absorption features at around 6300 and 7000 $\rm\AA$ remain noticeable in SN 2017hpa but they are barely seen in the comparison SNe Ia. The $R$(Si ii), defined as the line-strength ratio of Si ii $\lambda$5972 to Si ii $\lambda$6355 (Nugent et al., 1995), can be used as indicator of the photospheric temperature. A lower value of $R$(Si ii) corresponds to a higher photospheric temperature for the SNe Ia. At around maximum light, $R$(Si ii) is measured to be $0.18\pm 0.03$, comparable to that of SN 2018oh ($R$(Si ii) $=0.15\pm 0.04$), suggesting that these two SNe have similar photoshperic temperature around maximum light. The relatively larger ratio indicates a lower photospheric temperature for SN 2017hpa. The pseudo-equivalent widths (pEWs) of Si ii $\lambda$5972 and Si ii $\lambda$6355 near maximum light are measured to be $15.5\pm 0.6$ $\rm\AA$ and $83.9\pm 2.2$ $\rm\AA$, respectively, putting SN 2017hpa into the CN subtype of Branch et al. (2006) classification. Figure 11(d) shows the spectral evolution at $t\approx 30$ d. The C ii $\lambda$6580 absorption line disappeared in this phase in all of our objects. With the receding of the photosphere, the Fe ii features gain strength and gradually dominate at wavelengths between 4700 and 5000 $\rm\AA$. The absorption profiles of SN 2017hpa and the comparison sample are well developed and tend to have uniform morphologies. ### 4.2 Carbon Features The presence of C ii absorption can be easily identified in the early-time spectra of SN 2017hpa around 6300 and 7000 $\rm\AA$. The C ii absorption features of SN 2017hpa are stronger than in the comparison SNe Ia. The left panel of Figure 12 shows that the C ii $\lambda$6580 absorption lines persist in the spectra from $t\approx-14.1$ to $-7.9$ d. This absorption feature disappeared in the spectra approaching maximum light and then reemerged at $t\approx 9.5$ d. As a possible explanation, we propose that the C ii will be highly excited when the detonation front or the deflagration front propagates outward through the ejecta of SNe Ia (Ciaraldi-Schoolmann et al., 2013; Seitenzahl et al., 2013) and this will make the C ii absorption features disappear temporarily. With the receding and cooling of the photosphere of SNe Ia, the C ii absorption trough will reemerge in the spectra. The right panel of Figure 12 shows the relatively weak C ii $\lambda$7234 absorption; it is noticeable in the earliest four spectra, and it then became flattened. Inspection of the spectra does not reveal significant absorption of C ii $\lambda$4267 in SN 2017hpa. Both C ii $\lambda$6580 and C ii $\lambda$7234 became barely visible in spectra taken $\sim 10$ d after maximum light. Figure 12: The left panel shows the C ii $\lambda$6580 temporal evolution while the right panel shows that of C ii $\lambda$7234\. The dashed lines mark the Doppler velocity range from $-18,000$ km s-1 to $-5000$ km s-1. The SN Spectroscopic Evolution222https://mwvg-spec- evolve.readthedocs.io/en/latest/ package is employed to fit the absorption components of Si ii $\lambda$6355 and C ii $\lambda$6580\. For SN 2017hpa, the C ii $\lambda$6580 velocity is found to range from $\sim 13,000$ km s-1 at $t\approx-14.1$ d to $\sim 9,300$ km s-1 at $t\approx-7.9$ d. According to Silverman & Filippenko (2012), the average velocity ratio between C ii $\lambda$6580 and Si ii $\lambda$6355 is $\sim 1.05$ for SNe Ia with observations at least four days prior to $B$-band maximum light. However, the mean C ii $\lambda$6580 to Si ii $\lambda$6355 velocity ratio (hereafter $R$(C ii/Si ii)) measured for SN 2017hpa is only $\sim 0.81$ (as shown in Figure 13), suggesting that significant unburned carbon may have mixed deep into the ejecta. Figure 13: Top panel: temporal evolution of the C ii $\lambda$ expansion velocity for SNe Ia with carbon detections. Bottom panel: temporal evolution of the velocity ratio C ii $\lambda$6580 to Si ii $\lambda$6355\. The comparison data are taken from Silverman & Filippenko (2012). ### 4.3 Ejecta Velocity The ejecta velocities measured from the absorption lines, such as S ii $\lambda\lambda$5468,5640, Si ii $\lambda$6355, C ii $\lambda$6580, C ii $\lambda$7234, and O i $\lambda$7774, are shown in Figure 15. The photospheric velocity measured from Si ii $\lambda$6355 at $t\approx-13.5$ d is $\sim 16,000$ km s-1, which is comparable to that of the Ca ii NIR triplet ($\sim 16,200$ km s-1), but it is faster than the C ii velocity ($\sim 12,000$ km s-1 for both C ii $\lambda$6580 and C ii $\lambda$7234). The velocity of the C ii $\lambda$6580 absorption is roughly within the typical expansion velocity of normal SNe Ia (Silverman & Filippenko, 2012). At the time of $B$-band maximum, the velocity of Si ii $\lambda$6355 is estimated to be $\sim$ $9550\pm 170$ km s-1, which can put SN 2017hpa into the NV subclass in the Wang et al. (2009) classification scheme (the basic paramaters of SN 2017hpa are listed in Table 5), as shown in Figure 14. However, the Si ii $\lambda$6355 velocity of SN 2017hpa seems to have a large gradient, $\sim$ $130\pm 7$ km s-1 d-1, measured within about 10 d after maximum light. Table 5: Parameters of SN 2017hpa Parameter \| Value ---\|--- Photometric $B_{\rm max}$ \| $14.88\pm 0.02$ mag $B_{\rm max}-V_{\rm max}$ \| $0.005\pm 0.007$ mag $M_{\rm max}(B)$ \| $-19.12\pm 0.11$ mag $E(B-V)_{\rm host}$ \| $0.06\pm 0.06$ mag $\Delta m_{15}(B)$ \| $1.02\pm 0.07$ mag $s_{BV}$ \| $0.94\pm 0.03$ $t_{\rm max}(B)$ \| $58,066.64\pm 0.36$ d $t_{0}$ \| $58,050.22\pm 1.20$ d $\tau_{\rm rise}$ \| $18.26\pm 0.97$ d $L_{\rm bol}^{\rm max}$ \| $1.25\times 10^{43}$ erg s-1 $M_{{}^{56}\rm Ni}$ \| $0.63\pm 0.02\,M_{\odot}$ Spectroscopic $v\rm_{0}$(Si ii) \| $9550\pm 170$ km s-1 $\dot{v}$(Si ii) \| $130\pm 7$ km s-1 d-1 $R$(Si ii) \| $0.18\pm 0.03$ Figure 14: Velocity evolution of SN 2017hpa as derived from the absorption minimum of Si ii $\lambda$6355, compared with SNe 2005cf, 2011fe, 2013dy, and 2018oh. The average velocity curves obtained for SN 1991T-like and SN 1991bg- like SNe are overplotted in red and blue dashed lines, respectively. The normal subclass of SNe Ia is plotted with a black solid line. The shaded region represents the 1$\sigma$ uncertainty for the mean velocity curve of normal SNe Ia. Data for the comparison SNe and the region of normal SNe Ia are extracted from (Li et al., 2019a). Figure 15: Velocity evolution of different elements measured from spectra of SN 2017hpa. ### 4.4 High-Velocity Feature At early phases, the HVFs of the Ca ii NIR triplet can be clearly recognized from the coresponding absorption-line profiles in the spectra. We utilize a multi-Gaussian function to fit the absorption profile of the Ca ii NIR triplet following the method described by Zhao et al. (2015, 2016). For SN 2017hpa, the HVF of the Ca ii NIR triplet seen in the $t\approx-13.5$ d spectrum has a velocity of $\sim 24,000$ km s-1, comparable to that of SN 2011fe ($\sim\ 22,000$ km s-1; Zhang et al., 2016). The Ca ii NIR HVFs exhibit a velocity plateau of 20,000 km s-1 from $t\approx-10$ to $-2$ d, which is similarly seen in SN 2018oh but at different epochs (Li et al., 2019a). Note that there are no obvious Si ii HVFs in the early-phase spectra of SN 2017hpa. It is suggested that HVFs are more commonly detected in line profiles of the Ca ii NIR triplet than in Si ii (Maguire et al., 2012; Childress et al., 2014; Maguire et al., 2014; Pan et al., 2015; Silverman et al., 2015). Most SNe Ia are found to have strong Ca ii NIR HVFs in their spectra at $t<7$ d while no more than 30% of them have strong Si ii HVFs (Zhao et al., 2015). ## 5 Discussion ### 5.1 Distance and Quasibolometric Light Curve Applying a standard cosmological model and assuming H${}_{0}=73.5$ km s-1 Mpc-1, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$ (Riess et al., 2018), a distance modulus of $\sim 34.05$ mag can be obtained for the host galaxy of SN 2017hpa. We also utilize the latest $EBV$ model of SNooPy2 to fit the light curves of SN 2017hpa in several optical bands, and the best-fit result gives an average distance modulus of $34.00\pm 0.09$ mag. These two distance moduli agree well with each other within the uncertainties. Adopting the distance modulus as $34.00\pm 0.09$ mag and assuming $R_{V}=3.1$, we derive the absolute $B$-band peak magnitude to be $M_{\rm max}(B)$ = $-19.12\pm 0.11$ mag after correcting for both Galactic and host-galaxy extinction. This value agrees well with the typical value of normal SNe Ia ($M_{\rm max}(B)\approx-19.3$ mag; Phillips et al., 1999; Wang et al., 2009). Our extensive photometric observations are used to establish the quasibolometric light curve of SN 2017hpa. The spectral energy distribution (SED) includes flux contributions from the following bands: $uvw2$, $uvm2$, $uvw1$, $B$, $g$, $V$, $R$, $r$, $I$, and $i$. We adopt the procedure used for SN 2018oh to establish the SED at several epochs (Li et al., 2019a). The observed magnitudes are dereddened and converted into flux density. The flux densities are then integrated using Simpson’s rule (Rogers, 1920; Syam, 2003) through the effective wavelengths. To get better knowledge of the peak luminosity, we use the UV and optical observations to construct the quasibolometric light curves by assuming the NIR contribution to be 5% at maximum light (Leloudas et al., 2009; Wang et al., 2009; Zhang et al., 2016; Zhai et al., 2016). Applying a polynomial fitting, the maximum luminosity is estimated to be $L_{\rm peak}$ = $1.25\times 10^{43}$ erg s-1 at about 0.85 d prior to $B$-band maximum. This peak luminosity is comparable to that of SN 2011fe ($\sim 1.13\times 10^{43}$ erg s-1; Zhang et al., 2016) but lower than that of SN 2018oh ($\sim 1.49\times 10^{43}$ erg s-1; Li et al., 2019a). The modified radiation diffusion model of Arnett (Arnett, 1982; Chatzopoulos et al., 2012; Li et al., 2019a) is applied to evaluate the initial nickel mass together with other physical parameters of the SN ejecta. The Minim Code (Chatzopoulos et al., 2013) is used to fit the quasibolometric light curve with a constant opacity approximation. The model input parameters are the first-light time (FLT) $t_{0}$, the radioactive 56Ni ejecta mass $M_{\rm Ni}$, the timescale $t_{\rm lc}$ of the light curve, and the leaking timescale of gamma rays $t_{\gamma}$ (see, e.g., Chatzopoulos et al., 2012, 2013). We set all of these parameters free when performing the model fitting. The final best-fit result of the quasibolometric luminosity evolution of SN 2017hpa is shown in Figure 16. Based on $\chi^{2}$ minimization, a set of parameters is found: $t_{0}=-0.94\pm 1.06$ d, $t_{\rm lc}=\,$$15.86\pm 0.76$ d, $M_{\rm Ni}$= $0.63\pm 0.02\,M_{\odot}$, and $t_{\gamma}=\,$$28.37\pm 3.44$ d. The initial nickel mass is comparable to the estimates of $M_{\rm Ni}$$\approx 0.57$ $M_{\odot}$ for SN 2011fe (Zhang et al., 2016) and $0.55\pm 0.04$ $M_{\odot}$ for SN 2018oh (Li et al., 2019a), but smaller than that of $0.77\pm 0.11$ $M_{\odot}$ for SN 2005cf (Wang et al., 2009) and $0.68\pm 0.14$ $M_{\odot}$ for SN 2003du (Stanishev et al., 2007). Figure 16: The quasibolometric light curve (dots) with an Arnett (1982) radiation diffusion model (blue curve). Adopting the method used by Li et al. (2019a), the average opacity $\kappa$ is estimated to be $0.36\pm 0.15$ cm2 g-1. With the best-fit $t_{\rm lc}$ and $t_{\gamma}$, we then obtain the ejecta mass and kinetic energy as $M_{\rm ej}$$=0.70\pm 0.22$ $M_{\odot}$ and $E_{\rm kin}$$=(0.70\pm 0.50)\times 10^{51}$ erg. These values are within the range of typical SNe Ia as suggested by Scalzo et al. (2019). ### 5.2 High Velocity Gradient The ejecta velocity (i.e., $9550\pm 170$ km s-1) measured for SN 2017hpa near maximum light is comparable to that of normal SNe Ia. According to the velocity gradient of the Si ii, SNe Ia can be divided into LVG, HVG, and FAINT subtypes (Benetti et al., 2005). Most normal velocity (as opposed to HV) SNe Ia tend to be LVG or FAINT objects (Silverman et al., 2012). The left panel of Figure 17 shows the scatter plot of the ${\Delta}m_{15}(B)$ versus velocity gradient of SNe Ia, and the right panel displays that of the velocity gradient vs. velocity measured around the time of maximum light. It can be seen that SN 2017hpa should be classified in the HVG subcategory, contradicting the trend that SNe Ia showing prominent carbon features tend to be LVG objects (Parrent et al., 2011). According to previous studies, HV SNe Ia tend to have larger velocity gradients and vice versa, while this tendency seems to be broken by SN 2017hpa. Figure 17: Spectroscopic subclassification of SN 2017hpa (as marked with black dot) based on the scheme of Benetti et al. (2005). Left panel: ${\Delta}m_{15}(B)$ is plotted with respect to the velocity gradient which is measured from Si ii $\lambda$6355\. The SNe from different subtypes are taken from Benetti et al. (2005) and Chakradhari et al. (2018), the four transitional objects are from Pastorello et al. (2007) and Sahu et al. (2013), SN 2005cf is from Wang et al. (2009), and SN 2018oh is from Li et al. (2019a). Right panel: the scatter plot of the velocity measured from Si ii $\lambda$6355 near maximum light versus the velocity gradient. The velocities are taken from Silverman et al. (2012) and Wang et al. (2019). The horizontal dashed line in the left pannel marks the boundary between HVG and LVG, which is 70 km s-1 d-1 Benetti et al. (2005). Previous studies have shown that for the HVG and LVG subclasses, the difference in velocity gradient may be due to the different nature of the explosion or the mixing degree of heavy elements (Sahu et al., 2013). An off- center ignition will trigger SNe to explode asymmetrically. In this case, different viewing angles will cause the observed velocity gradient to vary greatly (Maeda et al., 2010). It is suggested that varying the criterion for deflagration-to-detonation transition (DDT; Woosley et al., 2009) in explosions also can result in a wide range of velocity gradients (Blondin et al., 2011). However, these two scenarios are not suited to SN 2017hpa, which has a low velocity but high velocity gradient. Alternatively, the effective mixing of heavy elements in the SN ejecta may lead to the high velocity gradient of the HVG subclass, while the inefficient mixing may cause the low velocity gradient of the LVG subclass (Blondin et al., 2012; Sahu et al., 2013). ### 5.3 Prominent Carbon Features Detection of unburned carbon is important for constraining the explosion mechanisms or progenitor systems of SNe Ia, and different explosion models predict the presence of unburned material in different regions of the ejecta (Fink et al., 2010; Pakmor et al., 2012; Sim et al., 2012; Seitenzahl et al., 2013; Shen et al., 2018; Li et al., 2021). The velocities of IMEs in DDT will increase with the explosion strength, leading to the diffusion of unburned materials farther outward (Fink et al., 2010; Blondin et al., 2011). The carbon and oxygen have distinct velocity distributions in the delayed- detonation model while a similar velocity distribution is predicted in the violent merger (Röpke et al., 2012). For those carbon-positive SNe Ia such as SNe 2005di, 2005el, 2005ki, and SNF20080514-002, the absorption notch due to C ii $\lambda$6580 usually disappears about one week before maximum light (Thomas et al., 2011). Recent studies suggest that some SNe Ia or peculiar SNe Ia exhibit unusually persistent carbon features in their spectra, such as SN 2002fk (Cartier et al., 2014), iPTF14atg (Cao et al., 2015), and SN 2018oh (Li et al., 2019a). SN 2017hpa is another example showing such a prominent C ii $\lambda$6580 feature at early epoches ($t\leq\ -7.9$ d), and this carbon feature disappeared in near-maximum-light spectra. However, unlike other SNe Ia, the carbon feature seems to reemerge at phases from $t\,\sim\,$+8.7 d to $\sim\,$+11.7 d after maximum light. The velocity ratio of C ii $\lambda$6580 to Si ii $\lambda$6355 is an important parameter for setting constraints on the explosion models of SNe Ia (Parrent et al., 2011; Folatelli et al., 2012). As shown in Figure 13, the typical value of such a velocity ratio is $\sim 1.05$ for SNe Ia (Silverman & Filippenko, 2012). However, the mean $R$(C ii/Si ii) is measured to be 0.81 for SN 2017hpa, much lower than the typical value. As noted by Silverman et al. (2012), for a given object, the $R$(C ii/Si ii) usually increases somewhat with time. This conclusion may be also supported by the data presented by Parrent et al. (2011). Scalzo et al. (2010) suggested that the prominent C ii $\lambda$6580 feature, concurrent with low velocities, could be associated with a pre-explosion envelope of progenitor material originated from the merger of two white dwarfs. With the receding of the photosphere of SNe Ia, the ejecta of the inner layer with more uniform velocity distribution begin to show up, which leads to the observed phenomenon that the $R$(C ii/Si ii) increases slowly with time. The abundance distribution inferred from the violent merger model indicates that both carbon and oxygen can be mixed deep into the inner layer of the ejecta (Röpke et al., 2012). The prominent carbon features and high velocity gradient may suggest that SN 2017hpa is reminiscent of a low-luminosity subclass like SN 1991bg. However, the light curves and color curves of SN 2017hpa are quite different from those of low-luminosity objects like SN 2005bl (Taubenberger et al., 2008) or even transitional objects like SN 2004eo (Pastorello et al., 2007). To investigate the abnormal behavior of SN 2017hpa, we perform a further comparison of C ii $\lambda$6580 absorption in Figure 18, where the sample includes SNe 2002fk, 2005el, 2005cf, 2009dc, 2011fe, 2012cg, 2013dy, and 2018oh. The spectra of the comparison SNe are taken from the (Wang et al., 2009; Yaron & Gal-Yam, 2012; Silverman et al., 2012; Zhai et al., 2016; Zhang et al., 2016; Guillochon et al., 2017; Li et al., 2019a; Stahl et al., 2020). In spectra of SN 2013dy, a strong C ii $\lambda$6580 absorption feature can be found with a velocity up to $\sim 16,300$ km s-1 at early epochs, but this absorption feature quickly fades away at $t\approx-12.9$ d, $\sim 3$ d after explosion (Zheng et al., 2013; Pan et al., 2015). SN 2012cg shows moderately strong C ii $\lambda$6580 at the same phase with respect to SN 2017hpa, and the C ii absorption feature lasts until $\sim 8$ d before maximum light (Silverman et al., 2012). The spectra of SN 2009dc exhibit very prominent C ii $\lambda$6580 absorption that lasts for a long time, and this SN is proposed to result from a super-Chandrasekhar mass progenitor system (Howell et al., 2006; Silverman et al., 2011; Taubenberger et al., 2011; Tanaka et al., 2010). SNe 2005cf, 2005el, and 2011fe show moderate C ii features along their spectra evolution, while SNe 2002fk and 2018oh have C ii absorption comparable to that of SN 2017hpa. All three of these normal SNe Ia show prominent C ii absorption feature. The C ii absorption lines could be detected even in the spectra at $\sim$7 days after $B$-band maximum light. Figure 18: The C ii $\lambda$6580 evolution of SN 2017hpa compared to some well-observed SNe Ia, including SN 2002fk, 2005el, 2005cf, 2009dc, 2011fe, 2012cg, 2013dy, and 2018oh. Previous studies suggest that carbon-positive SNe Ia tend to have bluer optical or UV colors (Thomas et al., 2011; Silverman et al., 2012; Milne et al., 2013). Swift/UVOT observations also suggest that SNe Ia with prominent carbon features are NUV-blue objects (Roming et al., 2005; Milne et al., 2010; Thomas et al., 2011). The only exception is SN 2005cf, which belongs to the NUV-red subgroup but with signatures of carbon features (Silverman et al., 2012; Milne et al., 2013). SN 2017hpa is also a NUV-red SN Ia and has carbon in its early-time spectra. Based on model comparisons, Brown et al. (2019) suggested that the physical origin of the NUV-blue and NUV-red subclasses are likely related to metal abundance. As suggested by Heringer et al. (2017), the C ii absorption features could be hidden by the emission from iron, leading to a lower metallicity within the outer layers of SNe Ia with carbon signatures. However, if metallicity is the dominant origin of the NUV differences and the presence of C ii features, a continuous distribution is expected in each of them (Brown et al., 2019). A large sample of SNe Ia with positive C ii features is needed for modeling metallicity effects on C ii absorption in the spectra. Based on the above discussions, SN 2017hpa shows prominent carbon features with distinct evolution, a low C ii $\lambda$6580 to Si ii $\lambda$6355 velocity ratio, and normal ejecta velocity but a high velocity gradient, all of which are unusual for known subtypes of normal SNe Ia. We suggest that SN 2017hpa could result from a violent merger explosion of two carbon-oxygen white dwarfs, which brings up the prominent and distinct C ii features in its spectra. The deep mixing of the SN ejecta may result in the high velocity gradient of SN 2017hpa. ## 6 Conclusion In this paper, we present extensive optical photometry and spectroscopy of the Type Ia SN 2017hpa, which was discovered at a relatively young phase. This object can be put into the category of normal and NUV-red SNe Ia, with ${\Delta}m_{15}(B)$= $1.02\pm 0.07$ mag and an absolute $B$-band magnitude $M_{\rm max}(B)$ = $-19.12\pm 0.11$ mag. The quasibolometric light curve of SN 2017hpa is established by using extensive UV/optical photometric observations. Arnett’s 56Ni and 56Co radioactive-decay-driven radiation diffusion model is utilized to fit the quasibolometric light curve, deriving a peak luminosity of SN 2017hpa as $L_{\rm peak}$ = $1.25\times 10^{43}$ erg s-1. The mass of nickel synthesized during the explosion is estimated to be $M_{\rm Ni}$= $0.63\pm 0.02\,M_{\odot}$, and the ejecta mass $M_{\rm ej}$$=0.70\pm 0.22$ $M_{\odot}$. The spectral evolution of SN 2017hpa is roughly the same as that of normal SN Ia such as SN 2018oh. However, prominent C ii absorption and abnormal velocity evolution distinguish it from other normal SNe Ia. The carbon and oxygen features appear stronger than in normal SNe Ia and lasted until about 10 d after maximum light, and both the carbon and oxygen have a lower velocity than intermediate-mass elements such as Si ii and Ca ii. Although SN 2017hpa has a typical ejecta velocity, $\sim$ $9550\pm 170$ km s-1 as measured near the maximum light, it has an unsually large velocity gradient ($\sim$ $130\pm 7$ km s-1 d-1) in comparison with other normal SNe Ia. The significant amount of unburned C and O in the ejecta, lower velocity relative to IMEs, and large velocity gradient are more consistent with the merger model. More observations and detailed modeling are needed to reveal the exact explosion physics in objects like SN 2017hpa. ## Acknowledgments We thank the anonymous referee for the suggestive comments, which improved the manuscript. Funding for this work was provided by the National Natural Science Foundation of China (NSFC, grants 11873081, U2031209, 12033002, 11633002, and 11761141001) and the National Program on Key Research and Development Project (grant 2016YFA0400803), the High Level Talent-Heaven Lake Program of Xinjiang Uygur Autonomous Region of China. This work is partially supported by the Scholar Program of Beijing Academy of Science and Technology (DZ:BS202002). We acknowledge the staffs of the Lijiang 2.4 m telescope (LJT), the Xinglong 2.16 m telescope (XLT), and Lick Observatory for their support. The Chinese Academy of Sciences and the People’s Government of Yunnan Province provide support for the LJT, which is corporately run and maintained by Yunnan Observatories and the Center for Astronomical Mega-Science (CAS). JuJia Zhang is supported by the National Natural Science Foundation of China (NSFC; grants 11773067, 11403096), the Youth Innovation Promotion Association of the CAS (grant 2018081), and the Ten Thousand Talents Program of Yunnan for Top-notch Young Talents. Support for A.V.F.’s group at U.C. Berkeley was provided by the TABASGO Foundation, the Christopher R. Redlich Fund, and the Miller Institute for Basic Research in Science (U.C. Berkeley). This work makes use of data from the Las Cumbres Observatory network. JB, DH, DAH, and CP were supported by NSF grant AST-1911225. The Swift/UVOT data were reduced by P.J. Brown and released in the Swift Optical/Ultraviolet Supernova Archive (SOUSA), which is supported by NASA’s Astrophysics Data Analysis Program (grant NNX13AF35G). Some of the observations with the Lick Observatory 1 m Nickel telescope were conducted by U.C. Berkeley undergraduate students Sanyum Channa, Edward Falcon, Nachiket Girish, Romain Hardy, Julia Hestenes, Andrew Hoffman, Evelyn Liu, Shaunak Modak, Costas Soler, Kevin Tang, Sameen Yunus, and Keto Zhang; we thank them for their excellent work. Lick/KAIT and its ongoing operation were made possible by donations from Sun Microsystems, Inc., the Hewlett-Packard Company, AutoScope Corporation, Lick Observatory, the U.S. National Science Foundation, the University of California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation. A major upgrade of the Kast spectrograph on the Shane 3 m telescope at Lick Observatory was made possible through generous gifts from the Heising-Simons Foundation as well as William and Marina Kast. Research at Lick Observatory is partially supported by a generous gift from Google. ## References * Abbott et al. (2019) Abbott, T. M. C., Allam, S., Andersen, P., et al. 2019, ApJ, 872, L30 * Aldering et al. 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# Unconventional Superfluidity in a model of Fermi-Bose Mixtures K Sheshadri1<EMAIL_ADDRESS>A Chainani2<EMAIL_ADDRESS>1226, Bagalur, Bangalore North, Karnataka State, India 562149 2Condensed Matter Physics Group, National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan ###### Abstract A finite-temperature ($T>0$) study of a model of a mixture of spin-zero hardcore bosons and spinless fermions, with filling fractions $\rho_{B}$ and $\rho_{F}$, respectively, on a two-dimensional square lattice with composite hopping $t$ is presented. The composite hopping swaps the locations of a fermion and a boson that occupy nearest-neighbor sites of the lattice. The superfluid order parameter $\psi$, the femion hopping amplitude $\phi$, the chemical potential $\mu$, the free energy minimum $\tilde{F}$ and entropy $S$ are calculated in the limit $\rho_{B}+\rho_{F}=1$ within a mean-field approximation, and lead to a phase diagram in the $\rho_{F}-T$ plane. This phase diagram consists of a metallic superfluid phase under a dome-shaped $T(\rho_{F})$, and insulating normal liquid and insulating normal gas phases outside the dome. These phases are separated by coupled discontinuous transitions as indicated by jumps in $\psi$ and $\phi$. The maximum critical transition temperature $T_{c}$ is observed very close to $\rho_{F}=1/2$. While $\tilde{F}(T)$ is continuous with a derivative discontinuity at $T=T_{c}(\rho_{F})$ for $0<\rho_{F}\leq 1/2$ (first-order transition), it becomes discontinuous for $\rho_{F}>1/2$ (zeroth-order transition), where the entropy becomes negative for a range of temperatures below $T_{c}$. The ratio of $T_{c}$ to Fermi band width agrees remarkably with the ratio of $T_{c}$/$T_{F}$ (where $T_{F}$ is the Fermi temperature) of unconventional superfluids and superconductors like Fermi-Bose mixtures, the high-$T_{c}$ cuprates, iron-based and hydride superconductors, that exhibit experimental values of $T_{c}$ spread over nine orders of magnitude from $\sim 200$nK to $\sim 260$K. ## I Introduction Fermi-Bose mixtures (FBMs) constitute an unusual and important state of matter, including well known examples like He3-He4 mixtures Ebner , the mixed phase of type-II superconductors, ultracold atom systemsTruscott ; Schreck ; Hadzibabic , unconventional superconductors which exhibit Bardeen-Cooper- Schrieffer to Bose-Einstein condensation (BCS-BEC) crossoverLubashevsky ; Okazaki ; Kasahara ; Rinott , and so on. Experimental and theoretical studies of FBMs have shown remarkable results, particularly in terms of the BCS-BEC crossover across a Feshbach resonance Regal , that have revealed their distinct aspects compared to the limiting cases of BCS superconductivity and BEC superfluidity Randeria ; Ketterle . The BCS-BEC crossover was originally predicted to occur for excitons in semiconductors Keldysh and quarks in high energy physicsKerbikov . However, it was first reported experimentally in ultracold fermionic atoms with s-wave interactions Bartenstein . Unusual and unexpected results include formation of a Feshbach moleculeCumby , and the role of three body physicsBloom in FBMs. On the other hand, the role of BCS-BEC crossover in condensed matter involves experimental results on iron-based superconductorsLubashevsky ; Okazaki ; Kasahara ; Rinott and its relation to well-accepted theoretical resultsRanderia ; Randeria2 ; quick . While interactions between fermions mediated by phonons define the BCS theory of superconductivity, several studies have also considered their importance in mixtures of ultracold atoms Bijlsma ; Viverit ; Capuzzi ; Albus . The Boson-Fermion (BF) model Schafroth , which preceded the BCS theory, discusses itinerant fermions hybridizing with bosons composed of bound pairs of fermions of opposite spins. The BF model was subsequently used to study electrons interacting with local lattice deformations Ranninger as well as high temperature superconductivity Ranninger2 ; Friedberg ; Gershkenbein ; Domanski . Recent studies have applied it for describing resonance superfluids in the BCS-BEC crossover regime Shin , as well as a temperature driven crossover in an FBM Maska2 . These studies have shown the importance and interplay of bosonic and fermionic degrees of freedom in various physical systems. Early studies on mixtures investigated the role of an attractive interaction between fermions and bosons. It was shown that an FBM with attractive interactions undergoes a collapse when the fermion number exceeded a critical valueUfrecht . The breakthrough in controlling a Feshbach resonance in FBMs allowed researchers to effectively tune the boson-fermion interaction and control the system from collapsing at high densities Ospelkaus . In contrast, theoretical studies employing repulsive interactions could describe various stable density configurations of FBMs. The role of finite temperatures and going beyond the mean-field approximation was also investigated Viverit . In the case of a strongly repulsive quasi one-dimensional FBM,Guan it was shown that the phase diagram as a function of applied magnetic field $H$ displays a pure boson phase for $H=0$, polarized fermions and bosons coexisting for $0<H<H_{c}$, and a fully polarized fermion phase for $H>H_{c}$. More interestingly, for an FBM on a 2D optical lattice in the framework of an extended single band Hubbard model with Coulomb interaction terms between bosons ($U_{BB}$), between fermions ($U_{FF}$) and an additional Coulomb interaction between bosons and fermions ($U_{BF}$), it was shown that the bosons can mediate an attractive interaction between fermions, leading to fermion paired states with different $s,~{}p$ and $d$ orbital symmetries Wang . Further, the phase diagram as a function of $U_{BF}$ versus fermion number also revealed the existence of spin density wave and charge density wave phases. The authors also predicted that for experimentally accessible regime of parameters, the 2D FBM would exhibit superfluidity with an unconventional fermion pairing having a transition temperature around a percent of the Fermi energy. On the other hand, the role of interaction-dependent temperature effects in an FBM were investigated by CramerCramer . It was shown that adiabatic temperature changes of the FBM occur which depend on the interaction between fermions and bosonsCramer . In addition, the dynamics of FBMs has also been investigated and it was shown that long range density wave phases can be obtained for fermions and bosons hopping independently in the presence of on-site boson-boson $U_{BB}$ and boson-fermion $U_{BF}$ Coulomb interactions Lewenstein ; Pollet . However, a composite hopping that exchanges a fermion with a boson, when they occupy neighboring sites, was not considered in earlier work. This form of hopping was proposed recently by us zeroTarxiv ; zeroTPRO and distinguishes our work from earlier work on FBMs. In this work, we calculate the thermodynamic properties of a model of FBM on a two-dimensional square lattice with composite hopping between neighboring spinless fermions and hardcore bosons, extending our earlier study of $T=0$ properties zeroTarxiv ; zeroTPRO . As in the previous work, we use a mean-field approximation and restrict ourselves to the case $\rho_{F}+\rho_{B}=1,$ (1) where $\rho_{F}$ and $\rho_{B}$ are the filling fractions of the fermions and bosons, respectively. To recall, at $T=0$, the model displays two distinct phases separated by coupled first-order transitions at Fermi filling fraction $\rho_{F}\simeq 0.3$: for $\rho_{F}<0.3$ the Fermi sector is insulating and the Bose sector is a normal liquid, while for $\rho_{F}>0.3$ the Fermi sector is metallic and the Bose sector is a superfluid. In the present work, we find that thermal fluctuations suppress superfluidity, and at a certain $T=T_{c}(\rho_{F})$ there is a discontinuous transition to an insulating non- superfluid phase as shown by the superfluid amplitude $\psi(T)$ and the fermion hopping amplitude $\phi(T)$. We further find that the transition occurring at $T_{c}(\rho_{F})$ is first order for $0.3<\rho_{F}\leq 1/2$ (the minimum free energy $\tilde{F}(T)$ is continuous with a discontinuity in its first derivative), but is zeroth order for $1/2<\rho_{F}<1$ (the minimum free energy $\tilde{F}(T)$ is discontinuous). In the latter regime, the entropy becomes negative for a range of temperatures below $T_{c}$. We compute the ratio of $T_{c}$ to the Fermi band width, and find remarkable agreement with measured values in the range of $0.02$ to $0.20$ for a wide variety of unconventional superfluids and superconductors, including Fermi-Bose mixtures, the high-$T_{c}$ cuprates, iron-based superconductors and hydrides, that have their $T_{c}$ spread over nine orders of magnitude from a few hundred nanokelvins to a few hundred kelvinszeroTPRO . Our estimate for the superconducting $T_{c}$ in the solid-state context with known experimental band widths or the Fermi temperature are consistent with observed $T_{c}$’s of the cuprates as well as iron-based superconductors. ## II The Composite-Hopping Model and Its Mean-Field Thermodynamics We consider the composite-hopping model with Hamiltonian $\displaystyle H$ $\displaystyle=$ $\displaystyle-\alpha\sum_{i}\left[b_{i}^{\dagger}b_{i}+f_{i}^{\dagger}f_{i}-1\right]-\mu\sum_{i}f_{i}^{\dagger}f_{i}$ (2) $\displaystyle-t\sum_{<ij>}f_{i}^{\dagger}f_{j}b_{j}^{\dagger}b_{i}$ that was proposed in a recent study for $T=0$zeroTarxiv ; zeroTPRO , where the notation used above is also explained. In this work also, we consider a FBM on a two-dimensional square lattice. The composite hopping term, the last term above, results in swapping of a hardcore boson and a spinless fermion when they occupy nearest-neighbor sites. Using the mean-field approximation, this term is transformed according to $\displaystyle f_{i}^{\dagger}f_{j}b_{j}^{\dagger}b_{i}$ $\displaystyle\simeq$ $\displaystyle\langle f_{i}^{\dagger}f_{j}\rangle~{}(\langle b_{j}^{\dagger}\rangle b_{i}+\langle b_{i}\rangle b_{j}^{\dagger}-\langle b_{j}^{\dagger}\rangle\langle b_{i}\rangle)$ (3) $\displaystyle+\langle b_{j}^{\dagger}\rangle\langle b_{i}\rangle f_{i}^{\dagger}f_{j}-\langle f_{i}^{\dagger}f_{j}\rangle\langle b_{j}^{\dagger}\rangle\langle b_{i}\rangle,$ so $H$ is approximated by a mean-field Hamiltonian $\displaystyle H^{MF}$ $\displaystyle=$ $\displaystyle H_{0}+H_{1}+H_{2},~{}~{}\mathrm{where}$ $\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle N(2\phi\psi^{2}+\alpha),$ $\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle-(\alpha+\mu)\sum_{i}f_{i}^{\dagger}f_{i}-\frac{1}{z}\psi^{2}\sum_{<ij>}f_{i}^{\dagger}f_{j},~{}\mathrm{and}$ $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle\sum_{i}\left[-\alpha b_{i}^{\dagger}b_{i}-\phi\psi(b_{i}+b_{i}^{\dagger})\right].$ (4) We have taken $zt=1$ ($z$ is the coordination number of the lattice), and introduced the thermodynamic expectation values $\phi=\langle f_{i}^{\dagger}f_{j}\rangle,~{}~{}\psi=\langle b_{i}\rangle=\langle b_{j}^{\dagger}\rangle.$ (5) We assume $\phi~{}\mathrm{and}~{}\psi$ to be real and homogeneous and consider $\psi$ to be the superfluid order parameter FisherWeichman89 ; shesh93 ; shesh95 ; gutz ; spin1_1 ; spin1_2 ; spin1_3 . For the hardcore bosons, we use the single-site boson occupation number basis $\\{\|0\rangle,~{}\|1\rangle\\}$ for diagonalizing the $2\times 2$ matrix $h_{2}$ of $H_{2}/N$, i.e., $h_{2}=\begin{bmatrix}0&-\phi\psi\\\ -\phi\psi&-\alpha\end{bmatrix}$ (6) that has the eigenvalues $\lambda_{\pm}=\frac{1}{2}\left[-\alpha\pm R\right],~{}~{}\mathrm{where}~{}~{}R=\sqrt{\alpha^{2}+4\phi^{2}\psi^{2}}.$ (7) Using the Fourier transform $f_{i}=\frac{1}{\sqrt{N}}\sum_{\bf k}e^{i{\bf k.r_{i}}}f_{\bf k},$ (8) the Hamiltonian $H_{1}$ of the fermion sector becomes $H_{1}=\sum_{\bf k}(\varepsilon_{\bf k}-\mu)f_{\bf k}^{\dagger}f_{\bf k},$ (9) where $\displaystyle\varepsilon_{\bf k}$ $\displaystyle=$ $\displaystyle-\alpha-\psi^{2}\gamma_{\bf k},~{}~{}\mathrm{and}$ $\displaystyle\gamma_{\bf k}$ $\displaystyle=$ $\displaystyle\frac{2}{z}(\cos k_{x}+\cos k_{y}).$ (10) The free energy per lattice site $F=F(\alpha,\psi)$ is now $\displaystyle F$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\alpha-R)+2\phi\psi^{2}-T\ln(1+e^{-R/T})$ (11) $\displaystyle-T\frac{1}{N}\sum_{k}\ln\left[1+e^{(\mu-\varepsilon_{\bf k})/T}\right].$ We take $k_{B}=1$ here and in the following. To calculate $\phi=\phi(\alpha,\psi)$, we use its definition in (5) and go over to $k$-space to get $\phi=\frac{1}{Nz}\sum_{<ij>}\langle f_{i}^{\dagger}f_{j}\rangle=\frac{1}{N}\sum_{\bf k}\gamma_{\bf k}\langle f_{\bf k}^{\dagger}f_{\bf k}\rangle.$ (12) By definition, $\langle f_{\bf k}^{\dagger}f_{\bf k}\rangle=\frac{\mathrm{Tr}(f_{\bf k}^{\dagger}f_{\bf k}e^{-H_{1}/T})}{\mathrm{Tr}(e^{-H_{1}/T})}=\frac{1}{1+e^{(\varepsilon_{\bf k}-\mu)/T}},$ (13) and so $\phi=\frac{1}{N}\sum_{\bf k}\frac{\gamma_{\bf k}}{1+e^{(\varepsilon_{\bf k}-\mu)/T}}.$ (14) Using $\gamma_{\bf k}=-(\alpha+\varepsilon_{\bf k})/\psi^{2}$ (the first equation in (II)) we obtain $\phi=-\frac{1}{N}\sum_{\bf k}\frac{\alpha+\varepsilon_{\bf k}}{\psi^{2}}\frac{1}{1+e^{(\varepsilon_{\bf k}-\mu)/T}}.$ (15) Introducing the density of states $\rho(E)=\frac{1}{N}\sum_{\bf k}\delta(E-\varepsilon_{\bf k}),$ (16) we can write $\phi=-\frac{1}{\psi^{2}}\int_{E_{0}}^{\mu}dE~{}\frac{\alpha+E}{1+e^{(E-\mu)/T}}\rho(E),$ (17) where $\mu$ is chosen such that the fermion filling fraction $\rho_{F}(\alpha,\psi)=\int_{E_{0}}^{\mu}dE\frac{1}{1+e^{(E-\mu)/T}}\rho(E)$ (18) has a desired value. Here, $E_{0}=-\alpha-\psi^{2}$ is the minimum value of fermion energy. To calculate the density of states (16), we convert the $k$-sum in to an integral according to $(1/N)\sum_{\bf k}\to(1/4\pi^{2})\int d{\bf k}$. Since $\varepsilon_{\bf-k}=\varepsilon_{\bf k}$, the $k$-space integral is four times the integral over the first quadrant of the Brillouin zone, and so we have $\rho(E)=\frac{1}{\pi^{2}}\int_{0}^{\pi}dk_{x}\int_{0}^{\pi}dk_{y}~{}\delta(E+\alpha+\psi^{2}\gamma_{\bf k}).$ (19) The integral over $k_{y}$ can be easily evaluated, and we get $\displaystyle\rho(E)$ $\displaystyle=$ $\displaystyle\frac{2}{\pi^{2}\psi^{2}}f\left(\frac{\alpha+E}{\psi^{2}}\right),~{}\mathrm{where}$ $\displaystyle f(u)$ $\displaystyle=$ $\displaystyle\int_{0}^{\pi}\frac{dk_{x}}{\sqrt{1-(2u+\cos k_{x})^{2}}}.$ (20) We can readily see that the function $f(u)$ is real only when $-1\leq u\leq 1$, and is non-negative. Therefore we have the inequality $-\alpha-\psi^{2}\leq E\leq-\alpha+\psi^{2}$ for the fermion energy $E$. We substitute the above expression for $\rho(E)$ in to equations (17) and (18) and transform the integrals to obtain $\displaystyle\rho_{F}$ $\displaystyle=$ $\displaystyle\frac{2}{\pi^{2}}\int_{-1}^{u_{F}}du\frac{f(u)}{1+e^{(u-u_{F})\psi^{2}/T}}~{}\mathrm{and}$ $\displaystyle\phi$ $\displaystyle=$ $\displaystyle-\frac{2}{\pi^{2}}\int_{-1}^{u_{F}}du\frac{uf(u)}{1+e^{(u-u_{F})\psi^{2}/T}},$ (21) where $u_{F}=(\alpha+\mu)/\psi^{2}$. This helps us choose the value of $\mu$ for a desired $\rho_{F}$, given the values of $(\alpha,\psi)$. For any $(\rho_{F},T)$, we determine $\psi$ and $\alpha$ by solving $\partial F/\partial\psi=0~{}\mathrm{and}~{}\partial F/\partial\alpha=0$ simultaneously. These two equations, and the two equations in (II) above, are solved iteratively to obtain $(\alpha,\psi)$ as well as $(\phi,\mu)$ for any chosen $(\rho_{F},T)$. In general, there could be multiple solutions $(\alpha,\psi)$. We substitute each solution in $F(\alpha,\psi)$ and denote the resulting free energy minimum for the solution by $\tilde{F}$; the correct solution is the one that corresponds to the lowest $\tilde{F}$. From equation (11), we obtain $\displaystyle\frac{\partial F}{\partial\psi}$ $\displaystyle=$ $\displaystyle 2\psi(\phi+\phi_{\psi}\psi)\left[1-\frac{\phi}{R}\chi\right],~{}\mathrm{and}$ $\displaystyle\frac{\partial F}{\partial\alpha}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[(1-2\rho_{F})-\frac{\chi}{R}\alpha\right]+2\phi_{\alpha}\psi^{2}\left[1-\frac{\chi}{R}\phi\right],$ where $\phi_{\psi}=\partial\phi/\partial\psi,~{}\phi_{\alpha}=\partial\phi/\partial\alpha$, and $\chi(R,T)=\frac{e^{R/T}-1}{e^{R/T}+1}.$ (23) Since $\phi~{}\mathrm{and}~{}\phi_{\psi}$ are positive (see equations (14) and (II)), we always have $\phi+\phi_{\psi}\psi>0$, so $\partial F/\partial\psi=0$ gives $\psi=0~{}~{}\mathrm{or}~{}~{}R=\phi\chi(R,T).$ (24) Using this in equation (II), $\partial F/\partial\alpha=0$ gives $\alpha=(1-2\rho_{F})R/\chi.$ (25) ## III Tractable Limits and Some General Observations Now that we have derived the implicit equations for $\psi$ and $\alpha$, we first analyze two simple limits: the $\psi=0$ (disordered phase) and the $T=0$ limits. We obtain closed-form solutions for $\alpha,~{}\psi$ and $\tilde{F}$ in these limits. For the general case, we make some observations before moving on to a discussion of the numerical results in the next section. From equation (25), we get $\alpha^{2}[\chi^{2}-(1-2\rho_{F})^{2}]=4\phi^{2}\psi^{2}(1-2\rho_{F})^{2}$. The solution $\psi=0$ corresponds to the disordered state, i.e., the Bose sector is in a non-superfluid, normal phase. We then get $\alpha=0$ or $\chi=\|1-2\rho_{F}\|$. In this state, $R=\|\alpha\|$, so using the definition of $\chi$ in equation (23) we get $\alpha=0,~{}T[\ln(1-\rho_{F})-\ln\rho_{F}]$ (26) in the disordered state ($\psi=0$). Of these two solutions for $\alpha$, we must pick the one corresponding to the lower $\tilde{F}$. Substituting the above in equation (11) for $F$, we obtain $\tilde{F}_{I}=-T(2\ln 2)$ for the first solution $\alpha=0$, and $\tilde{F}_{II}=-T[\ln 2-\ln(1-\rho_{F})]$ for the second solution $\alpha=T[\ln(1-\rho_{F})-\ln\rho_{F}]$. We can see that $\tilde{F}_{I}<\tilde{F}_{II}$ for $\rho_{F}<1/2$ and $\tilde{F}_{II}<\tilde{F}_{I}$ for $\rho_{F}>1/2$. This shows that in the disordered phase, $\alpha=0$ for $\rho_{F}\leq 1/2$ and $\alpha=T[\ln(1-\rho_{F})-\ln\rho_{F}]$ (which is the same as $\chi=\|1-2\rho_{F}\|$) for $\rho_{F}>1/2$. In Fig. (1) we show the behavior of $-\tilde{F}_{D}/T$ (which is the disordered-phase entropy $S_{D}$) as a function of $\rho_{F}$. At $T=0$, we get $\tilde{F}=0$ for the disordered state (since both $\tilde{F}_{I}$ and $\tilde{F}_{II}$ vanish). For the ordered state, $\chi=1$ so that $\psi_{0}^{2}=\rho_{F}(1-\rho_{F}),~{}~{}\alpha_{0}=(1-2\rho_{F})\phi_{0},$ (27) and the minimum free energy is $\tilde{F}_{0}=\rho_{F}^{2}[-\phi_{0}+u_{F}(\rho_{F}-1)]$ by taking the $T\rightarrow 0$ limit in equation (11) and substituting the above expressions for $\psi_{0},~{}\alpha_{0}$. Here $\psi_{0},~{}\phi_{0},~{}\alpha_{0}$ and $\tilde{F}_{0}$ are $T=0$ values. This gives coupled first-order transitions at $\rho_{F}\simeq 0.3$ at $T=0$ between a metallic superfluid for $\rho_{F}>0.3$ and an insulating normal liquid for $\rho_{F}\leq 0.3$. We referred to the latter as insulating normal gas in our earlier workzeroTarxiv ; zeroTPRO . However, since the destruction of superfluidity of the Bose sector in this regime is due to an interplay between correlation and quantum effects and not due to temperature, this unusual phase should be more appropriately called an insulating normal liquid rather than an insulating normal gas. As we increase the temperature at a fixed $\rho_{F}$, thermal fluctuations suppress superfluidity, reducing $\psi$. To see how this happens, we rewrite the ordered-phase self consistency equation $R=\phi\chi$ in the form $T=J(\psi)$ where $J(\psi)=\phi x/\ln[(1+x)/(1-x)]$; here $x=\sqrt{(1-2\rho_{F})^{2}+4\psi^{2}}$. The function $J(\psi)$ has zeros at $\psi=0$ (when $\phi=0$) and $\psi=\psi_{0}$ (when $x=1$). Since it is positive, it must have a maximum in the interval $(0,\psi_{0})$. This is graphically illustrated in Fig. (2). As $T$ increases from zero, the line $y=T$ intersects the curve $y=J(\psi)$ at two points $\psi=\psi_{1},\psi_{2}$. Since $J(\psi)$ is a single-valued function, we have $\psi_{2}<\psi_{1}<\psi_{0}$ and further, $\psi_{1}$ decreases and $\psi_{2}$ increases as $T$ is increased. The solution $\psi=\psi_{1}$ corresponds to the ordered minimum of the free energy. As the temperature increases further, $y=T$ increases, while the maximum of $J(\psi)$ decreases, and at a certain temperature $T_{2}(\rho_{F})$, the line $y=T$ becomes a tangent to $y=J(\psi)$ at the maximum, when $\psi_{1}=\psi_{2}$. For $T>T_{2}(\rho_{F})$, the self consistency equation $T=J(\psi)$ has no solution, and therefore $T_{c}\leq T_{2}$. For $T\leq T_{2}$, when the ordered solution exists, there are two possibilities. (1) The ordered free energy minimum ($\tilde{F}_{O}$) remains lower than the disordered free energy minimum ($\tilde{F}_{D}$) for all $T\leq T_{2}$. In this case the transition temperature is $T_{c}=T_{2}$, and occurs when the ordered minimum ceases to exist. The free energy minimum value is discontinuous at the transition, since the system switches to the only solution that exists, namely $\psi=0$. This corresponds to a zeroth-order transition. A zeroth-order phase transition has previously been considered in the theory of superfluidity and superconductivity Maslov2004 . More recently, the reentrant phase transition in black holes has been discussed as a zeroth- order transition.Gunasekaran ; Altamirano01 ; Zou ; Hennigar ; Altamirano02 ; Amin . Our numerical computations obtain this scenario for $\rho_{F}>1/2$. (2) There exists a certain temperature $T_{1}<T_{2}$ such that $\tilde{F}_{O}<\tilde{F}_{D}$ for $T<T_{1}$, and $\tilde{F}_{O}>\tilde{F}_{D}$ for $T>T_{1}$, with the two phases coexisting at $T_{c}=T_{1}$, which is a point of first-order transition. Our numerical results obtain this result for $\rho_{F}\leq 1/2$. We can show that the transition is indeed zeroth order for $\rho_{F}>1/2$ based on the disordered-phase behavior of $\alpha$ that we derived above. Formally, the solution of the self-consistency equation $R=\phi\chi$ is $\psi^{2}=(1/4)[\chi^{2}-(1-2\rho_{F})^{2}]$. As we saw above, $\chi=\|1-2\rho_{F}\|$ in the disordered phase when $\rho_{F}>1/2$, so the ordered solution $\psi>0$ does not exist in the disordred phase. If $T_{c}<T_{2}$, then we would have the ordered solution existing in the disordered phase for $T_{c}<T<T_{2}$, which is a contradiction. So we must have $T_{c}=T_{2}$ in this case, resulting in a zeroth-order transition. For $\rho_{F}\leq 1/2$, however, the ordered-phase self-consistency equation in the disordered phase (where $\alpha=0$) becomes $2\psi=(e^{2\phi\psi/T}-1)/(e^{2\phi\psi/T}+1)$, that has a finite $\psi$ solution for $T<T_{2}$. If $T_{c}=T_{2}$, then we would have an ordered solution existing in the disordered phase for $T>T_{c}=T_{2}$, a contradiction. We therefore have $T_{c}<T_{2}$ in this case. The transition criterion in this case is clearly $\tilde{F}_{O}=\tilde{F}_{D}$ since the ordered and disordered minima both exist at the transition. The transition for $\rho_{F}\leq 1/2$ is therefore first order. We now turn to the correlation function $C({\bf r})=(1/N)\sum_{i}\langle\delta\rho_{{\bf r}_{i}}\delta\rho_{{\bf r}_{i}+{\bf r}}\rangle$ where $\delta\rho_{{\bf r}_{i}}=f_{i}^{\dagger}f_{i}-\rho_{F}$. We obtain $C({\bf r})=-n^{2}({\bf r})$ where $n({\bf r})=(1/N)\sum_{\bf k}\langle f_{\bf k}^{\dagger}f_{\bf k}\rangle e^{i{\bf k}.{\bf r}}$, and $C_{\bf q}=-(1/N)\sum_{\bf k}\langle f_{\bf k}^{\dagger}f_{\bf k}\rangle\langle f_{\bf k+q}^{\dagger}f_{\bf k+q}\rangle$ for the Fourier transform of $C({\bf r})$zeroTarxiv ; zeroTPRO . In our earlier work, we could show that $n({\bf r})$ shows a periodicity of twice the lattice spacing for $\rho_{F}=1/2$, while no apparent periodicity was obtained for other values of $\rho_{F}$. Using the expression for the Fermi function in Eq. (13), we can readily see that in the disordered phase ($\psi=0$), we obtain $C_{\bf q}=-1/4$. This shows that there is no density wave (DW) order in the insulating normal gas phase, and the DW order obtained for the $T=0$ metallic superfluidzeroTarxiv ; zeroTPRO is stabilized by the nesting of the Fermi surface and the superfluidity of the Bose sector. ## IV Discussion Figure 1: The disordered-phase entropy $S_{D}=-\tilde{F}_{D}/T$ (equal to $-\tilde{F}_{I}/T$ for $\rho_{F}\leq 1/2$ and $-\tilde{F}_{II}/T$ for $\rho_{F}>1/2$) plotted as a function of $\rho_{F}$. The high-temperature entropy is $2\ln 2$, as expected, for $0<\rho_{F}\leq 1/2$. However, it is anomalously high for $1/2<\rho_{F}\leq 1$, the regime of zeroth-order transition. Figure 2: A figure illustrating the graphical solution of the self-consistency equation $T=J(\psi)$. The function $J(\psi)$ vanishes at $\psi=0,\psi_{0}$ and has a maximum in between. The horizontal lines show plots of $y=T$ (for $0<T<T_{2},~{}T=T_{2},~{}T>T_{2}$). For $0<T<T_{2}$, the line $y=T$ intersects the curve $y=J(\psi)$ (red curve) at two points $\psi=\psi_{1},\psi_{2}$, with $\psi_{0}\geq\psi_{1}>\psi_{2}$, that correspond, respectively, to a minimum and a maximum of $F$. For $y=T_{2}$, the two points merge and the line is a tangent to the curve $J(\psi)$ at its maximum. For $T>T_{2}$, the self-consistency equation has no solution for any real value of $\psi$. Figure 3: Panels (a)-(f) show the $T$-dependence plots, respectively, of $\psi,~{}\phi,~{}\mu,~{}\tilde{F},~{}\alpha$ and $S$. Each panel has plots at five different values of $\rho_{F}$, namely $0.35,1/2,0.70,0.81,0.95$. Figure 4: The phase diagram of model (2) in $(\rho_{F},T)$ plane showing the metallic superfluid ($\phi,~{}\psi>0$) and insulating non-superfluid phases ($\phi=\psi=0$) phases (the insulating normal liquid at $T=0$ and the insulating normal gas for $T>0$). These phases are separated by lines of discontinuous transitions: at the phase boundary, the minimum free energy $\tilde{F}$ is continuous with a derivative discontinuity for $\rho\leq 1/2$ (first-order transition, dashed line), while $\tilde{F}$ has a jump for $\rho>1/2$ (zeroth-order transition, dotted line). Figure 5: The temperatures $T_{1}$ and $T_{2}$ are plotted as functions of $\rho_{F}$. As explained in the text, $T_{1}$ is the critical temperature for first-order transition, where ${\tilde{F}}_{O}={\tilde{F}}_{D}$. It is not defined for $\rho_{F}>1/2$, where ${\tilde{F}}_{O}<{\tilde{F}}_{D}$ for $T\leq T_{2}$. The ordered minimum disappears at $T_{2}$. The critical temperature $T_{c}$ is $T_{1}$ for $\rho_{F}\leq 1/2$ (first-order transition) and $T_{2}$ for $\rho_{F}>1/2$ (zeroth-order transition). Note that $T_{2}>T_{1}$. This is responsible for a segment of the phase boundary in Fig. 4 being vertical at $\rho_{F}=1/2$. Figure 6: Plot of $\psi_{c}^{2}$, where $\psi_{c}$ is the jump in the superfluid order parameter at the discontinuous transition temperature $T_{c}$, as a function of $\rho_{F}$. Figure 7: Plot of $T_{c}/B_{0}$ as a function of $\rho_{F}$. Here, $B_{0}=2\psi_{0}^{2}$ is the Fermi band width at $T=0$zeroTarxiv ; zeroTPRO . Figure 8: Plot of $T_{c}/B_{c}$ as a function of $\rho_{F}$. Here, $B_{c}=2\psi_{c}^{2}$ is the $T=T_{c}$ analogue of the $T=0$ Fermi band widths $B_{0}$. Our numerical results are presented in figures 1-8. In Fig.1 we show the behavior of disordered-phase entropy as a function of $\rho_{F}$. The entropy in this case is $S_{D}=-\tilde{F}_{D}/T$, where $\tilde{F}_{D}$ is the disordered-phase free energy minimum. As we discussed above, this is $-\tilde{F}_{I}/T$ for $\rho_{F}\leq 1/2$ and $-\tilde{F}_{II}/T$ for $\rho_{F}>1/2$. We can see that the entropy is $2\ln 2$ for $\rho_{F}\leq 1/2$ as one might expect in the disordered phase. However, the entropy is anomalously large for $\rho_{F}>1/2$, which is also the regime where the temperature-driven transition is zeroth order. We believe this is a consequence of the filling constraint (1), that leads to the solution in equation (25) for $\alpha$ so that $\alpha=T[\ln(1-\rho_{F})-\ln\rho_{F}]$ in this case (see equation (26)). To understand how temperature suppresses the metallic-superfluid order, we solve the self-consistency equations (24) and (25) for $\psi,~{}\alpha$ using an approach graphically described in Fig.2. As shown in the figure, at a certain temperature $T<T_{2}$, the line $y=T$ intersects the curve $y=J(\psi)$ at two points $\psi_{1},\psi_{2}~{}(\psi_{0}\geq\psi_{1}>\psi_{2})$. The solution $\psi=\psi_{1}$ corresponds to the free energy minimum at this temperature. As the temperature increases, it is obvious from the figure that this solution moves to the left, i.e. decreases, leading to a thermal suppression of superfluidity. At a certain high temperature $T>T_{2}$, the line $y=T$ has no intersection with the curve $y=J(\psi)$. Therefore there is no solution to the self-consistency equation $T=J(\psi)$ at temperatures higher than a certain $T_{2}$, at which point the line $y=T$ becomes a tangent to the curve $y=J(\psi)$ at its maximum. By numerically implementing this graphical method of solution, we can obtain $\psi,~{}\phi,~{}\mu,~{}\tilde{F},~{}\alpha$ and $S$ (the entropy) as the temperature $T$ is varied; these results are plotted in panels (a)-(f) of Fig.3. Each panel shows temperature dependence of one of these quantities for five different values of $\rho_{F}=0.35,~{}1/2,~{}0.70,~{}0.81,~{}0.95$. This choice of $\rho_{F}$ values is the same as for the $T=0$ case, which span the superfluid metallic phase as reported earlier zeroTarxiv ; zeroTPRO . The figures show that at each of these fillings, there is a certain critical temperature $T_{c}$ where the model has a discrete phase transition: the quantities $\psi,\phi,\mu$ and $\alpha$ all show discontinuous changes at $T_{c}$ (figures 3(a, b, c, e)). We can observe that at $\rho_{F}=0.35$ and $1/2$, the free energy minimum $~{}\tilde{F}$ is continuous, but with a derivative discontinuity, whereas it is discontinuous at $\rho_{F}=0.70,~{}0.81$ and $0.95$ (Fig. 3(d)). In figure 3(f) we show plots of entropy $S(T)=-\partial\tilde{F}/\partial T$, computed by numerical differentiation of $\tilde{F}(T)$. The unusual feature that can be readily seen is that the entropy becomes negative for certain temperatures below $T_{c}$ when $\rho_{F}>1/2$. While the concept of negative entropy has been applied to quantum information systems earlier cerf ; delrio2011 , it’s relevance for physical systems has been discussed only recently Chatzi2020 . Cerf and Adami showed that unlike in classical information theory, quantum conditional entropies can be negative for quantum entangled systemscerf . Subsequently, del Rio et al. explained its thermodynamic meaning: negative entropy is related to a possible cooling of an environment connected to a quantum information system, when quantum information contained in the system is eraseddelrio2011 . In a very recent studyChatzi2020 , it has been proposed that the results of two independent inelastic neutron scattering experimentsOlsen ; Callear , which showed an anomalous scattering from H2 molecules in nanoscale confined geometries, can be explained on the basis of negative conditional entropy and quantum thermodynamics. Our numerical results for $\rho_{F}\leq 1/2$ show that at $T=T_{c}=T_{1}$, the ordered and disordered phases coexist and we have $\tilde{F}_{O}=\tilde{F}_{D}$. The ordered minimum survives well into the disordered phase, and disappears at a temperature $T_{2}>T_{1}$. We have therefore a standard first-order transition in this case. On the other hand, for $\rho_{F}>1/2$, the ordered minimum disappears abruptly at a certain temperature $T_{2}(\rho_{F})$. The system then assumes the only minimum available to it, namely the disordered minimum. The free energy obviously shows a discontinuous jump in this case. The transition is therefore zeroth order. These numerical results are consistent with the qualitative remarks we made above in Section III. As a result, we can define $T_{1}$ only for $\rho_{F}\leq 1/2$, where clearly $T_{c}=T_{1}<T_{2}$. These results are summarized in figures 4 and 5. Figure 4 shows the phase diagram of our model in $\rho_{F}-T$ plane, showing the metallic superfluid under the dome, and two insulating non-superfluid phases outside it. We refer to the insulating phase at $T=0$ for $0<\rho_{F}<0.3$ as an insulating normal liquid and the insulating phase for $T>0$ as an insulating normal gas. While the latter is dominated by thermal effects, the former is an unusual phase of bosons at $T=0$ that is a non-superfluid because of correlation and quantum effects. Figure 5 shows the plots of $T_{1}(\rho_{F})$ and $T_{2}(\rho_{F})$. The function $T_{2}(\rho_{F})$ can be defined for $0\leq\rho_{F}\leq 1$, and has a maximum at a certain $\rho_{F}<1/2$. The temperature $T_{1}$ on the other hand remains lower than $T_{2}$; it vanishes for $0\leq\rho_{F}\leq 0.3$, in agreement with the $T=0$ resultszeroTarxiv ; zeroTPRO . The line separating the two phases in Fig. 4 has a vertical segment at the van Hove point $\rho_{F}=1/2$. This is as a consequence of two facts: (a) $T_{c}$ is the lower of $T_{1}$ and $T_{2}$, and (b) $T_{1}<T_{2}$, for $\rho_{F}\leq 1/2$. As we saw in Fig.3, the superfluid order parameter $\psi$ is discontinuous with a jump $\psi_{c}$ at $T_{c}$. In Fig.6 we show the plot of $\psi_{c}^{2}(\rho_{F})$. The jump in the order parameter seems to be maximum around the same $\rho_{F}$ where $T_{2}(\rho_{F})$ is maximum. It shows an abrupt drop at $\rho_{F}=1/2$, and after a broad maximum, decreases slowly towards zero at $\rho_{F}=1$. Based on the zero-temperature Fermi band width of $B_{0}=2\psi_{0}^{2}$, the ratio $T_{c}/B_{0}$ agreed very well with measured values for several types of unconventional superfluids and superconductors in our earlier workzeroTarxiv ; zeroTPRO . This value of $T_{c}\simeq 0.12$ in our earlier work was not based on a calculation, but only an estimate based on the zero-temperature free energy minimum for $\rho_{F}=1/2$. However, in our present work we have performed explicit computation of $T_{c}$ based on $T_{1},~{}T_{2}$ calculations for the whole range of $\rho_{F}$ from $0$ to $1$, as shown in figures (4) and (5). Figure 7 presents a plot of the ratio $T_{c}/B_{0}$ as a function of $\rho_{F}$. We note that the calculated value of the ratio for $\rho_{F}=1/2$ is in almost exact agreement with our earlier estimatezeroTarxiv ; zeroTPRO . The ratio is between $0.01$ and $0.10$ for most $\rho_{F}$ values in the range 0.3 to 1.0. Figure 8 plots the ratio $T_{c}/B_{c}$ as a function of $\rho_{F}$, using $B_{c}=2\psi_{c}^{2}$, the $T=T_{c}$ analogue of the $T=0$ Fermi band width $B_{0}$. It is interesting to compare this with experimentally determined values of the ratio $T_{c}$/$T_{F}$, of the superfluid or superconducting transition temperature $T_{c}$ to the Fermi temperature $T_{F}$. This ratio also scales approximately with the ratio $\Delta$/$E_{F}$, the pairing strength of single-band superconductors, where $\Delta$ is the superconducting gap and $E_{F}$ is the Fermi energy. In particular, it has been recognizedUemura that the unconventional superconductors show a relatively large value of $T_{c}$/$T_{F}\sim 0.02-0.2$, while the conventional superconductors show a small value of $T_{c}$/$T_{F}$ $\sim$ $10^{-4}$ to $10^{-5}$. For example, elemental metal tin (Sn)Ashcroft has a $T_{c}$/$T_{F}$ $\sim$ 3$\times$$10^{-5}$, while the high-$T_{c}$-cupratesYamamoto ; Brookes ; Xie exhibit a $T_{c}$/$T_{F}$ $\sim$ 0.02-0.03. On the other hand, the iron based superconductors (estimated from experimental dataLubashevsky ; Okazaki ; Kasahara ; Rinott ; MonoFeSe ) and the newly discovered hydride superconductors (as estimated from experimental $T_{c}$ valuesDrozdov ; Drozdov2 ; Somayazulu and band structure calculationsBianconi ; Jarlborg ; Liu ) show $T_{c}$/$T_{F}$ $\sim$ 0.1 - 0.2. Surprisingly, the corresponding value for a Fermi-Bose mixtureFerrier estimated from experimental data is $T_{c}$/$T_{F}$ $\simeq 0.19$, although the $T_{c}$$\sim$ 200 nK. Even for a purely ultracold Fermi atomic systemChin , $T_{c}$/$T_{F}$ $\simeq 0.2$. Thus, the high-$T_{c}$ cuprates, iron-based superconductors, hydrides and ultracold atomic systems are clearly classified as unconventional superconductors/superfluids Millev90 . A very recent study on an iron-based superconductor has reported evidence for the first solid-state BEC superconductor, although the observed value of the ratio $\Delta$/$E_{F}$ ($\simeq 0.12$, corresponding to a $T_{c}$/$T_{F}$ $\simeq 0.025$) is reduced across the BCS-BEC crossover, and the authors interpret it as most probably arising from interband coupling effectsHashimoto . The $T_{c}$ values of these unconventional materials span nine orders of magnitude: while the ultracold FBMs have $T_{c}\simeq 200$nK, the hydrides have a $T_{c}\simeq 250$K. But the important point is that the ratio calculated in our model is remarkably close to these values for several classes of unconventional superfluids and superconductors. In particular, we obtain $T_{c}/B_{0}\sim 0.01-0.1~{}(T_{c}/B_{c}\sim 0.01-0.25)$; the value of this ratio is a very robust feature of our model, and compares well with the range $0.03-0.22$ for most unconventional materials (see the Table 1 of our previous paper for detailszeroTPRO ). An important question is that of a potential experimental realization of the composite hopping model. Consider a lattice with $N$ sites and $M$ electrons ($2N>M>N$) with nearest-neighbor hopping and on-site repulsion. In this case, by the Pigeonhole principle, at least one site must have more than one electron. Taking into account the repulsion between electrons, in the minimum energy configuration we have $M-N$ local pairs (that can be considered as hardcore bosons) and $2N-M$ electrons each occupiing a site. When one of the electrons of a pair hops to a neighboring site with one electron, we have a realization of composite hopping (see our earlier workzeroTarxiv ; zeroTPRO for a more detailed explanation). This situation is perhaps also well described by a one-band fermion Hubbard model above half filling, and the composite hopping model might offer a good effective description. Our model approximates the electrons by spinless fermions, but in subsequent work we plan to treat the spin half case. In this case, the composite hopping strength $t$ is just the single-electron hopping strength. For a narrow band Hubbard model, we can take $t\sim 0.1-0.2$eV, and if we further use $T_{c}\simeq 0.06zt$ (corresponding to its peak value in Fig.4 at the van Hove point $\rho_{F}=1/2$), we obtain a value of $T_{c}\simeq 250-500$K. If room-temperature superconductivity is theoretically possible within a composite-hopping framework, then it provides a strong reason to look for practical examples of it. In the pigeonhole context discussed above, external pressure might help overcome the repulsion between electrons and keep them paired. It is interesting to explore if high-pressure room-temperature superconductors like sulphidesdias2020 are indeed solid-state realizations of the composite-hopping model, and are superconducting because of such a ‘pigeonhole’ pairing mechanism. A second possibility might be realization of composite hopping in the context of ultracold atoms in an optical lattice. Quantum simulation is an exciting area of researchlewenstein2012 , and given the many unusual physical properties displayed by our model such as negative compressibility (discussed in our earlier workzeroTarxiv ; zeroTPRO ), negative entropycerf ; delrio2011 ; Chatzi2020 , zeroth-order transitionMaslov2004 ; Gunasekaran ; Altamirano01 ; Zou ; Hennigar ; Altamirano02 ; Amin ; kundu2020 and the remarkable agreement of the calculated ratio of $T_{c}$/$T_{F}$ with a wide range of unconventional superfluids and superconductors zeroTPRO , it would be interesting to explore the physics of composite hopping in this perspective. In our model, there is a clear distinction between the boson-dominated ($\rho_{F}\leq 1/2$) and the fermion-dominated ($\rho_{F}>1/2$) regimes: in the former, the temperature-driven transition is first-order and the entropy remains positive throughout, while in the latter regime the transition is zeroth order and the entropy is negative in the ordered phase close to $T_{c}$. In the latter regime, the zero-temperature bulk modulus becomes negative over a range of $\rho_{F}$zeroTPRO . An important question whether the zeroth order temperature-driven transition for $\rho_{F}>1/2$ is a consequence of mean-field approximation (3), or an inherent property of the composite-hopping model. This question assumes importance given that the model offers an irreducible nontrivial description of an FBM perhaps not explored before, and the circumstance that zeroth-order transitions are relatively rare in the solid stateMaslov2004 ; kundu2020 . It would therefore be of interest to study the model using other numerical techniques of quantum many-body theory like Quantum Monte-Carlo calculations (QMC)bhmqmc . ## V Conclusion In this work, we have extended the zero-temperature study of the composite hopping model of FBMs to explore finite-temperature thermodynamics. We computed the $\rho_{F}-T$ phase diagram and found that the temperature-driven metallic superfluid phase to insulating normal phase transition is discontinuous: for $\rho\leq 1/2$, it is first order, while for $\rho>1/2$ it is zeroth order. We calculated the the temperature dependent superfluid amplitude $\psi$, the fermion hopping amplitude $\phi$, the Fermi chemical potential, and free energy within a mean-field approximation. We also computed the entropy and found that it becomes negative for a certain range of temperatures below $T_{c}$ when $\rho_{F}>1/2$. 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11institutetext: Grupo de Astrofísica Molecular, Instituto de Física Fundamental, CSIC, C/ Serrano 123, 28006 Madrid, Spain 11email<EMAIL_ADDRESS>22institutetext: Observatorio Astronómico Nacional (IGN), C/ Alfonso XII 3, 28014 Madrid, Spain 33institutetext: Observatorio de Yebes (IGN). Cerro de la Palera s/n, 19141 Yebes, Guadalajara, Spain # A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS. N. Marcelino A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS. B. Tercero A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS. M. Agúndez A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS. J. Cernicharo A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS.A study of C4H3N isomers in TMC-1: line by line detection of HCCCH2CN††thanks: Based on observations with the 40-m radio telescope of the National Geographic Institute of Spain (IGN) at Yebes Observatory (projects 19A003 and 20A014). Yebes Observatory thanks the ERC for funding support under grant ERC-2013-Syg-610256-NANOCOSMOS. (Received ; accepted ) We present Yebes 40m telescope observations of the three most stable C4H3N isomers towards the cyanopolyyne peak of TMC-1. We have detected 13 transitions from CH3C3N (A and E species), 16 lines from CH2CCHCN, and 27 lines ($a$-type and $b$-type) from HCCCH2CN. We thus provide a robust confirmation of the detection of HCCCH2CN and CH2CCHCN in space. We have constructed rotational diagrams for the three species, and obtained rotational temperatures between $4-8$ K and similar column densities for the three isomers, in the range $(1.5-3)\times 10^{12}$ cm-2. Our chemical model provides abundances of the order of the observed ones, although it overestimates the abundance of CH3CCCN and underestimates that of HCCCH2CN. The similarity of the observed abundances of the three isomers suggests a common origin, most probably involving reactions of the radical CN with the unsaturated hydrocarbons methyl acetylene and allene. Studies of reaction kinetics at low temperature and further observations of these molecules in different astronomical sources are needed to draw a clear picture of the chemistry of C4H3N isomers in space. ###### Key Words.: Astrochemistry – ISM: abundances – ISM: clouds, TMC-1 – ISM: molecules – line: identification ## 1 Introduction Three C4H3N isomers have been detected in space to date. These are, in order of increasing energy, methylcyanoacetylene (CH3C3N), cyanoallene (CH2CCHCN), and propargyl cyanide (HCCCH2CN). Our knowledge of C4H3N isomers in the interstellar medium is the result of a nice multidisciplinary story with contributions from theoretical calculations, laboratory experiments, and astronomical observations. The presence of cyanoallene in cold interstellar clouds was predicted by Balucani et al. (2000, 2002) based on crossed molecular beam experiments and _ab initio_ calculations which indicated that the reaction of CN and CH3CCH would produce CH3C3N, already detected in TMC-1 (Broten et al., 1984), and CH2CCHCN in nearly equal amounts. Laboratory experiments indeed showed that the reaction CN + CH3CCH is rapid at low temperatures (Carty et al., 2001). These results motivated an astronomical search for cyanoallene in TMC-1, which turned out to be successful using the GBT (Lovas et al., 2006) and Effelsberg 100m (Chin et al., 2006) telescopes. In their combined crossed beam and _ab initio_ study, Balucani et al. (2000, 2002) studied also the reaction between CN and CH2CCH2 (allene), a non polar metastable isomer of CH3CCH which is thought to be also present in cold interstellar clouds. These authors found that the reaction should be rapid at low temperatures, something that was confirmed by Carty et al. (2001), producing cyanoallene and the third C4H3N isomer: HCCCH2CN. This isomer was not detected in TMC-1 by Lovas et al. (2006), although it was later on found toward this same source during a cm line survey with the GBT (McGuire et al., 2020). The detection of propargyl cyanide in TMC-1 by these authors relied on four individual lines detected at a modest signal-to-noise ratio (SNR) and was supported by line stacking of 68 transitions. Here we present an independent and robust detection of HCCCH2CN in TMC-1, with 10 lines detected with SNR above 10 plus 12 lines detected above 3$\sigma$, together with observations of the two other C3H4N isomers, CH3C3N and CH2CCHCN. The presence of the latter is confirmed by the detection of a significant number of rotational lines. The high sensitivity and number of lines detected allow us to derive precise abundances for the three isomers in a coherent and systematic way and to revisit the chemistry of C3H4N isomers in TMC-1. ## 2 Observations The data presented here are part of a deep spectral line survey in the Q band toward TMC-1, performed at the Yebes 40 m radiotelescope111http://rt40m.oan.es/rt40m$\\_$en.php (de Vicente et al., 2016), located at 990 m of altitude near Guadalajara (Spain). The observed position corresponds to the cyanopolyyne peak in TMC-1, at $\alpha_{J2000}=4^{\rm h}41^{\rm m}41.9^{\rm s}$ and $\delta_{J2000}=+25^{\circ}41^{\prime}27.0^{\prime\prime}$. We have covered the full Q band at the 40 m telescope, between 31.1 GHz and 50.4 GHz, using the recently installed NANOCOSMOS HEMT Q band receiver (Tercero et al., 2020b) and the fast Fourier transform spectrometers (FFTS) with 8$\times$2.5 GHz bands per lineal polarization, which allow a simultaneous scan of a band width of 18 GHz at a spectral resolution of 38 kHz ($\sim$0.27 km s-1). We observed two setups at different central frequencies in order to fully cover the lower and upper frequencies allowed by the Q band receiver, and to check for spurious signals and other technical artifacts. The observations were performed in several sessions, between November 2019 and February 2020, using the frequency switching technique with a frequency throw of 10 MHz. The intensity scale in the spectra obtained is T${}_{\rm A}^{}$, antenna temperature corrected for atmospheric absorption and spillover losses, which was calibrated using two absorbers at different temperatures and the atmospheric transmission model ATM (Cernicharo, 1985; Pardo et al., 2001). Pointing and focus were checked every hour through pseudo-continuum observations (see e.g. de Vicente et al. 2016; Tercero et al. 2020a) of the SiO $J=1-0$, $v=1$ maser emission towards the O-rich evolved star IK Tau, which is close to the target source. The pointing errors were always found within 2-3′′. System temperatures were in the range 50-250 K depending on the frequency, the particular weather conditions of each observing session (from 5 mm to 10 mm of precipitable water vapor), and the elevation of the source (from 15∘ to 80∘). The final rms obtained is in the range 0.5-1 mK, rising up to 3 mK at the highest frequencies. The main beam efficiency of the Yebes 40 m telescope ranges from 0.6 at 32 GHz to 0.43 at 49 GHz, and the half power beam width (HPBW) ranges from 55′′ at 32 GHz to 37′′ at 49 GHz. All the data were reduced and analyzed using the GILDAS222http://www.iram.fr/IRAMFR/GILDAS/ software. ## 3 Results Figure 1: Observed lines of HCCCH2CN ($a$-type) toward TMC-1 (CP). The vertical dashed green line marks a radial velocity of 5.7 km s-1. Figure 2: Observed lines of HCCCH2CN ($b$-type) toward TMC-1 (CP). Blue arrows show the position of the strongest three hyperfine components. Velocity axis refers to the frequency result of collapsing the hyperfine structure. The high sensitivity of this line survey allowed the detection of HCCCH2CN towards TMC-1 through 17 $a$-type lines up to quantum numbers $J=9-8$ and $K_{\rm a}=0,1,2$ ($E_{\rm u}\leq 13$ K), with 10 of them showing a SNR $>$10\. In addition, we detected 10 $b$-type lines harbouring hyperfine structure. These lines are shown in Fig. 1 and Fig. 2 and are listed in Table 2. Line identification was performed using the MADEX catalogue (Cernicharo 2012, see Table 2), which also includes predictions for the hyperfine structure. This detection confirms the presence of this species in space, recently claimed for the first time in TMC-1 by McGuire et al. (2020) using the Green Bank Telescope (GBT). These authors presented a 5$\sigma$ signal (18$\sigma$ in the response impulse function) obtained by an intensity and noise-weighted average (“stack”) of the data at the expected frequencies of the HCCCH2CN lines that could be present within the noise level. It is worth noting that our 40 m survey of TMC-1 in the Q band is complementary to that performed with the GBT between 8 GHz and 30 GHz. Although most of the individual lines of HCCCH2CN are below the detection limit of the GBT data, four of them are detected at 1-3$\sigma$ levels. Thanks to the high spectral resolution of these data (1.4 kHz) they distinguished three cloud components in the line profiles (see Fossé et al. 2001 for a detailed analysis of the velocity structure of this source). In this work, a single Gaussian function was fitted to the HCCCH2CN line profiles to obtain the observed line parameters (see Table 2). We derived a $V_{\rm LSR}=(5.70\pm 0.09)$ km s-1 and a line width ($\Delta$$v$, full width at half maximum) of $(0.66\pm 0.18)$ km s-1. The former is slightly different from the value $(5.83\pm 0.01)$ km s-1, obtained by Cernicharo et al. (2020a) from Gaussian fits to the 50 lines of HC5N and its 13C and 15N isotopologues detected in our line survey. Note we have a larger uncertainty due to the lower number of transitions and the weakness of some of the lines as compared to HC5N, in particular the $b$-type transitions. We also detected the other two C4H3N isomers, CH2CCHCN and CH3CCCN, using frequencies from the MADEX catalogue (Cernicharo 2012, see Table 2). The 16 lines of CH2CCHCN detected in our line survey are shown in Fig. 3 and are listed in Table 2. All of them are detected above a 10$\sigma$ level. This species was previously identified in TMC-1 through four lines between 20 GHz and 26 GHz (Lovas et al., 2006). Here we report the first detection of lines of CH2CCHCN above 30 GHz in TMC-1. Kaifu et al. (2004) did not detect lines above the noise limit at the CH2CCHCN frequencies in their line survey between 8.8 GHz and 50 GHz carried out with the Nobeyama 45 m telescope. As we mentioned in previous works (Cernicharo et al., 2020a, b, c; Marcelino et al., 2020), the sensitivity of our observations is a factor 5-10 better than that of Kaifu et al. (2004) at the same frequencies. The derived $V_{\rm LSR}$ for the CH2CCHCN lines, by fitting a single Gaussian, is $(5.66\pm 0.03)$ km s-1, which is similar, within errors, to the one obtained for HCCCH2CN. The isomer CH3CCCN, a well known species in TMC-1 (Broten et al., 1984; Kaifu et al., 2004), has been also identified in our line survey through 10 strong lines ($J_{\rm u}$ from 8 to 12 and $K=0,1$) plus five $K=2$ lines ($E_{\rm u}>29$ K) tentatively detected (see Fig. 4 and Table 2). These lines show a $V_{\rm LSR}$ of $(5.80\pm 0.02)$ km s-1 which matches that observed for HC5N. We can estimate rotational temperatures ($T_{\rm rot}$) and molecular column densities ($N$) for the detected species by constructing rotational diagrams (see e.g. Goldsmith & Langer 1999). This analysis assumes the Rayleigh-Jeans approximation, optically thin lines, and LTE conditions. The equation that derives the total column density under these conditions can be re-arranged as ${\rm\ln}\left(\frac{8\pi k_{\rm B}\nu^{2}\int{T_{\rm MB}dv}}{hc^{3}A_{\rm ul}g_{\rm u}b}\right)={\rm\ln}\left(\frac{N}{Q_{\rm rot}}\frac{T_{\rm rot}-T_{\rm bg}}{T_{\rm rot}}\right)-\frac{E_{\rm u}}{k_{\rm B}T_{\rm rot}},$ (1) where $g_{u}$ is the statistical weight in the upper level, $A_{\rm ul}$ is the Einstein $A$-coefficient for spontaneous emission, $Q_{\rm rot}$ is the rotational partition function which depends on $T_{\rm rot}$, $E_{\rm u}$ is the upper level energy, $\nu$ is the frequency of the transition, $b$ is the dilution factor, and $T_{\rm bg}$ is the cosmic microwave background radiation temperature. We assume a source diameter of 80′′ (see Fossé et al. 2001). The first term of Eq. (1), which depends only on spectroscopic and observational line parameters, is plotted as a function of $E_{\rm u}$/$k_{\rm B}$ for the different lines detected. Thus, $T_{\rm rot}$ and $N$ can be derived by performing a linear least squares fit to the points (see Fig. 5). Results for $T_{\rm rot}$ and $N$ using the population diagram procedure are shown in Table 1 and Fig. 5. The uncertainties were calculated using the statistical errors given by the linear least squares fit for the slope and the intercept. The individual errors of the data points are those derived by taking into account the uncertainty obtained in the determination of the observed line parameters (see Table 2). For HCCCH2CN ($a$-type transitions) and CH2CCHCN, different hyperfine structure components of the same $(J_{K_{\rm a},K_{\rm c}})_{\rm u}-(J_{K_{\rm a},K_{\rm c}})_{\rm l}$ transition are blended in a single line. Thus, to correctly determine $T_{\rm rot}$ and $N$, the Einstein $A$-coefficient for spontaneous emission and the statistical weight were assumed as the weighted average values of the sum of the hyperfine components, and the rotational partition function was calculated using this value for the statistical weight of each $(J_{K_{\rm a},K_{\rm c}})_{\rm u}-(J_{K_{\rm a},K_{\rm c}})_{\rm l}$ transition. For CH3CCCN we built independent rotational diagrams for each symmetry state $A$ and $E$. We obtained rotational temperatures between $4-8$ K for the three isomers (see Table 1), indicating they are subthermally excited, like most of the species in this region (see e.g. Cernicharo et al. 2020a, c; Marcelino et al. 2020). For the column density, we derived very similar values of the three isomers, in the range $(1.5-3)\times 10^{12}$ cm-2. Table 1: Derived rotational temperatures ($T_{\rm rot}$) and column densities ($N$) for the C4H3N isomers towards TMC-1 (CP). Species \| $T_{\rm rot}$ (K) \| $N$ (cm-2) ---\|---\|--- HCCCH2CN \| $4\pm 1$ \| $(2.8\pm 0.7)\times 10^{12}$ CH2CCHCN \| $5.5\pm 0.3$ \| $(2.7\pm 0.2)\times 10^{12}$ A-CH3CCCN \| $6.7\pm 0.2$ \| $(9.7\pm 0.3)\times 10^{11}$ E-CH3CCCN \| $8.2\pm 0.6$ \| $(7.7\pm 0.5)\times 10^{11}$ ## 4 Discussion The chemistry of C4H3N isomers in cold molecular clouds has been discussed by Balucani et al. (2000) and more specifically by Balucani et al. (2002), based on crossed molecular beam experiments and _ab initio_ calculations. In these studies it was pointed out that reactions of the CN radical with methyl acetylene and allene are barrierless and exothermic when producing CH3C3N and CH2CCHCN, in the methyl acetylene reaction, and CH2CCHCN and HCCCH2CN, in the reaction involving allene. Indeed, the reactions of CN with CH3CCH and CH2CCH2 were measured to be rapid at low temperatures (Carty et al., 2001). This chemical scheme was implemented in a chemical model by Quan & Herbst (2007) to explain the abundance of cyanoallene in TMC-1. Later on, Abeysekera et al. (2015) measured the product branching ratios of the reaction between CN and methyl acetylene at low temperature using a chirped-pulse uniform flow and found that HC3N is the major product, while CH3C3N accounts for 22 % of the products and CH2CCHCN is not formed. These results are in contrast with those obtained from crossed molecular beam experiments (Huang et al., 1999; Balucani et al., 2000, 2002), where CH2CCHCN is observed as product of the CN + CH3CCH reaction. Therefore, the most stable isomer CH3C3N can be formed in the reaction of CN and methyl acetylene, the second most stable isomer CH2CCHCN can be formed when CN reacts with CH2CCH2 and perhaps also with CH3CCH, depending on whether one gives credit to the chirped-pulse uniform flow experiment or to the crossed molecular beam ones, and the least stable isomer HCCCH2CN can only be formed in the reaction between CN and allene. These neutral-neutral reactions involving CN are therefore likely routes to the three C4H3N isomers in cold interstellar clouds like TMC-1, where abundant CN, CH3CCH, and probably CH2CCH2 (non polar and thus it cannot be detected at radio wavelengths) are present. Moreover, the presence of HCCCH2CN (and perhaps also CH2CCHCN) can be used as proxy of the non polar C3H4 isomer allene since this isomer is only formed from CH2CCH2 in the aforementioned reactions of CN. In the light of the recent discovery of HCCCH2CN in TMC-1 and the observational study of the three C4H3N isomers presented here, we have carried out chemical model calculations to review the chemistry of these species in cold clouds and evaluate whether the mechanism proposed by Balucani et al. (2002) is in agreement with observations. We adopt typical parameters of cold dark clouds, i.e., a gas kinetic temperature of 10 K, a volume density of H nuclei of $2\times 10^{4}$ cm-3, a visual extinction of 30 mag, a cosmic-ray ionization rate of H2 of $1.3\times 10^{-17}$ s-1, and the so-called ”low- metal” elemental abundances (Agúndez & Wakelam, 2013). We use the chemical network RATE12 from the UMIST database (McElroy et al., 2013), updated to include the C4H3N isomers CH2CCHCN and HCCCH2CN. The reactions $\displaystyle\rm CN+CH_{3}CCH$ $\displaystyle\rightarrow\rm HCN+CH_{2}CCH,$ (2a) $\displaystyle\rightarrow\rm HC_{3}N+CH_{3},$ (2b) $\displaystyle\rightarrow\rm CH_{3}C_{3}N+H,$ (2c) $\displaystyle\rightarrow\rm CH_{2}CCHCN+H,$ (2d) $\displaystyle\rm CN+CH_{2}CCH_{2}$ $\displaystyle\rightarrow\rm CH_{2}CCHCN+H,$ (3a) $\displaystyle\rightarrow\rm HCCCH_{2}CN+H,$ (3b) are included with the rate constants measured by Carty et al. (2001). For the branching ratios of reaction (2) we use either the values measured in the chirped-pulse uniform flow experiment by Abeysekera et al. (2015), 12 %, 66 %, 22 %, and 0 % for channels (a), (b), (c), and (d), respectively, or the values suggested by crossed molecular beam experiments and quantum chemical calculations Balucani et al. (2000), 50 % for channels (c) and (d). For reaction (3) we adopt branching ratios of 90 % and 10 % for channels (a) and (b), respectively, based on quantum chemical calculations by Balucani et al. (2002). The destruction processes of CH2CCHCN and HCCCH2CN are assumed to be the same as those of CH3C3N, which are basically reactions with abundant cations. The calculated abundances of the three C4H3N isomers are shown as a function of time in Fig. 6. It is seen that the three isomers reach their maximum abundance at early times, in the range $(1-4)\times 10^{5}$ yr, with CH3C3N being the most abundant and HCCCH2CN being the least abundant. According to the chemical model, the formation of CH3C3N occurs through two routes. The first and major involves the dissociative recombination of the precursor ion CH3C3NH+ with electrons and is the responsible of the larger calculated abundance of CH3C3N compared to the two other isomers. A second and minor route is provided by reaction (2c). Cyanoallene is formed through reaction (3), with reaction (2c) contributing to the same level if channel (2d) is assumed to be open. Propargyl cyanide is exclusively formed through reaction (3), with a lower abundance because it is formed with a branching ratio of just 10 %. The impact of using the branching ratios for reaction (2) of Balucani et al. (2000) or those of Abeysekera et al. (2015) is modest, with the main effect being a change of less than a factor of two in the abundance of CH2CCHCN (see Fig. 6). The fact that the observed abundances of the three isomers are remarkably similar provides clues on the underlying chemical processes at work. For example, the route to CH3C3N from the precursor ion CH3C3NH+ is probably overestimated in the chemical model, as indicated by the too large abundance calculated for this species. It has become clear in recent years that dissociative recombination of polyatomic ions usually results in a much larger fragmentation than previously believed (Larsson et al., 2012), meaning that it would not be strange than CH3C3N is a minor product in the dissociative recombination of CH3C3NH+. The low branching ratio adopted for HCCCH2CN formation in reaction (3) based on calculations by Balucani et al. (2002) seems also to be in conflict with the observational finding of similar abundances for CH2CCHCN and HCCCH2CN. It would be very interesting to measure the product branching ratios for the reaction of CN with allene, as was done for CN + CH3CCH (Abeysekera et al., 2015), to shed light on the formation routes of these two metastable C4H3N isomers. This will also allow to put tight constraints on the abundance of allene in cold dense clouds. In summary, the similar abundances observed for the three C4H3N isomers favors a common origin through reactions (2) and (3) with similar branching ratios in the latter reaction. If this scenario is correct, we can conclude that allene is as abundant as methyl acetylene in TMC-1. This is in fact predicted by the chemical model, where CH3CCH and CH2CCH2 are mostly formed during the dissociative recombination of the C3H${}_{7}^{+}$ ion (Larsson et al., 2005), with similar branching ratios assumed for the two C3H4 isomers. In addition to the three C4H3N isomers and the well known species HC3N and CH2CHCN, Balucani et al. (2000) predicted the presence of $c$-C6H5CN and the C5H5N isomer CH2CC(CN)CH3 in cold interstellar clouds. It is worth noting that all these species but CH2CC(CN)CH3 have been identified in TMC-1 (see McGuire et al. 2018 for the detection of cyanobenzene) and are also present in our survey. Another $-$CN species, cyanocyclopentadiene ($c$-C5H5CN), has been recently detected in this source (McCarthy et al., 2020). A complete study of the molecular rings $c$-C6H5CN and $c$-C5H5CN in our data will be published elsewhere. We searched in our data for the two C5H5N isomers CH3CH2CCCN and CH3CHCCHCN by performing a line stacking analysis (see, e.g., Cuadrado et al. 2016; Loomis et al. 2020). We added spectra at the expected frequency of several lines from these species that could be present within the noise level. More concreteley, we considered $a$-type transitions sharing similar upper level energies, up to 15 K, and Einstein coefficients. All spectra, in local standard of rest (LSR) velocity scale, are resampled to the same velocity channel resolution before stacking. Figure 7 shows the spectra obtained following this method. Whereas there is no evidence for the presence of CH3CH2CCCN in our data, the stacked spectrum of CH3CHCCHCN shows a 2$\sigma$ signal at the systemic velocity of the source. An observational effort at lowest frequencies has to be undertaken to confirm the presence of CH3CHCCHCN in space. ## 5 Conclusions Using a very sensitive line survey of TMC-1 in the Q band we have detected multiple transitions of the three C4H3N isomers CH3C3N, CH2CCHCN, and HCCCH2CN. The presence of the latter in TMC-1 is supported by 27 observed individual lines. We have constructed rotational diagrams for the three species and obtained similar rotational temperatures and column densities for the three isomers, in the range of $4-8$ K and $(1.5-3)\times 10^{12}$ cm-2, respectively. The observed abundances of the three isomers in TMC-1 suggest a similar chemical origin based on reactions of the radical CN with the isomers CH3CCH and CH2CCH2. There are still uncertainties in the network of reactions related to these species since our chemical model overestimates the abundance of CH3C3N and underestimates the production of HCCCH2CN. Further studies of these isomers in other sources could help in clarifying their chemical formation pathways. ###### Acknowledgements. We acknowledge funding support from the European Research Council (ERC Grant 610256: NANOCOSMOS). We also thank the Spanish MICIU for funding support under grants AYA2016-75066-C2-1-P, PID2019-106110GB-I00, and PID2019-107115GB-C21, and PID2019-106235GB-I00. M.A. thanks MICIU for grant RyC-2014-16277. ## References Abeysekera et al. (2015) Abeysekera, C., Joalland, B., Ariyasingha, N., et al. 2015, The Journal of Physical Chemistry Letters, 6, 1599 * Agúndez & Wakelam (2013) Agúndez, M. & Wakelam, V. 2013, Chemical Reviews, 113, 8710 * Balucani et al. (2000) Balucani, N., Asvany, O., Huang, L. C. L., et al. 2000, ApJ, 545, 892 * Balucani et al. 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(1986) Schwahn, G., Schieder, R., Bester, M., & Winnewisser, G. 1986, Journal of Molecular Spectroscopy, 116, 263 * Tercero et al. (2020a) Tercero, B., Cernicharo, J., Cuadrado, S., de Vicente, P., & Guélin, M. 2020a, A&A, 636, L7 * Tercero et al. (2020b) Tercero, F., López-Pérez, J. A., Gallego, J. D., et al. 2020b, arXiv e-prints, arXiv:2010.16224 ## Appendix A Additional figures and tables Table 2: Observed lines of C4H3N isomers towards TMC-1 (CP). Transition \| Rest Freq. \| $E{up}$ \| $A_{ij}$ \| $S_{ij}$ \| $\int T_{\rm A}^{}dv$ \| $V_{\rm LSR}$ \| $\Delta$v \| $T_{\rm A}^{}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- $(J_{K_{\rm a},K_{\rm c}})_{\rm u}-(J_{K_{\rm a},K_{\rm c}})_{\rm l}$ \| (MHz) \| (K) \| (10-6 s-1) \| \| (K km s-1) \| (km s-1) \| (km s-1) \| (K) HCCCH2CN, $a$-type, $\mu_{\rm a}=2.87$ D $6_{1,6}-5_{1,5}$ \| 31848.982(3) \| 6.2 \| 1.39 \| 5.83 \| 0.0093(10) \| 5.83( 2) \| 0.79( 6) \| 0.0111( 8) $6_{0,6}-5_{0,5}$ \| 32722.702(3) \| 5.5 \| 1.55 \| 5.99 \| 0.0099( 5) \| 5.67( 1) \| 0.78( 3) \| 0.0120( 4) $6_{2,5}-5_{2,4}$ \| 32876.187(3) \| 8.8 \| 1.40 \| 5.33 \| 0.0025( 6) \| 5.51( 6) \| 0.66(14) \| 0.0036( 5) $6_{2,4}-5_{2,3}$ \| 33048.726(3) \| 8.8 \| 1.42 \| 5.33 \| 0.0016( 6) \| 5.49(10) \| 0.70(20) \| 0.0021( 5) $6_{1,5}-5_{1,4}$ \| 33863.716(3) \| 6.5 \| 1.67 \| 5.83 \| 0.0072( 5) \| 5.66( 2) \| 0.79( 4) \| 0.0085( 4) $7_{1,7}-6_{1,6}$ \| 37139.207(4) \| 8.0 \| 2.24 \| 6.86 \| 0.0081( 5) \| 5.74( 2) \| 0.69( 3) \| 0.0110( 5) $7_{0,7}-6_{0,6}$ \| 38102.698(4) \| 7.3 \| 2.47 \| 6.99 \| 0.0110( 6) \| 5.70( 1) \| 0.74( 3) \| 0.0140( 5) $7_{2,6}-6_{2,5}$ \| 38342.339(4) \| 10.6 \| 2.32 \| 6.43 \| 0.0016( 6) \| 5.56( 9) \| 0.57(18) \| 0.0027( 7) $7_{2,5}-6_{2,4}$ \| 38616.702(4) \| 10.7 \| 2.37 \| 6.43 \| 0.0039( 5) \| 5.64( 3) \| 0.60( 5) \| 0.0061( 6) $7_{1,6}-6_{1,5}$ \| 39486.580(4) \| 8.4 \| 2.70 \| 6.86 \| 0.0075( 5) \| 5.66( 2) \| 0.60( 3) \| 0.0117( 6) $8_{1,8}-7_{1,7}$ \| 42421.779(4) \| 10.0 \| 3.39 \| 7.87 \| 0.0077( 8) \| 5.74( 2) \| 0.59( 5) \| 0.0122( 9) $8_{0,8}-7_{0,7}$ \| 43450.742(4) \| 9.4 \| 3.70 \| 7.99 \| 0.0109(11) \| 5.73( 2) \| 0.73( 5) \| 0.0140(10) $8_{2,7}-7_{2,6}$ \| 43802.419(4) \| 12.7 \| 3.55 \| 7.50 \| 0.0028( 8) \| 5.69( 4) \| 0.48(10) \| 0.0056(10) $8_{2,6}-7_{2,5}$ \| 44210.195(4) \| 12.8 \| 3.66 \| 7.50 \| 0.0018(16) \| 5.70( 5) \| 0.35(31) \| 0.0047(11) $8_{1,7}-7_{1,6}$ \| 45099.074(4) \| 10.6 \| 4.07 \| 7.87 \| 0.0125( 9) \| 5.69( 1) \| 0.61( 3) \| 0.0192(10) $9_{1,9}-8_{1,8}$ \| 47696.032(5) \| 12.3 \| 4.86 \| 8.89 \| 0.0036(13) \| 5.74( 6) \| 0.50(13) \| 0.0068(17) $9_{0,9}-8_{0,8}$ \| 48764.484(5) \| 11.8 \| 5.26 \| 8.98 \| 0.0078(16) \| 5.76( 4) \| 0.63(10) \| 0.0117(14) HCCCH2CN, $b$-type, $\mu_{\rm b}=2.19$ D $3_{1,3}-2_{0,2},F_{\rm u}-F_{\rm l}=3-2$ \| 32519.775(3) \| 2.3 \| 0.49 \| 1.79 \| }1.5[0.0031( 8)] \| 5.59( 9)∗ \| 1.06(20) \| 0.0028( 5) $3_{1,3}-2_{0,2},F_{\rm u}-F_{\rm l}=2-1$ \| 32519.815(3) \| 2.3 \| 0.46 \| 1.21 \| $3_{1,3}-2_{0,2},F_{\rm u}-F_{\rm l}=4-3$ \| 32519.916(3) \| 2.3 \| 0.55 \| 2.58 \| 0.0013( 4) \| 5.73( 7) \| 0.58(22) \| 0.0021( 4) $9_{0,9}-8_{1,8},F_{\rm u}-F_{\rm l}=8-7$ \| 36933.586(4) \| 11.8 \| 0.74 \| 4.44 \| }1.5[0.0034(17)] \| 5.80(11)∗ \| 1.16(31) \| 0.0027( 6) $9_{0,9}-8_{1,8},F_{\rm u}-F_{\rm l}=10-9$ \| 36933.621(4) \| 11.8 \| 0.75 \| 5.57 \| $9_{0,9}-8_{1,8},F_{\rm u}-F_{\rm l}=9-8$ \| 36933.800(4) \| 11.8 \| 0.74 \| 4.98 \| 0.0020( 8) \| 5.86(12) \| 0.84(20) \| 0.0023( 6) $4_{1,4}-3_{0,3},F_{\rm u}-F_{\rm l}=4-3$ \| 37340.269(4) \| 3.4 \| 0.77 \| 2.38 \| 0.0025( 4) \| 5.78( 7) \| 0.82(13) \| 0.0029( 6) $4_{1,4}-3_{0,3},F_{\rm u}-F_{\rm l}=3-2$ \| 37340.455(4) \| 3.4 \| 0.75 \| 1.81 \| }1.5[0.0030( 3)] \| 5.78( 4)∗ \| 0.60( 7) \| 0.0047( 5) $4_{1,4}-3_{0,3},F_{\rm u}-F_{\rm l}=5-4$ \| 37340.473(4) \| 3.4 \| 0.82 \| 3.10 \| $5_{1,5}-4_{0,4},F_{\rm u}-F_{\rm l}=5-4$ \| 42010.978(4) \| 4.6 \| 1.12 \| 2.96 \| 0.0017( 5) \| 5.85( 6) \| 0.46(19) \| 0.0035(10) $5_{1,5}-4_{0,4},F_{\rm u}-F_{\rm l}=6-5$ \| 42011.211(4) \| 4.6 \| 1.16 \| 3.65 \| }1.5[0.0028(13)] \| 5.80( 6)∗ \| 0.54(15) \| 0.0050(10) $5_{1,5}-4_{0,4},F_{\rm u}-F_{\rm l}=4-3$ \| 42011.215(4) \| 4.6 \| 1.10 \| 2.40 \| $6_{1,6}-5_{0,5},F_{\rm u}-F_{\rm l}=6-5$ \| 46545.801(5) \| 6.2 \| 1.55 \| 3.57 \| 0.0013( 5) \| 5.63(11) \| 0.48(20) \| 0.0026(12) $6_{1,6}-5_{0,5},F_{\rm u}-F_{\rm l}=7-6$ \| 46546.045(5) \| 6.2 \| 1.59 \| 4.24 \| }1.5*[0.0041( 7)] \| 5.69( 6)∗ \| 0.54( 9) \| 0.0071(11) $6_{1,6}-5_{0,5},F_{\rm u}-F_{\rm l}=5-4$ \| 46546.057(5) \| 6.2 \| 1.54 \| 3.00 \| CH2CCHCN, $\mu_{\rm a}=4.07$ D $6_{1,5}-5_{1,4}$ \| 31615.627(5) \| 6.4 \| 2.73 \| 5.83 \| 0.0247(12) \| 5.64( 1) \| 0.80( 3) \| 0.0289( 9) $7_{1,7}-6_{1,6}$ \| 35379.044(5) \| 7.9 \| 3.90 \| 6.86 \| 0.0213( 6) \| 5.68( 1) \| 0.77( 2) \| 0.0259( 5) $7_{0,7}-6_{0,6}$ \| 36064.688(5) \| 6.9 \| 4.22 \| 7.00 \| 0.0253( 4) \| 5.64( 1) \| 0.69( 1) \| 0.0347( 4) $7_{2,6}-6_{2,5}$ \| 36140.273(5) \| 11.4 \| 3.90 \| 6.43 \| 0.0086( 8) \| 5.61( 3) \| 0.90( 6) \| 0.0090( 6) $7_{2,5}-6_{2,4}$ \| 36222.501(5) \| 11.4 \| 3.93 \| 6.43 \| 0.0095( 6) \| 5.63( 2) \| 0.90( 4) \| 0.0099( 5) $7_{1,6}-6_{1,5}$ \| 36878.547(5) \| 8.2 \| 4.42 \| 6.86 \| 0.0226( 5) \| 5.63( 1) \| 0.70( 1) \| 0.0300( 5) $8_{1,8}-7_{1,7}$ \| 40425.712(6) \| 9.9 \| 5.90 \| 7.87 \| 0.0198( 8) \| 5.70( 1) \| 0.60( 2) \| 0.0311( 8) $8_{0,8}-7_{0,7}$ \| 41187.082(6) \| 8.9 \| 6.34 \| 8.00 \| 0.0245( 6) \| 5.66( 1) \| 0.59( 1) \| 0.0388( 7) $8_{2,7}-7_{2,6}$ \| 41297.656(6) \| 13.4 \| 5.99 \| 7.50 \| 0.0088( 7) \| 5.67( 2) \| 0.65( 4) \| 0.0128( 6) $8_{2,6}-7_{2,5}$ \| 41420.713(6) \| 13.4 \| 6.04 \| 7.50 \| 0.0084( 7) \| 5.64( 2) \| 0.65( 4) \| 0.0121( 6) $8_{1,7}-7_{1,6}$ \| 42138.451(6) \| 10.2 \| 6.68 \| 7.87 \| 0.0194( 9) \| 5.63( 1) \| 0.52( 2) \| 0.0348(10) $9_{1,9}-8_{1,8}$ \| 45469.519(6) \| 12.0 \| 8.48 \| 8.89 \| 0.0176( 9) \| 5.69( 1) \| 0.56( 2) \| 0.0293(11) $9_{0,9}-8_{0,8}$ \| 46297.882(6) \| 11.1 \| 9.06 \| 9.00 \| 0.0205(11) \| 5.67( 1) \| 0.55( 2) \| 0.0352(13) $9_{2,8}-8_{2,7}$ \| 46452.840(6) \| 15.6 \| 8.70 \| 8.56 \| 0.0097(10) \| 5.74( 2) \| 0.66( 5) \| 0.0138(10) $9_{2,7}-8_{2,6}$ \| 46628.069(6) \| 15.7 \| 8.80 \| 8.56 \| 0.0083( 9) \| 5.64( 2) \| 0.60( 4) \| 0.0129(11) $9_{1,8}-8_{1,7}$ \| 47394.848(6) \| 12.5 \| 9.60 \| 8.89 \| 0.0152(13) \| 5.63( 2) \| 0.62( 4) \| 0.0232(13) CH3CCCN, $\mu_{\rm a}=4.75$ D E $8_{2}-7_{2}$ \| 33050.3475(8) \| 29.4 \| 4.18 \| 7.50 \| 0.0044( 9) \| 5.14( 9) \| 1.41(16) \| 0.0029( 5) E $8_{1}-7_{1}$ \| 33051.3033(9) \| 6.9 \| 4.39 \| 7.88 \| 0.0472( 8) \| 5.79( 1) \| 0.76( 1) \| 0.0580( 6) A $8_{0}-7_{0}$ \| 33051.6219(9) \| 7.1 \| 4.46 \| 8.00 \| 0.0485( 8) \| 5.80( 1) \| 0.74( 1) \| 0.0616( 6) E $9_{2}-8_{2}$ \| 37181.5838(9) \| 31.2 \| 6.08 \| 8.56 \| 0.0013( 7) \| 5.77(10) \| 0.64(24) \| 0.0020( 7) E $9_{1}-8_{1}$ \| 37182.659(1) \| 8.7 \| 6.32 \| 8.89 \| 0.0421( 8) \| 5.79( 1) \| 0.68( 1) \| 0.0585( 7) A $9_{0}-8_{0}$ \| 37183.017(1) \| 8.9 \| 6.40 \| 9.00 \| 0.0455( 8) \| 5.79( 1) \| 0.68( 1) \| 0.0632( 7) E $10_{2}-9_{2}$ \| 41312.799(1) \| 33.2 \| 8.46 \| 9.60 \| 0.0025(11) \| 5.42( 8) \| 0.55(15) \| 0.0031(10) E $10_{1}-9_{1}$ \| 41313.994(1) \| 10.7 \| 8.73 \| 9.90 \| 0.0372(64) \| 5.81( 4) \| 0.58(10) \| 0.0604( 9) A $10_{0}-9_{0}$ \| 41314.393(1) \| 10.9 \| 8.81 \| 10.0 \| 0.0412(51) \| 5.83( 3) \| 0.57( 7) \| 0.0674( 9) E $11_{2}-10_{2}$ \| 45443.993(1) \| 35.4 \| 11.4 \| 10.6 \| … \| … \| … \| $\leq$0.0050(10) E $11_{1}-10_{1}$ \| 45445.307(1) \| 12.9 \| 11.7 \| 10.9 \| 0.0294(42) \| 5.80( 3) \| 0.57( 8) \| 0.0483(12) A $11_{0}-10_{0}$ \| 45445.745(1) \| 13.1 \| 11.8 \| 11.0 \| 0.0321(42) \| 5.82( 3) \| 0.62( 8) \| 0.0487(12) E $12_{2}-11_{2}$ \| 49575.162(1) \| 37.7 \| 14.9 \| 11.7 \| … \| … \| … \| $\leq$0.0060(20) E $12_{1}-11_{1}$ \| 49576.596(1) \| 15.3 \| 15.2 \| 11.9 \| 0.0242(52) \| 5.75( 5) \| 0.69(14) \| 0.0328(26) A $12_{0}-11_{0}$ \| 49577.073(1) \| 15.5 \| 15.4 \| 12.0 \| 0.0269(38) \| 5.81( 4) \| 0.64( 8) \| 0.0393(26) Notes. ∗ LSR velocity corresponds to the strongest hyperfine transition. Numbers in parentheses indicate the uncertainty in units of the last significant digits. For the observational parameters we adopted the uncertainty of the Gaussian fit provided by GILDAS. HCCCH2CN: Spectroscopic line parameters were obtained using MADEX by fitting the rotational lines reported by Demaison et al. (1985) and McNaughton et al. (1988). Dipole moments are from McNaughton et al. (1988). CH2CCHCN: Spectroscopic line parameters were obtained using MADEX by fitting the rotational lines reported by Bouchy et al. (1973) and Schwahn et al. (1986). Dipole moment is from Bouchy et al. (1973). CH3CCCN: Spectroscopic line parameters were obtained using MADEX by fitting the rotational lines reported by Moïses et al. (1982) and Bester et al. (1983). Rotation constants $A$ and $D_{\rm k}$ have been assumed to be the same as those of CH3CN. Some additional data have been taken from the CDMS (https://cdms.astro.uni-koeln.de/). Dipole moment is from Bester et al. (1984). Note that the E species is 7.8 K above the A species, and energies for the E species are referred to the lowest energy level (1,1). Figure 3: Observed lines of CH2CCHCN toward TMC-1 (CP). The vertical dashed green line marks a radial velocity of 5.7 km s-1. Figure 4: Observed lines from CH3CCCN towards TMC-1 (CP). Dashed green line marks a radial velocity of 5.8 km s-1. Figure 5: Rotational diagrams of the C4H3N isomers towards TMC-1 (CP). Derived values of the rotational temperature, $T_{\rm rot}$, column density, $N$, and their respective uncertainties are indicated for each molecule. Figure 6: Calculated fractional abundances of the three C4H3N isomers as a function of time. Solid and dashed lines correspond to two models in which we use branching ratios for the CN + CH3CCH reaction from Abeysekera et al. (2015) and from Balucani et al. (2000), respectively (see text). The abundances observed in TMC-1 for the three C4H3N isomers (from Table 1 adopting a H2 column density of 1022 cm-2; Cernicharo & Guelin 1987) are shown as horizontal dotted lines. Figure 7: Stacked spectra of CH3CH2CCCN and CH3CHCCHCN toward TMC-1.
∎ 11institutetext: J. Zeng 22institutetext: School of Computer and Information Engineering, Jiangxi Normal University, Nanchang, China. Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong. 22email<EMAIL_ADDRESS>33institutetext: W. Yin 44institutetext: Department of Mathematics, University of California, Los Angeles, CA. 44email<EMAIL_ADDRESS>55institutetext: D.X. Zhou 66institutetext: School of Data Science, Department of Mathematics, and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong. 66email<EMAIL_ADDRESS> # Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints††thanks: We thank Kaizhao Sun for discussions that help us complete this paper, as well as presenting to us an additional approach to ensure boundedness. The work of J. Zeng is partly supported by National Natural Science Foundation of China (No. 61977038) and the Thousand Talents Plan of Jiangxi Province (No. jxsq2019201124). The work of D.-X. Zhou is partly supported by Research Grants Council of Hong Kong (No. CityU 11307319), Laboratory for AI-powered Financial Technologies, and the Hong Kong Institute for Data Science. Jinshan Zeng Wotao Yin Ding-Xuan Zhou (Received: date / Accepted: date) ###### Abstract The augmented Lagrangian method (ALM) is one of the most useful methods for constrained optimization. Its convergence has been well established under convexity assumptions or smoothness assumptions, or under both assumptions. ALM may experience oscillations and divergence when the underlying problem is simultaneously nonconvex and nonsmooth. In this paper, we consider the linearly constrained problem with a nonconvex (in particular, weakly convex) and nonsmooth objective. We modify ALM to use a Moreau envelope of the augmented Lagrangian and establish its convergence under conditions that are weaker than those in the literature. We call it the Moreau envelope augmented Lagrangian (MEAL) method. We also show that the iteration complexity of MEAL is $o(\varepsilon^{-2})$ to yield an $\varepsilon$-accurate first-order stationary point. We establish its whole sequence convergence (regardless of the initial guess) and a rate when a Kurdyka-Łojasiewicz property is assumed. Moreover, when the subproblem of MEAL has no closed-form solution and is difficult to solve, we propose two practical variants of MEAL, an inexact version called iMEAL with an approximate proximal update, and a linearized version called LiMEAL for the constrained problem with a composite objective. Their convergence is also established. ###### Keywords: Nonconvex nonsmooth optimization augmented Lagrangian method Moreau envelope proximal augmented Lagrangian method Kurdyka-Łojasiewicz inequality ## 1 Introduction In this paper, we consider the following optimization problem with linear constraints $\begin{array}[]{ll}\mathrm{minimize}_{x\in\mathbb{R}^{n}}&f(x)\\\ \mathrm{subject\ to}&Ax=b,\end{array}$ (1) where $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a proper, lower- semicontinuous weakly convex function, which is possibly nonconvex and nonsmooth, $A\in\mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^{m}$ are some given matrix and vector, respectively. A function $f$ is said to be weakly convex with a modulus $\rho>0$ if $f(x)+\frac{\rho}{2}\\|x\\|^{2}$ is convex on $\mathbb{R}^{n}$, where $\\|\cdot\\|$ is the Euclidean norm. The class of weakly convex functions is broad Nurminskii73 , including all convex functions, smooth but nonconvex functions with Lipschitz continuous gradient, and their composite forms (say, $f(x)=h(x)+g(x)$ with both $h$ and $g$ being weakly convex, and $f(x)=g(h(x))$ with $g$ being convex and Lipschitz continuous and $h$ being a smooth mapping with Lipschitz Jacobian (Drusvyatskiy-Paquette19, , Lemma 4.2)). The augmented Lagrangian method (ALM) is a well-known algorithm for constrained optimization by Hestenes Hestenes69 and Powell Powell69 . ALM has been extensively studied and has a large body of literature (Bertsekas73 ; Birgin10 ; Conn91 ; Conn96 ; Rockafellar73-ALM just to name a few), yet _no ALM algorithm can solve the underlying problem ( 1) without at least one of the following assumptions_: convexity Bertsekas73 ; Bertsekas76 ; Fernadez12 ; Polyak-Tretyakov73 ; Rockafellar73-ALM , or smoothness Andreani08 ; Andreani10 ; Andreani19 ; Andreani18 ; Curtis15 , or solving nonconvex subproblems to their global minima Birgin10 ; Birgin18 , or an auto-updated penalty sequence staying bounded on the problem at hand Birgin20 ; Grapiglia-Yuan19 . Indeed, without these assumptions, ALM may oscillate and even diverge unboundedly on simple quadratic programs Wang19 ; Zhang-Luo18 on weakly convex objectives. An example is given Sec. 7.1 below. At a high level, we introduce a Moreau-envelope modification of the ALM for solving (1) and show the method can converge under weaker conditions. In particular, convexity is relaxed to weak convexity; nonsmooth functions are allowed; the subproblems can be solved inexactly to some extent; linearization can be applied to the Lipschitz-differential function in the objective; and, there is no assumption on the rank of $A$. On the other hand, we introduce two alternative subgradient properties in Definition 1 below as our main assumption. By also assuming either a bounded energy sequence or bounded primal-dual sequence, we derive certain subsequence rates of convergence. We introduce a novel way to establish those boundedness properties based on a feasible coercivity assumption and a local-stability assumption on the subproblem. Finally, with the additional assumption of Kurdyka-Łojasiewicz (KŁ) inequality, we establish global convergence. Overall, this paper shows that the Moreau envelope technique makes ALM applicable to more problems. ### 1.1 Proposed Algorithms To present our algorithm, define the augmented Lagrangian: ${\cal L}_{\beta}(x,\lambda):=f(x)+\langle\lambda,Ax-b\rangle+\frac{\beta}{2}\\|Ax-b\\|^{2},$ (2) and the _Moreau envelope_ of ${\cal L}_{\beta}(x,\lambda)$: $\phi_{\beta}(z,\lambda)=\min_{x}\left\\{{\cal L}_{\beta}(x,\lambda)+\frac{1}{2\gamma}\\|x-z\\|^{2}\right\\},$ (3) where $\lambda\in\mathbb{R}^{m}$ is a multiplier vector, $\beta>0$ is a penalty parameter, and $\gamma>0$ is a proximal parameter. The Moreau envelope applies to the primal variable $x$ for each fixed dual variable $\lambda$. We introduce Moreau Envelope Augmented Lagrangian method (dubbed MEAL) as follows: given an initialization $(z^{0},\lambda^{0})$, $\gamma>0$, a sequence of penalty parameters $\\{\beta_{k}\\}$ and a step size $\eta\in(0,2)$, for $k=0,1,\ldots,$ run $\mathrm{(MEAL)}\quad\left\\{\begin{array}[]{l}z^{k+1}=z^{k}-\eta\gamma\nabla_{z}\phi_{\beta_{k}}(z^{k},\lambda^{k}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}\nabla_{\lambda}\phi_{\beta_{k}}(z^{k},\lambda^{k}).\end{array}\right.$ (4) The penalty parameter $\beta_{k}$ can either vary or be fixed. Introduce $x^{k+1}=\mathrm{Prox}_{\gamma,{\cal L}_{\beta_{k}}(\cdot,\lambda^{k})}(z^{k}):=\operatorname{argmin}_{x}\left\\{{\cal L}_{\beta_{k}}(x,\lambda^{k})+\frac{1}{2\gamma}\\|x-z^{k}\\|^{2}\right\\},\ \forall k\in\mathbb{N},$ which yields $\nabla_{z}\phi_{\beta_{k}}(z^{k},\lambda^{k})=\gamma^{-1}(z^{k}-x^{k+1})$ and $\nabla_{\lambda}\phi_{\beta_{k}}(z^{k},\lambda^{k})=Ax^{k+1}-b$. Then, MEAL (4) is equivalent to: $\mathrm{(MEAL\ Reformulated)}\quad\left\\{\begin{array}[]{l}x^{k+1}=\mathrm{Prox}_{\gamma,{\cal L}_{\beta_{k}}(\cdot,\lambda^{k})}(z^{k}),\\\ z^{k+1}=z^{k}-\eta(z^{k}-x^{k+1}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b).\end{array}\right.$ (5) Next, we provide two practical variants of MEAL that do not require an accurate computation of $\mathrm{Prox}_{\gamma,{\cal L}_{\beta}}$. ##### Inexact MEAL (iMEAL) We call $x^{k+1}$ an $\epsilon_{k}$-accurate stationary point of the $x$-subproblem in (5) if there exists $s^{k}\in\partial_{x}{\cal L}_{\beta_{k}}(x^{k+1},\lambda^{k})+\gamma^{-1}(x^{k+1}-z^{k})\quad\text{such that}~{}\\|s^{k}\\|\leq\epsilon_{k}.$ (6) iMEAL is described as follows: given an initialization $(z^{0},\lambda^{0})$, $\gamma>0$, $\eta\in(0,2)$, and two positive sequences $\\{\epsilon_{k}\\}$ and $\\{\beta_{k}\\}$, for $k=0,1,\ldots,$ run $\mathrm{(iMEAL)}\quad\left\\{\begin{array}[]{l}\mathrm{find\ an}\ x^{k+1}\ \mathrm{to\ satisfy}\ \eqref{iMealCond},\\\ z^{k+1}=z^{k}-\eta(z^{k}-x^{k+1}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b).\end{array}\right.$ (7) ##### Linearized MEAL (LiMEAL) When problem (1) has the following form $\begin{array}[]{ll}\mathop{\mathrm{minimize}}_{x\in\mathbb{R}^{n}}&f(x):=h(x)+g(x)\\\ \mathrm{subject\ to}&Ax=b,\end{array}$ (8) where $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is Lipschitz-continuous differentiable and $g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is weakly convex and has an easy proximal operator (in particular, admitting a closed-form solution) Hajinezhad-Hong19 ; Wang19 ; Xu-Yin-BCD13 ; Zeng-DGD18 , we shall use $\nabla h$. Write $f^{k}(x):=h(x^{k})+\langle\nabla h(x^{k}),x-x^{k}\rangle+g(x)$ and ${\cal L}_{\beta,{f^{k}}}(x,\lambda):=f^{k}(x)+\langle\lambda,Ax-b\rangle+\frac{\beta}{2}\\|Ax-b\\|^{2}.$ We describe LiMEAL for (8) as: given $(z^{0},\lambda^{0})$, $\gamma>0$, $\eta\in(0,2)$ and $\\{\beta_{k}\\}$, for $k=0,1,\ldots,$ run $\mathrm{(LiMEAL)}\quad\left\\{\begin{array}[]{l}x^{k+1}=\mathrm{Prox}_{\gamma,{\cal L}_{\beta_{k},{f^{k}}}(\cdot,\lambda^{k})}(z^{k}),\\\ z^{k+1}=z^{k}-\eta(z^{k}-x^{k+1}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b).\end{array}\right.$ (9) Since one can choose to use $h$ or not in LiMEAL, LiMEAL is more general than MEAL. ### 1.2 Relation to ALM and Proximal ALM Like ALM, MEAL alternatively updates primal and dual variables; but unlike ALM, MEAL applies the update to the Moreau envelope of augmented Lagrangian. By Rockafellar-var97 , the Moreau envelope $\phi_{\beta_{k}}(z,\lambda^{k})$ provides a smooth approximation of ${\cal L}_{\beta_{k}}(x,\lambda^{k})$ from below and shares the same minima. The smoothness of Moreau envelope alleviates the possible oscillation that arises when ALM is applied to certain nonconvex optimization problems. For the problems satisfying the conditions in this paper, ALM may require a sequence of possibly unbounded $\\{\beta_{k}\\}$. When $\beta_{k}$ is large, the ALM subproblem is ill-conditioned. Therefore, bounding $\beta_{k}$ is practically desirable Birgin-book14 ; Conn91 . MEAL and its practical variants can use a fixed penalty parameter under a novel subgradient assumption in Definition 1 later. Proximal ALM was introduced in Rockafellar76-PALM . Its variants were recently studied in Hajinezhad-Hong19 ; Hong17-Prox-PDA ; Zhang-Luo20 ; Zhang-Luo18 . These methods add a proximal term to the augmented Lagrangian. Under the reformulation (5), proximal ALM Rockafellar76-PALM for problem (1) is a special case of MEAL with the step size $\eta=1$. In Hong17-Prox-PDA , a proximal primal-dual algorithm called Prox-PDA was proposed for problem (1). Certain non-Euclidean matrix norms were adopted in Prox-PDA to guarantee the strong convexity of the ALM subproblem. A proximal linearized version of Prox- PDA for the composite optimization problem (8) was studied in Hajinezhad- Hong19 . These methods are closely related to MEAL, but their convergence conditions in the literature are stronger. Recently, Zhang-Luo20 ; Zhang-Luo18 modified proximal inexact ALM for the linearly constrained problems with an additional bounded box constraint set or polyhedral constraint set, denoted by ${\cal C}$. Our method is partially motivated by their methods. Their problems are equivalent to the composite optimization problems (8) with $g(x)=\iota_{\cal C}(x)$, where $\iota_{\cal C}(x)=0$ when $x\in{\cal C}$ and $+\infty$ otherwise. In this setting, the methods in Zhang-Luo20 ; Zhang-Luo18 can be regarded as prox-linear versions of LiMEAL (9), that is, yielding $x^{k+1}$ via a prox-linear scheme Xu-Yin- BCD13 instead of the minimization scheme as used in LiMEAL (9), together with an additional dual step size and a sufficiently small primal step size in Zhang-Luo20 ; Zhang-Luo18 . Specifically, in the case of $g(x)=\iota_{\cal C}(x)$, the updates of $x^{k+1}$ in methods in Zhang-Luo20 ; Zhang-Luo18 are yielded by $\displaystyle x^{k+1}=\mathrm{Proj}_{\cal C}(x^{k}-s\nabla K(x^{k},z^{k},\lambda^{k})),$ where $K(x^{k},z^{k},\lambda^{k}))={\cal L}_{\beta^{k},f}(x,\lambda^{k})+\frac{1}{2\gamma}\\|x-z^{k}\\|^{2}$, and $\mathrm{Proj}_{\cal C}(x)$ is the projection of $x$ onto ${\cal C}$. Besides the difference, LiMEAL can handle proximal functions beyond the indicator function and permits a wider choice $\eta\in(0,2)$. ### 1.3 Other Related Literature On convex and constrained problems, locally linear convergence111Locally linear convergence means exponentially fast convergence to a local minimum from a sufficiently close initial point. of ALM has been extensively studied in the literature Bertsekas73 ; Bertsekas76 ; Bertsekas82 ; Conn00 ; Fernadez12 ; Nocedal99 ; Polyak-Tretyakov73 , mainly under the second order sufficient condition (SOSC) and constraint conditions such as the linear independence constraint qualification (LICQ). Global convergence (i.e., convergence regardless of the initial guess) of ALM and its variants were studied in Andreani07 ; Armand17 ; Birgin05 ; Birgin12 ; Birgin10 ; Conn91 ; Conn96 ; Rockafellar73-ALM ; Tretykov73 , mainly under constraint qualifications and assumed boundedness of nondecreasing penalty parameters. On nonconvex and constrained problems, convergence of ALM was recently studied in Andreani08 ; Andreani10 ; Andreani19 ; Andreani18 ; Birgin10 ; Birgin18 ; Curtis15 , mainly under the following assumptions: solving nonconvex subproblems to their approximate global minima or stationary points Birgin10 ; Birgin18 , or boundedness of the nondecreasing penalty sequence Birgin20 ; Grapiglia-Yuan19 . Most of them require Lipschitz differentiability of the objective. Convergence of proximal ALM and its variants was established under the assumptions of either convexity in Rockafellar76-PALM or smoothness (in particular, Lipschitz differentiablity) in Hajinezhad-Hong19 ; Hong17-Prox-PDA ; Jiang19 ; Xie-Wright19 ; Zhang-Luo20 ; Zhang-Luo18 . Besides proximal ALM, other related works for nonconvex and constrained problems include Bian15 ; Haeser19 ; Nouiehed18 ; ONeill20 , which also assume smoothness of the objective, plus either gradient or Hessian information. ### 1.4 Contribution and Novelty MEAL, iMEAL and LiMEAL achieve the same order of iteration complexity $o({\varepsilon^{-2}})$ to reach an $\varepsilon$-accurate first-order stationary point, slightly better than those in the ALM literature Hajinezhad- Hong19 ; Hong17-Prox-PDA ; Xie-Wright19 ; Zhang-Luo18 ; Zhang-Luo20 while also requiring weaker conditions. Our methods have convergence guarantees for a broader class of objective functions, for example, nonsmooth and nonconvex functions like the smoothly clipped absolute deviation (SCAD) regularization Fan-SCAD and minimax concave penalty (MCP) regularization Zhang-MCP , which are underlying the applications of statistical learning and beyond Wang19 . Note that we only assume the feasibility of $Ax=b$, which is weaker than the commonly-used hypotheses such as: the strict complementarity condition in Zhang-Luo18 , certain rank assumption (such as $\mathrm{Im}(A)\subseteq\mathrm{Im}(B)$ when considering the two- (multi-) block case $Ax+By=0$) in Wang19 , and the linear independence constrained qualification (LICQ) in Bertsekas82 ; Nocedal99 (which implies the full-rank assumption in the linear constraint case). Our analysis is noticeably different from those in the literature Rockafellar76-PALM ; Hajinezhad-Hong19 ; Hong17-Prox-PDA ; Jiang19 ; Zhang- Luo18 ; Zhang-Luo20 ; Xie-Wright19 ; Wang19 . We base our analysis on new potential functions. The Moreau envelope in the potential functions is partially motivated by Davis-Drusvyatskiy19 . Our overall potential functions are new and tailored for MEAL, iMEAL, and LiMEAL and include the augmented Lagrangian with additional terms. The technique of analysis may have its own value for further generalizing and improving ALM-type methods. ### 1.5 Notation and Organization We let $\mathbb{R}$ and $\mathbb{N}$ denote the sets of real and natural numbers, respectively. Given a matrix $A$, $\mathrm{Im}(A)$ denotes its image, and $\tilde{\sigma}_{\min}(A^{T}A)$ denotes the smallest positive eigenvalue of $A^{T}A$. $\\|\cdot\\|$ is the Euclidean norm for a vector. Given any two nonnegative sequences $\\{\xi_{k}\\}$ and $\\{\zeta_{k}\\}$, we write $\xi_{k}=o(\zeta_{k})$ if $\lim_{k\rightarrow\infty}\frac{\xi_{k}}{\zeta_{k}}=0$, and $\xi_{k}={\cal O}(\zeta_{k})$ if there exists a positive constant $c$ such that $\xi_{k}\leq c\zeta_{k}$ for all sufficiently large $k$. In the rest of this paper, Section 2 presents background and preliminary techniques. Section 3 states convergence results of MEAL and iMEAL. Section 4 presents the results of LiMEAL. Section 5 includes main proofs. Section 6 provides sufficient conditions for certain boundedness assumptions in above results along with comparisons with the related work. Section 7 provides some numerical experiments to demonstrate the effectiveness of proposed methods. We conclude this paper in Section 8. ## 2 Background and Preliminaries This paper uses extended-real-valued functions, for example, $h:\mathbb{R}^{n}\to\mathbb{R}\cup\\{+\infty\\}$. Write the domain of $h$ as $\mathrm{dom}(h):=\\{x\in\mathbb{R}^{n}:h(x)<+\infty\\}$ and its range as $\mathrm{ran}(h):=\\{y:y=h(x),\forall x\in\mathrm{dom}(h)\\}$. For each $x\in\mathrm{dom}(h)$, the Fréchet subdifferential of $h$ at $x$, written as $\widehat{\partial}h(x)$, is the set of vectors $v\in\mathbb{R}^{n}$ satisfying $\liminf_{u\neq x,u\rightarrow x}\ \frac{h(u)-h(x)-\langle v,u-x\rangle}{\\|x-u\\|}\geq 0.$ When $x\notin\mathrm{dom}(h),$ we define $\widehat{\partial}h(x)=\emptyset.$ The _limiting-subdifferential_ (or simply _subdifferential_) of $h$ Mordukhovich-2006 at $x\in\mathrm{dom}(h)$ is defined as $\partial h(x):=\\{v\in\mathbb{R}^{n}:\exists x^{t}\to x,\;h(x^{t})\to h(x),\;\widehat{\partial}h(x^{t})\ni v^{t}\to v\\}.$ (10) A necessary (but not sufficient) condition for $x\in\mathbb{R}^{n}$ to be a minimizer of $h$ is $0\in\partial h(x)$. A point that satisfies this inclusion is called limiting-critical or simply critical. The distance between a point $x$ and a subset ${\cal S}$ of $\mathbb{R}^{n}$ is defined as $\mathrm{dist}(x,{\cal S})=\inf_{u}\\{\\|x-u\\|:u\in{\cal S}\\}$. ### 2.1 Moreau Envelope Given a function $h:\mathbb{R}^{n}\rightarrow\mathbb{R}$, define its Moreau envelope Moreau65 ; Rockafellar-var97 : ${\cal M}_{\gamma,h}(z)=\min_{x}\left\\{h(x)+\frac{1}{2\gamma}\\|x-z\\|^{2}\right\\},$ (11) where $\gamma>0$ is a parameter. Define its associated proximity operator $\mathrm{Prox}_{\gamma,h}(z)=\operatorname{argmin}_{x}\left\\{h(x)+\frac{1}{2\gamma}\\|x-z\\|^{2}\right\\}.$ (12) If $h$ is $\rho$-weakly convex and $\gamma\in(0,\rho^{-1})$, then $\mathrm{Prox}_{\gamma,h}$ is monotone, single-valued, and Lipschitz, and ${\cal M}_{\gamma,h}$ is differentiable with $\nabla{\cal M}_{\gamma,h}(z)=\gamma^{-1}\left(z-\mathrm{Prox}_{\gamma,h}(z)\right)\in\partial h(\mathrm{Prox}_{\gamma,h}(z));$ (13) see (Rockafellar-var97, , Proposition 13.37). From Drusvyatskiy18 ; Drusvyatskiy-Paquette19 , we also have $\displaystyle{\cal M}_{\gamma,h}(\mathrm{Prox}_{\gamma,h}(z))\leq h(z),$ $\displaystyle\\|\mathrm{Prox}_{\gamma,h}(z)-z\\|=\gamma\\|\nabla{\cal M}_{\gamma,h}(z)\\|,$ $\displaystyle\mathrm{dist}(0,\partial h(\mathrm{Prox}_{\gamma,h}(z)))\leq\\|\nabla{\cal M}_{\gamma,h}(z)\\|.$ The first relation above presents Moreau envelope as a smooth lower approximation of $h$. By the second and third relations, small $\\|\nabla{\cal M}_{\gamma,h}(z)\\|$ implies that $z$ is near its proximal point $\mathrm{Prox}_{\gamma,h}(z)$ and $z$ is nearly stationary for $h$ Davis- Drusvyatskiy19 . Therefore, $\\|\nabla{\cal M}_{\gamma,h}(z)\\|$ can be used as a continuous stationarity measure. Hence, replacing the augmented Lagrangian with its Moreau envelope not only generates a strongly convex subproblem but also yields a stationarity measure. ### 2.2 Implicit Regularity Properties Let $h$ be a proper, lower semicontinuous, $\rho$-weakly convex function. Given a $\gamma\in(0,\rho^{-1})$, define the generalized inverse mapping of $\mathrm{Prox}_{\gamma,h}$: $\displaystyle\mathrm{Prox}_{\gamma,h}^{-1}(x):=\\{w:\mathrm{Prox}_{\gamma,h}(w)=x\\},\quad\forall x\in\mathrm{ran}(\mathrm{Prox}_{\gamma,h}).$ (14) In the definition below, we introduce two important regularity properties. ###### Definition 1 Let $h$ be a proper, lower semicontinuous and $\rho$-weakly convex function. 1. (a) We say $h$ satisfies the implicit Lipschitz subgradient property if for any $\gamma\in(0,\rho^{-1})$, there exists $L>0$ (depending on $\gamma$) such that for any $u,v\in\mathrm{ran}(\mathrm{Prox}_{\gamma,h})$, $\\|\nabla{\cal M}_{\gamma,h}(w)-\nabla{\cal M}_{\gamma,h}(w^{\prime})\\|\leq L\\|u-v\\|,\ \forall w\in\mathrm{Prox}_{\gamma,h}^{-1}(u),w^{\prime}\in\mathrm{Prox}_{\gamma,h}^{-1}(v);$ 2. (b) We say $h$ satisfies the implicit bounded subgradient property if for any $\gamma\in(0,\rho^{-1})$, there exists $\hat{L}>0$ (depending on $\gamma$) such that for any $u\in\mathrm{ran}(\mathrm{Prox}_{\gamma,h})$, $\\|\nabla{\cal M}_{\gamma,h}(w)\\|\leq\hat{L},\ \forall w\in\mathrm{Prox}_{\gamma,h}^{-1}(u).$ Since $\nabla{\cal M}_{\gamma,h}(x)\in\partial h(\mathrm{Prox}_{\gamma,h}(x))$ for any $x\in\mathbb{R}^{n}$, we have $\nabla{\cal M}_{\gamma,h}(w)\in\partial h(u),\forall u\in\mathrm{ran}(\mathrm{Prox}_{\gamma,h})$ and $w\in\mathrm{Prox}_{\gamma,h}^{-1}(u)$. Hence, the implicit Lipschitz subgradient and implicit bounded subgradient imply, respectively, the Lipschitz continuity and boundedness only on the components of $\partial h$ that are Moreau envelope gradients, but not on other components of $\partial h$. When $h$ is differentiable, implicit Lipschitz subgradient implies Lipschitz gradient. Having implicit bounded subgradients is weaker than having bounded $\partial h$, which is commonly assumed in the analysis of nonconvex algorithms (cf. Davis-Drusvyatskiy19 ; Hajinezhad-Hong19 ; Zeng-DGD18 ). Nonsmooth and nonconvex functions like the SCAD regularization and MCP regularization which appear in statistical learning Wang19 , have implicit bounded subgradients. ### 2.3 Kurdyka-Łojasiewicz Inequality The Kurdyka-Łojasiewicz (KŁ) inequality Bolte-KL2007a ; Bolte-KL2007b ; Kurdyka-KL1998 ; Lojasiewicz-KL1963 ; Lojasiewicz-KL1993 is a property that leads to global convergence of nonconvex algorithms in the literature (see, Attouch13 ; Bolte2014 ; Wang19 ; Xu-Yin-BCD13 ; Zeng-BCD19 ; Zeng-ADMM19 ). The following definition of Kurdyka-Łojasiewicz property is adopted from Bolte-KL2007a . ###### Definition 2 A function $h:\mathbb{R}^{n}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is said to have the Kurdyka-Łojasiewicz property at $x^{}\in\mathrm{dom}(\partial h)$ if there exist a neighborhood ${\cal U}$ of $x^{}$, a constant $\nu>0$, and a continuous concave function $\varphi(s)=cs^{1-\theta}$ for some $c>0$ and $\theta\in[0,1)$ such that the Kurdyka-Łojasiewicz inequality holds: for all $x\in{\cal U}\cap\mathrm{dom}(\partial h)$ and $h(x^{})<h(x)<h(x^{})+\nu$, $\varphi^{\prime}(h(x)-h(x^{}))\cdot\mathrm{dist}(0,\partial h(x))\geq 1,$ (15) (we use the conventions: $0^{0}=1,\infty/\infty=0/0=0$), where $\theta$ is called the KŁ exponent of $h$ at $x^{}$. Proper lower semicontinuous functions satisfying the KŁ inequality at every point of $\mathrm{dom}(\partial h)$ are called KŁ functions. This property was firstly introduced by Lojasiewicz-KL1993 on real analytic functions Krantz2002-real-analytic for $\theta\in\left[\tfrac{1}{2},1\right)$, was then extended to functions defined on the o-minimal structure in Kurdyka-KL1998 , and was later extended to nonsmooth subanalytic functions in Bolte-KL2007a . KŁ functions include real analytic functions Krantz2002-real-analytic , semialgebraic functions Bochnak- semialgebraic1998 , tame functions defined in some o-minimal structures Kurdyka-KL1998 , continuous subanalytic functions Bolte-KL2007a , definable functions Bolte-KL2007b , locally strongly convex functions Xu-Yin-BCD13 , as well as many deep-learning training models Zeng-BCD19 ; Zeng-ADMM19 . ## 3 Convergence of MEAL This section presents the convergence results of MEAL and iMEAL. We postpone their proofs to Section 5. ### 3.1 Assumptions and Stationarity Measure ###### Assumption 1 The set ${\cal X}:=\\{x:Ax=b\\}$ is nonempty. ###### Assumption 2 The objective $f$ in problem (1) satisfies: 1. (a) $f$ is proper lower semicontinuous and $\rho$-weakly convex; and for any $\gamma\in(0,\rho^{-1})$, either (b) or (c): 2. (b) $f$ satisfies the implicit Lipschitz subgradient property with a constant $L_{f}>0$ (possibly depending on $\gamma$); or, 3. (c) $f$ satisfies the implicit bounded subgradient property with a constant $\hat{L}_{f}>0$ (possibly depending on $\gamma$). We do not assume the following hypotheses: the strict complementarity condition used in Zhang-Luo18 , any rank assumption (such as $\mathrm{Im}(A)\subseteq\mathrm{Im}(\mathrm{B})$ when considering the two- (multi-)block case $Ax+By=0$) used in Wang19 , the linear independence constrained qualification (LICQ) used in Bertsekas82 ; Nocedal99 (implying the full-rank assumption in the linear constraint case). Assumption 2 is mild as discussed in Section 2.2. According to (3) and the update (4) of MEAL, we have $\displaystyle\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})=\left(\begin{array}[]{c}(\eta\gamma)^{-1}(z^{k}-z^{k+1})\\\ \beta_{k}^{-1}(\lambda^{k+1}-\lambda^{k})\end{array}\right)\in\left(\begin{array}[]{c}\partial f(x^{k+1})+A^{T}\lambda^{k+1}\\\ Ax^{k+1}-b\end{array}\right).$ (20) Let $\displaystyle\xi_{\mathrm{meal}}^{k}:=\min_{0\leq t\leq k}\\|\nabla\phi_{\beta_{t}}(z^{t},\lambda^{t})\\|,\ \forall k\in\mathbb{N}.$ (21) Then according to (20), the bound $\xi_{\mathrm{meal}}^{k}\leq\varepsilon$ implies $\displaystyle\min_{0\leq t\leq k}\mathrm{dist}\left\\{0,\left(\begin{array}[]{c}\partial f(x^{t+1})+A^{T}\lambda^{t+1}\\\ Ax^{t+1}-b\end{array}\right)\right\\}\leq\xi^{k}_{\mathrm{meal}}\leq\varepsilon,$ that is, MEAL achieves $\varepsilon$-accurate first-order stationarity for problem (1) within $k$ iterations. Hence, $\xi_{\mathrm{meal}}^{k}$ is a valid stationarity measure of MEAL. Define iteration complexity: $\displaystyle T_{\varepsilon}=\inf\left\\{t\geq 1:\\|\nabla\phi_{\beta_{t}}(z^{t},\lambda^{t})\\|\leq\varepsilon\right\\}.$ (22) Comparing $T_{\varepsilon}$ to the common iteration complexity $\displaystyle\hat{T}_{\varepsilon}=\inf\left\\{t\geq 1:\mathrm{dist}(0,\partial f(x^{t})+A^{T}\lambda^{t})\leq\epsilon\ \text{and}\ \\|Ax^{t}-b\\|\leq\varepsilon\right\\},$ we get $T_{\varepsilon}\geq\hat{T}_{\varepsilon}$. If $f$ is differentiable, $\mathrm{dist}(0,\partial f(x^{t})+A^{T}\lambda^{t})$ reduces to $\\|\nabla f(x^{t})+A^{T}\lambda^{t}\\|$. ### 3.2 Convergence Theorems of MEAL We present the quantities used to state the convergence results of MEAL. Let $\displaystyle{\cal P}_{\beta}(x,z,\lambda)={\cal L}_{\beta}(x,\lambda)+\frac{1}{2\gamma}\\|x-z\\|^{2},$ (23) for some $\beta,\gamma>0.$ Then according to (5), MEAL can be interpreted as a primal-dual update with respect to ${\cal P}_{\beta_{k}}(x,z,\lambda)$ at the $k$-th iteration, that is, updating $x^{k+1}$, $z^{k+1}$, and $\lambda^{k+1}$ by minimization, gradient descent, and gradient ascent respectively. Based on (23), we introduce the following Lyapunov functions for MEAL: $\displaystyle{\cal E}_{\mathrm{meal}}^{k}:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+2\alpha_{k}\\|z^{k}-z^{k-1}\\|^{2},\ \forall k\geq 1,$ (24) associated with the implicit Lipschitz subgradient assumption and $\displaystyle\tilde{\cal E}_{\mathrm{meal}}^{k}:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+3\alpha_{k}\\|z^{k}-z^{k-1}\\|^{2},\ \forall k\geq 1,$ (25) associated with the implicit bounded subgradient assumption, where $\displaystyle\alpha_{k}:=\frac{\beta_{k}+\beta_{k+1}+\gamma\eta(1-\eta/2)}{2c_{\gamma,A}\beta_{k}^{2}},\ \forall k\in\mathbb{N},$ (26) and $c_{\gamma,A}:=\gamma^{2}\tilde{\sigma}_{\min}(A^{T}A)$. When $\beta$ is fixed, we also fix $\displaystyle\alpha:=\frac{2\beta+\gamma\eta(1-\eta/2)}{2c_{\gamma,A}\beta^{2}}.$ (27) ###### Theorem 3.1 (Iteration Complexity of MEAL) Suppose that Assumptions 1 and 2(a) hold. Pick $\gamma\in(0,\rho^{-1})$ and $\eta\in(0,2)$. Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by MEAL (5). The following claims hold: 1. (a) Set $\beta$ sufficiently large such that in (27), $\alpha<\min\left\\{\frac{1-\gamma\rho}{4\gamma(1+\gamma L_{f})^{2}},\frac{1}{8\gamma}(\frac{2}{\eta}-1)\right\\}$. Under Assumption 2(b), if $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$ is lower bounded, then $\xi^{k}_{\mathrm{meal}}=o(1/\sqrt{k})$ for $\xi_{\mathrm{meal}}^{k}$ in (21). 2. (b) Pick any $K\geq 1$. Set $\\{\beta_{k}\\}$ so that in (26), $\alpha_{k}\equiv\frac{\alpha^{}}{K}$ for some positive constant $\alpha^{}\leq\min\left\\{\frac{1-\rho\gamma}{6\gamma},\frac{1}{12\gamma}\left(\frac{2}{\eta}-1\right)\right\\}$. Under Assumption 2(c), if $\\{\tilde{\cal E}_{\mathrm{meal}}^{k}\\}$ is lower bounded, then $\xi_{\mathrm{meal}}^{K}\leq\tilde{c}_{1}/\sqrt{K}$ for some constant $\tilde{c}_{1}>0$. Section 6.1 provides conditions sufficient for the lower-boundedness assumptions. Let us interpret the theorem. To achieve an $\varepsilon$-accurate stationary point, the iteration complexity of MEAL is $o(\varepsilon^{-2})$ assuming the implicit Lipschitz subgradient property and ${\cal O}(\varepsilon^{-2})$ assuming the implicit bounded subgradient property. Both iteration complexities are consistent with the existing results of ${\cal O}(\varepsilon^{-2})$ in Hajinezhad-Hong19 ; Hong17-Prox-PDA ; Xie- Wright19 ; Zhang-Luo20 . The established results of MEAL also hold for proximal ALM by setting $\eta=1$. We note that it is not our goal to pursue any better complexity (e.g., using momentum) in this paper. ###### Remark 1 Let $\bar{\alpha}:=\min\left\\{\frac{1-\gamma\rho}{4\gamma(1+\gamma L_{f})^{2}},\frac{1}{8\gamma}(\frac{2}{\eta}-1)\right\\}$. By (27), the requirement $0<\alpha<\bar{\alpha}$ in Theorem 3.1(a) is met by setting $\displaystyle\beta>\frac{1+\sqrt{1+\eta(2-\eta)\gamma c_{\gamma,A}\bar{\alpha}}}{2c_{\gamma,A}\bar{\alpha}}.$ (28) Similarly, the assumption $\alpha_{k}=\frac{\alpha^{}}{K}$ in Theorem 3.1(b) is met by setting $\displaystyle\beta_{k}=\frac{K\left(1+\sqrt{1+\eta(2-\eta)\gamma c_{\gamma,A}\alpha^{}/K}\right)}{2c_{\gamma,A}\alpha^{}},\ k=1,\ldots,K.$ (29) Next, we establish global convergence (whole sequence convergence regardless of initial points) and its rate for MEAL under the KŁ inequality (Definition 2). Let $\hat{z}^{k}:=z^{k-1}$, $y^{k}:=(x^{k},z^{k},\lambda^{k},\hat{z}^{k}),\ \forall k\geq 1,$ $y:=(x,z,\lambda,\hat{z})\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{n},$ and $\displaystyle{\cal P}_{\mathrm{meal}}(y):={\cal P}_{\beta}(x,z,\lambda)+3\alpha\\|z-\hat{z}\\|^{2}$ (30) where $\alpha$ is defined in (27). ###### Proposition 1 (Global convergence and rate of MEAL) Suppose that the assumptions required for Theorem 3.1(a) hold and that $\\{(x^{k},z^{k},\lambda^{k})\\}$ generated by MEAL (5) is bounded. If ${\cal P}_{\mathrm{meal}}$ satisfies the KŁ property at some point $y^{}:=(x^{},x^{},\lambda^{},x^{})$ with an exponent of $\theta\in[0,1)$, where $(x^{},\lambda^{})$ is a limit point of $\\{(x^{k},\lambda^{k})\\}$, then 1. (a) the whole sequence $\\{\hat{y}^{k}:=(x^{k},z^{k},\lambda^{k})\\}$ converges to $\hat{y}^{}:=(x^{},x^{},\lambda^{})$; and 2. (b) the following rate-of-convergence results hold: (1) if $\theta=0$, then $\\{\hat{y}^{k}\\}$ converges within a finite number of iterations; (2) if $\theta\in(0,\frac{1}{2}]$, then $\\|\hat{y}^{k}-\hat{y}^{}\\|\leq c\tau^{k}$ for all $k\geq k_{0}$, for certain $k_{0}>0,c>0,\tau\in(0,1)$; and (3) if $\theta\in(\frac{1}{2},1)$, then $\\|\hat{y}^{k}-\hat{y}^{}\\|\leq ck^{-\frac{1-\theta}{2\theta-1}}$ for all $k\geq k_{0}$, for certain $k_{0}>0,c>0$. In Proposition 1, the KŁ property of ${\cal P}_{\mathrm{meal}}$ defined in (30) plays a central role in the establishment of global convergence of MEAL. The KŁ exponent determines the convergence speed of MEAL; particularly, the exponent $\theta=1/2$ implies linear convergence so it is most desirable. Below we give some results on $\theta$, which are obtainable from (Shiota1997, , page 43), (Bolte-KL2007a, , Theorem 3.1), (Zeng-BCD19, , Lemma 5), and (Li- Pong-KLexponent18, , Theorem 3.6 and Corollary 5.2). ###### Proposition 2 The following claims hold: 1. (a) If $f$ is subanalytic with a closed domain and continuous on its domain, then ${\cal P}_{\mathrm{meal}}$ defined in (30) is a KŁ function; 2. (b) If ${\cal L}_{\beta}(x,\lambda)$ defined in (2) has the KŁ property at some point $(x^{},\lambda^{})$ with exponent $\theta\in[1/2,1)$, then ${\cal P}_{\mathrm{meal}}$ has the KŁ property at $(x^{},x^{},\lambda^{},x^{})$ with exponent $\theta$; 3. (c) If $f$ has the following form: $\displaystyle f(x)=\min_{1\leq i\leq r}\left\\{\frac{1}{2}x^{T}M_{i}x+u_{i}^{T}x+c_{i}+P_{i}(x)\right\\},$ (31) where $P_{i}$ are proper closed polyhedral functions, $M_{i}$ are symmetric matrices of size $n$, $u_{i}\in\mathbb{R}^{n}$ and $c_{i}\in\mathbb{R}$ for $i=1,\ldots,r$, then ${\cal L}_{\beta}$ is a KŁ function with an exponent of $\theta=1/2$. Claim (a) can be obtained as follows. The terms in ${\cal P}_{\mathrm{meal}}$ besides $f$ are polynomial functions, which are both real analytic and semialgebraic Bochnak-semialgebraic1998 . Since $f$ is subanalytic with a closed domain and continuous on its domain, by (Zeng-BCD19, , Lemma 5), ${\cal P}_{\mathrm{meal}}$ is also subanalytic with a closed domain and continuous on its domain. By (Bolte-KL2007a, , Theorem 3.1), ${\cal P}_{\mathrm{meal}}$ is a KŁ function. Claim (b) can be verified by applying (Li-Pong-KLexponent18, , Theorem 3.6) to ${\cal P}_{\mathrm{meal}}$. Claim (c) can be established as follows. The class of functions $f$ defined by (31) are weakly convex with a modulus $\rho=2\max_{1\leq i\leq r}\\|M_{i}\\|$. According to (Li-Pong- KLexponent18, , Sec. 5.2), this class covers many nonconvex functions such as SCAD Fan-SCAD and MCP Zhang-MCP in statistical learning. The function ${\cal L}_{\beta}(x,\lambda)=\frac{\beta}{2}\\|Ax+\beta^{-1}\lambda-b\\|^{2}+(f(x)-\frac{1}{2\beta}\\|\lambda\\|^{2})$. according to (Li-Pong-KLexponent18, , Corollary 5.2), is a KŁ function with an exponent of $1/2$. More results on the KŁ functions with exponent $1/2$ can be found in Li-Pong-KLexponent18 ; Yu-Li-Pong-KLexponent21 and the references therein. ### 3.3 Convergence of iMEAL When considering iMEAL, the Lyapunov functions need to be slightly modified into $\displaystyle{\cal E}_{\mathrm{imeal}}^{k}:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+3\alpha_{k}\\|z^{k}-z^{k-1}\\|^{2},\ \forall k\geq 1,$ (32) associated with the implicit Lipschitz subgradient assumption, and $\displaystyle\tilde{\cal E}_{\mathrm{imeal}}^{k}:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+4\alpha_{k}\\|z^{k}-z^{k-1}\\|^{2},\ \forall k\geq 1,$ (33) associated with the implicit bounded subgradient assumption, where $\alpha_{k}$ is defined in (26). ###### Theorem 3.2 (Iteration Complexity of iMEAL) Let Assumptions 1 and 2(a) hold, $\gamma\in(0,\rho^{-1})$, and $\eta\in(0,2)$. Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by iMEAL (7) with $\sum_{k=0}^{\infty}\epsilon_{k}^{2}<\infty$. The following claims hold: 1. (a) Set $\beta$ sufficiently large such that in (27), $\alpha<\min\left\\{\frac{1-\gamma\rho}{6\gamma(1+\gamma L_{f})^{2}},\frac{1}{12\gamma}(\frac{2}{\eta}-1)\right\\}$. Under Assumption 2(b), if $\\{{\cal E}_{\mathrm{imeal}}^{k}\\}$ is lower bounded, then $\xi^{k}_{\mathrm{meal}}=o(1/\sqrt{k})$ (cf. (21)). 2. (b) Pick $K\geq 1$. Set $\\{\beta_{k}\\}$ such that in (26), $\alpha_{k}\equiv\frac{\hat{\alpha}^{}}{K}$ for some positive constant $\hat{\alpha}^{}\leq\min\left\\{\frac{1-\rho\gamma}{8\gamma},\frac{1}{16\gamma}(\frac{2}{\eta}-1)\right\\}$. Under Assumption 2(c), if $\\{\tilde{\cal E}_{\mathrm{imeal}}^{k}\\}$ is lower bounded, then $\xi_{\mathrm{meal}}^{K}\leq\tilde{c}_{2}/\sqrt{K}$ for some constant $\tilde{c}_{2}>0$. By Theorem 3.2, the iteration complexity of iMEAL is the same as that of MEAL and also consistent with that of inexact proximal ALM Xie-Wright19 (when the stationary accuracy $\epsilon_{k}$ is square summable). Moreover, if the condition on $\epsilon_{k}$ is strengthened to be $\sum_{k=0}^{\infty}\epsilon_{k}<+\infty$ as required in the literature Rockafellar76-PALM ; Wang19 , then following a proof similar for Proposition 1, global convergence and similar rates of MEAL also hold for iMEAL under the assumptions required for Theorem 3.2(a) and the KŁ property. ## 4 Convergence of LiMEAL for Composite Objective This section presents the convergence results of LiMEAL (9) for the constrained problem with a composite objective (8). The proofs are postponed to Section 5 below. Similar to Assumption 2, we make the following assumptions. ###### Assumption 3 The objective $f(x)=h(x)+g(x)$ in problem (8) satisfies: 1. (a) $h$ is differentiable and $\nabla h$ is Lipschitz continuous with a constant $L_{h}>0$; 2. (b) $g$ is proper lower-semicontinuous and $\rho_{g}$-weakly convex; and either 3. (c) $g$ has the implicit Lipschitz subgradient property with a constant $L_{g}>0$; or 4. (d) $g$ has the implicit bounded subgradient property with a constant $\hat{L}_{g}>0$. In (c) and (d), $L_{g}$ and $\hat{L}_{g}$ may depend on $\gamma$. By the update (9) of LiMEAL, some simple derivations show that $\displaystyle x^{k+1}=\mathrm{Prox}_{\gamma,g}(z^{k}-\gamma(\nabla h(x^{k})+A^{T}\lambda^{k+1}))$ (34) and $\displaystyle g_{\mathrm{limeal}}^{k}:=\left(\begin{array}[]{c}\gamma^{-1}(z^{k}-x^{k+1})+(\nabla h(x^{k+1})-\nabla h(x^{k}))\\\ \beta_{k}^{-1}(\lambda^{k+1}-\lambda^{k})\end{array}\right)\in\left(\begin{array}[]{c}\partial f(x^{k+1})+A^{T}\lambda^{k+1}\\\ Ax^{k+1}-b\end{array}\right).$ (39) Actually, the term $\gamma^{-1}(z^{k}-x^{k+1})$ represents some prox-gradient sequence frequently used in the analysis of algorithms for the unconstrained composite optimization (e.g., Davis-Drusvyatskiy19 ). Thus, let $\displaystyle\xi_{\mathrm{limeal}}^{k}:=\min_{0\leq t\leq k}\\|g_{\mathrm{limeal}}^{t}\\|,\ \forall k\in\mathbb{N},$ (40) which can be taken as an effective stationarity measure of LiMEAL for problem (8). In the following, we present the iteration complexity of LiMEAL for problem (8). Since the prox-linear scheme is adopted in the update of $x^{k+1}$ in LiMEAL as described in (9), thus, the proximal term (i.e., $\\|x^{k}-x^{k-1}\\|^{2}$) should be generally included in the associated Lyapunov functions of LiMEAL, shown as follows: $\displaystyle{\cal E}^{k}_{\mathrm{limeal}}$ $\displaystyle:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+3\alpha_{k}(\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2})$ (41) associated with the implicit Lipschitz gradient assumption, and $\displaystyle\tilde{\cal E}_{\mathrm{limeal}}^{k}:={\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})+4{\alpha}_{k}(\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}),$ (42) associated with the implicit bounded subgradient assumption, where $\alpha_{k}$ is defined in (26). The iteration complexity of MEAL can be similarly generalized to LiMEAL as follows. ###### Theorem 4.1 (Iteration Complexity of LiMEAL) Take Assumptions 1 and 3(a)-(b). Pick $\eta\in(0,2)$ and $0<\gamma<\frac{2}{(\rho_{g}+L_{h})\left(1+\sqrt{1+\frac{2(2-\eta)\eta L_{h}^{2}}{(\rho_{g}+L_{h})^{2}}}\right)}$. Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by LiMEAL (9). The following claims hold: 1. (a) Set $\beta$ sufficiently large such that $\alpha<\min\left\\{\frac{1}{12\gamma}(\frac{2}{\eta}-1),\frac{1-\gamma(\rho_{g}+L_{h})-\eta(1-\eta/2)\gamma^{2}L_{h}^{2}}{6\gamma\left((1+\gamma L_{g})^{2}+\gamma^{2}L_{h}^{2}\right)}\right\\}$. Under Assumption 3(c), if $\\{{\cal E}^{k}_{\mathrm{limeal}}\\}$ is lower bounded, then $\xi^{k}_{\mathrm{limeal}}=o(1/\sqrt{k})$. 2. (b) Pick $K\geq 1$. Set $\\{\beta_{k}\\}$ such that $\alpha_{k}\equiv\frac{\bar{\alpha}^{}}{K}$ for some positive constant $\bar{\alpha}^{}\leq\min\Big{\\{}\frac{1-\gamma\left(\rho_{g}+L_{h})-\eta(1-\eta/2)\gamma^{2}L_{h}^{2}\right)}{8\gamma(1+\gamma^{2}L_{h}^{2})}$, $\frac{1}{16\gamma}\left(\frac{2}{\eta}-1\right)\Big{\\}}$. Under Assumption 3(d), if $\\{\tilde{\cal E}^{k}_{\mathrm{limeal}}\\}$ is lower bounded, then $\xi_{\mathrm{limeal}}^{K}\leq\tilde{c}_{3}/\sqrt{K}$ for some constant $\tilde{c}_{3}>0$. Similar to the discussions following Theorem 3.1, to yield an $\varepsilon$-accurate first-order stationary point, the iteration complexity of LiMEAL is $o(\varepsilon^{-2})$ under the implicit Lipschitz subgradient assumption and ${\cal O}(\varepsilon^{-2})$ under the implicit bounded subgradient assumption, as demonstrated by Theorem 4.1. The conditions on $\beta$ and $\beta_{k}$ in these two cases can be derived similarly to (28) and (29), respectively. In the following, we establish the global convergence and rates of LiMEAL under assumptions required for Theorem 4.1(a) and the KŁ property. Specifically, let $\hat{x}^{k}:=x^{k-1},\ \hat{z}^{k}:=z^{k-1},\ {y}^{k}:=(x^{k},z^{k},\lambda^{k},\hat{x}^{k},\hat{z}^{k}),\ \forall k\geq 1,$ ${y}:=(x,z,\lambda,\hat{x},\hat{z})\in\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{n}\times\mathbb{R}^{n},$ and $\displaystyle{\cal P}_{\mathrm{limeal}}({y}):={\cal P}_{\beta}(x,z,\lambda)+4\alpha\left(\\|z-\hat{z}\\|^{2}+\gamma^{2}L_{h}^{2}\\|x-\hat{x}\\|^{2}\right).$ (43) ###### Proposition 3 (Global convergence and rates of LiMEAL) Suppose that Assumptions 1 and 3(a)-(c) hold and that the sequence $\\{(x^{k},z^{k},\lambda^{k})\\}$ generated by LiMEAL (9) is bounded. If $\gamma\in(0,\frac{1}{\rho_{g}+L_{h}})$, $\eta\in(0,2)$, $0<\alpha<\min\left\\{\frac{1}{8\gamma}\left(\frac{2}{\eta}-1\right),\frac{1-\gamma(\rho_{g}+L_{h})}{8\gamma\left((1+\gamma L_{g})^{2}+\gamma^{2}L_{h}^{2}\right)}\right\\}$, and ${\cal P}_{\mathrm{limeal}}$ satisfies the KŁ property at some point $y^{}:=(x^{},x^{},\lambda^{},x^{},x^{})$ with an exponent of $\theta\in[0,1)$, where $(x^{},\lambda^{})$ is a limit point of $\\{(x^{k},\lambda^{k})\\}$, then 1. (a) the whole sequence $\\{\hat{y}^{k}:=(x^{k},z^{k},\lambda^{k})\\}$ converges to $\hat{y}^{}:=(x^{},x^{},\lambda^{})$; and 2. (b) all the rates of convergence results in Proposition 1(b) also hold for LiMEAL. ###### Remark 2 The established results in this section is more general than those in Zhang- Luo18 and done under weaker assumptions on $h$ and for more general class of $g$. Specifically, as discussed in Section 1.2, the algorithm studied in Zhang-Luo18 is a prox-linear version of LiMEAL with $g$ being an indicator function of a box constraint set. In Zhang-Luo18 , global convergence and a linear rate of proximal inexact ALM were proved for quadratic programming, where that the augmented Lagrangian satisfies the KŁ inequality with exponent $1/2$. Besides, the strict complementarity condition required in Zhang-Luo18 is also removed in this paper for LiMEAL. ## 5 Main Proofs In this section, we first prove some lemmas and then present the proofs of our main convergence results. ### 5.1 Preliminary Lemmas #### 5.1.1 Lemmas on Iteration Complexity and Global Convergence The first lemma concerns the convergence speed of a nonenegative sequence $\\{\xi_{k}\\}$ satisfying the following relation $\displaystyle\tilde{\eta}\xi_{k}^{2}\leq({\cal E}_{k}-{\cal E}_{k+1})+\tilde{\epsilon}_{k}^{2},\ \forall k\in\mathbb{N},$ (44) where $\tilde{\eta}>0$, $\\{{\cal E}_{k}\\}$ and $\\{\tilde{\epsilon}_{k}\\}$ are two nonnegative sequences, and $\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}<+\infty$. ###### Lemma 1 For any sequence $\\{\xi_{k}\\}$ satisfying (44), $\tilde{\xi}_{k}:=\min_{1\leq t\leq k}\xi_{t}=o(1/\sqrt{k})$. ###### Proof Summing (44) over $k$ from $1$ to $K$ and letting $K\rightarrow+\infty$ yields $\displaystyle\sum_{k=1}^{\infty}\xi_{k}^{2}\leq\tilde{\eta}^{-1}\left({\cal E}_{1}+\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}\right)<+\infty,$ which implies the desired convergence speed by $\frac{k}{2}\tilde{\xi}_{k}^{2}\leq\sum_{\frac{k}{2}\leq j\leq k}{\xi}_{j}^{2}\rightarrow 0$ as $k\rightarrow\infty$, as proved in (Deng- parallelADMM17, , Lemma 1.1). Then we provide a lemma to show the convergence speed of a nonenegative sequence $\\{\xi_{k}\\}$ satisfying the following relation instead of (44) $\displaystyle\tilde{\eta}\xi_{k}^{2}\leq({\cal E}_{k}-{\cal E}_{k+1})+\tilde{\epsilon}_{k}^{2}+{\alpha}_{k}\tilde{L},\ \forall k\in\mathbb{N},$ (45) where $\tilde{\eta}>0,$ $\tilde{L}>0$, $\\{{\cal E}_{k}\\}$, $\\{{\alpha}_{k}\\}$ and $\\{\tilde{\epsilon}_{k}\\}$ are nonnegative sequences, and $\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}<+\infty$. ###### Lemma 2 Pick $K\geq 1$. Let $\\{\xi_{k}\\}$ be a nonnegative sequence satisfying (45). Set $\alpha_{k}\equiv\frac{\tilde{\alpha}}{K}$ for some $\tilde{\alpha}>0$. Then $\tilde{\xi}_{K}:=\min_{1\leq k\leq K}\xi_{k}\leq\tilde{c}/\sqrt{K}$ for some constant $\tilde{c}>0$. ###### Proof Summing (45) over $k$ from $1$ to $K$ yields $\displaystyle\sum_{k=1}^{K}\xi_{k}^{2}\leq\frac{{\cal E}_{1}+\sum_{k=1}^{K}\tilde{\epsilon}_{k}^{2}+\tilde{L}\sum_{k=1}^{K}\alpha_{k}}{\tilde{\eta}}.$ From $\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}<+\infty$ and $\sum_{k=1}^{K}{\alpha}_{k}=\tilde{\alpha}$, we get $K\tilde{\xi}_{K}^{2}\leq\sum_{k=1}^{K}\xi_{k}^{2}\leq\frac{{\cal E}_{1}+\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}+\tilde{L}\tilde{\alpha}}{\tilde{\eta}}<+\infty$. The result follows with $\tilde{c}:=\sqrt{{\cal E}_{1}+\sum_{k=1}^{\infty}\tilde{\epsilon}_{k}^{2}+\tilde{L}\tilde{\alpha}}/\sqrt{{\tilde{\eta}}}$. In both Lemmas 1 and 2, the nonnegative assumption on the sequence $\\{{\cal E}_{k}\\}$ can be relaxed to its lower boundedness. The following lemma presents the global convergence and rate of a sequence generated by some algorithm for the nonconvex optimization problem, based on the Kurdyka-Łojasiewicz inequality, where the global convergence result is from (Attouch13, , Theorem 2.9) while the rate results are from (Attouch- Bolte09, , Theorem 5). ###### Lemma 3 (Existing global convergence and rate) Let ${\cal L}$ be a proper, lower semicontinuous function, and $\\{u^{k}\\}$ be a sequence that satisfies the following three conditions: 1. (P1) (Sufficient decrease condition) there exists a constant $a_{1}>0$ such that ${\cal L}(u^{k+1})+a_{1}\\|u^{k+1}-u^{k}\\|^{2}\leq{\cal L}(u^{k}),\ \forall k\in\mathbb{N};$ 2. (P2) (Bounded subgradient condition) for each $k\in\mathbb{N}$, there exists $v^{k+1}\in\partial{\cal L}(u^{k+1})$ such that $\\|v^{k+1}\\|\leq a_{2}\\|u^{k+1}-u^{k}\\|$ for some constant $a_{2}>0$; 3. (P3) (Continuity condition) there exist a subsequence $\\{u^{k_{j}}\\}$ and $\tilde{u}$ such that $u^{k_{j}}\rightarrow\tilde{u}$ and ${\cal L}(u^{k_{j}})\rightarrow{\cal L}(\tilde{u})$ as $j\rightarrow\infty$. If ${\cal L}$ satisfies the KŁ inequality at $\tilde{u}$ with an exponent of $\theta$, then 1. (1) $\\{u^{k}\\}$ converges to $\tilde{u}$; and 2. (2) depending on $\theta$, (i) if $\theta=0$, then $\\{u^{k}\\}$ converges within a finite number of iterations; (ii) if $\theta\in(0,\frac{1}{2}]$, then $\\|u^{k}-\tilde{u}\\|\leq c\tau^{k}$ for all $k\geq k_{0}$, for certain $k_{0}>0,c>0,\tau\in(0,1)$; and (iii) if $\theta\in(\frac{1}{2},1)$, then $\\|u^{k}-\tilde{u}\\|\leq ck^{-\frac{1-\theta}{2\theta-1}}$ for all $k\geq k_{0}$, for certain $k_{0}>0,c>0$. #### 5.1.2 Lemmas on controlling dual ascent by primal descent In the following, we establish several lemmas to show that the dual ascent quantities of proposed algorithms can be controlled by the primal descent quantities. ###### Lemma 4 (MEAL: controlling dual by primal) Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by MEAL (5). Take $\gamma\in(0,\rho^{-1})$. 1. (a) Under Assumptions 1, 2(a), and 2(b), we have for any $k\geq 1$, $\displaystyle\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|\leq(L_{f}+\gamma^{-1})\\|x^{k+1}-x^{k}\\|+\gamma^{-1}\\|z^{k}-z^{k-1}\\|,$ (46) $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 2c_{\gamma,A}^{-1}\left[(\gamma L_{f}+1)^{2}\\|x^{k+1}-x^{k}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}\right],$ (47) where $c_{\gamma,A}=\gamma^{2}\tilde{\sigma}_{\min}(A^{T}A)$. 2. (b) Alternatively, under Assumptions 1, 2(a), and 2(c), we have for any $k\geq 1$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 3c_{\gamma,A}^{-1}\left[4\gamma^{2}\hat{L}^{2}_{f}+\\|x^{k+1}-x^{k}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}\right].$ (48) ###### Proof The update (5) of $x^{k+1}$ implies $\displaystyle x^{k+1}=\operatorname{argmin}_{x}\left\\{f(x)+\langle\lambda^{k},Ax-b\rangle+\frac{\beta_{k}}{2}\\|Ax-b\\|^{2}+\frac{1}{2\gamma}\\|x-z^{k}\\|^{2}\right\\}.$ Its optimality condition and the update (5) of $\lambda^{k+1}$ in MEAL together give us $\displaystyle 0\in\partial\left(f+\frac{1}{2\gamma}\\|\cdot-(z^{k}-\gamma A^{T}\lambda^{k+1})\\|^{2}\right)(x^{k+1}).$ (49) Let $w^{k+1}:=z^{k}-\gamma A^{T}\lambda^{k+1},\ \forall k\in\mathbb{N}.$ The above inclusion implies $\displaystyle x^{k+1}=\mathrm{Prox}_{\gamma,f}(w^{k+1}),$ (50) and thus by (13), $\displaystyle A^{T}\lambda^{k+1}$ $\displaystyle=-\nabla{\cal M}_{\gamma,f}(w^{k+1})-\gamma^{-1}(x^{k+1}-z^{k}),$ (51) which further implies $\displaystyle\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|$ $\displaystyle=\\|(\nabla{\cal M}_{\gamma,f}(w^{k+1})-\nabla{\cal M}_{\gamma,f}(w^{k}))+\gamma^{-1}(x^{k+1}-x^{k})-\gamma^{-1}(z^{k}-z^{k-1})\\|.$ (a) With Assumption 2(b), the above equality yields $\displaystyle\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|\leq(L_{f}+\gamma^{-1})\\|x^{k+1}-x^{k}\\|+\gamma^{-1}\\|z^{k}-z^{k-1}\\|,$ which leads to (46). By Assumption 1 and the relation $\lambda^{k+1}-\lambda^{k}=\beta_{k}(Ax^{k+1}-b)$, $(\lambda^{k+1}-\lambda^{k})\in\mathrm{Im}(A)$. Thus, from the above inequality, we deduce $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|\leq\tilde{\sigma}_{\min}^{-1/2}(A^{T}A)\left[(L_{f}+\gamma^{-1})\\|x^{k+1}-x^{k}\\|+\gamma^{-1}\\|z^{k}-z^{k-1}\\|\right],$ and, further by $(u+v)^{2}\leq 2(u^{2}+v^{2})$ for any $u,v\in\mathbb{R}$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 2\tilde{\sigma}_{\min}^{-1}(A^{T}A)\left[(L_{f}+\gamma^{-1})^{2}\\|x^{k+1}-x^{k}\\|^{2}+\gamma^{-2}\\|z^{k}-z^{k-1}\\|^{2}\right].$ (b) From Assumption 2(c), we have $\displaystyle\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|\leq 2\hat{L}_{f}+\gamma^{-1}(\\|x^{k+1}-x^{k}\\|+\\|z^{k}-z^{k-1}\\|),$ which implies $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|\leq\tilde{\sigma}_{\min}^{-1/2}(A^{T}A)\left[2\hat{L}_{f}+\gamma^{-1}(\\|x^{k+1}-x^{k}\\|+\\|z^{k}-z^{k-1}\\|)\right],$ and further by $(a+c+d)^{2}\leq 3(a^{2}+c^{2}+d^{2})$ for any $a,c,d\in\mathbb{R}$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 3\tilde{\sigma}_{\min}^{-1}(A^{T}A)\left[4\hat{L}^{2}_{f}+\gamma^{-2}(\\|x^{k+1}-x^{k}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2})\right].$ The similar lemma also holds for iMEAL shown as follows. ###### Lemma 5 (iMEAL: controlling dual by primal) Let $(x^{k},z^{k},\lambda^{k})$ be a sequence generated by iMEAL (7). Take $\gamma\in(0,\rho^{-1})$. 1. (a) Under Assumptions 1 2(a), and 2(b), for any $k\geq 1$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 3c_{\gamma,A}^{-1}\left[(\gamma L_{f}+1)^{2}\\|x^{k+1}-x^{k}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}+\gamma^{2}(\epsilon_{k}+\epsilon_{k-1})^{2}\right].$ 2. (b) Alternatively, under Assumptions 1 2(a), and 2(c), for any $k\geq 1$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}\leq 4c_{\gamma,A}^{-1}\left[4\gamma^{2}\hat{L}^{2}_{f}+\\|x^{k+1}-x^{k}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}+\gamma^{2}(\epsilon_{k}+\epsilon_{k-1})^{2}\right].$ ###### Proof The proof is similar to that of Lemma 4, but with (49) being replaced by $\displaystyle 0\in\partial\left(f+\frac{1}{2\gamma}\left\\|\cdot-\Big{(}z^{k}-\gamma(A^{T}\lambda^{k+1}-s^{k})\Big{)}\right\\|^{2}\right)(x^{k+1}),$ and thus $w^{k+1}:=z^{k}-\gamma(A^{T}\lambda^{k+1}-s^{k}).$ ###### Lemma 6 (LiMEAL: controlling dual by primal) Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by LiMEAL (9). Take $\gamma\in(0,\rho_{g}^{-1})$. 1. (a) Under Assumptions 1, 3(a)-(b), and 3(c), for any $k\geq 1$, $\displaystyle\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|$ (52) $\displaystyle\leq(L_{g}+\gamma^{-1})\\|x^{k+1}-x^{k}\\|+L_{h}\\|x^{k}-x^{k-1}\\|+\gamma^{-1}\\|z^{k}-z^{k-1}\\|,$ $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}$ (53) $\displaystyle\leq 3c_{\gamma,A}^{-1}\left[(\gamma L_{g}+1)^{2}\\|x^{k+1}-x^{k}\\|^{2}+\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}\right].$ 2. (b) Alternatively, under Assumptions 1, 3(a)-(b), and 3(d), for any $k\geq 1$, $\displaystyle\\|\lambda^{k+1}-\lambda^{k}\\|^{2}$ (54) $\displaystyle\leq 4c_{\gamma,A}^{-1}\left[4\gamma^{2}\hat{L}^{2}_{g}+\\|x^{k+1}-x^{k}\\|^{2}+\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}\right].$ ###### Proof The proof is also similar to that of Lemma 4, but (49) needs to be modified to $\displaystyle 0\in\partial\left(g+\frac{1}{2\gamma}\\|\cdot-\Big{(}z^{k}-\gamma(A^{T}\lambda^{k+1}+\nabla h(x^{k}))\Big{)}\\|^{2}\right)(x^{k+1}),$ and thus $w^{k+1}:=z^{k}-\gamma(A^{T}\lambda^{k+1}+\nabla h(x^{k}))).$ #### 5.1.3 Lemmas on One-step Progress Here, we provide several lemmas to characterize the progress achieved by a single iterate of the proposed algorithms. ###### Lemma 7 (MEAL: one-step progress) Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by MEAL (4). Take Assumption 2(a), $\gamma\in(0,\rho^{-1})$, and $\eta\in(0,2)$. Then for any $k\in\mathbb{N}$, $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{(1-\gamma\rho)}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}$ (55) $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}+\frac{1}{4}\gamma\eta(2-\eta)\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-\alpha_{k}c_{\gamma,A}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},$ where $\alpha_{k}$ is presented in (26) and $c_{\gamma,A}=\gamma^{2}\tilde{\sigma}_{\min}(A^{T}A)$. ###### Proof By the update (5) of $x^{k+1}$ in MEAL, $x^{k+1}$ is updated via minimizing a strongly convex function ${\cal P}_{\beta_{k}}(x,z^{k},\lambda^{k})$ with modulus at least $(\gamma^{-1}-\rho)$, we have $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k}}(x^{k+1},z^{k},\lambda^{k})\geq\frac{\gamma^{-1}-\rho}{2}\\|x^{k+1}-x^{k}\\|^{2}.$ (56) Next, recall in (5), $z^{k+1}=z^{k}+\eta(x^{k+1}-z^{k})$ implies $\displaystyle 2x^{k+1}-z^{k}-z^{k+1}=(2\eta^{-1}-1)(z^{k+1}-z^{k}).$ (57) So we have $\displaystyle{\cal P}_{\beta_{k}}(x^{k+1},z^{k},\lambda^{k})-{\cal P}_{\beta_{k}}(x^{k+1},z^{k+1},\lambda^{k})=\frac{1}{2\gamma}(\\|x^{k+1}-z^{k}\\|^{2}-\\|x^{k+1}-z^{k+1}\\|^{2})$ $\displaystyle=\frac{1}{2\gamma}\langle z^{k+1}-z^{k},2x^{k+1}-z^{k}-z^{k+1}\rangle=\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}.$ Moreover, by the update $\lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b)$, we have $\displaystyle{\cal P}_{\beta_{k}}(x^{k+1},z^{k+1},\lambda^{k})-{\cal P}_{\beta_{k}}(x^{k+1},z^{k+1},\lambda^{k+1})=-\beta_{k}^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},$ and $\displaystyle{\cal P}_{\beta_{k}}(x^{k+1},z^{k+1},\lambda^{k+1})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})=\frac{\beta_{k}-\beta_{k+1}}{2\beta_{k}^{2}}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ Combining the above four terms of estimates yields $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})$ (58) $\displaystyle\geq\frac{(1-\rho\gamma)}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}+\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-\frac{\beta_{k}+\beta_{k+1}}{2\beta_{k}^{2}}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ Then, we establish (55) from (58). By the definition (20) of $\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})$, we have $\displaystyle\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}=(\eta\gamma)^{-2}\\|z^{k}-z^{k+1}\\|^{2}+\beta_{k}^{-2}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},$ which implies $\displaystyle(\eta\gamma)^{-2}\\|z^{k}-z^{k+1}\\|^{2}=\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-\beta_{k}^{-2}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ Substituting this into the above inequality yields $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{(1-\gamma\rho)}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}+\frac{1}{4}\gamma\eta(2-\eta)\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-\alpha_{k}c_{\gamma,A}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},$ where $\alpha_{k}=\frac{\beta_{k}+\beta_{k+1}+\gamma\eta(1-\eta/2)}{2c_{\gamma,A}\beta_{k}^{2}}$. This finishes the proof. Next, we provide a lemma for iMEAL (7). ###### Lemma 8 (iMEAL: one-step progress) Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by iMEAL (7). Take Assumptions 2(a) and (b), $\gamma\in(0,\rho^{-1})$, and $\eta\in(0,2)$. It holds that $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})$ (59) $\displaystyle\geq\frac{(1-\gamma\rho)}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}+\langle s^{k},x^{k}-x^{k+1}\rangle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}$ $\displaystyle+\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-\alpha_{k}c_{\gamma,A}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},\ \forall k\in\mathbb{N}.$ ###### Proof The proof of this lemma is similar to that of Lemma 7 and uses the descent quantity along the update of $x^{k+1}$. By the update (7) of $x^{k+1}$ in iMEAL and noticing that ${\cal L}_{\beta_{k}}(x,\lambda^{k})+\frac{\\|x-z^{k}\\|}{2\gamma}$ is strongly convex with modulus at least $(\gamma^{-1}-\rho)$, we have $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})$ $\displaystyle\geq{\cal P}_{\beta_{k}}(x^{k+1},z^{k},\lambda^{k})+\langle s^{k},x^{k}-x^{k+1}\rangle+\frac{\gamma^{-1}-\rho}{2}\\|x^{k+1}-x^{k}\\|^{2}.$ By replacing (56) in the proof of Lemma 7 with the above inequality and following the rest part of its proof, we obtain the following inequality $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{1-\gamma\rho}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}+\langle s^{k},x^{k}-x^{k+1}\rangle$ $\displaystyle+\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-\frac{\beta_{k}+\beta_{k+1}}{2\beta_{k}^{2}}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ We can establish (59) with a derivation similar to that in the proof of Lemma 7. Also, we state a similar lemma for one-step progress of LiMEAL (9) as follows. ###### Lemma 9 (LiMEAL: one-step progress) Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be a sequence generated by LiMEAL (9). Take Assumptions 3(a) and (b), $\gamma\in(0,\rho_{g}^{-1})$, and $\eta\in(0,2)$. We have $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})$ (60) $\displaystyle\geq\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-\frac{1}{4}\gamma(2-\eta)\eta L_{h}^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}+\frac{1}{4}\gamma(1-\eta_{k}/2)\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2}-\alpha_{k}c_{\gamma,A}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},\ \forall k\in\mathbb{N}.$ ###### Proof The proof of this lemma is similar to that of Lemma 7. By the update (9) of $x^{k+1}$ in LiMEAL, $x^{k+1}$ is updated via minimizing $(\gamma^{-1}-\rho_{g})$-strongly convex ${\cal L}_{\beta_{k},f^{k}}(x,\lambda^{k})+\frac{\\|x-z^{k}\\|}{2\gamma}$, so $\displaystyle{\cal L}_{\beta_{k},f^{k}}(x^{k},\lambda^{k})+\frac{\\|x^{k}-z^{k}\\|^{2}}{2\gamma}\geq{\cal L}_{\beta_{k},f^{k}}(x^{k+1},\lambda^{k})+\frac{\\|x-z^{k}\\|^{2}}{2\gamma}+\frac{\gamma^{-1}-\rho_{g}}{2}\\|x^{k+1}-x^{k}\\|^{2}.$ By definition, ${\cal L}_{\beta_{k},f^{k}}(x,\lambda)=h(x^{k})+\langle\nabla h(x^{k}),x-x^{k}\rangle+g(x)+\langle\lambda,Ax-b\rangle+\frac{\beta}{2}\\|Ax-b\\|^{2}$ and ${\cal P}_{\beta_{k}}(x,z,\lambda)=h(x)+g(x)+\langle\lambda,Ax-b\rangle+\frac{\beta_{k}}{2}\\|Ax-b\\|^{2}+\frac{\\|x-z\\|^{2}}{2\gamma}$, so the above inequality implies $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})$ $\displaystyle\geq{\cal P}_{\beta_{k}}(x^{k+1},z^{k},\lambda^{k})+\frac{\gamma^{-1}-\rho_{g}}{2}\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle-(h(x^{k+1})-h(x^{k})-\langle\nabla h(x^{k}),x^{k+1}-x^{k}\rangle)$ $\displaystyle\geq{\cal P}_{\beta_{k}}(x^{k+1},z^{k},\lambda^{k})+\frac{\gamma^{-1}-\rho_{g}-L_{h}}{2}\\|x^{k+1}-x^{k}\\|^{2},$ where the second inequality is due to the $L_{h}$-Lipschitz continuity of $\nabla h$. By replacing (56) in the proof of Lemma 7 with the above inequality and following the rest part of that proof, we obtain $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})$ (61) $\displaystyle\geq\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}\\|x^{k+1}-x^{k}\\|^{2}+\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-\frac{\beta_{k}+\beta_{k+1}}{2\beta_{k}^{2}}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ Next, based on the above inequality, we establish (60). By the definition (39) of $g_{\mathrm{limeal}}^{k}$ and noticing that $z^{k}-x^{k+1}=-\eta^{-1}(z^{k+1}-z^{k})$ by the update (9) of $z^{k+1}$, we have $\\|g_{\mathrm{limeal}}^{k}\\|^{2}\leq 2L_{h}^{2}\\|x^{k+1}-x^{k}\\|^{2}+2(\gamma\eta)^{-2}\\|z^{k+1}-z^{k}\\|^{2}+\beta_{k}^{-2}\\|\lambda^{k+1}-\lambda^{k}\\|^{2},$ which implies $(\gamma\eta)^{-2}\\|z^{k+1}-z^{k}\\|^{2}\geq\frac{1}{2}\\|g_{\mathrm{limeal}}^{k}\\|^{2}-\frac{1}{2}\beta_{k}^{-2}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}-L_{h}^{2}\\|x^{k+1}-x^{k}\\|^{2}.$ Substituting this inequality into (61) yields $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})$ $\displaystyle\geq\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-\frac{1}{4}\gamma(2-\eta)\eta L_{h}^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}+\frac{1}{4}\gamma(1-\eta/2)\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2}-\alpha_{k}c_{\gamma,A}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ This finishes the proof of this lemma. ### 5.2 Proofs for Convergence of MEAL Based on the above lemmas, we give proofs of Theorem 3.1 and Proposition 1. #### 5.2.1 Proof of Theorem 3.1 ###### Proof We first establish the $o(1/\sqrt{k})$ rate of convergence under the implicit Lipschitz subgradient assumption (Assumption 2(b)) and then the convergence rate result under the implicit bounded subgradient assumption (Assumption 2(c)). (a) In the first case, $\beta_{k}=\beta$ and $\alpha_{k}=\alpha$. Substituting (47) into (55) yields $\displaystyle{\cal P}_{\beta}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta}(z^{k},\lambda^{k})\\|^{2}$ $\displaystyle+\left(\frac{(1-\gamma\rho)}{2\gamma}-2\alpha(1+\gamma L_{f})^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}$ $\displaystyle-2\alpha\\|z^{k}-z^{k-1}\\|^{2}.$ By the definition (24) of ${\cal E}_{\mathrm{meal}}^{k}$, the above inequality implies $\displaystyle{\cal E}_{\mathrm{meal}}^{k}-{\cal E}_{\mathrm{meal}}^{k+1}$ $\displaystyle\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta}(z^{k},\lambda^{k})\\|^{2}+\left(\frac{1}{4\gamma}(\frac{2}{\eta}-1)-2\alpha\right)\\|z^{k+1}-z^{k}\\|^{2}$ $\displaystyle+\left(\frac{1-\gamma\rho}{2\gamma}-2\alpha(1+\gamma L_{f})^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}$ (62) $\displaystyle\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta}(z^{k},\lambda^{k})\\|^{2},$ where the second inequality holds due to the condition on $\alpha$. Thus, claim (a) follows from the above inequality, Lemma 1 with $\tilde{\epsilon}_{k}=0$ and the lower boundedness of $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$. (b) Similarly, substituting (48) into (55) and using the definition (25) of $\tilde{\cal E}_{\mathrm{meal}}^{k}$, we have $\displaystyle\tilde{\cal E}_{\mathrm{meal}}^{k}-\tilde{\cal E}_{\mathrm{meal}}^{k+1}\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-12\alpha_{k}\gamma^{2}\hat{L}_{f}^{2}$ $\displaystyle+\left(\frac{1-\gamma\rho}{2\gamma}-3\alpha_{k}\right)\\|x^{k+1}-x^{k}\\|^{2}+\left(\frac{1}{4\gamma}(\frac{2}{\eta}-1)-3\alpha_{k+1}\right)\\|z^{k+1}-z^{k}\\|^{2}.$ With $\alpha_{k}=\frac{\alpha^{}}{K}$, $\displaystyle\tilde{\cal E}_{\mathrm{meal}}^{k}-\tilde{\cal E}_{\mathrm{meal}}^{k+1}\geq\frac{1}{2}\gamma(1-\eta/2)\eta\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-12\alpha_{k}\gamma^{2}\hat{L}_{f}^{2},$ which yields claim (b) by Lemma 2 with $\tilde{\epsilon}_{k}=0$ and the lower boundedness of $\\{\tilde{\cal E}_{\mathrm{meal}}^{k}\\}$. #### 5.2.2 Proof of Proposition 1 ###### Proof With Lemma 3, we only need to check conditions $(P1)$-$(P3)$ hold for MEAL. (a) Establishing $(P1)$: With $a:=\frac{\gamma\eta(2-\eta)}{4\beta}$, we have $\frac{1+a}{\beta c_{\gamma,A}}=\alpha$ for $\alpha$ in (27). Substituting (47) into (58) with fixed $\beta_{k}$ yields $\displaystyle{\cal P}_{\beta}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta}(x^{k+1},z^{k+1},\lambda^{k+1})\geq(\frac{1-\rho\gamma}{2\gamma}-2\alpha(\gamma L_{f}+1)^{2})\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-2\alpha\\|z^{k}-z^{k-1}\\|^{2}+a\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ For the definition (30) of ${\cal P}_{\mathrm{meal}}$ and the assumption on $\alpha$, we deduce from the above inequality: $\displaystyle{\cal P}_{\mathrm{meal}}(y^{k})-{\cal P}_{\mathrm{meal}}(y^{k+1})\geq(\frac{1-\rho\gamma}{2\gamma}-2\alpha(\gamma L_{f}+1)^{2})\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\left(\frac{1}{2\gamma}(\frac{2}{\eta}-1)-3\alpha\right)\\|z^{k+1}-z^{k}\\|^{2}+\alpha\\|z^{k}-z^{k-1}\\|^{2}+a\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}$ $\displaystyle\geq c_{1}\\|y^{k+1}-y^{k}\\|^{2},$ (63) where $c_{1}:=\min\left\\{\frac{1-\rho\gamma}{2\gamma}-2\alpha(\gamma L_{f}+1)^{2},\alpha,a\beta^{-1}\right\\}$ by $\frac{1}{2\gamma}(\frac{2}{\eta}-1)-3\alpha\geq\alpha$. This yields $(P1)$ for MEAL. (b) Establishing $(P2)$: Note that ${\cal P}_{\mathrm{meal}}(y)=f(x)+\langle\lambda,Ax-b\rangle+\frac{\beta}{2}\\|Ax-b\\|^{2}+\frac{1}{2\gamma}\\|x-z\\|^{2}+3\alpha\\|z-\hat{z}\\|^{2}$. The optimality condition from the update of $x^{k+1}$ in (5) is $\displaystyle 0\in\partial f(x^{k+1})+A^{T}\lambda^{k+1}+\gamma^{-1}(x^{k+1}-z^{k}),$ which implies $\gamma^{-1}(z^{k}-z^{k+1})+A^{T}(\lambda^{k+1}-\lambda^{k})\in\partial_{x}{\cal P}_{\mathrm{meal}}(y^{k+1}).$ From the update of $z^{k+1}$ in (5), $z^{k+1}-x^{k+1}=-(1-\eta)\eta^{-1}(z^{k+1}-z^{k})$ and thus $\displaystyle\partial_{z}{\cal P}_{\mathrm{meal}}(y^{k+1})=\gamma^{-1}(z^{k+1}-x^{k+1})+6\alpha(z^{k+1}-z^{k})=\left(6\alpha-\frac{1-\eta}{\eta\gamma}\right)(z^{k+1}-z^{k}).$ The update of $\lambda^{k+1}$ in (5) yields $\partial_{\lambda}{\cal P}_{\mathrm{meal}}(y^{k+1})=Ax^{k+1}-b=\beta^{-1}(\lambda^{k+1}-\lambda^{k}).$ Moreover, it is easy to show $\partial_{\hat{z}}{\cal P}_{\mathrm{meal}}(y^{k+1})=6\alpha(z^{k}-z^{k+1}).$ Thus, let $v^{k+1}:=\left(\begin{array}[]{c}\gamma^{-1}(z^{k}-z^{k+1})+A^{T}(\lambda^{k+1}-\lambda^{k})\\\ \left(6\alpha-\frac{1-\eta}{\eta\gamma}\right)(z^{k+1}-z^{k})\\\ \beta^{-1}(\lambda^{k+1}-\lambda^{k})\\\ 6\alpha(z^{k}-z^{k+1})\end{array}\right),$ which obeys $v^{k+1}\in\partial{\cal P}_{\mathrm{meal}}(y^{k+1})$ and $\displaystyle\\|v^{k+1}\\|$ $\displaystyle\leq\left(\gamma^{-1}+\left\|6\alpha-\frac{1-\eta}{\eta\gamma}\right\|+6\alpha\right)\\|z^{k+1}-z^{k}\\|+\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|+\\|A^{T}(\lambda^{k+1}-\lambda^{k})\\|$ $\displaystyle\leq\left(\gamma^{-1}+\left\|6\alpha-\frac{1-\eta}{\eta\gamma}\right\|+6\alpha\right)\\|z^{k+1}-z^{k}\\|+\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|$ $\displaystyle+(L_{f}+\gamma^{-1})\\|x^{k+1}-x^{k}\\|+\gamma^{-1}\\|\hat{z}^{k+1}-\hat{z}^{k}\\|,$ where the second inequality is due to (46). This yields $(P2)$ for MEAL. (c) Establishing $(P3)$: $(P3)$ follows from the boundedness assumption of $\\{y^{k}\\}$, and the convergence of $\\{{\cal P}_{\mathrm{meal}}(y^{k})\\}$ is implied by $(P1)$. This finishes the proof. ### 5.3 Proof for Convergence of iMEAL In this subsection, we present the proof of Theorem 3.2 for iMEAL (7). ###### Proof (of Theorem 3.2) We first show the $o(1/\sqrt{k})$ rate of convergence under Assumption 2(b) and then the convergence rate result under Assumption 2(c). (a) In this case, we use a fixed $\beta_{k}=\beta$ and thus $\alpha_{k}=\alpha$. Substituting the inequality in Lemma 5(a) into (59) in Lemma 8 yields $\displaystyle{\cal P}_{\beta}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta}(z^{k},\lambda^{k})\\|^{2}$ (64) $\displaystyle+\left(\frac{1-\gamma\rho}{2\gamma}-3\alpha(1+\gamma L_{f})^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}+\langle s^{k},x^{k}-x^{k+1}\rangle-3\alpha\gamma^{2}(\epsilon_{k}+\epsilon_{k-1})^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-3\alpha\\|z^{k}-z^{k-1}\\|^{2}.$ Let $\delta:=2\left(\frac{(1-\gamma\rho)}{2\gamma}-3\alpha(1+\gamma L_{f})^{2}\right)$. By the assumption $0<\alpha<\min\big{\\{}\frac{1-\gamma\rho}{6\gamma(1+\gamma L_{f})^{2}}$, $\frac{1}{12\gamma}(\frac{2}{\eta}-1)\Big{\\}}$, we have $\delta>0$ and further $\displaystyle\langle s^{k},x^{k}-x^{k+1}\rangle\geq-\frac{\delta}{2}\\|x^{k+1}-x^{k}\\|^{2}-\frac{1}{2\delta}\\|s^{k}\\|^{2}\geq-\frac{\delta}{2}\\|x^{k+1}-x^{k}\\|^{2}-\frac{1}{2\delta}(\epsilon_{k}+\epsilon_{k-1})^{2}.$ Substituting this into (64) and noting the definition (32) of ${\cal E}_{\mathrm{imeal}}^{k}$, we have $\displaystyle{\cal E}_{\mathrm{imeal}}^{k}-{\cal E}_{\mathrm{imeal}}^{k+1}\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta}(z^{k},\lambda^{k})\\|^{2}-(3\alpha\gamma^{2}+\frac{1}{2\delta})(\epsilon_{k}+\epsilon_{k-1})^{2},$ which yields claim (a) by the assumption $\sum_{k=1}^{\infty}(\epsilon_{k})^{2}<+\infty$ and Lemma 1. (b) Then we establish claim (b) under Assumption 2(c). Substituting the inequality in Lemma 5(b) into (59) in Lemma 8 yields $\displaystyle{\cal P}_{\beta_{k}}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta_{k+1}}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{1}{2}\gamma\eta(1-\eta/2)\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}$ $\displaystyle+\left(\frac{(1-\gamma\rho)}{2\gamma}-4\alpha_{k}\right)\\|x^{k+1}-x^{k}\\|^{2}+\langle s^{k},x^{k}-x^{k+1}\rangle-4\gamma^{2}\alpha_{k}(\epsilon_{k}+\epsilon_{k-1})^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-4\alpha_{k}\\|z^{k}-z^{k-1}\\|^{2}-16\alpha_{k}\gamma^{2}\hat{L}_{f}^{2}.$ (65) Let $\hat{\alpha}^{}:=\min\left\\{\frac{1-\rho\gamma}{8\gamma},\frac{1}{16\gamma}(\frac{2}{\eta}-1)\right\\}$ and $\tilde{\delta}:=2\left(\frac{(1-\gamma\rho)}{2\gamma}-4\hat{\alpha}^{}\right)>0$. We have $\displaystyle\langle s^{k},x^{k}-x^{k+1}\rangle\geq-\frac{\tilde{\delta}}{2}\\|x^{k+1}-x^{k}\\|^{2}-\frac{1}{2\tilde{\delta}}\\|s^{k}\\|^{2}\geq-\frac{\tilde{\delta}}{2}\\|x^{k+1}-x^{k}\\|^{2}-\frac{1}{2\tilde{\delta}}(\epsilon_{k}+\epsilon_{k-1})^{2}.$ Substituting this into (5.3), and by the definition (33) of $\tilde{\cal E}_{\mathrm{imeal}}^{k}$ and setting of $\alpha_{k}$, we have $\displaystyle\tilde{\cal E}_{\mathrm{imeal}}^{k}-\tilde{\cal E}_{\mathrm{imeal}}^{k+1}$ $\displaystyle\geq\frac{1}{2}\gamma(1-\eta/2)\eta\\|\nabla\phi_{\beta_{k}}(z^{k},\lambda^{k})\\|^{2}-(4\alpha_{k}\gamma^{2}+\frac{1}{2\tilde{\delta}})(\epsilon_{k}+\epsilon_{k-1})^{2}-16\alpha_{k}\gamma^{2}\hat{L}_{f}^{2},$ which yields claim (b) by the assumption $\sum_{k=1}^{\infty}(\epsilon_{k})^{2}<+\infty$ and Lemma 2. ### 5.4 Proofs for Convergence of LiMEAL Now, we show proofs of main convergence theorems for LiMEAL (9). #### 5.4.1 Proof of Theorem 4.1 ###### Proof We first establish claim (a) and then claim (b) under the associated assumptions. (a) In this case, a fixed $\beta_{k}$ is used. Substituting (53) into (60) yields $\displaystyle{\cal P}_{\beta}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta}(x^{k+1},z^{k+1},\lambda^{k+1})\geq\frac{1}{4}\gamma(1-\eta/2)\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2}$ $\displaystyle+\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-\frac{1}{4}\gamma(2-\eta)\eta L_{h}^{2}-3(1+\gamma L_{g})^{2}\alpha\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\frac{1}{4\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-3\alpha(\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}+\\|z^{k}-z^{k-1}\\|^{2}).$ By the definition (41) of ${\cal E}^{k}_{\mathrm{limeal}}$, the above inequality implies $\displaystyle{\cal E}_{\mathrm{limeal}}^{k}-{\cal E}_{\mathrm{limeal}}^{k+1}\geq\frac{1}{4}\gamma(1-\eta/2)\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2}+\left(\frac{1}{4\gamma}(\frac{2}{\eta}-1)-3\alpha\right)\\|z^{k+1}-z^{k}\\|^{2}$ (66) $\displaystyle+\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-\frac{1}{4}\gamma(2-\eta)\eta L_{h}^{2}-3\alpha\left((1+\gamma L_{g})^{2}+\gamma^{2}L_{h}^{2}\right)\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle\geq\frac{1}{4}\gamma(1-\eta/2)\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2},$ where the second inequality holds under the conditions in Theorem 4.1(a). This shows the claim (a) by Lemma 1 and the lower boundedness of $\\{{\cal E}_{\mathrm{limeal}}^{k}\\}$. (b) Similarly, substituting (54) into (60) and using the definitions of ${\alpha}_{k}$ in (26) and $\tilde{\cal E}^{k}_{\mathrm{limeal}}$ in (42), we obtain $\displaystyle\tilde{\cal E}_{\mathrm{limeal}}^{k}-\tilde{\cal E}_{\mathrm{limeal}}^{k+1}$ $\displaystyle\geq\frac{1}{4}\gamma(1-\eta/2)\eta_{k}\\|g_{\mathrm{limeal}}^{k}\\|^{2}-16{\alpha}_{k}\gamma^{2}\hat{L}_{g}^{2}+\left(\frac{1}{4\gamma}(\frac{2}{\eta}-1)-4{\alpha}_{k+1}\right)\\|z^{k+1}-z^{k}\\|^{2}$ $\displaystyle+\left(\frac{(1-\gamma(\rho_{g}+L_{h}))}{2\gamma}-\frac{1}{4}\gamma(2-\eta)\eta L_{h}^{2}-4\alpha_{k}-4\gamma^{2}L_{h}^{2}\alpha_{k+1}\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle\geq\frac{1}{4}c\gamma\eta\\|g_{\mathrm{limeal}}^{k}\\|^{2}-16{\alpha}_{k}\gamma^{2}\hat{L}_{g}^{2},$ where the second inequality is due to the settings of parameters presented in Theorem 4.1(b). This inequality shows claim (b) by Lemma 2 and the lower boundedness of $\\{\tilde{\cal E}_{\mathrm{limeal}}^{k}\\}$. #### 5.4.2 Proof of Proposition 3 ###### Proof By Lemma 3, we only need to verify conditions $(P1)$-$(P3)$ hold for LiMEAL. (a) Establishing $(P1)$: Similar to the proof of Theorem 3.1, let $a:=\frac{\gamma\eta(2-\eta)}{4\beta}$. Then $\frac{1+a}{\beta c_{\gamma,A}}=\alpha$, where $\alpha$ is defined in (27). Substituting (53) into (61) with fixed $\beta_{k}$ yields $\displaystyle{\cal P}_{\beta}(x^{k},z^{k},\lambda^{k})-{\cal P}_{\beta}(x^{k+1},z^{k+1},\lambda^{k+1})$ $\displaystyle\geq\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-3\alpha(1+\gamma L_{g})^{2}\right)\\|x^{k+1}-x^{k}\\|^{2}-3\alpha\gamma^{2}L_{h}^{2}\\|x^{k}-x^{k-1}\\|^{2}$ $\displaystyle+\frac{1}{2\gamma}(\frac{2}{\eta}-1)\\|z^{k+1}-z^{k}\\|^{2}-3\alpha\\|z^{k}-z^{k-1}\\|^{2}+a\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}.$ By the definition (43) of ${\cal P}_{\mathrm{limeal}}$, the above inequality implies $\displaystyle{\cal P}_{\mathrm{limeal}}(y^{k})-{\cal P}_{\mathrm{limeal}}(y^{k+1})$ $\displaystyle\geq\left(\frac{1-\gamma(\rho_{g}+L_{h})}{2\gamma}-4\alpha\left((1+\gamma L_{g})^{2}+\gamma^{2}L_{h}^{2}\right)\right)\\|x^{k+1}-x^{k}\\|^{2}$ $\displaystyle+\left(\frac{1}{2\gamma}\left(\frac{2}{\eta}-1\right)-4\alpha\right)\\|z^{k+1}-z^{k}\\|^{2}+a\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|^{2}$ $\displaystyle+\alpha\left(\gamma^{2}L_{h}^{2}\\|\hat{x}^{k+1}-\hat{x}^{k}\\|^{2}+\\|\hat{z}^{k+1}-\hat{z}^{k}\\|^{2}\right),$ which, with the assumptions on the parameters, implies $(P1)$ for LiMEAL. (b) Establishing $(P2)$: Note that ${\cal P}_{\mathrm{limeal}}(y)=f(x)+\langle\lambda,Ax-b\rangle+\frac{\beta}{2}\\|Ax-b\\|^{2}+\frac{1}{2\gamma}\\|x-z\\|^{2}+4\alpha\gamma^{2}L_{h}^{2}\\|x-\hat{x}\\|^{2}+4\alpha\\|z-\hat{z}\\|^{2}$. The update of $x^{k+1}$ in (9) has the optimality condition $\displaystyle 0\in\partial g(x^{k+1})+\nabla h(x^{k})+A^{T}\lambda^{k+1}+\gamma^{-1}(x^{k+1}-z^{k}),$ which implies $\displaystyle(\nabla h(x^{k+1})-\nabla h(x^{k}))+8\gamma^{2}L_{h}^{2}\alpha(x^{k+1}-x^{k})$ $\displaystyle+\gamma^{-1}(z^{k}-z^{k+1})+A^{T}(\lambda^{k+1}-\lambda^{k})\in\partial_{x}{\cal P}_{\mathrm{limeal}}(y^{k+1}).$ The derivations for the other terms are straightforward and similar to those in the proof of Proposition 1. We directly show the final estimate: for some $v^{k+1}\in\partial{\cal P}_{\mathrm{limeal}}(y^{k+1})$, $\displaystyle\\|v^{k+1}\\|$ $\displaystyle\leq\left(L_{h}+L_{g}+\gamma^{-1}+16\alpha\gamma^{2}L_{h}^{2}\right)\\|x^{k+1}-x^{k}\\|$ $\displaystyle+\left(\gamma^{-1}+\left\|8\alpha-\frac{1-\eta}{\eta}\right\|+8\alpha\right)\\|z^{k+1}-z^{k}\\|$ $\displaystyle+\beta^{-1}\\|\lambda^{k+1}-\lambda^{k}\\|+L_{h}\\|\hat{x}^{k+1}-\hat{x}^{k}\\|+\gamma^{-1}\\|\hat{z}^{k+1}-\hat{z}^{k}\\|,$ which yields $(P2)$ for LiMEAL. (c) Establishing $(P3)$: $(P3)$ follows from the boundedness assumption of $\\{y^{k}\\}$ and the convergence of $\\{{\cal P}_{\mathrm{limeal}}(y^{k})\\}$ by $(P1)$. This finishes the proof. ## 6 Discussions on Boundedness and Related Work In this section, we discuss how to ensure the bounded sequences and then compare our results to related other work. ### 6.1 Discussions on Boundedness of Sequence Theorem 3.1 imposes the condition of lower boundedness of $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$ and Proposition 1 does with boundedness of the generated sequence $\\{(x^{k},z^{k},\lambda^{k})\\}$. In this section, we provide some sufficient conditions to guarantee the former and then the latter boundedness conditions. Besides the $\rho$-weak convexity of $f$ (implying the curvature of $f$ is lower bounded by $\rho$), we impose the coerciveness on the constrained problem (1) as follows. ###### Assumption 4 (Coercivity) The minimal value $f^{}:=\inf_{x\in{\cal X}}f(x)$ is finite (recall ${\cal X}:=\\{x:Ax=b\\}$), and $f$ is coercive over the set ${\cal X}$, that is, $f(x)\rightarrow\infty$ if $x\in{\cal X}$ and $\\|x\\|\rightarrow\infty$. The coercive assumption is a common condition used to obtain the boundedness of the sequence, for example, used in (Wang19, , Assumption A1) for the nonconvex ADMM. Particularly, let $(x^{0},z^{0},\lambda^{0})$ be a finite initial guess of MEAL and $\displaystyle{\cal E}^{0}:={\cal E}_{\mathrm{meal}}^{1}<+\infty.$ (67) By Assumption 4, if $x\in{\cal X}$ and $f(x)\leq{\cal E}^{0}$, then there exists a positive constant ${\cal B}_{0}$ (possibly depending on ${\cal E}^{0}$) such that $\\|x\\|\leq{\cal B}_{0}.$ Define another positive constant as $\displaystyle{\cal B}_{1}:={\cal B}_{0}+\sqrt{2\rho^{-1}\cdot\max\\{0,{\cal E}^{0}-f^{}\\}}.$ (68) Given a $\gamma\in(0,1/\rho)$ and $z\in\mathbb{R}^{n}$ with $\\|z\\|\leq{\cal B}_{1}$ and $u\in\mathrm{Im}(A)$, we define $\displaystyle x(u;z):=\operatorname{argmin}_{\\{x:Ax=u\\}}\left\\{f(x)+\frac{1}{2\gamma}\\|x-z\\|^{2}\right\\}.$ (69) Since $f$ is $\rho$-weakly convex by Assumption 2(a), then for any $\gamma\in(0,1/\rho)$, the function $f(x)+\frac{1}{2\gamma}\\|x-z\\|^{2}$ is strongly convex with respect to $x$, and thus the above $x(u;z)$ is well- defined and unique for any given $z\in\mathbb{R}^{n}$ and $u\in\mathrm{Im}(A)$. Motivated by (Boyd04, , Ch 5.6.3), we impose some local stability on $x(u;z)$ defined in (69). ###### Assumption 5 (Local stability) For any given $z\in\mathbb{R}^{n}$ with $\\|z\\|\leq{\cal B}_{1}$, there exist a $\delta>0$ and a finite positive constant $\bar{M}$ (possibly depending on $A$, ${\cal B}_{1}$ and $\delta$) such that $\\|x(u;z)-x(b;z)\\|\leq\bar{M}\\|u-b\\|,\ \forall u\in\mathrm{Im}(A)\cap\\{v:\\|v-b\\|\leq\delta\\}.$ The above local stability assumption is also related to the Lipschitz sub- minimization path assumption suggested in (Wang19, , Assumption A3). As discussed in Wang19 , the Lipschitz sub-minimization path assumption relaxes the more stringent full-rank assumption used in the literature (see the discussions in (Wang19, , Sections 2.2 and 4.1) and references therein). As $\\{z\in\mathbb{R}^{n}:\\|z\\|\leq{\cal B}_{1}\\}$ is a compact set, $\bar{M}$ can be taken as the supremum of these stability constants over this compact set. Based on Assumption 5, we have the following lemma. ###### Lemma 10 Let $\\{(x^{k},z^{k},\lambda^{k})\\}$ be the sequence generated by MEAL (5) with fixed $\beta>0$ and $\eta>0$. If $\gamma\in(0,1/\rho)$, $\\|z^{k}\\|\leq{\cal B}_{1}$ and $\\|Ax^{k+1}-b\\|\leq\delta$, there holds $\\|x^{k+1}-x(b;z^{k})\\|\leq\bar{M}\\|Ax^{k+1}-b\\|,\ \forall k\in\mathbb{N}.$ ###### Proof Let $u^{k+1}=Ax^{k+1}$. By the update of $x^{k+1}$ in (5), there holds ${\cal P}_{\beta}(x^{k+1},z^{k},\lambda^{k})\leq{\cal P}_{\beta}(x(u^{k+1};z^{k}),z^{k},\lambda^{k}).$ Noting that $Ax(u^{k+1};z^{k})=Ax^{k+1}$ due to its definition in (69), the above inequality implies $f(x^{k+1})+\frac{1}{2\gamma}\\|x^{k+1}-z^{k}\\|^{2}\leq f(x(u^{k+1};z^{k}))+\frac{1}{2\gamma}\\|x(u^{k+1};z^{k})-z^{k}\\|^{2}.$ By the definition of $x(u^{k+1};z^{k})$ in (69) again and noting that $Ax^{k+1}=u^{k+1}$, we have $f(x^{k+1})+\frac{1}{2\gamma}\\|x^{k+1}-z^{k}\\|^{2}\geq f(x(u^{k+1};z^{k}))+\frac{1}{2\gamma}\\|x(u^{k+1};z^{k})-z^{k}\\|^{2}.$ These two inequalities imply $f(x^{k+1})+\frac{1}{2\gamma}\\|x^{k+1}-z^{k}\\|^{2}=f(x(u^{k+1};z^{k}))+\frac{1}{2\gamma}\\|x(u^{k+1};z^{k})-z^{k}\\|^{2},$ which yields $x^{k+1}=x(u^{k+1};z^{k})=x(Ax^{k+1};z^{k})$ by the strong convexity of function $f(x)+\frac{1}{2\gamma}\\|x-z^{k}\\|^{2}$ for any $\gamma\in(0,1/\rho)$ and thus the uniqueness of $x(u^{k+1};z^{k})$. Then by Assumption 5, we yield the desired result. Based on the above assumptions, we establish the lower boundedness of $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$ and the boundedness of $\\{(x^{k},z^{k},\lambda^{k})\\}$ as follows. ###### Proposition 4 Let $\\{(x^{k},z^{k},\lambda^{k})\\}_{k\in\mathbb{N}}$ be a sequence generated by MEAL (5) with a finite initial guess $(x^{0},z^{0},\lambda^{0})$ such that $\\|z^{0}\\|\leq{\cal B}_{1}$, where ${\cal B}_{1}$ is defined in (68). Suppose that Assumptions 1, 2(a)-(b) and 4 hold and further Assumption 5 holds with some $0<\bar{M}<\frac{2}{\sqrt{\sigma_{\min}(A^{T}A)}}$. If $\gamma\in(0,\rho^{-1})$, $\eta\in(0,2)$ and $\beta>\max\left\\{\frac{1+\sqrt{1+\eta(2-\eta)\gamma c_{\gamma,A}\alpha_{\max}}}{2c_{\gamma,A}\alpha_{\max}},\frac{a_{2}+\sqrt{a_{2}^{2}+4a_{1}a_{3}}}{2a_{1}}\right\\},$ where $\alpha_{\max}:=\min\left\\{\frac{1-\gamma\rho}{4\gamma(1+\gamma L_{f})^{2}},\frac{1}{8\gamma}(\frac{2}{\eta}-1)\right\\}$, $c_{\gamma,A}=\gamma^{2}\sigma_{\min}(A^{T}A)$, $a_{1}=4-\bar{M}^{2}\sigma_{\min}(A^{T}A)$, $a_{2}=4(\bar{L}+\gamma^{-1})\bar{M}^{2}-\gamma\eta(2-\eta)$, $a_{3}=(1+\gamma\bar{L})\eta(2-\eta)\bar{M}^{2}$ and $\bar{L}=\rho+2L_{f}$, then the following hold: 1. (a) $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$ is lower bounded; 2. (b) $\\{(x^{k},z^{k})\\}$ is bounded; and 3. (c) if further $\lambda^{0}\in\mathrm{Null}(A^{T})$ (the null space of $A^{T}$) and $\\|\nabla{\cal M}_{\gamma,f}(w^{1})\\|$ is finite with $w^{1}=z^{0}-\gamma A^{T}\lambda^{1}$, then $\\{\lambda^{k}\\}$ is bounded. ###### Proof In order to prove this proposition, we firstly establish the following claim for sufficiently large $k$: Claim A: If $\\|z^{k-1}\\|\leq{\cal B}_{1},\\|Ax^{k}-b\\|\leq\delta,\forall k\geq k_{0}$ for some sufficiently large $k_{0}$, then ${\cal E}_{\mathrm{meal}}^{k}\geq f^{}$, and $\\|z^{k}\\|\leq{\cal B}_{1}$ and $\\|x^{k}\\|\leq{\cal B}_{2}$. By Theorem 3.1(a), such $k_{0}$ does exist due to the lower boundedness of $\\{{\cal E}_{meal}^{k}\\}$ for all finite $k$ and thus $\xi_{\mathrm{meal}}^{k}\leq\hat{c}/\sqrt{k}$ for some constant $\hat{c}>0$ (implying $\\|Ax^{k}-b\\|$ is sufficiently small with a sufficiently large $k$ ). In the next, we show Claim A. By the definition (24) of ${\cal E}_{\mathrm{meal}}^{k}$, we have $\displaystyle{\cal E}_{\mathrm{meal}}^{k}$ $\displaystyle=f(x^{k})+\langle\lambda^{k},Ax^{k}-b\rangle+\frac{\beta}{2}\\|Ax^{k}-b\\|^{2}+\frac{1}{2\gamma}\\|x^{k}-z^{k}\\|^{2}+2\alpha\\|z^{k}-z^{k-1}\\|^{2}$ $\displaystyle=f(x^{k})+\langle A^{T}\lambda^{k},x^{k}-\bar{x}^{k}\rangle+\frac{\beta}{2}\\|Ax^{k}-b\\|^{2}+\frac{1}{2\gamma}\\|x^{k}-z^{k}\\|^{2}+2\alpha\\|z^{k}-z^{k-1}\\|^{2},$ where $\bar{x}^{k}:=x(b;z^{k-1})$ as defined in (69). Let $\bar{\lambda}^{k}$ be the associated optimal Lagrangian multiplier of $\bar{x}^{k}$ and $\bar{w}^{k}=z^{k-1}-\gamma A^{T}\bar{\lambda}^{k}$. Then we have $\bar{x}^{k}=\mathrm{Prox}_{\gamma,f}(\bar{w}^{k}),$ and $\nabla{\cal M}_{\gamma,f}(\bar{w}^{k})\in\partial f(\bar{x}^{k})$. By (51) in the proof of Lemma 4, we have $\displaystyle A^{T}\lambda^{k}=-\nabla{\cal M}_{\gamma,f}(w^{k})-\gamma^{-1}(x^{k}-z^{k-1}),$ and $\nabla{\cal M}_{\gamma,f}(w^{k})\in\partial f(x^{k})$, where $w^{k}=z^{k-1}-\gamma A^{T}\lambda^{k}$. Substituting the above equation into the previous equality yields $\displaystyle{\cal E}_{\mathrm{meal}}^{k}$ $\displaystyle=f(x^{k})+\langle\nabla{\cal M}_{\gamma,f}(w^{k}),\bar{x}^{k}-x^{k}\rangle+\frac{\beta}{2}\\|Ax^{k}-b\\|^{2}$ (70) $\displaystyle+\gamma^{-1}\langle x^{k}-z^{k-1},\bar{x}^{k}-x^{k}\rangle+\frac{1}{2\gamma}\\|x^{k}-z^{k}\\|^{2}+2\alpha\\|z^{k}-z^{k-1}\\|^{2}.$ Noting that $\nabla{\cal M}_{\gamma,f}(\bar{w}^{k})\in\partial f(\bar{x}^{k})$ and by the $\rho$-weak convexity of $f$, we have $f(x^{k})\geq f(\bar{x}^{k})+\langle\nabla{\cal M}_{\gamma,f}(\bar{w}^{k}),x^{k}-\bar{x}^{k}\rangle-\frac{\rho}{2}\\|x^{k}-\bar{x}^{k}\\|^{2},$ which implies $\displaystyle f(x^{k})+\langle\nabla{\cal M}_{\gamma,f}(w^{k}),\bar{x}^{k}-x^{k}\rangle$ $\displaystyle\geq f(\bar{x}^{k})-\frac{\rho}{2}\\|\bar{x}^{k}-x^{k}\\|^{2}-\langle\nabla{\cal M}_{\gamma,f}(\bar{w}^{k})-\nabla{\cal M}_{\gamma,f}(w^{k}),\bar{x}^{k}-x^{k}\rangle$ $\displaystyle\geq f(\bar{x}^{k})-\frac{\rho}{2}\\|\bar{x}^{k}-x^{k}\\|^{2}-\\|\nabla{\cal M}_{\gamma,f}(\bar{w}^{k})-\nabla{\cal M}_{\gamma,f}(w^{k})\\|\cdot\\|\bar{x}^{k}-x^{k}\\|.$ By the implicit Lipschitz subgradient assumption (i.e., Assumption 2 (b)) and the definition of $\bar{L}:=\rho+2L_{f}$, the above inequality yields $\displaystyle f(x^{k})+\langle\nabla{\cal M}_{\gamma,f}(w^{k}),\bar{x}^{k}-x^{k}\rangle\geq f(\bar{x}^{k})-\frac{\bar{L}}{2}\\|\bar{x}^{k}-x^{k}\\|^{2}.$ (71) Moreover, it is easy to show that $\displaystyle\gamma^{-1}\langle x^{k}-z^{k-1},\bar{x}^{k}-x^{k}\rangle+\frac{1}{2\gamma}\\|x^{k}-z^{k}\\|^{2}+2\alpha\\|z^{k}-z^{k-1}\\|^{2}$ (72) $\displaystyle=\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}-\frac{1}{2\gamma}\\|\bar{x}^{k}-x^{k}\\|^{2}+\gamma^{-1}\langle z^{k}-z^{k-1},\bar{x}^{k}-x^{k}\rangle+2\alpha\\|z^{k}-z^{k-1}\\|^{2}$ $\displaystyle=\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}-\left(\frac{1}{2\gamma}+\frac{1}{8\alpha\gamma^{2}}\right)\\|\bar{x}^{k}-x^{k}\\|^{2}+2\alpha\left\\|(z^{k}-z^{k-1})+\frac{1}{4\alpha\gamma}(\bar{x}^{k}-x^{k})\right\\|^{2}.$ Substituting (71)-(72) into (70) and by Lemma 10, we have $\displaystyle{\cal E}_{\mathrm{meal}}^{k}$ $\displaystyle\geq f(\bar{x}^{k})+\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}+2\alpha\left\\|(z^{k}-z^{k-1})+\frac{1}{4\alpha\gamma}(\bar{x}^{k}-x^{k})\right\\|^{2}$ $\displaystyle+\frac{1}{2}\left[\beta-\left(\frac{1}{4\alpha\gamma^{2}}+\bar{L}+\gamma^{-1}\right)\bar{M}^{2}\right]\\|Ax^{k}-b\\|^{2}$ $\displaystyle\geq f(\bar{x}^{k})+\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}+2\alpha\left\\|(z^{k}-z^{k-1})+\frac{1}{4\alpha\gamma}(\bar{x}^{k}-x^{k})\right\\|^{2}$ (73) $\displaystyle\geq f^{}+\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}+2\alpha\left\\|(z^{k}-z^{k-1})+\frac{1}{4\alpha\gamma}(\bar{x}^{k}-x^{k})\right\\|^{2}$ (74) $\displaystyle>-\infty,$ (75) where the second inequality follows from the definition of $\alpha=\frac{2\beta+\gamma\eta(1-\eta/2)}{2\gamma^{2}\sigma_{\min}(A^{T}A)\beta^{2}}$ and the condition on $\beta$, the third inequality holds for $\bar{x}^{k}:=x(b;z^{k-1})$ and thus $A\bar{x}^{k}=b$ and $f(\bar{x}^{k})\geq f^{}$, and the final inequality is due to Assumption 4. The above inequality yields the lower boundedness of $\\{{\cal E}_{\mathrm{meal}}^{k}\\}$ in Claim A. Thus, clam (a) in this proposition holds. Then, we show the boundedness of $\\{(x^{k},z^{k})\\}$ in Claim A. By (73) and (5.2.1), we have $\displaystyle f(\bar{x}^{k})\leq{\cal E}^{0}:={\cal E}_{\mathrm{meal}}^{1},$ which implies $\\|\bar{x}^{k}\\|\leq{\cal B}_{0}$ by Assumption 4. By (74) and the condition on $\gamma\in(0,\rho^{-1})$, we have $f^{}+\frac{\rho}{2}\\|\bar{x}^{k}-z^{k}\\|^{2}\leq f^{}+\frac{1}{2\gamma}\\|\bar{x}^{k}-z^{k}\\|^{2}\leq{\cal E}^{0},$ which implies $\displaystyle\\|z^{k}\\|\leq{\cal B}_{0}+\sqrt{2({\cal E}^{0}-f^{})/\rho}={\cal B}_{1}.$ By (74) again, we have $\left\\|(z^{k}-z^{k-1})+\frac{1}{4\alpha\gamma}(\bar{x}^{k}-x^{k})\right\\|^{2}\leq\frac{{\cal E}^{0}-f^{}}{2\alpha},$ which, together with these existing bounds $\\|z^{k-1}\\|\leq{\cal B}_{1}$, $\\|z^{k}\\|\leq{\cal B}_{1}$ and $\\|\bar{x}^{k}\\|\leq{\cal B}_{0}$, yields $\displaystyle\\|x^{k}\\|\leq{\cal B}_{0}+4\alpha\gamma\left(2{\cal B}_{1}+\sqrt{\frac{{\cal E}^{0}-f^{}}{2\alpha}}\right)=:{\cal B}_{2}.$ (76) Thus, we have shown Claim A. Recursively, we can show that $\\{x^{k}\\}$ and $\\{z^{k}\\}$ are respectively bounded by ${\cal B}_{2}$ and ${\cal B}_{1}$ for any $k\geq 1$, that is, claim (b) in this proposition holds. In the following, we show claim (c) of this proposition. By the update of $\lambda^{k+1}$ in (5), it is easy to show $\lambda^{k}=\lambda^{0}+\hat{\lambda}^{k}$, where $\hat{\lambda}^{k}=\beta\sum_{t=1}^{k}(Ax^{t}-b)\in\mathrm{Im}(A)$ by Assumption 1. Furthermore, by the assumption that $\lambda^{0}\in\mathrm{Null}(A^{T})$, we have $\displaystyle\langle\lambda^{0},\hat{\lambda}^{k}\rangle=0,\ \forall k\geq 1.$ (77) By (51), for any $k\geq 1$, we have $\displaystyle A^{T}\lambda^{k}=-(\nabla{\cal M}_{\gamma,f}(w^{k})-\nabla{\cal M}_{\gamma,f}(w^{1}))-\nabla{\cal M}_{\gamma,f}(w^{1})-\gamma^{-1}(x^{k}-z^{k-1}),$ where $w^{k}=z^{k-1}-\gamma A^{T}\lambda^{k}$. By Assumption 2(b) and the boundedness of $\\{(x^{k},z^{k})\\}$ shown before, the above equation implies $\displaystyle\\|A^{T}{\lambda}^{k}\\|$ $\displaystyle\leq L_{f}\\|x^{k}-x^{1}\\|+\\|\nabla{\cal M}_{\gamma,f}(w^{1})\\|+\gamma^{-1}\\|x^{k}-z^{k-1}\\|$ $\displaystyle\leq\gamma^{-1}{\cal B}_{1}+(2L_{f}+\gamma^{-1}){\cal B}_{2}+\\|\nabla{\cal M}_{\gamma,f}(w^{1})\\|<+\infty.$ By the relation $\lambda^{k}=\lambda^{0}+\hat{\lambda}^{k}$ and (77), the above inequality implies $\displaystyle\\|A^{T}\hat{\lambda}^{k}\\|\leq\gamma^{-1}{\cal B}_{1}+(2L_{f}+\gamma^{-1}){\cal B}_{2}+\\|\nabla{\cal M}_{\gamma,f}(w^{1})\\|.$ Since $\hat{\lambda}^{k}\in\mathrm{Im}(A)$, the above inequality implies $\displaystyle\\|\hat{\lambda}^{k}\\|\leq\tilde{\sigma}_{\min}^{-1/2}(A^{T}A)\\|A^{T}\hat{\lambda}^{k}\\|\leq\tilde{\sigma}_{\min}^{-1/2}(A^{T}A)\left[\gamma^{-1}{\cal B}_{1}+(2L_{f}+\gamma^{-1}){\cal B}_{2}+\\|\nabla{\cal M}_{\gamma,f}(w^{1})\\|\right],$ which yields the boundedness of $\\{\lambda^{k}\\}$ by the triangle inequality. This finishes the proof. The proof idea of claim (c) of this proposition is motivated by the proof of (Zhang-Luo18, , Lemma 3.1). Based on Proposition 4, we show the lower boundedness of the Lypunov function sequence and the boundedness of the sequence generated by MEAL. Following the similar analysis of this section, we can obtain the similar boundeness results for both iMEAL and LiMEAL. ### 6.2 Discussions on Related Work When compared to the tightly related work Hajinezhad-Hong19 ; Hong17-Prox-PDA ; Jiang19 ; Rockafellar76-PALM ; Xie-Wright19 ; Zhang-Luo20 ; Zhang-Luo18 , this paper provides some slightly stronger convergence results under weaker conditions. The detailed discussions and comparisons with these works are shown as follows and presented in Tables 1 and 2. Table 1: Convergence results of our and related algorithms for problem (1) Algorithm \| MEAL (our) \| iMEAL (our) \| Prox-PDA Hong17-Prox-PDA \| Prox-ALM Xie-Wright19 ---\|---\|---\|---\|--- Assumption \| $f$: weakly convex, imp-Lip or imp-bound \| $\nabla f$: Lipschitz Iteration \| imp-Lip: $o(\varepsilon^{-2})$ \| imp-Lip: $o(\varepsilon^{-2})$ \| ${\cal O}(\varepsilon^{-2})$ \| ${\cal O}(\varepsilon^{-2})$ complexity \| imp-bound: ${\cal O}(\varepsilon^{-2})$ \| imp-bound: ${\cal O}(\varepsilon^{-2})$ Global \| $\checkmark$ under KŁ \| – \| – \| – Convergence $\bullet$ imp-Lip: the implicit Lipschitz subgradient assumption 2(b); $\bullet$ imp-bound: the implicit bounded subgradient assumption 2(c); $\bullet$ Xie-Wright19 considers a nonlinear equality constraints $c(x)=0$ where $\nabla c$ is Lipschitz and bounded. Table 2: Convergence results of our and related algorithms for the composite optimization problem (8). Algorithm \| LiMEAL (our) \| PProx-PDA Hajinezhad-Hong19 \| Prox-iALM Zhang-Luo18 \| S-prox-ALM Zhang-Luo20 ---\|---\|---\|---\|--- Assumption \| $\nabla h$: Lipschitz, \| $\nabla h$: Lipschitz, \| $\nabla h$: Lipschitz, \| $\nabla h$: Lipschitz, $g$: weakly convex, \| $g$: convex, \| $g:\iota_{\cal C}(x)$, \| $g:\iota_{\cal P}(x)$, imp-Lip or imp-bound \| $\partial g$: bounded \| ${\cal C}$: box constraint \| ${\cal P}$: polyhedral set Iteration \| imp-Lip: $o(\varepsilon^{-2})$ \| ${\cal O}(\varepsilon^{-2})$ \| ${\cal O}(\varepsilon^{-2})$ \| ${\cal O}(\varepsilon^{-2})$ complexity \| imp-bound: ${\cal O}(\varepsilon^{-2})$ Global \| $\checkmark$ under KŁ \| – \| $\checkmark$ for quadratic \| – Convergence \| programming When reduced to the case of linear constraints, the proximal ALM suggested in Rockafellar76-PALM is a special case of MEAL with $\eta=1$, and the Lipschitz continuity of certain fundamental mapping at the origin (Rockafellar76-PALM, , p. 100) generally implies the KŁ property of the proximal augmented Lagrangian with exponent $1/2$ at some stationary point, and thus, the linear convergence of proximal ALM can be directly yielded by Proposition 1(b). Moreover, the proposed algorithms still work (in terms of convergence) for some constrained problems with nonconvex objectives and a fixed penalty parameter. In Hong17-Prox-PDA , a proximal primal-dual algorithm (named Prox-PDA) was proposed for the linearly constrained problem (1) with $b=0$. Prox-PDA is shown as follows: $\displaystyle\text{(Prox-PDA)}\ \left\\{\begin{array}[]{l}x^{k+1}=\operatorname{argmin}_{x\in\mathbb{R}^{n}}\ \left\\{f(x)+\langle\lambda^{k},Ax\rangle+\frac{\beta}{2}\\|Ax\\|^{2}+\frac{\beta}{2}\\|x-x^{k}\\|^{2}_{B^{T}B}\right\\},\\\ \lambda^{k+1}=\lambda^{k}+\beta Ax^{k+1},\end{array}\right.$ where $B$ is chosen such that $A^{T}A+B^{T}B\succeq\mathrm{I}_{n}$ (the identity matrix of size $n$). To achieve a $\sqrt{\varepsilon}$-accurate stationary point, the iteration complexity of Prox-PDA is ${\cal O}(\varepsilon^{-1})$ under the Lipschitz differentiability of $f$ (that is, $f$ is differentiable and has Lipschitz gradient) and the assumption that there exists some $\underline{f}>-\infty$ and some $\delta>0$ such that $f(x)+\frac{\delta}{2}\\|Ax\\|^{2}\geq\underline{f}$ for any $x\in\mathbb{R}^{n}$. Such iteration complexity of Prox-PDA is consistent with the order of ${\cal O}(\varepsilon^{-2})$ to achieve an $\varepsilon$-accurate stationary point. On one hand if we take $B=\mathrm{I}_{n}$ in Prox-PDA, then it reduces to MEAL with $\gamma=\beta^{-1}$ and $\eta=1$. On the other hand, by our main Theorem 3.1(a), the iteration complexity of the order of $o(\varepsilon^{-2})$ is slightly better than that of Prox-PDA, under weaker conditions (see, Assumption 2(a)-(b)). Moreover, we established the global convergence and rate of MEAL under the KŁ inequality, while such global convergence result is missing (though obtainable) for Prox-PDA in Hong17-Prox- PDA . A prox-linear variant of Prox-PDA (there dubbed PProx-PDA) was proposed in the recent paper Hajinezhad-Hong19 for the linearly constrained problem (8) with a composite objective. Besides Lipschitz differentiability of $h$, the nonsmooth function $g$ is assumed to be convex with bounded subgradients. These assumptions used in Hajinezhad-Hong19 are stronger than ours in Assumption 3(a), (b) and (d), while the yielded iteration complexity of LiMEAL (Theorem 4.1(b)) is consistent with that of PProx-PDA in (Hajinezhad-Hong19, , Theorem 1). Moreover, we establish the global convergence and rate of LiMEAL (Proposition 3), which is missing (though obtainable) for PProx-PDA. In Xie-Wright19 , an ${\cal O}(\varepsilon^{-2})$-iteration complexity of proximal ALM was established for the constrained problem with nonlinear equality constraints, under assumptions that the objective is differentiable and its gradient is both Lipschitz continuous and bounded, and that the Jacobian of the constraints is also Lipschitz continuous and bounded and satisfies a full-rank property (see (Xie-Wright19, , Assumption 1)). If we reduce their setting to linear constraints, their iteration complexity is slightly worse than ours and their assumptions are stronger (of course, except for the part on nonlinear constraints). In Zhang-Luo18 , a very related algorithm (called Proximal Inexact Augmented Lagrangian Multiplier method, dubbed Prox-iALM) was introduced for the following linearly constrained problem $\displaystyle\min_{x\in\mathbb{R}^{n}}\ h(x)\quad\mathrm{subject\ to}\quad Ax=b,\ x\in{\cal C},$ where ${\cal C}$ is a box constraint set. Subsequence convergence to a stationary point was established under the following assumptions: (a) the origin is in the relative interior of the set $\\{Ax-b:x\in{\cal C}\\}$; (b) the strict complementarity condition Nocedal99 holds for the above constrained problem; (c) $h$ is differentiable and has Lipschitz continuous gradient. Moreover, the global convergence and linear rate of this algorithm was established for the quadratic programming, in which case, the augmented Lagrangian satisfies the KŁ inequality with exponent $1/2$, by noticing the connection between Luo-Tseng error bound and KŁ inequality Li-Pong- KLexponent18 . According to Theorem 4.1 and Proposition 3, the established convergence results in this paper are more general and stronger than that in Zhang-Luo18 but under weaker assumptions. Particularly, besides the weaker assumption on $h$, the strict complementarity condition (b) is also removed in this paper for LiMEAL. The algorithm studied in Zhang-Luo18 has been recently generalized to handle the linearly constrained problem with the polyhedral set in Zhang-Luo20 (dubbed S-prox-ALM). Under the Lipschitz differentiability of the objective, the iteration complexity of the order ${\cal O}(\varepsilon^{-2})$ was established in Zhang-Luo20 for the S-prox-ALM algorithm. Such iteration complexity is consistent with LiMEAL as shown in Theorem 4.1. Besides these major differences between this paper and Zhang-Luo20 ; Zhang-Luo18 , the step sizes $\eta$ are more flexible for both MEAL and LiMEAL (only requiring $\eta\in(0,2)$), while the step sizes used in the algorithms in Zhang-Luo20 ; Zhang-Luo18 should be sufficiently small to guarantee the convergence. Meanwhile, the Lyapunov function used in this paper is motivated by the Moreau envelope of the augmented Lagrangian, which is very different from the Lyapunov function used in Zhang-Luo20 ; Zhang-Luo18 . Based on the defined Lyapunov function, our analysis is much simpler than that in Zhang-Luo20 ; Zhang-Luo18 . ## 7 Numerical Experiments We use two experiments to demonstrate the effectiveness of the proposed algorithms: 1. 1. The first experiment is based on a nonconvex quadratic program on which ALM with any bounded penalty parameter diverges (Wang19, , Proposition 1) but LiMEAL converges. 2. 2. The second experiment borrows a general quadratic program from (Zhang-Luo18, , Sec. 6.2) and LiMEAL outperforms Prox-iALM suggested in Zhang-Luo18 . The source codes can be accessed at https://github.com/JinshanZeng/MEAL. ### 7.1 ALM vs LiMEAL Consider the following optimization problem from (Wang19, , Proposition 1): $\displaystyle\min_{x,y\in\mathbb{R}}\ x^{2}-y^{2},\quad\text{subject to}\ x=y,\ x\in[-1,1].$ (78) ALM with any bounded penalty parameter $\beta$ diverges on this problem. By Theorem 4.1 and Proposition 3, LiMEAL converges exponentially fast since its augmented Lagrangian is a KŁ function with an exponent of $1/2$. For both ALM and LiMEAL, we set the penalty parameter $\beta$ to 50. We set LiMEAL’s proximal parameter $\gamma$ to $1/2$ and test three different values $\eta^{\prime}s$: $0.5,1,1.5$. The curves of objective $f(x^{k},y^{k})=(x^{k})^{2}-(y^{k})^{2}$, constraint violation error $\|x^{k}-y^{k}\|$, multiplier sequences $\\{\lambda^{k}\\}$, and the norm of gradient of Moreau envelope in (39), which is the stationarity measure, are depicted in Fig. 1. Observe that ALM diverges: its multiplier sequence $\\{\lambda^{k}\\}$ oscillates between two distinct values (Fig. 1 (a)) and the constraint violation converges to a positive value (Fig. 1 (b)). Also observe that LiMEAL converges exponentially fast (Fig. 1 (c)–(e)) and achieves the optimal objective value of 0 in about 10 iterations (Fig. 1 (f)) with all $\eta$ values. This verifies Proposition 3. (a) Divergent $\\{\lambda^{k}\\}$ of ALM (b) Constraint violation of ALM (c) Convergent $\\{\lambda^{k}\\}$ of LiMEAL (d) Constraint violation of LiMEAL (e) convergence rate of LiMEAL (f) objective sequence of LiMEAL Figure 1: Apply ALM and LiMEAL to problem (78). ALM diverges. LiMEAL quickly converges. ### 7.2 Quadratic Programming Consider the quadratic program with box constraints: $\displaystyle\min_{x\in\mathbb{R}^{n}}\ \frac{1}{2}x^{T}Qx+r^{T}x\quad s.t.\quad Ax=b,\ \ell_{i}\leq x_{i}\leq u_{i},\ i=1,\ldots,n,$ (79) where $Q\in\mathbb{R}^{n\times n}$, $r\in\mathbb{R}^{n}$, $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^{m}$, and $\ell_{i},u_{i}\in\mathbb{R}$, $i=1,\ldots,n$. Let ${\cal C}:=\\{x:\ell_{i}\leq x_{i}\leq u_{i},i=1,\ldots,n\\}$. Applying LiMEAL yields: initialize $(x^{0},z^{0},\lambda^{0})$, $\gamma>0$, $\eta\in(0,2)$ and $\beta>0$, for $k=0,1,\ldots,$ run $\mathrm{(LiMEAL)}\quad\left\\{\begin{array}[]{l}\tilde{x}^{k}=(\beta A^{T}A+\gamma^{-1}{\bf I}_{n})^{-1}(\gamma^{-1}z^{k}+\beta A^{T}b-r- Qx^{k}-A^{T}\lambda^{k}),\\\ x^{k+1}=\mathrm{Proj}_{\cal C}(\tilde{x}^{k}),\\\ z^{k+1}=z^{k}-\eta(z^{k}-x^{k+1}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b).\end{array}\right.$ Applying Prox-iALM from (Zhang-Luo18, , Algorithm 2.2) yields: initialize $(x^{0},z^{0},\lambda^{0})$, parameters $\beta,p,\alpha,s,\eta>0$, for $k=0,1,\ldots,$ run $\mathrm{(Prox-iALM)}\quad\left\\{\begin{array}[]{l}\bar{x}^{k}=(\beta A^{T}A+p{\bf I}_{n})x^{k}+Qx^{k}+A^{T}\lambda^{k}-pz^{k}-(\beta A^{T}b-r),\\\ x^{k+1}=\mathrm{Proj}_{\cal C}(x^{k}-s\bar{x}^{k}),\\\ z^{k+1}=z^{k}-\eta(z^{k}-x^{k+1}),\\\ \lambda^{k+1}=\lambda^{k}+\beta_{k}(Ax^{k+1}-b).\end{array}\right.$ When $\eta=1$, then Prox-iALM reduces to Algorithm 2.1 in Zhang-Luo18 , which we name iALM. The experimental settings are similar to (Zhang-Luo18, , Sec. 6.2): set $m=5,n=20$, generate the entries of $Q$, $A$, $b$, and $\tilde{x}$ by sampling from the uniform distribution, and set $b=A\tilde{x}$. For LiMEAL, we set $\beta=50,\gamma=\frac{1}{2\\|Q\\|_{2}}$ and test three values of $\eta^{\prime}s$: $0.5,1,1.5$. For Prox-iALM, we use the parameter settings in (Zhang-Luo18, , Sec. 6.2): $p=2\\|Q\\|_{2},\beta=50,\alpha=\frac{\beta}{4},s=\frac{1}{2(\\|Q\\|_{2}+p+\beta\\|A\\|_{2}^{2})}$. Moreover, we test two values of $\eta^{\prime}s$: $1$ and $0.5$ for Prox-iALM. Prox-iALM with $\eta=1$ reduces to iALM. The curves of the objective sequence, $\\|Ax^{k}-b\\|$, $\\|x^{k+1}-z^{k}\\|$ and the norm of gradient of the Moreau envelope are depicted in Fig. 2. We observe that LiMEAL converges faster than both iALM and Prox-iALM. By Fig. 2(d), LiMEAL converges exponentially fast with all three values of $\eta^{\prime}s$. These results verify the results in Proposition 3(b) since the augmented Lagrangian of problem (79) is a KŁ function with an exponent of $1/2$. (a) Objective sequence (b) $\\|Ax^{k}-b\\|$ (c) $\\|x^{k+1}-z^{k}\\|$ (d) Convergence rates of LiMEAL Figure 2: Performance of LiMEAL and Prox-iALM for the quadratic programming problem (79). ## 8 Conclusion This paper suggests a Moreau envelope augmented Lagrangian (MEAL) method for the linearly constrained weakly convex optimization problem. By leveraging the implicit smoothing property of Moreau envelope, the proposed MEAL generalizes the ALM and proximal ALM to the nonconvex and nonsmooth case. To yield an $\varepsilon$-accurate first-order stationary point, the iteration complexity of MEAL is $o(\varepsilon^{-2})$ under the implicit Lipschitz subgradient assumption and ${\cal O}(\varepsilon^{-2})$ under the implicit bounded subgradient assumption. The global convergence and rate of MEAL are also established under the further Kurdyka-Łojasiewicz inequality. Moreover, an inexact variant (called iMEAL), and a prox-linear variant (called LiMEAL) for the composite objective case are suggested and analyzed for different practical settings. The convergence results established in this paper for MEAL and its variants are generally stronger than the existing ones, but under weaker assumptions. One future direction of this paper is to get rid of the implicit Lipschitz subgradient and implicit bounded subgradient assumptions, which in some extent limit the applications of the suggested algorithms, though these two assumptions are respectively weaker than the Lipschitz differentiable and bounded subgradient assumptions commonly used in the literature. Another direction is to generalize this work to the constrained problem with nonlinear constraints. 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# Out-of-Distribution Generalization Analysis via Influence Function Haotian Ye Yuanpei College Peking University Beijing, China <EMAIL_ADDRESS> &Chuanlong Xie Huawei Noah’s Ark Lab Hong Kong, China <EMAIL_ADDRESS> Yue Liu∗ Huawei Noah’s Ark Lab Beijing, China <EMAIL_ADDRESS> &Zhenguo Li Huawei Noah’s Ark Lab Hong Kong, China <EMAIL_ADDRESS> ###### Abstract The mismatch between training and target data is one major challenge for current machine learning systems. When training data is collected from multiple domains and the target domains include all training domains and other new domains, we are facing an Out-of-Distribution (OOD) generalization problem that aims to find a model with the best OOD accuracy. One of the definitions of OOD accuracy is worst-domain accuracy. In general, the set of target domains is unknown, and the worst over target domains may be unseen when the number of observed domains is limited. In this paper, we show that the worst accuracy over the observed domains may dramatically fail to identify the OOD accuracy. To this end, we introduce Influence Function, a classical tool from robust statistics, into the OOD generalization problem and suggest the variance of influence function to monitor the stability of a model on training domains. We show that the accuracy on test domains and the proposed index together can help us discern whether OOD algorithms are needed and whether a model achieves good OOD generalization. ## 1 Introduction Most machine learning systems assume both training and test data are independently and identically distributed, which does not always hold in practice (Bengio et al. (2019)). Consequently, its performance is often greatly degraded when the test data is from a different domain (distribution). A classical example is the problem to identify cows and camels (Beery et al. (2018)), where the empirical risk minimization (ERM, Vapnik (1992)) may classify images by background color instead of object shape. As a result, when the test domain is “out-of-distribution” (OOD), e.g. when the background color is changed, its performance will drop significantly. The OOD generalization is to obtain a robust predictor against this distribution shift. Suppose that we have training data collected from $m$ domains: $\displaystyle{\mathbb{S}}=\\{{\mathbb{S}}^{e}:e\in\mathcal{E}_{tr},\|\mathcal{E}_{tr}\|=m\\},\quad{\mathbb{S}}^{e}=\\{{\bm{z}}^{e}_{1},{\bm{z}}^{e}_{2},\ldots,{\bm{z}}^{e}_{n^{e}}\\}\,\,\text{with}\,\,{\bm{z}}^{e}_{i}\sim P^{e},$ (1) where $P^{e}$ is the distribution corresponding to domain $e$, $\mathcal{E}_{tr}$ is the set of _all available domains, including validation domains_ , and ${\bm{z}}^{e}_{i}$ is a data point. The OOD problem we considered is to find a model $f_{\text{OOD}}$ such that $\displaystyle f_{\text{OOD}}=\operatorname{arg\,min}_{f}\sup_{P^{e}\in\mathcal{E}_{all}}\ell(f,P^{e}),$ (2) where $\mathcal{E}_{all}$ is the set of all target domains and $\ell(f,P^{e})$ is the expected loss of $f$ on the domain $P^{e}$. Recent algorithms address this OOD problem by recovering invariant (causal) features and build the optimal model on top of these features, such as Invariant Risk Minimization (IRM, Arjovsky et al. (2019)), Risk Extrapolation (REx, Krueger et al. (2020)), Group Distributionally Robust Optimization (gDRO, Sagawa et al. (2019)) and Inter-domain Mixup (Mixup, Xu et al. (2020); Yan et al. (2020); Wang et al. (2020)). Most works evaluate on Colored MNIST (see 5.1 for details) where we can directly obtain the worst domain accuracy over $\mathcal{E}_{all}$. Gulrajani & Lopez-Paz (2020) has assembled many algorithms and multi-domain datasets, and finds that OOD algorithms can’t outperform ERM in some domain generalization tasks (Gulrajani & Lopez-Paz (2020)), e.g. VLCS (Torralba & Efros (2011)) and PACS (Li et al. (2017)). This is not surprising, since these tasks only require high performance on certain domains, while an OOD algorithm is expected to learn truly invariant features and be excellent on a large set of target domains $\mathcal{E}_{all}$. This phenomenon is described as “accuracy-vs-invariance trade-off” in Akuzawa et al. (2019). Two questions arise in the min-max problem (2). First, previous works assume that there is sufficient diversity among the domains in $\mathcal{E}_{all}.$ Thus the supremacy of $\ell(f,P^{e})$ may be much larger than the average, which implies that ERM may fail to discover $f_{OOD}.$ But in reality, we do not know whether it is true. If not, the distribution of $\ell(f,P^{e})$ is concentrated on the expectation of $\ell(f,P^{e})$, and ERM is sufficient to find an invariant model for $\mathcal{E}_{all}.$ Therefore, we call for a method to judge whether an OOD algorithm is needed. Second, how to judge a model’s OOD performance? Traditionally, we consider test domains $\mathcal{E}_{test}\subset\mathcal{E}_{tr}$ and use the worst-domain accuracy over $\mathcal{E}_{test}$ (which we call test accuracy) to approximate the OOD accuracy. However, test accuracy is a biased estimate of the OOD accuracy unless $\mathcal{E}_{tr}$ is closed to $\mathcal{E}_{all}$. More seriously, It may be irrelevant or even _negatively correlated_ to the OOD accuracy. This phenomenon is not uncommon, especially when there are features virtually spurious in $\mathcal{E}_{all}$ but show a strong correlation to the target in $\mathcal{E}_{tr}$. We give a toy example in Colored MNIST when the test accuracy fails to approximate the OOD accuracy. For more details, please refer to Section 5.1 and Appendix A.4. We choose three domains from Colored MNIST and use cross- validation (Gulrajani & Lopez-Paz (2020)) to select models, i.e. we take turns to select a domain $S\in\mathcal{E}_{tr}$ as the test domain and train on the rest, and select the model with max average test accuracy. Figure 1 shows the comparison between ERM and IRM. One can find that no matter which domain is the test domain, ERM model uniformly outperforms IRM model on the test domain. However, IRM model achieves consistently better OOD accuracy. Shortcomings of the test accuracy here are obvious, regardless of whether cross-validation is used. In short, the naive use of the test accuracy may result in a non-OOD model. Figure 1: Experiments in Colored MNIST to show test accuracy is not enough to reflect a model’s OOD accuracy. The top left penal shows the test accuracy of ERM and IRM. The other three panels present the relationship between test accuracy (x-axis) and OOD accuracy (y-axis) in three setups. To address this obstacle, we hope to find a metric that correlates better with model’s OOD property, _even when $\mathcal{E}_{tr}$ is much smaller than $\mathcal{E}_{all}$ and the “worst” domain remains unknown_. Without any assumption to $\mathcal{E}_{all}$, our goal is unrealistic. Therefore, we assume that features that are invariant across $\mathcal{E}_{tr}$ should also be across $\mathcal{E}_{all}$. This assumption is necessary. Otherwise, the only thing we can do is to collect more domains. Therefore, we need to focus on what features the model has learnt. Specifically, we want to check whether the model learns invariant features and avoid varying features. The influence function (Cook & Weisberg (1980)) can serve our purpose. Influence function was proposed to measures the parameter change when a data point is removed or upweighted by a small perturbation (details in 3.2). When modified it to domain-level, it measures the influence of a domain instead of a data point on the model. Note that we are not emulating the changes of the parameter when a domain is removed. Instead, we are exactly caring about upweighting the domain by $\delta\rightarrow 0^{+}$ (will be specified later). Base on this, the variance of influence function allows us to measure OOD property and solve the obstacle. ##### Contributions we summarize our contributions here: (i) We introduce influence function to domain-level and propose index $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ (formula 6) based on influence function of the model $f_{\bm{\theta}}$. Our index can measure the OOD extent of available domains, i.e. how different these domains (distributions) are. This measurement provides a basis for whether to adopt an OOD algorithm and to collect more diverse domains. See Section 4.1 and Section 5.1.1 for details. (ii) We point out that the proposed index $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ can solve the weakness of test accuracy. Specifically, under most OOD generalization problems, using test accuracy and our index together, we can discern the OOD property of a model. See Section 4.2 for details. (iii) We propose to use only a small but important part of the model to calculate the influence function. This overcomes the huge computation cost of solving the inverse of Hessian. It is not merely for calculation efficiency and accuracy, but it coincides with our understanding that only these parameters capture what features a model has learnt (Section 4.3). We organize our paper as follows: Section 2 reviews related works and Section 3 introduces the preliminaries of OOD methods and influence function. Section 4 presents our proposal and detailed analysis. Section 5 shows our experiments. The conclusion is given in Section 6. ## 2 Related work The mismatch between the development dataset and the target domain is one major challenge in machine learning (Castro et al. (2020); Kuang et al. (2020)). Many works assume that the ground truth can be represented by a causal Direct Acyclic Graph (DAG), and they use the DAG structure to discuss the worst-domain performance (Rojas-Carulla et al. (2018); Peters et al. (2016); Subbaswamy et al. (2019); Bühlmann et al. (2020); Magliacane et al. (2018)). All these works employ multiple domain data and causal assumptions to discover the parents of the target variable. Rojas-Carulla et al. (2018) and Magliacane et al. (2018) also apply this idea to Domain Generalization and Multi-Task Learning setting. Starting from multiple domain data rather than model assumptions, Arjovsky et al. (2019) proposes Invariant Risk Minimization (IRM) to extract causal (invariant) features and learn invariant optimal predictor on the top of the causal features. It analyzes the generalization properties of IRM from the view of sufficient dimension reduction (Cook (2009); Cook et al. (2002)). Ahuja et al. (2020) considers IRM as finding the Nash equilibrium of an ensemble game among several domains and develops a simple training algorithm. Krueger et al. (2020) derives the Risk Extrapolation (REx) to extract invariant features and further derives a practical objective function via variance penalization. Xie et al. (2020) employs a framework from distributional robustness to interpret the benefit of REx comparing to robust optimization (Ben-Tal et al. (2009); Bagnell (2005)). Besides, Adversarial Domain Adaption (Li et al. (2018); Koyama & Yamaguchi (2020)) uses discriminator to look for features that are independent of domains and uses these features for further prediction. Influence function is a classic method from the robust statistics literature (Robins et al. (2008; 2017); Van der Laan et al. (2003); Tsiatis (2007)). It can be used to track the impact of a training sample on the prediction. Koh & Liang (2017) proposes a second-order optimization technique to approximate the influence function. They verify their method with different assumptions on the empirical risk ranging from being strictly convex and twice-differentiable to non-convex and non-differentiable losses. Koh et al. (2019) also estimates the effect of removing a subgroup of training points via influence function. They find out that the approximation computed by the influence function is correlated with the actual effect. Influence function has been used in many machine learning tasks. Cheng et al. (2019) proposes an explanation method, Fast Influence Analysis, that employs influence function on Latent Factor Model to solve the lack of interpretability of the collaborative filtering approaches for recommender systems. Cohen et al. (2020) uses influence function to detect adversarial attacks. Ting & Brochu (2018) proposes an asymptotically optimal sampling method via an asymptotically linear estimator and the associated influence function. Alaa & Van Der Schaar (2019) develops a model validation procedure that estimates the estimation error of causal inference methods. Besides, Fang et al. (2020) leverages influence function to select a subset of normal users who are influential to the recommendations. ## 3 Preliminaries ### 3.1 ERM, IRM and REx In this section, we give some notations and introduce some recent OOD methods. Recall the multiple domain setup (1) and OOD problem (2). For a domain $P^{e}$ and a hypothetical model $f$, the population loss is $\ell(f,P^{e})=\mathbb{E}_{{\mathbf{z}}\sim P^{e}}[L(f,{\mathbf{z}})]$ where $L(f,{\mathbf{z}})$ is the loss function on ${\mathbf{z}}$. The empirical loss, which is the objective of ERM, is $\ell(f,{\mathbb{S}})=(1/m)\sum_{e\in\mathcal{E}_{tr}}\ell(f,{\mathbb{S}}^{e})$ with $\ell(f,{\mathbb{S}}^{e})=(1/n)\sum_{i=1}^{n}L(f,{\bm{z}}^{e}_{i}).$ Recent OOD methods propose some novel regularized objective functions in the form: $\mathcal{L}(f,{\mathbb{S}})=\ell(f,{\mathbb{S}})+\lambda R(f,{\mathbb{S}})$ (3) to discover $f_{\text{OOD}}$ in (2). Here $R(f,{\mathbb{S}})$ is a regularization term and $\lambda$ is the tuning parameter which controls the degree of penalty. Note that ERM is a special case by setting $\lambda=0$. For simplicity, we will use $\mathcal{L}(f,{\mathbb{S}})$ to represent the total loss in case of no ambiguity. Arjovsky et al. (2019) focuses on the stability of $f_{\text{OOD}}$ and considers the IRM regularization: $R(f,{\mathbb{S}})=\sum_{e\in\mathcal{E}_{tr}}\\|\nabla_{w}\ell\big{(}{\color[rgb]{0,0,0}wf}),{\mathbb{S}}^{e}\big{)}\big{\|}_{w=1.0}\\|^{2}$ (4) where $w$ is a scalar and fixed “dummy” classifier. Arjovsky et al. (2019) shows that the scalar fixed classifier $w$ is sufficient to monitor invariance and responds to the idealistic IRM problem which decomposes the entire predictor into data representation and one shared optimal top classifier for all training domains. On the other hand, Krueger et al. (2020) encourages the uniform performance of $f_{\text{OOD}}$ and proposes the V-REx penalty: $\displaystyle R(f,{\mathbb{S}})=\sum_{e\in\mathcal{E}_{tr}}(\ell(f,{\mathbb{S}}^{e})-\ell(f,{\mathbb{S}}))^{2}.$ Krueger et al. (2020) derives the invariant prediction by the robustness to spurious features and figure out that REx is more robust than group distributional robustness (Sagawa et al. (2019)). In this work, we also decompose the entire predictor into a feature extractor and a classifier on the top of the learnt features. As we will see, different from Arjovsky et al. (2019) and Krueger et al. (2020), we directly monitor the invariance of the top model. ### 3.2 Influence function and group effect Consider a parametric hypothesis $f=f_{\bm{\theta}}$ and the corresponding solution: $\hat{\bm{\theta}}=\operatorname{arg\,min}_{\bm{\theta}}\mathcal{L}(f_{\bm{\theta}},{\mathbb{S}}).$ By a quadratic approximation of $\mathcal{L}(f_{\bm{\theta}},{\mathbb{S}})$ around $\hat{\bm{\theta}}$, the influence function takes the form $\displaystyle\mathcal{IF}(\hat{\bm{\theta}},{\bm{z}})=-{\bm{H}}^{-1}_{\hat{\bm{\theta}}}\nabla_{\bm{\theta}}L(f_{\hat{\bm{\theta}}},{\bm{z}})\quad\text{with}\quad{\bm{H}}_{\hat{\bm{\theta}}}=\nabla^{2}_{\bm{\theta}}{\color[rgb]{0,0,0}\mathcal{L}(f_{\hat{\bm{\theta}}},{\mathbb{S}})}.$ When the sample size of ${\mathbb{S}}$ is sufficiently large, the parameter change due to removing a data point $z$ can be approximated by $-\mathcal{I}(z)/\sum_{e\in\mathcal{E}_{tr}}\|{\mathbb{S}}^{e}\|$ without retraining the model. Here $\|{\mathbb{S}}^{e}\|=n^{e}$ stands for the cardinal of the set ${\mathbb{S}}^{e}$. Furthermore, Koh et al. (2019) shows that the influence function can also predict the effects of large groups of training points (i.e. $\mathcal{Z}=\\{z_{1},...,z_{k}\\}$), although there are significant changes in the model. The parameter change due to removing the group can be approximated by $\mathcal{IF}(\hat{\bm{\theta}},\mathcal{Z})=-{\bm{H}}^{-1}_{\hat{\bm{\theta}}}\nabla_{\bm{\theta}}\frac{1}{\|\mathcal{Z}\|}\sum_{z\in\mathcal{Z}}L(f_{\hat{\bm{\theta}}},z).$ Motivated by the work of Koh et al. (2019), we introduce influence function to OOD problem to address our obstacles. ## 4 Methodology ### 4.1 Influence of domains We decompose a parametric hypothesis $f_{\bm{\theta}}(x)$ into a top model $g$ and a feature extractor $\Phi$, i.e. $f_{\bm{\theta}}(x)=g(\Phi({\bm{x}},{\bm{\beta}}),{\bm{\gamma}})$ and ${\bm{\theta}}=({\bm{\gamma}},{\bm{\beta}}).$ Such decomposition coincides the understanding of most DNN, i.e. a DNN extracts the features and build a top model based on the extracted features. When upweighting a domain $e$ by a small perturbation $\delta$, we do not upweight the regularized term, i.e. $\displaystyle\mathcal{L}_{+}({\bm{\theta}},\mathbb{S},\delta)=\mathcal{L}({\bm{\theta}},\mathbb{S})+\delta\cdot\ell(f,{\mathbb{S}}^{e}),$ since the stability across different domains, which is encouraged by the regularization, should not depend on the sample size of a domain. For a learnt model $f_{\hat{\bm{\theta}}}$, fixing the feature extractor $\Phi$, i.e. fixing ${\bm{\beta}}=\hat{\bm{\beta}}$, the change of top model $g$ caused by upweighting the domain is $\mathcal{IF}(\hat{\bm{\gamma}},{\mathbb{S}}^{e}\|\hat{\bm{\theta}}){\color[rgb]{0,0,0}:=\lim_{\delta\rightarrow 0^{+}}\frac{\Delta{\bm{\theta}}}{\delta}}=-{\bm{H}}^{-1}_{\hat{\bm{\gamma}}}\nabla_{\bm{\gamma}}\ell(f_{\hat{\bm{\theta}}},{\mathbb{S}}^{e}),\quad e\in\mathcal{E}_{tr}.$ (5) Here ${\bm{H}}_{\hat{\bm{\gamma}}}=\nabla^{2}_{\hat{\bm{\gamma}}}\mathcal{L}(f_{\hat{\bm{\theta}}},{\mathbb{S}})$, and we assume $\mathcal{L}$ is twice-differentiable in ${\bm{\gamma}}$. Please see Appendix A.3 for detailed derivation and why ${\bm{\beta}}$ should be fixed. For a regularized method, e.g. IRM and REx, the influence of their regularized term is reflected in ${\bm{H}}$ and in learnt model $f_{\hat{\bm{\theta}}}$. As mentioned above, $\mathcal{IF}(\hat{\bm{\gamma}},{\mathbb{S}}^{e}\|\hat{\bm{\theta}})$ measures change of model caused by upweighting domain $e$. Therefore, if $g(\Phi,\hat{\bm{\gamma}})$ is invariant across domains, the entire model $f_{\hat{\bm{\theta}}}$ treats all domains equally. As a result, a small perturbation on different domains should cause _the same model change_. This leads to our proposal. ### 4.2 Proposed Index and its Utility On basis of the domain-level influence function $\mathcal{IF}(\hat{\bm{\gamma}},{\mathbb{S}}^{e}\|\hat{\bm{\theta}})$, we propose our index to measure the fluctuation of the parameter change when different domains are upweighted: $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}:=\ln\Big{(}\\|\mathrm{Cov}_{e\in\mathcal{E}_{tr}}\big{(}\mathcal{IF}(\hat{\bm{\gamma}},{\mathbb{S}}^{e}\|\hat{\bm{\theta}})\big{)}\\|_{2}\Big{)}.$ (6) Here $\\|\cdot\\|_{2}$ is the 2-norm for matrix, i.e. the largest eigenvalue of the matrix, $\mathrm{Cov}_{e\in\mathcal{E}_{tr}}(\cdot)$ refers to the covariance matrix of the domain-level influence function over $\mathcal{E}_{tr}$ and $\ln(\cdot)$ is a nonlinear transformation that works well in practice. ##### OOD Model Under the OOD problem in (2), a good OOD model should (i) learn invariant and useful features; (ii) avoid spurious and varying features. Learning useful and invariant features means the model should have high accuracy over a set of test domains $\mathcal{E}_{test}$, no matter which test domain it is. In turn, high accuracy over $\mathcal{E}_{test}$ also means the model truly learns some useful features for the test domains. However, this is not enough, since we do not know whether the useful features are invariant features across $\mathcal{E}_{all}$ or just spurious features on $\mathcal{E}_{test}.$ On the other hand, avoiding varying features means that different domains are _actually the same_ to the learnt model, so according to the arguments in Section 4.1, $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ should be small. Combined this, we derive our proposal: if a learnt model $f_{\hat{\bm{\theta}}}$ manage to simultaneously achieve small $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ and high accuracy over $\mathcal{E}_{test}$, it should have good OOD accuracy. We prove our proposal in a simple but illuminating case, and we conduct various experiments (Section 5) to support our proposal. Several issues should be clarified. First, not all OOD problems demand models to learn invariant features. For example, the set of all target domains is small such that the varying features are always strongly correlated to the labels, or the objective is the mean of the accuracy over $\mathcal{E}_{all}$ rather than the worst-domain accuracy. But to our concern, we regard the OOD problem in (2) as a bridge to causal discover. Thus the set of the target domains is large, and the “weak” OOD problems are out of our consideration. To a large extent, invariant features are still the major target and our proposal is still a good criterion to model’s OOD property. Second, we admit that the gap between being stable in $\mathcal{E}_{tr}$ (small $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$) and avoiding all spurious features on $\mathcal{E}_{all}$ does exist. However, to our knowledge, for features that are varying in $\mathcal{E}_{all}$ but are invariant in $\mathcal{E}_{tr}$, demanding a model to avoid them is somehow unrealistic. Therefore, we make a step forward that we measure whether the learnt model successfully avoids features that vary across $\mathcal{E}_{tr}$. We leave index about varying features over $\mathcal{E}_{all}$ in our future work. ##### The Shuffle $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ As mentioned above, smaller metric $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ means strong stablility across $\mathcal{E}_{tr}$, and hence should have better OOD accuracy. However, the proposed metric depends on the dataset ${\mathbb{S}}$ and the learnt model $f_{\hat{\bm{\theta}}}.$ Therefore, there is no uniform baseline to check whether the metric is “small” enough. To this end, we propose a baseline value of the proposed metric by shuffling the multi-domain data. Consider pooling all data points in ${\mathbb{S}}$ and randomly redistributed to $m$ new synthetic domains $\\{\tilde{\mathbb{S}}^{1},\tilde{\mathbb{S}}^{2},...,\tilde{\mathbb{S}}^{m}\\}:=\tilde{\mathbb{S}}$. We compute _the shuffle version_ of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ for a learnt model $f_{\hat{\bm{\theta}}}$ over the shuffled data $\tilde{\mathbb{S}}$: $\tilde{\mathcal{V}}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}:=\ln\Big{(}\\|\mathrm{Cov}_{e\in\mathcal{E}_{tr}}\big{(}\mathcal{IF}(\hat{\bm{\gamma}},{\mathbb{S}}^{e}\|\hat{\bm{\theta}})\big{)}\\|_{2}\Big{)}.$ (7) and denote the standard version and shuffle version of the metric as $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ and $\tilde{\mathcal{V}}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ respectively. For any algorithm that obtains relatively good test accuracy, if $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ is much larger than $\tilde{\mathcal{V}}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$, $f_{\hat{\bm{\theta}}}$ has learnt features that vary across $e\in\mathcal{E}_{tr}$, and cannot treat domains in $\mathcal{E}_{tr}$ equally. This implies that $f_{\hat{\bm{\theta}}}$ may not be an invariant predictor over $\mathcal{E}_{all}.$ Otherwise, if the two values are similar, the model has avoided varying features in $\mathcal{E}_{tr}$ and maybe invariant across $\mathcal{E}_{tr}$. Therefore, either the model capture the invariance over the diverse domains, or the domains are not diverse at all. Note that this process is suitable for any algorithm, hence providing a baseline to see whether $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ is small. Here we also obtain a method to judge whether an OOD algorithm is needed. Consider $f_{\hat{\bm{\theta}}}$ learnt by ERM. If $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ is relatively larger than $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\theta}}}$, then ERM fails to avoid varying features. In this case, one should consider an OOD algorithm to achieve better OOD generalization. Otherwise, ERM is enough, and any attempt to achieve better OOD accuracy should start with finding more domains instead of using OOD algorithms. This coincides experiments in Gulrajani & Lopez-Paz (2020) (Section5.2). Our understanding is that domains in $\tilde{\mathbb{S}}$ are similar. Therefore, the difference between shuffle and standard version of the metric reflects how much varying features a learnt model uses. We show how to use the two version of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ in Section 5.1.1 and Section 5.2. ### 4.3 Influence Calculation There is a question surrounding the influence function: how to efficiently calculate and inverse Hessian? Koh & Liang (2017) suggests Conjugate Gradient and Stochastic estimation solve the problem. However, when $\hat{\bm{\theta}}$ is obtained by running SGD, it could hardly arrive at the global minimum. Although adding a damping term (i.e. let $\hat{\bm{H}}_{\hat{\bm{\theta}}}={\bm{H}}_{\hat{\bm{\theta}}}+\lambda I$) can moderately alleviate the problem by transforming it into a convex situation, under large neural-network with non-linear activation function like ReLU, this method may still work poorly since the damping term in order to satisfy the transform is so large that it will influence the performance significantly. Most importantly, the variation of the eigenvalue of Hessian is huge, making the convergence of influence function calculation quite slow and inaccurate (Basu et al. (2020)). In our metric, we circumvent the problem by excluding most parameters ${\bm{\beta}}$ and directly calculate Hessian of ${\bm{\gamma}}$ to get accurate influence function. This modification not only speed up the calculation, but it also coincides our expectation, that an OOD algorithm should learn invariant features _does not mean that_ the influence function of _all_ parameters should be identical across domains. For example, if $g(\Phi)$ wants to extract the same features in different domains, the influence function should be different on $\Phi(\cdot)$. Therefore, if we use all parameters to calculate the influence, given that ${\bm{\gamma}}$ is relatively insignificant in size compared with ${\bm{\beta}}$, the information of learnt features provided by ${\bm{\gamma}}$ is hard to be captured. On the contrary, only considering the influence of the top model will manifest the influence of different domains in the aspect of _features_ , thus enabling us to achieve our goal. As our experiments show, after this modification, the influence function calculation speed can be 2000 times faster, and the utility (correlation with OOD property) could be even higher. One may not feel surprised given the huge number of parameters in the embedding model $\Phi(\cdot)$. They slow down the calculation and overshadow the top model’s influence value. ## 5 Experiment In this section, we experimentally show that: (1) A model $f_{\hat{\bm{\theta}}}$ reaches small $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ if it has good OOD property, while a non-OOD model won’t. (2) The metric $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ provides additional information on the stability of a learnt model, which overcomes the weakness of the test accuracy. (3) The comparison of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ can check whether a better OOD algorithm is needed. We consider experiments in Bayesian Network, Colored MNIST and VLCS. The synthetic data generated by Bayesian Network includes domain-dependent noise and fake associations between features and response. For Colored MNIST, we already know that the digit is the causal feature and the color is non-causal. The causal relationships help us to determine the worst domain and obtain the OOD accuracy. VLCS is a real dataset, in which we show utility of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ step by step. Due to the space limitation, we put the experiments in Bayesian Network to the appendix. Generally, cross-validation (Gulrajani & Lopez-Paz (2020)) is used to judge a model’s OOD property. In the introduction, we have already shown that the leave-one-domain-out cross-validation may fail to discern OOD properties. We also consider another two potential competitors: conditional mutual information and IRM penalty. The comparison between our metric and the two competitors are postponed into Appendix. Figure 2: The index $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ is highly correlated to $x$. The plot contains 501 learnt ERM models with $x=2\times 10^{-4}i$, $i=0,1,...,500.$ The dashed line is the baseline value when the difference between domains is eliminated by pooling and redistributing the training data. The blue solid line is the linear regression of $x$ versus $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$. ### 5.1 Colored MNIST Colored MNIST (Arjovsky et al. (2019)) introduces a synthetic binary classification task. The images are colored according to their label, making color a spurious feature in predicting the label. Specifically, for a domain $e$, we assign a preliminary binary label $\tilde{y}=\mathbf{1}_{\text{digits}\leq 4}$ and randomly flip $\tilde{y}$ with $p=0.25$. Then, we color the image according to $\tilde{y}$ but with a flip rate of $p^{e}$. Clearly, when $p^{e}<0.25$ or $p^{e}>0.75$, color is more correlated with $\tilde{y}$ than real digit. Therefore, the oracle OOD model $f_{\text{OOD}}$ will attain accuracy $0.75$ in all domains while an ERM model may attain high training accuracy and low OOD property if $p^{e}$ in training domains is too small or too large. Throughout the Colored MNIST experiments, we use three-layer MLP with ReLU activation and hidden dimension 256. Although our MLP model has relatively many parameters and is non-convex due to the activation layer, due to the technique mentioned in Section 4.3, the influence calculation is still fast and accurate, with directly calculating influence once spends less than 2 seconds. #### 5.1.1 Identify OOD Problem In this section, we show that $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ can discern whether the training domains are sufficiently diverse as mentioned in Section 4.2. Assume $\mathcal{E}_{tr}$ has five training domains with $p^{e}\in\\{0.2-2x,0.2-x,0.2,0.2+x,0.2+2x\\},$ where $x\in[0.0,0.1]$ is positively related to the diversity among the training domains. If $x$ is zero, all data points are generated from the same domain ($p^{e}=0.2$) and so the learning task on $\mathcal{E}_{tr}$ is not an OOD problem. On the contrary, larger $x$ means that the training domains are more diverse. We repeat 501 times to learn the model with ERM. Given the learnt model $f_{\hat{\bm{\theta}}}$ and the training data, we compute $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ and check the correlation between $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ and $x$. Figure 2 presents the results. Our index $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ is highly related to $x.$ The Pearson coefficient is 0.9869, and the Spearman coefficient is 0.9873. Also, the benchmark of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ that learns on the same training domains ($\tilde{\mathbb{S}}$ in 4.2) can be derived from the raw data by pooling and redistributing all data points, and we mark it by the black dashed line. If $\mathcal{V}_{\hat{\bm{\gamma}}\|\hat{\bm{\theta}}}$ is much higher than the benchmark, indicating that $x$ is not small, an OOD algorithm should be considered if better OOD generalization is demanded. Otherwise, the present algorithm (like ERM) is sufficient. The results coincide our expectation that $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ can discern whether $P^{e}$ is different. #### 5.1.2 Relationship between $\mathcal{V}$ and OOD Accuracy In this section, we use an experiment to support our proposal in Section 4.2. As previously proposed, if a model shows high test accuracy and small $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ simultaneously, it captures invariant features and avoids varying features, so it deserves to be an OOD model. In this experiment, we consider a model with high test accuracy and show that smaller $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ generally corresponds to better OOD accuracy, which supports our proposal. Figure 3: The relationship between $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ and OOD accuracy in REx (left) and IRM (right) with $\lambda\in\\{0,50,100,500,1000\\}.$ We train 400 models for each $\lambda$. The OOD accuracy and $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ enjoy high Pearson coefficient: -0.9745 (up-left), -0.9761 (down-left), -0.8417 (up- right), -0.9476 (down-right). The coefficients are negative because lower $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ forebodes better OOD property. Consider two setups: $p^{e}\in\\{0.0,0.1\\}$ and $p^{e}\in\\{0.1,0.15,0.2,0.25,0.3\\}.$ We implement IRM and REx with different penalty (note that ERM is $\lambda=0$) to check relationship between $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ and OOD accuracy. For IRM and REx, we run $190$ epochs pre-training with $\lambda=1$ and use early stopping to prevent over-fitting. With this technique, all models successfully achieve good test accuracy (within 0.1 of the oracle accuracy) and meet our requirement. Figure 3 presents the results. We can see that $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ are highly correlated to OOD accuracy in IRM and REx, with the absolute of Pearson Coefficient never less than $0.8417$. Those models learned with larger $\lambda$ present better OOD property, learning less varying features, and showing smaller $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}.$ The results are consistent with our proposal, except that when $\lambda$ is large in IRM, $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ is a little bit unstable. We have carefully examined the phenomenon and found that it is caused by computational instability when inversing Hessian with eigenvalue quite close to 0. The problem of unstable inversing happens with a low probability and can be addressed by repeating the experiment once or twice. ### 5.2 Domain Generalization: VLCS In this section, we implement the proposed metric for 4 algorithms: ERM, gDRO, Mixup and IRM on the VLCS image dataset, which is widely used for doamin generalization. We emulate a real scenario with $\mathcal{E}_{all}=\\{V,L,C,S\\}$ and $\mathcal{E}_{tr}=\mathcal{E}_{all}\backslash\\{S\\}$. As mentioned in Gulrajani & Lopez-Paz (2020), we use “training-domain validation set” method, i.e. we split a validation set for each $S\in\mathcal{E}_{tr}$ and the test accuracy is defined as the average accuracy amount the three validation sets. Note that, our goal is to use the test accuracy and $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ to measure the _OOD generalization_ , rather than to tune for the SOTA performance on a unseen domain $\\{S\\}$. Therefore, we do not apply any model selection method and just use the default hyper-parameters in Gulrajani & Lopez-Paz (2020). #### 5.2.1 Step 1: Test accuracy comparison Table 1: Step1: Test Accuracy ($\%$) Domain \| C \| L \| V \| Mean ---\|---\|---\|---\|--- ERM \| 99.29 \| 73.62 \| 77.07 \| 83.34 Mixup \| 99.32 \| 74.36 \| 78.84 \| 84.17 gDRO \| 95.79 \| 70.95 \| 75.25 \| 80.66 IRM \| 49.44 \| 44.76 \| 41.17 \| 45.12 For each algorithm, we run the naive training process 12 times and show the average of test accuracy of each algorithm in Table 1. Before calculating $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$, the learnt model should at least arrive a good test accuracy. Otherwise, there is no need to discuss its OOD performance since OOD accuracy is smaller than test accuracy. In the table, the test accuracy of ERM, Mixup and gDRO is good, but that of IRM is not. In this case, IRM will be eliminated. If an algorithm fails to reach high test accuracy first, we should first change the hyper-parameters until we observe a relatively high test accuracy. #### 5.2.2 Step 2: shuffle and standard metric comparison Now we are ready to check whether the learnt models are invariant across $\mathcal{E}_{tr}$. As mentioned in 4.2, the difference of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ represents whether how much a model is invariant across $\mathcal{E}_{tr}$. We calculate the value and the results are in Figure 4. For ERM and Mixup, the two value is nearly the same. In this case, we expect that ERM and Mixup models are invariant and should have a relatively high OOD accuracy, so no more algorithm is needed. For gDRO, we can clearly see that $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ is uniformly smaller than $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$. Therefore, gDRO models don’t treat different domains equally, and hence we predict that the OOD accuracy will be relatively low. In this case, one who starts with gDRO should turn to other algorithms if a better OOD performance is demanded. Note that, in the whole process, we know nothing about $\\{S\\}$, so the OOD accuracy is unseen. However, from the above analysis, we know that (1) in this settings, ERM and Mixup is better than gDRO; (2) one who uses gDRO can turn to other algorithms (like Mixup) for better OOD performance; (3) one who uses ERM should consider collecting more environments if he (she) still wants to improve OOD performance. So far, we finish the judgement using test accuracy and the proposed metric. Figure 4: The standard and shuffle version of the metric, i.e. $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ for ERM, Mixup and gDRO. For each algorithm, each version of the metric, we run the experiments more than 12 times in case of statistical error. Similar $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ represents invariance across $\mathcal{E}_{tr}$, which is the case of ERM and Mixup. For gDRO, $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ is clearly smaller. #### 5.2.3 Step 3: OOD accuracy results (oracle) Table 2: Step3: OOD Accuracy ($\%$) \| ERM \| Mixup \| gDRO \| IRM ---\|---\|---\|---\|--- Mean \| 62.76 \| 63.91 \| 60.17 \| 31.33 Std \| 1.16 \| 1.57 \| 2.56 \| 13.44 In this step, we fortunately obatin $\mathcal{E}_{all}$ and can check whether our judgement is reasonable. Normally, this step will not happen. We now show the OOD accuracy of four algorithms in table 2. Similar to our judgement, ERM and Mixup models achieve a higher OOD accuracy than gDRO. The performance of IRM (under this hyper-parameters) is lower than test accuracy. During the above process, we can also compare the metric of the model from the same algorithm but with different hyper-parameters (as the same in section 5.1.2). Besides, one may notice that even the highest OOD accuracy is just $63.91\%$. That is to say, to obtain OOD accuracy larger than $70\%$, we should consider collecting more environments. In the appendix A.6, we continue our real scenario to see that, if initially $\mathcal{E}_{tr}$ is more diverse, what will our metric lead us to. The whole results in VLCS can also be found in the same appendix, and the comparison of the proposed metric with the IRM penalty in formula 4 can be found there too. Besides, we show the comparison with Conditional Mutual Information in the appendix A.5. In summary, we use a realistic task to see how to judge the OOD property of learnt model using the proposed metric and test accuracy. The judgement coincides well with the real OOD performance. ## 6 Conclusion In this paper, we focus on two presently unsolved problems, that how can we discern the OOD property of multiple domains and of learnt models. To this end, we introduce influence function into OOD problem and propose our metric to help solve these issues. Our metric can not only discern whether a multi- domains problem is OOD but can also judge a model’s OOD property when combined with test accuracy. To make our calculation more meaningful, accurate and efficient, we modify influence function to domain-level and propose to use only the top model to calculate the influence. Our method is proved in simple cases and it works well in experiments. We sincerely hope that, with the help of this index, our understanding of OOD generalization will become more and more precise and thorough. ## References * Ahuja et al. (2020) Kartik Ahuja, Karthikeyan Shanmugam, Kush Varshney, and Amit Dhurandhar. 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We assume the data is generated from the following Bayesian network: $\displaystyle{\mathbf{x}}_{1}\leftarrow\mathcal{N}(0,\sigma^{2}_{e}),\quad{\mathbf{y}}\leftarrow{\mathbf{x}}_{1}^{e}{\bm{W}}_{1\rightarrow{\mathbf{y}}}+\mathcal{N}(0,1),\quad{\mathbf{x}}_{2}\leftarrow{\mathbf{y}}^{e}{\bm{W}}_{{\mathbf{y}}\rightarrow 2}+\mathcal{N}(0,\sigma^{2}_{e}).$ (8) where ${\mathbf{x}}_{1},{\mathbf{x}}_{2}\in{\mathbb{R}}^{5}$ are the features, ${\mathbf{y}}\in{\mathbb{R}}^{5}$ is the target vector, ${\bm{W}}_{1\rightarrow{\mathbf{y}}}\in{\mathbb{R}}^{5\times 5}$ and ${\bm{W}}_{{\mathbf{y}}\rightarrow 2}\in\mathbb{R}^{5\times 5}$ are the underlying parameters that are invariant across domains. The variance of gaussian noise is $\sigma^{2}_{e}$ that depends on domain. For simplicity, we denote $e=\sigma_{e}$ to represent a domain. The goal here is to linearly regress the response y on the input vector $({\bm{x}}_{1},{\bm{x}}_{2})$, i.e. $\hat{\bm{y}}={\bm{x}}_{1}\hat{\bm{W}}_{1}+{\bm{x}}_{2}\hat{\bm{W}}_{2}.$ According to the Bayesian network (8), ${\mathbf{x}}_{1}$ is the invariant feature, while the correlation between ${\mathbf{x}}_{2}$ and ${\mathbf{y}}$ is spurious and unstable since $e=\sigma_{e}$ varies across domains. Clearly, the model based only on ${\mathbf{x}}_{1}$ is an invariant model. Any invariant estimator should achieve $\hat{\bm{W}}_{1}\approx{\bm{W}}_{1\rightarrow{\mathbf{y}}}$ and $\hat{\bm{W}}_{2}\approx{\bf 0}$. Table 3: Average parameter error $\\|\hat{\bm{W}}-{\bm{W}}\\|^{2}$ and the stable measurement $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ of 500 models from ERM, IRM and REx. Here, “Causal Error” represents $\\|\hat{\bm{W}}_{1}-{\bm{W}}_{1\rightarrow{\mathbf{y}}}\\|^{2}$ and “Non-causal Error” represents $\\|\hat{\bm{W}}_{2}\\|^{2}$. Method \| $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ \| Causal Error \| Non-causal Error ---\|---\|---\|--- ERM \| 15.844 \| 0.582 \| 0.581 IRM \| 5.254 \| 0.122 \| 0.109 REx \| 1.341 \| 0.042 \| 0.033 Now consider five training domains $e\in\mathcal{E}_{tr}=\\{0.2,0.7,1.2,1.7,2.2\\}$ , each containing 1000 data points. We estimate three linear models using ERM, IRM and REx respectively and record the parameter error as well as $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ (note that ${\bm{\gamma}}$ is ${\bm{\theta}}$ here). Table 3 presents the results among 500 repetitions. As expected, IRM and REx learn more invariant relationships than ERM (smaller causal error) and better avoid non-causal variables ($\hat{\bm{W}}_{2}\approx{\bf 0}$). Furthermore, the proposed measurement $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ is highly related to invariance, i.e. model with better OOD property achieves smaller $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$. This results coincides our understanding. ### A.2 Proof of an Example In this section, we use a simple model to illuminate the validity of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ proposed in Section 4. Consider a structural equation model (Wright (1921)): $\displaystyle{\textnormal{x}}_{1}\sim P_{x}^{e},\quad{\textnormal{y}}\leftarrow{\textnormal{x}}_{1}+\mathcal{N}(0,1),\quad{\textnormal{x}}_{2}\leftarrow{\textnormal{y}}+\mathcal{N}(0,\sigma_{e}^{2})$ where $P_{x}^{e}$ is a distribution with a finite second-order moment, i.e. $\mathbb{E}{\textnormal{x}}_{1}^{2}<+\infty$, and $\sigma_{e}^{2}$ is the variance of the noise term in ${\textnormal{x}}_{2}.$ Both $P_{x}^{e}$ and $\sigma_{e}^{2}$ vary across domains. For simplicity, we assume there are infinite training data points collected from two training domains $\mathcal{E}_{tr}=\\{(P_{x}^{1},\sigma_{1}^{2}),(P_{x}^{2},\sigma_{2}^{2})\\}$. Our goal is to predict y from ${\mathbf{x}}:=({\textnormal{x}}_{1},{\textnormal{x}}_{2})^{\top}$ using a least-squares predictor $\hat{y}={\bm{x}}^{\top}\hat{\bm{\beta}}:=x_{1}\hat{\beta}_{1}+x_{2}\hat{\beta}_{2}$. Here we consider two algorithms: ERM and IRM with $\lambda\rightarrow+\infty$. According to Arjovsky et al. (2019), using IRM we obtain ${\bm{\beta}}_{\text{IRM}}\rightarrow(1,0)^{\top}$. Intuitively, ERM will exploit both ${\textnormal{x}}_{1}$ and ${\textnormal{x}}_{2}$, thus achieving a better regression model. However, since relationship between y and ${\textnormal{x}}_{2}$ varies across domains, our index will be huge in such condition. Conversely, $\beta_{\text{IRM}}$ only uses invariant features ${\textnormal{x}}_{1}$, thus $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}\rightarrow-\infty$. Note that we do not have an embedding model here, so $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}=\mathcal{V}_{{\bm{\beta}}}$. ERM we denote $\displaystyle\ell({\bm{\beta}})=\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\ell_{e}({\bm{\beta}})\quad\text{with}\quad\ell_{e}({\bm{\beta}})=\mathbb{E}_{e}({\textnormal{y}}-{\mathbf{x}}{\bm{\beta}})^{2}.$ Note that in $\mathbb{E}_{e}$, ${\textnormal{x}}_{1}$ is sample from $P_{x}^{e}$. We then have $\displaystyle\frac{\partial\ell({\bm{\beta}})}{{\bm{\beta}}}=-\sum_{e\in\mathcal{E}_{tr}}\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]=-\frac{2}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\left(\begin{array}[]{c}\mathbb{E}_{e}[{\textnormal{x}}_{1}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\\\ \mathbb{E}_{e}[{\textnormal{x}}_{2}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\\\ \end{array}\right)$ To proceed further, we denote $\displaystyle\bar{d}=\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\mathbb{E}_{e}{\mathbf{x}}_{1}^{2},\quad s=\sum_{e\in\mathcal{E}_{tr}}\sigma_{e}^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}.$ By solving the following equations: $\displaystyle\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\mathbb{E}_{e}[{\textnormal{x}}_{1}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]=\bar{d}(1-\beta_{1}-\beta_{2})=0$ and $\displaystyle\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\mathbb{E}_{e}[{\textnormal{x}}_{2}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]=(\bar{d}+1)(1-\beta_{1}-\beta_{2})+\beta_{1}-\frac{s}{\|\mathcal{E}_{tr}\|}\beta_{2}=0$ we have $\hat{\bm{\beta}}=(\hat{\beta}_{1},\hat{\beta}_{2})^{\top}$ with $\displaystyle\hat{\beta}_{1}=\frac{s}{s+2},\quad\hat{\beta}_{2}=\frac{2}{s+2}.$ Now we calculate our index. It is easy to see that $\displaystyle\frac{\partial\ell_{e}({\bm{\beta}})}{\beta_{1}}$ $\displaystyle=$ $\displaystyle-2\mathbb{E}_{e}[{\textnormal{x}}_{1}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]=-2\mathbb{E}_{e}{\textnormal{x}}_{1}^{2}(1-\beta_{1}-\beta_{2})$ $\displaystyle\frac{\partial\ell_{e}({\bm{\beta}})}{\beta_{2}}$ $\displaystyle=$ $\displaystyle-2\mathbb{E}_{e}[{\textnormal{x}}_{2}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]=-2[(\mathbb{E}_{e}x_{1}^{2}+1)(1-\beta_{1}-\beta_{2})+\beta_{1}-\sigma_{e}^{2}\beta_{2}].$ Therefore, $\displaystyle\nabla\ell_{1}({\bm{\beta}})-\nabla\ell_{2}({\bm{\beta}})=\left(\begin{array}[]{cc}0\\\ 2\beta_{2}(\sigma_{1}^{2}-\sigma_{2}^{2})\end{array}\right)\quad\text{and}\quad\nabla\ell_{1}(\hat{\bm{\beta}})-\nabla\ell_{1}(\hat{\bm{\beta}})=\left(\begin{array}[]{cc}0\\\ \frac{4(\sigma_{1}^{2}-\sigma_{2}^{2})}{s+2}\end{array}\right)$ (14) On the other hand, calculate the hessian and we have $\displaystyle{\bm{H}}_{\text{ERM}}=\left(\begin{array}[]{cc}2\bar{d}&2\bar{d}\\\ 2\bar{d}&2\bar{d}+s+2\\\ \end{array}\right)\quad\text{and}\quad{\bm{H}}^{-1}=\frac{1}{2\bar{d}(s+2)}\left(\begin{array}[]{cc}2\bar{d}+s+2&-2\bar{d}\\\ -2\bar{d}&2\bar{d}\\\ \end{array}\right).$ Then we have (note that $\mathcal{IF}(\hat{\bm{\beta}},{\mathbb{S}}^{e})={\bm{H}}^{-1}\nabla\ell_{e}(\hat{\bm{\beta}})$) $\displaystyle\mathcal{V}_{\hat{\bm{\beta}}}$ $\displaystyle=$ $\displaystyle\ln(\\|\mathrm{Cov}_{e\in\mathcal{E}}(\mathcal{IF}(\hat{\bm{\beta}},{\mathbb{S}}^{e}))\\|_{2})$ $\displaystyle=$ $\displaystyle\ln(\frac{1}{4}\\|(\mathcal{IF}_{1}-\mathcal{IF}_{2})(\mathcal{IF}_{1}-\mathcal{IF}_{2})^{\top}\\|_{2})$ $\displaystyle=$ $\displaystyle\ln(\frac{1}{4}\\|\mathcal{IF}_{1}-\mathcal{IF}_{2}\\|^{2})$ $\displaystyle=$ $\displaystyle 2\ln(\frac{1}{2}\\|{\bm{H}}^{-1}(\nabla\ell_{1}(\hat{\bm{\beta}})-\nabla\ell_{2}(\hat{\bm{\beta}}))\\|)$ $\displaystyle=$ $\displaystyle 2\ln(\frac{1}{4\bar{d}(s+2)}\\|\left(\begin{array}[]{cc}2\bar{d}+s+2&-2\bar{d}\\\ -2\bar{d}&2\bar{d}\\\ \end{array}\right)\left(\begin{array}[]{cc}0\\\ \frac{4(\sigma_{1}^{2}-\sigma_{2}^{2})}{s+2}\end{array}\right)\\|)$ $\displaystyle=$ $\displaystyle 2\ln(\frac{2\sqrt{2}\|\sigma_{1}^{2}-\sigma_{2}^{2}\|}{(s+2)^{2}})$ where the third equation holds because the rank of matrix is $1$. Clearly, when $\|\sigma_{1}^{2}-\sigma_{2}^{2}\|\rightarrow 0$ (means two domains become identical), our index $\mathcal{V}_{\bm{\beta}}\rightarrow-\infty$. Otherwise, given $\sigma_{1}\not=\sigma_{2}$, we have $\mathcal{V}_{\bm{\beta}}>-\infty$, showing that ERM captures varied features. IRM We now turn to IRM model and show that $\mathcal{V}_{\bm{\beta}}\rightarrow-\infty$ when $\lambda\rightarrow+\infty$, thus proving IRM learnt model $\hat{\bm{\beta}}_{\text{IRM}}$ does achieve smaller $\mathcal{V}_{\bm{\beta}}$ compared with $\hat{\bm{\beta}}$ in ERM. Under IRM model, assuming the tuning parameter is $\lambda$, we have $\displaystyle\mathcal{L}({\bm{\beta}})=\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\mathbb{E}_{e}[({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})^{2}]+4\lambda\\|\mathbb{E}_{e}[{\mathbf{x}}^{\top}{\bm{\beta}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\\|^{2}.$ Then we have the gradient with respect ${\bm{\beta}}$: $\displaystyle\nabla\mathcal{L}({\bm{\beta}})$ $\displaystyle=$ $\displaystyle\frac{1}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\big{(}-2\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]+8\lambda\mathbb{E}_{e}[{\mathbf{x}}^{\top}{\bm{\beta}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-2{\mathbf{x}}^{\top}{\bm{\beta}})]\big{)},$ and the Hessian matrix $\displaystyle{\bm{H}}$ $\displaystyle=$ $\displaystyle{\bm{H}}_{\text{ERM}}+\frac{8\lambda}{\|\mathcal{E}\|}\sum_{e\in\mathcal{E}}\big{(}\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-2{\mathbf{x}}^{\top}{\bm{\beta}})]\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-2{\mathbf{x}}^{\top}{\bm{\beta}})]^{\top}-2\mathbb{E}_{e}[{\mathbf{x}}^{\top}{\bm{\beta}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\mathbb{E}_{e}[{\mathbf{x}}{\mathbf{x}}^{\top}]\big{)}.$ (17) Denote ${\bm{\beta}}_{\lambda}$ the solution of IRM algorithm on $\mathcal{E}_{tr}$ when penalty is $\lambda$. From Arjovsky et al. (2019) we know ${\bm{\beta}}_{\lambda}\rightarrow{\bm{\beta}}_{\text{IRM}}:=(1,0)^{\top}$. To show $\lim_{\lambda\rightarrow+\infty}\mathcal{V}_{{\bm{\beta}}_{\lambda}}=-\infty$, we only need to show that $\displaystyle\lim_{\lambda\rightarrow+\infty}{\bm{H}}^{-1}(\nabla\ell_{1}({\bm{\beta}}_{\lambda})-\nabla\ell_{2}({\bm{\beta}}_{\lambda}))=\mathbf{0}$ We prove this by showing that $\displaystyle\lim_{\lambda\rightarrow+\infty}{\bm{H}}^{-1}(\lambda)=\mathbf{0}\quad\text{and}\quad\lim_{\lambda\rightarrow+\infty}\nabla\ell_{1}({\bm{\beta}}_{\lambda})-\nabla\ell_{2}({\bm{\beta}}_{\lambda})=\mathbf{0}$ (18) simultaneously. We add $(\lambda)$ after ${\bm{H}}^{-1}$ to show that ${\bm{H}}^{-1}$ is a continuous function of $\lambda$. Rewrite ${\bm{H}}$ in formula 17 as $\displaystyle{\bm{H}}(\lambda,{\bm{\beta}}_{\lambda})={\bm{H}}_{\text{ERM}}+\lambda F({\bm{\beta}}_{\lambda})$ where $\displaystyle F({\bm{\beta}})$ $\displaystyle=$ $\displaystyle\frac{8}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\big{(}\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-2{\mathbf{x}}^{\top}{\bm{\beta}})]\mathbb{E}_{e}[{\mathbf{x}}({\textnormal{y}}-2{\mathbf{x}}^{\top}{\bm{\beta}})]^{\top}-2\mathbb{E}_{e}[{\mathbf{x}}^{\top}{\bm{\beta}}({\textnormal{y}}-{\mathbf{x}}^{\top}{\bm{\beta}})]\mathbb{E}_{e}[{\mathbf{x}}{\mathbf{x}}^{\top}]\big{)}$ $\displaystyle\lim_{{\bm{\beta}}_{\lambda}\rightarrow{\bm{\beta}}_{\text{IRM}}}F({\bm{\beta}}_{\lambda})$ $\displaystyle=$ $\displaystyle\frac{4}{\|\mathcal{E}_{tr}\|}\sum_{e\in\mathcal{E}_{tr}}\left(\begin{array}[]{c}-\mathbb{E}_{e}{\textnormal{x}}_{1}^{2}\\\ 1-\mathbb{E}_{e}{\textnormal{x}}_{1}^{2}\end{array}\right)\left(\begin{array}[]{cc}-\mathbb{E}_{e}{\textnormal{x}}_{1}^{2}&1-\mathbb{E}_{e}{\textnormal{x}}_{1}^{2}\end{array}\right)=F({\bm{\beta}}_{\text{IRM}})\text{ exists.}$ Obviously, $F({\bm{\beta}}_{\text{IRM}})$ is positive definite. Therefore, we have $\displaystyle\lim_{\lambda\rightarrow+\infty}{\bm{H}}(\lambda,{\bm{\beta}}_{\lambda})^{-1}$ $\displaystyle=$ $\displaystyle\lim_{\lambda\rightarrow+\infty}\lim_{{\bm{\beta}}_{\lambda}\rightarrow{\bm{\beta}}_{\text{IRM}}}[{\bm{H}}_{\text{ERM}}+\lambda F({\bm{\beta}}_{\lambda})]^{-1}$ $\displaystyle=$ $\displaystyle\lim_{\lambda\rightarrow+\infty}[{\bm{H}}_{\text{ERM}}+\lambda F({\bm{\beta}}_{\text{IRM}})]^{-1}$ $\displaystyle=$ $\displaystyle\mathbf{0}$ The first equation holds because $\lim_{\lambda\rightarrow+\infty}F({\bm{\beta}}_{\lambda})=F({\bm{\beta}}_{\text{IRM}})$ has the limit and is not $\mathbf{0}$, and the last equation holds because the eigenvalue of ${\bm{H}}$ goes to $+\infty$ when $\lambda\rightarrow+\infty$. Now consider $\nabla\ell_{1}({\bm{\beta}}_{\lambda})-\nabla\ell_{2}({\bm{\beta}}_{\lambda})$. According to formula 14, we have $\displaystyle\lim_{\lambda\rightarrow+\infty}\nabla\ell_{1}({\bm{\beta}}_{\lambda})-\nabla\ell_{2}({\bm{\beta}}_{\lambda})$ $\displaystyle=$ $\displaystyle\lim_{{\bm{\beta}}_{\lambda}\rightarrow{\bm{\beta}}_{\text{IRM}}}\nabla\ell_{1}({\bm{\beta}}_{\lambda})-\nabla\ell_{2}({\bm{\beta}}_{\lambda})$ $\displaystyle=$ $\displaystyle\nabla\ell_{1}({\bm{\beta}}_{\text{IRM}})-\nabla\ell_{2}({\bm{\beta}}_{\text{IRM}})$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}0\\\ 2\beta_{2}(\sigma_{1}^{2}-\sigma_{2}^{2})\end{array}\right)$ $\displaystyle=$ $\displaystyle\mathbf{0}$ Hence we finish proof of formula 18 and show that $\mathcal{V}_{\bm{\beta}}\rightarrow-\infty$ in IRM. ### A.3 Formula (5) This section shows the derivation of the expression (5). Recall that the training dataset $\mathbb{S}=\\{\mathbb{S}^{1},...,\mathbb{S}^{m}\\}$ and the objective function $\displaystyle\mathcal{L}(f,{\mathbb{S}})=\ell(f,{\mathbb{S}})+\lambda R(f,{\mathbb{S}}),$ where the second term on the right hand side is the regularization. As to ERM, the regularization term is zero. With the feature extractor (${\bm{\beta}}$) fixed, we upweight a domain $\mathbb{S}^{e}$. The new objective function is $\displaystyle\mathcal{L}_{+}({\bm{\theta}},\mathbb{S},\delta)=\mathcal{L}({\bm{\theta}},\mathbb{S})+\delta\cdot\ell({\bm{\theta}},{\mathbb{S}}^{e})$ Notice that when upweight an domain, we only upweight the empirical loss on the corresponding domain. Further, we denote $\hat{\bm{\gamma}},\hat{\bm{\gamma}}_{+}$ as the optimal solutions before and after upweighting a domain. It is easy to see that $\\|\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}\\|\rightarrow 0$ when $\delta\rightarrow 0$. Following the derivation in Koh & Liang (2017), according to the first-order Taylor expansion of $\nabla_{\bm{\gamma}}\mathcal{L}_{+}({\bm{\theta}},\mathbb{S},\delta)$ with respect to ${\bm{\gamma}}$ on $\hat{\bm{\gamma}}$, $\displaystyle\mathbf{0}$ $\displaystyle=$ $\displaystyle\nabla_{\bm{\gamma}}[\mathcal{L}(\hat{\bm{\theta}}_{+},\mathbb{S})+\delta\ell(\hat{\bm{\theta}}_{+},\mathbb{S}^{e})]$ $\displaystyle=$ $\displaystyle\nabla_{\bm{\gamma}}(\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})+\delta\ell(\hat{\bm{\theta}},\mathbb{S}^{e}))+\nabla_{\bm{\gamma}}^{2}[\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})+\delta\ell(\hat{\bm{\theta}},\mathbb{S}^{e})](\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}})+o(\\|\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}\\|)$ $\displaystyle=$ $\displaystyle\delta\nabla_{\bm{\gamma}}\ell(\hat{\bm{\theta}},\mathbb{S}^{e})+\nabla_{\bm{\gamma}}^{2}[\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})+\delta\ell(\hat{\bm{\theta}},\mathbb{S}^{e})](\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}})+o(\\|\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}\\|)$ Assume that $\nabla^{2}[\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})+\delta\ell(\hat{\bm{\theta}},\mathbb{S}^{e})]$ is invertible, we have $\displaystyle\frac{\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}}{\delta}$ $\displaystyle=$ $\displaystyle[\nabla_{\bm{\gamma}}^{2}\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})+\delta\ell(\hat{\bm{\theta}},\mathbb{S}^{e})]^{-1}\nabla_{\bm{\gamma}}\ell(\hat{\bm{\theta}},\mathbb{S}^{e})+o(\\|\frac{\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}}{\delta}\\|)$ $\displaystyle\lim_{\delta\rightarrow 0}\frac{\hat{\bm{\gamma}}_{+}-\hat{\bm{\gamma}}}{\delta}$ $\displaystyle=$ $\displaystyle[\nabla_{\gamma}^{2}\mathcal{L}(\hat{\bm{\theta}},\mathbb{S})]^{-1}\nabla_{\bm{\gamma}}\ell(\hat{\bm{\theta}},\mathbb{S}^{e})$ Note that this derivation is not fully rigorous. Please refer to Van der Vaart (2000) for more rigorous discussions about influence function. The reason that ${\bm{\beta}}$ should be fixed is as follows. First, if ${\bm{\beta}}$ can be varied, then the change of ${\bm{\theta}}$ will become: $\displaystyle\left(\begin{matrix}H_{{\bm{\gamma}}{\bm{\gamma}}}&H_{{\bm{\gamma}}{\bm{\beta}}}\\\ H_{{\bm{\beta}}{\bm{\gamma}}}&H_{{\bm{\beta}}{\bm{\beta}}}\end{matrix}\right)^{-1}\left(\begin{matrix}\nabla_{{\bm{\gamma}}}l(\hat{\bm{\theta}},{\mathbb{S}}^{e})\\\ \nabla_{{\bm{\beta}}}l(\hat{\bm{\theta}},{\mathbb{S}}^{e})\end{matrix}\right).$ Therefore, the computational cost is similar to calculate and inverse the whole hessian matrix. Most importantly, without fixing ${\bm{\beta}}$, the change of ${\bm{\gamma}}$ is somehow useless. Say when upweighting ${\mathbb{S}}^{e}$, the use of a feature decreases. It’s possible, however, that the parameter in ${\bm{\gamma}}$ corresponding to the feature increases while ${\bm{\beta}}$ decreases a larger scale. In this case, the use of the feature decreases but ${\bm{\gamma}}$ increases. Without fixing ${\bm{\beta}}$, the change of ${\bm{\gamma}}$ calculated by influence function may provide no information about the use a feature. Therefore, we argue that, fixing ${\bm{\beta}}$ is a “double-win” choice. ### A.4 Accuracy is not enough In Introduction, we have given an example where test accuracy misleads us. In this section, we will first supplement some examples where test accuracy not only misjudge different algorithms, but it also misjudges the OOD property of models learnt with different penalty within the same algorithm. After that, we will show the universality of these problems and why test accuracy fails. Figure 5: Experiments in Colored MNIST to show test accuracy (x-axis) cannot be used to judge model learnt with different penalty. Consider two test domains with $p^{\text{test}}=0.2$ (up penals) and $p^{\text{test}}=0.3$ (down penals). For each $\lambda$, we run IRM and REx 500 times. We can see that when $\lambda$ increases from $\lambda=0$ to $\lambda=1000$, the OOD accuracy also increases, but test accuracy does not. When $p^{test}=0.3$, their relationship becomes more perplexed. Consider two training domains $p^{e}\in\\{0.0,0.1\\},$ and a test domain with flip rate denoted by $p^{\text{test}}$. We implement IRM and REx with penalty $\lambda\in\\{0,50,100,500,1000\\}$ to check the relationship between test accuracy and OOD accuracy. The training process is identical to the experiment in section 5.1.2. As results showed in Figure 5, when OOD property of model gradually improves (caused by gradually increasing $\lambda$), its relationship with test accuracy is either completely (when $p^{\text{test}}$ is 0.2) or partly (when $p^{\text{test}}$ is 0.3) negatively correlated. This phenomenon reveals the weakness of test accuracy. If one wants to select a $\lambda$ when $p^{\text{test}}$ is 0.3, judged by test accuracy, $\lambda=50$ may be the best choice, no matter in IRM or REx. However, the model learnt with $\lambda=50$ has OOD accuracy even _less than a random guess model_. Whether test accuracy is positively, negatively correlated or irrelevant to model’s OOD property mainly depends on the “distance” between test domain and the “worst” domain for the model. If test accuracy happens to be the lowest among all the domains, we directly have OOD accuracy equals to test accuracy. In practice, however, their distance may be huge, and this is precisely the difficulty of OOD generalization. For example, we are accessible to images of cows in grasslands, woods and forests, but cows in desert are rare. At this point, the “worst” domain is certainly far from what we can get. If we expect a model to capture the real feature of cows, the model should avoid any usage of background color. However, a model based on color will perform consistently well (better than any OOD model) no matter in grasslands, woods and forests since all of the available domains are green background in general. In Colored MNIST, test accuracy fails in the same way. Such situations are quite common. Generally, within domains we have, there may be some features that are strongly correlated to the prediction but are slightly varied across domains. These features are spurious, given that their relationship with prediction is significantly disparate in other domains to which we want to generalize. However, using these features in prediction will easily achieve high test accuracy. Consequently, it will be extremely risky to judge models merely by test accuracy. ### A.5 Conditional mutual information A possible alternative of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ may be Conditional Mutual Information (CMI). For three continuous random variables $X$, $Y$, $Z$, the CMI is defined as $\displaystyle I(X;Y\|Z)=\int\int\int p(x,y,z)\log\frac{p(x,y,z)}{p(x,z)p(y\|z)}dxdydz$ (21) where $p(\cdot)$ is the probability density function. Consider $I(e;y\|\Phi(x))$ or $I(e;y\|\hat{y})$, i.e. the mutual information of $e$ and true label $y$, given the features or the prediction $\hat{y}$ of $x$. The insight is that, if the model is invariant across different domains, then little information about $e$ should be contained in $y$ given $\Phi(x)$. Otherwise, if the prediction $\hat{y}$ is highly correlated to $e$, then the mutual information will be high. Figure 6: Experiments of the relationship between OOD accuracy and CMI (true or estimated using the method in Sen et al. (2017)). Models are trained by REx (left) and IRM (right) with $\lambda\in\\{0,10,100,1000\\}$. We train 50 models for each $\lambda$ and calculate the true CMI $I(e;y\|\hat{y})$ or CCIT value. As analyzed in the appendix A.5, true CMI enjoys a highly correlated relationship to OOD accuracy, with Pearson Coefficient $-0.9923$ (left) and $-0.9858$ (right). However, the estimated value shows a completely different picture, with Pearson Coefficient $-0.0768$ (left) and $-0.1193$ (right). This metric seems to be promising. However, the numerical estimation of CMI remains a challenge. To this end, previous works have done a lot to solve this problem, including CCMI proposed in Mukherjee et al. (2020) and CCIT proposed in Sen et al. (2017). In this part, we will first calculate true $I(e;y\|\hat{y})$ in a simple Colored MNIST experiment to show that if there is no estimation problem, CMI could be a potential metric to judge the OOD property of the learnt model, at least in a simple, discrete task. We then run the code provided by Sen et al. (2017) (https://github.com/rajatsen91/CCIT) to show that even in this simple task, the estimation of CMI may severely influence its performance. Specifically, the experimental setting is similar to that in subsection 5.1.2, with two OOD algorithm and number of training domains in $\\{2,5\\}$. For each algorithm, we consider the penalty weight $\lambda\in\\{0,10,100,1000\\}$, run the algorithm 50 times, and record their OOD accuracy as well as true CMI value or CCIT value. The results are shown in Figure 6. We can see that in the case when true CMI can be easily calculated, especially in the case when the number of domains is small and the task is discrete (not continuous), CMI is highly correlated to OOD accuracy. However, in a regression task or in a task when directly calculating the value of CMI becomes impractical, the estimation process may severely destroy the correlation, and may also result in an inverse correlation. Therefore, we summarized the estimation of CMI has limited its utility. We leave the fine-grained analysis of the relationship between CMI, estimated CMI and OOD property to future works. ### A.6 Results on VLCS #### A.6.1 Continued Scenario Table 4: Domain $C$ out: Test Accuracy ($\%$) Domain \| L \| S \| V \| Mean ---\|---\|---\|---\|--- ERM \| 73.43 \| 73.87 \| 79.15 \| 75.48 Mixup \| 73.74 \| 74.54 \| 78.65 \| 75.64 gDRO \| 71.40 \| 71.95 \| 77.19 \| 73.51 IRM \| 49.61 \| 38.64 \| 45.35 \| 44.53 This is a continuation of section 5.2. Say in this task, $\mathcal{E}_{all}$ remains the four domains but ${}_{c}E_{tr}=\\{L,S,V\\}$ (empirically we find it more diverse). Similarly, we start with test accuracy shown in table 4. In this step, the situation is the same, i.e. IRM should be eliminated until proper hyper-parameters are found. In step 2, we show the comparison between $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ of three algorithms in Figure 7. As we can see, this time the two value are similar for all three algorithms, including gDRO. This is different from the case when $S$ is unseen. In this case, we predict that all of the three algorithms should achieve high OOD accuracy. In fact, if we act as the oracle and calculate their OOD performance, we will find that our judgement is close to the reality: ERM, Mixup and gDRO achieve OOD accuracy from $70.55\%$ to $72.87\%$. According to the confidence interval, they difference are not satistically significant. As for IRM, the OOD accuracy is $38.64\%$. One who use ERM, Mixup or gDRO should be satisfied for the performance, since higher demand is somehow impractical! Figure 7: The standard and shuffle version of the metric, i.e. $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$ for ERM, Mixup and gDRO. This time, all three algorithms show similar $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\beta}}}$ and $\tilde{\mathcal{V}}_{{\bm{\gamma}}\|{\bm{\beta}}}$. #### A.6.2 Full results and comparison with IRM penalty As mentioned in Section 5.2, we consider ERM, gDRO, Mixup and IRM on VLCS image dataset. We report the full results here, and compare the performance of out metric $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ with IRM penalty in formula 4. Thorough the whole experiments, $\mathcal{E}_{all}=\\{V,L,C,S\\}$. We construct four experimental settings. In each setting, one domain is removed and the rest consists of $\mathcal{E}_{tr}$. For each domain in $\mathcal{E}_{tr}$, we split a validation set, and test accuracy is the average accuracy amount validation sets. The results are shown in Table 5. First, our results coincide with Gulrajani & Lopez-Paz (2020) that ERM nearly outperforms any algorithms. We can see that the OOD accuracy of ERM is either the highest or only slightly lower than Mixup. Meanwhile, it has a relatively small $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$. Second, higher OOD accuracy corresponds to lower $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$. In addition, we notice that IRM has a relatively low test accuracy and OOD accuracy. We explain the phenomenon by an improper hyper-parameters in IRM, although we didn’t change the default hyper-parameters in the code of Gulrajani & Lopez- Paz (2020) (https://github.com/facebookresearch/DomainBed). No matter what, this phenomenon provides a good example in which we can compare our metric with IRM penalty and discuss their advantages and disadvantages. Table 5: Experiments in VLCS with 4 algorithms. OOD accuracy means the min accuracy in $\mathcal{E}_{all}$. We use training-domain validation method mentioned in Gulrajani & Lopez-Paz (2020), so test accuracy is the average accuracy of three split validation set. “Domain” means which domain is excluded, i.e. which domain is in $\mathcal{E}_{all}\backslash\mathcal{E}_{tr}$. In each setting, we run each algorithm 12 times and report the mean and (std). Note that in a real implementation, IRM penalty can be negative. OOD accuracy (%) \| Test accuracy (%) ---\|--- Domain \| C \| L \| S \| V \| Domain \| C \| L \| S \| V ERM \| 72.54 \| 61.48 \| 62.76 \| 65.59 \| ERM \| 75.48 \| 84.55 \| 83.33 \| 81.49 \| (2.62) \| (2.31) \| (1.16) \| (2.27) \| \| (3.37) \| (10.61) \| (11.64) \| (12.83) Mixup \| 72.87 \| 62.10 \| 63.91 \| 63.81 \| Mixup \| 75.65 \| 84.92 \| 84.17 \| 81.02 \| (2.04) \| (3.10) \| (1.57) \| (3.64) \| \| (2.80) \| (10.54) \| (11.07) \| (13.52) gDRO \| 70.55 \| 61.64 \| 60.17 \| 62.35 \| gDRO \| 73.51 \| 82.82 \| 80.66 \| 80.03 \| (1.91) \| (3.92) \| (2.56) \| (2.11) \| \| (3.27) \| (9.41) \| (11.17) \| (11.50) IRM \| 38.64 \| 38.84 \| 31.33 \| 39.50 \| IRM \| 44.53 \| 48.83 \| 45.13 \| 51.94 \| (0.54) \| (0.31) \| (13.44) \| (2.35) \| \| (4.94) \| (10.53) \| (16.53) \| (11.66) $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ (our metric) \| IRM penalty (e-4) Domain \| C \| L \| S \| V \| Domain \| C \| L \| S \| V ERM \| 2.0468 \| 1.9084 \| 1.8476 \| 1.9811 \| ERM \| 1.78 \| 1.48 \| 1.43 \| 1.63 \| (0.3474) \| (0.3231) \| (0.2887) \| (0.3955) \| \| (1.88) \| (1.03) \| (0.75) \| (2.04) Mixup \| 2.6996 \| 2.4417 \| 2.5810 \| 2.8780 \| Mixup \| 75.4 \| 57.7 \| 65.5 \| 48.3 \| (0.1926) \| (0.2003) \| (0.1492) \| (0.2304) \| \| (32.6) \| (26.4) \| (37.2) \| (30.6) gDRO \| 3.3371 \| 4.8520 \| 5.0915 \| 5.1675 \| gDRO \| 9.42 \| 2.13 \| 2.6 \| 1.94 \| (0.1385) \| (0.2515) \| (0.278) \| (0.3507) \| \| (10.1) \| (3.41) \| (2.46) \| (4.37) IRM \| 8.1820 \| 6.8329 \| 7.6234 \| 8.1288 \| IRM \| 2.59 \| 0.96 \| 0 \| 2.71 \| (0.9523) \| (0.6646) \| (0.6792) \| (0.974) \| \| (3.31) \| (3.31) \| (4.77) \| (9.2) Despite that IRM could be a good OOD algorithm, using IRM penalty as the metric to judge the OOD property of a learnt model still has much weakness, and some are severe. First, in different tasks, the value of $\lambda$ to obtain an OOD model may be different, so as other hyper-parameters like “anneal_steps” in IRM code. Without exhaustive search on the proper value of the hyper-parameters, it’s easy that IRM overfits on the penalty term (which is the situation in VLCS). When IRM overfits, the IRM penalty will become quite small (higher $\lambda$ often leads to smaller penalty), but absolutely overfitting on penalty term will not result in good OOD accuracy. Therefore, the balance between loss and penalty is important. However, how to find a balanced point? This is a model selection problem, and Gulrajani & Lopez-Paz (2020) propose that an OOD algorithm without model selection is not complete. No matter what to be used as the metric, it cannot be IRM penalty since we cannot use what is included in the training process as the metric to select training hyper-parameters. Second, IRM penalty shows a bias on different algorithms. In the Table 5, the IRM penalty of IRM is smaller than most algorithms. Besides, although the OOD accuracy of Mixup is similar to ERM, its IRM penalty is significantly higher. This is not strange but will limit the usage of IRM penalty. As for our metric, we mention that small $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ is better. However, the understanding of “smallness” is based on the relative value of the shuffle version and standard version of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$. As mentioned in section 5.2, when $\mathcal{E}_{all}\backslash\mathcal{E}_{tr}=\\{S\\}$, we can see that shuffle section 5.2 is obviously smaller than standard version in gDRO, but in ERM and Mixup, these value are relatively close or indistinguishable. In this case, we know that gDRO captures less invariant features and is not OOD than the other two algorithms. During the whole process, we can circumvent the direct comparison of $\mathcal{V}_{{\bm{\gamma}}\|{\bm{\theta}}}$ in different algorithms, which is quite important. In summary, IRM penalty makes IRM a good algorithm, but using it as the general metric of OOD performance is completely another picture.
Modelling and discretization of flow in porous media with thin, full-tensor permeability inclusions M. Starnoni1,2, I. Berre1, E. Keilegavlen1, & J.M. Nordbotten1 1Department of Mathematics, University of Bergen, Bergen, Norway 2Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy ## Abstract When modelling fluid flow in fractured reservoirs, it is common to represent the fractures as lower-dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modelling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this paper, we introduce a generalized dimensional reduced model which preserves full-tensor permeability effects also in the out-of-plane direction of the inclusion. The governing equations of flow for the lower-dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed-dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both 2D and 3D. Key points * • Existing local discretizations of flow in fractured porous media fail in modelling out-of plane anisotropic properties of thin inclusions * • We present a new framework to modelling and discretizing flow in porous media with thin, full-tensor permeability inclusions * • We show convergence of our method in both 2D and 3D faulted porous media Keywords discretization, faults, permeability, mixed-dimensional, flow, porous media ## 1 Introduction Modeling and simulation of flow in porous media with faults, fractures, and other thin inclusions representing discontinuities is central to a wide range of subsurface engineering applications, including geothermal energy exploitation (Bödvarsson and Tsang,, 1982), shale gas extraction (Cao et al.,, 2016), carbon sequestration (Johnson et al.,, 2009), and energy storage (Nagelhout and Roest,, 1997). The inclusions are characterized by a high aspect ratio, and permeability significantly different from that of the host medium; hence, they severely affect flow patterns. This poses a challenge for traditional simulation models, which are based on upscaling of fine-scale details into an equivalent permeability (Oda,, 1985; Farmer,, 2002; Liu et al.,, 2016; Sævik et al.,, 2013). We instead focus on an alternative approach, which explicitly represents the inclusions in the mathematical and simulation models and thereby to a large degree avoids challenges relating to parameter upscaling. To avoid elongated cells at the inclusion in the computational grid, it is common to represent the inclusions as co-dimension one objects embedded in the host medium (Boon et al.,, 2018; Nordbotten et al.,, 2019). The intersection of inclusions further gives rise to line and point intersections of co- dimension two and three. Each of these objects (matrix, inclusions, and intersection points and lines) are represented as independent subdomains separated by interfaces. We refer to this representation of the geometry as mixed-dimensional. Governing equations for fluid flow in lower-dimensional representation of the inclusion can be derived by integration in the direction orthogonal to the inclusion. This leads to a decomposition of the governing equations into an in-plane component that represents flow within the inclusion, and an out-of- plane component that couples flow between the inclusion and the host medium. While the in-plane flow has been modeled with both linear and non-linear, as well as both isotropic and non-isotropic flow models (Martin et al.,, 2005; Reichenberger et al.,, 2006; Brenner et al.,, 2017, 2018), existing models for the coupling term are limited by an assumption on orthogonal flow between inclusion and host. Reduced order models for flow were also developed for aquifers, leading to the same set of equations, see for instance Bear, (1979), Yortsos, (1995), and Nordbotten and Celia, (2011). These existing models will be denoted as ”local” in the following, meaning that each partial differential equation (PDE) contains only quantities associated with the subdomain where the PDE is defined. Local models generally work well when the inclusion is a joint (tensile fracture). However, inclusions with a more complex geological history may have significantly more complex flow properties in the out-of-plane direction. For instance, the damage zone in the vicinity of faults may exhibit shear fractures, slip surfaces, and/or deformation bands, as summarized in Fossen et al., (2007). These features introduce secondary permeability anisotropy in the damage zone as they tend to have preferred orientations, as shown by both field studies (Fossen et al.,, 2005; Johansen and Fossen,, 2008) and core analysis (Hesthammer et al.,, 2000). This leads to preferential flow directions that are neither parallel nor orthogonal to the main plane. This type of flow cannot be represented by existing models that employ dimension reduction. To the Authors’ best knowledge, the only attempt to modeling faults and their surrounding damage zones in a mixed-dimensional framework can be found in Fumagalli and Scotti, (2019). However, they still apply local formulations to model the damage zones as lower-dimensional objects which are connected on one side to the fault and on the other side to the rock matrix, hence conceptually seeing the whole fault zone as a multilayer object. An alternative approach would be to implement the fault core as a transmissibility multiplier and the damage zone by modifying the grid permeability in the cells adjacent to the model faults, as illustrated in Wilson et al., (2020). In the following, we will consistently refer to the thin inclusions as faults, notwithstanding that all methods presented herein can be applied to models of fractures and other thin inclusions, however, we expect that the methods proposed are of more importance for faults. The contribution of this paper is two-fold: First, we present a generalized dimensional reduced model that can preserve full-tensor permeability effects also in the out-of-plane direction of the fault. The resulting reduced equations have a form similar to that of traditional models, however the more general coupling structure leads to additional terms both in the in-plane and out-of-plane equations. These terms, as well as our whole novel formulation, will be denoted as ”semi-local” in the following, emphasizing the fact that the new PDEs will contain quantities that, while physically in the same location, from a modeling perspective reside outside the subdomain where the PDE is defined, specifically the internal boundary between the subdomain and its higher dimensional neighbor. Multiple discretization schemes have been proposed for the local dimensionally-reduced models, including methods based on finite volumes (Helmig et al.,, 1997; Karimi-Fard et al.,, 2003; Sandve et al.,, 2012), mixed finite elements (Martin et al.,, 2005; Boon et al.,, 2018), virtual elements (Fumagalli and Keilegavlen,, 2019) and mimetic methods (Formaggia et al.,, 2018). A comparison study for all these discretizations of flow in fractured media can be found in Flemisch et al., (2018) and Berre et al., (2020) for 2D and 3D flow, respectively. However, the additional terms arising in our formulation bring the semi-local model outside the scope of previously proposed discretization methods. The second contribution of the paper is therefore the derivation of discretization schemes for semi-local models. We achieve this in two stages: First, based on the unified framework for discretization of mixed-dimensional problems with local interface laws presented in Nordbotten et al., (2019), we present conditions under which any standard discretization scheme for fixed-dimensional problems can be extended to mixed-dimensional problems with semi-local interface laws. Second, we present a concrete discretization approach based on finite volume methods. The paper is organized as follows. In Sec. 2, the mathematical model is presented, first for a domain with a single fault, and then for a general faults configuration. Thereafter, in Sec. 3, the unified discretization is formulated. After presenting simulation results in Sec. 4, concluding remarks are given in Sec. 5. ## 2 Flow modelling in faulted porous media In this section, the mathematical model for flow in faulted porous media is presented, first for a porous domain containing a single fault (Sections 2.1 and 2.2), and then for a general network of faults (Section 2.3). For the general case, we also provide the weak formulation of the interface problem (Sections 2.4-2.5), which will be useful from the perspective of implementation. \begin{overpic}[width=325.215pt]{upscaling} \put(25.0,20.0){\small$\Psi_{3}$} \put(30.0,5.0){\small$\Psi_{2}$} \put(30.0,35.0){\small$\Psi_{1}$} \put(80.0,23.0){\small$\Omega_{3}$} \put(85.0,5.0){\small$\Omega_{2}$} \put(85.0,35.0){\small$\Omega_{1}$} \put(23.0,37.0){\small${\bm{n}}_{3}$} \put(23.0,5.0){\small${\bm{n}}_{3}$} \put(10.0,30.0){\small$\partial_{\Psi_{3}}\Psi_{1}$} \put(10.0,12.0){\small$\partial_{\Psi_{3}}\Psi_{2}$} \put(7.0,21.0){\small$a$} \end{overpic} Figure 1: Representation of the fault as a thin three- dimensional domain $\Psi_{3}$ (left) and as a two-dimensionl manifold $\Omega_{3}$ (right). The boundary of $\Psi_{j}$ adjacent to $\Psi_{3}$ is denoted by $\partial_{\Psi_{3}}\Psi_{j}$, for $j=1,2$, while ${\bm{n}}_{i}$ is the normal vector which is always pointing outwards from $\Psi_{i}$, for $i=1,2,3$. ### 2.1 Domain with a single fault We start by considering two three-dimensional porous media $\Psi_{1}$ and $\Psi_{2}$, each of them with its Neumann and Dirichlet boundaries $\partial_{N}$ and $\partial_{D}$, respectively. The two three-dimensional domains are separated by a fault $\Psi_{3}$, which is a thin, almost two- dimensional object of thickness $a$ (in the following $a$ will be denoted as the aperture), but which is currently represented as three-dimensional. We note that $\Psi_{3}$ need not be planar, i.e. $a$ need not be constant. We denote by $\partial_{\Psi_{3}}\Psi_{j}$, for $j=1,2$, the boundary of $\Psi_{j}$ adjacent to $\Psi_{3}$. Furthermore, let ${\bm{n}}_{i}$ be the normal vector which is always pointing outwards from $\Psi_{i}$. It thus follows that ${\bm{n}}_{3}=-{\bm{n}}_{j}$ on $\partial_{\Psi_{3}}\Psi_{j}$. A representation of the fault as a thin three-dimensional domain $\Psi_{3}$ is illustrated in the left of Fig. 1. Darcy flow in the three-dimensional porous medium is then governed by the following set of equations ($i=1,2,3$): $\displaystyle\nabla\cdot{\bm{q}}_{i}+f_{i}=0\quad$ $\displaystyle on\quad\Psi_{i}$ (1) $\displaystyle{\bm{q}}_{i}=-{\bm{K}}_{i}\nabla p_{i}\quad$ $\displaystyle on\quad\Psi_{i}$ (2) $\displaystyle\lambda_{3,j}={\bm{q}}_{3}\cdot{\bm{n}}_{3}=-{\bm{q}}_{j}\cdot{\bm{n}}_{j}=-\lambda_{j,3}\quad\quad(j=1,2)\quad$ $\displaystyle on\quad\partial_{\Psi_{3}}\Psi_{j}$ (3) $\displaystyle{\bm{q}}_{i}\cdot{\bm{n}}_{i}=g_{i}\quad$ $\displaystyle on\quad\partial_{N}\Psi_{i}$ (4) $\displaystyle\text{tr }p_{i}=0\quad$ $\displaystyle on\quad\partial_{D}\Psi_{i}$ (5) Here, $p$ is pressure, $\mathbf{q}$ is the Darcy flux, $f$ is a source, and $\bm{K}$ is a second-order tensor representing the absolute permeability divided by fluid viscosity. Equation (1) represents mass conservation, while equation (2) is Darcy’s law. Equation (3) represents flux continuity conditions on $\partial_{\Psi_{3}}\Psi_{j}$, where $\lambda_{3,j}$ represents flow from $\Psi_{3}$ to $\Psi_{j}$, thus by flux continuity it follows that $\lambda_{3,j}=-\lambda_{j,3}$. Finally, equations (4)-(5) are boundary conditions on $\partial_{N}\Psi_{i}$ and $\partial_{D}\Psi_{i}$, repectively. Figure 2: Illustration of possible structures of the permeability of a fault embedded in a porous domain indicated by the principal axis of the permeability tensor: (a) orthogonal permeability, (b) homogeneous full- permeability structure, (c) different structure on each half of the fault. Before deriving the governing equations for the lower-dimensional manifold, we discuss the decomposition of the permeability tensor within the fault. Existing local laws for faults as embedded thin inclusions assume that the principal directions of the local permeability tensor are aligned with the fault orientation, as illustrated in Fig. 2.a. Hence, more general orientations of the principal permeability directions, shown in Fig. 2.b-2.c, cannot be represented by existing models. To be concrete, we let the permeability on $\Psi_{3}$ have the following decomposition in terms of a coordinate system aligned with the fault orientation: ${\bm{K}}_{3}=\begin{bmatrix}{\bm{K}}_{3,\parallel}&{\bm{k}}_{3,t}\\\ {\bm{k}}^{T}_{3,t}&k_{3,\bot}\end{bmatrix}$ (6) Here, ${\bm{K}}_{3,\parallel}$ is a $2\times 2$ second-order tensor representing the within-fault permeability and $k_{3,\bot}$ is a scalar representing the normal permeability. The off-diagonal term ${\bm{k}}_{3,t}$ is a two-vector representing the symmetric off-diagonal components of ${\bm{K}}_{3}$; for local interface laws, these off-diagonal components are assumed to be negligible, i.e. ${\bm{k}}_{3,t}=0$ (Nordbotten and Celia,, 2011; Berre et al.,, 2020). The inclusion of this anisotropic term leads to significant complications in the modeling and discretization, and is the main topic of this work. With this structure of the fault permeability, the Darcy flux for the fault can be decomposed as ${\bm{q}}_{3}=[{\bm{q}}_{3,\parallel},q_{3,\bot}]$, where the 2-vector tangential component ${\bm{q}}_{3,\parallel}$ and the scalar normal component $q_{3,\bot}$ have the following form: $\displaystyle{\bm{q}}_{3,\parallel}=-{\bm{K}}_{3,\parallel}\nabla_{\parallel}p_{3}-{\bm{k}}_{3,t}\nabla_{\bot}p_{3},$ (7) $\displaystyle q_{3,\bot}=-{\bm{k}}_{3,t}\cdot\nabla_{\parallel}p_{3}-k_{3,\bot}\nabla_{\bot}p_{3}.$ (8) Here, $\nabla_{\parallel}$ and $\nabla_{\bot}=\dfrac{\partial}{\partial n}$ represent the in-plane and out-of-plane components of the gradient for the fault, respectively. \begin{overpic}[width=216.81pt]{sketch_interfaces} \put(32.0,50.0){\small$\Omega_{1}$} \put(32.0,25.0){\small$\Omega_{2}$} \put(27.0,40.0){\small$\Omega_{3}$} \put(102.0,37.0){\small$\Omega_{3}$} \put(102.0,29.0){\small$\Gamma_{2,3}$} \put(102.0,45.0){\small$\Gamma_{1,3}$} \put(85.0,56.0){\small$\partial_{\Omega_{3}}\Omega_{1}$} \put(85.0,18.0){\small$\partial_{\Omega_{3}}\Omega_{2}$} \put(75.0,65.0){\small$\Omega_{1}$} \put(75.0,5.0){\small$\Omega_{2}$} \end{overpic} Figure 3: Illustration of the mixed-dimensional geometry. $\Omega_{3}$ is connected to the higher dimensional neighbors $\Omega_{j}$ through the interfaces $\Gamma_{j,3}$, for $j=1,2$. Note that $\Omega_{3}$, $\Gamma_{j,3}$ and $\partial_{\Omega_{3}}\Omega_{j}$ are all coinciding in physical space. ### 2.2 Model reduction To proceed, we apply integration over the perpendicular direction to achieve a dimension reduction of the fault. This replaces $\Psi_{3}$ with a lower- dimensional domain $\Omega_{3}$ (see right of Fig. 1). Note that we use $\Psi$ to represent the equi-dimensional geometry, that is all $\Psi_{j}$ are 3D, and $\Omega$ to denote the mixed-dimensional geometry. We also introduce two interfaces $\Gamma_{j,3}$ on each side $j=1,2$ of $\Omega_{3}$, as illustrated in Fig. 3. The interfaces physically represent the half zone comprised between the fault and either side of the surrounding matrix. In a mixed-dimensional setting, they have no perpendicular extent, and serve as connectors between two objects of different dimensions. Note that, due to the dimension reduction of the model, $\Omega_{3}$, $\Gamma_{1,3}$, $\Gamma_{2,3}$, $\partial_{\Omega_{3}}\Omega_{1}$ and $\partial_{\Omega_{3}}\Omega_{2}$ are all coinciding in physical space. Furthermore, we define the integrated Darcy flux ${\bm{q}}_{3}^{(2)}$ and the average pressure $p^{(2)}_{3}$, respectively as ${\bm{q}}_{3}^{(2)}=\int_{-a/2}^{a/2}{\bm{q}}^{(3)}_{3,\parallel}dn,\quad\quad p^{(2)}_{3}=\dfrac{1}{a}\int_{-a/2}^{a/2}p_{3}^{(3)}dn.$ (9) Here, we use subscripts to index the domains, and superscripts (when necessary for clarity) to emphasize the effective topological dimension of the domain, e.g. $p_{3}^{(3)}$ and $p_{3}^{(2)}$ are the pressures within the fault in the 3D (on $\Psi_{3}$) and 2D (on $\Omega_{3}$) representations, respectively. When passing to a mixed-dimensional representation of the geometry, i.e. when integrating eqs. (1) and (7) along the perpendicular direction, the out-of- plane component of the gradient is converted into a jump operator as follows: $\int_{-a/2}^{a/2}\nabla_{\bot}p_{3}^{(3)}dn=(\text{tr }p_{1}-\text{tr }p_{2}).$ (10) The governing equations for the fault are then obtained from equations (1), (7), (4) and (5) by integrating in the perpendicular direction. Moreover, since the fault is assumed to be thin, we assume that the permeability is constant across the perpendicular direction. Together with the definitions above, this results in $\displaystyle\nabla_{3}\cdot{\bm{q}}_{3}^{(2)}-(\lambda_{1,3}+\lambda_{2,3})+f_{3}^{(2)}=0\quad$ $\displaystyle on\quad\Omega_{3}$ (11) $\displaystyle{\bm{q}}_{3}^{(2)}=-a{\bm{K}}_{3,\parallel}\nabla_{3}p_{3}^{(2)}+{\bm{\mu}}_{1,3}+{\bm{\mu}}_{2,3}\quad$ $\displaystyle on\quad\Omega_{3}$ (12) $\displaystyle{\bm{q}}_{3}^{(2)}\cdot{\bm{n}}_{3}^{(2)}=g_{3}^{(2)}\quad$ $\displaystyle on\quad\partial_{N}\Omega_{3}$ (13) $\displaystyle\text{tr }p_{3}^{(2)}=0\quad$ $\displaystyle on\quad\partial_{D}\Omega_{3}$ (14) where we have also introduced the integrated source term and boundary flux $f_{3}^{(2)}=\int_{-a/2}^{a/2}f_{3}^{(3)}dn,\quad\quad g_{3}^{(2)}=\int_{-a/2}^{a/2}g_{3}^{(3)}dn.$ (15) We emphasize that the differential operator $\nabla_{3}$ in eqs. (11)-(12) operates on the manifold $\Omega_{3}$. Compared to traditional upscaled models, see for instance Nordbotten et al., (2019), additional terms ${\bm{\mu}}_{j,3}$ appear in equation (12), analogous to the flux terms $\lambda_{j,3}$ in equation (11). This two-vector term, which is not present in previous work, represents the within-fault flux induced by pressure differences between the fault and the surrounding matrix, and is defined for either side of the fault as ${\bm{\mu}}_{j,3}=\epsilon_{j,3}{\bm{k}}_{3,t}(p_{3}^{(2)}-\text{tr }p_{j}),$ (16) where the permutation variable $\epsilon_{j,3}$ is positive if the coordinate systems of $\Omega_{3}$ and $\partial_{\Omega_{3}}\Omega_{j}$ coincide, and negative otherwise. To complete the model, we derive a constitutive law for $\lambda_{j,3}$. This is obtained by integrating equation (8) in the perpendicular direction, that is $\int_{-a/2}^{a/2}q_{3,\bot}^{(3)}dn=-\int_{-a/2}^{a/2}{\bm{k}}_{3,t}\cdot\nabla_{\parallel}p_{3}^{(3)}dn-\int_{-a/2}^{a/2}k_{3,\bot}\nabla_{\bot}p_{3}^{(3)}dn.$ (17) The left hand side of equation (17) is approximated using the trapeizodal rule, that is $\int_{-a/2}^{a/2}q_{3,\bot}^{(3)}dn\approx\dfrac{a}{2}(\epsilon_{1,3}\lambda_{1,3}+\epsilon_{2,3}\lambda_{2,3}),$ (18) where continuity of the flux across the boundary between the fault and the surrounding matrix is applied. The first term at the right hand side of equation (17) is approximated as $\int_{-a/2}^{a/2}{\bm{k}}_{3,t}\cdot\nabla_{\parallel}p_{3}^{(3)}dn={\bm{k}}_{3,t}\cdot\int_{-a/2}^{a/2}\nabla_{\parallel}p_{3}^{(3)}dn\approx{\color[rgb]{1,0,0}a}{\bm{k}}_{3,t}\cdot\nabla_{3}p_{3}^{(2)}.$ (19) Finally, the second term at the right hand side of (17) is resolved using the jump operator defined in equation (10) as follows: $\int_{-a/2}^{a/2}k_{3,\bot}\nabla_{\bot}p_{3}^{(3)}dn=\epsilon_{1,3}k_{3,\bot}(p_{3}^{(2)}-\text{tr }p_{1})+\epsilon_{2,3}k_{3,\bot}(p_{3}^{(2)}-\text{tr }p_{2}).$ (20) By incorporating eqs. (18), (19) and (20) into equation (17), we identify the flux $\lambda_{j,3}$ having the following form: $\lambda_{j,3}=-k_{3,\bot}\dfrac{2(p_{3}^{(2)}-\text{tr }p_{j})}{a}-\epsilon_{j,3}{\bm{k}}_{3,t}\cdot\nabla_{3}p_{3}^{(2)}.$ (21) Here, the first term on the right-hand side represents the local component of the constitutive law, while the second part is the semi-local contribution that induces a flux across $\Gamma_{j,3}$ due to the pressure gradient within the lower-dimensional manifold $\Omega_{3}$. Inspecting equations (16) and (21), we see that both the normal permeability $k_{3,\bot}$ and the off-diagonal permeability ${\bm{k}}_{3,t}$ are in the reduced model naturally interpreted as properties of the interface $\Gamma_{j,3}$. In the continuation, we will thus generalize the model as derived above, and index these quantities with the interface, i.e. ${\bm{k}}_{3,t}\rightarrow{\bm{k}}_{3,j,t}$ and $k_{3,\bot}\rightarrow k_{3,j,\bot}$ are assigned independently to either side of the fault. In summary, omitting superscripts for the sake of clarity, we can write the mixed-dimensional equations (1)-(5), (11)-(14), (16), and (21) in a unified way, that is for $i=\\{1,2,3\\}$ $\displaystyle\nabla_{i}\cdot{\bm{q}}_{i}-\sum_{j\in{\hat{S}}_{i}}\lambda_{j,i}+f_{i}=0\quad$ $\displaystyle on\quad\Omega_{i}$ (22) $\displaystyle{\bm{q}}_{i}=-{\bm{\kappa}}_{i,\parallel}\nabla_{i}p_{i}+\sum_{j\in{\hat{S}}_{i}}\epsilon_{j,i}{\bm{\kappa}}_{i,j,t}(p_{i}-\text{tr }p_{j})\quad$ $\displaystyle on\quad\Omega_{i}$ (23) $\displaystyle{\bm{q}}_{i}\cdot{\bm{n}}_{i}=\lambda_{i,3}\quad\quad(i\neq 3)\quad$ $\displaystyle on\quad\partial_{\Omega_{3}}\Omega_{i}$ (24) $\displaystyle\lambda_{j,3}=-\kappa_{3,j,\bot}(p_{3}-\text{tr }p_{j})-{\bm{\kappa}}_{3,j,t}\cdot\nabla_{3}p_{3}\quad\quad(j=1,2)\quad$ $\displaystyle on\quad\Gamma_{j,3}$ (25) $\displaystyle{\bm{q}}_{i}\cdot{\bm{n}}_{i}=g_{i}\quad$ $\displaystyle on\quad\partial_{N}\Omega_{i}$ (26) $\displaystyle\text{tr }p_{i}=0\quad$ $\displaystyle on\quad\partial_{D}\Omega_{i}$ (27) where ${\hat{S}}_{i}$ is the set of neighbors of $\Omega_{i}$ of higher dimension, e.g. ${\hat{S}}_{3}=\\{\Omega_{1},\Omega_{2}\\}$. Equations (22)-(27) are complemented with the natural convention that there is no four- dimensional domain in the model, thus ${\hat{S}}_{i}=\emptyset$ for $i=1,2$, and one clearly has for these three-dimensional domains also that $a=1$, ${\bm{K}}_{\parallel}=\bm{K}$ and $\nabla_{i}=\nabla$. We remark that due to the model reduction, the within-fault permeability ${\bm{K}}_{3,\parallel}$ and the normal permeability $k_{3,j,\bot}$ scale with the aperture $a$ and its inverse, respectively, while the off-diagonal permeability ${\bm{k}}_{3,j,t}$ remains as in the equi-dimensional model. In order to present equations (22)-(27) without reference to this small parameter, these scalings have been incorporated directly into the material constants. Thus, the mixed-dimensional permeability ${\bm{\kappa}}_{3}$ is related to the equi-dimensional ${\bm{K}}_{3}$ as follows ${\bm{\kappa}}_{3}=\begin{bmatrix}{\bm{\kappa}}_{3,\parallel}&{\bm{\kappa}}_{3,j,t}\\\ {\bm{\kappa}}^{T}_{3,j,t}&\kappa_{3,j,\bot}\end{bmatrix}=\begin{bmatrix}a{\bm{K}}_{3,\parallel}&{\bm{k}}_{3,j,t}\\\ {\bm{k}}^{T}_{3,j,t}&2a^{-1}k_{3,j,\bot}\end{bmatrix}.$ (28) We point out that, when one reduces multiple dimensions at once, these scalings get exponents corresponding to the number of dimensions below the ambient dimension. We also emphasize that the normal and off-diagonal permeabilities are in principle not a property of the fault itself, but instead a property which belongs to the internal interface $\Gamma_{j,3}$ between the fault and either side of the higher-dimensional neighbors. This represents an important extension of the existing local laws for fractured porous media, making the model also applicable to faulted porous media, since it allows for capturing the anisotropic character of the fault damage zone. Moreover, since different values of ${\bm{k}}_{3,j,t}$ and $k_{3,j,\bot}$ can be assigned to each side of the fault, our model can represent different permeability structures on each side of the fault. A schematic illustration of the different quantities and their domain of definition for the local and semi-local formulations is shown in Fig. 4. \begin{overpic}[width=216.81pt]{local_vs_semi_local} \put(-35.0,70.0){Semi- local} \put(110.0,70.0){Local} \put(50.0,15.0){\small${\bm{q}}_{j},p_{j}$} \put(-45.0,55.0){\small$\lambda_{j,3}\sim(\nabla_{3}p_{3},p_{j},p_{3})$} \put(-45.0,17.0){\small${\bm{q}}_{3}\sim(\nabla_{3}p_{3},p_{3},p_{1},p_{2})$} \put(103.0,60.0){\small$\lambda_{j,3}\sim(p_{j},p_{3})$} \put(105.0,32.0){\small${\bm{q}}_{3}\sim\nabla_{3}p_{3}$} \end{overpic} Figure 4: Illustration of the quantities associated with the local and semi-local formulations. ### 2.3 Domain with a general network of faults Following the theory by Boon et al., (2018), equations (22)-(27) can be generalized also to faults intersections, both the one-dimensional (1D) line intersections between two faults and the zero-dimensional (0D) point intersections of three faults (see Fig. 5 for an illustration of the mixed- dimensional geometry). To this end, we use subscripts to index each domain (matrix, fault, or intersection) by number as in the previous section, and let $I$ denote the index set of all domains. Superscripts for the topological dimension associated with each individual domain will be consistently omitted, keeping in mind that the dimension is always a property of the domain, i.e. $d=d_{i}$. Hence, we can write for all $i\in I$ the equations $\displaystyle\nabla_{i}\cdot{\bm{q}}_{i}-\sum_{j\in{\hat{S}}_{i}}\lambda_{j,i}+f_{i}=0\quad$ $\displaystyle on\quad\Omega_{i}$ (29) $\displaystyle{\bm{q}}_{i}=-{\bm{\kappa}}_{i,\parallel}\nabla_{i}p_{i}+\sum_{j\in{\hat{S}}_{i}}\epsilon_{j,i}{\bm{\kappa}}_{i,j,t}(p_{i}-\text{tr }p_{j})\quad$ $\displaystyle on\quad\Omega_{i}$ (30) $\displaystyle{\bm{q}}_{i}\cdot{\bm{n}}_{i}=\lambda_{i,j}\quad\quad(j\in{\check{S}}_{i})\quad$ $\displaystyle on\quad\partial_{\Omega_{j}}\Omega_{i}$ (31) $\displaystyle\lambda_{j,i}=-\kappa_{i,j,\bot}(p_{i}-\text{tr }p_{j})-\epsilon_{j,i}{\bm{\kappa}}_{i,j,t}\cdot\nabla_{i}p_{i}\quad\quad(j\in{\hat{S}}_{i})\quad$ $\displaystyle on\quad\Gamma_{j,i}$ (32) $\displaystyle{\bm{q}}_{i}\cdot{\bm{n}}_{i}=g_{i}\quad$ $\displaystyle on\quad\partial_{N}\Omega_{i}$ (33) $\displaystyle\text{tr }p_{i}=0\quad$ $\displaystyle on\quad\partial_{D}\Omega_{i}$ (34) where ${\check{S}}_{i}$ is the set of neighbors of $\Omega_{i}$ of lower dimension, e.g. ${\check{S}}_{1}=\\{\Omega_{2},\Omega_{3},\Omega_{4}\\}$. It is easy to show that as long as the mixed-dimensional permeabilities are diagonally dominant in the sense of $\kappa_{i,j,\bot}\det{\bm{\kappa}}_{i,\parallel}>{\bm{\kappa}}_{i,j,t}\cdot{\bm{\kappa}}_{i,j,t},$ (35) then the coefficients are globally positive definite, and equations (29)-(34) are well-posed as long as $\partial_{D}\Omega_{i}$ has non-zero measure for at least one domain (Boon et al.,, 2020). \begin{overpic}[width=173.44534pt]{sketch_domain_network_inclusions_separate} \put(52.0,95.0){\small$\Omega_{1}$} \put(5.0,52.0){\small$\Omega_{2}$} \put(40.0,47.0){\small$\Omega_{3}$} \put(60.0,45.0){\small$\Omega_{4}$} \put(25.0,20.0){\small$\Omega_{5}$} \put(50.0,26.0){\small$\Omega_{7}$} \put(36.0,30.0){\small$\Omega_{6}$} \put(36.0,2.0){\small$\Omega_{8}$} \end{overpic} Figure 5: Illustration of a 3D domain $\Omega_{1}$ containing three faults $\Omega_{j}$ ($j=2,3,4$) with their respective three 1D line intersections $\Omega_{k}$ ($k=5,6,7$) and one 0D point intersection $\Omega_{8}$. ### 2.4 Mixed-dimensional formulation of the fault-matrix flows While equations (29)-(34) constitute a full semi-local model, they are stated in a form which is not immediately amenable for discretization. This and the following subsection explore the model in more detail, with a goal of rewriting the equations in a form that can be handled by standard discretization schemes with only minimal adaptations. A discretization approach based on this reformulation is then given in Section 3. In order to simplify the exposition, we will introduce a mixed-dimensional notation following Nordbotten et al., (2019). In particular, we will denote the collection of pressure functions as $\mathfrak{p}=\left(p_{1},...,p_{\|I\|}\right)$, and similarly the collection of all fluxes (both in domains and across boundaries) as $\mathfrak{q}=\left({\bm{q}}_{1},...,{\bm{q}}_{\|I\|},\lambda_{1,1},...,\lambda_{\|I\|,\|J\|}\right)$. It is sometimes convenient to refer explicitly to only the domain or boundary fluxes, and we will therefore sometimes abuse notation and simply write $\mathfrak{q}=(q,\lambda)$. We refer to these as mixed-dimensional functions, and consistently denote them with calligraphic font. We adopt the natural convention that when evaluating a mixed-dimensional function at a point, say $x\in\Omega_{i}$, then we simply evaluate the function on that domain, so that $\mathfrak{p}(x)=p_{i}(x)$. In a similar sense, we denote the disjoint union of domains as $\mathfrak{F}=\left(\amalg_{i}\Omega_{i}\right)\sqcup\left(\amalg_{j,i}\Gamma_{j,i}\right)$. With this notion of mixed-dimensional functions, the extension of the divergence and gradient operators to the mixed-dimensional setting is natural. First, we extend the concept of continuous functions by requiring that for $\mathfrak{q}$ to be continuous, then it must hold that, for all $\Gamma_{j,i}$, ${\bm{q}}_{i}\cdot{\bm{n}}_{i}=\lambda_{i,j}$. Then, for any point $x\in\Omega_{i}$ we define $\left(\mathfrak{D}\cdot\mathfrak{q}\right)(x)=\left[\nabla_{i}\cdot{\bm{q}}_{i}-\sum_{j\in{\hat{S}}_{i}}\lambda_{j,i}\right]_{x}\quad\quad\text{and}\quad\quad\left(\mathbb{D}\mathfrak{p}\right)(x)=\left[\nabla_{i}p_{i}\right]_{x},$ (36) while for any point on an interface $x\in\Gamma_{j,i}$ we define $\left(\mathbb{D}\mathfrak{p}\right)(x)=\left[p_{i}-\text{tr }p_{j}\right]_{x}.$ (37) Now we can write equations (29) - (34) simply as: $\displaystyle\mathfrak{D}\cdot\mathfrak{q}+\mathfrak{f}=0\quad$ $\displaystyle on\quad\mathfrak{F}$ (38) $\displaystyle\mathfrak{q}=-\mathfrak{K}\mathbb{D}p\quad$ $\displaystyle on\quad\mathfrak{F}$ (39) $\displaystyle\mathfrak{q}\cdot\mathfrak{n}=\mathfrak{g}\quad$ $\displaystyle on\quad\partial_{N}\mathfrak{F}$ (40) $\displaystyle\text{tr }\mathfrak{p}=0\quad$ $\displaystyle on\quad\partial_{D}\mathfrak{F}$ (41) where we have also introduced the collection of sources $\mathfrak{f}=\left(f_{1},...,f_{\|I\|}\right)$ and the collection of boundary fluxes $\mathfrak{g}=\left(g_{1},...,g_{\|I\|}\right)$. Here, the material coefficients are now all part of the mixed-dimensional permeability $\mathfrak{K}$, which is defined such as that for any mixed-dimensional gradient $\mathfrak{u}=\mathbb{D}p=(u,\mu)$, it holds that for any point $x\in\Omega_{i}$: $\left(\mathfrak{K}\mathfrak{u}\right)(x)={\bm{\kappa}}_{i,\parallel}u_{i}-\sum_{j\in{\hat{S}}_{i}}\epsilon_{j,i}{\bm{\kappa}}_{i,j,t}\mu_{j,i},$ (42) while for any point on an interface $x\in\Gamma_{j,i}$, it holds that $\left(\mathfrak{K}\mathfrak{u}\right)(x)=\kappa_{i,j,\bot}\mu_{j,i}+\epsilon_{j,i}{\bm{\kappa}}_{i,j,t}\cdot u_{i}.$ (43) It is then also sometimes convenient to write equation (39) in matrix form, that is for $\mathfrak{q}=(q,\lambda)$ and $\mathfrak{u}=\mathbb{D}\mathfrak{p}=(u,\mu)$, one has: $\begin{Bmatrix}q\\\ \lambda\end{Bmatrix}=-\begin{bmatrix}\mathfrak{K}_{\Omega\Omega}&\mathfrak{K}_{\Omega\Gamma}\\\ \mathfrak{K}_{\Gamma\Omega}&\mathfrak{K}_{\Gamma\Gamma}\end{bmatrix}\begin{Bmatrix}u\\\ \mu\end{Bmatrix}.$ (44) Equation (44) highlights the contribution from the semi-local terms in the mixed-dimensional version of Darcy’s law. ### 2.5 Weak formulation as an interface system The semi-local terms in equations (29)-(34) lead to coupling terms between domains that are local in physical space, but non-local in the mixed- dimensional representation of the geometry. A critical example are the fault and its sides, which, from the perspective of implementation, we would prefer to only interact via the interfaces $\Gamma_{j,i}$, and not directly, as is the case for the last term in equation (30). Thus we are motivated to consider a reformulation of the governing equations before considering numerical discretizations. We proceed by first performing an LU decomposition of equation (44) as follows: $\mathfrak{K}_{U}\begin{Bmatrix}q\\\ \lambda\end{Bmatrix}=-\mathfrak{K}_{L}\begin{Bmatrix}u\\\ \mu\end{Bmatrix},$ (45) where $\mathfrak{K}_{U}$ and $\mathfrak{K}_{L}$ are defined, respectively, as: $\mathfrak{K}_{U}=\begin{bmatrix}I&\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}\\\ 0&I\end{bmatrix}\quad\quad\text{and}\quad\quad\mathfrak{K}_{L}=\begin{bmatrix}A_{\Omega}&0\\\ \mathfrak{K}_{\Gamma\Omega}&\mathfrak{K}_{\Gamma\Gamma}\end{bmatrix},$ (46) and $A_{\Omega}$ is the Schur-complement defined as $A_{\Omega}=\mathfrak{K}_{\Omega\Omega}-\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}\mathfrak{K}_{\Gamma\Omega}.$ (47) Note that, since $\mathfrak{K}_{\Gamma\Gamma}$ consists only of scalar values $(\kappa_{i,j,\bot})$, this reformulation only depends on the trivial inversion of scalars. In the following it will be helpful to discuss the components of the mixed- dimensional gradient and divergence, and we therefore additionally define the ”full jump” $\mathbb{d}\mathfrak{q}$ such that for any point $x\in\Omega_{i}$ it holds that $\left(\mathbb{d}\mathfrak{q}\right)(x)=\left[-\sum_{j\in{\hat{S}}_{i}}\lambda_{j,i}\right]_{x},$ (48) while the ”half jump” $\mathbb{d}^{\star}\mathfrak{p}$ is simply the restriction of $\mathbb{D}\mathfrak{p}$ to $\Gamma_{j,i}$. We then write (with the natural extension of $\nabla$ and $\nabla\cdot$): $\mathfrak{D}\cdot\mathfrak{q}=\nabla\cdot q+\mathbb{d}\lambda\quad\quad\text{and}\quad\quad\mathbb{D}\mathfrak{p}=\left(\nabla p,\mathbb{d}^{\star}\mathfrak{p}\right).$ (49) We now proceed by (formally) eliminating internal domain variables, in order to obtain a problem only posed on interfaces. We note that equations (38) and (39) can now be written as the first order system: $\displaystyle\mathfrak{D}\cdot\mathfrak{q}=\mathfrak{f}$ (50) $\displaystyle\mathfrak{K}_{U}\mathfrak{q}=-\mathfrak{K}_{L}\mathfrak{D}\mathfrak{p}$ (51) where use of equation (45) has been made. By writing out equation (50) in local notation for each $\Omega_{i}$ and by stating equation (51) explicitly as two equations, we obtain the following set of equations: $\displaystyle\nabla\cdot q=f-\mathbb{d}\lambda$ (52) $\displaystyle q+A_{\Omega}\nabla p=-\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}\lambda$ (53) $\displaystyle\lambda=-\left(\mathfrak{K}_{\Gamma\Omega}\nabla p+\mathfrak{K}_{\Gamma\Gamma}\mathbb{d}^{\star}\mathfrak{p}\right)$ (54) This reveals that equations (52) and (53) form a locally well-posed system (of standard Darcy type) on each $\Omega_{i}$, and we can therefore consider $p=p(\lambda)$ for any given $\lambda$. We formalize this concept by introducing the (continuous) solution operators for the standard elliptic value problem on $\Omega_{i}$, $\mathcal{S}_{\Omega_{i}}^{K}$, defined as: $\left(\upsilon,\nabla\upsilon,\text{tr }\upsilon,F\right)=\mathcal{S}_{\Omega_{i}}^{K}\left(f,\chi,b,\upsilon_{0}\right),$ (55) where $\upsilon$ is the solution to $\displaystyle\nabla\cdot\varphi=f-F\quad\quad$ $\displaystyle\text{on}\quad\Omega_{i}$ (56) $\displaystyle\varphi=-K\left(\nabla\upsilon+\chi\right)\quad\quad$ $\displaystyle\text{on}\quad\Omega_{i}$ (57) $\displaystyle\varphi\cdot n=b\quad\quad$ $\displaystyle\text{on}\quad\partial\Omega_{i}\setminus\partial\Omega$ (58) $\displaystyle\upsilon=0\quad\quad$ $\displaystyle\text{on}\quad\partial\Omega_{i}\cap\partial\Omega$ (59) $\displaystyle\dfrac{1}{\|\Omega_{i}\|}\int_{\Omega_{i}}\upsilon=\upsilon_{0}\quad\quad$ $\displaystyle\text{if}\quad\partial\Omega_{i}\cap\partial\Omega\neq\oslash$ (60) where $\partial\Omega$ is the global boundary and $F=\dfrac{1}{\|\Omega_{i}\|}\left(\int_{\Omega_{i}}f-\int_{\Omega_{i}}b\right)$ if $\partial\Omega_{i}\cap\partial\Omega\neq\oslash$, and zero otherwise. Using this solution operator, we see that the solution to equations (52) and (53) can be stated as functions of $\lambda$ (and a set of number of numbers $p_{0}$ corresponding to the domains where $\partial\Omega_{i}\cap\partial\Omega\neq\oslash$) as: $\left(p,\nabla p,\text{tr }p,F\right)_{\Omega_{i}}\left(\lambda,p_{0}\right)=\mathcal{S}_{\Omega_{i}}^{A_{i}}\left(f_{i}-\left(\mathbb{d}\lambda\right)_{i},A_{i}^{-1}\left(\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}\lambda\right)_{i},\lambda_{\check{I}_{i}},p_{0}\right).$ (61) Inserting $p=p(\lambda,p_{0})$ etc. into equation (54), we have now reformulated the fault-matrix problem into a pure interface problem. From the perspective of implementation, we desire to consider the interface problem in the weak sense, and we therefore multiply by test functions $w$ and integrate to obtain the problem: Find $\lambda\in L^{2}(\Gamma)$ such that, for all $w\in L^{2}(\Gamma_{j})$ $\left(\mathfrak{K}_{\Gamma\Gamma}^{-1}\lambda,w\right)_{\Gamma_{j,i}}+\left(\mathfrak{K}_{\Gamma\Gamma}^{-1}\mathfrak{K}_{\Gamma\Omega}\nabla p(\lambda,p_{0}),w\right)_{\Gamma_{j,i}}+\left(\mathbb{d}^{\star}\mathfrak{p}(\lambda,p_{0}),w\right)_{\Gamma_{j,i}}=0$ (62) and $F_{i}(\lambda,p_{0})=0$ if $\partial\Omega_{i}\cap\partial\Omega\neq\oslash$. We point out that the inner products in equation (62) are bounded from a formal perspective, since for $\lambda\in L^{2}(\Gamma)$, then $p_{i}\in H^{1}(\Omega_{i})$, and both $\mathfrak{K}_{\Gamma\Gamma}^{-1}\mathfrak{K}_{\Gamma\Omega}\nabla p$ and $\text{tr }p$ will lie in (at least) $L^{2}(\Gamma_{j,i})$. Finally, we emphasize that equations (61)-(62) are attractive from the perspective of implementation, since the inner products appearing are easy to evaluate, and the solution operators $\mathcal{S}_{\Omega_{i}}^{A_{i}}$ can be approximated by any standard method, as we will detail in the next section. ## 3 Discretizations of flow for faulted porous media The equations derived in Section 2.5, and in particular the interface problem of equation (61), form the starting point for the discretization approach laid out in this section. We present the general discretization framework in Section 3.1, and discuss implementational aspects in Section 3.2. ### 3.1 Unified discretization Equation (61) provides a solution operator for the arbitrary standard method used to solve the elliptic boundary value problem (52)-(53) on $\Omega_{i}$. To be concrete, we consider each domain $\Omega_{i}$ and its Neumann boundary $\partial\Omega_{i}=\partial_{N}\Omega_{i}\cup\sum_{j\in\check{S}_{i}}\partial_{\Omega_{j}}\Omega_{i}$ as endowed with a numerical discretization. Then, the solution operator $\mathcal{S}_{i}$ can be stated as $\mathcal{S}_{i}:\left[N(\Omega_{i}),N^{d_{i}}(\Omega_{i}),N(\partial\Omega_{i})]\rightarrow[N(\Omega_{i}),N^{d_{i}}(\Omega_{i}),N(\partial\Omega_{i})\right],$ (63) where $N(\Omega_{i})$, $N^{d_{i}}(\Omega_{i})$, and $N(\partial\Omega_{i})$ are the discrete representations of $L^{2}(\Omega_{i})$, $\left(L^{2}(\Omega_{i})\right)^{d_{i}}$, and $L^{2}(\partial\Omega_{i})$, respectively, and ${d_{i}}$ is the topological dimension of $\Omega_{i}$. In particular, $\mathcal{S}_{i}$ takes as input sinks, vector sources, and Neumann data and returns as output pressures, pressure gradients, and pressure traces. Most discretization schemes for elliptic equations can provide such a solution operator; we discuss the concrete implementation in the next subsection. To discretize the flux coupling term $\lambda_{j,i}$, we introduce a mortar- like grid $\mathcal{T}_{j,i}$ on the interface $\Gamma_{j,i}$ on which the boundary flux $\lambda_{j,i}$ will be defined. The flux variables are represented as piecewise constant on the mortar grid $\mathcal{T}_{j,i}$, thus $\lambda_{j,i}\in P_{0}(\mathcal{T}_{j,i})\subset\L^{2}(\Omega_{i})$. In order to allow communications between subdomains, and thus explicitly relate the degrees of freedom of the numerical methods $\mathcal{S}_{i}$ and the mortar grids $\mathcal{T}_{j,i}$, we introduce projection operators, namely $\Pi_{N(\Omega_{i})}$ and $\Pi_{L^{2}(\Omega_{i})}$. The former is the compound operator projecting from the coupling variables on the mortar grids to the subdomain degrees of freedom, that is $\begin{split}\Pi_{N(\Omega_{i})}:&\left[L^{2}(\Omega_{i}),\left(L^{2}(\Omega_{i})\right)^{d_{i}},L^{2}\left(\Omega_{\check{S}_{i}}\right),L^{2}(\partial\Omega_{i})\right]\\\ &\rightarrow\left[N(\Omega_{i}),N^{d_{i}}(\Omega_{i}),N(\partial\Omega_{i})\right],\end{split}$ (64) while the latter conversely moves from the numerical variables to the coupling variables, that is $\begin{split}\Pi_{L^{2}(\Omega_{i})}:&\left[N(\Omega_{i}),N^{d_{i}}(\Omega_{i}),N(\partial\Omega_{i})\right]\\\ &\rightarrow\left[L^{2}(\Omega_{i}),\left(L^{2}(\Omega_{i})\right)^{d_{i}},L^{2}\left(\Omega_{\check{S}_{i}}\right),L^{2}(\partial\Omega_{i})\right].\end{split}$ (65) Now, following the variational formulation derived in Sec. 2.5, we exploit equation (62) in order to provide discretization-independent framework for faulted porous media. This takes the form: for given numerical discretizations $\mathcal{S}_{i}$, find $\lambda_{j,i}\in P_{0}(\mathcal{T}_{j,i})$, for all $i\in I$ and $j\in\hat{S}_{i}$ such that $\begin{split}&\left(\mathbb{d}^{\star}\mathfrak{p},w_{j}\right)_{\Gamma_{j,i}}+\left(\mathfrak{K}^{-1}_{\Gamma\Gamma}\left(\lambda_{j,i}+\mathfrak{K}_{\Gamma\Omega}\cdot\nabla p\right),w_{j}\right)_{\Gamma_{j,i}}=0\\\ &\quad\text{for all }w_{j}\in P_{0}(\mathcal{T}_{j,i})\\\ \end{split}$ (66) subject to discrete constraints (for all $i\in I$): $\displaystyle[p_{i},u_{i},t_{j}]$ $\displaystyle=\Pi_{L^{2}\left(\Omega_{i}\right)}\mathcal{S}_{i}(\psi_{i}+a_{i},b_{i},c_{i})$ (67) $\displaystyle[a_{i},b_{i},c_{i}]$ $\displaystyle=\Pi_{N(\Omega_{i})}\left[-\sum_{j\in\hat{S}_{i}}\lambda_{j,i},-\sum_{j\in\hat{S}_{i}}A_{i}^{-1}\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}\lambda_{j,i},\sum_{j\in\check{S}_{i}}\lambda_{i,j}\right]$ (68) where the dummy variables $a_{i}$, $b_{i}$ and $c_{i}$ have the interpretations of sources, forces, and fluxes due to interactions with other domains, respectively. In contrast, the variables $p_{i}$, $u_{i}$, and $t_{j}$ are the pressures, pressure gradients, and pressure traces after projection onto the grids $\mathcal{T}_{j,i}$. The interpretation of this scheme is as follows. Eq. (67) resolves the internal differential equations in each subdomain, eq.(68) is the projection of variables from the flux grids to the numerical boundary (and source) data, while equation (66) simply states that the flux $\lambda_{j,i}$ between the fault and its surrodundings should satisfy Darcy’s law. In the following section, we present the strategy for implementation of this approach and give details for a specific numerical scheme. ### 3.2 MPFA discretization It is of interest to consider the requirements put on the subdomain solution operators $\mathcal{S}_{i}$ in some more detail. From the variational formulations stated above, we see that for a discretization on a generic subdomain $\Omega_{i}$ to interact with the interface $\Gamma_{j}$, we need to provide operators which: 1. 1. Handle Neumann boundary data on the form $\Pi_{N(\Omega_{i})}\lambda_{j}$, for all interfaces $\Gamma_{j}$ where $\Omega_{i}$ is the higher-dimensional neighbor. 2. 2. Handle source terms $\Pi_{N(\Omega_{i})}\lambda_{j}$ from interfaces $\Gamma_{j}$ where $\Omega_{i}$ is the lower-dimensional neighbor. 3. 3. Provide a discrete operator $\text{tr }p_{i}$ so that $\Pi_{L^{2}(\Omega_{i})}$ can project the pressure trace from $\partial_{j}\Omega_{i}$ to $\Gamma_{j}$ where $\Omega_{i}$ is the higher- dimensional neighbor. 4. 4. Provide a pressure $p_{i}$ so that $\Pi_{L^{2}(\Omega_{i})}$ can project the pressure to all $\Gamma_{j}$ where $\Omega_{i}$ is the lower-dimensional neighbor. 5. 5. Handle the divergence of vector source terms $\Pi_{N(\Omega_{i})}(\nabla\cdot{\bm{\mu}}_{j,i})$ from interfaces $\Gamma_{j}$ where $\Omega_{i}$ is the lower-dimensional neighbor. 6. 6. Provide a pressure gradient $u_{i}$ so that $\Pi_{L^{2}(\Omega_{i})}$ can project the pressure gradient to all $\Gamma_{j}$ where $\Omega_{i}$ is the lower-dimensional neighbor. The four first requirements are readily available for any discretization scheme for elliptic equations. Specifically, we have based our solution operators on a cell-centered finite volume method termed the multi-point flux approximation (MPFA) (Aavatsmark,, 2002; Nordbotten and Keilegavlen,, 2020). Treatment of vector source terms (item 5) is not as natural in primal discretization schemes such as finite elements, but is easy to include in most flux-based discretization methods such as e.g. mixed finite elements. We have employed the approach introduced in Starnoni et al., (2019), which treats the vector source term as part of the discrete divergence operator, and thereby provides an expression of the fluxes in terms of jumps in cell-centers vector sources. Finally, the pressure gradients are discretized as piece wise constant on each cell from an interpolation of the face cells fluxes (item 6). We implemented our model in PorePy, an open-source software for simulation of multiphysics processes in fractured porous media (Keilegavlen et al.,, 2021). \begin{overpic}[width=108.405pt]{sketch_discretization} \put(30.0,76.0){\small$\Omega_{h}$} \put(50.0,60.0){\small$\partial_{j}\Omega_{h}$} \put(75.0,30.0){\small$\Gamma_{j}$} \put(75.0,0.0){\small$\Omega_{l}$} \put(60.0,12.0){\small$\Pi_{L^{2}(\Omega_{h})}$} \put(60.0,42.0){\small$\Pi_{L^{2}(\Omega_{l})}$} \put(20.0,12.0){\small$\Pi_{N(\Omega_{l})}$} \put(20.0,42.0){\small$\Pi_{N(\Omega_{h})}$} \end{overpic} Figure 6: Illustration of a coupling between subdomains. $\Omega_{h}$ and $\Omega_{l}$ are the higher and lower subdomains respectively, $\Gamma_{j}$ is the interface between the two subdomains, $\partial_{j}\Omega_{h}$ is the portion of the boundary of $\Omega_{h}$ as seen from $\Gamma_{j}$, $\Pi_{N(\Omega_{k})}$ is the projection operator from coupling variables on the mortar grid to each of the subdomains degrees of freedom ($k=h,l$), and $\Pi_{L^{2}(\Omega_{k})}$ is the projection operator from numerical variables to coupling variables. To better understand the structure of the discrete coupling, it is instructive to write out the coupled system for two subdomains $\Omega_{h}$ and $\Omega_{l}$ separated by an interface $\Gamma_{j}$ (see Fig. 6). Let $\overline{p}_{h}$ and $\overline{p}_{l}$, be the vectors of cell-center pressures in $\Omega_{h}$ and $\Omega_{l}$ respectively, and let $\overline{\lambda}_{j}$ be the vector of discrete mortar fluxes in $\Gamma_{j}$. The discrete coupled system in absence of external sources can then be represented on the generic form $\begin{bmatrix}A_{h}&0&G_{h}\Pi_{N(\Omega_{h})}\\\ 0&A_{l}&B_{l}\Pi_{N(\Omega_{l})}+J_{l}\Pi_{N(\Omega_{l})}T_{j}\\\ -\Pi_{L^{2}(\Omega_{h})}P_{h}&\Pi_{L^{2}(\Omega_{l})}P_{l}+T_{j}\Pi_{L^{2}(\Omega_{l})}R_{l}&D_{j}\\\ \end{bmatrix}\begin{bmatrix}\overline{p}_{h}\\\ \overline{p}_{l}\\\ \overline{\lambda}_{j}\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\ 0\end{bmatrix}.$ (69) The first two rows of the system (69) represent the discretised differential equations in each subdomain, while the third row is the discretized Darcy’s law in the direction perpendicular to the interface. Here, $A_{h}$ and $A_{l}$ are the fixed-dimensional discretizations on the subdomains, $G_{h}$ is the discretization of Neumann boundary conditions on $\Omega_{h}$, $B_{l}$ is the discretization of source terms in $\Omega_{l}$, $J_{l}$ is the discretization of the vector source term on $\Omega_{l}$, $T_{j}$ is the discretized $\mathfrak{K}_{\Omega\Gamma}\mathfrak{K}_{\Gamma\Gamma}^{-1}$ product on $\Gamma_{j}$, and $\Pi_{N(\Omega_{h})}$ and $\Pi_{N(\Omega_{l})}$ are the projection operators from coupling variables on the mortar grid to each of the subdomains degrees of freedom. Furthermore, $P_{h}$ provides a discrete representation of the pressure trace operator on $\Omega_{h}$, $P_{l}$ gives the pressure unknowns on $\Omega_{l}$, $R_{l}$ gives the reconstruction of the pressure gradient on $\Omega_{l}$, and $\Pi_{L^{2}(\Omega_{k})}$ is the projection operator from numerical variables to coupling variables. Finally, $D_{j}$ is the discretized inverse normal permeability on $\Gamma_{j}$. We conclude by making two remarks: firstly, there is no direct coupling between $\Omega_{h}$ and $\Omega_{l}$ and secondly, global boundary conditions are left out of the system. ## 4 Numerical examples We validate the semi-local model and our implementation by a suite of numerical examples. First, we consider a case with a single fault, and show how the semi-local model can capture the effects of anisotropic off-diangonal permeabilities, while the local model fails to do so. Second, we probe the robustness of our discretization on more complex geometries in 2D and 3D. ### 4.1 Comparison to the equi-dimensional model In this first example, we compare our reduced model to an equi-dimensional model. The aim is to highlight the enhanced modelling capabilities of our formulation with respect to the standard local formulation. With reference to this latter point, we present results of two test cases: the first one where the fault has the same off-diagonal permeability on both sides (see Fig. 2.b), and a second one where different permeability structures are assigned to each side of the fault (see Fig. 2.c). \begin{overpic}[width=151.76964pt]{sketch_equi_dim} \put(5.0,54.0){\small$\Omega_{3}$} \put(63.0,66.0){\small$\Omega_{1}$} \put(63.0,32.0){\small$\Omega_{2}$} \put(8.0,90.0){\small$h_{out}$} \put(83.0,90.0){\small$h_{out}$} \put(48.0,6.0){\small$h_{in}$} \end{overpic} Figure 7: Setup of Case 1. #### 4.1.1 Case 1: homogeneous permeability We consider a 2D square domain of side $L=1~{}m$ cut by a horizontal fault of aperture $a=1~{}cm$ located in the middle of the domain. In the mixed- dimensional setting we therefore have two 2D domains $\Omega_{1}$ and $\Omega_{2}$ and one 1D fault $\Omega_{3}$, as illustrated in Fig. 7. The hydraulic conductivity is isotropic homogeneous for the 2D matrix, that is ${\bm{K}}_{j}=K_{j}\mathbf{I}$, with $j=1,2$, while for the fault we consider the following equi-dimensional full tensor: ${\bm{K}}_{3}=\begin{bmatrix}K_{f,\parallel}&k_{f,t}\\\ k_{f,t}&k_{f,\bot}\end{bmatrix},$ (70) For simplicity, we take $K_{1}=K_{2}=K_{m}$. Boundary conditions consist of an applied difference in hydraulic head along the vertical direction and no-flow conditions elsewhere. In particular, the inlet pressure $h_{in}$ is specified on the portion of the bottom boundary where $0.25<x<0.75~{}m$, while the outlet pressure $h_{out}$ is specified on the portion of the top boundary where $x<0.25~{}\&~{}x>0.75~{}m$ (see Fig. 7). Data for the simulations are reported in Table 1. We consider as reference solution the solution obtained with an equi-dimensional model of $N=40k$ structured square cells (mesh size $dx=5~{}mm$), where the fault is discretized with two rows of 200 elements each. Then, for the reduced models, we consider triangular grids with approximately $N=[40,160,700,3k,11k]$ (respectively $N_{f}=[4,8,16,32,64]$ cells for the fault), and report the average $L^{2}$ error in pressure along the fault $\varepsilon_{p}=\dfrac{\sqrt{\sum_{i}\Delta_{i}(p_{i}-p_{i,eq})^{2}}}{\sqrt{\sum_{i}\Delta_{i}p_{i,eq}^{2}}},$ (71) where $\Delta_{i}$ is the size of the fault element in the reduced model, and $p_{i,eq}$ is calculated from the equi-dimensional model as the mean value of the two fault cells at each location $x_{i}$: $p_{i,eq}(x_{i})=\sum_{j=y_{1},y_{2}}p_{ij},$ (72) where $y_{j}=L/2\pm dx/2$. Convergence results are shown in Fig. 8a. As Fig. 8a clearly shows, our formulation presents about first-order convergence rate, while the local formulation does not converge. This is due to the strong anisotropy of the fault, which is not captured by the standard local formulation. As a result of the anisotropy of the fault, the flow will take a preferential direction towards one of the two inlets, therefore breakig the symmetry of the local formulation. This is better observed in Fig. 8b showing the pressure distribution along the fault for the three models. As Fig. 8b clearly shows, the semi-local and the equi-dimensional models coincide, while the local formulation exhibits an erroneous symmetric profile. Table 1: Data for Case 1. Values of the fault hydraulic conductivity are given for the equi-dimensional model, i.e. before scaling. Parameter \| Description \| Value ---\|---\|--- $K_{m}$ \| Matrix hydraulic conductivity \| $1~{}m/s$ $K_{f,\parallel}$ \| Fault tangential hydraulic conductivity \| $100~{}m/s$ $k_{f,\bot}$ \| Fault normal hydraulic conductivity \| $100~{}m/s$ $k_{f,t}$ \| Fault off-diagonal hydraulic conductivity \| $80~{}m/s$ $a$ \| Fault aperture \| $0.01~{}m$ $L$ \| Side of the square domain \| $1~{}m$ $h_{in}$ \| Hydraulic head at the bottom boundary \| $10~{}m$ $h_{out}$ \| Hydraulic head at the top boundary \| $1~{}m$ (a) (b) Figure 8: Case 1: (a) convergence of the average error in pressure within the fault and (b) pressure distribution along the fault for different methods. #### 4.1.2 Case 2: dual permeability As a further illustration of the enhanced modeling capabilities of the semi- local model, we modify the setup used in the previous section to have different permeability structures on the two sides of the fault. This is relevant for modeling of geological faults, where the two sides of the fault may undergo different damage processes. To that end, we divide the fault into an upper and lower part (see Fig. 9) and assign different permeability structures to the two sides, that is for $j=1,2$: ${\bm{K}}_{3,j}=\begin{bmatrix}K_{f,\parallel}&k_{f,j,t}\\\ k_{f,j,t}&k_{f,\bot}\end{bmatrix}.$ (73) In particular, values of $K_{m}$, $K_{f,\parallel}$ and $k_{f,\bot}$ are the same as those given in Table 1, while $k_{f,1,t}$ and $k_{f,2,t}$ take values of $50$ and $80~{}m/s$, respectively. The aperture of the fault is set to $a=2~{}cm$ and we use the same boundary conditions as in Case 1. Convergence results for the local and semi-local models are shown in Figs. 10a-10b, with the reference solution again computed from an equi-dimensional model with a grid with 40k cells. As in the previous case, the local model fails to converge, while the semi-local model exhibits first order convergence up to the last refinement step. Here, the mesh size is of the same order of the fault aperture, thus further error reduction cannot be expected due to the modeling error in the dimension reduction. \begin{overpic}[width=325.215pt]{sketch_dual_kt} \put(75.0,26.0){\small${K}_{1}$} \put(75.0,8.0){\small${K}_{2}$} \put(75.0,15.0){\small${K}_{3,2}$} \put(75.0,19.5){\small${K}_{3,1}$} \put(102.0,15.0){$\small{1~{}cm}$} \put(102.0,20.0){$\small{1~{}cm}$} \end{overpic} Figure 9: Setup of Case 2. (a) (b) Figure 10: Case 2: (a) convergence of the average error in pressure within the fault and (b) pressure distribution along the fault for different methods. ### 4.2 Self-convergence In this section, we test the robustness of the method on more challenging fault configurations in 2D and 3D. #### 4.2.1 2D case We consider the same test case as Case 1 in Boon et al., (2018). The domain is a unit square including a network of five faults (Fig. 11a). Of these five faults, one cuts the square domain into two 2D subdomains, denoted as $\Omega_{1}$ and $\Omega_{2}$, respectively. The faults are numbered for $j=3,..,7$ and are of two kinds: $\Omega_{3}$ and $\Omega_{4}$ are conductive, that is $K_{3}=K_{4}=K_{f,1}$, while the other three are blocking, that is $K_{5}=K_{6}=K_{7}=K_{f,2}$. The hydraulic conductivity is isotropic homogeneous for the 2D matrix, with $K_{1}=K_{2}=K_{m}$, while for the faults we consider an equi-dimensional full tensor with $k_{j,t}=0.1K_{j,\parallel}$, for $j=3,..,7$. Boundary conditions consist of an applied difference in hydraulic head along the vertical direction and no-flow conditions elsewhere. Data for the simulations are reported in Table 2. We consider as reference solution the solution obtained with approximately $N=133k$ cells for the 2D domain and a total number of $N_{f}=510$ cells for the faults. Then we consider grids with approximately $N=[300,1k,4k,17k,67k]$ (respectively $N_{f}=[26,48,93,183,363]$), and report the average $L^{2}$ error in pressure along the faults. The convergence results, shown in Fig. 11b, indicate a rate of at least first order. The test thus confirms the performance of our method also in cases that involve faults that are intersecting and have low permeability. Both these features are highly relevant in a geologic setting where fault may have complex geometry and reduced permeability compared to the host rock. Table 2: Data for the 2D self-convergence test. Values of the fault hydraulic conductivity are given for the equi-dimensional model, i.e. before scaling. Parameter \| Description \| Value ---\|---\|--- $K_{m}$ \| Matrix hydraulic conductivity \| $1~{}m/s$ $K_{f,1,\parallel}$ \| Fault tangential hydraulic conductivity \| $100~{}m/s$ $k_{f,1,\bot}$ \| Fault normal hydraulic conductivity \| $100~{}m/s$ $k_{f,1,t}$ \| Fault off-diagonal hydraulic conductivity \| $10~{}m/s$ $K_{f,2,\parallel}$ \| Fault tangential hydraulic conductivity \| $0.01~{}m/s$ $k_{f,2,\bot}$ \| Fault normal hydraulic conductivity \| $0.01~{}m/s$ $k_{f,2,t}$ \| Fault off-diagonal hydraulic conductivity \| $0.001~{}m/s$ $a$ \| Fault aperture \| $0.01~{}m$ $h_{in}$ \| Hydraulic head at the top boundary \| $1~{}m$ $h_{out}$ \| Hydraulic head at the bottom boundary \| $0~{}m$ \begin{overpic}[width=151.76964pt]{sketch_2d_intersections} \put(5.0,52.0){\small$\Omega_{1}$} \put(88.0,5.0){\small$\Omega_{2}$} \put(55.0,80.0){\small$\Omega_{3}$} \put(63.0,66.0){\small$\Omega_{4}$} \put(75.0,52.0){\small$\Omega_{5}$} \put(25.0,32.0){\small$\Omega_{6}$} \put(78.0,15.0){\small$\Omega_{7}$} \end{overpic} (a) (b) Figure 11: 2D self-convergence test: (a) mixed-dimensional geometry and (b) convergence of the average error in pressure within the faults. #### 4.2.2 3D case As a final verification, we consider a 3D case with multiple intersecting faults. The setup is based on Case 2 in the benchmark study described in Berre et al., (2020). The domain is a unit cube including a network of 9 faults, whose intersections divide the cubic domain into several subdomains, as illustrated in Fig. 12a. These 3D subdomains are grouped into two regions, where we assigned different permeabilities $K_{m,1}$ and $K_{m,2}$, both homogeneous and isotropic (see Berre et al., (2020) for a visualization of these two regions). For the faults we consider full tensors with tangential permeability ${\bm{K}}_{j,\parallel}=K_{f,\parallel}\mathbf{I}_{\parallel}$, normal permeability $k_{j,\bot}=k_{f,\bot}$, and off-diagonal permeability ${\bm{k}}_{j,t}=0.1k_{j,t}\mathbf{i}_{\parallel}$. Boundary conditions consist of an imposed normal flux $q_{in}$ on the portion of the boundary where $(x,y,z)<0.25~{}m$ and a constant hydraulic head $h_{out}$ on the portion of the boundary where $(x,y,z)>0.875~{}m$. Data for the simulations are reported in Table 3. We consider as reference solution the solution obtained with approximately $N_{3}=85k$ cells for the 3D domain and a total number of $N_{f}=8364$ cells for all faults. Then we consider $N_{3}=[500,1k,2k,4k,10k,20k,40k]$ (respectively $N_{f}=[148,282,384,814,1536,2298,3456]$) and report the average $L^{2}$ error in pressure along the faults. Convergence results are shown in Fig. 12b, indicating first order convergence on average. This confirms the consistency of our implementation also for 3D problems with complex fault geometries. Table 3: Data for the 3D self-convergence test. Values of the fault hydraulic conductivity are given for the equi-dimensional model, i.e. before scaling. Parameter \| Description \| Value ---\|---\|--- $K_{m,1}$ \| Matrix hydraulic conductivity \| $1~{}m/s$ $K_{m,2}$ \| Matrix hydraulic conductivity \| $0.1~{}m/s$ $K_{f,\parallel}$ \| Fault tangential hydraulic conductivity \| $1e^{4}~{}m/s$ $k_{f,\bot}$ \| Fault normal hydraulic conductivity \| $1e^{4}~{}m/s$ $k_{f,t}$ \| Fault off-diagonal hydraulic conductivity \| $1e^{3}~{}m/s$ $a$ \| Fault aperture \| $1e^{-4}~{}m$ $q_{in}$ \| Normal flux at the inflow boundary \| $-1~{}m/s$ $h_{out}$ \| Hydraulic head at the outflow boundary \| $1~{}m$ (a) (b) Figure 12: 3D self-convergence test: (a) mixed-dimensional geometry and (b) convergence of the average error in pressure within the faults. ## 5 Conclusions We presented an improved framework to modelling and discretizing flow in generally anisotropic porous media with thin inclusions, within the context of mixed-dimensional partial differential equations. Our model considers a full permeability tensor for the inclusions, resulting in additional terms arising in our formulation as compared to existing local discretizations. We expect our model to be important for modeling of flow in faulted porous media, however the methods proposed herein can be in any case applied to models of fractures, in fact our full-permeability model naturally reduces to the existing models of fracture-matrix flow when the off-diagonal components of the inclusion permeability tensor are set to zero. We provided numerical examples showing convergence of the method for both 2D and 3D faulted porous media. In particular, we provided numerical evidence that, as opposed to existing local discretizations, our model is capable of simulating the anisotropic behaviour of the faults near damage zone. 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# Variational quantum solver employing the PDS energy functional Bo Peng<EMAIL_ADDRESS>Physical and Computational Science Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States of America Karol Kowalski<EMAIL_ADDRESS>Physical and Computational Science Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States of America ###### Abstract Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese- Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eigensolver (VQE) and the original static PDS approach, this new variational quantum solver offers an effective approach to navigate the dynamics to be free from getting trapped in the local minima that refer to different states, and achieve high accuracy at finding the ground state and its energy through the rotation of the trial wave function of modest quality, thus improves the accuracy and efficiency of the quantum simulation. We demonstrate the performance of the proposed variational quantum solver for toy models, H2 molecule, and strongly correlated planar H4 system in some challenging situations. In all the case studies, the proposed variational quantum approach outperforms the usual VQE and static PDS calculations even at the lowest order. We also discuss the limitations of the proposed approach and its preliminary execution for model Hamiltonian on the NISQ device. ## 1 Introduction Quantum computing (QC) techniques attract much attention in many mathematics, physics, and chemistry areas by providing means to address insurmountable computational barriers for simulating quantum systems on classical computers.[47, 61, 53, 3, 2, 43] One of the focus areas for quantum computing is quantum chemistry, where Hamiltonians can be effectively mapped into qubit registers. In this area, several quantum computing algorithms, including quantum phase estimator (QPE) [38, 15, 5, 12, 58, 69, 26, 52] and variational quantum eigensolver (VQE), [50, 44, 55, 60, 31, 32, 24, 16, 29] have been extensively tested on benchmark systems corresponding to the description of chemical reactions involving bond-forming and breaking processes, excited states, and strongly correlated molecular systems. In more recent applications, several groups reported quantum algorithms for imaginary time evolution,[42, 46] quantum filter diagonalization,[48] quantum inverse iteration algorithms,[36] and quantum power/moments methods. [59, 67] The main thrust that drives this field is related to the efficient encoding of the electron correlation effects that are needed to describe molecular systems. Basic methodological questions related to an efficient way of incorporating large degrees of freedom required to capture a subtle balance between static and dynamical correlations effects still need to be appropriately addressed. A typical way of addressing these challenges in VQE approaches is by incorporating more and more parameters (usually corresponding to excitation amplitudes in a broad class of unitary coupled-cluster methods [27, 4, 64, 35, 37, 17]). Unfortunately, this brute force approach is quickly stumbling into insurmountable problems associated with the resulting quantum circuit complexity and problems with numerical optimization procedures performed on classical machines (the so-called barren plateau problem reported in Refs.[45, 11, 68, 10, 51, 41, 66]). In this paper, we propose a new solution to these problems. Instead of adding more parameters to the trial wave function, we choose to optimize a new class of energy functionals (or quasi-functionals, where the energy is calculated as a simple equation solution) that already encompasses information about high- order static and dynamical correlation effects. An ideal choice for such high- level functional is based on the Peeters, Devreese, and Soldatov (PDS) formalism,[49, 62] where variational energy is obtained as a solution of simple equations expressed in terms of the Hamiltonian’s moments or expectations values of the powers of the Hamiltonians operator defined for the trial wave function. In Ref. [34] we demonstrated that in such calculations high-level of accuracy can be achieved even with very simple parametrization of the trial wave functions (capturing only essential correlation effects) and low-rank moments. We believe that merging the PDS formalism with the quantum gradient based variational approach would be considered as a more interesting alternative for by-passing main problems associated with the excessive number of amplitudes that need to be included to reach the so-called chemical accuracy. In the following sections we will briefly introduce the PDS formalism and describe how the PDS energy functional can be incorporated with the minimization procedures that are based on the quantum gradient approach [25, 57, 45, 42, 71] to produce a new class of variational quantum solver (which is called PDS($K$)-VQS for short in the rest of the paper) to target the ground state and its energy in a quantum-classical hybrid manner. Furthermore, we will test its performance, in particular the performance of the more affordable lower order PDS($K$)-VQS ($K=2,3,4$) approaches combining with the trial wave function expressed in low-depth quantum circuits, at finding the ground state and its energy for the Hamiltonians describing toy models and H2 molecular system, as well as the strongly correlated planar H4 system, in some challenging situations where the barren plateau problem precludes the effective utilization of the standard VQE approach. ## 2 Method ### 2.1 PDS formalism In this section we will give a brief description of the PDS formalism. The detailed discussion of the PDS methodology and highly relevant connected moment expansion (CMX) formalisms have been given in the original work[49, 62] as well as our recent work[34, 14] and many earlier literatures (see for example Refs. [33, 54, 39, 65, 40, 18, 19]). The many-body techniques used in the derivation of PDS expansions originate in the effort to provide upper bounds for the free energies, and to provide alternative re-derivation of the Bogolubov’s [8] and Feynman’s [20] inequalities. Since the Gibbs-Bogolubov inequality reduces to the Rayleight-Ritz variational principle in zero temperature limit, these formulations can be directly applied to quantum chemistry. Here we only provide an overview of basic steps involved in the derivations of the PDS formulation. A starting point of the studies of upper bounds for the exact ground-state energy $E_{0}$ is the analysis of function $\Gamma(t)$ (defined for trial wave function $\|\phi\rangle$ having non-zero overlap with the ground-state wave function) $\Gamma(t)=\langle\phi\|e^{-tH}\|\phi\rangle\;,$ (1) and its Laplace transform $f(s)$ $f(s)=\int_{0}^{+\infty}e^{-st}\Gamma(t)dt\;.$ (2) It can be proved that, for a complex scalar $s$, Eq. (2) exists if the real part of $s$ $\Re(s)>-E_{0}$. Under this condition, for Hamiltonian $H$ defined by discrete energy levels $E_{i}$ and corresponding eigenvectors $\|\Psi_{i}\rangle$ ($i=0,1,\ldots,M$) $H=\sum_{i=0}^{M}E_{i}\|\Psi_{i}\rangle\langle\Psi_{i}\|\;,$ (3) $f(s)$ takes the form $f(s)=\sum_{i=0}^{M}\frac{\omega(E_{i})}{s+E_{i}}$ (4) where $\omega(E_{i})=\|\langle\Psi_{i}\|\phi\rangle\|^{2}$. The PDS formalism is based on introducing parameters into expansion (4) using a simple identity (with a real parameter $a$) $\displaystyle\frac{1}{s+E_{n}}$ $\displaystyle=$ $\displaystyle\frac{1}{s+a}-\frac{E_{n}-a}{(s+a)^{2}}+\frac{(E_{n}-a)^{2}}{(s+E_{n})(s+a)^{2}}\;.$ (5) When the above identity is applied for the first time to Eq. (4) (introducing the first parameter $a_{1}$) one gets the following expression for the $f(s)$ function $\displaystyle f(s)$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{M}\omega(E_{i})\left[\frac{1}{s+a_{1}}-\frac{E_{i}-a_{1}}{(s+a_{1})^{2}}+\frac{(E_{i}-a_{1})^{2}}{(s+E_{i})(s+a_{1})^{2}}\right]$ (6) The transformation (6) can be repeated $K$ times (with each time introducing a new parameter $a_{i}$, $i=1,\ldots,K$) to reformulate the $f(s)$ function as $f(s)=R_{K}(s,a_{1},\ldots,a_{K})+W_{K}(s,a_{1},\ldots,a_{K})\;,$ (7) where $\displaystyle R_{K}(s,a_{1},\ldots,a_{K})$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{M}\left[\frac{\omega(E_{i})}{s+E_{i}}\prod_{j=1}^{K}\frac{(E_{i}-a_{j})^{2}}{(s+a_{j})^{2}}\right]\geq- E_{0}\leavevmode\nobreak\ \leavevmode\nobreak\ (\text{if}\leavevmode\nobreak\ \leavevmode\nobreak\ \Re(s)>-E_{0}),$ (8) $\displaystyle W_{K}(s,a_{1},\ldots,a_{K})$ $\displaystyle=$ $\displaystyle\sum_{i=0}^{M}\left\\{\omega(E_{i})\sum_{j=1}^{K}\left[\Big{(}\frac{1}{s+a_{j}}-\frac{E_{i}-a_{j}}{(s+a_{j})^{2}}\Big{)}\prod_{n=1}^{j-1}\frac{(E_{i}-a_{n})^{2}}{(s+a_{n})^{2}}\right]\right\\}\;.$ (9) The $K$-th order PDS formalism (PDS($K$) for short henceforth) is then associated with defining the introduced $K$ real parameters $(a_{1},\ldots,a_{K})$ that minimize the value of $R_{K}(s,a_{1},\ldots,a_{K})$. In this minimization process the necessary extreme conditions are given by the system of equations $\frac{\partial R_{K}(s,a_{1},\ldots,a_{K})}{\partial a_{i}}=0,\leavevmode\nobreak\ \leavevmode\nobreak\ (i=1,\ldots,K),$ (10) which can be alternatively represented by the matrix system of equations for an auxiliary vector $\mathbf{X}=(X_{1},\cdots,X_{K})^{T}$ $\mathbf{MX}=-\mathbf{Y}.$ (11) Here, the matrix elements of $\bf M$ and vector $\bf Y$ are defined as the expectation values of Hamiltonian powers (i.e. moments), $M_{ij}=\langle\phi\|H^{2K-i-j}\|\phi\rangle$, $Y_{i}=\langle\phi\|H^{2K-i}\|\phi\rangle$ ($i,j=1,\cdots,K$) (for simplicity, we will use the notation $\langle H^{n}\rangle\equiv\langle\phi\|H^{n}\|\phi\rangle$). It can be shown that the optimal parameters in the PDS($K$) formalism, $(a_{1}^{(K)},\ldots,a_{K}^{(K)})$, are the roots of the polynomial $P_{K}(\mathcal{E})$, $P_{K}(\mathcal{E})=\mathcal{E}^{K}+\sum_{i=1}^{K}X_{i}\mathcal{E}^{K-i},$ (12) and these roots provide upper bounds for the exact ground and excited state energies, e.g., for the ground state energy we have $E_{0}\leq{\rm min}(a_{1}^{(K)},\ldots,a_{K}^{(K)})\leq\langle\phi\|H\|\phi\rangle\;.$ (13) Note that, as shown in Refs.[49, 62] the PDS formalism also applies to the Hamiltonian characterized by discrete and continuous spectral resolutions together. ### 2.2 PDS($K$)-VQS formalism In the variational method, we approximate the quantum state using parametrized trial state $\|\Psi\rangle\approx\|\phi\rangle$. Using a quantum circuit, the trial state can be prepared by applying a sequence of parametrized unitary gates on the initial state $\|0\rangle$, $\displaystyle\|\phi\rangle=\|\phi(\vec{\theta})\rangle=\cdots U_{k}(\theta_{k})\cdots U_{1}(\theta_{1})\|0\rangle$ (14) ($\vec{\theta}=\\{\theta_{1},\cdots,\theta_{n}\\}$). Here $U_{k}(\theta_{k})$ is the $k$-th unitary single- or two-qubit gate that is controlled by parameter $\theta_{k}$. The goal is to approach the ground-state energy of a many-body Hamiltonian, $H$, by finding the values of these parameters, $\vec{\theta}$, that minimize the expectation value of the Hamiltonian $\displaystyle E_{\min}=\min_{\vec{\theta}}\langle\phi(\vec{\theta})\|H\|\phi(\vec{\theta})\rangle.$ (15) To do this, the conventional VQE starts by constructing the ansatz $\|\phi(\vec{\theta})\rangle$ and measuring the corresponding expectation value of the Hamiltonian using a quantum computer, and then relies on a classical optimization routine to obtain new $\vec{\theta}$. During the parameter optimization (or dynamics), the set of parameters that is updated at the $k$-th step ($k>1$) can be written as $\displaystyle\vec{\theta}_{k}=\vec{\theta}_{k-1}-\eta\mathcal{R}^{-1}(\vec{\theta})\nabla\mathcal{E}(\vec{\theta}).$ (16) where $\nabla\mathcal{E}(\vec{\theta})=\partial\mathcal{E}/\partial\vec{\theta}$ is the energy gradient vector, and $\eta$ is the step size (or learning rate). $\mathcal{R}(\vec{\theta})$ is the Riemannian metric matrix at $\vec{\theta}$ that is flexible to characterize the singular point in the parameter space and is essentially related to the indistinguishability of $\mathcal{E}(\vec{\theta})$.[71] It is worth mentioning that Eq. (5) originates from natural gradient learning method in the general nonlinear optimization framework especially targeting machine learning problems.[1] Here, the natural gradient is the optimizer that accounts for the geometric structure of the parameter space. For the curved (or nonorthonormal) parameter manifold that exhibits the Riemannian character (e.g. in large neural networks), natural gradient learning method is often employed to avoid the plateaus in the parameter space.[42, 71, 63] Note that when the parameter space is a Euclidean space with orthonormal coordinate system the Riemannian metric tensor will reduce to the unity matrix (see Tab. 1). In VQE setting, one can define the Riemannian metric as the quantum Fubini-Study metric, which is the quantum analog of the Fisher information matrix in the classical natural gradient,[1] to measure the distance in the space of pure quantum states. The quantum Fubini-Study metric describes the curvature of the ansatz class rather than the learning landscape, but often performs as well as Hessian based methods (e.g. BFGS optimizer that approximates the Hessian of the cost function using first-order gradient, see Ref. [70] for a recent detailed discussion). There are also some other options for the Riemannian metric including imaginary-time evolution (ITE) or even classical Fisher metric that have been discussed in some recent reports.[42, 71, 63] In Tab. 1, three commonly used flavors of the Riemannian metric matrix $\mathcal{R}(\vec{\theta})$ are listed and will be used in the following case studies. Remarkably, as pointed out in Refs. [73, 42], the difference between natural gradient descent (NGD) and ITE accounts for the global phase, and if introducing a time-dependent phase gate to the trial state, the Riemannian metric employing NGD will be equivalent to the metric employing ITE. Riemannian metric $\mathcal{R}_{ij}(\vec{\theta})$ GD $\delta_{ij}$ NGD $\Re\Big{(}\frac{\partial\langle\phi(\vec{\theta})\|}{\partial\theta_{i}}\frac{\partial\|\phi(\vec{\theta})\rangle}{\partial\theta_{j}}\Big{)}-\frac{\partial\langle\phi(\vec{\theta})\|}{\partial\theta_{i}}\|\phi(\vec{\theta})\rangle\langle\phi(\vec{\theta})\|\frac{\partial\|\phi(\vec{\theta})\rangle}{\partial\theta_{j}}$ ITE $\Re\Big{(}\frac{\partial\langle\phi(\vec{\theta})\|}{\partial\theta_{i}}\frac{\partial\|\phi(\vec{\theta})\rangle}{\partial\theta_{j}}\Big{)}$ Table 1: Three Rimannian metric forms, ordinary gradient descent (GD), natural gradient descent (NGD), and imaginary time evolution (ITE), exploited in the present study. Figure 1: The workflow of variational quantum solver employing the PDS energy functional. To get the energy gradient in the PDS framework, take the derivative w.r.t. $\theta_{i}$ on both sides of Eq. (12), and after reorganizing the terms we can express the energy derivative as $\displaystyle\frac{\partial\mathcal{E}}{\partial\theta_{i}}=\frac{-1}{K\mathcal{E}^{K-1}+\displaystyle\sum_{i=1}^{K-1}(K-i)X_{i}\mathcal{E}^{K-i-1}}\left(\begin{array}[]{c}\mathcal{E}^{K-1}\\\ \vdots\\\ 1\end{array}\right)^{T}\frac{\partial\mathbf{X}}{\partial\theta_{i}},$ (20) where $\frac{\partial\mathbf{X}}{\partial\theta_{i}}$ is associated with the $\theta_{i}$-derivative of Eq. (11), $\displaystyle\mathbf{M}\frac{\partial\mathbf{X}}{\partial\theta_{i}}=-\frac{\partial\mathbf{Y}}{\partial\theta_{i}}-\frac{\partial\mathbf{M}}{\partial\theta_{i}}\mathbf{X},$ (21) and can be obtained by solving Eq. (21) as a linear equation with $\partial Y_{i}/\partial\theta_{k}=\partial\langle H^{2K-i}\rangle/\partial\theta_{k}$ and $\partial M_{ij}/\partial\theta_{k}=\partial\langle H^{2K-i-j}\rangle/\partial\theta_{k}$. Fig. 1 summarizes the workflow of PDS($K$)-VQS, where on the classical side the PDS($K$) module includes two steps, (i) solving two consecutive linear problems to get $\mathbf{X}$ and $\partial\mathbf{X}/\partial\theta_{i}$, and (ii) solving for roots of polynomial (12) and computing Eq. (20). On the quantum side, in comparison with the conventional VQE, the present PDS($K$)-VQS infrastructure relies on quantum circuits to measure $\langle H^{n}\rangle$ and their $\vec{\theta}$-derivatives. In the present work, due to the relatively small system size, we directly exploit the Hadamard test to compute the real part of $\langle H^{n}\rangle$ for the Hamiltonians that are represented as a sum of Pauli strings. It is worth mentioning that for typical molecular systems that can be represented by $N$ qubits, the number of $\langle H^{n}\rangle$ measurement scales as $\mathcal{O}(N^{4n})$, which nevertheless can be reduced once the Pauli strings are multiplied and their expectation values are re-used as the contributions to the higher order moments.[34] For example, as we will show later for the H4 system that comprises 184 Pauli strings in the Hamiltonian, the effective number of Pauli strings required for arbitrary $\langle H^{n}\rangle$ ($n=2,3,4$) measurements can be dropped from 1842, 1843, and 1844 to 1774, 3702, and 4223, respectively, after the Pauli reduction, and the 4223 strings will not be changed for more complex $\langle H^{n}\rangle$’s ($n>4$). Similar findings have also been reported in Ref. [67], where by grouping the Pauli strings into tensor-product basis sets the authors examined the operator counts for $\langle H^{4}\rangle$ of Heisenberg model defined on different lattice geometries for the number of qubits ranging from 2 up to 36, and found that the effective number of Pauli strings to be measured drops by several orders of magnitude with sub-linear scaling in a given number of qubits. For larger systems, the number of measurements can be further reduced by introducing active space and local approximation. Alternatively, one can approximate $\langle H^{n}\rangle$ by a linear combination of the time- evolution operators as introduced in some recent reports.[59, 7] For the estimation of $\partial\langle H^{n}\rangle/\partial\theta_{k}$, in the present work we limit $U_{k}(\theta_{k})$ exploited in the state preparation to be only one-qubit rotations. Then, following Ref. [57], $\partial\langle H^{n}\rangle/\partial\theta_{k}$ can be obtained by measuring $\langle H^{n}\rangle$ twice using the same circuit but shifting $\theta_{k}$ by $\pm\frac{\pi}{2}$ separately, i.e. $\displaystyle\frac{\partial\langle H^{n}\rangle_{(\cdots,\theta_{k},\cdots)}}{\partial\theta_{k}}=\frac{1}{2}\Big{(}\langle H^{n}\rangle_{(\cdots,\theta_{k}+\frac{\pi}{2},\cdots)}-\langle H^{n}\rangle_{(\cdots,\theta_{k}-\frac{\pi}{2},\cdots)}\Big{)}.$ (22) If $\theta_{k}$ parametrizes more than one one-qubit rotations in the circuit, then based on the product rule $\partial\langle H^{n}\rangle/\partial\theta_{k}$ will have contributions from all one-qubit $\theta_{k}$ rotations, each of which will be obtained by applying (22) on the corresponding rotation. ## 3 Numerical examples In this section, with several examples, we will demonstrate how the PDS($K$)-VQS performs in some challenging situations, and its difference in comparison to the conventional VQE and static PDS($K$) expansions. ### 3.1 Toy Hamoltonians We first test the PDS($K$)-VQS on two toy Hamiltonians $\displaystyle H_{A}$ $\displaystyle=1.5I_{4\times 4}+0.5(I_{2\times 2}\otimes\sigma_{z}-2\sigma_{z}\otimes\sigma_{z})$ $\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&2&0&0\\\ 0&0&3&0\\\ 0&0&0&0\end{array}\right),$ (27) $\displaystyle H_{B}$ $\displaystyle=I_{4\times 4}+0.5(I_{2\times 2}\otimes\sigma_{z}-\sigma_{z}\otimes\sigma_{z})$ $\displaystyle=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&0&2&0\\\ 0&0&0&0\end{array}\right),$ (32) with ansatze $\displaystyle\|\phi_{A}(\theta_{1},\theta_{2})\rangle=\tilde{R}_{Y}^{0,1}(\theta_{2})R_{X}^{0}(\theta_{1})\|00\rangle,$ $\displaystyle\|\phi_{B}(\theta_{1},\theta_{2})\rangle=\tilde{R}_{Y}^{0,1}(\theta_{2})R_{X}^{0}(\theta_{1})R_{X}^{1}(\theta_{1})\|01\rangle,$ that have been exploited by McArdle et al. [42] to demonstrate the performance of different Riemannian metrics in the conventional VQE approach for finding the ground-state energy of the same Hamiltoinans. Here, $\tilde{R}_{Y}^{p,q}(\theta)$ is a controlled $Y$ rotation of $\theta$ with control qubit $p$ and target qubit $q$, and $R_{X}^{p}(\theta)$ is a rotation of $\theta$ on qubit $p$ around the $x$-axis. The rotation about the $j$-axis is defined as $R_{\sigma_{j}}(\theta)=e^{-\text{i}\theta\sigma_{j}/2}$ with $\sigma_{j}$ being one of the Pauli spin matrices. Figure 2: Variational trajectories on the PDS(2) energy surface (left panels) and original potential energy surface (right panels) discovering the ground state energy of Hamiltonian, $H_{A}$, explored by gradient descent (top panels) and natural gradient descent/imaginary time evolution (bottom panels). On the background energy surfaces, the dark blue and white colors correspond to the global maximum and minimum energies, respectively. The arrows indicate the trajectories of the dymanics, and are colored green if the trajectory converges to the ground state energy, and red otherwise. The step size $\eta=0.05$ in all the calculations. Figure 3: Variational trajectories on the PDS(2) energy surface (left panels) and original potential energy surfaces (right panels) discovering the ground state energy of Hamiltonian, $H_{B}$, explored by gradient descent (top panels), natural gradient descent (middle panels), and variational imaginary time (bottom panels). On the background energy surfaces, the dark blue and white colors correspond to the global maximum and minimum energies, respectively. The arrows indicate the trajectories of the methods, and are colored green if the trajectory converges to the true ground state energy, and red otherwise. The step size $\eta=0.05$ in all the calculationss. Figs. 2 and 3 show the performances of the proposed PDS($K$)-VQS ($K=2$, i.e. PDS(2)-VQS) and the conventional VQE approaches for finding the ground state energy of the toy Hamiltonians. As can be seen, the ability of VQE navigation to avoid the local minima on the conventional PES depends on the Riemannian metric exploited. For system A, in comparison to GD, the NGD (or equivalently ITE in this case) is able to avoid the local minimum at $(\theta_{1},\theta_{2})=(0,0)$. This is because the Riemannian metric, $\displaystyle\mathcal{R}=\left(\begin{array}[]{cc}\frac{1}{4}&0\\\ 0&\frac{1}{4}\sin^{2}(\frac{\theta_{1}}{2})\end{array}\right),$ (35) used in the NGD/ITE correctly characterizes any rotation pair with $\theta_{1}=0$ as a singular point (i.e. $\det\|R\|=0$) such that $\mathcal{R}^{-1}$ will numerically navigate the dynamics (e.g. via singular value decomposition) to avoid collapsing in this local minimum once the trajectory is getting close. Therefore, if the metric is unable to characterize the local minima as singular points, the VQE would still get trapped. This can be observed from the VQE performance for system B, where both NGD and ITE fail to escape the local minima, $(\theta_{1},\theta_{2})\sim(\pm\frac{3\pi}{8},0)$, in the dynamics due to the fact that the local minima are not the singular points of $R$ in either NGD or ITE. In contrast, the PDS(2)-VQS robustly converge to the true ground state for both systems regardless of the employed Riemannian metric. The success of PDS($K$)-VQS in these toy examples can be essentially attributed to the fact that, in comparison to the original PES where the local minima corresponding to a non-ground state, the entire PDS($K$) energy surface, except the singular areas (see the infinitesimal white strips on the left panels of Fig. 2 at $\theta_{1}=0$) where the fidelity of the trial wave function w.r.t the target state is strictly zero, provides an upper bound energy surface for the same (ground) state. This state-specific nature makes the PDS($K$)-VQS essentially explore a lower upper bound of the ground state at a given PDS order, and therefore the dynamics will not be trapped at a location that is associated with a different state. It worth mentioning that, a lower bound of the ground state energy can also be obtained from a static, and more costly, higher order PDS($K$) standalone calculation as demonstrated in our previous work.[34] From this perspective, the PDS($K$)-VQS approach provides an effective way to explore the possibility of pushing the low order PDS($K$) results towards high accuracy that would otherwise require higher-order and more expensive PDS($K$) calculations. Besides, since generalized variational principle applies in the PDS framework,[49, 62] if other roots of Eq. (12) are concerned, the PDS($K$)-VQS will also be able to navigate the dynamics to give lower upper bounds for excited states as long as the fidelity of the trial wave function with respect to the target state is non-zero. ### 3.2 $H_{2}$ and $H_{4}$ systems We further employ the proposed PDS($K$)-VQS approach to find the ground state energy of H2 and H4 molecular systems. For H2 molecule, we exploit an effective Hamiltonian and an ansatz exploited by Yamamoto [71] and Bravyi et al.[9], $\displaystyle H=0.4(\sigma_{z}\otimes I+I\otimes\sigma_{z})+0.2\sigma_{x}\otimes\sigma_{x}$ $\displaystyle\|\phi(\vec{\theta})\rangle=R_{Y}^{0}(2\theta_{3})R_{Y}^{1}(2\theta_{4})\tilde{U}_{N}^{0,1}R_{Y}^{0}(2\theta_{1})R_{Y}^{1}(2\theta_{2})\|00\rangle$ where $\tilde{U}_{N}^{p,q}$ denotes the CNOT gate with control qubit $p$ and target qubit $q$. Figure 4: The computed ground state energy (top panels), energy deviation w.r.t. exact energy (middle panels), and fidelity of the trial state (bottom panels) of the H2 molecule iterate in the conventional VQE and PDS($K$)-VQS ($K=2,3,4$) infrastructures employing gradient descent (left panels) and natural gradient descent/imaginary time evolution (right panels). The initial rotation is given by $\vec{\theta}=(7\pi/32,\pi/2,0,0)$. The step size $\eta=0.05$ in all the calculations. Fig. 4 compares the VQE and PDS($K$)-VQS performances exploiting the above- mentioned ansatz to find the ground state energy of the H2 Hamiltonian. As can be seen, starting from the given initial rotation, the VQE is unable to converge to the ground state energy within 100 iterations, but rather drops to an excited state energy ($-0.2$ a.u. in this case). Actually, it has been shown that,[71] starting from the same initial rotation, the VQE needs to go through a “plateau” that resides at this energy value and spreads over $\sim$400 iterations before hitting the ground state energy ($\sim$$-0.8$ a.u. in this case) regardless of the employed Riemannian metric. To achieve a higher level of accuracy (e.g. chemical accuracy $\\|E(\vec{\theta})-E_{exact})\\|<1.5\times 10^{-3}$ a.u.), low order PDS($K$)-VQS typically needs more iterations than high order PDS($K$)-VQS. As shown in the middle left panel of Fig. 4, by employing GD in the dynamics, it takes the PDS(4)-VQS $<$10 iterations to converge to the ground state energy with energy deviation being $<10^{-14}$ a.u. regardless of the employed Riemannian metric, while it takes the PDS(2)/PDS(3)-VQS almost 100 steps to bound the deviation to be $<10^{-3}$ a.u. Remarkably, the performance can be improved when GD is replaced by NGD/ITE in the PDS(2)/PDS(3)-VQS dynamics. In particular, within 80 iterations the PDS(3)-VQS employing NGD/ITE can converge to the accuracy level that is almost same as that of PDS(4)-VQS. On the other hand, the quality of the trial wave function is more significantly improved in the low order PDS($K$)-VQS dynamics than in the high order PDS($K$)-VQS dynamics. For example, the fidelity of the trial wave function w.r.t the exact ground state gradually increases from almost zero to $\sim$0.35 within 100 iterations using PDS(2)-VQS regardless of the employed Riemannian metric, and this change is significantly steeper than the almost flat curves of PDS(3)/PDS(4)-VQS as shown at the bottom of Fig. 4. However, in comparison to GD, employing NGD/ITE in the PDS(3)/PDS(4)-VQS quickly improves the fidelity of the trial wave function from $<$0.02 to 0.2$\sim$0.3 within 10 iterations. It is worth mentioning that since the fidelity of the trial wave function at the initial rotation is almost zero, both VQE and the static PDS($K$) ($K=2,3,4$) calculations alone cannot help identify the ground state energy in this case, which makes PDS($K$)-VQS a necessary and effective approach to target ground state energy and improve the trial wave function. Remarkably, the improvement of the trial wave function employing PDS($K$)-VQS approach might be limited. This can be seen from the flat fidelity curves of the trial state driven in the PDS(3/4)-VQS dynamics after first several iterations as shown at the bottom right of Fig. 4. This is due to the fact that the PDS($K$) formalism does not require the ansatz to sufficiently approximate the target state, while is still able to provide systematically improvable upper bounds of the expectation value of the target state by exploring the Krylov subspace. The benefit is the great simplification of the state preparation. The limitation is also obvious in that it sometimes would be challenging to further improve the quality of the trial state within the PDS($K$)-VQS framework if the energies were well converged already, which would then compromise the accuracy of the property calculations that usually requires a sufficiently accurate description of the target wave function. Figure 5: The performance of VQE and PDS($K$)-VQS ($K=2,3,4$) employing ordinary gradient descent (GD) to compute the ground state energy of a planar H4 system with $R_{H-H}=2.0$ a.u. (Top left) The circuit used to generate the ansatz with 16 rotation parameters that is inspired by the basis rotation ansatz for a linear hydrogen chain in Ref. [2]. Here, we consider the planar H4 system in 3-21G basis. The generated Hamiltonian acts on eight qubits, and considers an active space of four electrons in eight spin-orbitals. With this Hamiltonian, the ground state of the planar H4 system is a triplet state with the exact energy $E_{\text{exact}}=-2.00591266$ a.u. (Bottom left) the deviations of the VQE and PDS($K$)-VQS energies and (bottom right) the fidelity change of the trial wave function w.r.t. true ground state during the PDS($K$)-VQS calculations. The initial values of all the rotations are set to 0.001. The step size $\eta=1.0$ in all the calculations. We also test the proposed PDS($K$)-VQS approach for a slightly larger system, the planar H4 system, where a 8-qubit circuit with 16 rotation parameters shown at the top of Fig. 5 is employed to prepare the trial wave function for finding the ground state energy. The state preparation circuit is inspired by the similar circuit that has been reported being successfully applied for preparing the trial state for the Hartree Fock state of the linear hydrogen chain systems.[2] For the planar H4 system whose ground state is a triplet, the circuit with close-to-zero initial rotations would generate a trial state that is almost singlet, which makes the conventional VQE and the static PDS($K$) ($K=2,3,4$) simply fail. On the other hand, as shown at the bottom of Fig. 5, the PDS($K$)-VQS ($K=2,3,4$) are capable of dealing with such a tough situation and again outperform. As can be seen, within 200 iterations, PDS($K$)-VQS ($K=2,3,4$) are able to converge to the ground state energy well below chemical accuracy and improving the fidelity of the trial wave function to be $>$0.96. It is worth noting that even though the converged rotations obtained from the PDS($K$)-VQS calculations generate a high fidelity state, the expectation value of the generated state is still $\sim 0.02$ a.u. above the exact energy, and it then becomes challenging to further improve the fidelity employing the same circuit infrastructure through varying the rotations. Therefore, the circuit used here might not be sufficient for preparing true ground state in practice if higher fidelity is desired. We here intentionally employ the circuit to artificially generate an extreme challenging case to show the performance difference between conventional VQE and PDS($K$)-VQS approaches. ## 4 Discussion From Section III, it has been seen that the PDS($K$)-VQS approach bears the potential of speeding up the iterations in comparison with the conventional VQE approach. However, it is worth noting that the measurement effort of evaluating $\langle H^{n}\rangle$’s ($n>1$) and their derivatives are usually more expensive than that of $\langle H\rangle$ and its derivative, and the actual cost saving will therefore be compromised. To have a close look at the measurement of the $\langle H^{n}\rangle$ (and its impact on the total cost), we employ the following metric to give an estimate for the number of measurements, $M$,[23, 56, 69] $\displaystyle M=\Bigg{(}\frac{\sum_{G}\sqrt{\sum_{i,j,\in G}h_{i}h_{j}\text{cov}\big{(}P_{i},P_{j}\big{)}}}{\epsilon}\Bigg{)}^{2},$ (36) where $\epsilon$ is the desired precision and $h_{i}$’s and $P_{i}$’s are the coefficients and Pauli strings representing a moment (i.e. $H^{n}=\sum_{i}h_{i}P_{i}$) and having been partitioned into certain groups, $G$’s, in which simultaneous measurement can be performed. $\text{cov}\big{(}P_{i},P_{j}\big{)}$ is the covariance between two Pauli strings bounded by $\displaystyle\text{cov}\big{(}P_{i},P_{j}\big{)}\leq\|\sqrt{\text{var}(P_{i})\cdot\text{var}(P_{j})}\|$ (37) with the variance being computed from $\text{var}(P_{i})=1-\langle P_{i}\rangle^{2}$. Here, we assume the covariances between different Pauli strings to be zero for the brevity of the discussion. We can apply the above metric to, for example, estimate the number of measurements of $H^{n}$ ($n=1,2,3$) required by the PDS(2)-VQS calculation for the complete active space (4 electrons, 4 spin-orbitals) of the planar H4 system. Given $\epsilon\sim 0.5$mHartree, since $H^{n}$ ($n=1,2,3$) can be generated from at most $\sim 3700$ Pauli strings, the estimated number of measurements needs to be done is $\sim 4.8\times 10^{9}$, which is one order of magnitude higher than that for $\langle H\rangle$ ($\sim 1.2\times 10^{8}$). Thus, given the same trial state in this H4 case, if the number of conventional VQE iterations is no more than one order of magnitude larger than that of the PDS(2)-VQS iterations, VQE would outperform PDS(2)-VQS in terms of total number of measurements, and PDS(2)-VQS outperforms otherwise. It is worth mentioning that, during the PDS($K$)-VQS process for the ground state and energy, the excited state energies can also be estimated directly from the higher roots of the polynomial (12) without any additional measurement (although accurate excited state energies would require higher order PDS($K$)-VQS calculations). In contrast, the conventional VQE would need distinct trial states, and thus different measurements, for targeting different states. Generally speaking, as long as the relatively large number of measurements of the Pauli strings becomes manageable, the PDS($K$)-VQS approach can be potentially applied for targeting the exact solutions for the system sizes that are not classically tractable, in particular for the systems whose true ground and excited states we have little knowledge of, or are challenging to obtain classically. To reduce the measurement demand, typical strategy is to partition the Pauli strings (that contribute to the moments) into commuting subsets that follow a certain rule, e.g. qubit-wise commutativity (QWC)[31, 44], general commutativity[22, 72], unitary partitioning[30], and/or Fermionic basis rotation grouping[28] to name a few. The applications of these commuting rules to the single Hamiltonian have shown that, at a cost of introducing additional one-/multi-qubit unitary transformation before the measurement, the total number of required measurements can be significantly reduced from $\mathcal{O}(N^{4})$ to $\mathcal{O}(N^{2\sim 3})$, or for simpler cases even $\mathcal{O}(N)$. For higher order moments, as we mentioned in the method section, early study of applying QWC bases to Heisenberg models represented by up to 36 qubits exhibits a sub-linear scaling of the number of measurements in the number of the qubits (Ref. [67]), which then leads us to expect similar scaling behaviors of the number of required measurements for evaluating moments for molecular systems. Beside exploring the commutativity of Pauli strings, other approaches including the linear combinations of unitary operators (LCU) technique[13], direct block-encoding[21, 6], and quantum power methods[59] might also be worth studying for reducing the number of measurements at the cost of circuit depth. In the light of that, we plan to perform a comprehensive benchmark as a follow-up work. Since the PDS($K$)-VQS formalism involves solving linear system of equations and polynomial, there is a concern of numerical instability when applying the PDS($K$)-VQS approach in optimization. Theoretically, the numerical instability of the PDS($K$)-VQS approach might come from two sources, (a) the singularity and ill-conditioning of the matrix $M$ in Eq. (11) that might consist of high order moments, and (b) the singularity of the Riemannian metric ($\mathcal{R}$) used in the dynamics (16). In particular, the singularity of matrix $M$ can be easily observed if the trial vector becomes very close to the exact wave function ($\rm det\|M\|=0$ if we replace the trial vector with exact vector). Numerically, the singularity problem can be avoided by adding a small positive number (e.g. $10^{-6}$) to the eigenvalue of the matrix $M$ or $\mathcal{R}$ via singular value decomposition (SVD). However, it is worthing noting that adding small perturbation to $M$ might violate the variationality of the PDS approach, and would not be recommended to use if the strict upper-bounds to the true energy are concerned. The ill-conditioning of matrix $M$ could occur in the high order PDS calculations, where high order moments could make the condition number of matrix $M$ very large. Thus, from the practical point of view, due to the potentially larger number of measurements and ill-conditioning arising from high Hamiltonian powers, lower oder PDS($K$)-VQS approaches are usually more feasible. Figure 6: The performance of VQE and PDS(2)-VQS employing ordinary gradient descent (GD) to compute the ground state energy of the four-site 2D Heisenberg model. (Top right) The circuit employed to generate the trial vector, where only the first rotation in RY gate is treated as a variational parameter $\theta$, and other three rotations are fixed to $(0,3,3)$. Two initial rotations $\theta_{0}=-2.0$ and $\theta_{0}=-3.0$ are chosen for performance comparison. The exact ground state energy of the 2D Heisenberg model is $E_{\text{exact}}=-3.6$ a.u. (Center) The VQE and PDS(2)-VQS energies and (bottom) the corresponding fidelity changes of the trial vectors w.r.t. true ground state in the first ten iterations in the conventional VQE and PDS(2)-VQS noise-free calculations. The step size $\eta=1.0/\text{Iteration}$ in all the calculations. Figure 7: The computed ground state energy (top left) and magnetization (top right) of the four-site 2D Heisenberg model and the corresponding changes of the fidelity (bottom left) and variational parameter $\theta$ (bottom right) in the first ten PDS($K$)-VQS ($K=2,3,4$) iterations running on IBM Toronto quantum hardware. The physical setup, error sources, and computed expectation values of Hamiltonian moments (up to $\langle H^{7}\rangle$) and the associated standard deviations are shown in Fig. 8. In all the calculations ordinary gradient descent (GD) is employed. The initial rotation $\vec{\theta}_{0}=-3.0$. The exact ground state energy and magnetization of the 2D Heisenberg model are $E_{\text{exact}}=-3.6$ a.u. and $\sum_{i}\langle\sigma_{z_{i}}\rangle=-4.0$ a.u., respectively. The step size $\eta=1.0/\text{Iteration}$ in all the calculations. Ultimately, one would be concerned about how the PDS($K$)-VQS applies to general models and how it performs on the real quantum hardware subject to the device noise. To address these concerns and explore the potential of the PDS($K$)-VQS approach, we have started to launch the PDS($K$)-VQS calculations for more general Hamiltonians on both simulator and the real quantum hardware. Figs. 6 and 7 exhibit some preliminary results for a four-site 2D Heisenberg model with external magnetic field, $H=J\sum_{\langle ij\rangle}\big{(}X_{i}X_{j}+Y_{i}Y_{j}+Z_{i}Z_{j}\big{)}+B\sum_{i}Z_{i}$ with $J/B=0.1$. The simple circuit employed for the state preparation in both VQE and PDS($K$)-VQS simulations is shown in Fig. 6, where, for the brevity of our discussion, we only treat one rotation in the state preparation as the variational parameter, and fix all other three rotations. As can be seen from the noise-free simulations in Fig. 6, the PDS(2)-VQS results quickly converge within five iterations achieving $\sim 0.99$ fidelity, while the performance of VQE exhibits strong dependence on the initial rotation (for $\vec{\theta}_{0}=-2.0$, the conventional VQE is able to converge in 10 iterations with $\Delta E<0.05$ a.u. and Fidelity $\sim 0.97$). When running the PDS($K$)-VQS simulations for the same model on the IBM Toronto quantum hardware, as shown in Fig. 7, in comparison to the ideal curves, the PDS(2/3)-VQS optimization curves on the real hardware significantly slows down, and deviate from the exact solutions due to the error from the real machine. However, if we increase the PDS order to perform PDS(4)-VQS calculations, the accuracy of the results systematically improves. For example, in the PDS(4)-VQS approach both the computed ground state energy and the trial state (and thus the magnetization) converge within 10 iterations being very close to the exact solutions. ## 5 Conclusion In summary, we propose a new variational quantum solver that employs the PDS energy gradient. In comparison with the usual VQE, the PDS($K$)-VQS helps identify an upper bound energy surface for the ground state, and thus frees the dynamics from being trapped at local minima that refer to non-ground states. In comparison with the static PDS($K$) expansions, the PDS($K$)-VQS guides the rotation of the trial wave function of modest quality, and is able to achieve high accuracy at the expense of low order PDS($K$) expansions. We have demonstrated the capability of the PDS($K$)-VQS approach at finding the ground state and its energy for toy models, H2 molecule, and strongly correlated planar H4 system in some challenging situations. In all the case studies, the PDS($K$)-VQS outperforms the standalone VQE and static PDS($K$) calculations in terms of efficiency even at the lowest order. We also discussed the limitations of the PDS($K$)-VQS approach at the current stage. In particular, the PDS($K$)-VQS approach may suffer from large amount of measurements for large systems, which can nevertheless be reduced at the cost of circuit depth by working together with some measurement reduction methods. Finally, we have started to launch PDS($K$)-VQS simulations for more general Hamiltonians on IBM quantum hardware. Preliminary results for Heisenberg model indicate that higher order PDS($K$)-VQS approach exhibits better noise- resistance than the lower order ones. The discussed approach can be extended to any variational formulation based on the utilization of $\langle H^{n}\rangle$ moments (e.g. Krylov subspace algorithms). Figure 8: (a) Quantum processor device map for ibmq_toronto showing the four qubits (Qn, $n=0-3$) used in the present computation. (b) Average CNOT error, 1-qubit readout assignment error, and thermal relaxation time constant (T1) and dephasing time constant (T2) in the four qubits used in the present computation. (c) The expectation values of the Hamiltonian moments, $\langle H^{n}\rangle$ ($n=1-7$), assembled from the measurements of the expectation values of 21 QWC bases for four-site 2D Heisenberg model $H=J\sum_{\langle ij\rangle}\big{(}X_{i}X_{j}+Y_{i}Y_{j}+Z_{i}Z_{j}\big{)}+B\sum_{i}Z_{i}$ with $J/B=0.1$. The data points correspond to mean value from the calculations on IBM Quantum processor ibmq_toronto with statistical error bars corresponding to $5\times 8192$ shots (per point). The trial state is constructed using the circuit given in Fig. 6 with initial rotation $\theta_{0}=-3.0$. ## 6 Acknowledgement B. P. and K. 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